JP2005183411A - Interface tracking simulation method and program - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide an interface tracking simulation method in which a trouble that the order of vertexes of respective lines in a line array representing the interface profile is replaced to cause a looping phenomenon can be eliminated. <P>SOLUTION: Euler image writing design idea is taken into interface tracking simulation method. The distance along an interface is defined to be derived naturally from a two-dimensional Euclidean space, length of a micro line element of an interface curve is made invariant for deformation of interface, and the properties of a function on an isometric interface is utilized in the calculation of a speed field representing deformation of interface. Consequently, tracking of an interface profile under time development can be simulated at an arbitrary time without causing lateral asymmetry. <P>COPYRIGHT: (C)2005,JPO&NCIPI

Description

  The present invention relates to an interface tracking simulation method, and a program for executing the simulation. More specifically, the present invention relates to a temporal process of these interface shapes in a semiconductor process, crystal growth, an interface between two fluids, an interface with a redox reaction, and the like. The present invention relates to a method for simulating development using a computer, and an execution program thereof.

  In order to understand a physical phenomenon, it may be necessary to analyze how the interface formed in a specific two-dimensional plane evolves with time by computer simulation. A typical example is a semiconductor process simulation (see, for example, Patent Document 1 and Patent Document 2), which is a technology having extremely important significance from an industrial viewpoint.

  For example, in Patent Document 1, even when an oxide film having a fine unevenness with a slightly different height is overlapped with the growth of the oxide film in the oxide film growth process which is one process of the semiconductor process, An oxidation simulation method is described in which the time increment of the oxide film growth is not unnecessarily subdivided, and therefore no stagnation occurs in the temporal progression. However, such a conventional simulation method has the following problems.

  FIGS. 11A and 11B show the interface shape by a line segment sequence including a plurality of line segments having different lengths. The black circles in each figure are points (vertices) for dividing the interface shape into multiple line segments. Line segments obtained by connecting points (vertices) adjacent to each other with straight lines are used as line segment sequences. The shape of the interface is approximated.

  The shape of the interface changes from time to time, and this time change is characterized by a movement vector given to each vertex. That is, a vector having a norm corresponding to the moving distance per unit time is defined in the bisecting direction of the angle formed by the two line segments at each vertex, and this is used as the moving vector of this vertex. . As a result, the interface after the unit time increment has a shape indicated by a thin line in the figure.

FIGS. 12A to 12F are diagrams for explaining the change of the interface shape with time, and show the interface shape at times t ₁ , t ₂ ′, and t ₂ . The interface having the shape shown in FIG. 12A at time t ₁ changes to the shape of FIG. 12B at time t ₂ ′ and FIG. 12C at time t ₂ . Similarly, the interface having the shape shown in FIG. 12D at time t ₁ changes to the shape of FIG. 12E at time t ₂ ′ and FIG. 12F at time.

  In this case, in the interface shape shown in FIG. 12 (c), a phenomenon (Phenomenon 1) in which a loop is formed because the order of the vertices of the neighboring line segment sequence is changed occurs, while FIG. In the interface shape shown in (2), a phenomenon (phenomenon 2) occurs in which line segments constituting two different line segment arrays cross each other.

Further, according to the simulation method described in Patent Document 2 described above, when the line segment crossing phenomenon occurs at time t _2, first, the time step is a simulation parameter resetting small, FIG. 12 (a ) Or (d) based on the interface shape at time t ₁ , the simulation is executed until t ₂ ′, which is the time preceding time t ₂ , and no line segment crossing occurs (FIG. 12B). Alternatively, the interface shape shown in (e) is obtained. Then, after performing the initialization process by rearranging each vertex representing the interface shape shown in FIG. 12B or 12E, the simulation is executed again.

Japanese Patent Laid-Open No. 5-85891 Japanese Patent Application Laid-Open No. 9-199492 Arnold V. I. , “Mathematical Methods of Classical Mechanics”, Second Edition, Springer (1989). R. C. Browne et al. "Geometrical models of interface evolution" Physical Review A, vol. 29, p. 1335-1342 (1984). S. Saito et al. , Journal of Physics A: Mathematical and General, vol. 30, pp. 6951 (1997).

  However, according to the simulation method as described above, it is necessary to reset the time step as a simulation parameter and restart the simulation every time an intersection of the line segments constituting the line segment representing the interface shape occurs. There is a problem that the simulation takes a long time.

  The essence of the phenomenon of line segment crossing is that the conventional simulation method uses a calculation system based on the Lagrangian flow function widely used in fluid flow analysis (Lagrange picture). is there.

  There are two approaches to analyzing fluid flow: Lagrangian and Euler. Specifically, the Lagrangian picture starts from the equation of motion describing the trajectory of particles moving in the fluid, whereas the Euler picture assumes the “velocity field” formed in the fluid and the time of this velocity field. We start with an equation of motion that describes the dynamic change.

  When this is interpreted mathematically, the Lagrange picture traces temporally the trajectory of the subject affected by a group defined in the phase space, while the Euler picture itself is the group action itself. This means tracking changes over time.

  For example, the Euler equation for a two-dimensional perfect fluid is: velocity field is v, pressure is p, time is t

Given in.

  Reinterpreting this Euler equation from the point of view of group action in phase space, it constitutes a set of smooth functions defined over a region formed in a two-dimensional space. It can be regarded as an equation that expresses the action of a differential in-phase group that acts from one element to another (for example, Non-Patent Document 1).

  Based on such mathematical interpretation, assuming the axis of time t that is a parameter of deformation as a base space, and assuming a fiber bundle in a desired region in a two-dimensional Euclidean space as a fiber, these Consider a manifold consisting of a base space and fibers.

  If the action on each fiber is differential in-phase, the fiber bundle can be handled as a frame bundle of the differential in-phase group, and can also be handled as a main fiber bundle of the differential in-phase group. Handling this as a frame bundle corresponds to a Lagrange image, and handling it as a main fiber bundle corresponds to an Euler image.

  In such a treatment, the Euler equation described above assumes that the differential in-phase group in the two-dimensional Euclidean space is an energy functional in the harmonic mapping theory (or nonlinear sigma model), and applies the minimum principle to this energy functional. It will be understood as an equation of motion derived as a result. Further, the velocity field formed in the two-dimensional Euclidean space described above is understood as a generator of the Lie ring of the differential in-phase group which is an infinite dimensional Lie group. Furthermore, Lagrangian differentiation is understood as a geometric connection in a two-dimensional Euclidean space.

  Adopting a Lagrangian image has the advantage of making it easier to intuitively understand the events that occur, while also making it difficult to define the phase. For example, depending on the coordination state of the velocity field, a phenomenon may occur in which the neighboring point of the focused fluid particle does not move to the neighboring system after a certain period of time.

  FIG. 13 is a diagram for explaining the above-described problem. In order to facilitate understanding, a two-dimensional plane which is a motion field of fluid particles is divided by an equidistant grid.

  The thick arrows shown in the figure are vector representations of the velocity field at each lattice point. The thin line curve means the trajectory of each fluid particle. That is, with the passage of time, the fluid particle a moves to a ′, the fluid particle b moves to b ′, and the fluid particle c moves to c ′. When attention is paid to the temporal change in the relative distance relationship between each fluid particle, in the initial state, a is in the vicinity of b and c is far from b, but after a certain period of time, a ′ is b. C ′ exists in the vicinity of b ′, not in the vicinity of ′, and the relative distance relationship of each fluid particle changes with time.

  However, if the Euler image is adopted, such unnaturalness is eliminated. That is, the phase of the fiber becomes a metric space derived from the two-dimensional Euclidean space, and the velocity field can be handled as a field formed on the two-dimensional Euclidean space (where the velocity field is arranged). What kind of phase is given to the position space is another problem).

  The above-mentioned two phenomena (phenomenon 1 and phenomenon 2) that occur in the conventional simulation method are caused by the problem that the Lagrange picture contains.

  However, among these phenomena, phenomenon 2 is a phenomenon that occurs because the design philosophy of the conventional interface tracking simulation is based on a Lagrangian image in a broad sense. That is, phenomenon 2 should be considered as a characteristic of the design concept of the conventional interface tracking simulation. Further, it is known that a physical phenomenon such as phenomenon 2 may not easily occur in a certain kind of interface phenomenon.

  On the other hand, the phenomenon 1 can be considered to be caused by immaturity of the simulation method. That is, phenomenon 1 means that the conventional simulation method cannot mathematically express the shape of the interface that changes with time, and is a fatal defect.

  The present invention has been made in view of such problems, and an object of the present invention is to provide an interface tracking simulation method that overcomes the phenomenon 1 described above by incorporating the Euler pictorial design philosophy into the simulation method. There is to do. Therefore, the distance along the interface is defined as naturally derived from the two-dimensional Euclidean space according to the Euler picture. In this case, the length of the microwire element of the interface curve is not changed with respect to the interface deformation, and the velocity field expressing the interface deformation is utilized as a function on the isometric interface.

  In order to achieve the above object, the present invention provides a computer simulation method for tracking the temporal evolution of an interface formed in a two-dimensional plane, comprising: Is approximated by a line segment sequence obtained by dividing the curve into equal-length line segments, and the interface shape of the next time step is determined while maintaining the length of the line segments.・ Simulation method.

The invention according to claim 2 is the computer simulation method according to claim 1, wherein each vertex of the line segment sequence is designated by integers n = 1,. Is calculated from the contact angle {φ _n | n = 0,..., N, φ ₀ = φ ₁ , φ _N = φ _N−1 } using the contact angle

Tangent vector t _n and normal vector n _n of each line segment,

And the behavior of the line segment sequence around n = 0 and n = N corresponding to the velocity {U _n | n = 0,..., N} in the normal direction of each line segment is known analytically. The value of V ₀ is determined so that it can be approximated to the asymptotic behavior given, and corresponding to the velocity {U _n | n = 0,..., N} in the normal direction,

, The tangential speed {V _n | n = 0,..., N} is determined so as to satisfy the above equation, the normal direction speed {U _n | n = 0,..., N} and the tangential speed Based on {V _n | n = 0,..., N}, the n-th new tangent angle is calculated for the minute time interval ε.

A computer simulation method characterized by the following.

The invention according to claim 3 is the computer simulation method according to claim 1, wherein each vertex of the line segment sequence is designated by integers n = 1,..., N, and a reference angle of each line segment is specified. Is calculated from the contact angle {φ _n | n = 0,..., N, φ ₀ = φ ₁ , φ _N = φ _N−1 } using the contact angle

Seeking
The tangent vector t _n and normal vector n _n of each line segment are

The tangential speed V ₀ corresponding to the line segment corresponding to n = 0 is appropriately determined, and the tangential direction corresponding to the normal speed {U _n | n = 0,. Speed {V _n | n = 0,..., N},

, And the fineness based on the normal speed {U _n | n = 0,..., N} and the tangential speed {V _n | n = 0,. For the time interval ε, the nth intermediate tangent angle is

And the interface is updated using the intermediate contact angle, and the intermediate contact angle is replaced with a new contact angle {φ _n | n = 0,..., N, φ ₀ = φ ₁ , φ _N = φN ₋₁ } is redefined, and the velocity {U _n | n = 0,..., N} in the normal direction of each line segment that approximates the updated interface is calculated, and n = 0 The line segment tangential velocity V ₀ is appropriately determined, and the tangential velocity {V _n | n = 0,...

, And the fineness based on the normal speed {U _n | n = 0,..., N} and the tangential speed {V _n | n = 0,. For the time interval ε, the nth new tangent angle is

And the interface shape is updated based on the tangent vector t _n , the normal vector n _n , and the tangent angle.

  The invention according to claim 4 is a computer simulation method for tracking the temporal development of an interface formed in a two-dimensional plane, the curve representing the interface is oriented, and the interface curve is Dividing and approximating to equal-length line segments, maintaining the length of the line segments, determining the interface shape of the next time step, and changing the direction of the curve at each time step .

The invention according to claim 5 is the computer simulation method according to claim 4, wherein each vertex of the line segment sequence is designated by integers n = 1,..., N, and a reference angle of each line segment is specified. Is calculated from the contact angle {φ _n | n = 0,..., N, φ ₀ = φ ₁ , φ _N = φ _N−1 } using the contact angle

Tangent vector t _n and normal vector n _n of each line segment,

And the value of V ₀ is zero,

, The tangential speed {V _n | n = 0,..., N} is determined so as to satisfy the above equation, the normal direction speed {U _n | n = 0,..., N} and the tangential speed Based on {V _n | n = 0,..., N}, the n-th new tangent angle is calculated for the minute time interval ε.

The interface shape is obtained by replacing the number of each vertex of the line segment sequence so that the direction of the number assigned to the contact angle is reversed at each time step.

The invention according to claim 6 is the computer simulation method according to claim 4, wherein each vertex of the line segment sequence is designated by integers n = 1,. Is calculated from the contact angle {φ _n | n = 0,..., N, φ ₀ = φ ₁ , φ _N = φ _N−1 } using the contact angle

Tangent vector t _n and normal vector n _n of each line segment,

The tangential speed V ₀ corresponding to the line segment corresponding to n = 0 is appropriately determined, and the tangential direction corresponding to the normal speed {U _n | n = 0,. Speeds {V _n | n = 0,..., N} in ascending order of n values

, And the fineness based on the normal speed {U _n | n = 0,..., N} and the tangential speed {V _n | n = 0,. For the time interval ε, the nth intermediate tangent angle is

And the interface is updated using the intermediate contact angle, and the intermediate contact angle is replaced with a new contact angle {φ _n | n = 0,..., N, φ ₀ = φ ₁ , φ _N = φ _N−1 } is redefined, and the velocity {U _n | n = 0,..., N} in the normal direction of each line segment that approximates the updated interface is calculated and corresponds to n = N The tangential speed V _N of the line segment is appropriately determined, and the tangential speed {V _n | n = 0,...

, And the fineness based on the normal speed {U _n | n = 0,..., N} and the tangential speed {V _n | n = 0,. For the time interval ε, the nth new tangent angle is

The interface shape is updated based on the tangent vector t _n , the normal vector n _n , and the tangent angle.

The invention according to claim 7 is a program for performing computer simulation for tracking the temporal development of an interface formed in a two-dimensional plane using a computer, and approximates the interface. Each vertex of the line segment sequence is designated by integers n = 1,..., N, and the tangent angle {φ _n | n = 0,..., N, φ ₀ = φ ₁ , φ _N from the reference angle of each line segment. = Φ _N−1 }, and using the tangent angle

Tangent vector t _n and normal vector n _n of each line segment,

And the asymptotic behavior of the line segment corresponding to the velocity {U _n | n = 0,..., N} in the normal direction of each line segment is symmetric with respect to the permutation of n−1 and n + 1. Determine the value of V ₀ so that

, The tangential speed {V _n | n = 0,..., N} is determined so as to satisfy the above equation, the normal direction speed {U _n | n = 0,..., N} and the tangential speed Based on {V _n | n = 0,..., N}, the n-th new tangent angle is calculated for the minute time interval ε.

It is a program characterized by that.

The invention according to claim 8 is a program for performing computer simulation for tracking temporal development of an interface formed in a two-dimensional plane using a computer, and approximates the interface. Each vertex of the line segment sequence is designated by integers n = 1,..., N, and the tangent angle {φ _n | n = 0,..., N, φ ₀ = φ ₁ , φ _N from the reference angle of each line segment. = Φ _N−1 }, and using the tangent angle

Tangent vector t _n and normal vector n _n of each line segment,

The tangential speed V ₀ corresponding to the line segment corresponding to n = 0 is appropriately determined, and the tangential direction corresponding to the normal speed {U _n | n = 0,. Speed {V _n | n = 0,..., N},

, And the fineness based on the normal speed {U _n | n = 0,..., N} and the tangential speed {V _n | n = 0,. For the time interval ε, the nth intermediate tangent angle is

And the interface is updated using the intermediate contact angle, and the intermediate contact angle is replaced with a new contact angle {φ _n | n = 0,..., N, φ ₀ = φ ₁ , φ _N = φN ₋₁ } is redefined, and the velocity {U _n | n = 0,..., N} in the normal direction of each line segment that approximates the updated interface is calculated, and n = 0 The line segment tangential velocity V ₀ is appropriately determined, and the tangential velocity {V _n | n = 0,...

, And the fineness based on the normal speed {U _n | n = 0,..., N} and the tangential speed {V _n | n = 0,. For the time interval ε, the nth new tangent angle is

The interface shape is updated based on the tangent vector t _n , the normal vector n _n , and the tangent angle.

The invention according to claim 9 is a program for performing a computer simulation for tracking the temporal development of an interface formed in a two-dimensional plane using a computer, and approximates the interface. Each vertex of the line segment sequence is designated by integers n = 1,..., N, and the tangent angle {φ _n | n = 0,..., N, φ ₀ = φ ₁ , φ _N from the reference angle of each line segment. = Φ _N−1 }, and using the tangent angle

Tangent vector t _n and normal vector n _n of each line segment,

And the value of V ₀ is zero,

, The tangential speed {V _n | n = 0,..., N} is determined so as to satisfy the above equation, the normal direction speed {U _n | n = 0,..., N} and the tangential speed Based on {V _n | n = 0,..., N}, the n-th new tangent angle is calculated for the minute time interval ε.

The interface shape is obtained by replacing the number of each vertex of the line segment sequence so that the direction of the number assigned to the contact angle is reversed at each time step.

The invention according to claim 10 is a program for performing computer simulation for tracking the temporal development of an interface formed in a two-dimensional plane using a computer, and approximates the interface. Each vertex of the line segment sequence is designated by integers n = 1,..., N, and the tangent angle {φ _n | n = 0,..., N, φ ₀ = φ ₁ , φ _N from the reference angle of each line segment. = Φ _N−1 }, and using the tangent angle

Tangent vector t _n and normal vector n _n of each line segment,

The tangential speed V ₀ corresponding to the line segment corresponding to n = 0 is appropriately determined, and the tangential direction corresponding to the normal speed {U _n | n = 0,. Speeds {V _n | n = 0,..., N} in ascending order of n values

, And the fineness based on the normal speed {U _n | n = 0,..., N} and the tangential speed {V _n | n = 0,. For the time interval ε, the nth intermediate tangent angle is

And the interface is updated using the intermediate contact angle, and the intermediate contact angle is replaced with a new contact angle {φ _n | n = 0,..., N, φ ₀ = φ ₁ , φ _N = φ _N−1 } is redefined, and the velocity {U _n | n = 0,..., N} in the normal direction of each line segment that approximates the updated interface is calculated and corresponds to n = N The tangential speed V _N of the line segment is appropriately determined, and the tangential speed {V _n | n = 0,...

, And the fineness based on the normal speed {U _n | n = 0,..., N} and the tangential speed {V _n | n = 0,. For the time interval ε, the nth new tangent angle is

The interface shape is updated based on the tangent vector t _n , the normal vector n _n , and the tangent angle.

  Furthermore, the invention described in claim 11 is a computer-readable storage medium storing the program described in claims 7-10.

  In the interface tracking simulation method of the present invention, the distance along the interface is defined as being naturally derived from the two-dimensional Euclidean space by incorporating the Euler pictorial design philosophy, and the length of the micro-line element of the interface curve is defined. Is used as a function on the isometric interface in calculating the velocity field expressing the interface deformation, and the vertex of each line segment in the line segment sequence expressing the interface shape is used. It is possible to provide a simulation method that does not cause the inconvenience that a phenomenon of creating a loop appears due to the change of the order.

  First, for the sake of simplicity, the case where the shape of the interface to be handled is handled as one continuous curve will be described. In this case, research from the viewpoint of analytical handling of the interface has been made, and has already been reported in Non-Patent Document 2.

  FIG. 1 is a diagram representing an interface shape by a single smooth curve. In this figure, if the position vector of each point of the curve is X, this curve is characterized by a tangent vector t and a normal vector n determined for each point. Here, when s is an arc length, the normal vector is given by n = dX / ds.

  These vectors are the so-called Fresne-Cere relations,

Related through. Here, κ is the curvature of the curve, and the curvature is defined as the reciprocal of the radius of curvature. Therefore, if the angle of the tangent from a certain reference angle is φ, the tangent vector t and the normal vector n are given by t = (cosφ, sinφ) and n = (− sinφ, cosφ), respectively. The curvature is given by κ = dφ / ds.

  Next, consider isometric deformation of this curve. Let t be the transformation parameter.

It shall be given by
At this time,

If the above condition is imposed, the deformation in a state where the local length of the curve is invariable is given. here,

The abbreviation is used.
When differentiation is applied to equation (3)

It becomes.
Therefore, given the “velocity” U in the normal direction,

If we consider the deformation of the curve using the “velocity” V defined in (1), this deformation means an isometric deformation.

  Next, discretization (also referred to as difference) of an object to be handled is performed. That is, the target to be handled is approximated by a finite (or addition) set. Various methods are known for this discretization, and it is known that a differential equation that becomes a differential equation by taking the continuous limit exists in the number of continuous concentrations.

  For example, it is known that the difference equation (recurrence formula) called the logistic map, which is famous for forming chaos, becomes a logistic equation that is a differential equation that does not form chaos when the continuous limit is taken. It is also known that there is a differential equation that becomes this differential equation after the limit and does not generate a chaotic signal (see, for example, Non-Patent Document 3). That is, the process of discretization is a process of generating very abundant information, and means that the system can be destabilized by performing “bad” discretization of behavior. (Of course, an unstable system may be stabilized at the time of continuation by discretization.)

  Therefore, in order to achieve the object of the present invention, it is necessary to discretize the equation (5), but it is not necessary to discretize simply by the center difference or the forward difference. As described below, each geometric quantity obtained by continuous imaging is a quantity that can be defined based on the geometric background, and is discretized or discretized by ignoring the background. It is expected that the simulation result will become unstable after a certain finite time development.

  Therefore, in the simulation method of the present invention, the above-described continuous geometric concept is discretized from the standpoint of discrete geometry so as not to cause contradiction in the simulation results, and the deformation of the curve is described. Gives the scheme of the difference equation.

FIG. 2 shows a line segment obtained by discretizing a single curve. As shown in this figure, each line segment has a length L, and is distinguished by a sequence number n (n = 0, 1, 2,... N). Each line segment is characterized by parameters of a position vector r _n , a tangent vector t _n , a normal vector n _n , and a tangent angle φ _n determined at each vertex.

  These parameters satisfy the following discrete Fressen-Sele relation:

In this line segment sequence, the following minute deformation is virtually considered. (With respect to _{U n} are noted in Figure 18.)

  Also,

Define

Is a condition for maintaining the length of each line segment.
From the relational expression of Expression (7) and Expression (10),

Is obtained.
Further, (t _n−1 , n _n−1 ) of the last term on the left side of the formula (11) is decomposed into (t _n , n _n ),

^Assuming that φ _n ^ε is introduced, the deformation that does not change the length of the line segment is

It becomes.
Furthermore, when the left side of the equation (13) is approximated by the order ε,

It becomes.
This is a discretization of Equation (5), and a solution can be obtained by an explicit method.

If a velocity distribution {U _n } corresponding to the deformation in the normal direction is given to a certain interface, V ₀ is appropriately determined,

{V _n } can be determined by solving.
And {U _n } {V _n } and {φ _n } give a new angle,

The shape after a certain time step is determined.

  However, as is clear from the equation (14), these equations are not invariant to the exchange between n-1 and n + 1. That is, the left-right asymmetry that should not be inherent in the handled physical system is caused by the discretization.

In the following embodiment, a method of solving a difference equation that does not cause this left-right asymmetry will be described.
The solution of the differential equations of the present invention described below, by some physical conditions, speed U _n of the direction perpendicular to the interface it is assumed that the given. Then, the movement of the interface to be expressed Eulerian imaged, the relative speed given U _n, performs sequential calculation based on equation (15) is to obtain the velocity V _n of the tangential surface.

  In order for such calculation to be meaningful, the asymptotic behavior of the interface in a region where the sign is sufficiently large must be uniform and known for an arbitrary arc length s.

3, in the region 51 and 53 region sufficiently separated away from the focused region 52, a U _n is constant, and showing the state of the line segments in the case where the curvature k _n was assumed to be zero FIG. Such a state of the line segment is sufficiently general to express the change of the interface, and is a boundary condition that frequently occurs when simulating the change of the interface that occurs in a semiconductor process or the like.

(Example)
FIG. 4 shows an example of a simulation system for executing the interface tracking simulation of the present invention. This simulation system includes a computer system that manages simulation operations and a storage medium that stores an execution file of a program. The computer system includes a computer 111 that is a computer main body, a floppy (registered trademark) disk drive 113 built in the computer 111, a display 112 that is a display device, a mouse 114 and a keyboard 115 that are input devices, A floppy (registered trademark) disk 110 is used as the storage medium.

  FIG. 5 is a block diagram for explaining the mutual relationship among the display device, the input device, the storage device, and the arithmetic device that constitutes the simulation system.

  A display 112 that is a display device, and a mouse 114 and a keyboard 115 that are input devices are connected to a CPU 116 that is an arithmetic device built in the computer 111, and a signal input via the input device is input thereto. The calculation result is output to the display 112. Also, a hard disk 117 and memory 118 built in the computer 111 and a floppy (registered trademark) disk 110 that is read and written via a floppy (registered trademark) disk drive 113 are connected to the CPU 116 and are necessary for computation. Various information is read by the CPU 116 and the calculation result is stored.

  An execution file of a program based on the interface tracking algorithm of the present invention is stored in advance in a hard disk 117 built in the computer 111 or a floppy (registered trademark) disk 110 which is an external storage device. In order to cause the CPU 116 to execute the contents of the interface tracking simulation program, it is loaded onto the memory 118 by an instruction input from the keyboard 115 or the mouse 114, and the interface tracking simulation of the present invention is executed.

  In this embodiment, a difference equation calculation method in the simulation method of the present invention will be described.

The problem with the conventional calculation method is that the interface shape that occurs after a certain time step has left-right asymmetry. In order to overcome this problem, in the calculation method of this embodiment, the tangential velocity V ₀ of the interface is set so as to cancel the asymmetry.

  FIG. 6 is a flowchart showing the algorithm of the calculation method of the present invention. Moreover, FIG. 7 is a figure which shows the mode of the time change of the interface shape obtained by solving a difference equation according to this algorithm.

First, initialization is performed, an execution program is loaded from the hard disk 117 onto the memory 118, a necessary calculation area is secured, and an initial interface shape input by a desired method is formed. Also, the initial value V ₀ which is tangential velocity of the interface to zero (S21).

Next, based on physical considerations, the normal direction of the interface, calculates the discretized velocity field U _n (S22). For example, if the simulation to be executed is related to the phenomenon of temporal change of the interface due to oxidation, the velocity field _Un is calculated by a difference equation for describing the physical phenomenon of oxidation. In general, since the velocity field is a quantity of one form, even when discretized, it is defined on the line segment. In this embodiment, in consideration of the equation (8), the velocity field defined on the line segment determined by the position vector r _n and r _n-1 as shown in FIG. 18 was U _n. As a result of these calculations, {U _n | n = 1,..., N} is determined.

Further, U ₀ is set as U ₀ = U ₁ (S23). Thereby, {U _n | n = 0,..., N} is set.

Subsequently, {V _n } is determined by sequentially solving Equation (15) for a given value of V ₀ (S24), and a new tangent distribution {φ _n ^ε } according to Equation (16). Is redefined as {φ _n } (S25).

The interface shape is reconstructed based on these parameter settings (S26). In the case of the present embodiment, since the time evolution at r ₀ is known, r ₀ + εU ₀ n ₀ + εV ₀ t ₀ is reconfigured from data of the shape {φ _n }.

Then, in FIG. 7, δs, which is an asymptotic length difference at time t + ε, is obtained, and the value of V _{0 in} the next step is set to −δs (S27).

Finally, an end determination (S28) is performed. If it is determined to end (S28: Yes), the process ends (S29). If it is determined to continue (S28: No), steps S22 to S28 are repeated. At this time, since the calculation is performed with V ₀ = −δs, left-right asymmetry does not occur as shown in FIG.

  The interface tracking simulator or the subroutine including the difference scheme shown in the first embodiment is stored in a storage unit such as the floppy (registered trademark) disk 110 or the hard disk 117 shown in FIG. Thereby, a simulation can be easily performed.

  The problem that left-right asymmetry appears in the calculation result obtained by the conventional calculation method is due to the fact that the velocity vector to be defined on the line segment is defined on the vertex. In order to overcome this problem, the present embodiment cancels the left-right asymmetry by changing the assignment of the velocity vector to the vertex.

  FIG. 8 is a flowchart showing the algorithm of the calculation method of this embodiment.

  First, initialization is performed, an execution program is loaded from the hard disk 117 onto the memory 118, a necessary calculation area is secured, and an initial interface shape input by a desired method is formed (S221).

Next, based on physical considerations, the normal direction of the interface, calculates the discretized velocity field U _n (S222). For example, if the simulation to be executed is about the phenomenon of the change time of the interface due to oxidation, the velocity field _Un is calculated by a difference equation for describing the physical phenomenon of oxidation. In general, since the velocity field is a quantity of one form, even when discretized, it is defined on the line segment.

FIG. 18 is a diagram illustrating a state in which the velocity field U _n is defined on the position vectors r _n and r _n−1 by modifying Expression (8) and performing natural allocation to the vertices of the velocity vector. It is. This assignment determines the speed of {U _n | n = 1,..., N}.

Further, U ₀ is set as U ₀ = U ₁ (S223). Thereby, {U _n | n = 0,..., N} is set.

Subsequently, {V _n } is determined by sequentially solving Equation (15) for a given value of V ₀ (S224), and a new tangent distribution {φ _n ^ε } according to Equation (16). Is redefined as {φ _n } (S225).

The interface shape is reconstructed based on these parameter settings (S226). In the case of this embodiment, since the time evolution at r ₀ is known, r ₀ + εU ₀ n ₀ + εV ₀ t ₀ is reconfigured from data of the shape {φ _n }.

Further, as the determination 2 of U _n , the discretized velocity field U _n in the normal direction is calculated again by physical consideration (S227). Determining to assign how velocity field U _n shown in different Figures 9 and 2 in Figure 18. That is, since the velocity field U _n defined on the position vectors r _n and r _n−1 is considered, the interface shape is

Will be satisfied. With this assignment, the speed of {U _n | n = 1,..., N−1} is determined.

Next, the _{U N,} is set as _{U N = U N-1 (} S228). Thereby, {U _n | n = 0,..., N} is set.

Further, an equation for preserving the length of each line segment constituting the line segment sequence shown in FIG. 9 with V ₀ being the initial velocity being zero is calculated by the equation (16), and the calculation result is reduced,

{V _n } is determined by solving (S229).
Furthermore,

Then, a new tangent distribution {φ _n ^ε } is determined, and the result is redefined as {φ _n } (S230), and r ₀ + εU ₀ n ₀ + εV ₀ t _{0 is used} as the interface from the data of {φ _n } The shape is reconfigured (S231).

  Finally, an end determination (S232) is performed. If it is determined to end (S232: Yes), the process ends (S233). If it is determined to continue (S232: No), steps S222 to S232 are repeated.

  FIG. 10 shows how the interface shape changes over time obtained by solving the difference equation according to the algorithm described above. As is apparent from this figure, by solving the difference equation according to the above-described algorithm, left-right asymmetry does not occur in the region of interest.

  By storing the interface tracking simulator or subroutine including the difference scheme shown in the third embodiment in a storage medium such as the floppy (registered trademark) disk 110 or the hard disk 117, the simulation can be easily performed.

  In this embodiment, a difference equation calculation method in the simulation method of the present invention will be described in which the asymmetry of the interface shape is canceled by changing the direction of the curve representing the interface shape at each step. Specifically, handling is performed to change the grid order for each step.

  FIG. 14 is a flowchart showing the algorithm of the calculation method of the present invention. FIG. 15 is a diagram showing the temporal change of the interface shape obtained by solving the difference equation according to this algorithm.

  First, initialization is performed, an execution program is loaded from the hard disk 117 onto the memory 118, a necessary calculation area is secured, and an initial interface shape input by a desired method is formed (S31).

Next, based on physical considerations, the normal direction of the interface, calculates the discretized velocity field U _n (S32). For example, if the simulation to be executed is related to the phenomenon of temporal change of the interface due to oxidation, the velocity field _Un is calculated by a difference equation for describing the physical phenomenon of oxidation. In general, since the velocity field is a quantity of one form, even when discretized, it is defined on the line segment. In this embodiment, considering the equation (8), the velocity field defined on the line segment determined by the position vectors r _n and r _n−1 is defined as U _n . As a result of these calculations, {U _n | n = 1,..., N} is determined.

Further, U ₀ is set as U ₀ = U ₁ (S33). Thereby, {U _n | n = 0,..., N} is set.

Subsequently, {V _n } is determined for the given value of V ₀ by sequentially solving equation (15) in ascending order of the n value (S34), and a new tangent distribution according to equation (16). {Φ _n ^ε } is determined and redefined as {φ _n } (S35).

The interface shape is reconstructed based on these parameter settings (S36). In the case of the present embodiment, since the time evolution at r ₀ is known, r ₀ + εU ₀ n ₀ + εV ₀ t ₀ is reconfigured from data of the shape {φ _n }.

As shown in FIG. 15, there is an asymptotic length difference δs at time t + ε. In the calculation of the difference equation used in the simulation of the present invention, as shown in FIG. 15, the names of the vertex rows are reversed and the direction of the curve is reversed. In accordance with the reverse operation, the variables {φ _n }, {U _n }, and {V _n } also reverse the subscripts (S37). In mounting the program, the memory areas for the respective orientations may be taken and replaced to be used in order.

  Finally, an end determination (S38) is performed. If it is determined that the process is ended (S38: Yes), the process ends (S39). If it is determined that the process is continued (S38: No), steps S32 to S38 are repeated.

  By performing the above-described difference equation calculation, left-right asymmetry does not occur.

  By storing the interface tracking simulator or subroutine including the difference scheme shown in the fifth embodiment in a storage medium such as the floppy (registered trademark) disk 110 and the hard disk 117, the simulation can be easily performed.

  The problem that left-right asymmetry appears in the calculation result obtained by the conventional calculation method is due to the fact that the velocity vector to be defined on the line segment is defined on the vertex. In order to overcome this problem, in this embodiment, the left-right asymmetry is canceled by changing the assignment of the velocity vectors to the vertices and changing the solving order.

  FIG. 16 is a flowchart showing the algorithm of the calculation method of this embodiment.

  First, initialization is performed, an execution program is loaded from the hard disk 117 onto the memory 118, a necessary calculation area is secured, and an initial interface shape input by a desired method is formed (S321).

Next, based on physical considerations, the normal direction of the interface, calculates the discretized velocity field U _n (S322). For example, if the simulation to be executed is about the phenomenon of the change time of the interface due to oxidation, the velocity field _Un is calculated by a difference equation for describing the physical phenomenon of oxidation. In general, since the velocity field is a quantity of one form, even when discretized, it is defined on the line segment.

FIG. 18 is a diagram illustrating a state in which the velocity field U _n is defined on the position vectors r _n and r _n−1 by modifying Expression (8) and performing natural allocation to the vertices of the velocity vector. It is. This assignment determines the speed of {U _n | n = 1,..., N}.

Further, U ₀ is set as U ₀ = U ₁ (S323). Thereby, {U _n | n = 0,..., N} is set. Subsequently, it sets the _{U N} as _{U N = U N-1.} This sets {U _n | n = 0,..., N}.

Further, the value of V ₀ is set to zero, and {V _n } is determined by sequentially solving Equation (15) in ascending order of the n value (S324). Further, according to Equation (16), a new tangent distribution {φ _n ^ε } And redefine it as {φ _n } (S325).

Based on these parameter settings, the interface shape is reconstructed (S326). In the case of this embodiment, since the time evolution at r ₀ is known, r ₀ + εU ₀ n ₀ + εV ₀ t ₀ is reconfigured from data of the shape {φ _n }.

In addition, as the determination 2 of U _n , the discretized velocity field U _n in the normal direction is calculated again by physical consideration (S327). In this embodiment, as shown in FIG. 9, since the velocity field U _n defined on the position vectors r _n and r _n−1 is considered, the interface shape is

Will be satisfied. With this assignment, the speed of {U _n | n = 1,..., N−1} is determined.

Next, the _{U N,} is set as _{U N = U N-1 (} S328). Thereby, {U _n | n = 0,..., N} is set.

Further, an equation for preserving the length of each line segment constituting the line segment sequence shown in FIG. 9 with V _N being zero is calculated by the equation (16), and the calculation result is reduced,

{V _n } is determined by solving sequentially from the larger n value (S329).
Furthermore,

Then, the new tangent distribution {φ _n ^ε } is determined, and the result is redefined as {φ _n } (S330), and r ₀ + εU ₀ n ₀ + εV ₀ t _{0 is used} as the interface from the data of {φ _n }. The shape is reconfigured (S331). This is substantially the same as the direction reversal shown in the fifth embodiment.

  Finally, an end determination (S332) is performed. If it is determined to be ended (S332: Yes), the process ends (S333), and if it is determined to be continued (S332: No), steps S322 to S332 are repeated.

  FIG. 17 shows how the interface shape changes over time obtained by solving the difference equation according to the algorithm described above. As is apparent from this figure, by solving the difference equation according to the above-described algorithm, left-right asymmetry does not occur in the region of interest.

  By performing the above-described difference equation calculation, left-right asymmetry does not occur.

  By storing the interface tracking simulator or the subroutine including the difference scheme shown in the seventh embodiment in a storage medium such as the floppy (registered trademark) disk 110 or the hard disk 117, the simulation can be easily performed.

  It is possible to provide a simulation method that does not cause inconvenience that a phenomenon of creating a loop appears because the order of the vertices of each line segment in the line segment sequence expressing the interface shape is changed.

It is explanatory drawing of the classic geometry about a smooth curve. It is a figure explaining the line segment sequence obtained by giving discretization to one curve. In sufficiently distant region from the focused region in the constant U _n, and is a diagram for explaining the interface curvature k _n is zero. It is a figure for demonstrating the simulation means to perform the interface tracking method of this invention. It is a block diagram for demonstrating the simulation means of this invention. It is a flowchart for demonstrating the algorithm of the calculation method of this invention. It is a figure for demonstrating the mode of the time change of the interface shape obtained by solving a difference equation according to the algorithm of the calculation method of this invention. It is a flowchart for demonstrating the algorithm of the calculation method of this invention. By performing natural assignment to the apex of the velocity vector, which is a diagram showing a state that defines the velocity field U _n on the position vector r _n and r _n-1. It is a mode of the time change of the interface shape obtained by solving the difference equation according to the algorithm of the calculation method of the present invention. The interface shape is expressed by a line segment sequence composed of a plurality of line segments having different lengths. It is a figure for demonstrating the mode of a time-dependent change of an interface shape. It is a figure for demonstrating the problem which arises with the conventional simulation method. It is a flowchart for demonstrating the algorithm of the calculation method of this invention. It is a mode of the time change of the interface shape obtained by solving the difference equation according to the algorithm of the calculation method of the present invention. It is a flowchart for demonstrating the algorithm of the calculation method of this invention. It is a mode of the time change of the interface shape obtained by solving the difference equation according to the algorithm of the calculation method of the present invention. By performing natural assignment to the apex of the velocity vector, which is a diagram showing a state that defines the velocity field U _n on the position vector r _n and r _n-1.

Explanation of symbols

51 A region away from the region of interest 52 A region of interest 53 A region away from the region of interest 110 Floppy (registered trademark) disk 111 Computer 112 Display 113 Floppy (registered trademark) disk drive 114 Mouse 115 Keyboard 116 CPU
117 hard disk 118 memory

Claims (11)

A computer simulation method for tracking the temporal evolution of an interface formed in a two-dimensional plane, comprising:
Approximate the curve representing the interface with a line segment divided into equal-length line segments,
Determining the interface shape for the next time step while maintaining the length of the line segment;
A computer simulation method characterized by the above. Specify each vertex of the line segment sequence by integers n = 1,..., N;
The tangent angles {φ _n | n = 0,..., N, φ ₀ = φ ₁ , φ _N = φ _N−1 } from the reference angle of each line segment are calculated,
Using the tangent angle

Seeking
The tangent vector t _n and normal vector n _n of each line segment are

Defined by
Asymptotically known behavior of the line segment around n = 0 and n = N corresponding to the velocity {U _n | n = 0,..., N} in the normal direction of each line segment The value of V ₀ is determined so that the behavior can be approximated, and corresponding to the velocity in the normal direction {U _n | n = 0,..., N},

Tangential speed {V _n | n = 0,..., N} so as to satisfy
Based on the normal speed {U _n | n = 0,..., N} and the tangential speed {V _n | n = 0,. The new new contact angle

To calculate by
The computer simulation method according to claim 1. Each vertex of the line segment sequence is designated by integers n = 1,.
The tangent angles {φ _n | n = 0,..., N, φ ₀ = φ ₁ , φ _N = φ _N−1 } from the reference angle of each line segment are calculated,
Using the tangent angle

Seeking
The tangent vector t _n and normal vector n _n of each line segment are

Defined by
appropriately determine the tangential velocity V ₀ of the line segment corresponding to n = 0,
The tangential speed {V _n | n = 0,..., N} corresponding to the normal speed {U _n | n = 0,.

Determined to satisfy
Based on the normal speed {U _n | n = 0,..., N} and the tangential speed {V _n | n = 0,. The intermediate intermediate angle

Calculated by
Update the interface using the intermediate contact angle,
Redefine the intermediate contact angle as a new contact angle {φ _n | n = 0,..., N, φ ₀ = φ ₁ , φ _N = φ _N−1 };
Calculating the velocity {U _n | n = 0,..., N} in the normal direction of each line segment approximating the updated interface;
appropriately determine the tangential velocity V ₀ of the line segment corresponding to n = 0,
Tangential velocity {V _n | n = 0,..., N}

Determined to satisfy
Based on the normal speed {U _n | n = 0,..., N} and the tangential speed {V _n | n = 0,. The new new contact angle,

As sought
Updating the interface shape based on the tangent vector t _n , the normal vector n _n , and the tangent angle;
The computer simulation method according to claim 1. A computer simulation method for tracking the temporal evolution of an interface formed in a two-dimensional plane, comprising:
Orient the curve representing the interface,
Dividing and approximating the interface curve into equal-length line segments,
While maintaining the length of the line segment, determine the interface shape for the next time step,
The direction of the curve is changed at each time step.
A computer simulation method characterized by the above. Each vertex of the line segment sequence is designated by integers n = 1,.
The tangent angles {φ _n | n = 0,..., N, φ ₀ = φ ₁ , φ _N = φ _N−1 } from the reference angle of each line segment are calculated,
Using the tangent angle

Seeking
The tangent vector t _n and normal vector n _n of each line segment are

Defined by
The value of V ₀ is zero,

Tangential speed {V _n | n = 0,..., N} so as to satisfy
Based on the normal speed {U _n | n = 0,..., N} and the tangential speed {V _n | n = 0,. The new new contact angle

Calculated by
For each time step, obtain the interface shape by replacing the number of each vertex of the line segment sequence so that the direction of the number assigned to the contact angle is reversed,
The computer simulation method according to claim 4. Each vertex of the line segment sequence is designated by integers n = 1,.
The tangent angles {φ _n | n = 0,..., N, φ ₀ = φ ₁ , φ _N = φ _N−1 } from the reference angle of each line segment are calculated,
Using the tangent angle

Seeking
The tangent vector t _n and normal vector n _n of each line segment are

Defined by
appropriately determine the tangential velocity V ₀ of the line segment corresponding to n = 0,
The tangential speed {V _n | n = 0,..., N} corresponding to the normal speed {U _n | n = 0,.

Determined to satisfy
Based on the normal speed {U _n | n = 0,..., N} and the tangential speed {V _n | n = 0,. The intermediate intermediate angle

Calculated by
Update the interface using the intermediate contact angle,
Redefine the intermediate contact angle as a new contact angle {φ _n | n = 0,..., N, φ ₀ = φ ₁ , φ _N = φ _N−1 };
Calculating the velocity {U _n | n = 0,..., N} in the normal direction of each line segment approximating the updated interface;
appropriately determine the tangential velocity V _N of the line segment corresponding to n = N;
Tangential velocity {V _n | n = 0,..., N} in ascending order of n value

Determined to satisfy
Based on the normal speed {U _n | n = 0,..., N} and the tangential speed {V _n | n = 0,. The new new contact angle,

As sought
Updating the interface shape based on the tangent vector t _n , the normal vector n _n , and the tangent angle;
The computer simulation method according to claim 4. A computer simulation program for tracking the temporal evolution of an interface formed in a two-dimensional plane using a computer,
Each vertex of the line segment approximating the interface is designated by integers n = 1,.
The tangent angles {φ _n | n = 0,..., N, φ ₀ = φ ₁ , φ _N = φ _N−1 } from the reference angle of each line segment are calculated,
Using the tangent angle

Seeking
The tangent vector t _n and normal vector n _n of each line segment are

Defined by
Asymptotically known behavior of the line segment around n = 0 and n = N corresponding to the velocity {U _n | n = 0,..., N} in the normal direction of each line segment The value of V ₀ is determined so that the behavior can be approximated, and corresponding to the velocity in the normal direction {U _n | n = 0,..., N},
,

Tangential speed {V _n | n = 0,..., N} so as to satisfy
Based on the normal speed {U _n | n = 0,..., N} and the tangential speed {V _n | n = 0,. The new new contact angle

And
A program characterized by that. A computer simulation program for tracking the temporal evolution of an interface formed in a two-dimensional plane using a computer,
Each vertex of the line segment approximating the interface is designated by integers n = 1,.
The tangent angles {φ _n | n = 0,..., N, φ ₀ = φ ₁ , φ _N = φ _N−1 } from the reference angle of each line segment are calculated,
Using the tangent angle

Seeking
The tangent vector t _n and normal vector n _n of each line segment are

Defined by
appropriately determine the tangential velocity V ₀ of the line segment corresponding to n = 0,
The tangential speed {V _n | n = 0,..., N} corresponding to the normal speed {U _n | n = 0,.

Determined to satisfy
Based on the normal speed {U _n | n = 0,..., N} and the tangential speed {V _n | n = 0,. The intermediate intermediate angle

Calculated by
Update the interface using the intermediate contact angle,
Redefine the intermediate contact angle as a new contact angle {φ _n | n = 0,..., N, φ ₀ = φ ₁ , φ _N = φ _N−1 };
Calculating the velocity {U _n | n = 0,..., N} in the normal direction of each line segment approximating the updated interface;
appropriately determine the tangential velocity V ₀ of the line segment corresponding to n = 0,
Tangential velocity {V _n | n = 0,..., N}

Determined to satisfy
Based on the normal speed {U _n | n = 0,..., N} and the tangential speed {V _n | n = 0,. The new new contact angle,

As sought
Updating the interface shape based on the tangent vector t _n , the normal vector n _n , and the tangent angle;
A program characterized by that. A computer simulation program for tracking the temporal evolution of an interface formed in a two-dimensional plane using a computer,
Each vertex of the line segment approximating the interface is designated by integers n = 1,.
The tangent angles {φ _n | n = 0,..., N, φ ₀ = φ ₁ , φ _N = φ _N−1 } from the reference angle of each line segment are calculated,
Using the tangent angle

Seeking
The tangent vector t _n and normal vector n _n of each line segment are

Defined by
The value of V ₀ is zero,

Tangential speed {V _n | n = 0,..., N} so as to satisfy
Based on the normal speed {U _n | n = 0,..., N} and the tangential speed {V _n | n = 0,. The new new contact angle

Calculated by
For each time step, obtain the interface shape by replacing the number of each vertex of the line segment sequence so that the direction of the number assigned to the contact angle is reversed,
A program characterized by that. A computer simulation program for tracking the temporal evolution of an interface formed in a two-dimensional plane using a computer,
Each vertex of the line segment approximating the interface is designated by integers n = 1,.
The tangent angles {φ _n | n = 0,..., N, φ ₀ = φ ₁ , φ _N = φ _N−1 } from the reference angle of each line segment are calculated,
Using the tangent angle

Seeking
The tangent vector t _n and normal vector n _n of each line segment are

Defined by
appropriately determine the tangential velocity V ₀ of the line segment corresponding to n = 0,
The tangential speed {V _n | n = 0,..., N} corresponding to the normal speed {U _n | n = 0,.

Determined to satisfy
Based on the normal speed {U _n | n = 0,..., N} and the tangential speed {V _n | n = 0,. The intermediate intermediate angle

Calculated by
Update the interface using the intermediate contact angle,
Redefine the intermediate contact angle as a new contact angle {φ _n | n = 0,..., N, φ ₀ = φ ₁ , φ _N = φ _N−1 };
Calculating the velocity {U _n | n = 0,..., N} in the normal direction of each line segment approximating the updated interface;
appropriately determine the tangential velocity V _N of the line segment corresponding to n = N;
Tangential velocity {V _n | n = 0,..., N} in ascending order of n value

Determined to satisfy
Based on the normal speed {U _n | n = 0,..., N} and the tangential speed {V _n | n = 0,. The new new contact angle,

As sought
Updating the interface shape based on the tangent vector t _n , the normal vector n _n , and the tangent angle;
A program characterized by that.   A computer-readable storage medium storing the program according to claim 7.

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