JP4457186B2 - Mesh correction method using quasi-statistical model - Google Patents

Description

  The present invention relates to a method for correcting a created mesh, and more particularly, to a method for correcting a mesh using a quasi-statistical model.

  Triangular meshes, quadrilateral and hexahedral meshes are essential for rapid prototyping such as CAE (Computer-Aided Engineering), CAD (Computer-Aided Designing), structural reconstruction, stereolithography, etc. It has become something that can not be.

  According to the current mesh creation software, a mesh can be obtained instantaneously, but an irregular element as shown in FIG. 1 is generated, and the accuracy of analysis deteriorates. In this specification, “element” refers to a polygonal shape surrounded by nodes, and a set of elements is a mesh. The closer the mesh element is to a regular polygon or regular polyhedron, the better the analysis accuracy, but the extremely thin element as shown in FIG. 1 deteriorates the analysis accuracy. If a finer mesh is generated, it is possible to suppress the occurrence of parts with extremely poor analysis accuracy, such as extremely thin elements. However, in this case, the number of elements increases, so the analysis time is increased. It will be very expensive.

  Because of the problems described above, it is necessary to correct an annoying element. As research on conventional mesh correction methods, there are a method of changing the topology by deleting or adding nodes, and a method of improving elements without changing the number of nodes. The latter method includes a Laplacian method, a Taubin method, a bubble method, a subdivision method, and the like. In the CG world, there are a method based on a signal process and the like.

F. J. Bossen and P. S. Heckbert, A Pliant Method for Anisotropic Mesh Generation, Proceedings of the 5th International Meshing Roundtable, Sandia National Laboratories, 1996, 63-74. D. A. Field, Laplacian Smoothing and Delaunay Triangulations, J. Communications in Applied Numerical Methods, vol. 4, 1998, 709-712. T. Zhou and K. Shimada, An Angle-Based Approach to Two-dimensional Mesh Smoothing, Proceedings of the 9th International Meshing Roundtable, 2000, 373-384. L. A. Freitag, On Combining Laplacian and Optimization-Based Mesh Smoothing Techniques, AMD-vol. 220 Trends in Unstructured Mesh Generation, 1997, 37-43. L.A.Freitag and C.Olliver-Gooch, A Comparison of Tetrahedral Mesh Improvement Techniques, Proceedings of the 5th International Roundtable, Sandia National Lab., Albuquerque NM, 1996, 87-106. V. Parthasarathy and S. Kodiyalam, A Constrained Optimization Approach to Finite Element Mesh Smoothing, J. Finite Elements in Analysis and Design, vol.9, 1991, 309-320. S. A. Canann, J. R. Tristano, and M. L. Staten, An approach to Combined Laplacian and Optimization-Based Smoothing for Triangular, Quadrilateral, and Quad-Dominant Meshes, Proceedings of the 7th International Meshing Roundtable, 1998, 479-494. O. P. Jacquotte and G. Coussement, Structured Mesh Adaptation: Space Accuracy and Interpolation Methods, Computer Methods in Applied Mechanics and Engineering, vol. 101, 1992, 397-432. S. A. Canann, M.B. Stephenson, and T.D.Blacker, Optismoothing: An Optimization-Driven Approach to Mesh Smoothing, J. Finite Elements in Analysis and Design, vol. 13, 1993, 185-190. R. Lohner, K. Morgan, and O. C. Zienkiewicz, Adaptive Grid Refinement for the Euler and Compressible Navier Stokes Equation, Proceedings of the International Conference on Accuracy Estimates and Adaptive Refinement in Finite Element Computations.Lisbon, 1984. I. Babuska, O. C. Zienkiewicz, J. Gago, and E. R. de A. Oliviera (eds.), Accuracy Estimates and Adaptive Refinements in Finite Element Computations, John Wiley & Sons, Chichester, 1986, 281-297. K. Shimada and D. C. Gossard, Bubble Mesh: Automated Triangular Meshing of Non-manifold Geometry by Sphere Packing, Proceedings of the ACM Third Symposium on Solid Modeling and Applications, 1995, 409-419. K. Shimada, Anisotropic Triangular Meshing of Parametric Surfaces via Close Packing of Ellipsoidal Bubbles, Proceedings of the 6th International Meshing Roundtable, Sandia National Laboratories, 1997, 375-390. F. J. Bossen and P. S. Heckbert, A Pliant Method for Anisotropic Mesh Generation, Proceedings of the 5th International Meshing Roundtable, Sandia National Laboratories, 1996, 63-74. H. N. Djidjev, Force-directed Methods for Smoothing Unstructured Triangular and Tetrahedral Meshes, Proceedings of the 9th International Meshing Roundtable, Sandia National Laboratories, 2000, 395-406. G. Taubin, A Signal Processing Approach to Fair Surface Design, Proceedings of SIGGRAPH'95, 29, 1995, 351-358. M. Desbrun, M. Meyer, P. Schroder, and A.H. Barr, Implicit Fairing of Irregular Meshes using Diffusion and Curvature Flow, Proceedings of SIGGRAPH'99, 33, 1999, 317-32. J. Warren and H. Weimer, Subdivision Methods for Geometric Design, Academic Press, 2002.

  FIG. 2 shows a mesh used for a vehicle collision model. Such a mesh itself can be obtained instantaneously by the mesh generation software. However, it is necessary to find an irregular element from the obtained mesh and correct it manually. As a result, it took a considerable number of days to correct. Moreover, none of the various methods mentioned in the above conventional example has a standard, and cannot be corrected according to the purpose. In addition, each of them has a problem that the shape itself is smooth and the original structural shape is not maintained.

  In view of such a situation, the present invention can easily modify a mesh so as to improve an analysis accuracy by correcting an irregular element so as to approximate the shape of an ideal element. It is intended to provide a mesh correction method using a quasi-statistical model that can be applied to correction of the above.

The present invention is a mesh correction method for correcting, with a computer, an analysis mesh composed of a set of elements surrounded by nodes generated in advance ,
(A) a node selection process in which the computer selects an arbitrary node of the analysis mesh ;
(A) a node selection process in which the computer selects an arbitrary node of the analysis mesh ;
(B) a local region defining step in which the computer defines a local region including a plurality of elements including the node centered on the node selected in the node selecting step;
(C) an evaluation parameter calculation process in which the computer calculates, as a quasi-statistic model, a parameter for evaluating the quality of each element in the local area defined in the local area definition process;
(D) a vertex coordinate movement process in which the computer moves the coordinates of the vertices of each element that is the selected node so that the parameter distribution calculated in the evaluation parameter calculation process becomes a predetermined distribution; ,
(E) a center determination processing step in which the computer determines a corrected new center of the local region by averaging the coordinates of each vertex moved in the vertex coordinate movement step;
In the evaluation parameter calculation process, the aspect ratio of the element is calculated as the parameter,
In the vertex coordinate movement process, a new aspect ratio value a * is obtained from the following equation (1) as a quasi-statistical model so that the individual aspect ratio values calculated in the evaluation parameter calculation process have a Gaussian distribution. i is calculated,
However, i = 1,. . . , N and D are deviation values of the aspect ratio value a _i , F _i is a pseudorandom number assigned to the i-th element of the mesh, μ is an ideal value, and the aspect ratio value a _i from the average value M The deviation value D, which is a statistical parameter, is obtained by the following equation (2).

The analysis mesh may be a triangular mesh or a quadrilateral mesh.

Furthermore, the analysis mesh may be a mixed mesh of a triangular mesh and a quadrangular mesh.

Further, the method may include a process of repeating the processes (a) to (e), and in the node selection process, a contact point may be randomly selected for all the nodes in the analysis mesh. .

Furthermore, the process of repeating the steps (a) to (e) is performed, and in the node selection process, the next contact point is selected from the nodes near the previously selected node based on a predetermined rule. it may be.

In the node selection process, all nodes included in the selected element are selected by selecting an arbitrary element, and (b)-(e) are selected for all the selected nodes. The process of repeating may be included.

After the process of repeating, further said node selection process is performed, in the node selection process, elements to be subsequently selected, based from the elements in the vicinity of the selected element prior to a predetermined rule selection You may make it do.

The program for correcting a mesh according to the present invention causes a computer to execute the mesh correction method based on the quasi-statistical model .

  The mesh correction method using the quasi-statistical model of the present invention has an advantage that the mesh can be easily and quickly corrected. Also, various correction methods can be combined depending on the application.

  The best mode for carrying out the present invention will be described below with reference to the drawings. First, the “kernel” which is the central scheme of the mesh correction method of the present invention will be described. In the following description, a method for optimizing a triangular mesh is basically described. However, the present invention is not limited to this, and can be similarly applied to a quadrangular or hexahedral mesh. Furthermore, even a mixed mesh in which these various mesh shapes are combined is applicable.

FIG. 3 is a diagram for explaining a mesh correction method, and FIG. 3A shows a part of a mesh in an initial state, that is, before the correction method according to the present invention is applied. First, select any node P ₁ in the mesh. Then, a local region composed of a plurality of triangular elements including the node P ₁ , specifically, a gray portion in FIG. 3, is defined with the selected node P ₁ as the center. Next, parameters for evaluating the quality of each element in the local region are calculated as quasi-statistical models. Then, as shown in FIG. 3B, for example, the vertices of the respective elements are moved so that the calculated parameter distribution becomes a predetermined distribution. Finally, the coordinates of the moved vertices are averaged to determine a new corrected center of the local region as shown in FIG. This will be described in more detail below.

The closer the mesh element is to a regular polygon or regular polyhedron, the better the analysis accuracy. Therefore, the element aspect ratio can be considered as a parameter for evaluating the quality of the element. The aspect ratio is the maximum length with respect to the minimum length of the sides constituting the element. The aspect ratio of the triangular element is obtained from the longest and shortest sides of the three sides of the triangular element. In the case of a regular triangle, the aspect ratio is 1.0. Further, as another evaluation parameter, the vertex of the element, that is, the inner angle of the node P ₁ can be used. The internal angle is the angle of each element in the portion of the nodes P _1. 60 degrees for the regular triangle and 90 degrees for the square are the easiest to analyze. These various parameters may be used in combination.

  As the parameter distribution, various distributions such as a Gaussian distribution, a Poisson distribution, and a Poincare distribution can be used in accordance with the application. In addition, any other distribution can be applied. For example, when the aspect ratio of a triangle is used as a parameter, the analysis accuracy is best when all the elements are 1, so that the aspect ratio distribution is the largest and decreases as it becomes larger. Things are desired. Therefore, for example, it is convenient to use a Gaussian distribution. If the elements are modified so that the aspect ratio distribution is a Gaussian distribution, the analysis accuracy is improved.

Hereinafter, derivation of a new value of the mesh quality parameter will be described.
Theorem 1 Random number γ: γ ₁ , γ ₂ , ...
Random number ξ: ξ ₁ , ξ ₂ , ...
Distribution function F (ξ) = γ
Has a density function P (ξ).

According to Theorem 1, ξ has a density function P (ξ). If, for example, a Gaussian distribution is selected as P (ξ), the distribution function F (ξ) is expressed by equation (1).
However, e is an exponent, and t is to calculate ξ = G (γ) from the integral variable equation (1) (that is, ξ is to be calculated by the equation of γ), but the equation (1) is directly solved by Cartesian coordinates It is difficult. Therefore, in order to express Cartesian coordinates (x, y) with polar coordinates (r, φ), rewriting expression (1) with polar coordinates, the following expression (2) is obtained. Even if coordinate transformation is performed, the properties of ξ are not changed.
From the formula (2), the following formula (3) is obtained.

Equation (3) is used to generate a Gaussian random variable ξ value from a given pseudorandom number or random number γ, and the average value (ideal value) μ is 0 and the deviation value D is 1. Considering a triangular mesh process, the aspect ratio is defined by the ratio of the maximum side length to the minimum side length of the triangular mesh elements. The definition of the aspect ratio ensures the range of the value from the predetermined value μ to the maximum value. Here, this range indicated by X = [μ, a _max ] is called a probabilistic position. The pseudo-random number F _i is assigned to the i-th element of the mesh. The random number F _i which is the value of the aspect ratio of the i-th element is equal to the number of empirical distributions corresponding to the value a _i (probability value). By using the following formula (4), it is possible to generate a new aspect ratio value a ^* ₁ for each random number F _i .
However, i = 1,. . . , N, and D are deviation values, _Fi is a pseudorandom number, and μ is an ideal value. The deviation value D, which is a statistical parameter of the aspect ratio value a _i from the average value M, is obtained by the following equation (5).

The procedure of the mesh correction method will be described in order with reference to FIG. For each of the elements around the selected node P _1, to calculate the aspect ratio. When there are N elements around the node P ₁ , the aspect ratio values are a _{1 to} a _N. New aspect ratio values a ^* _{1 to} a ^* _N are calculated from Equation (4) as a quasi-statistical model so that the individual aspect ratio values have a Gaussian distribution.

  Here, the ideal value μ is μ = 1 when the aspect ratio is a parameter for a triangle, μ = 2 when the aspect ratio is a parameter for a quadrangle, μ = π / 2 when the interior angle is a parameter for a quadrangle, and so on. It depends on the used elements and parameters. In the example of FIG. 3, since it is a triangle and the aspect ratio is a parameter, μ = 1.

  When the vertex coordinates are moved so as to obtain this aspect ratio according to the new aspect ratio thus obtained, the vertex coordinates are set as new vertices of each element having the selected node as one vertex in FIG. ). However, since the elements must be continuous as a mesh, this is an incomplete mesh. Therefore, by averaging the coordinates of the moved vertices, the corrected new center of the local region is determined as shown in FIG.

  In the above description, an example in which a kernel is applied to a local region has been described. In the following description, an example in which the kernel of the present invention is applied to the entire mesh will be described.

  The first embodiment is a method in which the kernel according to the present invention is applied to the entire mesh by selecting nodes at random. In the present specification, this method is referred to as Gentle Enhancement Algorithm (GEA). In GEA, first, an arbitrary node is selected from the mesh, and a kernel is executed to perform element optimization on a local region centered on the node. When local region optimization is finished, another node is randomly selected and the kernel is executed again. Repeat this to run the kernel for all nodes.

  The result of applying GEA to a certain mesh sample is shown in FIG. FIG. 4 is a distribution graph in which the x-axis is the aspect ratio value and the y-axis is the number of elements. 4A shows the mesh and its distribution in the initial state, and FIG. 4B shows the mesh and its distribution after execution of the GEA of this embodiment. As can be seen from the figure, the distribution in the initial state is a Gaussian distribution according to this embodiment.

  The second embodiment is a method of applying a kernel according to the present invention to a mesh by selecting nodes based on a predetermined rule. In the present specification, this method is referred to as Wave Algorithm. First, an arbitrary node is selected from the mesh, the kernel is executed, and element optimization is performed on the local region centered on the node, similar to GEA. In Wave Algorithm, when the optimization of the local region is completed, the node to be selected next is selected from the nodes near the previously selected node based on a predetermined rule. For example, after the optimization by the center node of the local region is finished, a predetermined rule is determined in advance such that the next node to be selected is a neighboring node clockwise, and the kernel is determined based on the rule. Apply. In the case of Wave Algorithm, the node selection order can be specified in advance, so that the kernel can be applied only to a specific portion of the mesh, for example, a portion with low analysis accuracy.

An example of Wave Algorithm will be described with reference to FIG. First, select a node _P ^{1 0.} This is arbitrarily determined by the operator. For local region (a gray portion in FIG. 5) around the nodal point P ₁ ^0, optimized by applying the kernel of the present invention. Next, the kernels are sequentially applied with the nodes P ₁ ^{1 to} P ₅ ¹ as the centers. Further, around the nodal point _P ^{1 2} _~P ⁹ 2 respectively, apply the kernel in order. Thus, around the nodal point P ₁ ⁰ that is initially selected, based on rules that apply kernel so as to spread outwardly in order, so as to select a node.

  The result of applying Wave Algorithm to a certain mesh sample is shown in FIG. FIG. 6 is a distribution graph in which the x-axis is the aspect ratio value and the y-axis is the number of elements. FIG. 6A shows the mesh and its distribution in the initial state, and FIG. 6B shows the mesh and its distribution after execution of Wave Algorithm of this embodiment. As can be seen from the figure, the distribution in the initial state is a Gaussian distribution according to this embodiment.

  The third embodiment is a method of applying the kernel according to the present invention to nodes included in an element by selecting the element. In the present specification, this method is referred to as Triangle-tail Algorithm. First, an arbitrary element is selected from the mesh, for example, an element that deteriorates the analysis accuracy. Then, for all nodes included in the selected element, optimization of the local area centered on each node is performed. In this way, by selecting an element instead of selecting a node, for example, if an element with a large aspect ratio is directly selected, it is possible to optimize by applying a kernel around that element Become.

  Of course, the triangle-tail algorithm can be applied to the mesh as a whole. In this case, the element to be selected next may be selected at random, or may be a predetermined one like the wave algorithm. Neighboring elements may be selected based on this rule.

An example of the triangle-tail algorithm will be described with reference to FIG. First, as shown in FIG. 7, as shown in the node _P ^{1 0} _~P ³ 0, selects the high aspect ratio elements. Around these nodal _P ^{1 0} _~P ³ 0 respectively, apply the kernel in order. Thereafter, similarly to the Wave Algorithm, the kernels are sequentially applied with the nodes P ₁ ^{1 to} P ₅ ¹ as the centers. In this way, an element is first selected, and a node is selected based on a rule that a kernel is applied so that the element is spread outward in order.

  The result of applying Triangle-tail Algorithm to a certain mesh sample is shown in FIG. FIG. 8 is a distribution graph in which the x-axis is the aspect ratio value and the y-axis is the number of elements. FIG. 8A shows the mesh and its distribution in the initial state, and FIG. 8B shows the mesh and its distribution after execution of the triangle-tail algorithm of this embodiment. As can be seen from the figure, the distribution in the initial state is a Gaussian distribution according to this embodiment.

  FIG. 9 shows a distribution comparison diagram for explaining the results of applying these three types of examples in which the kernel of the present invention is applied to the entire mesh to the same mesh sample. Various methods can be selected depending on the purpose of use of the corrected mesh, such as whether to improve only the analysis accuracy or to improve the analysis accuracy while preventing the original shape from changing as much as possible. In addition, GEA is usually used, but various methods can be selected depending on the correction method, such as using Wave Algorithm, especially when the location to be corrected is clear.

  The above-described kernel and the method of applying it to the entire mesh can be executed by a computer by creating a program that performs these processes.

  Note that the mesh correction method using the quasi-statistical model of the present invention is not limited to the illustrated example described above, and it is needless to say that various modifications can be made without departing from the scope of the present invention.

FIG. 1 is a diagram for explaining irregular elements of a mesh obtained by mesh creation software. FIG. 2 is a diagram illustrating a mesh used in a vehicle collision model. FIG. 3 is a diagram for explaining a mesh correction method according to the present invention. FIG. 3A is an initial state, and FIG. 3B is a potential shape in which each element is statistically most easily analyzed. FIG. 3C is a diagram showing a mesh state of a final local region obtained by averaging the coordinates of the vertices. FIG. 4 is a distribution graph of aspect ratios of a certain mesh and its elements, FIG. 4 (a) shows the mesh and its distribution in the initial state, and FIG. 4 (b) is the mesh after execution of the GEA of this embodiment. It is a figure which shows the distribution. FIG. 5 is a diagram for explaining a node selection method in the Wave Algorithm of the present invention. FIG. 6 is a distribution graph of aspect ratios of a certain mesh and its elements, FIG. 6 (a) shows the mesh and its distribution in the initial state, and FIG. 6 (b) is after execution of Wave Algorithm of this embodiment. It is a figure which shows a mesh and its distribution. FIG. 7 is a diagram for explaining a method for selecting an element in the triangle-tail algorithm of the present invention. FIG. 8 is a distribution graph of the aspect ratio of a certain mesh and its elements, FIG. 8A shows the mesh in the initial state and its distribution, and FIG. 8B is the execution of the triangle-tail algorithm of this embodiment. It is a figure which shows the subsequent mesh and its distribution. FIG. 9 is a comparison diagram of aspect ratio distributions when GEA, Wave, and Triangle-tail are executed on the same mesh.

Claims (8)

A mesh correction method for correcting an analysis mesh composed of a set of elements surrounded by nodes generated in advance by a computer ,
(A) a node selection process in which the computer selects an arbitrary node of the analysis mesh ;
(B) a local region defining step in which the computer defines a local region including a plurality of elements including the node centered on the node selected in the node selecting step;
(C) an evaluation parameter calculation process in which the computer calculates, as a quasi-statistic model, a parameter for evaluating the quality of each element in the local area defined in the local area definition process;
(D) a vertex coordinate movement process in which the computer moves the coordinates of the vertices of each element that is the selected node so that the parameter distribution calculated in the evaluation parameter calculation process becomes a predetermined distribution; ,
(E) a center determination processing step in which the computer determines a corrected new center of the local region by averaging the coordinates of each vertex moved in the vertex coordinate movement step;
In the evaluation parameter calculation process, the aspect ratio of the element is calculated as the parameter,
In the vertex coordinate movement process, a new aspect ratio value a ^{* is obtained} from the following equation (1) as a quasi-statistical model so that the individual aspect ratio values calculated in the evaluation parameter calculation process have a Gaussian distribution ^. _i is calculated,
However, i = 1,. . . , N and D are deviation values of the aspect ratio value a _i , F _i is a pseudorandom number assigned to the i-th element of the mesh, μ is an ideal value, and the aspect ratio value a _i from the average value M The mesh correction method using the quasi-statistical model, wherein the deviation value D, which is a statistical parameter, is obtained by the following equation (2).
  The mesh correction method according to claim 1, wherein the analysis mesh is a triangular mesh or a quadrilateral mesh.   The mesh correction method according to claim 1, wherein the analysis mesh is a mixed mesh of a triangular mesh and a quadrilateral mesh.   4. The mesh correction method according to claim 1, further comprising a step of repeating the steps (a) to (e), wherein the node selection step A mesh correction method based on a quasi-statistical model, characterized in that contacts are randomly selected for all nodes.   The mesh correction method according to any one of claims 1 to 3, further comprising a step of repeating the steps (a) to (e), wherein the node selection step has been selected before. A mesh correction method using a quasi-statistical model, wherein a next contact point is selected based on a predetermined rule from a node near the node.   6. The mesh correction method according to claim 1, wherein in the node selection process, all nodes included in the selected element are selected by selecting an arbitrary element, and the selection is further performed. A method of correcting a mesh using a quasi-statistical model, comprising the step of repeating the steps (b) to (e) for all the nodes that have been made.   The mesh correction method according to claim 6, wherein after the repeating process, the node selection process is further performed, and in the node selection process, a predetermined rule is applied from a node near the previously selected node. A mesh correction method using a quasi-statistical model, wherein the next contact point is selected based on the quasi-statistic model.   A mesh correction program using a quasi-statistic model, which causes a computer to execute the mesh correction method using a quasi-statistic model according to claim 1.