WO2020064761A1 - Procédé, produit-programme informatique et dispositif informatique pour l'évaluation de modèles de subdivision volumétrique - Google Patents

Procédé, produit-programme informatique et dispositif informatique pour l'évaluation de modèles de subdivision volumétrique Download PDF

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WO2020064761A1
WO2020064761A1 PCT/EP2019/075750 EP2019075750W WO2020064761A1 WO 2020064761 A1 WO2020064761 A1 WO 2020064761A1 EP 2019075750 W EP2019075750 W EP 2019075750W WO 2020064761 A1 WO2020064761 A1 WO 2020064761A1
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subdivision
irregular
evaluated
location
edge
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PCT/EP2019/075750
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German (de)
English (en)
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Daniel Weber
Joel Müller
André STORK
Christian ALTENHOFEN
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Fraunhofer Gesellschaft zur Förderung der angewandten Forschung e.V.
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
    • G06T17/00—Three dimensional [3D] modelling, e.g. data description of 3D objects
    • G06T17/20—Finite element generation, e.g. wire-frame surface description, tesselation
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/20—Design optimisation, verification or simulation
    • G06F30/23—Design optimisation, verification or simulation using finite element methods [FEM] or finite difference methods [FDM]
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2111/00—Details relating to CAD techniques
    • G06F2111/10—Numerical modelling

Definitions

  • the invention relates to a method, a computer program product and a
  • Computer-Aided Design creates a virtual 3D model of an object. This model is then examined in a simulation phase to determine how e.g. acting forces, temperatures or other influences, for which numerical simulation approaches are usually used.
  • the simulation phase can also be called CAE [Computer-Aided Engineering].
  • Adjustments can be simulated.
  • the transition between design and simulation usually means converting the continuous CAD representation into a CAE or simulation-compatible, discrete geometry (e.g. in the form of discrete finite elements). This conversion is also known as “discretization” or “meshing". On the one hand, this step is time-consuming and often involves manual intervention and error handling. On the other hand, CAD geometry information is irretrievably lost. A conversion in the opposite direction to, for example, automated conclusions Drawing the simulation results is usually also not possible with the current approaches.
  • FEM finite element method
  • the invention particularly relates to the last variant.
  • One advantage of subdivision or cell decomposition is the avoidance of loss of information between a CAD model and a model used for the simulation. Often the same object description (or the same discrete control point network) can be used as the basis for both models, for example if the CAD models are also generated as a volumetric subdivision model. With the previously known modeling and simulation approaches, however, this is not always possible.
  • checkpoint networks are also generally referred to as control networks.
  • the invention presented here is based on a subdivision, which can also be referred to as a subdivision and / or cell division.
  • a subdivision which can also be referred to as a subdivision and / or cell division.
  • an output volume model and / or an output checkpoint network is used in accordance with
  • predetermined rules further subdivided, in particular in such a way that the resulting control point network is successively further refined.
  • a so-called limit volume can be achieved, which represents the greatest possible refinement. Since carrying out such a maximum limit refinement is computationally complex, a direct mathematical evaluation of the limit is preferred instead, ie a mathematical calculation and evaluation of points of the limit volume without actually having to form the limit volume completely.
  • Catmull-Clark algorithm for volumes (Catmull Clark Solids), the rules of which are summarized, for example, in the following scientific work, which is also referred to below as “Burkhart et al.”: Daniel Burkhart, Bernd Hamann and Georg Umlauf: Iso-geometric Finite Element Analysis Based on Catmull-Clark: Subdivision Solids In Computer Graphics Forum, Vol. 29.Wiley Online Library, 1575-1584. Reference is also made to the following
  • the Catmull-Clark algorithm generally provides for a refined network to be calculated on the basis of an initial network of control points connected by edges, new control points being determined in each subdivision step, which subdivide the network into cells with a smaller area (or with a smaller volume in three dimensions).
  • a problem with such subdivision models is often that they merely approximate the shape of an object volume (or even just the object surface) via a grid of control points.
  • the individual control points can be defined as so-called nodes or control nodes by functions describing the object shape and in particular surface shape (for example by B-spline functions). Therefore, one would like to evaluate a specific location of the object, for example, on its surface or in its volume (i.e.
  • Control points 100 of defined B-spline functions but without being part of the object 102 itself or lying within its area.
  • DE 699 15 837 T2 describes the problem of directly evaluating any location in the subdivision model in such a way that the coordinates of the control points of a subdivision model can define basic B-spline functions to describe the contour of individual patches within a two-dimensional (patch) model of an object.
  • the checkpoints can be general
  • Space coordinates can be defined (for example, as X, Y and Z values). Since the
  • control points do not always match actual locations within the area
  • another parameter set is used that actually describes locations within the area or is defined in a corresponding coordinate system of the area.
  • a so-called u-v coordinate system or a u-v parameter set is defined for this.
  • the u-v parameters can be translated into three-dimensional spatial coordinates in space.
  • the corresponding X, Y and Z values can be determined for a location defined in the area by means of u-v parameters using the basic functions.
  • a relationship between the parameters for the locations in space and the area definition is defined by the B-spline basic functions in order to finally be able to specify the coordinates of any object location in a preferably ordinary Cartesian spatial coordinate system (for example in an ordinary X-, Y and Z coordinate system).
  • the basic functions can do more than just the object shape
  • object properties such as a material density or a material type. Consequently, the described parameterization can also be used to determine specific properties of a given u-v location using the basic functions.
  • each object location is mathematically clearly defined and / or its coordinates (preferably in the form of XYZ spatial coordinates) as well as gradient (derivation of the coordinate according to u and v for surfaces, or according to u, v, w for subdivision volumes) or properties using the B -Spline basic functions of the subdivision model can be determined, this model can be evaluated at any location.
  • this model can be evaluated at any location.
  • the matrix C contains control points of a surface element under consideration and the matrix contains the coefficients of the bicubic B-spline basic functions on the
  • this equation would be expanded by a further dimension w.
  • checkpoints In particular, it is a piece of land that is spanned by a network of 4 x 4 control points and only regular
  • Control points contains (i.e. the control points within the network must be regular, whereas control points on the edges or outside edges of the network do not necessarily have to be regular).
  • a checkpoint is said to be regular if it has a valence (or even a degree or node degree) of four.
  • the valence corresponds to the number of edges that originate from or meet at a control point.
  • a network with an irregular checkpoint becomes a regular network by (arithmetically) performing a certain number of further subdivision steps.
  • new control points are defined and / or calculated in the course of these subdivision steps, which ultimately lead to a piece of land (in particular a piece of land defined by a network (output network)) with an originally irregular control point being divided into pieces of land with only regular control points .
  • the subdivision is carried out in such a way that the area or its output network is subdivided until the control point to be evaluated lies in a regular 4 x 4 control point network which is smaller than the output network.
  • the areal portion of the checkpoint network containing the irregular checkpoint and resulting from the subdivision is reduced. More specifically, the subdivision of the irregular starting network or
  • Area that contains the irregular control point is subdivided by calculation, but not necessarily the entire model, this can be called a local subdivision.
  • some matrix expressions can be calculated in advance and for the arithmetic execution of the further subdivision steps can only be read from a memory.
  • this calculation can be carried out essentially with a constant regardless of the position of a location to be examined and also the additional subdivision steps required
  • any location within the area can be evaluated in a simulation using a set of B-spline basic functions.
  • a detailed description of the calculation steps required for this can be found, for example, in paragraphs [0042-0088] of DE 699 15 837 T2, for example in paragraph [0080] dealing with information that can be calculated and stored in advance.
  • volume-metric subdivision models have so far not been able to be used in the desired manner for simulating object properties, for example to calculate forces or stresses acting in the object and to suitably adapt the construction of the object. There is therefore no satisfactory way to take advantage of the real advantage of subdivision models in the form of the described slight loss of information during the transition between CAD and CAE in the context of corresponding simulations.
  • the object of the present invention is therefore to better coordinate a computer-aided design phase and simulation phase in the development of technical objects or systems, in particular in such a way that precise determinations of relevant properties are made possible with a limited computing time.
  • This object is achieved by a method, a computer program product and a computer device according to the attached independent claims.
  • the invention solves the technical problem that computing times of a computer for simulations of the above type should not be arbitrarily high.
  • the technical means proposed for this purpose is to align the irregular edges
  • the inventors have generally recognized that it is possible to use a volumetric
  • irregular structures can be converted into so-called layered structures, which are explained below and which are regular in at least one dimension and are irregular in planes or layers extending to this dimension. These can then be in constant time (i.e.
  • a surprising finding of the invention is in particular that irregular elements of a volumetric subdivision model taking into account the
  • the Dimension can be strung together or layered.
  • the third dimension preferably extends along the irregular edge, the orientation of which is determined and taken into account as an essential new feature according to the invention.
  • a coordinate system used for evaluation can therefore preferably be aligned along this edge with one of its dimensions.
  • equation (17) this makes it possible to reuse the self-basis functions cp (u, v) known from equation (11) of the 2D case and for the most part previously calculated, and with a further basis function, which is also aligned along the irregular edge, to multiply. This is discussed below by way of example using equations (17) - (19) and, inter alia, FIG. 7b and FIG. 9.
  • control points are preferably numbered according to a fixed scheme and are consistent for the particular irregular configuration found in the cell. This consistent numbering allows a calculation of the own structure of the subdivision matrix and the transfer of Stams evaluation algorithm to such irregular ones volumetric subdivision models, which is why irregular cells with more than one irregular edge can be evaluated in constant time.
  • Finite element analysis FEA
  • IGA isogeometric analysis
  • Calculation of the simulation results or to solve the simulation scenario must be calculated at different points in the simulation algorithm integrals via finite elements (e.g. to calculate a stiffness matrix or a volume of the component to be simulated). If volumetric subdivision models are used as the basis for the simulation, as generally provided in the present case, the procedure according to the invention can accelerate the calculation of these integrals and thus the calculation of the entire simulation, since several locations (ie integration locations) have to be evaluated for each cell (or each element) . Such an evaluation of any location succeeds
  • Additive manufacturing processes especially 3D printing with multiple materials, e.g. using the PolyJet process.
  • 3D printing processes that can process several materials in one component, the material value to be printed is required for each location within each slice.
  • volumetric subdivision models as commonly used herein, multi-material 3D printing models can be efficiently represented and even graded
  • All of the measures listed can be carried out in a computer-assisted manner, for example by means of a computing or processor unit of a conventional PC, for example a desktop or laptop, a server, a handheld device, for example a smartphone, and / or the computer device disclosed herein.
  • a combination of such computers for example a computer network
  • the measures according to at least one of the last four structuring points of the preceding list can be carried out by such a computer or a combination of such computers.
  • Intermediate and / or final results can be stored in at least one storage device of the computer or the computer network.
  • the method can include a step of storing the evaluated location in a
  • Storage means or, in other words, in a computer memory.
  • one control point can have several neighboring ones
  • Control points must be connected via individual edges, ie there are no further control points on the edges and at the ends of the edges.
  • a plurality of control points connected to one another by means of edges can define a so-called (virtual) face in the subdivision model. More precisely, an area can be delimited and / or spanned by a plurality of control points, one of the control points being connected to two others from the plurality of control points by means of edges. If this area lies in the area of the surface of the subdivision model, it can accordingly be one
  • the subdivision model can generally be used to describe and / or model an object, a workpiece, an assembly, a technical system or the like.
  • a volumetric subdivision model can generally define and / or include a three-dimensional volume and also model the interior of the volume using the cells explained above.
  • Depth dimension i.e. in depth and / or inside the volume
  • Depth dimension i.e. in depth and / or inside the volume
  • control points can be defined in the manner described above as control points or, in other words, control points or control nodes of functions which describe properties of the subdivision model.
  • functions can describe a form of the subdivision model, for example its so-called limit volume.
  • the latter corresponds to the continuous volumetric body, which is generated by conceptually infinite refinements of the checkpoint network structure of the subdivision model and can be mathematically determined for many subdivision methods. A comparatively high or the maximum possible number of subdivisions of the subdivision model results in an approximation to this limit volume.
  • a location in the volume of the subdivision model to be evaluated can also be understood to mean locations in or on a surface of the volume or of the subdivision model.
  • volume element (English: cell or element) can generally be understood to mean any volume portion of the subdivision model including the edges, control points, etc. contained therein. In particular, it can be a volume fraction that can be evaluated using the regular calculation approaches explained below or that can generally be evaluated using preferred evaluation options.
  • An example is a volume element that is defined or spanned by 4 x 4 x 4 control points and that can include 3 x 3 x 3 cells.
  • Evaluating can generally identify those 4 x 4 x 4 checkpoint arrangements that contain a location to be evaluated.
  • the volume element can be a local (control point) neighborhood of a location to be evaluated and, for example, a cell which contains the location to be evaluated. More precisely, the location to be evaluated lies in a specific cell, with this cell and its neighbors and the associated control points being considered for the evaluation of the location.
  • the cell containing the location to be evaluated can accordingly be arranged centrally or centrally (for example centrally in a regular 4 x 4 x 4 control point network or also centrally in an initially irregular one
  • a volume element can include at least one cell.
  • a cell can in particular be defined and / or surrounded by eight surfaces which are opposite one another in pairs and preferably parallel to one another in pairs.
  • a cell can be defined by any number of areas that completely surround and / or enclose a volume within the subdivision model. The corresponding volume can then form the cell. It it goes without saying that the cell is also delimited by corresponding control points and edges of the areas or comprises them.
  • the information regarding the location to be evaluated can relate to coordinates or parameter values of the location. In particular, it can be
  • Act coordinate values that indicate a position of the location in the model volume for example in a spatial coordinate system, which can also be referred to as an X, Y, Z coordinate system.
  • a cell containing the location to be evaluated can be identified and parameter values can be specified which define the position of the location in a local coordinate system of this cell.
  • This local coordinate system can, for example, be referred to as a u, v, w coordinate system.
  • the subdivision model can be described and / or stored using known data formats or general computer-aided formats. Properties of the subdivision model sought can be determined from the corresponding data sets using conventional algorithms. For example, the valence of control points and consequently also the presence and / or a position of an irregular edge can be determined. Extentions, positions or orientations of individual volume elements or cells can also be determined or defined, and in particular those cells which contain the location to be evaluated. In principle, however, it is also possible to represent the subdivision model virtually and to enable an operator to manually identify irregular properties such as control points or edges and / or individual volume elements or cells within the
  • volume elements which have or comprise irregular properties can be determined, for example on the basis of
  • An irregular edge can also be defined to have one of four different valences, which valence can be defined as the number of faces adjacent to the edge or, in other words, the number of faces originating from the edge and / or are connected via or with this.
  • the evaluation can be understood to mean the parameterization described above, in particular the determination of location coordinates or properties of or at the location to be evaluated using functions which are defined by the control points. These functions can generally be basic functions or, in the case of regular cells, (in particular trivial) B-spline functions (modified B-spline functions can be used instead for irregular cells).
  • the alignment of the edge can be used to identify areas (e.g.
  • a volume element under consideration that can be evaluated using preferred and / or conventional approaches (e.g. using the two-dimensional approaches from the scientific work of Jos Stam or from the DE 699 15 837 T2).
  • preferred and / or conventional approaches e.g. using the two-dimensional approaches from the scientific work of Jos Stam or from the DE 699 15 837 T2.
  • These can generally be irregular or regular areas or levels, wherein the irregularity can in turn depend on whether there are control points with predetermined valences and in the case of two-dimensional levels with one of four different valences.
  • the location to be evaluated is not in a volume element and / or a cell that has an irregular edge
  • a location evaluation can be carried out using regular approaches.
  • the volume element can then be defined in the form of a local neighborhood around the location to be evaluated (or the cell which contains this location), for example in the form of a 4 x 4 x 4 control point network or an arrangement of 3 c 3 c 3 cells.
  • One possibility for evaluation is then to determine trivariate basic functions from products of univariate basic functions of the control points of this volume element.
  • volume element with the location to be evaluated has a completely irregular structure, ie no dimension or plane can be defined, along or in which there are regular properties (ie regular edges and control points); and if this is the case (if not there is already a layered structure and the following step is skipped):
  • a number of (arithmetic) subdivision steps are determined (e.g.
  • an alignment of the (still) remaining irregular edge is determined and then evaluated based on the location under consideration, in particular by means of subsequent (subdivision) basic functions and / or a calculated intrinsic structure of the subdivision matrix.
  • irregular edge and in particular such that the irregular edge extends along a (e.g. predetermined and / or preferred) axis of the coordinate system.
  • Defining the coordinate system according to the orientation of the irregular edge means that the orientation of the irregular edge when defining the
  • Coordinate system is taken into account.
  • This coordinate system can be the same coordinate system to which the information relating to the location to be evaluated also relates.
  • it can be the coordinate system explained above for defining locations in a single cell, for example a so-called u, v, w coordinate system.
  • Such cell-related coordinate systems can be distinguished above all by the fact that their axes, which preferably each run orthogonally to one another, each have values from 0 to 1 and the cell is parameterized according to these axes.
  • it can be the same coordinate system in which the control points are defined, or a different one
  • Coordinate systems can be defined as Cartesian coordinate systems. It goes without saying that the definition of the coordinate system can in turn be virtual or arithmetic and generally computer-aided and / or by manual input by the user.
  • the irregular edge can be defined, for example, in such a way that the irregular edge extends, for example, along a corresponding u, v or w axis.
  • Coordinate system can be determined.
  • the coordinate system or the subdivision model can then be realigned comprehensively around the irregular edge, so that the irregular edge extends along the predetermined axis.
  • the coordinates of the control points and / or edges of the subdivision model can be transformed in a manner known per se in accordance with the adjustment to be carried out.
  • a volume element which contains a cell with the location to be evaluated, can be divided at least after at least one further (computational) division into regular planes along the irregular edge and irregular planes orthogonally thereto.
  • this enables a layer-by-layer structure and / or a layer-by-layer evaluation of the volume element to be achieved, in which two-dimensional node networks or layers of essentially or completely identical construction are strung together along the irregular edge.
  • these layers can comprise irregular control points at the same positions in each case.
  • volume elements are preferably used (ie defined and / or selected in the subdivision model), which are spanned in at least one dimension (preferably even two or three dimensions) by 4 nodes and / or 3 cells, this state in turn after at least one further (computational) subdivision can be achieved.
  • the subdivision (at least locally, that is, in the area of the cell) is carried out computationally in order to create an evaluable environment around the location to be evaluated.
  • the arithmetic subdivision can take place with a (predetermined) constant calculation time.
  • the evaluable cell is characterized in that it does not comprise a control point with one of six different valences. This applies, for example, to the case of a regular cell described below.
  • teaching disclosed herein can also be used to evaluate cells that are present in a layered structure explained below or (at most) have an irregular edge. The number n of necessary subdivisions for a location to be evaluated at the position u, v, w in one
  • the number of subdivisions required can be determined by means of the relationships explained below with reference to FIG. 9, which in turn are based on a cell-related coordinate system and in particular a predetermined alignment of the irregular edge therein.
  • the cell can be divided into eight smaller ones in a single step
  • Subdivision steps depending on a distance of the location to be evaluated from the irregular control point can be determined beforehand and stored (see also illustration of the corresponding relationships in the following FIG. 9), so that depending on the specific parameters of the location to be evaluated, a computational local division of the subdivision model is carried out immediately can be.
  • the evaluation of the cell which contains the location to be evaluated can be carried out using special subdivision basis functions, as have already been explained above for the two-dimensional case. This offers the advantage that at least some of the previously calculated matrix expressions and / or matrix-matrix multiplication can be used and the required computing time can thus be reduced and in particular kept essentially constant.
  • the given location can be evaluated with little effort, in particular with knowledge of the alignment of the irregular edge and preferably by means of partially already pre-calculated matrix expressions and / or matrix-matrix multiplication (for example as part of the subdivision basic functions).
  • Overriding can also be provided that trivariate basic functions are used for the evaluation at the given location or the given location, which are based on basic functions along an axis of the coordinate system that runs along the irregular edge.
  • the basic functions for the evaluation are also based on bivariate subdivision basic functions that are determined in planes spanned by the remaining axes of the coordinate system.
  • j denotes the respective control points in an i-th u, v plane or layer, which are each orthogonal to the irregular edge along the w axis.
  • j denotes the respective control points in an i-th u, v plane or layer, which are each orthogonal to the irregular edge along the w axis.
  • the layered structure can be created as part of a preparatory computational subdivision.
  • a subdivision matrix can be defined in a predetermined manner in order to achieve subdivision in a constant calculation time.
  • the volume element is subdivided and evaluated by means of a subdivision matrix (see e.g. the following example for subdivision matrix A) and the entries in the subdivision matrix depend on at least one structural property of the checkpoint network.
  • Entries can generally be the subdivision-based weighting factors for checkpoints or checkpoint coordinates. These should be entered in the subdivision matrix (i.e. with preferred column and row positions) in order to enable evaluation in a constant calculation time.
  • An example of this is described below, in which numbering or indexing and, based on this, entry in the subdivision matrix takes place in a specific sequence according to three different groups of control points. In particular, so-called changes of direction can be prevented during successive subdivisions, which would increase the calculation time linearly with the required subdivision steps.
  • the structural property can be a position of a checkpoint with one of six different valences (also referred to herein as irregular checkpoint) within the checkpoint network. More specifically, an irregular checkpoint can be determined and, for example, according to a distance from this irregular checkpoint
  • Checkpoint numbering or indexing of the other checkpoints is carried out in order to enter them in the subdivision matrix.
  • a control point which is an end point of the irregular edge can be selected as the irregular control point for specifying the structural properties and the entries in the division matrix.
  • a control point which is connected to all irregular edges can be selected as the control point for determining the structural properties and the entries in the division matrix.
  • the Checkpoint network can be divided once to achieve the above-mentioned state.
  • volumetric subdivision model is also described in
  • a boundary region can generally be understood to mean a region and / or volume element of the subdivision model that is positioned near an interface of the model to the surroundings and / or that contains such an interface.
  • the border region can be a neighborhood of at least 3 c 3 c 3 control points or at least 3 x 2 x 2 cells around the irregular property (for example the irregular edge)
  • At least one basic function used for the evaluation can be determined based on a sharp edge function along the axis along which the irregular edge also runs. This can be seen, for example, from equation (3), which is explained in more detail below and below, and which can be used in equation (15) below
  • Basic functions B are considered for a solid element considered at individual nodes i.
  • conventional basis functions are considered, while in the direction along the irregular edge (in the example below w) and in Direction of the edge or the boundary, a sharp edge function C, which is explained in more detail in the context of the exemplary embodiments, is considered.
  • the square brackets are a round-up operator that rounds up the content of the brackets in the case of fractions to the next higher integer, whereas the% sign indicates a modulo operator:
  • equation (3) The evaluation described above and explained on the basis of equation (3) can take place in particular if it is determined that the irregular edge is orthogonal to or in an interface of the subdivision model with the surroundings. In particular, there should be regular structures in the u and v directions, and if this is not the case, the equation (19) explained below can be used.
  • the location to be evaluated is located in a border area of the subdivision model to the surroundings and the location to be evaluated lies in a cell of the subdivision model which comprises more than one interface to the surroundings, the cell is evaluated at least once (computationally) before the evaluation. be divided.
  • the inventors have recognized that, at the latest after such a division, it is possible to use one of the evaluation options explained above for border areas, for example on the basis of sharp edge functions.
  • the invention further relates to a computer program product comprising
  • the computer program product in particular a computer program, is therefore designed to carry out the method according to the invention in one of the embodiments which are described in this description.
  • the method can be carried out using software.
  • at least parts of the method can alternatively be carried out by a combination of software and hardware designed to carry out the method, in particular with at least one integrated circuit (such as an FPGA) designed to carry out the method, or exclusively by such hardware .
  • the invention relates to a computer device comprising a processor unit and a memory unit in which a computer program product according to the
  • Fig. 1 is a schematic diagram for explaining a relationship of
  • Fig. 3 is an illustration for explaining the necessary number of
  • 5a-b are a diagram for explaining sharp edge functions
  • Fig. 6 is an illustration of a regular volume element of a volumetric
  • FIGS. 2a-c show examples of regular and irregular checkpoint networks 10, 11 for modeling two-dimensional surface pieces, such as those from one
  • the solid lines and points show an output network 10 and the dashed lines and points show a network 11 after performing a subdivision step in accordance with the Catmull-Clark algorithm (also referred to below as subdivision network 11).
  • the subdivision model can comprise a plurality of surface pieces and thus also a number of corresponding control point networks 10, 11, which then
  • FIG. 2b shows a regular checkpoint network 10 that of 4 ⁇ 4
  • Control points 12 is clamped, which are connected to one another by means of edges 14.
  • a plurality of edges 14 and control points 12 each close two-dimensional cells 13 between them.
  • the control point network 10 serves to model or mathematically describe the shape of a two-dimensional surface piece, which is not shown separately.
  • Each point within the control point network 10 (i.e. with the exception of its edge) has a valence of four and is therefore regular.
  • a 4 c 4 control point network 10 which contains only regular control points 12 (i.e. with the exception of the edge), defines a so-called regular bicubic B-spline surface piece or can be described mathematically as such. Coordinates and properties of locations within such a piece of land can be determined with little effort using the B-spline basic functions described at the beginning. An evaluation can therefore be carried out at any location within such a regular area.
  • control point network 11 is also shown in dashed lines after a further subdivision in the form of a subdivision network 11 for reasons of clarity. It can be seen that this has a finer subdivision, since its control points 12 are much closer together.
  • the fact that the subdivision network 11 and the output network 10 have common points can be attributed to the fact that the output network comprises control points 12 which are equidistant from one another and results generally from the
  • FIG. 2a shows a control point network 10, again shown with solid lines, for modeling a two-dimensional surface piece, which is not shown separately.
  • the network 10 contains an irregular or extraordinary point 18, which has a valence of three.
  • the checkpoint network 10 is therefore irregular and cannot be evaluated easily using basic B-spline functions.
  • no regular 4x4 checkpoint network 10 can be defined that contains only regular checkpoints 12 in its interior (partially overlaid by dashed checkpoints 12 of a subdivided checkpoint network explained below). Instead, it would only be possible to define a 3 c 3 network of control points 12 which are correspondingly provided with a reference symbol and which are the irregular ones Item 18 includes.
  • this network does not define a bicubic B-spline patch or cannot be mathematically described and evaluated as such.
  • control point network 10 contains an irregular point 18 which has one of four different valences, namely a valence of five.
  • the output network 10 shown again by means of solid lines (but also no possible partial network included hereof) thus does not define a regular 4 ⁇ 4 checkpoint network 10 which contains only regular points.
  • the parameter space is generally shown in the form of a cell-related u-v coordinate system 17 of a two-dimensional cell 13 or a two-dimensional subnetwork that encloses such a cell 13 (cf. FIG. 2a).
  • Each two-dimensional cell 13 can be described by means of such a two-dimensional cell-related u-v coordinate system 17, where u and v each take values between 0 and 1 (i.e. the cell 13 is parameterized in a corresponding value range).
  • n denotes the number of subdivision steps to be carried out in succession and k the number of steps
  • Subdivision step each obtained regular cells or networks.
  • any locations within the checkpoint networks 10 or cells 13, which are in themselves irregular, can be evaluated based on this with a substantially constant calculation time by reading out and using the previously stored matrix expressions .
  • FIG. 4 the illustration there to explain the indexing of individual control points 12 before and after an additional subdivision. Again one can see one shown with solid lines
  • Exit checkpoint network 10 containing an irregular point 18.
  • the exit control point network 10 further comprises a plurality of control points 12 (not all with one corresponding reference numerals), which are denoted by 1 to 2N + 8.
  • N denotes the valence of the irregular point 18.
  • a subdivision network 11 is shown in dashed lines which, as a result of the subdivision, contains control points 12 which are numbered 2N + 9 to 2N + 17. Because of their distance from the irregular point 18, these are no longer required for further local subdivision and are therefore shown in dashed lines.
  • the starting point is a matrix Co, which contains the checkpoints 12 of the starting checkpoint network 10. More precisely, the location vectors of the individual control points 12 defined in an X, Y, Z coordinate system are combined as (K x 3) - matrix Co, where K is the number 2N + 8 of control points 12 that define the output control point network 10 . The numbering of the
  • Control points 12 are also used to define the content of the matrix Co. More precisely, this defines the line number at which the coordinates of a respective point 12 are sorted into the matrix Co.
  • the x, y, z coordinates each result in a column value (i.e. the first column contains the x, the second the y and the third z coordinate).
  • a matrix Ci can be determined, the 2N + 17 location vectors of the individual control points 12 of the
  • Subdivision network 1 1 contains. More precisely, the connection applies
  • Ci ⁇ Co (4).
  • Subdivision steps are calculated in advance and stored (i.e. before the actual evaluation and / or an actual simulation). For any number of
  • Equation (5) If the diagonal form of A is inserted into equation (5) above, the following relationship is obtained, which only requires a limited number of matrix multiplications and exponentiation of a diagonal matrix, the exponentiation of the diagonal matrix against the matrix multiplications in particular that Equation (5) requires significantly less computing power and computing time:
  • the resulting regular network 20 can be evaluated on the basis of the control points 12 defining the network 20 by means of a bicubic B-spline area piece evaluation.
  • These points which are outlined in a wave shape for example in FIG. 2c for one of the regular networks 20, can be extracted from the matrix C n by means of a so-called selection matrix P k (also a recording matrix or English “picking matrix”), which is multiplied by C n outputs the required points in the correct order
  • Control points of the associated regular network 20 or regular area are multiplied with the B-spline basic functions N (u, v) at these control points:
  • equation (9) can be used to evaluate the location p an u, v in the sense of determining its coordinates and / or properties are made.
  • the expression from equation (1 1) can also be referred to as a bivariate subdivision basis function.
  • any location in a possibly also irregular exit control point network 10 is made possible, for which purpose only the number of required subdivisions n has to be determined.
  • This can be done, for example, using FIG. 3 and in particular a mathematical description or tabular summary of the relationships shown there (for example a table in which the required subdivisions n and optionally also a relevant area piece k are stored as a function of the u, v coordinates of a location to be evaluated are).
  • the number of required subdivisions n can be determined using the following equation: n [min
  • equation (1 1) or (10) can be calculated in advance and stored in a memory.
  • DE 699 15 837 T2 describes a computer-assisted implementation of the calculations described, in particular from paragraph [0080]. Accordingly, the potentiation of the diagonal matrix L as a function of the subdivision steps n is mainly or exclusively decisive for the actual calculation time, which, however, represents a comparatively low-cost arithmetic operation with an essentially constant arithmetic time. The total calculation time required can therefore be independent of the number n of local subdivision steps required
  • Three-dimensional subdivision models represent so-called sharp edges (also known as folds, English: crease edges). These can be defined by an operator, for example when creating an output network. They generally describe edges 14 within a control point network 10 at which no smoothing (i.e. finer subdivision) usually associated with advancing subdivisions or subdivisions is desired. Details on the definition of sharp edges and the special subdivision rules to be found there can be found in the following scientific work (see in particular pages 102-105 and 122 ff.): Sven Havemann: “Generative Mesh Modeling”, doctoral thesis 2005, available at https : //publikationsserver.tu-braunschweig.de/receive/dbbs_mods_00000008).
  • a B-spline curve shown in dashed lines and defined by control points pi-p 4 is shown in the left case, the control points pi-p 3 being collinearly and equally spaced from one another or also equally spaced from one another.
  • the control point pi represents the result of the extrapolation from p 3 to p 2 .
  • the edge 14 extending from pi to p 2 is an extrapolation of the edge 14 which extends from P 3 to p 2 .
  • the corresponding B-spline curve or sharp-edge functions c can therefore be formulated as follows, with the B-spline base function (one per control point pi-p) in the final column vector:
  • Control points are defined as shown in Figure 5b.
  • volumetric subdivision models consist of control points, edges and areas delimited by them. They also include cells, each delimited by a set of areas.
  • a volume fraction of the model in which an evaluation is to take place is also referred to below as
  • Volume element called. A special case is also represented by cells that lie in the outer edge areas of the model and in which at least some of the surfaces
  • extraordinary or irregular control points are defined as points that have one of six different valences.
  • extraordinary or irregular edges that have one of four different valences.
  • the valence is defined as the number of surfaces adjoining the edge or, in other words, as the number of surfaces which start from and / or are connected by the edge.
  • Subdivision model occurs when the location to be evaluated lies in a regular cell 52 without irregular control points and / or irregular edges.
  • a neighborhood of the model can be considered in the form of a volume element 50 that is spanned by 4 x 4 x 4 control points 12 or 3 c 3 c 3 cells 52.
  • FIG. 6 Such a case is shown in FIG. 6, again not all of the above or elements mentioned below are provided with a corresponding reference symbol (the same also applies to the following further figures).
  • FIG. 6 shows a central cell 52 which is delimited by six surfaces 54, each surface 54 being spanned by four control points 12 together with edges 14 connecting these control points 12.
  • Each control point 12 of a surface 54 is connected to two of the other control points 12 of this surface 54 by means of edges 14.
  • the cell to be evaluated in the example shown is located in this cell 52, which is why the corresponding neighborhood of 4 x 4 x 4 control points 12 or 3 x 3 x 3 cells 52 in the form of the volume element 50 is considered around this cell 30.
  • the subdivision model can consist of numerous further volume elements 50, which can adjoin the volume element 50 shown in FIG. 6 accordingly.
  • any location e in the central cell 52 can be evaluated as follows, taking into account the basic functions defined by the control points p:
  • N (u, v, w) at the i-th control point are defined as a product of three univariate B-spline basic functions according to the following equation (16), which also corresponds to the aforementioned equation (3), where the square brackets again denote a round-up operator and the% sign a modulo operator:
  • the evaluation of a location in the cell 30 shown in FIG. 5 is thus possible without considerable computational effort and per se in a constant calculation time.
  • the basic functions can specify coordinates and / or properties of or at the corresponding control points p by interpolation.
  • Such a structure of a volume element 50 which is only irregular in two dimensions, is also referred to below as a layered structure. If a plurality of irregular edges are present, the necessary number n of subdivision steps must be determined using FIG. 9 explained below until the layered structure is obtained, these subdivisions by calculating the intrinsic structure of the subdivision matrix in a constant calculation time
  • control points are preferably carried out, so that the topology of the control point network containing the evaluating cell for each
  • Checkpoint network (also called local checkpoint network) can be reached or, in other words, objected to by an edge or an edge length of the irregular edge.
  • These control points can also be called direct neighbors of the extraordinary edge and are numbered 1 to 3 in FIG. 13b. Metaphorically speaking, points 1 to 3 share a common edge with irregular control point 0.
  • checkpoints are numbered in ascending order, which are made by traversing two consecutive or adjacent edges in the checkpoint network containing the irregular cell (also local
  • Checkpoint network can be reached or, in other words, objected to by two edges or edge lengths of the irregular edge (see FIGS. 4 to 10 in FIG. 13c). Furthermore, these control points assume that they share an area with the irregular edge (that is, they delimit or can be assigned to a common area with the irregular edge). 3. Beginning with the number following the last checkpoint in group 2, all checkpoints are numbered in ascending order, which are made by traversing three consecutive or adjacent edges in the checkpoint network containing the irregular cell (also local
  • Called control point network can be reached or, in other words, are objected to by three edges or edge lengths of the irregular edge (see FIGS. 11 to 14 in FIG. 13d). Furthermore, these control points assume that they share a cell with the irregular edge (that is, they delimit or can be assigned to a common cell with the irregular edge).
  • control points of the lower layer are then numbered 15 to 21.
  • the control point of this layer lying along the w axis from FIG. 13 a, viewed on a line with the control point 0, is numbered 15.
  • the points surrounding this point 15, which are not necessarily connected directly via an edge are numbered 16 to 21. Metaphorically speaking, starting from point 15, the surrounding points are numbered in a star shape.
  • control point of the right (outer) layer lying along the v-axis on a line with the control point 0 is numbered 22.
  • a control point of the rear (outer) layer lying along the u-axis on a line with the control point 0 is numbered 32.
  • control point network only comprises hexahedral cells
  • control points numbered in the order 1 to 3 can also be assumed for the control points numbered in the order 1 to 3:
  • the corresponding control point in the subdivided control point network is accordingly an edge point
  • edge point or a cell point, depending on which element is shared with the irregular edge.
  • Zelltician derive from the conventional subdivision rules for Catmull-Clark surfaces or volumes, which were discussed above and for which corresponding quotations are also mentioned.
  • An edge point is a new point that is created when you perform a subdivision step on an edge to subdivide it. The same applies to area and cell points. For each cell e.g. creates a cell point in the arithmetic center of the cell when subdividing according to the Catmull-Clark rules.
  • the weights of the corresponding subdivided control points are in row i of the subdivision matrix, where i is the assigned index of the
  • control points form the three outer layers of the control point network, each layer forming a two-dimensional control point network.
  • a lowermost layer 56a (the base layer in FIG. 8b)
  • a frontal layer 56b which in the figure directly faces the viewer
  • an outer layer 56c on the right in the figure are shown as outer layers are.
  • the lowermost and frontal layers 56a, b each comprise a control point 18 with the valence 5 (see in each case marked irregular points 18 with a corresponding valence for these layers 56a, b).
  • For these layers 56a, b one can proceed as in the two-dimensional case from FIG. 4 and one
  • the right layer 56c does not include an irregular checkpoint and defines a regular two-dimensional checkpoint network.
  • the corresponding lines in the subdivision matrix can then be generated for the control points 12 of the outer layers 56a, 56b in a conventional manner. It must be taken into account that these three outer layers each share four checkpoints 12 in the exit checkpoint network and five in the subdivided checkpoint network in order to prevent these checkpoints from being taken into account twice. Furthermore, the considered (local) control network along the neighborhood relationships (which is also referred to as local
  • Neighborhoods or topological information is traversed, e.g. using a volumetric data structure created for this purpose. When creating the numbering described above, this information is taken into account and the control network is traversed based on the neighborhoods.
  • the weights and their corresponding column index can be determined by examining the local neighborhood of the corresponding control point 12.
  • the control points 12, which influence the position of a divided control point 12 can be used by using the topological Information is determined that is contained in the volumetric data structure.
  • the topological information can, for example, indicate which control points are connected to one another or which share a surface, cell or edge.
  • the diagonal matrix can be used to carry out the subdivisions, which contains the eigenvalues, as described above in
  • the M x K subdivision matrix is converted into a square matrix by removing the last M - K (ie M less K) rows, where K is the absolute number of control points in the original and M is the absolute number of control points is in the divided checkpoint network.
  • the removed rows correspond to the three outer layers of the divided checkpoint network. Taking into account that these layers are five each
  • MK 2 (Ni + N 2 + N 3 ) + 37, where Ni is the valence of the irregular edge that is adjacent to the i-th outer layer.
  • FIG. 7b shows an example of a volumetric control point network 10 which has irregular edges 15 with a valence of three. This applies in particular to the central cell 52, which contains the location to be evaluated. You can see immediately that there is no regular neighborhood for this cell 32, as shown in FIG. 6.
  • FIG. 8b shows an example of a volumetric network that has irregular edges 15 (and in particular a central cell 52 with such an edge 15) with a valence of five.
  • This example can be understood as a volumetric version of the irregular two-dimensional surface network shown in FIG. 8a.
  • an evaluation is to take place at a location which is in the inner central cell 52 and for which there is a new one
  • Volume element 50 is defined in the form of a local neighborhood around the inner cell 52. In the state shown, the volume element 50 or this neighborhood does not yet represent a regular arrangement of 4 x 4 x 4 control points 12 or 3 c 3 c 3 cells. This is particularly the case because the cell 52 still contains an irregular point 18 .
  • one of the irregular control points 18 contained therein must first be selected. This is one of the end points of an irregular edge 15 which is encompassed by the cell 52, and preferably that irregular control point 18 at which several or all of the irregular edges 15 converge. More specifically, in the case of a plurality of irregular edges 15 which delimit a common cell 52, the irregular control point 18 at which all irregular edges 15 of the cell 52 converge is generally selected.
  • an edge 14, 15 is defined in the example shown in such a way that it extends between two control points 12 but not through a plurality of control points 12.
  • the layered structure can then be obtained based on the above rules for defining a subdivision matrix and this can then be selected further.
  • a u, v, w coordinate system 57 for describing locations in the cell 52 is again defined in FIG. 7b. It can be seen that the irregular edge 15 extends along the w axis of this coordinate system 57.
  • the coordinate system 57 can be defined from the outset in such a way that its w-axis runs along the irregular edge 15.
  • the state shown in FIG. 7b represents the end result of such measures.
  • the w axis of the coordinate system 57 being inclined analogously to the slightly oblique course of the selected irregular edge 15 which delimits the cell 52 to be evaluated (in of the merely schematic FIG. 8b may not be clearly recognizable).
  • the planes extending orthogonally to the irregular edge 15 in u and v are referred to below as individual layers 56 of the volume element 50.
  • the cell 52 to be evaluated must also be locally subdivided in the three-dimensional case, since it initially contains the irregular point 18 and thus cannot be evaluated in the usual way by means of basic functions (ie not according to
  • Equation (15) can be evaluated).
  • FIG. 9 explained below is used for the corresponding subdivision.
  • the Catmull-Clark subdivision rules which are used to further subdivide the irregular cell 52, can be summarized in a subdivision matrix (or subdivision matrix) with the size 5M x 4K, at least if there are already layered structures, where K is the number of control points 12 per layer 56 of the control point network 10 or volume element 50 and M the
  • the additional local subdivision results in six sub-elements (or cells 52) that become regular, while two irregular cells remain that still contain the irregular edge 15 or adjoin it. Additional subdivisions of these irregular sub-elements would increase the volume fraction of the regular cells, but not to a complete one
  • FIG. 9 shows the volumetric variant or volumetric extension of Figure 3.
  • the formula cited in the general description section can be used.
  • the irregular control point 18 is at the position 0, 0, 0.
  • the irregular edge 15 extends from this along the w-axis. This assignment is achieved by the definition of the coordinate system 57, which is based on the previously determined alignment of the irregular edge 15. An at least theoretical possibility to evaluate any location in an irregular cell 52 would be this or its neighborhood in the form of the
  • Subdivide volume element 50 until the location to be evaluated lies within a completely regular cell 52. If a location to be evaluated is located near an irregular edge 15, but not close to an extraordinary point 18, the direction of the subdivision would have to be changed during the evaluation process.
  • Numbering scheme of the subdivision matrix can also maintain the intrinsic structure of the subdivision matrix and a constant calculation time can be achieved by exponentiating the eigenvalue matrix.
  • FIG. 10a-b show a representation of the irregular volume elements 50 from FIG. 7b and FIG. 8b after a further local division of the model (ie a division of its volume element 50) (in FIG. 10a the network from FIG. 7b and in FIG. 10b the Network from Figure 8b).
  • the case k 3 from FIG. 9 is assumed here.
  • the control point networks 10 of the respective volume elements 52 each consist of four layers 56, which each extend in the u-v plane (i.e. orthogonal to the irregular edge 15), and are strung together or, in other words, layered.
  • the inner cells 52 in which the location to be evaluated is located are also marked.
  • Pk is a picking matrix, which is analogous to the
  • Expression ⁇ p (u, v, w) is also referred to herein as a trivariate subdivision basis function (or trivariate division basis function).
  • the basic function N from equation (17) can also be found in equation (18), but there as a function of
  • Equation (18) contains the complete process described here for evaluating or calculating the limit point p, including setting up the subdivision matrix and its own structure, determining the number of subdivision steps n required and the ranges k etc.
  • Edge 15 is merely an example and that any of the other u-v axes or axes with entirely different names could also be assigned to the irregular edge 15.
  • the assignment of the u, v, w parameters in equation (17) would then have to be adapted accordingly.
  • Boundary elements 33 (a boundary element 33 comprises a plurality of cells 52, at least one of which is a boundary cell 34) are to be distinguished from the cases shown in FIGS. 1 a-e. More precisely, two first examples of border cells 34 are shown in FIG. 1 a and 11 b, and a corresponding one in FIGS. 1 a-e.
  • 1 1 d and 1 1 e show further examples of borderline cases, for which a corresponding two-dimensional configuration is shown in FIG. 1 1 f.
  • FIG. 1 a shows a first borderline case, in which an inherently regular structure of a single border layer 32 in the form of the topmost layer 56 in this figure has (the remaining edge surfaces of the border element 33 shown adjoin surfaces of adjacent volume elements 52, not shown, and therefore do not form a corresponding boundary layer 32).
  • the boundary layer 32 can generally be treated similarly to sharp edges in the bivariate (ie two-dimensional) case, which is why the boundary layer 32 in the corresponding two-dimensional configuration from FIG. 11 c is shown as a corresponding sharp edge 17 .
  • FIGS. 1 1 c and 1 1 f show that, analogously to the case of FIGS. 5a, 5b, the sharp edges 17 can be extrapolated in order to define the regular 4 ⁇ 4 (for the analog two-dimensional case shown)
  • To add neighborhood missing control points 12 see correspondingly shown control points 12 and edges in Figures 1 1 c, 1 1 f).
  • Such an actual addition of control points 12 by extrapolation does not have to take place, but legitimizes the use of the sharp-edge basic functions.
  • the B-spline basic functions from the above
  • Equation (16) replaced with sharp-edge basis functions in the w direction, since there the edge or border area to the environment is present.
  • Each of the three individual sharp edge functions from equation (14) corresponds to one of the layers 56 of the
  • the second borderline case which is shown in FIG. 11b, is characterized in that an irregular edge 15 runs orthogonally to the border layer 32.
  • a coordinate system 57 is defined such that its w-axis runs along the irregular edge 15 and the u-v plane parallel to layers 56 of the shown
  • Boundary element 33 runs orthogonal to this edge 15. For one
  • Bi j (u, V, w) ipfi III V) Ci (w) i 1, 3, j-1, K (21).
  • an irregular edge 15 extends within or along an interface.
  • the irregular edge 15 is contained in a corresponding boundary layer 32.
  • FIG. 11 d this relates to the vertical irregular edge 15 provided with a corresponding reference symbol, which runs in an interface 36 facing away from the viewer.
  • the corresponding two-dimensional case is shown in Figure 1 1 f.
  • the coordinate system 57 of the boundary cell 34 containing the location to be evaluated, which applies to both of FIGS. 1 1 d and 1 1 e, is again defined in such a way that the w axis runs along the irregular edge 15.
  • the layers 56 of the boundary elements 33 which run orthogonally to this edge 15, extend parallel to the uv plane of this coordinate system 57.
  • Subdivision basis functions in u and v are calculated and multiplied by the regular B-spline basis functions in the w dimension, which corresponds to equation (21) above. More precisely, the two-dimensional basic functions including sharp edges for the irregular surface piece, as shown in FIG. 1 1 f, are calculated and multiplied by the function in the w dimension.
  • the border cell 34 to be evaluated is positioned in such a way that two of its surfaces within an interface 36 or
  • Boundary layer 32 of the boundary element 33 lie. These are the upper boundary surface 36 marked in FIG. 1 e and the boundary surface 36 facing away from the viewer, each of which contains corresponding surfaces of the boundary element 33.
  • the irregular edge 15 is defined as an edge divided by the two interfaces 36 of the boundary cell 34 facing away from the viewer and the irregular point 18 as an end point of this irregular edge 15.
  • the number of subdivision steps n is defined, which result in a cell 34 containing the location to be evaluated and obtained after a corresponding subdivision still contains a single surface in a corresponding interface 36 of the boundary element 33.
  • the size of the Catmull-Clark subdivision matrix is reduced to 4M x 3K, where M corresponds to the number of control points 12 in each layer 56 of the subdivided network 11 and K corresponds to the corresponding number in the output network 10.
  • M corresponds to the number of control points 12 in each layer 56 of the subdivided network 11
  • K corresponds to the corresponding number in the output network 10.
  • Control points 12 with respect to a cell arranged in the interior of a volume are missing. Subsequently, one of the first to third limit cases explained above can be used and the relevant location can be evaluated directly accordingly. If a cell 34 to be evaluated has more than two surfaces which lie in boundary surface 36 or two such surfaces but which do not adjoin one another, the output network 10 of the associated boundary element 33 must be subdivided once. A resulting cell 34, which contains the location to be evaluated, then automatically falls under one of the above first to fourth limit cases and can be evaluated accordingly.
  • the solution shown thus also provides options for evaluating the subdivision model in its border or border areas in a constant time, so that in fact any locations in the entire model with a constant
  • Figure 12 is an illustration to illustrate advantages of the solution disclosed herein.
  • a first straight line 106 and a second straight line 108 are shown in detail, each of which shows a relationship between a required number of subdivision steps (i.e.
  • an output network 10 which, according to the fourth limit case above, comprises a cell 34 with at least two areas in the interfaces of the volume element 52 under consideration and an irregular point 18 which has a valence of five.
  • the first straight line 106 results from the application of a solution algorithm according to the prior art and more precisely from Burkhart et al. (Daniel Burkhart, Bernd Hamann and Georg Umlauf: Iso-geometric Finite Element Analysis Based on Catmull-Clark: Subdivision Solids In Computer Graphics Forum, Vol. 29.Wiley Online Library, 1575-1584). In the case of locations to be evaluated in irregular cells and / or volume elements, further local subdivisions are often carried out there until the location to be evaluated ultimately lies in a completely regular range.
  • the calculation time primarily results from the fact that for a given output network, essential matrix expressions such as the subdivision matrix and a calculation of its own structure are calculated in advance (i.e. before carrying out evaluations) and e.g. be stored in an electronic storage unit, and furthermore from the fact that one does not have to multiply n and with each subdivision step increasing subdivision matrices, but only potentiates the diagonal matrix.
  • the calculation can be carried out using a computer, in particular one
  • Computer device which can also include the aforementioned electronic storage unit. Subsequently, as explained above, only a diagonal matrix has to be potentiated depending on the number n of subdivision steps to be carried out, but this is characterized by generally short and, above all, essentially unchangeable calculation times.
  • Coordinate determination and / or property determination for any location of a subdivision model take place, in particular also if the basic functions define or specify material properties at individual control points.
  • the described solution can be used in particular in the context of simulations, e.g. when properties (especially physical properties) of an object that has been modeled using a volumetric subdivision model are to be simulated. Based on this, the construction of the object can be selected and / or adapted appropriately.
  • FEM finite element method
  • finite element methods investigate displacements of finite elements into which a given object has been divided. The resulting strains and stresses (and consequently also locally acting forces) are then determined from these displacements using a stiffness matrix likewise determined on the basis of the division. For a volumetric subdivision model, each cell can be understood and simulated as a finite element.
  • a displacement function u for individual control points i can therefore be based on the basic functions N defined in the respective control points, in accordance with the following
  • the basic functions N can be evaluated at any location and with a substantially constant calculation time on the basis of the above approaches, and the simulation and force determination can thus be carried out in a particularly computationally efficient manner.

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Abstract

L'invention concerne un procédé pour l'évaluation assistée par ordinateur d'un modèle de subdivision volumétrique, qui est décrit par un réseau (10) de points de contrôle composé de points de contrôle (12) reliés au moyen d'arêtes (14), le procédé comprenant : − l'obtention d'informations en ce qui concerne un lieu à évaluer situé dans le volume du modèle de subdivision ; − le fait de déterminer si le lieu à évaluer se situe dans un élément volumique (52), qui présente une arête (15) irrégulière, laquelle est une arête (15) qui relie un point de contrôle (12) à une valence différente de six, − et si c'est le cas : − la définition d'une orientation de l'arête irrégulière (15) ; − l'évaluation du modèle de subdivision volumétrique au lieu à évaluer sur la base de l'orientation déterminée. L'invention concerne en outre un produit-programme informatique pour la mise en œuvre d'un tel procédé et un dispositif informatique pour la mise en œuvre du produit-programme informatique.
PCT/EP2019/075750 2018-09-28 2019-09-24 Procédé, produit-programme informatique et dispositif informatique pour l'évaluation de modèles de subdivision volumétrique WO2020064761A1 (fr)

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