WO2015068784A1 - 流体遷移経路取得装置、流体遷移経路取得方法、および、プログラム - Google Patents
流体遷移経路取得装置、流体遷移経路取得方法、および、プログラム Download PDFInfo
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Definitions
- the present invention relates to a fluid transition path acquisition device, a fluid transition path acquisition method, and a program.
- the conventional structure design optimization method leads to an increase in calculation time and design cost due to repeated large-scale calculation, but the search range is inevitably limited due to these restrictions. There is a problem that the possibility that the optimal structure is locally optimal cannot be excluded. In other words, traditionally, the search range has to be relied on the experience and intuition of engineers, and the optimization result of the derived structure depends on where the search range is set. Had.
- the present invention has been made in view of the above problems, and an object of the present invention is to provide a fluid transition path acquisition device, a fluid transition path acquisition method, and a program capable of obtaining knowledge about the transition of a flow pattern. .
- a fluid transition path acquisition device of the present invention is a fluid transition path acquisition device including a storage unit and a control unit, wherein the storage unit has a topological two-dimensional flow structure.
- the control unit acquires transition information related to a transition route from the designated structurally stable flow pattern to another topologically stable flow pattern that can be taken topologically. , Provided.
- the fluid transition path acquisition device is the fluid transition path acquisition device described above, wherein the transition information is information on a flow pattern of an unstable structure in an intermediate state leading to the another structure stable flow pattern. It is characterized by including.
- the fluid transition path acquisition device of the present invention is the above-described fluid transition path acquisition device, wherein the transition information acquisition unit is configured to specify the two structurally stable flow patterns specified based on the pattern transition information. The transition information from one structure-stable flow pattern to the other structure-stable flow pattern is acquired.
- the fluid transition path acquisition apparatus of the present invention is the fluid transition path acquisition apparatus described above, wherein the transition information acquisition means performs one transition through the flow pattern of one intermediate state that is unstable in structure.
- the transition information including the number of transitions from the one structurally stable flow pattern to the other structurally stable flow pattern is acquired.
- the fluid transition path acquisition device is the fluid transition path acquisition device described above, wherein the transition information acquisition means is configured such that the number of transitions is minimized from the one structure stable flow pattern to the other. The transition information up to the structure-stable flow pattern is acquired.
- the fluid transition path acquisition device of the present invention is the fluid transition path acquisition device described above, wherein the structure-stable flow pattern has two types that can be taken topologically in a single connected external region having one hole.
- the pattern words that define a total of three types of flow patterns with the addition of a pattern that does not have a suction-and-out pair in a double-connected external region having two holes, It is identified by a word expression formed by adding any one of the five operation words that define the topological operations that can be taken topologically when adding holes to the number of added holes.
- the pattern transition information includes the structurally stable flow patterns that can transition to each other via the flow pattern of one intermediate state that is unstable.
- the transition information acquisition means acquires the transition information based on the pattern transition information on the basis of the word representation of the specified structure-stable flow pattern. .
- the fluid transition path acquisition device of the present invention is the fluid transition path acquisition device described above, wherein the pattern transition information is a structure that is capable of transitioning to each other via a flow pattern of one intermediate state that is unstable in the structure. Including transition information of the operation word of the word expression between the stable flow patterns, the transition information acquisition means, based on the change of the operation word of the word expression between the two specified structural stable flow patterns, The transition information is acquired based on the change information.
- the transition information acquisition means has a plurality of transition route candidates acquired based on the pattern transition information on the basis of the word representation of the designated structure-stable flow pattern.
- a single transition route is specified by using a regular expression corresponding to a one-to-one correspondence with a flow pattern having a stable structure.
- the transition information acquisition unit acquires the transition information based on the change information on the basis of the change of the operation word of the word expression between the two designated structurally stable flow patterns.
- the regular expressions corresponding to the word expressions of the two structurally stable flow patterns are respectively acquired, and the partial expressions for the transition to occur for the plurality of transition route candidates are If it is not included in the regular expression corresponding to the word expression of the structure-stable flow pattern, it is excluded from the transition route candidates.
- the transition information acquisition unit calculates a regular expression corresponding to the plurality of transition route candidates that are not excluded.
- the transitional pattern is specified as a transition pattern whose regular expression matches the regular expression corresponding to the word expression of the other structure-stable flow pattern.
- the regular expression is created based on the created graph expression in which a graph expression corresponding to the flow pattern is created one-to-one.
- the fluid transition path acquisition apparatus of the present invention further sets an upper limit and a lower limit of the design parameter when selecting a design parameter candidate for an object in the fluid, and sets the upper limit and the lower limit of the design parameter.
- a plurality of parameters are selected from the defined parameter area, and a flow experiment and / or numerical calculation is performed on each of the selected parameters, and the above word expression and Simulation that assigns the regular expression and / or selects the design parameter having the word expression and / or the regular expression indicating the optimum state as the candidate for the design parameter among the assigned word expression and / or the regular expression. Means are provided.
- the fluid transition path acquisition device of the present invention is such that the flow pattern is 1 even when the assigned word expression and / or the regular expression does not have the word expression and / or the regular expression indicating an optimum state.
- a simulation unit is provided for selecting as a candidate for the design parameter.
- the fluid transition path acquisition method of the present invention is a fluid transition path acquisition method executed in a computer including a storage unit and a control unit, wherein the storage unit is an entire topological two-dimensional flow structure or Stores pattern transition information related to whether or not a structurally stable flow pattern can transition to another structurally stable flow pattern via a structurally unstable flow pattern in one intermediate state, and the control unit Transition information acquisition for acquiring transition information on a transition route from a specified structural stable flow pattern to another structurally stable flow pattern that can be taken topologically based on the pattern transition information executed in A step.
- the program of the present invention is a program for causing a computer including a storage unit and a control unit to execute the program, wherein the storage unit is structurally stable in all or part of the topological two-dimensional flow structure.
- the control unit the pattern transition information is stored in the pattern transition information.
- a transition information acquisition step for acquiring transition information on a transition route from a designated structurally stable flow pattern to another structurally stable flow pattern that can be taken topologically based on the specified structurally stable flow pattern .
- the present invention also relates to a recording medium, and is characterized by recording the above-described program.
- a structurally stable flow pattern in all or part of a topological two-dimensional flow structure is transferred from another structurally stable flow through a structurally unstable intermediate state flow pattern.
- this invention has the effect that the knowledge regarding the transition of a flow pattern can be acquired.
- the transition order of the phase structure can be specified by this transition information, for example, in order to guide it to an ideal flow pattern in the design of an aircraft or the like, the structural design is performed using the transition order of the flow pattern as an index be able to.
- the transition information includes information on the flow pattern of the structurally unstable intermediate state leading to another structurally stable flow pattern, so that the intermediate state of the transition between the structurally stable flow patterns is determined.
- the transition information from one structural stable flow pattern to the other structural stable flow pattern is acquired for two designated structural stable flow patterns. Therefore, when transition is possible between two structurally stable flow patterns, it is possible to obtain knowledge about the flow pattern transition route, the intermediate state flow pattern, and the like.
- a transition from one structurally stable flow pattern to the other structurally stable flow pattern is defined as a transition through one intermediate state flow pattern that is structurally unstable. Since the transition information including the number of times is acquired, a distance index such as the distance of the flow structure is obtained by acquiring how many transitions are required to move two flow patterns as the number of transition times. There is an effect that can be obtained.
- the structure-stable flow pattern is a double-connected external having two holes in addition to two types of flow patterns that can be taken topologically in a single-connected external region having one hole.
- a word that defines a total of three types of flow patterns with the addition of a pattern that does not have a pair of suction and outflow in the region five types of operations that can be taken topologically when adding one hole to the flow pattern Any one of the defined words is identified by a word expression formed by adding the number of added holes.
- the pattern transition information is information in which the structurally stable flow patterns that can transition with each other through the flow pattern of one intermediate state that is structurally unstable are described in word expression, and when the transition information is acquired Obtains transition information based on the pattern transition information with the word representation of the designated structurally stable flow pattern as a reference.
- the present invention has an effect that the computer can easily handle using the word expression when determining the topological transition possibility of the flow pattern.
- the pattern transition information includes change information of an operation word in a word expression between structurally stable flow patterns that can transition to each other via a flow pattern of an unstable intermediate state.
- the transition information is acquired based on the change information with reference to the change of the operation word of the word expression between the two designated structurally stable flow patterns.
- the flow pattern has a one-to-one correspondence.
- One transition route is specified using a regular expression.
- the two structures Regular expressions corresponding to word expressions of stable flow patterns are obtained, and for a plurality of transition route candidates, a partial expression for the transition to occur is a regular expression corresponding to the word expression of one structure stable flow pattern. If it is not included, it is excluded from transition route candidates. Thus, there is an effect that it is possible to exclude transition route candidates that have no possibility of actual transition simply by comparing the partial expressions of transition route candidates with regular expressions.
- each of the regular expressions corresponding to the plurality of transition route candidates that are not excluded is obtained, and a plurality of selections that are not excluded are selected.
- the route candidates those whose regular expression matches the regular expression corresponding to the word expression of the other structurally stable flow pattern are specified as transition patterns. Thereby, it is possible to specify one transition route only by comparing the regular expressions.
- the regular expression is created based on the created graph expression in which the graph expression corresponding to the flow pattern is created one-to-one.
- the structure-stable Hamilton vector field can be provided as a graph representation of a tree with a unique root, a labeled tree, and a direction visualized as a planar graph.
- an upper limit and a lower limit of the design parameter are set, and a parameter region defined by the upper limit and the lower limit of the design parameter.
- Select a plurality of parameters perform flow experiments and / or numerical calculations on the selected parameters, and assign word expressions and / or regular expressions to the results of the experiments and / or numerical calculations.
- a design parameter having the word expression and / or regular expression indicating the optimum state is selected as a design parameter candidate.
- the word expression and / or regular expression can be converted in one transition. If it has a word expression and / or a regular expression indicating the optimum state, it is selected as a design parameter candidate. This produces an effect that it is possible to adopt a word expression and / or a regular expression indicating an optimal state in one transition as a parameter candidate.
- FIG. 1 is a flowchart showing an outline of a word expression algorithm used in this embodiment.
- FIG. 2 is a diagram schematically showing a flow pattern.
- FIG. 3 is a diagram describing all the characteristic trajectories (streamlines) that perform topological classification of structurally stable flows in a region.
- FIG. 4 is a diagram schematically showing three types of structure-stable flow patterns serving as an initial structure.
- FIG. 5 is a diagram schematically showing five types of operations for forming a structurally stable flow by adding one hole.
- FIG. 6 is a diagram showing all classifications of flow patterns that are stable when there are two structures and a uniform flow.
- FIG. 7 is a diagram illustrating an example of an unstable track.
- FIG. 1 is a flowchart showing an outline of a word expression algorithm used in this embodiment.
- FIG. 2 is a diagram schematically showing a flow pattern.
- FIG. 3 is a diagram describing all the characteristic trajectories (streamlines) that perform topological classification of
- FIG. 8 is a diagram illustrating ten operations for generating an h-unstable streamline pattern in D z (M) by adding one hole from the structure stable streamline pattern in D z (M ⁇ 1). It is.
- FIG. 9A is a diagram (part 1) illustrating an h-unstable streamline pattern generated by the operation D 0, s .
- FIG. 9-2 is a diagram (part 2) illustrating an h-unstable streamline pattern generated by the operation D 0, s .
- FIG. 10 is a diagram showing h-unstable streamline patterns generated by the operation D 0 and perturbations thereof.
- FIG. 11 is a diagram showing h-unstable streamline patterns generated by operations E 0, s and E 0 and perturbations thereof.
- FIG. 9A is a diagram (part 1) illustrating an h-unstable streamline pattern generated by the operation D 0, s .
- FIG. 9-2 is a diagram (part 2) illustrating an h-unstable streamline pattern generated by the operation D 0, s .
- FIG. 12 is a diagram showing h-unstable streamline patterns generated by operations ⁇ 0, s and ⁇ 0 and perturbations thereof.
- FIG. 13 is a diagram showing h-unstable streamline patterns generated by operation D 2, s and perturbations thereof.
- Figure 14 is a diagram showing the perturbation and h- unstable streamline pattern generated by the operation D 2.
- FIG. 15 is a diagram showing h-unstable streamline patterns generated by operations E 2, s and E 2 and perturbations thereof.
- FIG. 16 is a diagram showing four operations for generating an h-unstable streamline pattern from a structure-stable streamline pattern by adding two or more holes.
- FIG. 17 is a diagram showing h-unstable streamline patterns generated by the operations ⁇ s (2) and ⁇ (2) and perturbations thereof.
- FIG. 18 is a diagram showing h-unstable streamline patterns generated by the operation ⁇ (2) and perturbations thereof.
- FIG. 19 is a diagram showing h-unstable streamline patterns generated by the operations ⁇ s (2n-1) and ⁇ (2n-1) and perturbations thereof.
- FIG. 20 is a diagram showing a p-unstable pattern ⁇ and its perturbation.
- FIG. 21 is a diagram illustrating an operation for generating a p-unstable streamline pattern by adding one pinching orbit to a structurally stable flow field in D z (M ⁇ 1).
- Figure 22 is a diagram showing the generated p- unstable streamline patterns and their perturbation in operation delta 1 (Part 1).
- Figure 23 is a diagram showing operation delta 1 p-labile streamline pattern generated by and its perturbation (Part 2).
- FIG. 24 is a diagram showing a p-unstable streamline pattern generated by the operation M 1, s and its perturbation.
- FIG. 25 is a diagram showing a p-unstable streamline pattern generated by operations M 1, i and M 1, o and perturbation thereof.
- FIG. 26 is a diagram showing a list of h-unstable streamline patterns that appear as intermediate states between two structurally stable streamline patterns.
- FIG. 27 is a diagram showing a list of h-unstable streamline patterns that appear as intermediate states between two structurally stable streamline patterns.
- FIG. 28 is a diagram showing a list of p-unstable streamline patterns that appear as intermediate states between two structurally stable streamline patterns.
- FIG. 29 is a block diagram illustrating an example of the fluid transition path acquisition apparatus 100 to which the present embodiment is applied.
- FIG. 30 is a flowchart illustrating an example of processing of the fluid transition path acquisition device 100.
- FIG. 31 is a flowchart illustrating an example of a transition route search process using a difference in exponents in the fluid transition route acquisition apparatus 100.
- FIG. 32 schematically illustrates a case where a thin flat plate having a finite thickness is tilted at a constant angle with respect to a uniform flow as an object in the uniform flow within a two-dimensional external region.
- FIG. FIG. 33 is a diagram illustrating a numerical calculation result of the flow around the object.
- FIG. 34 is a diagram illustrating a numerical calculation result of the flow around the object.
- FIG. 36 is a flowchart illustrating an example of pattern word assignment processing.
- FIG. 37 is a flowchart illustrating an example of an I-sequence operation word assignment process in algorithm B.
- FIG. 38 is a flowchart illustrating an example of an I-sequence operation word assignment process in algorithm B.
- FIG. 39 is a flowchart illustrating an example of II-series operation word assignment processing in algorithm B.
- FIG. 40 is a flowchart illustrating an example of O-sequence operation word assignment processing in algorithm B.
- FIG. 41 is a flowchart showing another example of assignment processing of I and II series operation words in algorithm B (I, II-Word algorithm).
- FIG. 42 is a flowchart showing another example of processing for assigning operation words for I and II sequences in algorithm B (I, II-Word algorithm).
- FIG. 43 is a flowchart illustrating another example of an O-sequence operation word assignment process in algorithm B (O-Word algorithm).
- FIG. 44 is a flowchart for explaining a flow pattern regular expression creation method according to the second embodiment.
- FIG. 45 is a flowchart for explaining an example of the graph expression creating process of FIG.
- FIG. 46 is a diagram showing all the flow patterns generated by the operations B 0 , B 2 , and C, and the parent-child relationships excited correspondingly to the connected subset (vertex set).
- FIG. 47-A is a flowchart for explaining a conversion process of a saddle connection diagram tree in the O series.
- FIG. 47-B is a flowchart for explaining a conversion process of a saddle connection diagram in the O series to a tree.
- FIG. 47-C is a flowchart for explaining a conversion process of a saddle connection diagram tree in the O series.
- FIG. 47-D is a flowchart for explaining a conversion process of a saddle connection diagram tree in the O series.
- FIG. 47-E is a flowchart for explaining a conversion process of a saddle connection diagram to a tree in the O series.
- FIG. 49 is a diagram showing the local structure of the ss-saddle connection diagram generated by the operations A 0 , A 2 , and C, and the visualization of the graph expression obtained corresponding thereto.
- FIG. 50A is a flowchart for explaining the conversion process to the ss-saddle connection diagram tree in the I and II series.
- FIG. 50-B is a flowchart for explaining the conversion process to the tree of the ss-saddle connection diagram in the I and II series.
- FIG. 50-C is a flowchart for explaining conversion processing to a tree of ss-saddle connection diagram in I and II series.
- FIG. 50-D is a flowchart for explaining the conversion process of the ss-saddle connection diagram tree to the I and II series.
- FIG. 50A is a flowchart for explaining the conversion process to the ss-saddle connection diagram tree in the I and II series.
- FIG. 50-B is a flowchart for explaining the conversion process
- FIG. 51-A is a diagram showing a graph expression of a structurally stable streamline pattern having the word expression IA 0 C and its regular expression.
- FIG. 51-B is a flowchart for explaining conversion processing of a tree into a regular expression.
- FIG. 52 is a block diagram illustrating an example of the fluid transition path acquisition apparatus 101 to which the second embodiment is applied.
- FIG. 53 is a flowchart illustrating an example of a transition route search process using a difference in exponents in the fluid transition route acquisition apparatus 101.
- FIG. 56 is a flowchart for explaining a method of designing an object in a fluid using word expressions and regular expressions.
- FIG. 57 is a diagram for explaining an example of a specific example for explaining a method of designing an object in a fluid using word expressions and regular expressions.
- FIG. 58 is a diagram for explaining an example of a specific example for explaining a method for designing an object in a fluid using word expressions and regular expressions.
- one intermediate state in which a structurally stable flow pattern (hereinafter also referred to as a “streamline pattern”) in a whole or part of a topological two-dimensional flow structure is structurally unstable.
- the pattern transition information related to whether or not transition to another structure-stable flow pattern is possible is stored. A method for comprehensively acquiring such pattern transition information will be described later.
- this Embodiment acquires the transition information regarding the transition route from the designated structural stable flow pattern to another structural stable flow pattern which can be taken topologically based on the pattern transition information. That is, in this embodiment, a transition route to another structurally stable flow pattern is searched for using the designated structurally stable flow pattern as a starting point. In the present embodiment, all possible routes may be derived. For example, with the specified structural stable flow pattern as the starting point, referring to the pattern transition information, all the second structural stable flow patterns via the intermediate structural flow pattern are written out. Good. Then, starting from each of the second structurally stable flow patterns, refer to the pattern transition information and write out all the third structurally stable flow patterns via one intermediate state flow pattern that is structurally unstable. May be.
- transition routes may be calculated by repeating the above-described processing.
- route search for roads etc.
- a known route search is considered by considering a structure-stable flow pattern as a node and a single transition through a flow pattern in an unstable state as a link.
- a transition route may be searched using a technique or the like.
- the transition information may include information on a flow pattern in an intermediate state that is unstable in structure, leading to another structure stable flow pattern. That is, in this embodiment, not only the transition order of the structure stable flow pattern is acquired as the transition route, but the transition information of the intermediate state flow pattern in between is also acquired as a part of the transition route. May be.
- this embodiment acquires transition information from one structure stable flow pattern to the other structure stable flow pattern for two designated structure stable flow patterns based on the pattern transition information. May be. More specifically, in the present embodiment, a transition route between them may be searched using one designated structurally stable flow pattern as a starting point and the other structurally stable flow pattern as a destination. . If there is no transition route between two structurally stable flow patterns, a result indicating that transition is impossible is acquired as transition information.
- the transition from one structurally stable flow pattern to the other structurally stable flow pattern is considered as a transition through one structurally unstable flow pattern of an intermediate state.
- Transition information including the number of times may be acquired. That is, the number of transitions may be acquired as a distance measure between two flow patterns, such as the distance of the flow structure.
- transition information from one structurally stable flow pattern to the other structurally stable flow pattern having the minimum number of transitions may be acquired. That is, in this embodiment, the shortest route of the transition route may be calculated using the number of transitions as an index.
- the flow pattern may be identified (labeled) by word expression. For example, in addition to two types of flow patterns that can be taken topologically in a single connected outer region having one hole, a pattern that does not have a suction-and-out pair in a double connected outer region having two holes has been added. , Any one of the five operation words defining the topological operations that can be taken topologically when adding one hole to the flow pattern with respect to the pattern words defining three types of flow patterns.
- the structure-stable flow pattern may be identified by the word expression formed by adding the number of added holes.
- the structure-stable flow patterns that can transition to each other through the flow pattern of one intermediate state that is structurally unstable are expressed in the word expression.
- the transition information can be acquired on the basis of the word expression of the designated structure-stable flow pattern based on the pattern transition information described in (1).
- transition information may be acquired based on change information with reference to a change in an operation word of a word expression between two designated structure-stable flow patterns.
- an algorithm for forming this word expression is referred to as a “word conversion algorithm”.
- the mathematical details of the “word conversion algorithm” can be found in the published paper (T. Yokoyama and T. Sakajo, “Word repres- sion of streamlined strategy for structuringly stubbornly-stable. 469 (2013) doi: 10.1098 / rspa.2012.558) (hereinafter referred to as “published paper (2013)”).
- FIG. 1 is a flowchart showing an outline of the word expression algorithm of the present embodiment.
- a single connected external region having one hole is formed in order to form a word expression of a flow pattern in a connected external region having topologically N holes.
- a pattern word that defines a total of three types of flow patterns, in addition to two types of flow patterns that can be taken topologically, plus a pattern that does not have a pair of suction and outflow in a double-connected external region having two holes Any one word (for example, I, II, O) is assigned (step SA-1).
- step SA-1 for the word given in step SA-1, an operation word that defines five types of operations that can be taken topologically when adding one hole to the flow pattern (for example, Any one of A 0 , A 2 , B 0 , B 2 , C) is assigned (step SA-2), and the process of step SA-2 is repeated until the number of holes becomes N.
- step SA-3 a word expression corresponding to the connected external region having N holes is formed.
- pattern word assignment step SA-1 is performed first, and operation word assignment (step SA-2) is performed later.
- step SA-1 is performed first
- step SA-2 operation word assignment
- the “connected external region” is an expression including a single connected external region and a multiple connected external region
- the “single connected external region” is a region in two dimensions (plane).
- the term “multiple connected outer region” refers to a region in two dimensions (a plane) that has a plurality of holes.
- the expression “hole” here is a mathematical abstract expression, but various expression forms may be applied. For example, when focusing on a uniform flow that occurs relatively when a vehicle moves, if there is one or more structures on the cross-section of the vehicle along the uniform flow, the region is simply It can be treated as a connected external region or a multiple connected external region.
- one or more holes and a single connection or multiple connection Can be treated as a connected external region.
- a flow such as “one or more obstacles in the flow” is treated as a flow in the connected external region in this embodiment.
- an isolated vortex structure or a flow structure (such as an elliptical stopping point) having a periodic orbit around it can be regarded as a “hole”.
- Topological is a technical term in mathematics and refers to a field of geometry, also called topology (topology).
- topology topology
- triangles and quadrilaterals are considered to be different figures due to the difference in the number of their corners, but from the viewpoint of topological geometry, triangles and quadrilaterals can move to each other by deforming rubber bands, for example.
- topology topology
- the connected external region is expressed as D ⁇ (M) with respect to the number of holes M + 1. For example, when there is only one hole, it is a single connected external region D ⁇ (0), and when there are two holes, it is a double connected external region D ⁇ (1).
- topological structure a specific structure that characterizes the flow
- Those that cannot be deformed by continuous (ie, not cut or stuck) deformation of a particular structure are considered different flows.
- the fluid in this embodiment may be handled as a non-viscous fluid.
- An inviscid fluid can treat the boundary condition as a “slip condition”.
- the boundary condition is the “zero boundary condition”.
- the word conversion algorithm for non-viscous fluids may be described in particular, but this embodiment can also be applied to viscous fluids by appropriate boundary expansion.
- FIG. 2 is a diagram schematically showing a flow pattern.
- FIG. 2C schematically shows a boundary having four saddle points (definition will be described later).
- a vortex is an element that creates a flow that rotates around it.
- the uniform flow is a basic flow in terms of river flow, and is a flow that crosses the entire region.
- the uniform flow is a flow such as a relative air flow or a water flow as viewed from an observer riding on the vehicle. That is, in the coordinate system of the moving object, even if air or water is actually stationary, it is possible to imagine the relative flow that flows from the infinity point.
- a plane may be projected onto a spherical surface by a projection method called stereographic projection (stereo projection / stereo projection) in mathematics.
- stereographic projection stereo projection / stereo projection
- the uniform flow has a flow structure such as a flow source and a suction pair in the spherical north pole, and it can be mathematically shown to correspond to the flow field as shown in FIG.
- the position of the north pole and the south pole can be appropriately shifted using the fact that the spherical surface has a highly symmetric shape, so the point at infinity is the south pole,
- a flow as shown in FIG. 2B is produced near the origin corresponding to the south pole.
- the flow field of the entire plane can be expressed as a bounded region having a shape as shown in FIG.
- D ⁇ (M) is moved to the unit circle outer region by selecting the pair of suction and spring at infinity point as the origin and one of M + 1 boundaries by this conversion method.
- a connected region inside a unit circle including M boundaries inside is sometimes written as D z (M). All subsequent descriptions are made using this D z (M). Therefore, the expression as shown in FIG. 2B is equivalent to a flow in which a uniform flow is contained in the entire plane through an appropriate projection method. In the description of the present embodiment, it is convenient to schematically show the flow, and therefore, it may be expressed in the drawing using such a projection method.
- FIG. 3 is a diagram describing all the characteristic trajectories (streamlines) for performing topological classification of the structure-stable flow in such a connection region D z (M).
- ss-orbit the trajectory returning from the pair of suction and outflow and returning to itself.
- Fig. 3 (b) the trajectory that comes out of the pair of suction and outflow and connects to the boundary is called ss- ⁇ -saddle connection, and the trajectory is connected as shown in Fig. 3 (c).
- the point on the border is called ss- ⁇ -saddle.
- FIG. 3 (e) the trajectory leading from a point above a certain boundary and leading to a point on the same boundary, not from a pair of suction and outflow, is called a saddle connection, and FIG. 3 (d) The points on the boundary connected by this are called ⁇ -saddle.
- a hyperbolic stop point that is not on the boundary as shown in FIG. 3 (h) is called a saddle point (saddle point), but as shown in FIG.
- the trajectory leading to this saddle point is called ss-saddle connection.
- a closed curve orbit that creates a boundary or a vortex is called a closed orbit
- Fig. 3 (i) a trajectory that leaves the saddle point and returns to itself. It is called homoclinic saddle connection. It can be mathematically proved that the target structurally stable flow can only be expressed by a combination of these trajectories.
- step SA-2 in the above-described step SA-2, one hole and the flow structure associated therewith are added to the flow of the connected external region D z (M ⁇ 1) having M holes.
- the flow field of many multiple connected external regions D z (M) is constructed recursively. Therefore, the simplest hole is a single connected external region D z (0) or a double connected external region D z (1), which is the initial structure of these inductive structures given in step SA-1 ing.
- FIG. 4 is a diagram schematically showing three types of structure-stable flow patterns as an initial structure.
- FIGS. 4A and 4B there are two types of flows, a pattern I and a pattern II, in which a hole is in one single connected external region D z (0). Both of these patterns have a pair of suction and spring, and it can be proved mathematically that there are only these two types.
- the double-coupled outer region D z (1) is composed of these, but does not have a suction-and-out pair. Is not constructed from here, the initial flow necessary for constituting the flow is the pattern O schematically shown in FIG.
- step SA-2 the five types of operations that can be taken topologically are: 1) A 0 operation that replaces one ss-orbit with one saddle point, a homoclinic saddle connection that connects the ss-orbit and a hole inside, and two ss-saddle connections. 2) A 2 operation to replace one ss-orbit with two s- ⁇ -saddle connections and two ⁇ -saddles on the newly added boundary, 3) B 0 operation that replaces one closed orbit with two homoclinic saddle connections in which one hole and saddle point are added to form a figure 8 (this pattern is hereinafter referred to as “figure eight pattern”).
- FIG. 5 is a diagram schematically showing five types of operations for forming a structurally stable flow by adding one hole.
- operation A 0 and A 2 is performed for a single ss-orbit. Further, as shown in FIG. 5 (b), the operation B 0 and B 2 is performed for a single closed orbit. Further, as shown in FIG. 5C, the operation C is performed on the boundary already having ⁇ -saddle. It is possible to mathematically prove that there are only five kinds of operations that enable such a process while maintaining the structural stability (Theorem 3.1, Collarly 3.1 in the published paper (2013)). , Theorem 3.2). A flow pattern diagram obtained by repeating this step is referred to as ss-saddle connection diagram when starting from patterns I and II, and as a saddle connection diagram when starting from pattern O.
- FIG. 6 is a diagram illustrating the entire classification of the structure-stable flow pattern when there are two structures and a uniform flow. As shown in FIG. 6, all the flows in the double connected external region D z (1) are given to the patterns I and II of the initial structure in the single connected external region D z (0). A pattern can be described. However, all the flow patterns shown in FIG.
- the constraints will be described as follows. That is, as described above with reference to FIG. 5, since the operations A 0 and A 2 are performed on one ss-orbit, the existence of one ss-orbit is indispensable as a premise for performing this operation. It becomes. Further, since the operations B 0 and B 2 are performed on one closed orbit, the presence of one closed orbit is indispensable as a premise for performing this operation. Further, since the operation C is performed on a boundary having ⁇ -saddle, the existence of a boundary having ⁇ -saddle is indispensable as a premise for performing this operation. Therefore, the rules for arranging the pattern words are different depending on where the pattern words are started from I, II, and O. The rules for arranging the character strings starting from the pattern words I, II, and O, which are derived based on the above constraint conditions, will be described below.
- Such a character string is called an O-series word (O-Word), and the correctness of the rule can be mathematically proved (see Lemma 3.1 in the published paper (2013)).
- I-1 The operations that can be performed are all of A 0 , A 2 , B 0 , B 2 , and C, and as a result, the word expression starting with I lists these five types of operation words.
- I-2) In order to include the word B 0 or B 2 in the word expression of the operation sequence, C or A 0 must always exist before that.
- I-Word I-series word
- Such a character string is called a II-series word (II-Word), and the correctness of the rule can be mathematically proved (see Lemma 3.4 in the published paper (2013)).
- word expressions obtained by excluding overlapping word expressions and included word expressions are called maximal word representations.
- the maximal word expression may be replaced with a word expression according to the inclusion relation shown in the following table to form a word expression without duplication or inclusion relation.
- “ ⁇ ” or “ ⁇ ” indicates an inclusion relationship.
- B 2 and C Since these cannot be interchanged even if they are interchanged, the two cannot be interchanged. This is represented symbolically as B 2 C
- a word expression is assigned to allow duplication and inclusion relation (for example, a plurality of corresponding word expressions are assigned to one structure-stable flow pattern). ), It is not essential to use maximal expression. Further, if one algorithm is determined, such as algorithm B and its variants shown later, a word expression can be uniquely assigned to a given flow.
- pattern transition information Next, pattern transition information according to the present embodiment will be described.
- the flow field may already be labeled with a unique maximal word expression using the algorithm described above.
- the present embodiment is not limited to using the maximal word expression as long as the word transition is assigned while allowing the pattern transition information to be overlapped or included, as described above.
- Pattern transition information of the present embodiment when the respective maximum word representation into two structural stability streamline pattern W 1 and W 2 are given, two streamlines patterns thereof, a certain one intermediate state It is information that identifies whether or not the streamline pattern candidate of the intermediate state can be determined by determining whether or not they can move to each other.
- W For the maximal word expression W, a 0 (W), a 2 (W), b 0 (W), b 2 (W) and c (W) are respectively included in W A 0 , A 2 , Let it be the number of characters B 0 , B 2 , C.
- W is an O-series maximal word (maximal O-word)
- an I-series maximal word maximal I-word
- an II-series maximal word maximal II-word
- W subword a character substring (not necessarily adjacent) S included in the maximal word expression W is referred to as a W subword.
- W O 0 O 1 . . .
- O 0 any one of O, I, and II
- O i (1 ⁇ i ⁇ k) is any one of A 0 , A 2 , B 0 , B 2 , and C.
- S O i1 . . . O il (0 ⁇ i 1 ⁇ ... ⁇ I l ⁇ k) is referred to as a subword of W and is described as S ⁇ W.
- the partial word S can include characters representing three types of pattern words such as O, I, and II.
- FIG. 7 is a diagram illustrating an example of an unstable track.
- a trajectory connecting two different saddle points is called a heteroclinic orbit.
- FIG. 7B shows (two) s- ⁇ -saddle connections connecting the saddle point and two ⁇ -saddles on the boundary.
- the s- ⁇ -saddle connection is a trajectory connecting the saddle point and the ⁇ -saddle on the boundary.
- FIG. 7 (c) shows (two) heteroclinic or-orbits connecting ⁇ -saddle on two different boundaries.
- a heteroclinic ⁇ -orbit is a trajectory connecting ⁇ ⁇ -saddles on two different boundaries.
- heteroclinic pair The set of saddle points and circle boundaries connected by such trajectories will be referred to as “heteroclinic pair” hereinafter.
- heteroclinic pair there is a non-viscous and incompressible flow field that has only one such heteroclinic pair and is otherwise structurally stable. (H-unstable) ", or its streamline pattern is called” h-unstable streamline pattern ".
- FIG. 7D shows a homoclinic pinching orbit on the boundary.
- a non-regular stop point on the boundary and connected to four branching orbits (separatrix) is called “pinching point”.
- the homoclinic pinching orbit represents a trajectory such that two of the four branched trajectories are connected to each other on the boundary.
- An inviscid / incompressible flow field that has only one such pinching point and is otherwise structurally stable is called “p-unstable” or its streamline pattern is “p-unstable”. Is called a “p-unstable streamline pattern”.
- a streamline pattern in an unstable state between two structurally stable flow fields is expressed using these h-unstable or p-unstable flow fields.
- the structurally unstable flow field may include not only p-unstable flow and h-unstable flow, but strictly more various and complicated flow fields.
- FIG. 8 is a diagram illustrating ten operations for generating an h-unstable streamline pattern in D z (M) by adding one hole from the structure stable streamline pattern in D z (M ⁇ 1). It is.
- the hatched circle in the figure represents a newly added hole (the same applies to the following figures).
- operations D 0, s and D 0 are operations for attaching one homoclinic saddle on each of ss- ( ⁇ -) saddle connection and ⁇ -saddle connection.
- s subscript appearing in the character D 0, s representing an operation this structure means that are connected directly to the 1-source-sink point representative of the presence of a uniform flow.
- FIGS. 9A to 9C are diagrams showing h-unstable streamline patterns generated by the operation D 0, s , perturbations thereof, and subword expressions.
- FIG. 9 (d) shows two h-unstable flow patterns.
- the h-unstable streamline pattern and its partial word expression are perturbed in the middle, and this pattern is perturbed to the left and right. It expresses two structurally stable streamline patterns and their partial word expressions.
- FIG. 9 (a) is a diagram showing h-unstable flow pattern ID 0, s and its perturbation pattern. Two patterns are obtained depending on whether the homoclinic saddle is attached to the upper side or the lower side of the ss- ⁇ -saddle connection. Although both do not change the pattern of the streamline after perturbation, the position of the hole moves from the top to the bottom of the ss- ⁇ -saddle connection. The subword expressions after perturbation are both IA 0 . Thus, this transition does not change the index between the maximal word representations representing the respective structurally stable flow fields.
- FIG. 9B shows the h-unstable flow pattern IID 0, s and its perturbation pattern.
- two types of h-unstable patterns are obtained depending on the direction in which the homoclinic saddle is added, but both transition to two structurally stable flow patterns having the subword expression IIA 0 by perturbation. This transition does not change the index between the maximal word representations representing the respective structurally stable flow fields.
- FIG. 9 (c) is a diagram showing an h-unstable flow pattern A 0 D 0, s and its perturbation pattern.
- three types of h-unstable patterns are obtained depending on the structure of A 0 on which the operation is performed and the direction in which the homoclinic saddle is added.
- the two sub-word expressions A 0 A 0 are obtained by perturbation. Transition to one structurally stable flow pattern. This transition does not change the index between the maximal word representations representing the respective structurally stable flow fields.
- FIG. 9 (d) is a diagram showing an h-unstable flow pattern A 2 D 0, s .
- two types of h-unstable patterns are obtained depending on the direction in which homoclinic saddle is added, but these streamline patterns are made to be the same in phase as IDs 0 and s by appropriate variable transformation. Therefore, the same transition as in FIG. The index between word expressions does not change.
- two types of h-unstable patterns can be obtained with ID 0, s , for example, depending on the combination of the direction of operation. For all subsequent operations, multiple patterns corresponding to the same combination can be obtained.However, in any case, when the perturbation is performed, the transition to a structure having exactly the same word expression is made. Since the change is the same, there is no essential difference. Therefore, for the sake of convenience, only a representative one of these patterns will be illustrated and described below. These are hereinafter referred to as “representative patterns”.
- FIG. 10 is a diagram showing h-unstable streamline patterns generated by the operation D 0 and perturbations thereof.
- FIG. 10 (a) shows an h-unstable streamline pattern B 2 D 0 and its perturbation. By perturbing this pattern, a structure-stable streamline pattern having the same subword representation B 0 B 2 (or B 2 B 0 ) is obtained although the structure is different. Therefore, there is no change in the index between maximum word expressions at this time.
- FIG. 10 (c) shows an h-unstable streamline pattern ICCD 0 and its perturbation.
- Two structurally stable streamline patterns having the same subword expression ICCB 0 (or ICB 0 C) are obtained due to the perturbation of this pattern. Therefore, there is no change in the index.
- FIG. 10 (d) shows an h-unstable streamline pattern B 2 CD 0 and its perturbation.
- Two structurally stable streamline patterns having the same subword expression B 2 CB 0 (or B 0 B 2 C, B 2 B 0 C) are obtained due to perturbation of this pattern. Therefore, there is no change in the index.
- FIG. 10 (e) shows an h-unstable streamline pattern B 2 CD 0 and its perturbation.
- Two structurally stable streamline patterns having the same subword expression B 2 CB 0 (or B 0 B 2 C, B 2 B 0 C) are obtained due to perturbation of this pattern. Therefore, there is no change in the index.
- FIG. 11 is a diagram showing h-unstable streamline patterns generated by the operations E 0, s and E 0 and perturbations thereof.
- FIG. 11 (a) shows an h-unstable streamline pattern IIE 0, s and its perturbation.
- structurally stable streamline patterns whose partial word expressions correspond to IIA 0 and IIB 0 , respectively, are obtained.
- FIG. 11 (b) shows an h-unstable streamline pattern A 0 E 0, s and its perturbation.
- structurally stable streamline patterns whose partial word expressions correspond to A 0 A 0 and A 0 B 0 respectively are obtained.
- FIG. 11 (c) shows an h-unstable streamline pattern B 0 E 0 and its perturbation.
- a structure with the same subword representation B 0 B 0 is obtained by perturbation. Therefore, there is no change in the index.
- FIG. 8 (Operation ⁇ 0, s and ⁇ 0 )
- the operations ⁇ 0, s and ⁇ 0 both divide one saddle point into two and connect these points with two new heteroclinic orbits (enclosing one new hole) in FIG. 8 (c).
- the operation is as shown. This symbol is used because a streamline pattern having the shape of ⁇ is created by this operation.
- these operations yield three typical h-unstable streamline patterns and their perturbations as shown in FIG.
- FIG. 12 is a diagram showing h-unstable streamline patterns generated by operations ⁇ 0, s and ⁇ 0 and perturbations thereof.
- FIG. 12 (a) shows an h-unstable streamline pattern II ⁇ 0, s and its perturbation.
- a structure with the same subword representation IIA 0 is obtained by perturbation. Therefore, there is no change in the index.
- FIG. 12B shows an h-unstable streamline pattern A 0 ⁇ 0, s and its perturbation.
- a structure having the same subword expression A 0 A 0 is obtained by perturbation. Therefore, there is no change in the index.
- FIG. 12 (c) shows an h-unstable streamline pattern B 0 ⁇ 0 and its perturbation.
- a structure with the same subword representation B 0 B 0 is obtained by perturbation. Therefore, there is no change in the index.
- FIG. 13 is a diagram showing h-unstable streamline patterns generated by operation D 2, s and perturbations thereof.
- FIG. 13 (a) shows h-unstable streamline pattern ID 2, s and its perturbation.
- a structure having the same subword expression IA 2 is obtained by perturbation. Therefore, there is no change in the index.
- FIG. 13 (b) shows an h-unstable streamline pattern IID 2, s and its perturbation.
- a structure having the same subword expression IA 0 is obtained by perturbation. Therefore, there is no change in the index.
- Figure 14 is a diagram showing the perturbation and h- unstable streamline pattern generated by the operation D 2.
- FIG. 14 (a) shows an h-unstable streamline pattern B 2 D 2 and its perturbation.
- a structure having the same partial word representation B 2 B 2 is obtained by perturbation. Therefore, there is no change in the index.
- the h-unstable streamline pattern A 2 CD 2 is the same as ICD 2 under appropriate variable transformation, so its perturbation is considered. Perturbation results in different structurally stable streamline patterns with subword representations IA 2 C and ICB 2 .
- FIG. 14 (c) shows the h-unstable streamline pattern ICCD 2 and its perturbation.
- a structure having the same subword expression ICCB 2 (or ICB 2 C) is obtained by perturbation. Therefore, there is no change in the index.
- FIG. 14 (d) shows an h-unstable streamline pattern B 2 CD 2 and its perturbation.
- a structure having the same subword expression B 2 CB 2 (or B 2 B 2 C) is obtained by perturbation. Therefore, there is no change in the index.
- FIG. 14 (e) shows an h-unstable streamline pattern B 2 CCD 2 and its perturbation. This pattern B 2 because CD 2 into the same as those subjected to C B 2 CD 2 to give the same partial word changes (see FIG. 14 (d)).
- FIG. 15 is a diagram showing h-unstable streamline patterns generated by operations E 2, s and E 2 and perturbations thereof.
- FIG. 15 (a) shows an h-unstable streamline pattern IIE 2, s and its perturbation.
- IIE 2 h-unstable streamline pattern
- FIG. 15B shows an h-unstable streamline pattern IA 0 E 2, s and its perturbation.
- the h-unstable streamline pattern IIA 0 E 2, s is the same as that obtained by performing operation A 0 on IIE 2, s (see FIG. 15 (a)). .
- the h-unstable streamline pattern B 0 E 2 is phase-same as B 2 D 0 (see FIG. 10A).
- FIG. 16 is a diagram showing four operations for generating an h-unstable streamline pattern from a structure-stable streamline pattern by adding two or more holes.
- FIG. 17 is a diagram showing h-unstable streamline patterns generated by the operations ⁇ s (2) and ⁇ (2) and perturbations thereof.
- FIG. 18 is a diagram showing h-unstable streamline patterns generated by the operation ⁇ (2) and perturbations thereof.
- FIG. 17A shows an h-unstable streamline pattern I ⁇ s (2) and its perturbation.
- a structure having the same subword expression IA 0 C is obtained by perturbation. Therefore, there is no change in the index.
- the h-unstable streamline pattern A 2 ⁇ s (2) is topologically the same as I ⁇ s (2) (see FIG. 17A).
- FIG. 17 (c) shows an h-unstable streamline pattern B 2 ⁇ (2) and its perturbation.
- a structure having the same subword expression B 0 B 2 C (or B 2 CB 0 , B 2 B 0 C) is obtained by perturbation. Therefore, there is no change in the index.
- FIG. 18 (a) shows an h-unstable streamline pattern IC ⁇ (2) and its perturbation.
- perturbation By perturbation, different structurally stable streamline patterns with subword representations IA 0 CC and ICCB 0 (or ICB 0 C) are obtained.
- FIG. 18B shows an h-unstable streamline pattern ICC ⁇ (2) and its perturbation. Perturbed by the same sub-word representations ICCCB 0 (or ICCB 0 C, ICB 0 CC) structure having obtained. Therefore, there is no change in the index.
- FIG. 18 (c) shows an h-unstable streamline pattern B 2 C ⁇ (2) and its perturbation.
- B 2 CCB 0 or B 0 B 2 CC, B 2 B 0 CC, B 2 CB 0 C
- FIG. 18 (d) shows an h-unstable streamline pattern B 2 CC ⁇ (2) and its perturbation. Since this pattern is the same as B 2 C ⁇ (2) with C, it gives the same subword expression change as B 2 C ⁇ (2) (see FIG. 18C).
- FIG. 19 is a diagram showing h-unstable streamline patterns generated by the operations ⁇ s (2n-1) and ⁇ (2n-1) and perturbations thereof.
- FIG. 19 (a) shows an h-unstable streamline pattern I ( s (2n-1) and its perturbation. Perturbation yields a structure with the same subword representation IA 2 C 2n-2 . Therefore, there is no change in the index.
- FIG. 19B shows an h-unstable streamline pattern B 2 ⁇ (2n ⁇ 1) and its perturbation.
- a structure having the same subword representation B 2 C k B 2 C 2n-2-k (0 ⁇ k ⁇ n ⁇ 1) is obtained. Therefore, there is no change in the index.
- FIG. 19 (c) shows an h-unstable streamline pattern IC (2n-1) and its perturbation.
- IC (2n-1) By perturbation, different structurally stable streamline patterns with subword representations IA 2 C 2n-1 and IC k B 2 C 2n-1-k (1 ⁇ k ⁇ n) are obtained.
- FIG. 19 (d) shows an h-unstable streamline pattern ICC2 (2n-1) and its perturbation.
- IC k B 2 C 2n ⁇ k (1 ⁇ k ⁇ n) is obtained. Therefore, there is no change in the index.
- FIG. 20 is a diagram showing a p-unstable pattern ⁇ and its perturbation.
- the basic pattern ⁇ is a p-unstable streamline pattern in D z (0) and is given in FIG. Transition is made to basic patterns I and II that perturb this pattern. Therefore, this pattern changes the character at the head of the maximal word expression.
- FIG. 20B shows a streamline pattern ⁇ C in which the structure formed by ⁇ is surrounded by one saddle connection and its perturbation. Different structures stable streamline pattern subword representations by perturbation has the IC and IIB 2 is obtained.
- the p-unstable flow field obtained by repeating the operations A 0 , A 2 , B 0 , B 2 , and C for creating a structure-stable flow field was applied. Since the maximal word expression obtained by arranging the operations in a row can be called maximal ⁇ -word (maximum word of ⁇ sequence), this ⁇ -word perturbation causes a transition between I-word and II-word. .
- FIG. 21 is a diagram illustrating an operation of generating a p-unstable streamline pattern by adding one pinching orbit to the structurally stable flow field in D z (M ⁇ 1).
- Operation delta 1 as shown in FIG. 21 (a), represents the operation of increasing the hole by adding homoclinic pinching orbit any circle boundaries.
- FIGS. 22 and 23 are diagrams showing operation delta 1 p-labile streamline pattern generated by and their perturbation.
- FIG. 22 (a) shows p- unstable streamline pattern delta 1 and its perturbation.
- perturbation By perturbation, different structurally stable streamline patterns with subword representations B 0 and B 2 are obtained.
- FIG. 22 (c) shows a p- unstable streamline pattern IIderuta 1 and their perturbation. Perturbation results in different structurally stable streamline patterns with subword representations IIB 0 and IIB 2 .
- FIG. 22 (d) shows a p-unstable streamline pattern A 0 ⁇ 1 and its perturbation, which is equivalent to the one with A 0 applied to the boundary with ⁇ 1 (FIG. 22 (a)). )reference).
- FIG. 22 (e) shows the p-unstable streamline pattern B 0 ⁇ 1 and its perturbation, which is equivalent to B 0 applied to the boundary with ⁇ 1 (FIG. 22 (a)). )reference).
- FIG. 23 (b) shows the perturbation p- unstable streamline pattern B 2 delta 1 and.
- B 0 B 2 or B 2 B 0 ) and B 2 C are obtained.
- the p-unstable streamline pattern A 2 C ⁇ 1 is topologically equal to IC ⁇ 1 .
- the perturbation results in different structurally stable streamline patterns having subword expressions ICB 0 and ICC.
- Figure 23 (d) is, p- unstable streamline pattern B 2 C delta 1 does not need to consider it is equal to B 2 delta 1 to those subjected to C (see FIG. 23 (b)).
- FIG. 21 (b) an operation for adding ⁇ -saddle connection that shares an end point with ⁇ -saddle connected to 1-source-sink point by ss- ⁇ -saddle connection on the same boundary is performed as M 1, Call it s .
- M 1 an operation for adding ⁇ -saddle connection that shares an end point with ⁇ -saddle connected to 1-source-sink point by ss- ⁇ -saddle connection on the same boundary
- M 1 Call it s .
- FIG. 24 is a diagram showing a p-unstable streamline pattern generated by the operation M 1, s and its perturbation.
- FIG. 25 (a) shows p-unstable streamline patterns B 2 M 1, o and B 2 M 1, i and their perturbations. Both perturbations produce different structurally stable streamline patterns with subword representations B 2 B 0 and B 2 C.
- FIG. 25B shows the p-unstable streamline pattern ICM 1, o and its perturbation.
- ICM 1 p-unstable streamline pattern
- ICB 0 and ICC subword expressions
- FIG. 25 (c) shows a p-unstable streamline pattern ICCM 1, i and its perturbation.
- ICCB 0 or ICB 0 C
- the above is one basic pattern and six operations for generating a p-unstable streamline pattern, and 13 types of p-unstable streamline patterns generated by them, perturbations and subword expressions. It can also be mathematically proved that there is no other operation that generates a p-unstable streamline pattern.
- FIG. 26 and FIG. 27 are diagrams showing a list of h-unstable streamline patterns that appear as an intermediate state between two structurally stable streamline patterns.
- the transition having a partial structure with h-unstable streamline pattern X is the condition S 1 ⁇ W 1 , S 2 ⁇ W for two structurally stable flow fields having maximal expression W 1 and W 2.
- ind * (W 1 ) ⁇ ind * (W 2 ) d (* is one of O, I, and II).
- FIG. 28 is a diagram showing a list of p-unstable streamline patterns that appear as intermediate states between two structurally stable streamline patterns.
- FIG. 29 is a block diagram illustrating an example of the fluid transition path acquisition apparatus 100 to which the present embodiment is applied, and conceptually illustrates only a portion related to the present embodiment in the configuration.
- the fluid transition path acquisition apparatus 100 in the present embodiment schematically includes at least a control unit 102 and a storage unit 106.
- the fluid transition path acquisition device 100 further controls communication with the input / output control interface unit 108.
- An interface unit 104 is provided.
- the control unit 102 is a CPU or the like that comprehensively controls the entire fluid transition path acquisition apparatus 100.
- the communication control interface unit 104 is an interface connected to a communication device (not shown) such as a router connected to a communication line or the like, and the input / output control interface unit 108 is connected to the input device 112 or the output device 114.
- the storage unit 106 is a device that stores various databases and tables.
- route acquisition apparatuses 100 is connected so that communication is possible via arbitrary communication paths. Furthermore, the fluid transition path acquisition device 100 is communicably connected to the network 300 via a communication device such as a router and a wired or wireless communication line such as a dedicated line.
- a communication device such as a router and a wired or wireless communication line such as a dedicated line.
- the various databases and tables (such as the simulation result file 106a, streamline diagram file 106b, and pattern transition file 106c) stored in the storage unit 106 are storage means such as a fixed disk device.
- the storage unit 106 stores various programs, tables, files, databases, web pages, and the like used for various processes.
- the simulation result file 106a is a simulation result storage unit that stores data indicating simulation results that are simulated experimentally or mathematically by the simulation unit 102a.
- the simulation result file 106a may store experimental results or numerical calculation results regarding the force received by the object and the flow around the object.
- the simulation result file 106a includes a value of a design variable indicating the shape of a structure, and a hydrodynamic simulation result of a predetermined fluid (such as an ocean current or an air current) with respect to the structure (fluid pressure or flow at each spatial coordinate). Or the like.
- the simulation result file 106a may store, as simulation results, data input in advance via the input device 112 through model measurement in a laboratory such as a wind tunnel experiment.
- the data stored in the simulation result file 106a is not limited to numerical calculation results, but may be data obtained through model measurement in a laboratory. Even experimental results can be expressed as snapshots that capture the flow state of streamlines by visualizing these data or processing them with a computer.
- the streamline diagram file 106b is streamline data storage means for storing data indicating streamlines such as a streamline diagram.
- the streamline data stored in the streamline diagram file 106b may be data indicating streamlines analyzed by the streamline analysis unit 102b based on data indicating simulation results.
- the pattern transition file 106c is a correspondence information storage unit that stores pattern transition information.
- the pattern transition file 106c can be used to convert a structurally stable flow pattern from one or more topological two-dimensional flow structures to another structurally stable one-state flow pattern. Pattern transition information relating to whether or not a transition to a simple flow pattern is possible.
- Each flow pattern may be identified by a word expression.
- the word expression data stored in the pattern transition file 106c is a character string made up of a combination of pattern words and operation words.
- a structurally stable flow pattern can be drawn in a double connected outer region with two holes in addition to two types of flow patterns that can be topologically taken in a single connected outer region with one hole.
- a pattern word for example, I, II, O
- a pattern word that defines a total of three types of flow patterns to which a pattern having no pattern is added
- a word expression formed by assigning any one of operation words for example, A 0 , A 2 , B 0 , B 2 , C
- any one of operation words for example, A 0 , A 2 , B 0 , B 2 , C
- the pattern transition information stored in the pattern transition file 106c includes change information of operation words in the word expression between structurally stable flow patterns that can transition with each other via a flow pattern in an intermediate state that is unstable in structure.
- Index difference d or the like
- the pattern transition information stored in the pattern transition file 106c includes structurally stable flow patterns that can transition with each other via a flow pattern X in an intermediate state that is unstable as shown in FIGS. Together with the partial words S 1 and S 2 , the exponent difference d is stored in association with each other.
- the route search method by the link and the node it corresponds to the case where a structure stable flow pattern is set in the node and transition is possible between two structure stable flow patterns through one intermediate state.
- a link may be set between nodes.
- the pattern transition information in the pattern transition file 106c is not limited to identifying the flow pattern by word expression, but may be identified by other identification information (sequential number or symbol). For example, in a connected external region having a finite number of N holes, all the identification numbers are assigned to the flow pattern diagrams, and when the flow patterns can be transitioned at one time, the identification numbers of the two flow patterns are set. It is good also as a list
- the input / output control interface unit 108 controls the input device 112 and the output device 114.
- the output device 114 in addition to a monitor (including a home television), a speaker can be used (hereinafter, the output device 114 may be described as a monitor).
- the input device 112 a keyboard, a mouse, a microphone, and the like can be used.
- control unit 102 has an internal memory for storing a control program such as an OS (Operating System), a program defining various processing procedures, and necessary data. And the control part 102 performs the information processing for performing various processes by these programs.
- the control unit 102 is functionally conceptually provided with a simulation unit 102a, a streamline analysis unit 102b, a word expression formation unit 102c, and a transition information acquisition unit 102g.
- the simulation unit 102a is a simulation unit that performs a fluid simulation on an object such as a structure.
- the simulation unit 102a may experimentally or mathematically simulate a fluid with respect to a structure to obtain an experimental result or numerical calculation result of a force received by the object and a flow around the object.
- the simulation unit 102a is not limited to a simulation in a two-dimensional plane, and may perform a fluid simulation in a three-dimensional space.
- the simulation unit 102a stores data indicating the simulation result in the simulation result file 106a.
- the simulation unit 102a determines the value of a design variable indicating the shape of the structure, and the hydrodynamic simulation result of a predetermined fluid (such as an ocean current or an air current) with respect to the structure (the pressure or flow direction of the fluid in each spatial coordinate). Or data indicating resistance) may be stored.
- the simulation unit 102a may use a Navier-Stokes equation, a finite element method, a finite difference method, a finite volume method, or the like as a known hydrodynamic calculation method.
- the simulation unit 102a may optimize the structure using a known optimization method. For example, the simulation unit 102a performs a fluid simulation on the structure while repeatedly changing the design variable for determining the shape of the structure using an annealing method, a genetic algorithm method, or the like, thereby obtaining an appropriate shape of the structure. (For example, the shape of a pier with little resistance to water flow, etc.) may be obtained. At that time, the simulation unit 102a may perform optimization with reference to the transition route based on the transition information acquired by the transition information acquisition unit 102g.
- the simulation unit 102a By changing the shape and arrangement of the structure so that the next streamline pattern can be found from the current streamline pattern in the route, the shape and arrangement of the structure can be optimized by approaching the ideal streamline pattern. You may go.
- the streamline analysis unit 102b is streamline analysis means for performing streamline analysis.
- the streamline analysis unit 102b may derive a streamline diagram by performing streamline analysis on the simulation result of the simulation unit 102a.
- the streamline analysis unit 102b uses a well-known method from the flow experiment result (experiment data, etc.) or the numerical calculation result (numerical simulation result, etc.) around the object stored in the simulation result file 106a. Create a diagram.
- the streamline analysis unit 102b calculates all the saddle points, ⁇ -saddles, etc. from the numerical simulation results, and then draws all the contours of the flow function having the same value as the value of the flow function at that point.
- a flow diagram can be created by drawing all the contour lines of the flow function having the same value as the value of the flow function on the boundary.
- the streamline analysis unit 102b may perform streamline analysis after converting it to two-dimensional data in a cross section of the structure.
- the plane used as a cross section is arbitrary, but preferably, the streamline analysis unit 102b may convert it into two-dimensional data in a cross section along the direction of the fluid flow (uniform flow). For example, in a vehicle such as a train, an automobile, or an aircraft, a cross section may be generated along the traveling direction.
- the streamline analysis unit 102b may extract a characteristic structure satisfying a condition from the flow field using a technique described in Computational Homology (“Computational Homology by Tomasz Kaczynski, Konstantin Miskaikou, Marian Mrzek”). “See Springer, 2000”).
- the streamline analysis unit 102b stores the created streamline diagram data in the streamline diagram file 106b.
- the word expression forming unit 102c defines two types of flow patterns that can be taken topologically in a single connected external region having one hole based on the streamline diagram acquired by the streamline analyzing unit 102b.
- a word (operation words A 0 , A 2 ) defining five types of operations that can be taken topologically when adding one hole to the flow pattern together with any one of the words (pattern words I, II).
- B 0 , B 2 , C) is a word expression forming means for forming a word expression indicating a flow pattern of the flow around the object by giving one word of the number of the added holes. .
- the word expression forming unit 102c uses a pattern word ( ⁇ ) or an operation word (D 0 , E 0, s , ⁇ 0 , D 2 or the like for expressing a flow pattern in an intermediate state. Or M1 , i, etc.) may be used.
- the word expression forming unit 102c includes a pattern word adding unit 102d, an operation word adding unit 102e, and a maximal word expressing unit 102f.
- the pattern word giving unit 102d is one of words (for example, pattern words I and II) that define two types of flow patterns that can be taken topologically in a single connected outer region having one hole. It is a pattern word giving means for giving one word.
- the pattern word assigning unit 102d defines a total of three types of flow patterns (i.e., a pattern having no suction spring-out pair in the double connected external region having two holes (i.e., Any one of the pattern words I, II, and O) may be given.
- Pattern I which has a pair of suction springs and has two ss- ⁇ -saddle connections
- Pattern O having a suction spring-out pair, one saddle point, a homoclinic saddle connection connecting the same and two ss-saddle connections
- 3) a pattern O having no suction spring-out pair It is.
- the pattern word assigning unit 102d determines whether or not ss- ⁇ -saddle connection exists in the streamline diagram in an algorithm for obtaining a word expression from the streamline diagram (referred to as “algorithm B”).
- Algorithm B an algorithm for obtaining a word expression from the streamline diagram.
- Ss-saddle connection exists in streamline diagram when ss- ⁇ -saddle connection exists, I classification step that gives word of pattern I, and ss- ⁇ -saddle connection does not exist in I classification step
- An II / O classification step that gives a pattern II word when a ss-saddle connection exists, and gives a pattern O word when a ss-saddle connection does not exist, Also good.
- the pattern word assigning unit 102d can appropriately determine the series (I series, II series, O series) to which the streamline diagram belongs.
- the operation word assigning unit 102e defines words (for example, operation words A 0 , A 2 , B 0 , B 2 ) that define topological operations when one hole is added to the flow pattern.
- C) is an operation word assigning means for forming a word expression corresponding to a multi-connected external region having N holes by repeatedly assigning any one of the words. More specifically, the five types of operations are: 1) A 0 operation that replaces one ss-orbit with one saddle point, a homoclinic saddle connection that connects the ss-orbit and a hole inside, and two ss-saddle connections.
- a 2 operation to replace one ss-orbit with two s- ⁇ -saddle connections and two ⁇ -saddles on the newly added boundary 2) A 2 operation to replace one ss-orbit with two s- ⁇ -saddle connections and two ⁇ -saddles on the newly added boundary, 3) B 0 operation that replaces one closed orbit with two homoclinic saddle connections with the addition of one hole and saddle point and a figure of 8. 4) B 2 operation that replaces one closed orbit with a trajectory that connects two saddles on the boundary of the newly added hole and connects them with a single saddle connection; and 5) A C operation that adds two new ⁇ -saddles to a boundary that already has 2k (k> 0) ⁇ -saddles, and places a newly added hole inside by connecting with one ⁇ -saddle connection, It is.
- the operation word giving unit 102e may give the operation word according to the following constraint conditions. That is, the operation word giving unit 102e gives five types of operation words. 1) Provide the operation word A 0 or A 2 on condition that one ss-orbit exists, 2) Provide the operation word B 0 or B 2 on condition that one closed orbit exists. 3) The operation word C may be given on the condition that a boundary having ⁇ -saddle exists. Note that details of specific processing of the operation word assigning unit 102e according to the constraint condition will be described later with reference to a flowchart.
- the operation word assigning unit 102e may store a word expression that is a character string of the operation word assigned to the pattern word in the pattern transition file 106c.
- the maximal word expression unit 102f is a maximal word expression means for converting the word expression (a character string formed by a combination of a pattern word and an operation word) formed by the pattern word adding unit 102d and the operation word adding unit 102e into a maximal word. is there. That is, in an algorithm for writing all possible flow patterns (hereinafter referred to as “algorithm A”), the word expression group formed by the pattern word assigning unit 102d and the operation word assigning unit 102e writes all the flow patterns. However, there are word expressions that prescribe the same flow patterns and word expressions that prescribe the flow patterns that are inclusive of each other.
- the maximal word representation unit 102f performs a process for obtaining a maximal word representation on these word representation groups by eliminating redundant word representations and included word representations.
- the maximal word expression unit 102f replaces the word expressions according to the inclusion relationship shown in the following table to form the maximal word expression.
- “ ⁇ ” or “ ⁇ ” indicates an inclusion relationship.
- B 2 and C Since these cannot be interchanged even if they are interchanged, the two cannot be interchanged. This is represented symbolically as B 2 C
- the transition information acquisition unit 102g relates to a transition route from a specified structural stable flow pattern to another structurally stable flow pattern that can be taken topologically based on the pattern transition information of the pattern transition file 106c. It is a transition information acquisition means for acquiring transition information. For example, the transition information acquisition unit 102g may search for a transition route that leads to another structurally stable flow pattern, starting from the structurally stable flow pattern designated by the user via the input device 112. Further, the transition information acquisition unit 102g may search for a transition route that leads to another structurally stable flow pattern, starting from the structurally stable flow pattern specified by the simulation result of the simulation unit 102a.
- the word expression formed by the word expression formation unit 102c is used as the starting point of the specified flow pattern.
- An ideal flow pattern designated in advance may be used as the reaching point.
- the transition information acquisition unit 102g may return the transition information of the transition route obtained by the search to the simulation unit 102a, and may be used as an optimization technique index by the simulation unit 102a.
- the transition information acquisition unit 102g may acquire all possible transition routes, or may acquire only the shortest transition route for two designated structurally stable flow patterns.
- the transition information acquisition unit 102g takes a transition through one intermediate state flow pattern that is unstable in structure as one transition, from one structure stable flow pattern to the other structure stable flow pattern. Transition information including the number of transitions may be acquired.
- the transition information acquisition unit 102g may search for a transition route using a graph theory with links and nodes, similarly to a route search for a road or the like. In that case, the transition information acquisition unit 102g may treat a structure-stable flow pattern as a node, and treat a single transition through a flow pattern of one unstable intermediate state as a link. For example, a case will be described in which an ideal flow pattern is set as an arrival point with the current flow pattern according to the shape and arrangement of the structure as a starting point. As an example, the transition information acquisition unit 102g refers to the pattern transition information with the designated structural stable flow pattern as a starting point, and the second structural stability via the flow pattern of one intermediate state that is structurally unstable. Write out all the flow patterns.
- the transition information acquisition unit 102g refers to the pattern transition information with each of the second structure stable flow patterns as a starting point, and the third structure through one intermediate state flow pattern that is structurally unstable. Write all stable flow patterns. In this way, by sequentially expanding the search range, when the ideal flow pattern set as the arrival point is reached, the transition information acquisition unit 102g ends the iterative process until the arrival point is reached.
- the transition route may be output as the shortest route.
- the transition information acquisition unit 102g performs transition from one structurally stable flow pattern to the other structurally stable flow pattern for two designated structurally stable flow patterns based on the pattern transition information. Information may be acquired.
- the transition information acquisition unit 102g may output the acquired transition information to the output device 114 such as a monitor, may be referred to for optimization by the simulation unit 102a, and may store a simulation result file 106a or the like. You may store in the part 106.
- the output device 114 such as a monitor
- the transition information acquisition unit 102g may perform an efficient transition route search by using, as an index, a change in the number of operation words in the word expression (increase or decrease in the number of operation words). That is, in the pattern transition file 106c, the change information of the operation word in the word expression between the structurally stable flow patterns that can be shifted to each other through the flow pattern of the intermediate state that is unstable is stored in the pattern transition information. In the case, the transition information acquisition unit 102g may inquire the pattern transition information for the change in the operation word of the two specified word expressions and extract the corresponding flow pattern.
- the transition information obtaining unit 102g determines the difference d of the index of the formed maximum word represented by the word representation forming portion 102c
- the transition information may be acquired by searching for a transition pattern having the difference d from the pattern transition information as shown in FIGS.
- the fluid transition path acquisition apparatus 100 may be connected to the external system 200 via the network 300.
- the communication control interface unit 104 performs communication control between the fluid transition path acquisition device 100 and the network 300 (or a communication device such as a router). That is, the communication control interface unit 104 has a function of communicating data with other terminals via a communication line.
- the network 300 has a function of connecting the fluid transition path acquisition apparatus 100 and the external system 200 to each other, such as the Internet.
- the external system 200 is mutually connected to the fluid transition path acquisition device 100 via the network 300, and an external database regarding various data such as simulation result data such as experimental results and numerical calculation results, streamline diagram data, It has a function of providing a program or the like for causing the connected information processing apparatus to execute the fluid transition path acquisition method.
- the external system 200 may be configured as a WEB server, an ASP server, or the like.
- the hardware configuration of the external system 200 may be configured by an information processing apparatus such as a commercially available workstation or a personal computer and its attached devices.
- Each function of the external system 200 is realized by a CPU, a disk device, a memory device, an input device, an output device, a communication control device, and the like in the hardware configuration of the external system 200 and a program for controlling them.
- FIG. 30 is a flowchart illustrating an example of processing of the fluid transition path acquisition device 100.
- the transition information acquisition unit 102g performs control so that a structure-stable flow pattern is designated (step S-1).
- the transition information acquisition unit 102g may control the user to specify a flow pattern via the input device 112.
- the transition information acquisition unit 102g may use a streamline diagram obtained by the streamline analysis unit 102b based on a simulation result by the simulation unit 102a as a specified flow pattern.
- the transition information acquisition part 102g converts the designated streamline figure and flow pattern into word expression by the word expression formation part 102c, when performing a route search based on word expression.
- the transition information acquisition unit 102g refers to the pattern transition information corresponding to the designated structurally stable flow pattern (step S-2). For example, when a structure-stable flow pattern that can transition through one intermediate state is stored in the pattern transition file 106c in association with a word expression, the transition information acquisition unit 102g has the designated structure-stable flow pattern.
- the pattern transition information that matches the word expression of the flow pattern may be searched to sequentially search for a flow pattern that can be shifted.
- the transition information acquisition unit 102g may sequentially search from one structurally stable flow pattern to the other structurally stable flow pattern. Alternatively, the search may be based on a topological difference (eg, exponential difference) between two structurally stable flow patterns. Details of the latter processing will be described later.
- the transition information acquisition unit 102g may search for a transition route using a known route search method or a known shortest route search method (Dijkstra method or the like).
- the transition information acquisition unit 102g acquires transition information regarding the searched transition route (step S-3). For example, when one flow pattern is designated, the transition information acquisition unit 102g displays transition information regarding a transition route from the designated structurally stable flow pattern to another structurally stable flow pattern over a predetermined number of transitions. May be obtained. Note that the transition information acquisition unit 102g may include not only information related to the structurally stable flow pattern but also information related to an intermediate structurally unstable flow pattern in the transition information. When two flow patterns are designated, the transition information acquisition unit 102g displays the transition information from one structure stable flow pattern (W 1 ) to the other structure stable flow pattern (W 2 ). You may get it.
- the transition information acquisition unit 102g may acquire a transition route that minimizes the number of transitions as transition information.
- the transition information acquisition unit 102g may acquire transition information in which a transition route is a stream diagram or a flow pattern graphic, or may acquire transition information described in word expression. Further, the transition information acquisition unit 102g may display and output the acquired transition information on the output device 114, store it in the storage unit 106, or refer to it for optimization processing by the simulation unit 102a. .
- FIG. 31 is a flowchart illustrating an example of a transition route search process using a difference in exponents in the fluid transition route acquisition apparatus 100.
- the list of FIGS. 26 to 28 obtained for perturbation of h-unstable and p-unstable streamline patterns and their partial word expressions is used as the pattern transition information.
- it outputs are designated its maximum word representation W 1 and W 2 and two structural stability streamline patterns, a list of potential X transitions is determined from the maximum word represented as transition information It will be explained on the assumption.
- the transition information acquisition unit 102g designates two structure-stable streamline patterns as in step S-1 described above (step S-11).
- the transition information acquisition unit 102g acquires the word expressions W 1 and W 2 corresponding to the two structure stable streamline patterns (step S-12). For example, when a streamline diagram is designated, the transition information acquisition unit 102g acquires the corresponding word expressions W 1 and W 2 by the processing of the word expression forming unit 102c.
- the transition information obtaining unit 102g acquires the partial sequence S 1 and S 2 corresponding to each transition pattern X (step S- 15).
- partial sequences S 11 , S 21 to S 1k, and S 2k are obtained for transition patterns X 1 , X 2 ,..., X k , respectively.
- the transition information obtaining unit 102g to the subsequence S 1 and S 2, they investigated whether is W 1 and W 2 of the sub-words respectively, if you are a subword, as a transition candidate It is included in the transition information (step S-16). That is, it is checked whether S 1 ⁇ W 1 and S 2 ⁇ W 2 . If this is true, X is acquired as a transition candidate, otherwise it is rejected. In addition, when assuming the above-mentioned multiple times of transition, rearrangement is performed so that adjacent streamline patterns at the time of transition share partial columns. If there is a sequence sharing a partial sequence, the permutation is acquired as a transition candidate, and if there is no sequence sharing a partial sequence, it is rejected. If an arrangement sharing a partial sequence is made, it is possible to sequentially determine the entire word expression in the transition route by sequentially tracing from the end W 1 or W 2 .
- the transition candidate pattern X is It only gives candidates for possible transitions (requirements), and whether or not the corresponding intermediate state can actually be realized depends on whether the structure obtained for the subword actually exists (sufficiently) Need to be confirmed. If confirmed, it is possible to actually write down a p-unstable or h-unstable streamline pattern as an intermediate state between two structurally stable streamline patterns.
- streamline pattern matching may be performed manually or using a known pattern matching technique.
- a thin flat plate having a finite thickness is tilted at a constant angle with respect to the uniform flow as an object in the uniform flow. It is the figure which showed the case typically.
- the symbol U in the figure indicates a uniform flow
- cl is the length of the long side of the flat plate
- ⁇ is the angle to the uniform flow
- Lx and Ly are the lengths in the x-axis direction and the y-axis direction.
- a rectangular area is shown.
- FIG. 32 for example, in an object such as an aircraft, a train, or an automobile, a uniform flow from a relatively infinite point is generated along the traveling direction. Assuming a cross-section, let us consider obtaining, by simulation, the force applied to the object and the flow around the object in a certain period for an object placed in a uniform flow.
- FIG. 33 and FIG. 34 are diagrams showing the numerical calculation results of the flow around the object. Note that the details of the numerical calculation conditions in FIG. 34 (for example, Reynolds number: 1370, ⁇ : 15 °, analysis range (Resolution): 1024 ⁇ 256) are not important here, and will be omitted.
- FIG. 33 illustrates the flow around the object when the uniform flow does not hit the object
- the flow of this is illustrated.
- the counterclockwise (counterclockwise) flow has a positive value
- the clockwise flow has a negative value
- a straight flow is indicated by zero. That is, the figure represents the vorticity component of the flow, where the positive value part shows the region with the counterclockwise positive vortex component, and the negative value part shows the clockwise negative vortex component. Represents a region with components.
- a large time fluctuation is observed in the lift (lift) and lift-drag ratio (l / d ratio), and it vibrates up and down periodically.
- the optimal arrangement is known from the viewpoint of the efficiency of interaction between various fluids and structures
- the next step is performed.
- the state of the power flow pattern is known.
- the shortest route to the optimum arrangement can be grasped through continuous changes in the shape and arrangement of the structure, and an efficient guideline for optimization is given.
- the shortest control can be considered. For example, an efficient wing control method for preventing a stall in an aircraft or the like can be understood.
- step SA-1 pattern word assignment
- step SA-2 operation word assignment
- step SA-2 operation word assignment
- step SA-2 operation word assignment
- the present invention is not limited to this, and operation word assignment is performed first.
- the pattern word may be added later.
- the pattern word assigning unit 102d of the word expression forming unit 102c assigns one of the pattern words I and II (step SA-1).
- the pattern word assigning unit 102d may assign any one of a total of three types of pattern words I, II, and O, to which the pattern O is added in addition to the above.
- the three types of pattern words are: 1) A pattern word I that defines a pattern having a pair of suction springs and two ss- ⁇ -saddle connections, 2) A pattern word II having a suction spring-out pair, one saddle point, a pattern having a homoclinic saddle connection and two ss-saddle connections, and 3) a pattern without a suction spring-out pair
- the defined pattern word O It is.
- step SA-2 the operation word assigning unit 102e of the word expression forming unit 102c, among the five types of operation words A 0 , A 2 , B 0 , B 2 , C, with respect to the pattern words given by the pattern word giving unit 102d Is given (step SA-2).
- the five operation words are: 1) An operation word A 0 stipulating an operation of replacing one ss-orbit with one saddle point, a homoclinic saddle connection having a hole inside and two ss-saddle connections, 2) An operation word A 2 that defines an operation for replacing one ss-orbit with two ss- ⁇ -saddle connections and two ⁇ -saddles on the newly added boundary, 3) An operation word B 0 stipulating an operation for replacing one closed orbit with two homoclinic saddle connections having a shape of 8 by adding one hole and a saddle point.
- Operation word B 2 stipulating the operation of replacing one closed orbit with a trajectory in which two ⁇ -saddles are attached on the boundary of the newly added hole and connected with one ⁇ -saddle connection, and , 5) Specifying the operation of adding two new ⁇ -saddles to a boundary that already has 2k (k> 0) ⁇ -saddles and placing a newly added hole inside one ⁇ -saddle connection.
- Manipulated word C It is. Since the above operation defines an operation for adding a hole, in Algorithm B in which an operation word is assigned to a stream diagram, the reverse of the above operation is performed on the stream diagram. Will be performed (reverse replacement operation).
- the operation word assigning unit 102e may assign the operation word in accordance with the following constraints. That is, the operation word giving unit 102e gives five types of operation words. 1) Provide the operation word A 0 or A 2 on condition that one ss-orbit exists, 2) Provide the operation word B 0 or B 2 on condition that one closed orbit exists. 3) The operation word C may be given on the condition that a boundary having ⁇ -saddle exists.
- the operation word giving unit 102e of the word expression forming unit 102c determines whether the number of holes has reached N (step SA-3). For example, the operation word assigning unit 102e assigns (N ⁇ 1) operation words to the pattern words I and II in the single connected external region D ⁇ (0) with one hole until there are N holes. It is determined whether or not it has been completed. The operation word assigning unit 102e assigns (N-2) operation words to the pattern word O in the double connected external region D ⁇ (1) with two holes until there are N holes. Determine if finished. In Algorithm B, the number of holes is not set in advance. In the case of algorithm B, the operation word assigning unit 102e determines whether the streamline diagram has reached the initial pattern (I, II, or O) by the replacement operation according to the operation word for the streamline diagram. It may be determined whether the number has reached N.
- Step SA-3 When the number of holes has not reached N (No in step SA-3), the operation word assigning unit 102e of the word expression forming unit 102c assigns further operation words to the character strings created so far. (Step SA-2).
- step SA-3 Yes
- the operation word assigning unit 102e of the word expression forming unit 102c stores the character string created so far in the pattern transition file 106c as a word expression. Store and finish processing.
- the above is an example of the basic processing of the word expression algorithm of the fluid transition path acquisition apparatus 100 in the present embodiment.
- By forming the word expression in this way it is possible to express with a word expression which flow pattern the simulation result corresponds to.
- FIG. 36 is a flowchart illustrating an example of pattern word assignment processing.
- the given streamline diagram is described as a region in a circle as shown in FIG. 6 for convenience of drawing, but this algorithm is an algorithm for determining a flow in an unbounded region that originally has a uniform flow. It is.
- the flow obtained by numerical simulation and actual measurement is a streamline diagram of an unbounded flow field (the necessary part is cut out from it), so it is a problem whether it is necessary to convert it to an area within a circle.
- it has been found that such conversion is unnecessary in applying this algorithm. Based on this, if the following processing is performed on a given stream diagram, it can be determined whether the stream diagram is the I series, II series or O series. That is, when the following processing is performed, it is not necessary to convert the stream diagram into a region within a circle by a stereo projection method or the like.
- the pattern word assigning unit 102d determines whether or not ss- ⁇ -saddle connection exists in the given streamline diagram in the algorithm B for obtaining the word representation from the streamline diagram ( Step SA-11).
- step SA-11 If it is determined that ss- ⁇ -saddle connection exists (step SA-11, Yes), the pattern word assigning unit 102d assigns the pattern word I (step SA-12). That is, the given streamline diagram is determined as the I series.
- step SA-11, No the pattern word assigning unit 102d determines whether or not the ss-saddle connection exists in the stream diagram (step SA-13). ).
- step SA-13 If it is determined that the ss-saddle connection exists (step SA-13, Yes), the pattern word assigning unit 102d assigns the pattern word II (step SA-15). That is, the given streamline diagram is determined as the II series.
- step SA-14 the pattern word assigning unit 102d assigns the pattern word O (step SA-14). That is, the given streamline diagram is determined as the O series.
- FIG. 37 and FIG. 38 are flowcharts showing an example of an I-sequence operation word assignment process in algorithm B.
- the streamline analysis unit 102b creates a streamline diagram by streamline analysis from the numerical simulation and experimental data stored in the simulation result file 106a (step SB-1). Specifically, the streamline analysis unit 102b calculates all the saddle points, ⁇ -saddles, etc. from the numerical simulation results, and then draws all the contours of the flow function having the same value as the value of the flow function at that point. In addition, a flow diagram can be created by drawing all the contour lines of the flow function having the same value as the value of the flow function on the boundary. In addition, the streamline analysis unit 102b may extract a characteristic structure that satisfies a condition from the flow field using a technique described in Computational Homology (Non-Patent Document 1).
- the streamline analysis unit 102b stores the created streamline diagram data in the streamline diagram file 106b.
- the streamline analysis unit 102b may handle streamline diagram data by assigning numbers to all the obtained lines, saddle point, ss-orbits, ⁇ -saddle, and boundary. This makes it easy to handle the subsequent preprocessing and algorithms of each series on the computer by assigning the inclusion relations and assigning the rankings to determine the data structure.
- the streamline analysis unit 102b performs preprocessing on the streamline diagram (step SB-2).
- the streamline analysis unit 102b performs the following three preprocessing on a given streamline diagram.
- the number of times these operations are performed is referred to as the number of errors, and is an index of how much streamlines can be expressed in word expression.
- the ocean current is moving not only in the fluid movement on the plane but also in the vertical direction of the plane.
- a gas or the like that hits an obstacle may also flow in a direction that is not horizontal with a two-dimensional cross section along a uniform flow.
- the operation word assigning unit 102e determines whether there is a saddle-saddle in the streamline diagram preprocessed by the streamline analyzing unit 102b (step SB-3).
- step SB-3 If ⁇ -saddle exists (step SB-3, Yes), the operation word assigning unit 102e determines whether there is a boundary having ss- ⁇ -saddle connection at the boundary where exactly two ⁇ -saddles exist. Determination is made (step SB-4).
- Step SB-4 Just when the one with the ss- ⁇ -saddle connection including a boundary that there are two ⁇ -saddle is k pieces (Step SB-4, Yes), the operation word imparting unit 102e, the operation term A 2 k pieces grant (Step SB-5), the streamline analysis unit 102b deletes these two ⁇ -saddles and boundaries on the streamline diagram and replaces them with one ss-orbit k times. (Step SB-6).
- step SB-4 when there is no ss- ⁇ -saddle connection including the boundary where exactly two ⁇ -saddles exist (step SB-4, No), the operation word assigning unit 102e has exactly two ⁇ -saddles. It is determined whether there is a boundary having a saddle connection on the same boundary (step SB-7).
- step SB-7 If those just having ⁇ -saddle connection including the boundary of the two ⁇ -saddle there is the k (step SB-7, Yes), the operation word imparting unit 102e, the operation term B 2 and k pieces grant ( In step SB-8), the streamline analysis unit 102b deletes the ⁇ -saddle and the boundary connected by the ⁇ -saddle connection on the same boundary on the streamline diagram, and replaces it with a single closed orbit. The operation is performed k times (step SB-9).
- step SB-7 if there is no one having a saddle connection including a boundary where exactly two saddles exist (step SB-7, No), the operation word assigning unit 102e has a boundary having four or more saddles. It is determined whether or not there is a ad-saddle connection on the same boundary with a center / disk inside (step SB-10).
- step SB-10 When the ⁇ -saddle connection on the boundary including four or more ⁇ -saddles has k boundaries whose inner side is center / disk (step SB-10, Yes), the operation word assigning unit 102e K words C are assigned (step SB-11), and the streamline analysis unit 102b, on the streamline diagram, ⁇ -saddle connection on the same boundary and two ⁇ -saddles connected thereby, The operation of deleting the center / disk surrounded by them is performed k times (step SB-12).
- step SB-10 No
- the innermost FIG. ⁇ -saddle in connection since at least one of its inside there is one) and center / disk, and the number and k-number, the operation word imparting unit 102e, the operation term B 0 and the k grant (step SB ⁇ 13)
- the streamline analysis unit 102b deletes one sad point and boundary from the innermost figure eye on the streamline diagram, and performs k operations to replace it with a closed orbit surrounding the center / disk.
- “FIGURE EIGHT” is composed of one saddle point and two homoclinic saddle connections connecting them, and each homoclinic saddle connection encloses a center / disk (such as the figure 8). It refers to the flow structure.
- step SB-3 when there is still a saddle-saddle (step SB-3, Yes), the fluid transition path acquisition apparatus 100 repeats the above processing. .
- step SB-3 when ⁇ -saddle disappears (step SB-3, No), as shown in FIG. 38, the operation word assigning unit 102e determines whether or not a saddle point (saddle point) exists (step SB3). SB-15).
- step SB-15 If there is a saddle point (step SB-15, Yes), the operation word assigning unit 102e determines whether there is a saddle point existing in the FIG. Right eye pattern (step SB-16).
- step SB-16 If saddle point present in Figure eight pattern is k pieces (step SB-16, Yes), the operation word imparting unit 102e, the operation term B 0 and the k grant (step SB-17), streamline analysis unit 102b Performs the operation of replacing the innermost figure eight pattern on the streamline diagram with a closed orbit around one center / disk k times (step SB-18).
- step SB-16, No when there is no saddle point that exists in the FIG. Eight pattern (step SB-16, No), there is a ss-saddle connection that has a homoclinic saddle connection and that has a center / disk inside.
- the operation word assigning unit 102e assigns the operation word A 0 (step SB-19), the streamline analysis unit 102b assigns a saddle point and a homoclinic saddle connection connecting them, and The operation of deleting the center / disk surrounded by is performed k times (step SB-20).
- step SB-15 when there is still a saddle point (step SB-15, Yes), the fluid transition path acquisition apparatus 100 repeats the above-described processing.
- step SB-15, No the fluid transition path acquisition apparatus 100 finishes the I-sequence operation word assignment process in the algorithm B, and the word formed by the word expression forming unit 102c.
- the expression is stored in the pattern transition file 106c in association with the force received by the object at that time under the control of the transition information acquisition unit 102g.
- the maximum word representation portion 102f of the word representations forming unit 102c select one A 2 from the resulting string, it is moved to the beginning of the string is replaced with I, the rest of the string If rearranged according to the rules, it can be converted into a maximal word.
- FIG. 39 is a flowchart illustrating an example of II-series operation word assignment processing in algorithm B. It should be noted that, similar to step SB-1 and step SB-2 of the I-sequence operation word assignment process, streamline diagram creation and pre-processing may be performed by streamline analysis.
- the operation word assigning unit 102e determines whether there is a saddle-saddle for the given streamline diagram (step SC-1).
- step SC-1 If ⁇ -saddle exists (step SC-1, Yes), the operation word assigning unit 102e has a boundary where there are exactly two ⁇ -saddles, and has a ⁇ -saddle connection on the boundary. Is determined (step SC-2).
- Step SC-2 If those just having ⁇ -saddle connection including the boundary of the two ⁇ -saddle there is k pieces (Step SC-2, Yes), the operation word imparting unit 102e, the operation term B 2 and k pieces grant ( In step SC-3), the streamline analysis unit 102b performs an operation of replacing ⁇ -saddle and ⁇ -saddle connection on the same boundary with one closed orbit on the streamline diagram k times (step SC-4). .
- step SC-2 when there is no one having a saddle connection including a boundary where exactly two saddles exist (step SC-2, No), the operation word assigning unit 102e has a boundary having four or more saddles. It is determined whether or not there is a s-saddle connection on the same boundary with a center / disk inside (step SC-5).
- step SC-5 When the ⁇ -saddle connection on the boundary including four or more ⁇ -saddles has k boundaries that are center / disk inside (step SC-5, Yes), the operation word assigning unit 102e K words C are assigned (step SC-6), and the streamline analysis unit 102b has a saddle connection on the same boundary on the streamline diagram, and two saddle-saddles connected thereby, The operation of deleting the center / disk surrounded by them is performed k times (step SC-7).
- step SC-5 No
- the innermost figure eight exists. since, with the number and k-number, the operation word imparting unit 102e, the operation term B 0 and the k grant (step SC-8), streamline analysis unit 102b, in the drawing flow lines is most inwardly
- the operation of replacing the figure eight pattern with a closed orbit surrounding one center / disk is performed k times (step SC-9).
- step SC-1 the processing returns to step SC-1, and when there is still a saddle-saddle (step SC-1, Yes), the fluid transition path acquisition apparatus 100 repeats the above processing. .
- step SC-10 determines whether or not a saddle point (saddle point) exists.
- step SC-10 If there is a saddle point (step SC-10, Yes), the operation word assigning unit 102e determines whether there is a saddle point existing in the figure eight pattern (step SC-11).
- step SC-11 If saddle point present in Figure eight pattern is k pieces (step SC-11, Yes), the operation word imparting unit 102e, the operation term B 0 and the k grant (step SC-12), streamline analysis unit 102b Performs the operation of replacing the innermost figure eight pattern on the streamline diagram with a closed orbit around one center / disk k times (step SC-13).
- step SC-11 when there is no saddle point that exists in the FIG. Eight pattern (step SC-11, No), there is an ss-saddle connection that has a homoclinic saddle connection and that has a center / disk inside.
- step SC-15 The operation of deleting the center / disk surrounded by them is performed k times (step SC-15).
- step SC-10 the fluid transition path acquisition apparatus 100 repeats the above-described processing.
- step SC-10, No the fluid transition path
- FIG. 40 is a flowchart illustrating an example of O-sequence operation word assignment processing in algorithm B. It should be noted that, similar to step SB-1 and step SB-2 of the I-sequence operation word assignment process, streamline diagram creation and pre-processing may be performed by streamline analysis.
- the operation word assigning unit 102e determines whether there is a saddle-saddle for the given streamline diagram (step SD-1).
- step SD-1 When ⁇ -saddle exists (step SD-1, Yes), the operation word assigning unit 102e determines whether or not there is a ⁇ -saddle connection on the same boundary among the boundaries where two ⁇ -saddles exist. Is determined (step SD-2).
- step SD-2 If those just having ⁇ -saddle connection boundary where two ⁇ -saddle there is the k (step SD-2, Yes), the operation word imparting unit 102e, the operation term B 2 to the k grant (step SD-3), the streamline analysis unit 102b deletes the ⁇ -saddle and the boundary connected by the ⁇ -saddle connection on the same boundary on the streamline diagram, and replaces it with a single closed orbit. Is performed k times (step SD-4).
- step SD-2 when there is no one having a ⁇ -saddle connection including a boundary where exactly two ⁇ -saddles exist (step SD-2, No), the operation word assigning unit 102e has a boundary having four or more ⁇ -saddles. It is determined whether or not there is a s-saddle connection on the same boundary with a center / disk inside (step SD-5).
- step SD-5 When the k-saddle connection on the boundary including four or more ⁇ -saddles has k boundaries such that the inside is center / disk (step SD-5, Yes), the operation word assigning unit 102e performs the operation K words C are assigned (step SD-6), and the streamline analysis unit 102b has a saddle connection on the same boundary on the streamline diagram, and two saddle-saddles connected thereby, The operation of deleting the center / disk surrounded by them is performed k times (step SD-7).
- step SD-5, No if there is no boundary that is center / disk inside the ⁇ -saddle connection on the boundary including 4 or more ⁇ -saddles (step SD-5, No), there is the innermost figure eight since, with the number and k-number, the operation word imparting unit 102e, the operation term B 0 and the k grant (step SD-8), streamline analysis unit 102b, in the drawing flow lines is most inwardly
- step SD-9 The operation of replacing the figure eight pattern with a closed orbit surrounding one center / disk is performed k times (step SD-9).
- step SD-1 the processing returns to step SD-1, and when there is still a saddle-saddle (step SD-1, Yes), the fluid transition path acquisition apparatus 100 repeats the above processing. .
- step SD-10 the operation word assigning unit 102e determines whether or not a saddle point (saddle point) exists (step SD-10).
- step SD-10 If the saddle point is the k exists (step SD-10, Yes), the operation word imparting unit 102e, the operation term B 0 and the k grant (step SD-11), streamline the analysis unit 102b, streamline diagram
- step SD-12 the operation of replacing the innermost figure eight pattern with a closed orbit around one center / disk is performed k times (step SD-12).
- step SD-10 the fluid transition path acquisition apparatus 100 repeats the above-described processing.
- step SD-10, No the fluid transition path acquisition apparatus 100 finishes the O-sequence operation word assignment process in the algorithm B. If the obtained character string is prefixed with O and rearranged according to the rule, it can be converted into a maximal word.
- FIG. 41 and FIG. 42 are flowcharts showing another example of I and II sequence operation word assignment processing in algorithm B (I, II-Word algorithm).
- the creation and preprocessing of the streamline diagram may be performed by streamline analysis.
- the following operation word assignment process when an operation word is given, it is given from the back to the front.
- the operation word assigning unit 102e is not connected to the 1-source-sink point for the streamline diagram preprocessed by the streamline analysis unit 102b. It is determined whether there is a saddle or ⁇ -saddle connection (step SI-1).
- the operation word assigning unit 102e When there is a saddle that is not connected to a suction-out pair (1-source-sink point) or there is a saddle-saddle connection (step SI-1, Yes), the operation word assigning unit 102e has exactly two It is determined whether there is a boundary having ⁇ ⁇ -saddle connection at the boundary where ⁇ -saddle exists (step SI-2).
- Step SI-2 Just at the boundary where two ⁇ -saddle there, if one with the ⁇ -saddle connection is k pieces (Step SI-2, Yes), the operation word imparting unit 102e, the operation term B 2 and k pieces grant ( In step SI-3), the streamline analysis unit 102b deletes the ⁇ -saddle and the boundary connected by the ⁇ -saddle connection on the same boundary on the streamline diagram, and replaces it with a single closed orbit. The operation is performed k times (step SI-4).
- step SI-2 In the case where there is no boundary having ⁇ -saddle connection in the boundary where exactly two ⁇ ⁇ ⁇ ⁇ -saddles exist (step SI-2, No), and there is the innermost figure eight pattern (step SI-5) yes), the operation word imparting unit 102e impart operation term B 0, operation term imparting unit 102e, the drawing flow lines, replacing the Figure eight patterns in most inwardly single center / disk around the closed orbit An operation is performed (step SI-6). This operation (operation that assigns the operation word B 0 and replaces it with closed orbit) is repeatedly executed as much as possible.
- step SI-5 when there is no saddle point that exists in the innermost figure eight pattern (step SI-5, No), that is, ⁇ -saddle connection on the boundary including four or more ⁇ -saddles, and the inside is center / disk
- the operation word assigning unit 102e assigns the operation word C
- the streamline analysis unit 102b is connected to the saddle connection on the boundary on the streamline diagram.
- the operation of deleting the two saddles-saddle and the center / disk surrounded by them is repeated (step SI-7). This operation (operation for assigning operation word C and deleting center / disk) is repeated as much as possible.
- step SI-1 the processing is returned to step SI-1, and there is a saddle that is not yet connected to the suction source pair (1-source-sink point), or there is a saddle-connection.
- step SI-1, Yes the fluid transition path acquisition apparatus 100 repeats the above-described processing.
- step SI-1, No when there is only a saddle connected to a suction-outflow pair (1-source-sink point) and there is no longer a saddle-saddle connection (step SI-1, No), as shown in FIG.
- the operation word assigning unit 102e determines whether ss- ⁇ -saddle exists (step SH-1).
- step SH-1 When ss- ⁇ -saddle does not exist (step SH-1, No), the streamline analysis unit 102b calculates a saddle point (saddle point), a homoclinic saddle connection connecting it, and a center / disk surrounded by them.
- the erase operation is performed once (step SH-2).
- step SH-3 When the ss-saddle is the k, the operation word imparting unit 102e, the operation term A 0 to the k grant (step SH-3).
- the streamline analysis unit 102b performs k times the operation of deleting the saddle point (saddle point), the homoclinic saddle connection connecting the saddle point, and the center / disk surrounded by them (step SH-4).
- the pattern word assigning unit 102d assigns the pattern word II (step SH-5), and the fluid transition path acquisition apparatus 100 assigns the operation words of the I and II series in the algorithm B (I, II-Word algorithm). Finish the process.
- the obtained character string is a maximal word.
- step SH-1 when ss- ⁇ -saddle exists in step SH-1 (step SH-1, Yes), the streamline analysis unit 102b has a boundary between these two ⁇ -saddles on the streamline diagram ( The operation of deleting (boundary) and replacing it with one ss-orbit is performed once (step SH-7).
- step SH-8 When the ss- ⁇ -saddle is the k, the operation word imparting unit 102e, the operation term A 2 to the k grant (step SH-8).
- the streamline analysis unit 102b deletes these two ⁇ -saddles and the boundary on the streamline diagram and replaces them with one ss-orbit k times (step SH-9). .
- step SH-10 the operation word imparting unit 102e, when the ss-saddle is the k, the operation term A 0 to the k grant (step SH-10).
- the streamline analysis unit 102b performs k times the operation of deleting the saddle point (saddle point), the homoclinic saddle connection connecting the saddle point, and the center / disk surrounded by them (step SH-11).
- the pattern word assigning unit 102d assigns the pattern word I (step SH-12), and the fluid transition path acquisition apparatus 100 assigns the operation words of the I and II series in the algorithm B (I, II-Word algorithm). Finish the process.
- the obtained character string is a maximal word.
- FIG. 43 is a flowchart illustrating another example of the O-sequence operation word assignment process in algorithm B (O-Word algorithm). It should be noted that, as in the above-described allocation processing step SB-1 and step SB-2, the creation and preprocessing of the streamline diagram may be performed by streamline analysis. In addition, in the following operation word assignment process, when an operation word is given, it is given from the back to the front.
- the operation word assigning unit 102e determines whether there is a saddle or a saddle-saddle for the given streamline diagram (step SJ-1).
- step SJ-1 When saddle or ⁇ -saddle exists (step SJ-1, Yes), the operation word assigning unit 102e has a boundary where two ⁇ ⁇ ⁇ ⁇ -saddles exist and has ⁇ ⁇ ⁇ -saddle connection on the same boundary. (Step SJ-2).
- step SJ-2 If those having a ⁇ -saddle connection connecting boundary just two ⁇ -saddle there is the k (step SJ-2, Yes), the operation word imparting unit 102e, the operation term B 2 to the k grant ( Step SJ-3).
- the streamline analysis unit 102b deletes k-saddle and boundary bound by the ⁇ -saddle connection on the boundary on the streamline diagram, and replaces it with a single closed orbit k times. Perform (Step SJ-4).
- step SJ-2 if there is no ⁇ -saddle connection on the boundary connecting the boundaries where exactly two ⁇ -saddles exist (step SJ-2, No), the operation word assigning unit 102e has the figure eight at the innermost side. It is determined whether or not there is (step SJ-5).
- step SJ-5 If there is an innermost figure eye (that is, a saddle connection and at least one center / disk inside it) (step SJ-5, Yes), the operation word assigning unit 102e performs the operation The word B 0 is assigned, and the streamline analysis unit 102b performs an operation of replacing it with a closed orbit surrounding one center / disk on the streamline diagram (step SJ-6).
- This operation (operation that assigns the operation word B 0 and replaces it with closed orbit) is repeatedly executed as much as possible.
- step SJ-5 when there is no inner figure eight (step SJ-5, No), that is, there is a boundary that is center / disk with a saddle connection on the boundary including four or more saddles.
- the operation word assigning unit 102e assigns the operation word C
- the streamline analysis unit 102b has a saddle connection on the boundary on the streamline diagram and two saddles connected thereby.
- the saddle and the center / disk surrounded by them are deleted (step SJ-7). This operation (operation for assigning operation word C and deleting center / disk) is repeated as much as possible.
- step SJ-1 the processing returns to step SJ-1, and if there is still saddle or saddle-saddle (step SJ-1, Yes), the fluid transition path acquisition apparatus 100 has been described above. Repeat the process.
- step SJ-1, No the fluid transition path acquisition apparatus 100 finishes the O-sequence operation word assignment processing in algorithm B (O-Word algorithm).
- O-Word algorithm O-Word algorithm
- the topological structures of all structurally stable flows can be expressed by enumerating the operation words.
- the set of patterns that it expresses may change.
- the set of patterns represented by the former has an inclusion relationship included in the set of patterns represented by the latter. That is, if a set symbol is used, (IB 0 A 0 ) ⁇ (IA 0 B 0 ) is obtained.
- the word expression on the side to be included is excluded and only a larger word expression is adopted.
- This relational expression is symbolically expressed as B 0 A 0 ⁇ A 0 B 0 .
- a combination of word replacements that gives rise to such an inclusive relationship is given by Lemma 3.7 (see published paper (2013)).
- Theorem 3.3 gives the standard form of the maximal word expression for the O series
- Theorem 3.4 gives the standard form of the maximal word expression for the I series
- the Theorem 3.5 for the II series.
- the transition information obtained in the optimization method of structure design can be utilized as an example.
- a flow pattern transition order, an intermediate state flow pattern, and the like can be obtained.
- Control may be performed so that an optimal structure is searched for by the simulation unit 102a under the constraints based on the above.
- the simulation unit 102a can escape the local minimum in the annealing method while the next flow pattern of the current flow pattern is not obtained by the simulation.
- the global parameter T temperature
- the design variable may be set high so as to allow a large change to the design variable.
- the simulation unit 102a has a search range that defines a set desired pattern in accordance with the match / mismatch with the word expression on the transition route. You may adjust so that it may become a pattern. For example, similarly to the above, in the annealing method, the simulation unit 102a sets the global parameter T (temperature) high so as to allow a large change to the design variable when the search range does not match the desired pattern. On the other hand, when the search range matches the desired pattern, the global parameter T (temperature) may be set low so as to obtain a small change in the design variable.
- the design of the pier arrangement the design of the breakwater arrangement, the efficient removal of pollutants in the harbor, the design of the shape of the wing and aircraft of the wind power generation, the oil fence, etc. Can be efficiently identified as transition paths.
- the present invention is not limited thereto, and may be applied to a two-dimensional fluid simulation. It ’s good.
- the fluid transition path acquisition apparatus 100 performs processing in response to a request from a client terminal, and the processing result is You may make it return to a client terminal.
- all or a part of the processes described as being automatically performed can be manually performed, or all of the processes described as being manually performed can be performed.
- a part can be automatically performed by a known method.
- each illustrated component is functionally conceptual and does not necessarily need to be physically configured as illustrated.
- each device of the fluid transition path acquisition device 100 is performed by a CPU (Central Processing Unit) and the CPU. It may be realized by a program to be interpreted and executed as hardware by wired logic. The program is recorded on a recording medium to be described later, and is mechanically read by the fluid transition path acquisition apparatus 100 as necessary. That is, in the storage unit 106 such as a ROM or HD, a computer program for performing various processes by giving instructions to the CPU in cooperation as an OS (Operating System) is recorded. This computer program is executed by being loaded into the RAM, and constitutes the control unit 102 in cooperation with the CPU.
- OS Operating System
- the computer program may be stored in an application program server connected to the fluid transition path acquisition apparatus 100 via an arbitrary network 300, and the computer program may be downloaded in whole or in part as necessary. Is also possible.
- the program according to the present invention may be stored in a computer-readable recording medium, or may be configured as a program product.
- the “recording medium” is any memory card, USB memory, SD card, flexible disk, magneto-optical disk, ROM, EPROM, EEPROM, CD-ROM, MO, DVD, Blu-ray Disc, etc. Of “portable physical media”.
- program is a data processing method described in an arbitrary language or description method, and may be in any form such as source code or binary code. Note that the “program” is not necessarily limited to a single configuration, but is distributed in the form of a plurality of modules and libraries, or in cooperation with a separate program typified by an OS (Operating System). Including those that achieve the function.
- OS Operating System
- a well-known structure and procedure can be used about the specific structure for reading a recording medium in each apparatus shown in embodiment, a reading procedure, or the installation procedure after reading.
- Various databases and the like (simulation result file 106a, streamline file 106b, pattern transition file 106c, etc.) stored in the storage unit 106 are a memory device such as a RAM and a ROM, a fixed disk device such as a hard disk, a flexible disk, and
- the storage means such as an optical disk stores various programs, tables, databases, web page files, and the like used for various processes and website provision.
- the fluid transition path acquisition apparatus 100 may be configured as an information processing apparatus such as a known personal computer or workstation, or may be configured by connecting an arbitrary peripheral device to the information processing apparatus.
- the fluid transition path acquisition apparatus 100 may be realized by installing software (including programs, data, and the like) that causes the information processing apparatus to realize the method of the present invention.
- the specific form of distribution / integration of the devices is not limited to that shown in the figure, and all or a part of them may be functional or physical in arbitrary units according to various additions or according to functional loads. Can be distributed and integrated. That is, the above-described embodiments may be arbitrarily combined and may be selectively implemented.
- Embodiment 2 A fluid transition path acquisition device, a fluid transition path acquisition method, and a program according to Embodiment 2 will be described with reference to FIGS. 44 to 66.
- a plurality of flow patterns may correspond to word expressions. Therefore, as in the first embodiment, the method of specifying the transition pattern X using only the word expression only gives candidates (necessary conditions) of possible transitions, and the corresponding intermediate state is actually Whether it can be realized or not must be confirmed by streamline structure pattern matching to check whether the structure obtained corresponding to the subword actually exists.
- a regular expression which is a new expression method that completely corresponds to the flow pattern
- the transition pattern X is specified using the regular expression in addition to the word expression.
- a method for automatically specifying the transition pattern X without pattern matching is proposed.
- the streamline pattern and the regular expression correspond completely one-to-one, and the local streamline structure corresponding to the transition pattern X is changed. Note that they can be described in advance as changes in a part of a regular expression (referred to as a partial expression).
- FIG. 44 is a flowchart for explaining the outline of the flow pattern regular expression method according to the second embodiment.
- the flow pattern regular expression method is the topological pattern of the flow pattern in the multiple connected outer region having N holes (where N is an integer of 1 or more).
- a regular expression is created, and roughly divided from a graph expression creating step (step S21) for creating a graph expression corresponding to the flow pattern on a one-to-one basis, and a graph expression created in the graph expression creating step in step S12.
- V is called a vertex
- a set of points, E is a set of edges connecting vertices) and may be visualized as a planar graph.
- the graph representation is such that the parent vertex is v, the child vertex is w, the label assigned to the parent vertex v is l (v), and the label assigned to the child vertex w is l (w), v.
- the child vertex set ⁇ (v) is used, the child vertex set ⁇ (v) of v is rearranged according to a rule of a predetermined order relation, and from w ( ⁇ ) to l (w) for w ⁇ (v). You may include what was drawn with the arrow arranged from left to right.
- the flow patterns are (1) two ss- ⁇ -saddle connections, which have a suction-and-out pair of two types of flow patterns that can be taken topologically in a single connected outer region having one hole.
- One or more pattern words consisting of a pattern II having a homoclinic saddle connection and two ss-saddle connections, and (3) a pattern O having no suction spring-out pair in a double-connected external region having two holes. It may be.
- the word representation does not have a suction-out pair in a double connected outer region with two holes.
- the operation words that define the five types of operations that can be taken topologically when adding one hole to the flow pattern with respect to the pattern words that define a total of three types of flow patterns with added patterns It is good also as a symbol word formed by giving any one word by the number of the added holes.
- FIG. 45 is a flowchart for explaining an example of the graph expression creating process of FIG.
- the graph expression creation step first, when the flow pattern has a suction-and-out pair, the region including the counterclockwise ss-orbit closest to the suction-and-out pair is the outermost region. (Step S31).
- the flow pattern starts from one or more of the above-described pattern words, and is one of the operation words that define five types of operations that can be taken topologically when a hole is added to the flow pattern. It is good also as a flow pattern figure with the number of the holes created by repeating the operation which provides a word until the number of holes becomes N pieces.
- the route of the ss-saddle connection diagram (flow pattern diagram) of I-word and II-word is converted as shown in FIG.
- the trajectory that appears in the (ss-) saddle connection diagram of the converted flow pattern is extracted from the entire region (step S32).
- a vertex is set to the connected component obtained by excluding all trajectories appearing in the (ss-) saddle connection diagram from the entire region, and the outermost connected component is set as the root (step S33).
- the current component is set as the root (step S34).
- a connected component that touches the boundary with the current component is a child of the current component, labels are assigned according to the trajectory corresponding to the boundary, and they are arranged in accordance with a predetermined label order relationship (step S35).
- the child of the current component is set as the current component, and step S35 is repeatedly executed until there are no more children (step S36).
- the graph T (V, E) is a set given as a pair of a set of points (vertex set) called “vertex” V and a set E called “edge” connecting the vertices. It is.
- a graph can have various structures, but in the graph representation theory of the present embodiment, the following set of graphs having a specific structure in the entire set of graphs is considered.
- Tree and tree refer to a graph in which two arbitrary vertices are connected only by one edge.
- a routed graph refers to a graph in which a specific vertex (hereinafter referred to as a root) exists.
- a root a specific vertex
- “Directed” means that all edges have a parent-child order.
- the edge from the vertex v ⁇ V to w ⁇ V in the directed graph is expressed as v ⁇ w ⁇ E.
- v is called the parent of w and w is called the child of v.
- Write ⁇ (v) to represent the set of all children of vertex v ⁇ V. That is, ⁇ (v): ⁇ w ⁇ V
- the number of child vertices # ⁇ (v) included in the set is called v out-degree, and conversely, the number of edges entering v is called v in-degree.
- a graph is labeled as being labeled with a specific label assigned to all vertices.
- a tree with root, label, and orientation is considered. Further, the root is considered to be a vertex where in-degree is zero, that is, there is no incoming edge.
- the labels of the vertices, ⁇ o ⁇ , o 0, o 2, + ⁇ , + 0, + 2, - ⁇ , - 0, - 2 ⁇ shall assign one of.
- Vo Vo ⁇ UVo 0 UVo 2
- V + V + ⁇ UV + 0 UV + 2
- V ⁇ V ⁇ ⁇ UV ⁇ 0 UV ⁇ 2
- V 0 Vo 0 ⁇ V + 0 -V ⁇ 0
- V 2 Vo 2 UV + 2 UV ⁇ 2
- U disjoint unit.
- ⁇ (v) can be divided into a child vertex set of vertex v as follows.
- H a structurally stable Hamiltonian vector field expressed by O-word on the two-dimensional region D z (M), and let D be its saddle connection diagram.
- a set C H D z (M) ⁇ D consisting of several connected components (including infinite periodic orbits) is an open set. it can.
- Each connected component is set as a vertex set V. Apex of the root is a connected component in the outermost of these connected components, when periodic orbits included as the label is counterclockwise + phi, when a clockwise - assign phi.
- FIG. 46 shows all streamline patterns generated by the operations B 0 , B 2 , and C and the parent-child relationships that are excited by the connected subset (vertex set) correspondingly.
- (A) is B 0
- (b) is B 0
- (c) is B 0 B 0
- (d) is B 2 C k ⁇ 1 , k ⁇ 1
- (e) is B 2 C 1 , l ⁇ 1
- (F) is B 2 2 C k + l ⁇ 1 , k, l ⁇ 1
- (g) is B 0 B 2 C k , k ⁇ 1
- (h) is B 0 B 2 C l , l ⁇ 1.
- v represents a parent connected subset
- its child connected subset is expressed as w, y, z, and the like. Since the broken line represents a set of parent connected parts, only one periodic trajectory included therein is drawn with a direction.
- FIG. 46 (a) can be constructed character pattern outward 8 made by the operation B 0.
- the number of elements included in the set of child connected components is # ⁇ + 0 (v), # ⁇ ⁇ 0 (v) ⁇ 2, and any non-negative integer can be selected for the original numbers # ⁇ + 2 (v) and # ⁇ 2 (v) included in the child vertex set.
- the order is arranged cyclically with respect to the connected portion written as j in FIG. That is, a specific connected portion is selected as y 1 from these, and thereafter the connected components are arranged counterclockwise.
- the child connected components written as j j can be simply arranged counterclockwise.
- N a set of non-negative integers
- N ⁇ a set of finite sequences of non-negative integers
- ⁇ ⁇
- G a tree
- s, s ′, t, u vertex ids
- T a subset of vertex ids.
- FIGS. 47A to 47E are flowcharts for explaining the conversion process of a saddle connection diagram to a tree in the O series. 47.
- a conversion process to a tree of saddle connection diagrams in the O series shown in FIGS. 47-A to 47-E can be executed by a device such as a computer.
- step S101 input is set as saddle connection diagram D
- step S101 saddle connection diagram D is converted so that there is root in outermost
- step S106 it is determined whether or not root 0 has the shape (pattern) shown in FIG. If root 0 has the form shown in FIG. 46 (a) (“Yes” in step 106), the vertices id of w 1 and w 2 are set to 00, 01, and the tree G is changed to FIG. 46 (a).
- step S106 if root 0 does not have the shape of FIG. 46 (a) (“No” in step 106), the process proceeds to step S108.
- step S108 it is determined whether or not root 0 has the left shape of FIG. If root 0 has the left shape of FIG. 46 (b) (“Yes” in step 108), the vertex id of w is set to 00, and the tree G is labeled as the tree of FIG. 46 (b). Is replaced with a set of id and label (ie, G is ((0, ⁇ ⁇ ) ⁇ (00, ⁇ 0 ))), and ⁇ 00 ⁇ is added to the set T (step S109). That is, T ⁇ TU ⁇ 00 ⁇ . Thereafter, the process proceeds to step S114.
- step S108 if root 0 does not have the shape on the left in FIG. 46B ("No" in step S108), the process proceeds to step S110.
- step S110 it is determined whether or not root 0 has the shape shown in FIG. If root 0 has the form of FIG. 46D (“Yes” in step S110), y 1 ,. . . , Y k with vertex ids 00,. . . , 0k ⁇ 1, and the tree G is defined as a tree in which the label of the tree in FIG. 46D is replaced with a set of id and label, and ⁇ 00,. . . , 0k-1 ⁇ is added (step S111). That is, T ⁇ TU ⁇ 00,. . . , 0k-1 ⁇ . Thereafter, the process proceeds to step S114.
- step S110 if root 0 does not have the shape of FIG. 46D in step S110 (“No” in step S11), the tree G is defined as ⁇ ⁇ (step S112), and this tree G is output. (Step S113), the process ends.
- step S114 it is determined whether or not an element t greater than s exists in T.
- s is the largest element in T (“No” in step S114)
- all the vertex ids of the tree G are removed, the remaining tree is set as G (step S115), and the tree G is output (step S113). ), The process is terminated.
- step S117 it is determined whether or not the outer boundary of the vertex s is composed of one homoclinic saddle connection and saddle, and the inner boundary s of the vertex s has the shape shown in FIG.
- the outer boundary of the vertex s is composed of one homoclinic saddle connection and saddle, and the inner boundary of the vertex s has the shape of FIG. 46A (“Yes” in step 117)
- w 1 , w 2 Are defined as s0 and s1
- the tree G is defined as a tree obtained by replacing the label of the tree in FIG. 46A with a set of id and label, and ⁇ s0, s1 ⁇ is added to T (step S118). . Thereafter, the process returns to step S114.
- step S117 if the outer boundary of the vertex s is not one homoclinic saddle connection and saddle in step S117, or the inner boundary of the vertex s does not have the shape of FIG. 46A ("No" in step S117). The process proceeds to step S119.
- step S119 it is determined whether or not the outer boundary of the vertex s is composed of one homoclinic saddle connection and saddle, and the inner boundary of the vertex s has the left shape of FIG. 46 (b). If the outer boundary of the vertex s is composed of one homoclinic saddle connection and saddle and the inner boundary of the vertex s has the left shape of FIG. 46B (“Yes” in step S119), the vertex id of w Is defined as s0, the tree G is defined as a tree in which the label of the left tree in FIG. 46B is replaced with a set of id and label, and ⁇ s0 ⁇ is added to T (step S120). Thereafter, the process returns to step S114.
- step S119 if the outer boundary of the vertex s is not one homoclinic saddle connection and saddle in step S119, or if the inner boundary of the vertex s does not have the shape on the left in FIG. '), The process proceeds to step S121.
- step S121 it is determined whether or not the outer boundary of the vertex s is composed of one homoclinic saddle connection and saddle, and the inner boundary of the vertex s has the shape shown in FIG.
- the outer boundary of the vertex s is composed of one homoclinic saddle connection and saddle, and the inner boundary of the vertex s has the shape of FIG. 46D (“Yes” in step 121)
- the tree G is defined as a tree in which the label of the tree in FIG. 46D is replaced with a pair of id and label, and ⁇ s0,. . . , Sk-1 ⁇ is added (step S122). Thereafter, the process returns to step S114.
- step S121 when the outer boundary of the vertex s is not one homoclinic saddle connection and saddle in step S121, or the inner boundary of the vertex s does not have the shape of FIG. 46 (d) (“No” in step S121). The process proceeds to step S123.
- step S123 it is determined whether or not the vertex s has the left shape of FIG.
- the process proceeds to step S124.
- step S124 the one sigma label sigma 0 vertex s is + (i.e., flow in the s is either a counter-clockwise) to determine.
- ⁇ of the label ⁇ 0 of the vertex s is + (ie, the flow in s is counterclockwise) (“Yes” in step S124)
- the vertex ids of w 1 , w 2 , and w 3 are set.
- s0, s1, s2 are determined, and the tree G is determined as a tree in which the label of the left tree in FIG. 46C is replaced with a set of id and label, and ⁇ s0, s1, s2 ⁇ is added to T (step S125). Thereafter, the process returns to step S114.
- step S124 when ⁇ of label ⁇ 0 of vertex s is not +, that is, when ⁇ of label ⁇ 0 of vertex s is ⁇ (that is, the flow in s is clockwise) (
- step S124, “No”) the vertices id of w 1 , w 2 , w 3 are defined as s1, s2, s0, and the tree G is set to the pair of id and label on the left tree in FIG. 46 (c). The replaced tree is determined, and ⁇ s0, s1, s2 ⁇ is added to T (step S126). Thereafter, the process returns to step S114.
- step S123 when the vertex s does not have the left shape of FIG. 46 (c) (“No” in step S123), the process proceeds to step S127.
- step S127 it is determined whether or not the vertex s has the right shape of FIG. If the vertex s has the right shape in FIG. 46C (“Yes” in step S127), the process proceeds to step S128.
- step S1208 it is determined whether or not ⁇ of label ⁇ 0 of vertex s is + (that is, the flow in s is counterclockwise).
- ⁇ of label ⁇ 0 of vertex s is + (ie, the flow in s is counterclockwise) (“Yes” in step S128)
- the vertex ids of w 1 and w 2 are set to s0 and s1.
- the tree G is defined as a tree in which the label of the right tree in FIG. 46C is replaced with a set of id and label, and ⁇ s0, s1 ⁇ is added to T (step S129). Thereafter, the process returns to step S114.
- step S1208 when ⁇ of label ⁇ 0 of vertex s is not +, that is, when ⁇ of label ⁇ 0 of vertex s is ⁇ (that is, the flow in s is clockwise) ( In step S128, “No”), the vertex ids of w 1 and w 2 are defined as s1 and s0, and the tree G is defined as a tree in which the label of the right tree in FIG. , T is added to ⁇ s0, s1 ⁇ (step S130). Thereafter, the process returns to step S114.
- step S127 when the vertex s does not have the right shape of FIG. 46C (“No” in step S127), the process proceeds to step S131.
- step S131 it is determined whether or not the vertex s has the shape shown in FIG.
- y 1 ,. . . , Y l , w are the vertex ids s1,. . . , Sl, s0
- the tree G is defined as a tree in which the label of the tree in FIG. 46 (h) is replaced with a set of id and label, and ⁇ s0,. . . , Sl ⁇ are added (step S132). Thereafter, the process returns to step S114.
- step S131 when the vertex s does not have the shape of FIG. 46 (h) (“No” in step S131), the process proceeds to step S133.
- step S133 when the vertex s does not have the left shape of FIG. 46 (g) (“No” in step S133), the process proceeds to step S135.
- step S135 it is determined whether or not the vertex s has the right shape of FIG.
- w, z 1 ,. . . , Z k vertices id are s0, s1,. . . , Sk
- the tree G is defined as a tree in which the label of the right tree in FIG. 46G is replaced with a set of id and label, and ⁇ s0,. . . , Sk ⁇ is added (step S136). Thereafter, the process returns to step S114.
- step S135 when the vertex s does not have the right shape of FIG. 46G (“No” in step S135), the process proceeds to step S138 (in this case, as shown in step S137, the vertex s has the form of FIG. 46 (f)).
- step S138 it is determined whether or not ⁇ of label ⁇ 0 of vertex s is + (that is, the flow in s is counterclockwise).
- ⁇ of label ⁇ 0 of vertex s is + (ie, the flow in s is counterclockwise) (“Yes” in step S138)
- the tree G is defined as a tree in which the label of the tree in FIG. 46F is replaced with a set of id and label, and ⁇ s0,. . . , Sl + k ⁇ 1 ⁇ are added (step S139). Thereafter, the process returns to step S114.
- step S138 when ⁇ of label ⁇ 0 of vertex s is not +, that is, when ⁇ of label ⁇ 0 of vertex s is ⁇ (that is, the flow in s is clockwise) (step S138).
- “No”) y 1 ,. . . , Y k , z 1 ,. . . , Z l with the vertex ids sl,. . . , Sl + k-1, s0,. . . , Sl-1 and the tree G is defined as a tree in which the label of the tree in FIG. 46 (f) is replaced with a set of id and label, and ⁇ s0,. . . , Sl + k-1 ⁇ is added (step S140). Thereafter, the process returns to step S114.
- the above operation is the conversion process to the tree of the saddle connection diagram in the O series.
- D is an ss-saddle connection diagram that can be expressed in I-word or II-word.
- the broken line is one ss-orbit included in the connected component that is the root, along with its direction.
- (a) is a ss-saddle connection diagram represented by the word expression IA 0 A 0 , and the flow direction of the ss- ⁇ -saddle connection is from right to left.
- (B) is a ss-saddle connection diagram having the word expression IA 0 A 0, and the direction of the ss- ⁇ -saddle connection is from left to right.
- (C) is a ss-saddle connection diagram having the word expression IIA 0 A 0, and the direction of the periodic orbit included in the outermost connected component is clockwise.
- the word expression IIA 0 A 0 is the same as (c), but the direction of the periodic orbit is opposite.
- each ss-saddle connected diagram an image of a ss-saddle connection diagram is shown which is formed by an equiangular mapping that infinitely maps the points in the connected component that is the root.
- this mapped image all ss-orbits included in the route are counterclockwise.
- the connected component (the curve drawn with a broken line is directly above 1-source-sink point). Included area) becomes the root. If the same ss-saddle connection diagram has the same shape but the direction of flow is reversed as shown in FIG. 48 (b), the root becomes a connected component immediately below 1-source-sink point from the definition. .
- the ss-saddle connection diagram with the word representation IIA 0 A 0 in FIG. 48 (c) when the connected component immediately above the 1-source-sink point reverses the direction (FIG. 48 (d)). The connected component directly below is the root.
- the connected component of the root in the image is the outermost component. Note that the ss-orbit is always counterclockwise (see the lower part of each figure in FIG. 48).
- the orientation edge between the vertices is defined as shown in FIG. 48 using a conformal mapping obtained by mapping the root connected component to the outermost side.
- a connected component w ⁇ V with ss-orbit if cl (v) ⁇ cl (w) contains ss-saddle connection (or ss- ⁇ -saddle connection), its child vertex w has o 0 ( Or assign a label of o 2 ).
- FIG. 49 shows the visualization of the local structure of the ss-saddle connection diagram generated by the operations A 0 , A 2 , and C and the corresponding graph representation.
- the parent connected component v is expressed as o ⁇ (o 0 or o 2 ).
- the label of the parent connected component is expressed as o * (where * is either ⁇ , 0, or 2). Since the route is determined so that the direction of the ss-orbit is always counterclockwise, the direction in which the trajectory included in the connected component as a child flows is automatically determined.
- (2-2-3-1. Conversion process of ss-saddle connection diagram in I and II series to tree) 50-A to 50-D are flowcharts for explaining the conversion process of the ss-saddle connection diagram to the tree in the I and II sequences.
- the ss-saddle connection diagram tree conversion process in the I and II series shown in FIGS. 50-A to 50-D can be executed by an apparatus such as a computer.
- step S143 it is determined whether or not root 0 has the shape shown in FIG. If root 0 has the form of FIG. 49A (“Yes” in step S143), the vertex ids of w and y are set to 1, 00, and the tree G is the tree of FIG. 49A.
- a tree in which a label is replaced with a set of id and label that is, G is (1, o 0 ) ⁇ (0, o ⁇ ) ⁇ (00, + 0 )
- step S143 if root 0 does not have the form of FIG. 49A (“No” in step S143), the process proceeds to step S145.
- step S145 it is determined whether or not root 0 has the left shape of FIG. If root 0 has the left shape of FIG. 49B (“Yes” in step S145), the vertex id of w is set to 1, the tree G is the label of the tree of FIG. 49B. The tree replaced with the set of id and label (that is, G is defined as ((0, o ⁇ ) ⁇ (1, o 0 )), and ⁇ 1 ⁇ is added to T (step S146). The process proceeds to step S149.
- step S145 if root 0 does not have the left shape of FIG. 49B (“No” in step S145), the process proceeds to step S148 (in this case, as shown in step S147, root). 0 has the shape of FIG. 49 (d)).
- step S148 w, z 1 ,. . . , Z k vertices id are set to 1, 00,. . . , 0k ⁇ 1, and the tree G is defined as a tree obtained by replacing the label of the tree in FIG. 49D with a set of id and label, and ⁇ 1,0,. . . , 0k ⁇ 1 ⁇ is added, and the process proceeds to step S149.
- step S149 it is determined whether or not an element t larger than u exists in T. If u is the largest element in T (“No” in step S149), all vertex ids of the tree G are removed, the remaining tree is set as G (step S150), and the tree G is output (step S150). S151), the process ends.
- step S153 it is determined whether u is a natural number (that is, corresponding to the ids of o ⁇ , o 0 , o 2 ). If u is not a natural number (“No” in step S153), the process proceeds to S114 in FIG. 47-B, which is an O-word process.
- step S153 if u is a natural number in step S153 (“Yes” in step S153), the process proceeds to step S155.
- step S155 the outer boundary of the vertex u is composed of an outward homoclinic ss-saddle connection and a circle S (see FIG. 4), and the inner boundary of u has the shape of FIG. 49 (a). Determine whether or not.
- the outer boundary of the vertex u is composed of one homoclinic ss-saddle connection and a circle S, and the inner boundary of u has the shape shown in FIG. 49A (“Yes” in step S155)
- w, y Vertices id is defined as u + 1, u0
- the tree G is a tree in which the tree label in FIG.
- step S156 the process returns to step S149.
- step S155 if the outer boundary of the vertex u does not consist of the outward homoclinic ss-saddle connection and the circle S, or if the inner boundary of u does not have the shape of FIG. (“No” in S155), the process proceeds to step S157.
- step S157 it is determined whether or not the outer boundary of the vertex u is composed of the outward homoclinic ss-saddle connection and the circle S, and the inner boundary of u has the left shape of FIG. 49 (b).
- the outer boundary of the vertex u is composed of one homoclinic ss-saddle connection and a circle S, and the inner boundary of u has the left shape of FIG. 49 (b) (“Yes” in step S157)
- the tree G is determined as a tree in which the label of the left tree in FIG. 49B is replaced with a set of id and label, and ⁇ u + 1 ⁇ is added to T (step S158). Thereafter, the process returns to step S149.
- step S157 if the outer boundary of the vertex u does not consist of the outward homoclinic ss-saddle connection and the circle S, or the inner boundary of u does not have the left shape of FIG. (“No” in step S157), the process proceeds to step S159.
- step S159 it is determined whether or not the outer boundary of the vertex u is composed of an outward homoclinic ss-saddle connection and a circle S, and the inner boundary of u has the shape shown in FIG.
- the outer boundary of the vertex u is composed of one homoclinic ss-saddle connection and a circle S, and u is in the shape of FIG. 49 (d) (“Yes” in step S159)
- the tree G is defined as a tree in which the label of the tree in FIG. . . , Uk-1 ⁇ is added (step S160). Thereafter, the process returns to step S149.
- step S159 if the outer boundary of the vertex u does not consist of the outward homoclinic ss-saddle connection and the circle S in step S159, or if the inner boundary of u does not have the shape of FIG. ("No" in S159), the process proceeds to step S161.
- step S161 it is determined whether or not the vertex u has the left shape of FIG. If the vertex u has the left shape of FIG. 49C (“Yes” in step S161), the vertex ids of w, y 1 , y 2 are defined as u + 1, u0, u1, and the tree G is The tree in FIG. 49 (c) is determined as a tree in which the label of the left tree is replaced with a set of id and label, and ⁇ u + 1, u0, u1 ⁇ is added to T (step S162). Thereafter, the process returns to step S149.
- step S161 when the vertex u does not have the left shape of FIG. 49C (“No” in step S161), the process proceeds to step S163.
- step S163 it is determined whether or not the vertex u has the right shape in FIG.
- the vertex ids of w 1 and w 2 are determined as u + 1, u0, and the tree G is transformed into FIG. ) To the right and the tree is replaced with a set of id and label, and ⁇ u + 1, u0 ⁇ is added to T (step S164). Thereafter, the process proceeds to step S149.
- step S163 when the vertex u does not have the right shape of FIG. 49C (“No” in step S163), the process proceeds to step S165.
- step S165 it is determined whether or not the vertex u has the shape shown in FIG.
- the vertex u has the shape of FIG. 49G (“Yes” in step S165)
- the tree G is defined as a tree in which the label of the tree in FIG. 49 (g) is replaced with a set of id and label, and ⁇ u + 1, u0,. . . , Uk ⁇ is added (step S166). Thereafter, the process returns to step S149.
- step S165 when the vertex u does not have the shape of FIG. 49 (g) (“No” in step S165), the process proceeds to step S167.
- step S167 it is determined whether or not the vertex u has the left shape of FIG.
- the vertex id of y l u + 1, u0, u1 ,. . . , Ul, and the tree G is defined as a tree in which the label of the left tree in FIG. 49 (h) is replaced with a set of id and label, and ⁇ u + 1, u0, u1,. . . , Ul ⁇ is added (step S168).
- the process returns to step S149.
- step S167 when the vertex u does not have the left shape of FIG. 49 (h) (“No” in step S167), the process proceeds to step S169.
- step S169 it is determined whether or not the vertex u has the right shape in FIG.
- the vertex u has the right shape of FIG. 49 (h) (“Yes” in step S169)
- the tree G is defined as a tree in which the label of the tree on the right side of FIG. 49 (h) is replaced with a set of id and label, and ⁇ u + 1, u0,. . . , Ul-1 ⁇ is added (step S170). Thereafter, the process returns to step S149.
- step S169 when the vertex u does not have the right shape of FIG. 49 (h) (“No” in step S169), the process proceeds to step S171.
- step S171 it is determined whether or not the vertex u has the shape shown in FIG.
- the vertex u has the shape of FIG. 49 (f) (“Yes” in step S171)
- the vertex ids are u + 1, u0,. . . , Uk-1, uk,. . . , Ul + k ⁇ 1
- the tree G is defined as a tree in which the label of the tree in FIG. 49F is replaced with a set of id and label, and T is set to ⁇ u + 1, u0,. . . , Ul + k-1 ⁇ is added (step S172). Thereafter, the process returns to step S149.
- step S171 when the vertex u does not have the shape of FIG. 49 (f) (“No” in step S171), the process proceeds to step S173.
- step S173 it is determined whether or not the vertex u has the right shape in FIG.
- the vertex id of y is set as u0, and the tree G is changed to the right tree in FIG. 49B.
- the tree is determined by replacing the label with a set of id and label, and ⁇ u0 ⁇ is added to T (step S174). Thereafter, the process returns to step S149.
- step S173 when the vertex u does not have the right shape of FIG. 49B in step S173 (“No” in step S173), the process proceeds to step S175.
- step S175 it is determined whether or not the vertex u has the shape shown in FIG.
- the tree G is defined as a tree in which the label of the tree in FIG. 49 (e) is replaced with a set of id and label, and ⁇ u0,. . . , Ul-1 ⁇ is added (step 176). Thereafter, the process returns to step S149.
- step S175 when the vertex u does not have the shape of FIG. 49 (e) (“No” in step S175), the process returns to step S149 (in this case, as shown in step S177, the vertex u is (It has the shape of FIG. 49 (a)).
- the above operation is the conversion process from ss-saddle connection diagram in I / II series to tree.
- Proposition 3.1 A streamlined phase structure of a stable Hamilton vector field with each 1-source-sink point corresponds to a tree expression with a root, a label, and an orientation.
- the route selection method is not unique. Because, for any connected component that includes a circular boundary that does not have ⁇ -saddle inside, there is a continuous mapping, so that the connected component can always be the outermost connected component. is there. Therefore, for the saddle connection diagram, there are different graph representations as many as the number of connected components. In order to remove such ambiguity, the graph representations that can be transferred by these continuous maps are different.
- V 1 and V 2 For the two structural stability Hamiltonian vector field V 1 and V 2, V a 1 ⁇ V 2 becomes equivalence relation, V 1 exist homeomorphism on certain homeomorphism D z (M) and V 2 Are defined by the fact that the outer boundaries of V 1 and V 2 correspond to each other without changing their orientation.
- ⁇ defines an equivalence relation, and a tree with a unique root, a label, and a direction can be given to this equivalence class.
- Proposition 3.2 Equivalent class streamline phase structure of ⁇ in a structure-stable Hamiltonian vector field corresponds to tree representation with unique root, labeled, and orientation.
- FIG. 51-A shows visualization of a graph expression of a structure-stable streamline pattern given by the same word expression IA 0 C and its regular expression.
- IA 0 C the same word expression
- FIG. 51-A shows a graph representation of a structurally stable streamline pattern having the word representation IA 0 C and its regular expression.
- the route corresponds to the connected component immediately below 1-source-sink point from its definition.
- the root connected component is copied to the outermost connected component on the right side.
- FIG. 51-B is a flowchart for explaining conversion processing of a tree into a regular expression.
- the conversion process of a tree into a regular expression shown in FIG. 51-B can be executed by a device such as a computer.
- id is given to all points of the tree as preprocessing. However, ids of different points are different.
- input is set as a tree G (step S178).
- V is set as the vertex set of tree G
- X is set as empty set ⁇
- regular expression N (s, ⁇ ⁇ ) () (step S179).
- step S180 the result of reading the children of s from the left is expressed as (s (1) , ⁇ (1) ),. . . , (S (h) , ⁇ (h) ).
- (s, ⁇ * ) () in the regular expression N is changed to ⁇ * (s (1) , ⁇ (1) ) (),. . . , (S (h) , ⁇ (h) ) ()), and replace T with ⁇ s (1) ,. . . , S (h) ⁇ and ⁇ s ⁇ are added to X.
- the above operation is the process of converting the tree into a regular expression.
- FIG. 52 is a block diagram showing an example of the fluid transition path acquisition apparatus 101 to which the second embodiment is applied, and conceptually shows only a part related to the second embodiment in the configuration.
- parts having the same functions as those of the fluid transition path acquisition apparatus 100 of Embodiment 1 shown in FIG. 29 are denoted by the same reference numerals, description of common parts is omitted, and only different points will be described.
- the fluid transition path acquisition device 101 of the second embodiment is different from the fluid transition path acquisition device 100 of the first embodiment in that it includes a regular expression forming unit 120, and the transition information acquisition unit 102g2 is shown in FIG.
- the simulation unit 102a2 In addition to the processing of the transition information acquisition unit 102g, the point that the transition information is acquired using a regular expression, and the simulation unit 102a2 further uses the word expression and the regular expression in addition to the processing of the simulation unit 102a of FIG.
- a design parameter process selection process is performed in which design parameter candidates are selected and used.
- the storage unit 106 further includes various types of data (for example, FIGS. 2 to 6, FIG. 46, FIG. 48, and the like) that are necessary for the fluid transition path acquisition apparatus 101 to execute the above-described regular expression creation method. 49, FIG. 51-A converted into data), a database, a table, and the like are stored.
- the control unit 102 further includes a regular expression forming unit 120 that executes the above-described regular expression creating method.
- the regular expression forming unit 120 includes a graph expression forming unit 120a and a regular expression creating unit 120b.
- the graph representation forming unit 120a is a graph representation creating unit for executing the above-described graph representation creating step (step S21).
- the sad representation connection diagram in the O series shown in FIGS. 47-A to 47-E is used.
- 50 A to 50-D
- N topologically N (however, by converting the ss-saddle connection diagram to the tree in the I and II series shown in FIGS. 50-A to 50-D).
- N is an integer greater than or equal to 1)
- a graph representation corresponding to the flow pattern in the multi-connected outer region having a number of holes is created in a one-to-one correspondence.
- the child vertex set ⁇ (v) When the child vertex set ⁇ (v) is used, the child vertex set ⁇ (v) of v is rearranged according to a rule of a predetermined order relation, and from w ( ⁇ ) to l (w) for w ⁇ (v). You may include what was drawn by arranging the arrows from left to right.
- the flow pattern includes (1) two ss- ⁇ -saddle connections, which have a suction-out flow pair of two types of flow patterns that can be taken topologically in a single connected outer region having one hole. (2) Of the two types of flow patterns that can be taken topologically in the single connected outer region having the above-mentioned one hole, it has a suction / outflow pair, and a single saddle point is connected.
- It may be one or more of a pattern II having a homoclinic saddle connection and two ss-saddle connections, and (3) a pattern O having no suction spring-out pair in a double-connected external region having two holes. If the flow pattern has a suction-and-out pair, the flow pattern is converted so that the region containing the counterclockwise ss-orbit closest to the suction-and-out pair is the outermost region, and the converted flow The trajectory that appears in the (ss-) saddle connection diagram of the pattern is extracted from the entire region, and the vertex is set to the connected component obtained by excluding all the trajectories that appear in the (ss-) saddle connection diagram from the entire region.
- the connected component at is the root, the connected component that touches the boundary with the current component is the child of the current component, labels are assigned according to the trajectory corresponding to the boundary, the labels are arranged according to the predetermined order relationship, and the children of the current component To the current component and repeat until there are no more children You may decide to do it.
- the flow pattern starts from one or more of pattern words, and is one of operation words defining five types of operations that can be taken topologically when adding one hole to the flow pattern. May be a flow pattern diagram created by repeating the operation to give N until the number of holes reaches N.
- the regular expression creating unit 120b is a regular expression creating means for realizing the above-described regular expression creating step (step S22).
- the regular expression creating unit 120b performs, for example, a tree conversion process shown in FIG. 51-B.
- a regular expression is created from the graph expression created by the graph expression creating unit 120a.
- the simulation unit 102a2 further executes a design parameter candidate selection process (see FIG. 56).
- a design parameter candidate selection process when selecting a design parameter candidate for an object in a fluid, an upper limit and a lower limit of the design parameter are set, and a plurality of parameters are defined from the parameter region defined by the upper limit and the lower limit of the design parameter. Select parameters, perform flow experiments and / or numerical calculations on the selected parameters, and assign and assign word expressions and / or regular expressions to the results of experiments and / or numerical calculations.
- a design parameter having a word expression and / or a regular expression indicating an optimal state may be selected as the design parameter candidate.
- the regular expression may be selected as a design parameter candidate.
- optimization design may be further performed using the selected design parameter.
- the transition information acquisition unit 102g2 may acquire a transition route by using a regular expression that completely corresponds to the flow pattern in addition to the word expression.
- the transition information acquisition unit 102g2 has a one-to-one correspondence with the structure-stable flow pattern. You may decide to specify one transition route using a corresponding regular expression.
- the transition information acquisition unit 102g2 determines that if there are a plurality of transition route candidates acquired based on the change information based on the change of the operation word of the word expression between the two designated structurally stable flow patterns, 2 Regular expressions (for example, N 1 , N 2 ) corresponding to word expressions (for example, W 1 , W 2 ) of two structurally stable flow patterns are acquired, and the transition occurs for a plurality of transition route candidates. Is not included in the regular expression (for example, N 1 ) corresponding to the word expression (for example, W 1 ) of one structure-stable flow pattern, it is excluded from the transition route candidates. Also good.
- the transition information acquisition unit 102g2 acquires regular expressions corresponding to the plurality of transition route candidates that are not excluded, Among the selection route candidates, a regular expression whose regular expression matches a regular expression (N 2 ) corresponding to a word expression (for example, W 2 ) of the other structure-stable flow pattern is specified as a transition pattern. Also good.
- FIG. 53 is a flowchart showing an example of a transition route search process using a difference in exponents in the fluid transition route acquisition apparatus 101 according to the second embodiment.
- steps that perform the same processing as in FIG. 31 are given the same step numbers.
- the list of FIGS. 26 to 28 obtained for perturbation of h-unstable and p-unstable streamline patterns and their partial word expressions is used as the pattern transition information.
- the transition information acquisition unit 102g2 designates two structure-stable streamline patterns (step S-11).
- the transition information acquisition unit 102g2 acquires the word expressions W 1 and W 2 corresponding to the two structure stable streamline patterns (step S-12). For example, when a streamline diagram is specified, the transition information acquisition unit 102g2 acquires the corresponding word expressions W 1 and W 2 by the processing of the word expression forming unit 102c.
- the transition information acquisition unit 102g2 includes transition patterns X of out looking all the list (step S-14) with the difference d, acquires the partial sequence S 1 and S 2 corresponding to each transition pattern X (step S- 15).
- step S- 15 When the above-described multiple transitions are assumed, partial sequences S 11 , S 21 to S 1k, and S 2k are obtained for transition patterns X 1 , X 2 ,..., X k , respectively.
- transition information acquisition unit 102g2 to the subsequence S 1 and S 2, when they checked whether is W 1 and W 2 of the sub-words respectively, has become a subword, transition patterns X (Step S-16). That is, it is checked whether S 1 ⁇ W 1 and S 2 ⁇ W 2 . If this is true, X is acquired as a transition candidate, otherwise it is rejected.
- the transition information acquisition unit 102g2 performs streamline structure regular expressions N 1 and N 2 having word expressions of W 1 and W 2 by processing of the regular expression forming unit 120. Is acquired (step S-17).
- Transition information acquisition unit 102g2 for candidates of the plurality of transition patterns, excluding those portions representation for the transition occurs is not present in the regular expression N 1 from the candidate (step S-18).
- the transition information acquisition unit 102g2 determines whether or not there is one transition pattern candidate (step S-19). If there is one transition pattern candidate (“Yes” in step S-19), the transition pattern acquisition unit 102g2 determines that there is one transition pattern candidate. (Step S-20).
- the transition information acquisition unit 102g2 uses the regular expression forming unit 120 to process the regular expression for the transition pattern X candidate. Each is acquired (step S-21).
- Transition information acquisition unit 102g2 among the candidates for the transition pattern X, the regular expression, to identify those that match the regular expression N 2 as a transition pattern (step S-22).
- the comparison of the partial expressions included in the regular expression of N 1 , the regular expression of the transition pattern, and N 2 without performing pattern matching. It can be determined whether or not a transition has actually occurred, and a transition pattern can be specified as one.
- design method is used to determine the optimum parameters for a design object through the shape and arrangement of objects (hereinafter referred to as design objects) such as bridges and piers, and the control of the flow around them.
- design objects such as bridges and piers
- Assumption 1 There are design parameters that can be changed in the design object (shape, arrangement, flow control device, etc.).
- Assumption 2 In designing a design object, an “optimal state” corresponding to the problem is set, and the optimal state can be described as a feature of the streamline structure. For example, when the state where the separation vortex is confined is set to the optimum state, it is expected that the wings maximize the lift, and the piers minimize the drag.
- Assumption 3 In these designs, experiment and numerical calculation (and word expression / regular expression of streamline pattern obtained in it) can be performed by changing design parameters. Hereinafter, it is limited to a two-dimensional flow. In the case of 3D, it may be possible to design by taking a cross-section and making it 2D, etc. (Note that not all cases are possible in 3D).
- an ideal situation is set as a word expression or regular expression of a streamline pattern from the beginning for such a situation, and a parameter area is searched so that the situation is realized. Will be able to. In this way, it becomes possible to search for as many and as quickly as possible design parameters “candidates” for realizing optimum settings. Starting with those candidates, performing a known optimization method increases the possibility that many and highly feasible design parameters can be obtained.
- FIG. 56 is a flowchart for explaining a method for designing an object in a fluid using word expressions and regular expressions, that is, the design parameter candidate selection process described above.
- the flowchart shown in FIG. 56 is executed by the simulation unit 102a2.
- an upper limit and a lower limit of possible design parameters are determined (step S311).
- Word expressions and regular expressions are assigned to the obtained results (step S314). It is determined whether or not the word expression has an IC partial word representing the optimum state or is a structural expression having a regular expression corresponding thereto (step S315). If the word expression has an IC partial word representing the optimum state or a structural expression having a regular expression corresponding to the IC partial word (“Yes” in step S315), the word expression is adopted as a design parameter candidate (step S316).
- the word expression is optimal in one transition. It is determined whether or not it is a structural expression having an IC partial word representing a state or having a regular expression corresponding thereto (step S320). If the word expression has an IC partial word representing the optimal state in one transition, or a structural expression having a corresponding regular expression (“Yes” in step S320), the word expression is adopted as a design parameter candidate. (Step S316). A regular expression whose word expression has an IC subword representing the optimal state or corresponding to it in one transition (one transition from a structurally stable flow pattern to a structurally unstable intermediate state flow pattern) Is not adopted as a design parameter candidate.
- step S317 it is determined whether there is a design parameter candidate (step S317). If there is a design candidate parameter (“Yes” in step S317), an existing optimization is performed on the design parameter candidate. A method is applied (step S319). In addition to applying the existing optimization method, it is ideal not only to quantitatively optimize the flow, but also to calculate and monitor word expressions and regular expressions at each stage of the optimization process. It is possible to optimize the streamline structure at the same time.
- step S317 if there is no design parameter candidate (“No” in step S317), the parameter area is further divided (step S318), and the processes of steps S313 to S317 are executed.
- 57 and 58 are diagrams for explaining an example of a specific example for explaining a method for designing an object in a fluid using the above-described word expression and regular expression.
- FIG. 57 it is assumed that there are two vortex structures (vortex 1 and vortex 2) behind an object 400 placed in a uniform flow. It is assumed that the uniform flow speed (or the traveling speed of the object) is U, and the object 400 can rotate at the angular velocity G.
- An apparatus in which the strength and position of the vortex is already given in the flow, of which the vortex 1 is “closed” to maximize the lift on the object 400 and the vortex 2 can be moved away from the object along the flow Is optimized by searching for “U and G parameters”. Since the word representation of the vortex confinement state vortex away by the "C" can therefore represent "the A 0", the parameter area by the application of this design approach those that achieve IA 0 C as word representation of situations to be targeted Narrow down.
- U can vary from 0 to 1.1
- G can vary from ⁇ 1.6 ⁇ 2 ⁇ to 1.6 ⁇ 2 ⁇ (the upper and lower limits of the design parameters are determined in step S311). .
- These parameter sections are divided into five equal parts to divide the design parameter area (step S312 above).
- the results of executing steps S313 to S320 are (a) to (f) in FIG.
- a single transition route is specified by using a regular expression corresponding to the flow pattern one-to-one. Therefore, even when there are a plurality of selection route candidates, it is possible to specify a transition route without performing pattern matching.
- the regular expressions corresponding to the word expressions of the two structurally stable flow patterns are respectively acquired, and the transitions of the plurality of transition route candidates are obtained. If the partial expression for the occurrence of the error is not included in the regular expression corresponding to the word expression of one of the structurally stable flow patterns, it is excluded from the transition route candidates. It is possible to exclude transition route candidates that have no possibility of actual transition simply by comparing the regular expression with.
- the regular expressions corresponding to the plurality of transition route candidates that are not excluded are respectively acquired and not excluded.
- the regular expression that matches the regular expression corresponding to the word expression of the other structure-stable flow pattern is specified as the transition pattern, so the regular expressions are compared with each other. Only one transition route can be specified.
- a regular expression creation method for creating a regular expression of a flow pattern in a multi-connected outer region having topologically N (where N is an integer of 1 or more) holes.
- N is an integer of 1 or more
- a graph expression creating process for creating a graph expression corresponding to the flow pattern on a one-to-one basis and a regular expression creating process for creating a regular expression from the graph expression created in the graph expression creating process are included.
- the graph representation is such that the parent vertex is v, the child vertex is w, the label assigned to the parent vertex v is l (v), and the label is assigned to the child vertex w.
- the l (w), v child vertex set ⁇ (v), the child vertex set ⁇ (v) of v is rearranged according to a rule of a predetermined order relation, and l (v ) To l (w) are drawn side by side from left to right, so that it is possible to provide a graph expression in which the parent-child connection relationship is easy to visually recognize.
- the flow pattern has (1) a suction / outflow pair of two types of flow patterns that can be taken topologically in a single connected outer region having one hole, Pattern I with two ss- ⁇ -saddle connections, (2) One of two types of flow patterns that can be taken topologically in a single connected outer region having one hole, and having a pair of suction and spring One saddle point, a homoclinic saddle connection connecting it and a pattern II having two ss-saddle connections, and (3) one or a plurality of patterns O having no suction spring pair in the double connected external region having two holes All the basic flow patterns Can impart regular expression against emissions, it will handle the regular expression of a specific flow pattern.
- the regular expression created in the regular expression creating process is provided with the word expression conversion process, the regular expression can be converted into the word expression.
- the word expression includes a double-connected external region having two holes in addition to two types of flow patterns that can be taken topologically in a single-connected external region having one hole.
- a pattern word that defines a total of three types of flow patterns, to which a pattern that does not have a pair of suction and outflow is added, any one of operation words that define topological operations that can be taken topologically Since the word is a symbolic word formed by adding the number of added holes, regular operations can be performed with all five basic flow patterns and topologically five types of operations. It is possible to convert to a specified word expression.
- the flow pattern that can be taken for the structure can be easily handled without depending on experience or intuition.
- a regular expression creation method, regular expression creation device, and program for a flow pattern can be provided.
- the word expression and the regular expression include, for example, bridge pier design, breakwater arrangement, port contamination removal, wind power blade design, train pantograph structure, oil fence design, etc. It is extremely useful in various fields involving structure design and arrangement, such as optimal arrangement. Also, it can be applied to fields such as sports mechanics such as structural design of sports equipment.
- a fluid transition path acquisition device As described above in detail, according to the present invention, a fluid transition path acquisition device, a fluid transition path acquisition method, a program, and a recording medium that can obtain knowledge about the transition of a flow pattern are provided. can do.
- design and control of the shape and arrangement of structures such as flying object design, train pantograph structure, automobile structure such as racing cars, sports equipment design such as ski jump, high-speed ship design, weather forecast,
- This is extremely useful in various fields involving fluid change prediction.
- the flow field and many structures included in it are important areas (bridge piers, breakwater arrangements, port pollutant removal, wind power blade design, aircraft wing design, oil This is extremely useful in the optimal placement of fences).
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Abstract
Description
[本発明の実施形態の概要]
以下、本発明の実施形態の概要について図1~図28を参照して説明し、その後、本実施形態の構成および処理等について詳細に説明する。
ここで、上述した語表現を形成させる語変換アルゴリズムの概要について以下に説明する。ここで、図1は、本実施形態の語表現アルゴリズムの概要を示すフローチャートである。
これに限られず、操作語の付与を先に行い、後にパターン語の付与を行ってもよいものである。
ここで、本実施形態において扱う流れの構成要素について図2~図4を参照して説明する。本実施形態における「流れ」として非圧縮流れを扱ってもよい。流体の非圧縮性とは、力を加えてもほとんどその体積を変えないような性質をいう。通常の水や大気の流れは、日常生活のスケールで考える場合は、こうした流れの枠組みで考えても概ね差し支えはない。なお、本発明は、これに限られず、圧縮性のある流れを計算上取り扱ってもよい。
1)吸い込み湧き出し対をもち、二つのss-∂-saddle connectionをもつパターンI、
2)吸い込み湧き出し対をもち、一つのsaddle point、それを結ぶhomoclinic saddle connectionと二つのss-saddle connectionをもつパターンII、および
3)吸い込み湧き出し対をもたないパターンO、
である。ここで、図4は、初期構造となる3種類の構造安定な流れパターンを模式的に示す図である。
帰納的に流れを構成していくために、穴を一つとそれに伴う流れの構造を追加するという「操作」について、図5および図6を参照して説明する。すなわち、穴の数がM個ある連結外部領域Dz(M-1)の流れに、一つの穴を加えて多重連結外部領域Dz(M)の流れを求める操作について説明する。
1)一本のss-orbitを、一つのsaddle point、それを結び内部に穴をもつhomoclinic saddle connectionと二つのss-saddle connectionに置き換えるA0操作、
2)一本のss-orbitを、二つのss-∂-saddle connectionと新たに追加した境界上の二つの∂-saddleに置き換えるA2操作、
3)一本のclosed orbitを、一つの穴とsaddle pointを追加して8の字をした2本のhomoclinic saddle connectionに置き換えるB0操作(このパターンを以後Figure Eightパターンと呼ぶ。)、
4)一本のclosed orbitを、新たに追加した穴の境界上に二つ∂-saddleをつけて一本の∂-saddle connectionでつなぐような軌道に置き換えるB2操作、および、
5)既に2k個(k>0)の∂-saddleをもつ境界に、新たに二つの∂-saddleを付け加えて一本の∂-saddle connectionでつなぎ内部に新たに付け加えた穴を置くC操作、
である。ここで、図5は、穴を一つ付け加えて構造安定な流れを構成する5種類の操作を模式的に示した図である。
O-1)実際に施すことができる操作は、B0,B2,Cのみであり、その結果、Oから始まる語表現はこれら三つの語を列挙したものとなる。
O-2)操作列の語表現においてCの語が含まれるためには、それ以前に必ずB2が存在しなければならない。
I-1) 実施可能な操作はA0,A2,B0,B2,Cの全てであり、その結果、Iから始まる語表現はこれら5種類の操作語を列挙したものである。
I-2) 操作列の語表現において、B0あるいはB2の語が含まれるためには、それ以前に必ずCかA0が存在しなければならない。
II-1)実施可能な操作はA0,B0,B2,Cであり、その結果、IIから始まる語表現はこれら四つの語を列挙したものである。
II-2)操作列の語表現においてCの語が含まれるためには、それ以前に必ずB2が存在しなければならない。
つづいて、本実施の形態のパターン遷移情報について説明する。なお、流れ場には、上述したアルゴリズムを使って既に一意な極大語表現がラベル付けされているとしてよい。ただし、本実施の形態は、上述のようにパターン遷移情報に重複や包含関係を許して語表現の割り当てを行えば、極大語表現を用いることに限られない。
極大語表現Wに対して、a0(W),a2(W),b0(W),b2(W)およびc(W)をそれぞれWの中に含まれるA0,A2,B0,B2,Cの文字の数とする。このとき、Wが、O系列の極大語(maximal O-word),I系列の極大語(maximal I-word),II系列の極大語(maximal II-word)であるとき、その指数(index)をそれぞれ以下のように定義する。
indO(W)=(0,0,b0(W),b2(W),c(W))
indI(W)=(a0(W),a2(W)+1,b0(W),b2(W),c(W))
indII(W)=(a0(W)+1,a2(W),b0(W),b2(W),c(W))
極大語表現がW=O0O1...Okと与えられているとする。ただし、O0はO,I,IIのいずれか、Oi(1≦i≦k)はA0,A2,B0,B2,Cのいずれかである。このときWに含まれる文字列の一部、すなわちS=Oi1...Oil(0≦i1<・・・<il≦k)をWの部分語(subword)とよび、S⊆Wのように記述する。
以下に、一つのhomoclinic pairをもちそれ以外はすべて構造安定な流線パターンを持つh-不安定な流線構造を生成する14個の操作を定義し、それらが摂動を受けたときにどのような構造安定な流線パターンに遷移するかをその部分語表現とともに記述する。
まず、構造安定な連結外部領域Dz(M-1)内における流線パターンに対して、一つ穴を追加してDz(M)におけるh-不安定な流線パターンを生成する10個の操作を定義する。図8は、Dz(M-1)における構造安定な流線パターンから穴を一つ追加してDz(M)におけるh-不安定な流線パターンを生成する10個の操作を示す図である。図中の斜線の円が新たに追加された穴を表している(以下の図でも同様)。
操作D0,sとD0は、図8(a)に示すように、それぞれss-(∂-)saddle connectionおよび∂-saddle connectionの上にhomoclinic saddleを一つくっつける操作である。なお、操作を表す文字D0,sに現れる下付文字のsはこの構造が、一様流の存在を表す1-source-sink pointに直接つながっていることを意味する。
操作E0,sとE0は、図8(b)のように、homoclinic saddle connectionの軌道内に、新たにhomolclinic saddleを一つつける操作を表す。数学的な検討の結果、これらの操作によって得られる代表的なh-不安定な流線パターンとその摂動は以下の通りである。ここで、図11は、操作E0,sおよびE0によって生成されるh-不安定な流線パターンとその摂動を示す図である。
操作Φ0,sとΦ0は、ともに一つのサドル点を二つに分離して、それらの点を新たに二つの(新しい穴を一つ取り囲む)ヘテロクリニック軌道で結ぶ図8(c)に示したような操作である。なお、この操作によってΦの形をした形状をした流線パターンを作るので、この記号を用いている。数学的な検討の結果、これらの操作によって図12に示すような3種類の代表的なh-不安定な流線パターンとその摂動が得られる。図12は、操作Φ0,sとΦ0によって生成されるh-不安定な流線パターンとその摂動を示す図である。
ss-(∂-)saddle connectionあるいは∂-saddle connectionを切断して、その切断点を新しく付け加えた境界につなげる図8(d)のような操作をそれぞれ操作D2,sおよびD2と呼ぶ。まず、数学的な検討の結果、D2,sによって図13に示すような3種類の代表的なh-不安定な流線パターンが得られる。図13は、操作D2,sによって生成されたh-不安定な流線パターンとその摂動を示す図である。
操作E2,sとE2は、homoclinic saddle connectionを切断して、その切断点に新しい円境界をつなぐ図8(e)のような操作を表す。数学的な検討の結果、この操作により図15のような3種類の代表的なh-不安定な流線パターンが得られる。図15は、操作E2,sとE2によって生成されるh-不安定な流線パターンとその摂動を示す図である。
ここでは、図16に示すような一回の操作で二つ以上の穴を追加する操作を導入して、そこから生成されるh-不安定な流線パターンとその摂動について説明する。図16は、二つ以上の穴を追加して構造安定な流線パターンからh-不安定な流線パターンを生成する四つの操作を示す図である。
図16(a)に示すように、境界上にある一つのss-∂-saddleを選んで、それにつながる軌道の上に一つsaddle pointと境界に二つの∂-saddlesを追加して、それらの間を2本のs-∂-saddle connectionでつなぐ操作をΨs(2)と呼ぶ。一方、一つの∂-saddleを選んで同じ操作をすることをΨ(2)と呼ぶ。この操作により、3本のs-∂-saddle connectionが得られ、それらによって囲まれる新たな二つの穴が追加される。なお、操作の結果Ψの形の流線パターンが形成されるので、この記号を用いている。また、Ψs(2)における(2)の文字は、この操作によって追加される穴の数を明示的に表現している。
図16(b)に示すように操作Ξs(2n-1)とΞ(2n-1)は、三個以上の任意の奇数個の新しい穴を追加してh-不安定な流線パターンを生成する操作である。具体的には、∂-saddleをすでに持っている円境界を一つ選び、それとは別のところに新たに円境界を生成して、この二つの境界を2n-1本のheteroclinic ∂-orbitで結ぶ操作である。この結果として、∂-orbitで囲まれた2n-2個の円境界が追加され、全部で2n-1個の穴が生成される。なお、n=1のときは、これらの操作はD2,sおよびD2と一致するので、一般性を失うことなくn≧2としてよい。数学的な検討の結果、これらの操作は、図19に示すように4種類の代表的なh-不安定な流線パターンを生成する。図19は、操作Ξs(2n-1)とΞ(2n-1)によって生成されるh-不安定な流線パターンとその摂動を示す図である。
まず、内部に境界を持たない単連結外部領域Dz(0)において一つのpinching orbitを持つ基本パターンが存在するので、それに名前をつける。図20は、p-不安定なパターンΛとその摂動を示す図である。
基本パターンΛは、Dz(0)におけるp-不安定な流線パターンであり、図20(a)で与えられる。このパターンを摂動する基本パターンIとIIに遷移する。したがって、このパターンによって極大語表現の先頭部分の文字を変更することになる。このときの極大語表現の指数の変化はindI(I)-indII(II)=(-1,1,0,0,0)である。一方、図20(b)は、このΛの作る構造が一つの∂-saddle connectionに囲まれた流線パターンΛCとその摂動を表している。摂動により部分語表現がICおよびIIB2を持つ異なる構造安定な流線パターンが得られる。このときの極大語表現の指数の変化はindI(IC)-indII(IIB2)=(-1,1,0,-1,1)である。なお、このΛパターンを基本パターンとして、構造安定な流れ場を作る操作A0,A2,B0,B2,Cを繰り返して得られるp-不安定な流れ場に対して、その施した操作を列に並べて得られる極大語表現をmaximal Λ-word(Λ系列の極大語)と呼ぶことができるので、このΛ-wordの摂動によって、I-wordとII-wordの間の遷移が起こる。ただし、Λ-wordの中にA2の文字が含まれているときは、その摂動によってII-wordに変化したものはIA0=IIA2の関係式を使って、I-wordに常に変換されることに注意する。例えば、ΛA0A2B0Cで表現されるp-不安定な流れ場を摂動すると、IA0A2B0CとIIA0A2B0C=IA0A0B0Cの極大語表現を持つ構造安定な流れ場が得られる。
操作Δ1は、図21(a)に示すように、任意の円境界にhomoclinic pinching orbitを追加して穴を増やす操作を表している。数学的な検討の結果、操作Δ1により、図22や図23に示すような5種類の代表的なp-不安定な流線パターンが生成される。ここで、図22および図23は、操作Δ1で生成されるp-不安定な流線パターンとその摂動を示す図である。
図21(b)に示すように、1-source-sink pointにss-∂-saddle connectionによってつながっている∂-saddleと端点を共有する∂-saddle connectionを同じ境界上に付け加える操作をM1,sと呼ぶ。この操作によって穴が一つ増えると同時に、数学的な検討の結果、以下の図24に示すような1種類の代表的なp-不安定な流線パターンを生成する。図24は、操作M1,sによって生成されるp-不安定な流線パターンとその摂動を示す図である。
操作M1,iとM1,oは、それぞれ図21(c)に示すように境界上に存在する∂-saddle connectionの∂-saddleと端点を共有するような∂-saddle connectionを同じ境界上に新たに追加する操作である。M1,iは∂-saddle connectionを既存の∂-saddle connectionの内側に、M1,oは外側に追加するものである。数学的な検討の結果、これらの操作は、図25に示すような5種類の代表的なp-不安定な流線パターンを生成する。図25は、操作M1,iとM1,oによって生成されるp-不安定な流線パターンとその摂動を示す図である。
次に、本実施形態における流体遷移経路取得装置の構成について図29を参照して説明する。図29は、本実施形態が適用される流体遷移経路取得装置100の一例を示すブロック図であり、当該構成のうち本実施形態に関係する部分のみを概念的に示している。
1)吸い込み湧き出し対をもち、二つのss-∂-saddle connectionをもつパターンI、
2)吸い込み湧き出し対をもち、一つのsaddle point、それを結ぶhomoclinic saddle connectionと二つのss-saddle connectionをもつパターンII、および
3)吸い込み湧き出し対をもたないパターンO、
である。
1)一本のss-orbitを、一つのsaddle point、それを結び内部に穴をもつhomoclinic saddle connectionと二つのss-saddle connectionに置き換えるA0操作、
2)一本のss-orbitを、二つのss-∂-saddle connectionと新たに追加した境界上の二つの∂-saddleに置き換えるA2操作、
3)一本のclosed orbitを、一つの穴とsaddle pointを追加して8の字をした2本のhomoclinic saddle connectionに置き換えるB0操作、
4)一本のclosed orbitを、新たに追加した穴の境界上に二つ∂-saddleをつけて一本の∂-saddle connectionでつなぐような軌道に置き換えるB2操作、および、
5)既に2k個(k>0)の∂-saddleをもつ境界に、新たに二つの∂-saddleを付け加えて一本の∂-saddle connectionでつなぎ内部に新たに付け加えた穴を置くC操作、
である。
1)一本のss-orbitが存在することを条件に、操作語A0またはA2を付与し、
2)一本のclosed orbitが存在することを条件に、操作語B0またはB2を付与し、
3)∂-saddleを持つ境界が存在することを条件に、操作語Cを付与してもよい。なお、当該制約条件にしたがった操作語付与部102eの具体的な処理の詳細については、フローチャートを参照して後述する。また、本実施形態において、操作語付与部102eは、パターン語に対して付与した操作語の文字列である語表現をパターン遷移ファイル106cに格納してもよい。
次に、このように構成された本実施形態における流体遷移経路取得装置100の処理の一例について、以下に図面を参照して詳細に説明する。図30は、流体遷移経路取得装置100の処理の一例を示すフローチャートである。
ここで、指数の差を用いた遷移ルート探索処理の一例について以下に図面を参照して詳細に説明する。図31は、流体遷移経路取得装置100における、指数の差を用いた遷移ルート探索処理の一例を示すフローチャートである。なお、以下の例では、パターン遷移情報として、h-不安定およびp-不安定な流線パターンの摂動とその部分語表現に関して得られた図26~図28のリストを用いている。また、入力として、二つの構造安定な流線パターンとその極大語表現W1とW2が指定されており、極大語表現から判定される遷移の可能性Xのリストを遷移情報として出力することを前提として説明する。
ここで、シミュレーション結果を用いた実施例について説明する。前提として、一様流中におかれた物体について、ある期間における、物体が受ける力と物体周りの流れの実験結果または数値計算結果を記憶している。ここで、物体が受ける力は、一様流によって物体が受ける力であり、具体的には、揚力(lift)や抗力(drag)であり、一例として、揚抗比(揚力と抗力の比)であってもよい。
ここで、流体遷移経路取得装置100により実行される語表現アルゴリズムの基本処理の一例について、上述した図1を再び参照して説明する。なお、以下の例では、パターン語の付与(ステップSA-1)が先に行われ、操作語の付与(ステップSA-2)が後に行われるが、これに限られず、操作語の付与を先に行い、後にパターン語の付与を行ってもよいものである。
1)吸い込み湧き出し対をもち、二つのss-∂-saddle connectionをもつパターンを規定するパターン語I、
2)吸い込み湧き出し対をもち、一つのsaddle point、それを結ぶhomoclinic saddle connectionと二つのss-saddle connectionをもつパターンを規定するパターン語II、および
3)吸い込み湧き出し対をもたないパターンを規定するパターン語O、
である。
1)一本のss-orbitを、一つのsaddle point、それを結び内部に穴をもつhomoclinic saddle connectionと二つのss-saddle connectionに置き換える操作を規定した操作語A0、
2)一本のss-orbitを、二つのss-∂-saddle connectionと新たに追加した境界上の二つの∂-saddleに置き換える操作を規定した操作語A2、
3)一本のclosed orbitを、一つの穴とsaddle pointを追加して8の字をした2本のhomoclinic saddle connectionに置き換える操作を規定した操作語B0、
4)一本のclosed orbitを、新たに追加した穴の境界上に二つ∂-saddleをつけて一本の∂-saddle connectionでつなぐような軌道に置き換える操作を規定した操作語B2、および、
5)既に2k個(k>0)の∂-saddleをもつ境界に、新たに二つの∂-saddleを付け加えて一本の∂-saddle connectionでつなぎ内部に新たに付け加えた穴を置く操作を規定した操作語C、
である。なお、上記の操作は、穴を追加する場合の操作を規定したものであることから、ある流線図に操作語を割り当てていくアルゴリズムBにおいては、当該流線図に対して上記操作の逆の操作(逆の置き換え操作)を行うことになる。
1)一本のss-orbitが存在することを条件に、操作語A0またはA2を付与し、
2)一本のclosed orbitが存在することを条件に、操作語B0またはB2を付与し、
3)∂-saddleを持つ境界が存在することを条件に、操作語Cを付与してもよい。
つづいて、上述した語表現形成部102cによる基本処理を基礎として、流線図から語表現を求めるアルゴリズムBの処理の詳細について、以下に図36~図43を参照して説明する。
上述した基本処理のステップSA-1におけるパターン語の割り当て処理について図36を参照して以下に説明する。図36は、パターン語の割り当て処理の一例を示すフローチャートである。
つづいて、アルゴリズムBにおけるI系列の操作語の割り当て処理の一例について、図37および図38を参照して説明する。図37および図38は、アルゴリズムBにおけるI系列の操作語の割り当て処理の一例を示すフローチャートである。
1. ∂-saddleの安定多様体、不安定多様体あるいは極限集合が、sink,sourceのlimit cycleを含むならば、それを閉じてsink,source,limit cycleを消去する。
2. sinkの(吸引)領域を、center/diskとhomoclinic(∂-)saddle connectionに置き換える。
3. sourceの(発散)領域を、center/diskとhomoclinic(∂-)saddle connectionに置き換える。
つづいて、アルゴリズムBにおけるII系列の操作語の割り当て処理の一例について、図39を参照して説明する。図39は、アルゴリズムBにおけるII系列の操作語の割り当て処理の一例を示すフローチャートである。なお、I系列の操作語の割り当て処理のステップSB-1およびステップSB-2と同様に、流線解析により流線図の作成と前処理が行われていてもよい。
つづいて、アルゴリズムBにおけるO系列の操作語の割り当て処理の一例について、図40を参照して説明する。図40は、アルゴリズムBにおけるO系列の操作語の割り当て処理の一例を示すフローチャートである。なお、I系列の操作語の割り当て処理のステップSB-1およびステップSB-2と同様に、流線解析により流線図の作成と前処理が行われていてもよい。
ここで、アルゴリズムBにおける操作語I,IIの割り当て処理の他の例(亜種アルゴリズム)について、図41および図42を参照して説明する。図41および図42は、アルゴリズムB(I,II-Word algorithm)におけるI,II系列の操作語の割り当て処理の他の例を示すフローチャートである。なお、上述した割り当て処理のステップSB-1およびステップSB-2と同様に、流線解析により流線図の作成と前処理が行われていてもよい。また、以下の操作語の割り当て処理において、操作語を付与する場合は、後ろから前に付与するものとする。
つづいて、アルゴリズムBにおけるO系列の操作語の割り当て処理(亜種アルゴリズム)の一例について、図43を参照して説明する。図43は、アルゴリズムB(O-Word algorithm)におけるO系列の操作語の割り当て処理の他の例を示すフローチャートである。なお、上述した割り当て処理のステップSB-1およびステップSB-2と同様に、流線解析により流線図の作成と前処理が行われていてもよい。また、以下の操作語の割り当て処理において、操作語を付与する場合は、後ろから前に付与するものとする。
得られる語表現には冗長性がある場合があるので、パターン語の組み合わせから全文字列を生成したのち、各文字列が極大語になっているか否かを判定し、冗長性のない極大語表現を得る原理と方法について説明する。
さて、これまで本発明の実施形態について説明したが、本発明は、上述した実施形態以外にも、特許請求の範囲に記載した技術的思想の範囲内において種々の異なる実施形態にて実施されてよいものである。
実施の形態2に係る流体遷移経路取得装置、流体遷移経路取得方法、および、プログラムを図44~図66を参照して説明する。実施の形態2では、実施の形態1と共通する部分の説明を省略し、異なる点についてのみ説明する。
実施の形態2に係る流れパターンの正規表現作成方法について説明する。実施の形態2に係る流れパターンの正規表現作成方法では、実施の形態1の語表現理論を一部使用している。本実施の形態2に係る流れパターンの正規表現作成方法はコンピュータ等の装置により実行可能である。図44は、実施の形態2に係る流れパターンの正規表現方法の概略を説明するためのフローチャートである。
上記図44のグラフ表現作成工程(ステップS21)について詳細に説明する。まず、実施の形態2で使用しているグラフ理論について説明する。
グラフT=(V,E)とは、「vertex(頂点)」と呼ばれる点の集合(頂点集合)Vとその頂点の間を結ぶ「edge(エッジ)」と呼ばれる集合Eのペアとして与えられる集合である。一般にグラフは多様な構造を持ちうるが、本実施の形態のグラフ表現理論では、グラフ全体の集合におけるある特定の構造をもった以下のグラフの集合を考える。
2)ルート付き(Rooted)グラフとは、ある特定の頂点(以下、ルート(root)と呼ぶ)が存在しているグラフを指す。ルート付きグラフに対しては、このルートから各頂点v∈Vへのエッジの連結による最短経路を考えることができるので、これをvの高さ(height)と呼び、記号をht(v)と書く。これによりルート付きツリーTに対してはht(T):=maxv∈Vht(v)によって、ツリーT自体の高さを考えることができる。
3)グラフが向き有り(directed)であるとは、すべてのエッジに親子の順序が入っているようなものを指す。向き有りグラフにおける、頂点v∈Vからw∈Vへのエッジは、v⇒w∈Eと表わす。このとき、vはwの親、wはvの子と呼ぶ。Г(v)と書いて、頂点v∈Vの子供全体の集合を表す。すなわち、Г(v):={w∈V│v⇒w∈E}。また、その集合に含まれる子供頂点の数#Г(v)を持って,vのout-degreeとよび、逆にvに入ってくるエッジの数をvのin-degreeと呼ぶ。
本実施の形態のグラフ表現作成工程に係る二次元構造安定なハミルトンベクトル場のツリー表現について説明する。以下、O-wordで表現される流れパターンのグラフ表現と、I,II-wordで表現される流れパターンのグラフ表現について説明する。
Hを二次元領域Dz(M)上のO-wordで表現される構造安定なハミルトンベクトル場とし、Dをそのsaddle connection diagramとする。このハミルトンベクトル場Hに対して固有のルート付き、ラベル付き、及び向き有りのツリーTH=(V,E)を割り当てる方法とその平面グラフとしての可視化アルゴリズムを以下に説明する。
図47-A~図47-Eは、O系列におけるsaddle connection diagramのツリーへの変換処理を説明するためのフローチャートである。コンピュータ等の装置により、図47-A~図47-Eに示す、O系列におけるsaddle connection diagramのツリーへの変換処理を実行可能である。
HをDz(M)内に1-source-sink pointを持つ構造安定なハミルトンベクトル場、Dをそのss-saddle connection diagramとする。このとき、CH=Dz(M)\Dは、closed orbitを含む円環回領域またはss-orbitを含む開円板領域となる連結成分からなる。これに対して、ルート付き、ラベル付き向き有りツリーを以下のように構成する。O-wordの時と同様に、各CHの連結成分を頂点集合に対応づける。ルートとなる連結成分はCHの中で1-source-sink pointに最も近い開円板領域で、その閉包は1-source-sink pointを含みかつ内部に時計回りのss-orbitを含むものを選ぶ。この連結成分にラベルoφを割り当てる。このような決め方によりルートは一義的に決定することができる。
図50-A~図50-Dは、I,II系列におけるss-saddle connection diagramのツリーへの変換処理を説明するためのフローチャートである。図50-A~図50-Dに示す、I,II系列におけるss-saddle connection diagramのツリーへの変換処理は、コンピュータ等の装置によって実行可能である。
以上のことから、1-source-sink pointを持つss-saddle connection diagramに対して、そのルートを一義的に決定でき、また、そこから操作A0,A2,B0,B2,Cによって構成されるss-saddle connection diagramの局所構造と、そこから導入される連結成分の間の親子関係は図46と図49の中で全て表現されているので、以下のことが示されたことになる。
上記図2の正規表現作成工程(ステップS2)について詳細に説明する。さて、グラフ理論でよく知られた事実として、任意のルート付き、ラベル付き、向き有りツリーには固有の「正規表現(regular expression)」なるものを考えることができる。TH=(V,E)を上記方法で与えられた構造安定なハミルトンベクトル場Hに対して与えられたグラフ表現とする。このグラフ表現に対して、その正規表現は以下のようにして帰納的に与えられる。まず、ルート以外のすべての頂点のin-degreeはすべて1であることに注意する。もし、ht(TH)=0ならグラフはルートのみ、すなわちV={v0}であり、その正規表現はl(v0)である。いま、高さht(TH)=n-1の正規表現Nがあった時に、そこにある頂点集合Tn-1={v1,v2,...,vm}とすると、各頂点viの子頂点集合Г(vi)={vi1,vi2,...,vimi}とそれに対応するラベルli=l(vi)(i=1,...,m)を使うと、Nに対してliをli(li1,li2,...,limi)に置き換えることによって新しい正規表現が構成できる。ただし、lik=l(vik)(k=1,...,mi)である。このようにして構成されたTHの正規表現をNTHとすると、その構成手法から以下の命題を得る。
図51-Bは、ツリーの正規表現への変換処理を説明するためのフローチャートである。図51-Bに示す、ツリーの正規表現への変換処理は、コンピュータ等の装置により実行可能である。以下の処理では、前処理として,ツリーの全ての点にidが与える。ただし,異なる点のidは異なるものとする。図51-Bにおいて、inputをツリーGとする(ステップS178)。VをツリーGの頂点集合,s=0,T={0},Xを空集合φ,正規表現N=(s,σφ)()と置く(ステップS179)。
次に、実施の形態2に係る流体遷移経路取得装置の構成について図52を参照して説明する。図52は、本実施の形態2が適用される流体遷移経路取得装置101の一例を示すブロック図であり、該構成のうち本実施形態2に関係する部分のみを概念的に示している。図52において、図29に示す実施の形態1の流体遷移経路取得装置100と同等機能を有する部位には同一符号を付して、共通する部分の説明を省略し、異なる点についてのみ説明する。実施の形態2の流体遷移経路取得装置101が実施の形態1の流体遷移経路取得装置100と異なる点は、正規表現形成部120を備えている点と、遷移情報取得部102g2が、図29の遷移情報取得部102gの処理に加えて、正規表現も使用して遷移情報を取得する点と、シミュレーション部102a2が、図29のシミュレーション部102aの処理に加えて、さらに、語表現と正規表現を使用して設計パラメータの候補を選択する設計パラメータ工程選択処理を実行する点である。
図54および図55を参照して、図32~35で説明した実施例と同様の条件において、語表現と正規表現を使用して遷移パターンを特定する実施例を説明する。図54は、一様流中におかれた一枚平板のt=5.5からt=7.7までの時間発展とその流線パターンの位相構造、その極大語表現、およびその正規表現である。図55は、時刻t=5.5から7.7の時間発展の間に起こった遷移を示す図である。
例えば、橋や橋脚など流体中におかれた物体(以下設計対象と呼ぶ)の形状や配置、またその周囲の流れの制御などを通じて設計の対象にとって最適なパラメータを定めるような設計手法について,語表現と正規表現を使ってどのように設計を行うかを説明する。
前提2:設計対象を設計する上で、問題に応じた「最適な状態」が設定されており、その最適状態が流線構造の特徴として記述できているものとする。例えば、剥離渦を閉じこめる状態を最適状態とすると、翼であれば揚力の最大化,橋脚であれば抗力の最小化などが期待できる。
前提3:これらの設計において、設計パラメータを変えて実験および数値計算(そしてそれにおいて得られる流線パターンの語表現・正規表現)ができるものとする。以下、二次元の流れに限定する。三次元の場合は断面をとって考えたりして二次元化することなどで設計が可能になる場合もある(三次元の場合はすべての場合でできるとは限らないことに注意する)。
102 制御部
102a,102a2 シミュレーション部
102b 流線解析部
102c 語表現形成部
102d パターン語付与部
102e 操作語付与部
102f 極大語表現部
102g,102g2 遷移情報取得部
104 通信制御インターフェース部
106 記憶部
106a シミュレーション結果ファイル
106b 流線図ファイル
106c パターン遷移ファイル
108 入出力制御インターフェース部
112 入力装置
114 出力装置
120 正規表現形成部
120a グラフ表現作成部
120b 正規表現作成部
120c 語表現作成部
200 外部システム
300 ネットワーク
Claims (16)
- 記憶部と制御部とを備えた流体遷移経路取得装置において、
上記記憶部は、
位相幾何学的な二次元流れ構造の全体または一部における、構造安定な流れパターンが、構造不安定な1つの中間状態の流れパターンを介して、他の構造安定な流れパターンに遷移可能か否かに関するパターン遷移情報を記憶し、
上記制御部は、
上記パターン遷移情報に基づいて、指定された構造安定な流れパターンから、位相幾何学的に採り得る別の構造安定な流れパターンへ至る遷移ルートに関する遷移情報を取得する遷移情報取得手段、
を備えたことを特徴とする、流体遷移経路取得装置。 - 請求項1に記載の流体遷移経路取得装置において、
上記遷移情報は、
上記別の構造安定な流れパターンに至る、構造不安定な中間状態の流れパターンに関する情報を含むこと
を特徴とする、流体遷移経路取得装置。 - 請求項1または2に記載の流体遷移経路取得装置において、
上記遷移情報取得手段は、
上記パターン遷移情報に基づいて、指定された2つの上記構造安定な流れパターンについて、一方の上記構造安定な流れパターンから他方の上記構造安定な流れパターンに至るまでの上記遷移情報を取得すること
を特徴とする、流体遷移経路取得装置。 - 請求項3に記載の流体遷移経路取得装置において、
上記遷移情報取得手段は、
上記構造不安定な1つの中間状態の流れパターンを介した遷移を1回の遷移として、上記一方の上記構造安定な流れパターンから上記他方の上記構造安定な流れパターンに至るまでの遷移回数を含む上記遷移情報を取得すること
を特徴とする、流体遷移経路取得装置。 - 請求項4に記載の流体遷移経路取得装置において、
上記遷移情報取得手段は、
上記遷移回数が最小となる、上記一方の上記構造安定な流れパターンから上記他方の上記構造安定な流れパターンに至るまでの上記遷移情報を取得すること
を特徴とする、流体遷移経路取得装置。 - 請求項1乃至5のいずれか一つに記載の流体遷移経路取得装置において、
上記構造安定な流れパターンは、
一つの穴を有する単連結外部領域において位相幾何学的に採り得る2種類の流れパターンに加えて、二つの穴を有する二重連結外部領域において吸い込み湧き出し対を持たないパターンを追加した、合計3種類の流れパターンを規定するパターン語に対して、上記流れパターンに一つの穴を加える場合に位相幾何学的に採り得る5種類の操作を規定した操作語のうちのいずれか一語を、追加された穴の数だけ付与することにより形成された語表現によって識別されており、
上記パターン遷移情報は、
上記構造不安定な1つの中間状態の流れパターンを介して、互いに遷移可能な上記構造安定な流れパターンが、上記語表現にて記述された情報であり、
上記遷移情報取得手段は、
指定された構造安定な流れパターンの語表現を基準として、上記パターン遷移情報に基づいて、上記遷移情報を取得すること
を特徴とする、流体遷移経路取得装置。 - 請求項6に記載の流体遷移経路取得装置において、
上記パターン遷移情報は、
上記構造不安定な1つの中間状態の流れパターンを介して、互いに遷移可能な上記構造安定な流れパターン間の、上記語表現の上記操作語の変化情報を含み、
上記遷移情報取得手段は、
指定された2つの構造安定な流れパターン間の語表現の操作語の変化を基準として、上記変化情報に基づいて、上記遷移情報を取得すること
を特徴とする、流体遷移経路取得装置。 - 請求項6または7に記載の流体遷移経路取得装置において、
上記遷移情報取得手段は、上記指定された構造安定な流れパターンの語表現を基準として上記パターン遷移情報に基づいて取得した遷移ルートの候補が複数ある場合には、流れパターンに1対1に対応する正規表現を使用して遷移ルートを1つに特定すること
を特徴とする流体遷移経路取得装置。 - 請求項8に記載の流体遷移経路取得装置において、
上記遷移情報取得手段は、
上記指定された2つの構造安定な流れパターン間の語表現の操作語の変化を基準として上記変化情報に基づいて取得した遷移ルートの候補が複数ある場合は、上記2つの構造安定な流れパターンの語表現に対応する正規表現をそれぞれ取得し、複数の遷移ルートの候補について、その遷移が起こるための部分表現が、一方の構造安定な流れパターンの語表現に対応する正規表現に含まれていない場合は、遷移ルートの候補から除外することを特徴とする流体遷移経路取得装置。 - 請求項9に記載の流体遷移経路取得装置において、
上記遷移情報取得手段は、
除外されなかった遷移ルートの候補が複数ある場合は、当該除外されなかった複数の遷移ルートの候補に対応する正規表現をそれぞれ取得し、上記除外されなかった複数の選択ルートの候補のうち、その正規表現が、他方の構造安定な流れパターンの語表現に対応する正規表現と一致するものを遷移パターンとして特定することを特徴とする流体遷移経路取得装置。 - 請求項8~請求項10のいずれか1つに記載の流体遷移経路取得装置において、
上記正規表現は、上記流れパターンに1対1に対応するグラフ表現が作成され、作成されたグラフ表現に基づいて作成されたものであることを特徴とする流体遷移経路取得装置。 - 請求項11に記載の流体遷移経路取得装置において、
上記グラフ表現は、上記流れパターンで規定される構造安定なハミルトンベクトル場Hに対して、固有のルート付き、ラベル付き、及び向き有りのツリーTH=(V,E)を割り当て(但し、Vは頂点と呼ばれる点の集合、Eは、頂点の間を結ぶエッジの集合である)、平面グラフとして可視化したものであることを特徴とする流体遷移経路取得装置。 - 請求項8~請求項12のいずれか1つに記載の流体遷移経路取得装置において、
さらに、流体中の物体に対して設計パラメータの候補を選択する場合に、上記設計パラメータの上限及び下限を設定し、当該設計パラメータの上限と下限で規定されるパラメータ領域から複数のパラメータを選択し、選択した複数のパラメータに対して、それぞれ流れの実験及び/又は数値計算を行い、実験及び/又は数値計算の結果に対して、上記語表現及び/又は上記正規表現を割り当て、割り当てた上記語表現及び/又は上記正規表現のうち、最適状態を示す上記語表現及び/又は上記正規表現を有する設計パラメータを、上記設計パラメータの候補として選択するシミュレーション手段を備えたことを特徴とする流体遷移経路取得装置。 - 請求項13に記載の流体遷移経路取得装置において、
上記割り当てた上記語表現及び/又は上記正規表現が、最適状態を示す上記語表現及び/又は上記正規表現を有しない場合でも、1回の遷移で、上記語表現及び/又は上記正規表現が、最適状態を示す上記語表現及び/又は上記正規表現を有する場合は、上記設計パラメータの候補として選択するシミュレーション手段を備えたことを特徴とする流体遷移経路取得装置。 - 記憶部と制御部とを備えたコンピュータにおいて実行される流体遷移経路取得方法において、
上記記憶部は、
位相幾何学的な二次元流れ構造の全体または一部における、構造安定な流れパターンが、構造不安定な1つの中間状態の流れパターンを介して、他の構造安定な流れパターンに遷移可能か否かに関するパターン遷移情報を記憶し、
上記制御部において実行される、
上記パターン遷移情報に基づいて、指定された構造安定な流れパターンから、位相幾何学的に採り得る別の構造安定な流れパターンへ至る遷移ルートに関する遷移情報を取得する遷移情報取得ステップ、
を含むことを特徴とする、流体遷移経路取得方法。 - 記憶部と制御部とを備えたコンピュータに実行させるためのプログラムにおいて、
上記記憶部は、
位相幾何学的な二次元流れ構造の全体または一部における、構造安定な流れパターンが、構造不安定な1つの中間状態の流れパターンを介して、他の構造安定な流れパターンに遷移可能か否かに関するパターン遷移情報を記憶し、
上記制御部において、
上記パターン遷移情報に基づいて、指定された構造安定な流れパターンから、位相幾何学的に採り得る別の構造安定な流れパターンへ至る遷移ルートに関する遷移情報を取得する遷移情報取得ステップ、
を実行させるための、プログラム。
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JPWO2015068784A1 (ja) | 2017-03-09 |
US20160259864A1 (en) | 2016-09-08 |
EP3067805A1 (en) | 2016-09-14 |
CN105706090B (zh) | 2019-03-15 |
KR101816778B1 (ko) | 2018-01-09 |
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