JP2003223432A - Data construction method for meshless analysis - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To accurately and easily analyze through numerical analysis the problems of a complex area composed of materials with a plurality of material characteristics. <P>SOLUTION: Independently created individual mesh units are joined together on their respective mating surfaces while nodal points are unaligned. During the integration of the mesh units, not only the nodal points belonging to the mesh units but also other nodal points constituting the boundary shells of other mesh units included in an area where the boundary shells of the mesh units are offset are the nodal points for local approximate solution construction. <P>COPYRIGHT: (C)2003,JPO

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a data construction method for a meshless analysis method, and enables an accurate and easy analysis of a problem of a composite region composed of materials having a plurality of material characteristics in numerical analysis. The present invention relates to a data construction method for a meshless analysis method.

[0002]

[Problems to be solved by the prior art and invention] The current state of groundwater flow fields and changes in groundwater flow fields due to construction of underground facilities, such as groundwater pollution problems and safety evaluation of waste disposal sites and radioactive waste disposal sites. It is extremely important to predict In this case, the examination based on the geological survey and the numerical analysis represented by the finite element method (FEM) is often performed. At this time, a detailed field survey and
It is necessary to complement each other with elaborate three-dimensional analysis. However, in reality, the work of creating a mesh model (FE mesh) for finite element analysis corresponding to a three-dimensional hydrogeological model is a hindrance, and the analysis results cannot follow the updated geological survey results or In many cases, simplified analysis, such as two-dimensionalization, is performed to improve the efficiency of mesh creation, and it has been difficult to sufficiently perform complementary research and analysis.

Meshless analysis methods such as the element free Galerkin method (EFGM) and the free mesh method (FMM) have attracted attention as analysis methods for fundamentally solving this mesh formation problem, and in recent years, they have been rapidly used. The research and development movement has spread. The meshless method is an analysis method that does not require element information but only node information. However, the reality is that an analysis system that can handle even complex three-dimensional shapes composed of multiple materials (geospheres) such as geological structures has not been developed. Even in the meshless method, it is necessary to define the geometric shape of the analysis target. In addition, a rational algorithm for node collection for local approximation solution construction is a bottleneck.

For example, any analysis method requires definition of the geometric shape of the analysis target. Solid modeling is currently the most powerful and widely used three-dimensional geometric definition. In the case of solid modeling in the case of being composed of a plurality of materials (geospheres) such as geological structure, it is more efficient to divide and create at the boundary of the geological formation.
For example, when considering the geological structure model as shown in FIG. 10 (a), it is divided into 4 blocks having different strata as shown in FIG. 10 (b), and a solid model is created in each block. Easy to do. However, in the solid model as shown in FIG. 10B, since the boundary surface of the adjacent stratum blocks is not shared, when the FE mesh is generated by the mesh generation program, normally, the mesh is formed on the stratum boundary surface between the adjacent meshes.・ Inconvenience that pattern matching is not guaranteed.
In other words, for FEM analysis, it is necessary to modify the solid model so that the five planes shown by the thick lines in FIG. 10C are shared. This work is one of the major factors that make FE mesh creation difficult. Assuming an actual three-dimensional geological structure, each mesh model is processed by modifying the model of FIG. 10 (b) to the model of FIG. 10 (c) or by directly creating the model of FIG. 10 (c). There is a problem that it takes more than several times as much time as the model preparation of 10 (b).

Therefore, an object of the present invention is to solve the above-mentioned problems of the conventional technique and to provide a data construction method capable of easily constructing highly convenient three-dimensional analysis data for a meshless analysis method. To provide.

[0006]

In order to achieve the above object, the present invention uses an unmatched mesh aggregate in a meshless analysis method, and, when integrating each mesh unit, the mesh -Not only the nodes that belong to the unit but also the nodes that form the boundary shells of other mesh units included in the area offset from the boundary shell of the mesh unit should be the target nodes for local approximate solution construction. Characterize.

Further, it is characterized in that the mesh units are given a priority order to allow overlapping of meshes.

It is preferable to construct a local approximate solution by collecting the nodes in the influence area around the evaluation points from the list generated by the node collection.

With the above configuration, it is possible to realize a practical numerical analysis system that can handle even a composite region composed of a plurality of materials. The mesh units that make up the mesh aggregate used at this time allow non-conforming states of mesh patterns and overlapping of meshes at the joint surface of adjacent meshes, and thus, compared to conventional meshing in FEM etc. The construction of mesh data can be greatly simplified.

[0010]

BEST MODE FOR CARRYING OUT THE INVENTION An embodiment of a data construction method for a meshless analysis method of the present invention will be described below with reference to the accompanying drawings. The present invention relates to a data construction method for a meshless analysis method, that is, an inconsistent mesh aggregate (hereinafter, CMA: Consistency-free Mesh Assemb).
Write ly. ) And the data construction method as the node collection method in the meshless analysis method using the original mesh data.

The CMA in the following embodiment
The terms are defined as follows in the definition of the analysis region and the method of node collection when using for meshless analysis. Nodes: the vertices of the elements that make up the element (for higher-order elements
(Also on the element side) Element: Polyhedron that constitutes the mesh, eg tetrahedron, pentahedron, hexahedron Mesh unit: Mesh created independently for each layer CMA: Aggregate of mesh units

FIG. 1 shows an example of CMA. Each mesh unit (-) forming the CMA is in a non-conforming state at the adjoining joint surface. In these mesh units, mesh generation is performed independently for each of the four layers. Therefore, the mesh pattern of each mesh unit does not match at the boundary of the stratum. Furthermore, in addition to these mesh units, the underground facilities, etc. are made meshes, allowing overlap with the mesh units in the stratum. As a result, it is possible to know the influence of the added meshes in the analysis. It is relatively easy to create meshes independently for each block from a solid model by ignoring the conformity at the boundary surface of the strata using existing techniques such as commercial mesh generation software. In the present embodiment, the geological structure of a composite region composed of a plurality of materials will be described as a typical example of a compound male three-dimensional shape. However, the application of the present invention is applicable to an analysis targeting a geological structure. It goes without saying that it is not limited.

An embodiment of the content of the present invention will be described below by taking the above-mentioned EFGM as an example. With FEGM,
When integrating each mesh unit, the nodes in the influence area centering on the evaluation point (integration point) are collected, and a local approximate solution is constructed from these nodes to perform area integration.

[Data Construction Procedure for Target Mesh Unit] The data construction procedure for the mesh unit will be described below. Note that these procedures are specific examples, and the procedures are not limited. (Nodal point collection processing) First, for each mesh unit to be analyzed, nodal points for analysis are collected by the processing shown in (1) to (3) below. FIG. 2 is a schematic flowchart thereof. (1) Generate a boundary shell on the surface of each mesh unit. (2) Create a target node list for creating a local approximate solution for each mesh unit. The target node list includes the following node groups. -All nodes that belong to the relevant mesh unit-Nodes that form the boundary shell of other mesh units that are included in the area that is offset from the boundary shell of the relevant mesh unit (3) Collected nodes have priority over the relevant mesh unit It is determined whether or not it is inside a high-ranked mesh unit, and the internal nodes are removed from the node list (nodes on the boundary surface are not removed).

(Numerical integration process) The processes (4) and (5) are performed on each mesh unit. (4) Define numerical integration points in the mesh unit. At this time, each element forming the mesh unit may be a cell for integration. Further, a structured grid for integration or the like may be separately generated like a background cell. (5) Perform the following processing for each integration point. (5-1) Judge whether the integration point is inside the mesh unit, and if it is outside, skip the integration point. (5-2) Determine whether the integration point is inside a mesh unit with a higher priority than the relevant mesh, and if it is inside,
Skip the integration point. (5-3) Collect the nodes in the area of influence around the integration point from the node list and construct a local approximate solution. (5-4) Execute integration. At this time, the processes of (3) and (5-2) are not necessary for the mesh unit with the highest priority.

[Handling of Elements when Setting Influence Area of EFGM] For example, if the basis function of the locally approximated solution is the following perfect linear polynomial {lxyz}, there are 4 nodes not lying on the same plane in the influence area. Need more than points. An element including an evaluation point (integration point) can be used to consider as follows. Influence radius = a × d. (However, a: real number> 1) At this time, as shown in FIG. 3, if the element including the evaluation point is a tetrahedron, the distance from the evaluation point to the furthest node among the four nodes forming the tetrahedron. Be d. If the element is a pentahedron or a hexahedron, the distance from the evaluation point to the fifth closest node among the nodes forming the element is d. This is because, in the case of a pentahedron or a hexahedron, the four nodes closest to the evaluation point and the fourth node are vertices of a quadrangle and may be on the same plane.

[Collection of Nodes Around Evaluation Points for Constructing Local Approximate Solution] FIG. 4 shows each generated mesh unit (
6A to 6D are model diagrams showing a mesh unit showing an example of collecting nodes around evaluation points for constructing a local approximate solution. At this time, as shown in the figure, the nodes on the boundary surface between the adjacent mesh units must be shared by both mesh units. In addition, in the figure, when collecting the nodes in the area of influence represented by a circle centered on the evaluation point (marked with ● in the figure), it is necessary to also target the nodes on the boundary surface of the adjacent mesh units. There is a mark (○ in the figure). Therefore, in the present invention, a triangular or quadrangular mesh is generated on the surface of each mesh unit, and a polyhedron (hereinafter,
Described as a boundary shell. ) Represents the analysis area. On this occasion,
The face of the element that contacts the surface of the mesh unit can be utilized.

Then, as indicated by broken lines around the mesh unit shown in FIG. 4, the polyhedron is offset by an appropriate amount in the direction of the normal to each surface, and the polyhedron is inserted into the offset polyhedron. Nodes that make up the boundary shell of the mesh unit are also collected. Since the actual boundary surface is a complicated curved surface, it is difficult to judge numerically whether or not the nodes are on the same curved surface, but with this method, only the occurrence of the polyhedron, its offset, and the inside / outside judgment of the points for the polyhedron are possible. Should be done.

Next, the handling of data when mesh units overlap will be described. In FIG. 4, the mesh unit and the mesh unit overlap each other. These assume, for example, that an underground facility represented by a mesh unit was built within the formation of the mesh unit.

First, priorities are set among the mesh units. For example, a mesh unit as a structure is given a higher priority than a mesh unit as a formation. When the mesh unit is integrated here, the polyhedron generated on the surface of the mesh unit may be offset, and the nodes inside the polyhedron may be set as the collection target for constructing the local approximate solution. At this time, the constituent nodes of the mesh unit are also targeted (see FIG. 4). On the other hand, when integrating the mesh unit, the mesh
The unit needs to be removed from the integration area. This is to offset the polyhedron generated on the surface of the mesh unit and collect the nodes included in the mesh unit, and then the internal nodes of the mesh unit including the constituent nodes of the mesh unit (in Fig. 4, △ It can be dealt with by removing (mark). However, since the nodes on the boundary surface of the mesh unit must be included in the target node list of the mesh unit, the boundary shell constituent nodes of the mesh unit are left without being removed.

The example in which the present invention is applied to the meshless analysis method of EFGM has been described above, but it goes without saying that the present invention can also be applied to the meshless analysis method of FMM or the like.

[0022]

EXAMPLE Next, an example in which the above-mentioned geometrical shape definition as the data construction method in EFGM and the data construction method for the nodal point collection method are applied to the following permeation flow equations for permeation flow analysis will be shown. [Equation 1] φ represents velocity potential (total head), k represents hydraulic conductivity, and V represents analysis region. The boundary condition is given by the following equations (Equation-2 and Equation-3). [Equation 2] Here, the subscript ^ is a known quantity on the boundary, n is an outward normal line set on the boundary of the analysis region, and Sφ and S _n are boundaries of the analysis region. We introduce the variational principle to this problem. When the penalty method is applied to the basic boundary condition of Expression-2, the functional is expressed by the following Expression (Expression-4). [Equation 3] Here, α is a sufficiently large positive real number called a penalty number. In EFGM, the shape function is constructed by the moving least squares method. The unknown function φ is approximated by the following equation (Equation-5) at an arbitrary evaluation point within the region. [Equation 4] Here, {P (x) _m } is a basis function, and assuming a three-dimensional perfect first-order polynomial, [Equation 5] {A (x) _m } is an undetermined constant, which is a function of the spatial coordinate x. This is obtained by minimizing the following evaluation function. [Equation 6] Here, N is the number of nodes in the influence area around the evaluation point. {Φ _i } is the node value of φ at the node coordinate x = x _i , and r _i is the distance between the evaluation point and the node. As the weighting function, the following fourth-order spline (Equation-8) is introduced in the influence area of the radius d _m . [Equation 7] The following equation (Equation-9) is obtained from the stopping condition of Equation-7. [Equation 8] Therefore, [Equation 9] [Numerical equation 10] Therefore, the following formulas (Formula-16 and Formula-17) are obtained as the shape function. [Equation 11] Substituting Equation-16 into the functional of Equation-4 and stopping it yields the following linear algebraic equation. [Equation 12]

[Analysis Example] Next, an example of performing a permeation flow analysis in a three-dimensional ground permeation flow analysis model based on the derived permeation flow equation will be described. FIG. 5 shows a stratum model in which underground facilities are arranged in the stratum to be analyzed. The permeability coefficient k in each layer is set as shown in the figure, and the boundary condition of the model is φ = 2 in the plane of x = 0.
Φ = 0 on the plane of 0, x = -20, ∂φ / ∂n on other planes
= 0.

FIG. 6 shows an unmatched mesh aggregate (CMA) for EFGM analysis constructed based on the stratum model of FIG. The underground facility model is also shown outside the stratum model for understanding. As shown in the figure, the meshes do not match at the boundary surface of each layer, and the mesh of the underground facility overlaps the mesh in the stratum. Figure 7
Is a mesh model for FEM analysis generated for comparison. This mesh model for FEM analysis is constructed based on a complicated generation method so that the meshes of various layers and underground facilities are matched.

FIG. 8 shows the result of analysis performed based on the CMA shown in FIG. This analysis result represents the velocity potential distribution on the output cross section set so as to cut out a part of the stratum model shown in FIG. The thin solid line is the result of applying CMA (FIG. 6) to the EFGM analysis as the input data, and the light thick line is the result of the FEM analysis using the FEM mesh as the input data. The thick solid line is the stratum boundary appearing on the output cross section. EFGM results and FE
The result of M shows a good agreement, which confirms the validity of the present invention. FIG. 9 shows the analysis result when the position of the underground facility is moved within the model. With CMA, only the mesh of the underground facility is moved, but with the FEM analysis mesh, it is difficult to change the mesh. The contour line is doubled near the underground facility.
This is because it is plotted using the post processor for M. If necessary, you can easily avoid this by using commercially available visualization software.

[0026]

According to the present invention, the following various effects are exhibited. (1) Since non-conformity at the mesh joint surface and overlapping of meshes are allowed, the degree of freedom is remarkably higher than that of the conventional FEM analysis mesh, and mesh creation becomes easy. (2) Since meshes are allowed to overlap with each other, for example, when modeling an underground facility in a stratum, the mesh unit corresponding to the underground facility can be freely arranged without changing the mesh unit corresponding to the stratum. It becomes very easy to examine the position of. (3) Even in the case of a complex shape made of a single material, since the shape can be divided into simple shapes, it is very effective even when applied to a complex shape of a single material. (4) Peripheral software such as pre / post processor for FEM analysis can be used, and input data and analysis result processing are easy to use. (5) The element can be used as a cell for the numerical integration of EFGM. (6) Elements can be used when setting the influence area of EFGM.

[Brief description of drawings]

FIG. 1 is an analysis model diagram of a mesh unit according to the present invention.

FIG. 2 is a flowchart showing a schematic processing flow of a data construction method of the present invention.

FIG. 3 is a model diagram showing an example of a method of determining an influence radius in data construction.

FIG. 4 is a detailed analysis model diagram by the mesh unit of the present invention.

FIG. 5 is a schematic model diagram showing a stratum model as an example.

FIG. 6 is an analysis model diagram by CMA created based on the model of FIG.

7 is an FEM mesh model diagram as a comparative example with respect to the model of FIG.

8 is a result diagram showing an analysis result (permeation flow velocity potential diagram) of a target cross section in the analysis model shown in FIG.

9 is a result diagram showing an analysis result (permeation flow velocity potential diagram) in a target cross section in the analysis model in which a part of the mesh in FIG. 6 is moved.

FIG. 10 is a schematic analysis model diagram showing an example of a problem of conventional solid modeling.

[Explanation of symbols]   Stratum mesh unit (~ same)   Underground facility mesh

Claims (2)

[Claims] 1. Individual mesh units that are independently generated are joined in a state where the nodes are inconsistent at their joint surfaces, and the nodes that belong to the mesh unit when the mesh units are integrated. In addition to that, the nodes that constitute the boundary shells of other mesh units included in the area offset from the boundary shell of the relevant mesh unit are also set as the target nodes for constructing the local approximate solution. Data construction method for meshless analysis method. 2. The data construction method according to claim 1, wherein the mesh of the mesh unit is generated by giving priority to the mesh units to allow overlapping of the meshes.