JP2005044146A - Finite element analysis method, program and device - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To reduce the period of mesh design based on the numerical simulations of computer assisted engineering (CAE) diversified by finite element method, even if nodes between partial areas are discontinuous, and to increase the efficiency of mesh design. <P>SOLUTION: This finite element analysis method comprises inputting predetermined area shape data for numerical analysis; designating a division number for dividing a simple shape area that can be mapped, where the area shape data are broken up by the breaking up of the area shape data into the simple shape area; dividing the simple shape area into elements using a mapping method based on the designated division number; creating a discontinuous mesh of the entire predetermined area shape data based on each simple shape area subsequently divided into elements by the mapping method; executing high-precision finite element analysis that permits the generated discontinuous mesh and that does not bind discontinuous node behaviors with surrounding continuous node behaviors; and outputting the analysis results. <P>COPYRIGHT: (C)2005,JPO&NCIPI

Description

  The present invention relates to a finite element analysis method, a finite element analysis program, and a finite element analysis apparatus that allow a three-dimensional discontinuous mesh.

  In the three-dimensional finite element analysis, hexahedral finite elements are mainly used from the viewpoint of accuracy and calculation efficiency. Since a fully automatic division method for hexahedral finite elements has not been proposed, interactive automatic division using a mapping method as shown in FIG. 21 is used in the industry. In addition, the basic algorithm of the mapping method has been followed, in which the area of complex arbitrary shape is divided into simple shapes that can be mapped, and the number of divisions is adjusted so that discontinuous nodes do not occur, and element generation is performed for each partial area. Technology is disclosed.

  For example, a technology that can be applied only to a two-dimensional model, a 2.5-dimensional model obtained by sweeping a two-dimensional model, and a shell model in a three-dimensional space, but cannot be applied to a three-dimensional model is disclosed in the following patent document. 1-4. Further, a technique for improving the efficiency of interactive work for dividing an area having a complicated arbitrary shape into a simple shape that can be mapped or limiting the pattern to be automated is disclosed in Patent Document 5 below.

  Further, regarding the technology for improving the efficiency of interactive work for adjusting the number of divisions for each partial region so that discontinuous nodes do not occur, automation for limiting the pattern, or ignoring the calculation efficiency, the following Patent Document 6 ~ 15. In addition, technologies based on the nodal constraint method that allows the generation of discontinuous nodal points for each partial area but guarantees a reduction in accuracy are disclosed in Patent Documents 16 and 17 below.

  Moreover, the technique which introduces not only the hexahedron element but also the tetrahedron element whose accuracy is guaranteed to be reduced is disclosed in Patent Documents 18 and 19 below. In addition, a hexahedral element mesh can be generated in terms of topology, but a technique that has a problem in accuracy due to a poor element shape is disclosed in Patent Document 20 below.

JP-A-3-75970 JP-A-3-265978 JP-A-5-73647 Japanese Patent Laid-Open No. 6-266811 Japanese Patent Laid-Open No. 4-222061 JP-A-3-265978 JP-A-4-335475 Japanese Patent Laid-Open No. 4-241073 JP-A-5-73647 JP-A-8-320948 JP-A-9-282491 JP-A-11-96398 JP-A-11-224352 JP 2000-105842 A JP 2002-140373 A Japanese Patent Laid-Open No. 10-334276 JP 2000-40166 A JP-A-8-16629 JP 2001-76029 A JP-A-7-121579

  As described above, hexagonal finite elements are mainly used in 3D finite element analysis from the viewpoint of accuracy and computational efficiency, but a fully automatic segmentation method for hexahedral finite elements has not been proposed by academic societies. Interactive automatic division by law is used in industry. Therefore, the conventional techniques disclosed in Patent Documents 1 to 20 described above have a problem that it takes several months of work time for mesh generation, which is a bottleneck of the design process.

  In addition, the mapping method requires that complex arbitrary shaped regions be divided into simple shapes that can be mapped, and that the number of divisions be adjusted so that discontinuous nodes do not occur, and that element generation be performed for each partial region. This is the cause of the decline.

  In addition, a nodal linear constraint method is conventionally known as a method for handling discontinuous nodes in finite element analysis, but a problem of deterioration in accuracy due to linear constraints is a problem. In addition, the Element-free Galerkin method (EFGM) that does not require a mesh can handle discontinuous nodes. However, since the node distribution is inhomogeneous, it may be difficult to determine the effective influence range. It is clear that accuracy is not guaranteed.

  In order to solve the above-described problems caused by the prior art, the present invention provides a mesh design period based on numerical simulation of computational aided engineering (CAE) that is frequently used by the finite element method even if the nodes between the partial regions are discontinuous. An object of the present invention is to provide a finite element analysis apparatus capable of reducing the mesh size and improving the efficiency of mesh design.

  In order to solve the above-described problems and achieve the object, a finite element analysis method, a finite element analysis program, and a finite element analysis apparatus according to the present invention input predetermined region shape data to be numerically analyzed, and input regions When the shape data is decomposed into simple shape regions that can be mapped by the mapping method, the number of divisions for dividing the simple shape region into which the region shape data is decomposed is specified, and based on the specified number of divisions, The simple shape area is divided into elements by a mapping method, and a non-continuous mesh of the entire predetermined area shape data is generated based on each simple shape area divided into elements by the mapping method. Finite element analysis is performed, and the analyzed analysis result is output. In addition, the number of divisions may be specified under conditions where discontinuous nodes are generated.

  According to the present invention, the mapping type mesh generated for each of a plurality of partial areas is effective even when the nodes between the partial areas are discontinuous, and can be seen in a coupling application with EFGM. There is no parameter dependence and it can be made mathematically robust. In addition, a serendipity approximation function that behaves like a blending shape function and can be applied to discontinuous mesh portions between partial regions can be constructed through local meshing on the element surface and numerical integration by intra-element partitioning.

  According to the finite element analysis method, the finite element analysis program, and the finite element analysis apparatus according to the present invention, even if the nodes between the partial regions are discontinuous, the computer aided engineering (CAE) that is frequently used by the finite element method is used. The mesh design period by numerical simulation can be shortened, and the mesh design can be made more efficient.

  Exemplary embodiments of a finite element analysis method, a finite element analysis program, and a finite element analysis apparatus according to the present invention will be described below in detail with reference to the accompanying drawings. First, the influence range dependency of the Element-free Galerkin method (EFGM) that does not require a mesh will be described.

  The Element-free Galerkin method (EFGM) is conceptually defined by defining an approximate function using the moving least squires method (MLSM) for all positions within the analysis region. This is a method that applies the Galerkin method to continuous approximation functions. When an approximate function is constructed, unlike a normal finite element method (Finite Element Method, FEM), the existence of a finite element (mesh) is not assumed.

In MLSM, specifically, when the approximate function of the physical quantity at the integration point x is expressed by the following formula (1), the following formula (2) is obtained for a node group within a certain distance (influence range) from the integration point x. An EFG (Element-free Galerkin) approximation function Φ ^h _I is obtained by the coefficient a _J when the evaluation function J defined in (1) is minimized.

Here, p _J is the basis of the approximate function, m is the number of bases, n is the number of nodes in the influence range, x _I is the coordinate, Φ ^h _I is the EFG approximate function, and u _I is the physical quantity. , W _I is a weight function. EFGM conceptually defines a continuous approximation function within a region, but on the algorithm, MLSM is performed around numerical integration points in the construction of a stiffness matrix to approximately achieve continuity within the region. The accuracy of the approximate function is more than the basic order. The weight function w _I is a function of the integration point x and the node, and is expressed by, for example, the following expression (3), and is 0 on the boundary of the influence range due to the continuity condition. Here, d _mI is the radius of influence, and d _I is the distance between the integration point x and the node I.

Since this method is a kind of least square method, it has the following properties.
(1) For function derivation, the number of nodes with a base number of m or more is necessary for the influence range. MLSM is impossible when all the influence range nodes exist on a line / plane not including the integration point.
(2) The EFG approximation function does not satisfy the Kronecker delta condition, and becomes a smoothed function as the number of nodes in the influence range increases.

  In a discontinuous mesh, if all the influence range nodes are arranged on a line / plane that does not include integration points (see Fig. 1-1), or if the influence range is slightly expanded, more than necessary nodes enter at the same time. The case where the approximation function is smoothed (see FIG. 1-2) can easily occur. In addition, MLSM replaces the node distribution information in xyz space with one-dimensional information of distance from the integration point, and then derives the EFG approximation function, so that an approximate function construction that sufficiently reflects the inhomogeneity of the node distribution is constructed. It is difficult. As a result of various numerical experiments, it has been found that the MLSM condition for obtaining an effective EFG solution is “a minimum number of nodes having a basis number of m or more should be arranged in a balanced manner around the integration point”. Therefore, it can be said that EFG is not suitable for a discontinuous mesh model having a non-homogeneous node distribution.

  Next, the blending shape function will be described. The blending shape function is a polygonal shape function with singularity at the middle node. The basic concept of the one-dimensional line segment element is shown in FIG. Basically, in the hierarchical shape function construction algorithm, a blending shape function can be derived by introducing a polygonal function instead of a high-order function as an intermediate node function.

  A 1-singular type with one intermediate node has been proposed for 2D quadrilateral elements and 3D hexahedral elements. The blending shape function bends at intermediate nodes, so it is necessary to subdivide the elements locally and perform numerical integration. This subdivision is only for numerical integration, and no degree of freedom is added.

  Now, the above 1-singular type formulation can be applied only to the regular node distribution with respect to the local coordinate system, and the blending shape for a 3D element having two or more arbitrarily arranged intermediate nodes. It is impossible to express function in the same way. The edge nodes can be developed by developing a 1-singular type algorithm and incorporating integration point position determination. For example, in the case of QUAD6 shown in FIG. 3, the approximation function Ni can be derived by the following algorithm.

  As shown in FIG. 3, the abs function cannot be applied when the intermediate nodes are arranged at non-uniform intervals on the side. Therefore, the broken line between the intermediate node and the adjacent node is expressed by an if statement regarding the integration point position. It will be described individually (see FIG. 4). When intermediate nodes are distributed at arbitrary positions on the surface, even if an if statement is used, it cannot be expressed in a functional form, so local triangle groups are generated between the intermediate nodes and surrounding nodes. Do it.

  If the integration point is included in the triangle group, the height at the integration point is calculated for the polygonal pyramid consisting of the triangle group with the intermediate node as the vertex of height 1, and the hierarchical element approximation function generation algorithm It will be put in. In this case, the range of the triangle group with respect to the node can be considered as a kind of influence range. Therefore, in the triangulation, the right division is more preferable than the left in FIG.

As shown in FIG. 5, the following algorithm is conceivable as such a local triangulation algorithm.
(1) Search for the nearest node A from the central node P.
(2) The node B having the maximum angle of attack with respect to the line segment P-A is searched, and a triangle PA-B is generated.
(3) The process (2) is performed on the line segment P-B.
(4) The above processes (1) to (3) are performed until the triangulation is closed.

  The above local triangle group generation is performed for all nodes on the element plane. In general, the division pattern of local triangle groups is different. For subdivision within elements for numerical integration, triangle groups for all surface nodes are overlapped, and the intersected parts are subdivided triangle groups and element centroids. A tetrahedron is formed by and numerical integration is performed for each tetrahedron.

  Now, comparing the numerically formed blending shape function with the 1-singular type hexahedral blending shape function, the values are slightly different, but the shape function for the nodes on the surface is linear by triangulation. This is why the latter is described as a bilinear function, whereas it is a function.

  Therefore, when the proposed element surface meshing type arbitrary blending shape function construction method is used for 3D discontinuous mesh analysis, local areas around the same node on the element surfaces on both sides of the discontinuous mesh surface to which the arbitrary blending shape function is applied. The triangulation pattern must be the same.

This condition is essential for satisfying the C0 continuity of the discontinuous mesh surface. As a method for dealing with discontinuous meshes, a method of dividing the hexahedral element of the discontinuous part into tetrahedrons and using a tetrahedral element approximation function is also conceivable, but this method has the following advantages. Yes.
(1) To use the tetrahedral element approximation function, extra nodes having degrees of freedom are required inside the elements of the discontinuous mesh portion, but this method does not require the addition of extra degrees of freedom.
(2) In this method, the base function for forming the hexahedral element is tri-linear, but when the tetrahedral element approximation function is applied to the tetrahedral division group, it is simply linear.
(3) The accuracy of the tetrahedral approximation function is easily affected by the tetrahedral element shape, and when the elements of the discontinuous mesh portion are divided into tetrahedrons, the element shape tends to be defective.

  Next, the interpolation performance is checked. When an appropriate displacement was given to each node of a hexahedral element composed of 98 nodes as shown in FIG. 6, it was verified whether the displacement field could be expressed by a blending shape function. FIG. 7A is a displacement field obtained by interpolating a homogeneous node distribution with a function description including an if statement as shown in FIG.

  FIG. 7-2 is a displacement field interpolated by the algorithm of the element surface meshing type arbitrary blending shape function of the present invention. FIG. 7-3 is a displacement field obtained by interpolating the non-homogeneous node distribution with the algorithm of the present invention. As described above, function descriptions including if statements are not applicable to non-homogeneous node distributions. The mesh in the figure is used only for displaying the displacement field and is not used for deriving the interpolation function.

  Next, a finite element analysis processing procedure according to the embodiment of the present invention will be described. FIG. 8 is a flowchart showing a finite element analysis processing procedure according to the embodiment of the present invention. First, predetermined area shape data to be numerically analyzed is input (step S801). Next, the input region shape data is decomposed into simple shape regions that can be mapped (step S802). This operation is performed by an operator's operation.

  Next, for the simple shape region obtained by decomposing the region shape data, the number of divisions for dividing the simple shape region is designated (step S803). Then, based on the designated number of divisions, the simple shape region is divided into elements by the mapping method (step S804). Thereafter, a discontinuous mesh of the entire predetermined region shape data is generated based on each simple shape region divided into elements by the mapping method (step S805).

  Here, an analysis condition such as a boundary condition between the generated discontinuous meshes is added (step S806). Then, a finite element analysis that allows the generated discontinuous mesh is executed (step S807). Thereafter, the analyzed analysis result is output (step S808).

  Next, an example of finite element analysis incorporating the present invention will be shown. FIG. 9 shows an application example of a discontinuous mesh. FIG. 9A is a perspective view showing the three-dimensional shape data 901 of the elastic beam. FIG. 9B is an explanatory diagram illustrating an example in which the finite element method is used for the three-dimensional shape data 901 of the elastic beam in FIG. In FIG. 9-2, a continuous mesh is used, the number of nodes (number of nodes) is 341, and the number of elements is 200.

  FIG. 9C is an explanatory diagram illustrating an example in which the nodal linear constraint method (penalty constraint method, NLCM) is used for the three-dimensional shape data 901 of the elastic beam in FIG. FIG. 9-4 is an explanatory diagram illustrating an example in which EFGM is used for the three-dimensional shape data of the elastic beam in FIG. 9B, that is, an example in which FEM and EFGM are coupled (FE-EFGM). In FIG. 9-4, the mesh is discontinuous, the number of nodes (number of nodes) is 425, the number of elements is 232, and the number of discontinuous nodes is 54. According to FIGS. 9-1 to 9-4, it can be seen that NLCM and EFGM are not suitable for non-homogeneous node distribution.

  FIG. 10 and FIG. 11 are graphs showing the parameter dependence of the accuracy influence range when FEM and EFGM are coupled (FE-EFGM). 10 and 11, it can be seen that the accuracy of EFGM is affected by the number of nodes in the MLS.

  Next, an analysis example of the three-dimensional shape data of the plates to which the pipes are joined will be described. FIG. 12 is a perspective view showing three-dimensional shape data 1200 of a plate to which pipes are joined. An example in which a continuous mesh is applied to the three-dimensional shape data 1200 shown in FIG. 12 is shown in FIG. In FIG. 13, the number of nodes is 5125 and the number of elements is 3520.

  FIG. 14 shows an example in which a discontinuous mesh is applied to the three-dimensional shape data 1200 shown in FIG. In FIG. 14, the pipe has 3951 nodes, 2490 elements, and 238 constraints. As for the plate, the number of nodes is 4188, the number of elements is 2490, and the number of constraints is 677.

  Next, the case where the three-dimensional shape data shown in FIGS. 13 and 14 is deformed so as to lift the plate will be described. 15 and 16 are perspective views showing an example in which the three-dimensional shape data to which the continuous mesh shown in FIG. 13 is applied is deformed. 17 and 18 are perspective views showing an example in which the three-dimensional shape data to which the discontinuous mesh shown in FIG. 14 is applied is deformed by the FEM + penalty constraint method.

  FIGS. 19 and 20 are perspective views showing an example in which the three-dimensional shape data to which the discontinuous mesh shown in FIG. 14 is applied is deformed by the method of the present invention. The three-dimensional shape data shown in FIG. 12 (plates joined with pipes) are discontinuous meshes at strictly different positions, although the node positions between the partial areas are almost the same. It is a severe condition.

  As shown in FIGS. 17 and 18, the conventional nodal constraint method shows unrealistic rigid behavior, and the element-free Galerkin method (EFGM) and FEM coupling (FE-EFGM) cannot be analyzed. However, in FIGS. 19 and 20, it can be seen that the proposed method shows the same accuracy (deformation) as the continuous mesh (FIGS. 15 and 16).

  According to the embodiment of the present invention, the generation efficiency can be improved as compared with the conventional mapping method. In addition, regarding the processing of discontinuous meshes where the nodes between the partial areas do not match, the nodal linear constraint method (penalty constraint method, NLCM), the mesh-less Element-free Galerkin method (EFGM), and FEM coupling ( Higher accuracy than conventional methods such as FE-EFGM) can be achieved, and there is no parameter dependency, and a mathematically robust method can be provided.

  In addition, a serendipity approximation function that behaves like a blending shape function that can be applied to discontinuous mesh portions between partial regions can be constructed through local meshing on the element surface and numerical integration by intrapartition partitioning.

  Further, it is possible to reduce the mesh generation load for the three-dimensional hexahedral finite element analysis that is most often used for numerical analysis in the design process of the manufacturing industry. For example, conventionally, it took about 1-2 months to generate a hexahedral mesh of an engine block with a complicated shape. Therefore, conventionally, only 6 to 12 different types of engine blocks can be analyzed per year. This makes it impossible to perform finite element analysis by changing the shape conditions in various ways during design. According to the embodiment of the present invention, the labor required for generating the hexahedral mesh is less than half even if it is estimated, and the product development time can be shortened. This also makes it possible to reduce the price of the product.

  As described above, the finite element analysis method, the finite element analysis program, and the finite element analysis apparatus according to the present invention are useful for numerical analysis in the design process of the manufacturing industry.

It is explanatory drawing which shows the nodal distribution in a discontinuous mesh. It is explanatory drawing which shows the nodal distribution in a discontinuous mesh. It is explanatory drawing which shows blending shape function. It is explanatory drawing which shows a broken line type | mold QUAD6 element. It is explanatory drawing which shows the approximation function derivation method of a broken line type | mold QUAD6 element. It is explanatory drawing which shows the pattern of local triangulation. It is explanatory drawing which shows the hexahedral element which consists of 98 nodes (HEXA98). FIG. 5 is an explanatory diagram showing a displacement field obtained by interpolating a homogeneous node distribution with a function description including an if statement as shown in FIG. 4. It is explanatory drawing which shows the displacement field interpolated with the algorithm of the element surface meshing type arbitrary blending shape function of this invention. It is explanatory drawing which shows the displacement field interpolated with the algorithm of this invention with respect to nonhomogeneous nodal distribution. It is a flowchart which shows the finite element analysis process procedure concerning embodiment of this invention. It is a perspective view which shows the three-dimensional shape data of an elastic beam. It is explanatory drawing which shows the example which used the finite element method with respect to the three-dimensional shape data of the elastic beam of FIGS. It is explanatory drawing which shows the example which used the nodal linear constraint method (penalty constraint method, NLCM) with respect to the three-dimensional shape data of the elastic beam of FIGS. It is explanatory drawing which shows the example which used EFGM with respect to the three-dimensional shape data of the elastic beam of FIG. 9-2, ie, the example which coupled FEM and EFGM (FE-EFGM). It is a graph which shows the parameter dependence about the influence range of the precision at the time of coupling FEM and EFGM (FE-EFGM). It is a graph which shows the parameter dependence about the influence range of the precision at the time of coupling FEM and EFGM (FE-EFGM). It is a perspective view which shows the three-dimensional shape data of the board to which the pipe was joined. It is explanatory drawing which showed the example which applied the continuous mesh with respect to the three-dimensional shape data shown in FIG. It is explanatory drawing which shows the example which applied the discontinuous mesh with respect to the three-dimensional shape data shown in FIG. It is a perspective view which shows the example which deform | transformed the three-dimensional shape data which applied the continuous mesh shown in FIG. It is a perspective view which shows the example which deform | transformed the three-dimensional shape data which applied the continuous mesh shown in FIG. It is a perspective view which shows the example which deform | transformed the three-dimensional shape data to which the discontinuous mesh shown in FIG. 14 was applied by FEM + penalty restraint method. It is a perspective view which shows the example which deform | transformed the three-dimensional shape data to which the discontinuous mesh shown in FIG. 14 was applied by FEM + penalty restraint method. It is a perspective view which shows the example which deform | transformed the three-dimensional shape data to which the discontinuous mesh shown in FIG. 14 was applied with the method of this invention. It is a perspective view which shows the example which deform | transformed the three-dimensional shape data to which the discontinuous mesh shown in FIG. 14 was applied with the method of this invention. It is explanatory drawing which shows the element production | generation process by a mapping method.

Explanation of symbols

901 Three-dimensional shape data of elastic beam 1200 Three-dimensional shape data of plate joined with pipe

Claims (6)

An area shape input step for inputting predetermined area shape data to be numerically analyzed;
Division specifying the number of divisions for dividing the simple shape region into which the region shape data is decomposed by dividing the region shape data input in the region shape input step into simple shape regions that can be mapped by a mapping method Number designation process,
An element dividing step of dividing the simple shape region into elements by the mapping method based on the number of divisions specified by the division number specifying step;
A non-continuous mesh generating step for generating a non-continuous mesh of the entire predetermined region shape data based on each simple shape region divided by the mapping method by the element dividing step;
A finite element analysis execution step for executing a finite element analysis that allows the discontinuous mesh generated by the discontinuous mesh generation step;
An analysis result output step for outputting an analysis result analyzed by the finite element analysis execution step;
The finite element analysis method characterized by including.   2. The finite element analysis method according to claim 1, wherein the division number designation step designates the division number under a condition in which discontinuous nodes are generated. An area shape input step for inputting predetermined area shape data to be numerically analyzed;
A division number designation step for designating the number of divisions for dividing the simple shape area into which the area shape data has been decomposed by dividing the area shape data input in the area shape input step into mappable simple shape areas. When,
An element dividing step of dividing the simple shape region into elements by the mapping method based on the number of divisions specified by the division number specifying step;
A non-continuous mesh generating step for generating a non-continuous mesh of the entire predetermined region shape data based on each simple shape region divided by the mapping method by the element dividing step;
A finite element analysis execution step for performing a finite element analysis that allows the discontinuous mesh generated by the discontinuous mesh generation step;
An analysis result output step for outputting the analysis result analyzed by the finite element analysis execution step;
A finite element analysis program characterized by causing a computer to execute.   4. The finite element analysis program according to claim 3, wherein the division number designation step causes the division number to be designated under a condition in which discontinuous nodes are generated. Area shape input means for inputting predetermined area shape data to be numerically analyzed;
Division number designation means for designating the number of divisions for dividing the simple shape area into which the area shape data has been decomposed by dividing the area shape data input by the area shape input means into mappable simple shape areas. When,
Element division means for dividing the simple shape region into elements by the mapping method based on the division number designated by the division number designation step;
Non-continuous mesh generating means for generating a non-continuous mesh of the entire predetermined area shape data based on each simple shape area divided by the mapping method by the element dividing means;
A finite element analysis executing means for executing a finite element analysis that allows the discontinuous mesh generated by the discontinuous mesh generating means;
Analysis result output means for outputting an analysis result analyzed by the finite element analysis execution means;
A finite element analysis apparatus comprising: 6. The finite element analysis apparatus according to claim 5, wherein the division number designating unit designates the division number under a condition in which discontinuous nodes are generated.

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