JP5774404B2 - Analysis apparatus, method thereof and program thereof - Google Patents

Description

  The present invention relates to an analysis apparatus for analyzing a tire, a method thereof, and a program thereof.

  Conventionally, a deformation simulation method has been proposed that reproduces the deformation behavior of a rubber structure that occurs when the rubber structure is placed in a mold or the like and expanded toward the inner surface of the mold (see, for example, Patent Document 1). ).

  This deformation simulation uses a mesh-free method. That is, only a rubber member that undergoes a large deformation by the mold uses a mesh-free model, and nodes are arranged at regular intervals along the shape.

JP-A-2005-14301

  In the mesh-free method described in Patent Document 1, an approximation function is determined based on node information in the support. However, the nodes shown in Patent Document 1 are simple models arranged at regular intervals. On the other hand, the actual tire has a complicated shape, and the stability of the solution varies greatly depending on the shape and size of the support range, the integration method, and the characteristics of the approximate function.

  In view of the above problems, an object of the present invention is to provide an analysis apparatus, a method thereof, and a program for setting a support range so that a solution can be stably obtained.

The present invention relates to an analysis apparatus for analyzing a tire by a mesh-free method, and includes an input unit to which a three-dimensional tire model divided into subcell regions by a plurality of nodes is input, and three dimensions centering on each of the nodes. support range of oval set respectively, the weighting factor is determined according to the distance from the position information of the three-dimensional of the nodes included in the supported range, the nodes in each node and the center of the said support ranges And an analysis unit that obtains an approximate function indicating an approximate value of displacement within the support range using a kernel function.

  According to the present invention, the displacement of each node of the tire can be analyzed, and a solution can be obtained stably.

It is a block diagram of a model generation device of one embodiment of the present invention. It is a flowchart which shows the operation processing of a model production | generation apparatus. It is a figure which shows the cross-sectional data of a tire. It is sectional data of a carcass member. It is the figure which generated the node in the normalization area. It is a figure of the state which divided | segmented the normalization area | region into Delonie. It is a figure of the cross-sectional data which mapped the subcell area | region. It is explanatory drawing of the side view of the tire model which connected the prism and made it ring shape. It is a perspective view of the prism made from the cross-sectional data in which the subcell area was generated. It is a three-dimensional tire model made from cross-sectional data divided into subcell regions. It is the figure which set the elliptical support range to the tire model. It is the figure which set two types of support ranges. It is a flowchart which shows the operation | movement process of an analyzer.

  An analysis device 30 according to an embodiment of the present invention will be described with reference to the drawings.

  The analysis device 30 of the present embodiment is a device that uses the three-dimensional tire model 6 input from the model generation device 10 and analyzes the displacement at each node set in the tire model 6.

  The model generation device 10 is a device that generates a tire model used in a mesh-free method used for analysis or simulation from cross-sectional data representing a cross-sectional shape of a tire created by a CAD device.

  Before describing the model generation device 10 and the analysis device 30, the structure of the tire 100 will be described with reference to FIG.

  As shown in FIG. 3, the tire 100 includes a pair of left and right bead portions 102 and a side wall portion 104, and a toret portion 106 that straddles between the side wall portions 104 and 104.

  In the bead portion 102, an annular bead core 108 and a rubber bead filler 110 on the outer side in the radial direction are disposed. Between the pair of left and right bead cores 108, 108, a carcass member 112 is provided in which a large number of cords arranged at right angles to the circumferential direction of the tire 100 extend. A belt member 114 made of a non-extendable cord is provided on the outer side in the direction, and a tort rubber member 116 is provided on the outer side of the belt member 114 in the radial direction of the tire 100. Further, a groove 118 extending in the circumferential direction of the tire 100 is provided in the toret portion 106, and in this example, two mediates 118A near the center and two shoulder grooves 118B near the ends, a total of four. Is provided.

(1) Configuration of Model Generation Device 10 The configuration of the model generation device 10 of the present embodiment will be described based on the block diagram of FIG.

  As illustrated in FIG. 1, the model generation apparatus 10 includes an acquisition unit 12, a generation unit 14, a division unit 16, a mapping unit 18, a combination unit 22, a development unit 24, and an output unit 26.

  Hereinafter, functions of the respective units 12 to 26 will be described in order.

(2) Acquisition unit 12
First, the acquisition unit 12 will be described with reference to FIG.

  The acquisition unit 12 acquires cross-sectional data 1 related to the right half cross-sectional shape of the tire 100 in FIG. 3 from the CAD device, and acquires the cross-sectional data 1 for each material. FIG. 4 shows cross-sectional data 1 relating to the shape of the carcass member 112 of the tire 100, and the cross-sectional data 1 is constituted by a point sequence group representing the shape (contour) of the carcass member 112. Note that the cross-sectional data 1 needs to be a closed region. In addition to the cross-sectional data 1 of the carcass member 112, the acquisition unit 12 also receives cross-sectional data 1 such as the belt member 114 and the tort rubber member 116. In the following description, the cross-sectional data 1 representing the carcass member 112 will be described.

  The coordinates of each point representing the contour of the two-dimensional cross-sectional data 1 of the carcass member 112 are represented in the XY orthogonal coordinate system, which is the overall coordinate system, and input from the CAD device to the acquisition unit 12. As shown in FIG. 4, the acquisition unit 12 sets the position of the left end of the cross-section data 1 as a point P1, the right lower end as a point P2, the upper right end as a point P3, and the left upper end as a point P4.

(3) Generator 14
Next, the generator 14 will be described with reference to FIG.

  The generation unit 14 sets a convex normalization region 2 set in advance. The normalization area 2 must have a convex shape, and in this embodiment, the square normalization area 2 is set in advance. The convex normalization region 2 may be a rectangle or a triangle other than a square. In the square normalization region 2, a local coordinate system is set as shown in FIG. This local coordinate system is a u-w orthogonal coordinate system, in which the lower left corner of the normalized region 2 is the origin (0, 0), the lower right corner is (1, 0), and the upper right corner is (1, 1), the upper left corner is set to (0, 1).

  The generation unit 14 generates a plurality of nodes in the normalization region 2. The arrangement of the nodes may be random or regular. The generation density of the nodes is controlled by the state of the tire 100 to be analyzed. In the present embodiment, as shown in FIG. 5, the generation unit 14 generates a plurality of nodes in a vertical and horizontal grid pattern.

(4) Divider 16
Next, the dividing unit 16 will be described with reference to FIG.

  The dividing unit 16 generates a plurality of triangular divided regions 3 by performing Delaunay triangulation on the normalized region 2 where the node is generated by the generating unit 14. “Deroni division” is a method of forming a triangle between adjacent points based on a group of points on a two-dimensional plane and filling the plane with a set of triangular regions (divided regions 3).

  FIG. 6 is a diagram in which the normalized region 2 in which the node has been generated is divided by Delaunay.

  Next, the dividing unit 16 further subdivides the divided area 3 of the normalization area 2 and subdivides it into triangular subcell areas 4. A method of subdividing the triangular divided area 3 into the subcell areas 4 is as follows.

  First, the dividing unit 16 obtains the midpoint of each side of the triangular divided region 3.

  Second, the dividing unit 16 obtains the center of gravity of the triangular divided region 3.

  Third, the dividing unit 16 connects the three midpoints and the center of gravity.

  Fourth, the dividing unit 16 connects the vertex and the center of gravity of the triangular divided region 3.

  As a result, the inside of the triangular divided region 3 is subdivided into six triangular subcell regions. This triangular subcell region 4 is used for analysis of the mesh-free method. In Delaunay division, for example, 2960 nodes are provided. For this subcell region 4, 36600 division points are provided, and 58560 subcell regions 4 are provided.

(5) Mapping unit 18
Next, the mapping unit 18 will be described with reference to FIGS. 4, 6, and 7.

  The mapping unit 18 maps the normalized region 2 divided up to the subcell region 4 by the dividing unit 16 into the cross-sectional data 1 shown in FIG. 4, and generates the subcell region 4 in the cross-sectional data 1 as shown in FIG. . In this embodiment, the mapping unit 18 uses blending functions F0 and F1 as mapping functions used for this mapping.

  First, the relationship between the normalized region 2 in the local coordinate system and the cross-sectional data 1 in the global coordinate system will be described.

  As shown in FIG. 4, the cross-sectional data 1 in the global coordinate system has its contour represented by a point sequence group, and the coordinate position of each point is represented in the global coordinate system. As described above, the acquisition unit 12 sets the position of the lower left end as a point P1, the lower right end as a point P2, the upper right end as a point P3, and the upper left end as a point P4, as shown in FIG. .

  On the other hand, as shown in FIG. 6, the four corners of the normalized region 2 represented by the local coordinate system are the node Q1 = (0, 0), the node Q2 = (1, 0), and the node Q3 = ( 1, 1) and node Q4 = (0, 1). The normalized area 2 is Deloni divided using a plurality of nodes.

  When mapping unit 18 maps normalized region 2 to cross-section data 1, node Q1 is mapped to node P1, node Q2 is mapped to node P2, node Q3 is mapped to node P3, and node Q4 is mapped. Map to node P4. In the present embodiment, the origin in the global coordinate system is set to the node P1. The position at which the arbitrary node Q5 in the normalized region 2 is mapped in the cross-section data 1 is determined by the distance between the node Q5 and the origin (0, 0) in the local coordinate system. For example, if the node Q5 = (0.33, 0.4), the node P5 corresponding to the node Q5 in the cross-sectional data 1 is at the position of 0.33 of the distance P1-P2 in the X direction, and Y In the direction, it is at a position of 0.4 of the distance of P1-P4. Specifically, assuming that the coordinate position of the node P5 is (X5, Y5) in the global coordinate system, X5 / (P1-P2 distance) = 0.33, and Y5 / (P1-P4 distance) = 0. .4.

When coordinates (X, Y) in the global coordinate system are coordinates (u, w) in the local coordinate system, X = Ax (u, w) and Y = Ay (u, w). , Ax (u, w) is represented by the following formula (1), and Ay (u, w) is represented by the following formula (2). However, 0 <= u, w <= 1.

However, F1 (u), F0 (u), F1 (w), F0 (w) are blending functions, and are represented by the following formulas (3) to (6).

  Further, Ax (0,0) and Ay (0,0) in the expressions (1) and (2) indicate coordinate values of the node P1 in the entire coordinate system, and Ax (1,0) and Ay (1, 0) indicates the coordinate value of the node P2, Ax (1, 1) and Ay (1, 1) indicate the coordinate value of the node P3, and Ax (0, 1) and Ay (0, 1) are The coordinate value of the node P4 is shown.

  Furthermore, Ax (0, w) indicates the coordinate value of the global coordinate system when the node Q (0, w) in the local coordinate system is mapped to the cross-sectional data 1 in the global coordinate system. For example, w = 0.2. In this case, since the node P6 in FIG. 7 corresponds, the coordinate value (0, 0.2) of the node P6 is substituted. As described above, this calculation method is the same as the method for obtaining the coordinate value of the node P5 from the node Q5. Ax (1, w) is similarly determined for the coordinate value.

  As described above, the mapping unit 18 maps all the nodes in the local coordinate system to the cross-section data 1 in the overall coordinate system in FIG. 7, and also uses the mapped nodes as a reference for the triangle. Each vertex of the subcell area 4 (the midpoint of each side of the triangular divided area 3 and the center of gravity of the divided area 3) is also mapped, and the subcell area 4 is generated in the cross-section data 1.

  As described above, the mapping unit 18 can map the nodes and the subcell region 4 that are Deloni-divided in the normalized region 2 by the mapping function to the cross-sectional data 1 in the global coordinate system.

  The mapping unit 18 outputs the cross-sectional data 1 in which the subcell region 4 is generated to the combination unit 22.

(6) Combination unit 22
Next, the combination unit 22 will be described.

  The cross section data 1 described above is the cross section data 1 of the carcass member 112. Therefore, for example, the cross-sectional data 1 in which the plurality of subcell regions 4 are generated is generated in the same process for each member such as the belt member 114 and the tort rubber member 116. The combination unit 22 combines the cross-sectional data 1 generated for each member to generate the cross-sectional data 1 of the entire tire 100.

  First, the combining unit 22 combines the cross-sectional data 1 of each member in which the subcell region 4 output from the mapping unit 18 is generated, and the cross-sectional data 1 obtained by dividing the entire tire 100 illustrated in FIG. 9 into the subcell region 4. Is generated and output to the expansion unit 24.

(7) Deployment unit 24
Next, the expansion | deployment part 24 is demonstrated based on FIGS.

  The development unit 24 develops the entire two-dimensional cross-sectional data 1 of the tire 100 output from the combination unit 22 into a three-dimensional tire model 6 using the periodic symmetry of the tire 100.

  First, the expansion unit 24 expands the two-dimensional cross-section data 1 into a prism. That is, the development unit 24 stacks the two-dimensional cross-sectional data 1 by a predetermined angle (several radians) to generate a three-dimensional prism 5 as shown in FIGS.

  Second, since the entire cross-sectional data 1 of the tire 100 is the three-dimensional prism 5 having only the right half, the deploying unit 24 deploys to the symmetrical three-dimensional prism 5.

  Thirdly, as shown in FIGS. 8 and 10, the development unit 24 connects the three-dimensional left and right symmetrical prisms 5 in a ring shape with reference to the reference plane to generate a three-dimensional tire model 6. .

  The order of development is not limited to the order described above, and the cross-sectional data 1 may be developed symmetrically to generate the prism 5 or may be developed symmetrically after being developed in a ring shape.

(8) Output unit 26
Next, the output unit 26 will be described.

  The output unit 26 outputs the three-dimensional tire model 6 generated by the development unit 24 to the analysis device 30.

(9) Operation State of Model Generation Device 10 Next, the operation state of the model generation device 10 will be described based on the flowchart of FIG.

  In step S1, the acquisition unit 12 acquires cross-sectional data 1 for each member, and proceeds to step S2.

  In step S2, the generating unit 14 generates a plurality of nodes in a preset square normalization region 2, and the process proceeds to step S3.

  In step S <b> 3, the dividing unit 16 generates a triangular divided region 3 by Deloni dividing a plurality of nodes generated in the normalized region 2 using adjacent nodes. Next, the dividing unit 16 subdivides the divided region 3 to generate the subcell region 4. Then, the process proceeds to step S4.

  In step S4, the mapping unit 18 maps the normalized area 2 divided into the subcell areas 4 to the cross section data 1, generates the sub cell area 4 in the cross section data 1, and proceeds to step S5.

  In step S <b> 5, the combination unit 22 generates the overall cross-sectional data 1 of the tire 100 by combining the cross-sectional data 1 for each member divided into the subcell regions 4. Then, the process proceeds to step S6.

  In step S <b> 6, the development unit 24 develops the cross-sectional data 1 divided into the subcell regions 4 to generate a three-dimensional tire model 6. Then, the process proceeds to step S7.

  In step S <b> 7, the output unit 26 outputs the three-dimensional tire model 6 developed by the development unit 24 to the analysis device 30.

(10) Configuration of Analysis Device 30 The configuration of the analysis device 30 of this embodiment will be described based on the block diagram of FIG.

  As illustrated in FIG. 1, the analysis device 30 includes an input unit 32 and an analysis unit 34. Hereinafter, functions of the units 32 and 34 will be described in order.

(11) Input unit 32
First, the input unit 32 will be described.

  The input unit 32 receives the three-dimensional tire model 6 output from the output unit 26 of the model generation device 10. The three-dimensional tire model 6 may be directly input from the development unit 24.

(12) Analysis unit 34
Next, the analysis unit 34 will be described.

  In the three-dimensional tire model 6 input to the input unit 32, a plurality of nodes are set in the three-dimensional tire model 6 as described above, and a triangular subcell region 4 is generated. The nodes and the subcell area 4 are expressed in a three-dimensional global coordinate system, and the position information of each node is displayed as coordinate values in the three-dimensional global coordinate system.

  The analysis unit 34 sets a three-dimensional elliptical support range 7 around each node, and as shown in FIG. 12, the three-dimensional position information (coordinate values) of the nodes included in the support range 7 is set. Then, an approximate function indicating the distribution of displacement in the support range 7 is obtained using each node in the support range 7 and a kernel function whose weighting coefficient is determined according to the distance from the node at the center.

  Hereinafter, a method in which the analysis unit 34 obtains an approximate value of the displacement of each node and a method in which a distortion or the like is obtained using the method will be described in order.

(13) Analysis of Displacement Hereinafter, a method in which the analysis unit 34 obtains an approximate value of displacement at each node will be described using an RKPM approximate expression.

Non-Patent Document 1 (Chen, JS, Pan, C., Wu, CT, Liu, WK, “Reproducing kernel particle methods for large deformation analysis of non-linear structures,” Computer Methods in Applied Mechanics and Engineering, Vol. 139, 1996, pp. 195-227.), An approximate function indicating an approximate value of the displacement of X in the three-dimensional coordinate system is expressed by the following equation (7).

Where u ^a (x) is an approximate value of displacement at coordinate X, u _I is a generalized displacement at the I-th node, NP is the total number of nodes, and ψ _I (x) is a shape function. The shape function Ψ _I (x) is expressed as the following formula (8).

However,

And expressed by polynomials up to the Nth order. It is also a basis function

Represents a moment matrix.

  Is a kernel function, and a is a parameter representing the spread. Φ is a spline function, which will be described later.

  a is the support range 7 and represents the size of the range in which the kernel function takes a positive value. The above is a one-dimensional case, but the three-dimensional case is expanded based on this. In the present embodiment, an ellipse is used as the expansion method. That is, as shown in FIG. 11, the support range 7 is a three-dimensional ellipse centered on the node.

  FIG. 11A is an overall view of the tire model 6 as viewed from the side, and is a view in which a three-dimensional elliptical support range 7 is set at one node in the tire model 6; An elliptical support range 7 having radii R1, R2, and R3 is set. FIG. 11B is a diagram showing an elliptical support range 7 in the cross-sectional shape of the tire model 6, and the support range 7 has radii R1 and R2 when viewed from this direction. . . FIG. 11C shows an enlarged view of the tire model 6 viewed from the side, and the support range 7 has radii R2 and R3 when viewed from this direction.

  The reason for setting such an elliptical support range 7 is that a shape function can be created at adjacent nodes regardless of the density of the nodes.

When the support range 7 is set in an elliptical shape, Φ in the above can be expressed as in Expression (12) and Expression (13).

However, t is expressed as in Expression (14).

Further, M _I in the formula (14) is expressed as in the formula (15).

Incidentally, N _I denotes the set of adjacent nodes of the I-th node, beta _I represents a scaling factor.

  Since the RKPM approximate expression is not an interpolation form, special measures are required to impose the displacement boundary condition. For example, a Lagrangian multiplier method, a penalty method (fine method), and a variable conversion method have been proposed in the past. However, there are problems such as an increase in the number of degrees of freedom of equations and a process of inverse matrix conversion.

Instead, in the present embodiment, a reproducing kernel complementing format (complementing formula using a reconfigurable kernel function) is used, and is expressed by the following formula (16).

  The first term on the right side of Equation (16) is a primitive function (primitive function) for expressing the Kronecker delta characteristic, and the second term is for satisfying the re-production condition (reconfigurable condition) necessary for convergence. Enrich functions (fertilization functions), both of which give different support range 7 sizes. The support range 7 for the first term is set to a small support range 7 that does not include any other nodes. Here, the size of the support range 7 is represented by a in Expression (16). The two support ranges 7 in the equation (16) are shown in FIG. 12, and the inner (solid line) ellipse shown in FIG. 12 indicates the support range 7 for the first term, and the outer side shown in FIG. The oval shape of the dotted line indicates the support range 7 of the second term. Regarding this equation (16), Non-Patent Document 2 (Chen, JS, Han, W., You, Y., Meng, X., “A reproducing kernel method with nodal interpolation property,” International Journal for Numerical Methods in Engineering , Vol. 56, 2003, pp. 935-960.).

  As described above, the analysis unit 34 obtains an approximate value of displacement at each node from the shape function Ψ.

(14) Analysis of distortion etc. The analysis part 34 calculates | requires distortion etc. based on the approximate value of the displacement in each node calculated | required as mentioned above. An example in which the analysis unit 34 obtains distortion will be briefly described.

  First, the analysis unit 34 is described in Non-Patent Document 3 (Chen, JS, Yoon, S., Wu, CT, “Non-linear version of stabilized conforming nodal integration for Galerkin mesh-free methods,” International Journal for Numerical Methods. in Engineering, Vol. 53, 2002, pp. 2587-2615.), the deformation gradient averaged for each node is obtained using the SCNI integration method (The stabilized conforming nodal integration method).

  Secondly, the analysis unit 34 derives an incremental stiffness equation specialized in SCNI type from the deformation gradient averaged for each node. Finally, as shown in Non-Patent Document 3, the analysis unit 34 can derive an incremental stiffness equation that reflects the averaged deformation gradient and nodal integration. This incremental stiffness equation is an equation representing the strain to be obtained in the tire model 6.

(15) Operation State of Analysis Device 30 Next, the operation state of the analysis device 30 will be described based on the flowchart of FIG.

  In step S <b> 11, the input unit 32 acquires the three-dimensional tire model 6.

  In step S12, the analysis unit 34 sets a three-dimensional elliptical support range 7 for each node in the three-dimensional tire model 6, and obtains an approximate value of displacement at each node from the RKPM approximation formula.

  In step S <b> 13, the analysis unit 34 obtains a distortion or the like of the tire model 6 based on the approximate value of displacement at each node.

(16) Effect According to the present embodiment, the support range 7 having a three-dimensional ellipse is set for each node of the three-dimensional tire model 6, and the approximate value of the displacement at each node is obtained. A displacement solution can be obtained stably regardless of the arrangement of the nodes. In addition, the analysis speed can be increased.

(17) Modification 1
In the above-described embodiment, the analysis unit 30 acquires the tire model 6 from the model generation device 10. However, the tire model 6 generated by another model generation device is not limited thereto.

(18) Modification 2
In the above-described embodiment, the analysis unit 30 analyzes the distortion and the like using the SCNI integration method or the like. However, the analysis unit 30 is not limited thereto, and may be analyzed by another method.

(19) Modification 3
The model generation device 10 and the analysis device 30 of the above embodiment can be realized by using, for example, a general-purpose computer having a mouse and a keyboard as basic hardware. That is, the acquisition unit 12, the generation unit 14, the division unit 16, the mapping unit 18, the combination unit 22, the development unit 24, the output unit 26, the input unit 32, and the analysis unit 34 execute a program on a processor mounted on the computer. It can be realized by executing. At this time, the model generation device 10 and the analysis device 30 may be realized by installing the above-described program in a computer in advance, or may be stored in a storage medium such as an E-ROM, and this program may be appropriately installed in the computer. It may be realized by doing.

(20) Others While an embodiment of the present invention has been described above, this embodiment is presented as an example and is not intended to limit the scope of the invention. These novel embodiments can be implemented in various other forms, and various omissions, replacements, and changes can be made without departing from the spirit of the invention. These embodiments and modifications thereof are included in the scope and gist of the invention, and are included in the invention described in the claims and the equivalents thereof.

DESCRIPTION OF SYMBOLS 1 ... Cross-section data, 2 ... Normalization area | region, 3 ... Division | segmentation area | region, 4 ... Subcell area | region, 5 ... Square pillar, 6 ... Tire model 6, 10 ... Model production | generation apparatus , 12 ... acquisition unit, 14 ... generation unit, 16 ... division unit, 18 ... mapping unit, 22 ... combination unit, 24 ... development unit, 26 ... output unit, 30: Analysis device. 32 ... Input unit, 34 ... Analyzing unit, 100 ... Tire

Claims (4)

In the analysis device that analyzes tires by mesh-free method,
An input unit for inputting a three-dimensional tire model divided into subcell regions by a plurality of nodes;
A three-dimensional elliptical support range is set around each node, and the three-dimensional position information of the nodes included in the support range, each node in the support range, and the center An analysis unit for obtaining an approximate function indicating an approximate value of a displacement within the support range using a kernel function in which a weighting factor is determined according to a distance from a node ;
The analysis apparatus characterized by having. The analysis unit obtains the approximation function based on an RKPM approximation formula;
The analysis apparatus according to claim 1, wherein: In the analysis method performed by the analysis device that analyzes the tire by mesh free method,
An input step for inputting a three-dimensional tire model divided into sub-cell regions by a plurality of nodes;
A three-dimensional elliptical support range is set around each node, and the three-dimensional position information of the nodes included in the support range, each node in the support range, and the center An analysis step for obtaining an approximate function indicating an approximate value of a displacement within the support range using a kernel function in which a weighting factor is determined according to a distance from a node ;
The analysis method characterized by having. In an analysis program that analyzes tires using the mesh-free method,
On the computer,
An input function for inputting a three-dimensional tire model divided into subcell regions by a plurality of nodes;
A three-dimensional elliptical support range is set around each node, and the three-dimensional position information of the nodes included in the support range, each node in the support range, and the center An analysis function for obtaining an approximate function indicating an approximate value of displacement within the support range using a kernel function in which a weighting coefficient is determined according to a distance from the contact point;
Analysis program to execute.