JP4631319B2 - Simulation model creation method for heterogeneous materials - Google Patents

Description

  The present invention relates to a method for creating a simulation model of a heterogeneous material in which a plurality of material phases having different material properties are dispersedly arranged.

  Bulk properties of filler-containing rubber and reinforcing members used for tire tread rubber members, bead filler rubber members and the like affect tire performance. For this reason, simulation calculation of tire performance and simulation calculation of rubber material characteristics are important factors in tire development. Bulk property is a macro property in the state of a mass (bulk) of heterogeneous material in which material phases with different material properties are dispersed, and equivalent material property when the heterogeneous material is regarded as a homogeneous material. is there.

In order to evaluate the bulk properties of heterogeneous materials in which a plurality of material phases with different material properties such as the above-mentioned filler-blended rubber are dispersed by simulation, for example, a representative model of the heterogeneous material is cut into a rectangular parallelepiped shape and a simulation model (Patent Documents 1, 2, and 3).
Then, as if this simulation model is infinitely continuous in the vertical and horizontal directions and in the depth direction and forms a bulk shape, the periodic calculation is performed by assigning periodic boundary conditions to the opposing faces of the rectangular parallelepiped in the simulation model. Do.

On the other hand, in the above simulation model, the same rectangular parallelepiped unit cells (hereinafter also referred to as voxels) are used because mesh division for forming unit cells such as finite elements is easily performed (hereinafter also referred to as voxels). Non-patent document 1). Here, the voxel is characterized by a plurality of discrete points, and in the finite element model, it means a finite element.
That is, a simulation model composed of voxels is created, and after that, a model that can be simulated by associating one material phase for each voxel unit, that is, an overall matrix that can be simulated (the matrix components are digitized) A matrix in which element matrices representing each voxel are added). After this, a reduced matrix is created in the pre-processing by modifying the matrix that can be simulated using the periodic boundary conditions given to reproduce the bulk characteristics, and simulation computation (this calculation) is performed using this reduced matrix. I do.

  In the preprocessing, for example, in a rectangular parallelepiped-shaped representative region, a discrete expression that is constrained using a relational expression that constrains a discrete point of a voxel on one opposing end face to the behavior of the discrete point of the voxel on the other end face. A reduced matrix is created by eliminating the degree of freedom of the points from the entire matrix that can be simulated.

However, in this pre-processing, there are discrete points to be constrained for each voxel located at the boundary of the representative region. For example, when 256 simulations are arranged in the vertical, horizontal, and depth directions, The number of discrete points to be increased is extremely large. For this reason, the processing time required for the pre-processing is extremely long, and may be longer than the time required for the simulation calculation.
In this way, since one simulation calculation including pre-processing takes time, when performing the simulation calculation repeatedly by changing the dispersion arrangement of the material phase or changing the occupation ratio of the material phase, the calculation takes an extremely long time. In short, it is not practically preferable for early development of heterogeneous materials.

Japanese Patent Laid-Open No. 9-180002 Japanese Patent Laid-Open No. 9-288689 JP-A-10-11613 Takano, Zako, Transactions of the Japan Society of Mechanical Engineers (A), 61, 583, 1995, pp.905

  Therefore, the present invention provides a heterogeneous material simulation model that can reduce the time for one simulation operation including preprocessing without generating time when generating a matrix representing an executable simulation model. The purpose is to provide a creation method.

In order to achieve the above object, the present invention is a method for creating a simulation model for a heterogeneous material using a computer to create a simulation model for a heterogeneous material in which a plurality of material phases having different material properties are dispersedly arranged. the computer, discretization of a representative area of the heterogeneous material by a unit cell of the same shape characterized the shape of a plurality of discrete points, continuous plurality arranged along the two directions even without least A step of creating a model, wherein the computer has a discrete point corresponding to a behavior of a discrete point of a unit cell located at one end of each arrangement direction of the unit cell in the discretization model. The behavior of the discrete point located at the one end is changed to the behavior of the corresponding discrete point located at the other end so that it is smoothly connected to the behavior of the point. A step of determining a beam to periodic boundary conditions, the computer, the discrete points from the entire matrix representing the discretized model for simulation operation is restrained by applying the periodic boundary conditions in the discrete model the processing routine reduction matrix erased freedom, the steps that you create a processing routine to generate a simulation operation can become a matrix by inputting a material constant of the unit cell as a parameter in advance, which was created Is stored in a memory, and a material constant is input to the computer from an input operation system according to an instruction from an operator and determined for each unit cell of the discretization model based on a dispersion arrangement of material phases in the representative region. a step that is, the computer, the processing is stored and held in the memory Call the routine, by executing the processing routine gives the value of the material constant defined in the processing routine, the corresponding matrix elements in giving numerical simulation operation possible reduction matrix of the reduction matrix possess and generating a said simulation operation possible the reduction matrix by the computer, without simulation operation possible whole matrix is created, the said reduction matrix generated in the process routine The material constant of the unit cell is input in advance as a parameter to be generated in advance prior to the simulation calculation, and then used for the simulation calculation performed with a predetermined physical quantity from the outside. In the step to generate the reduced matrix In this case, by inputting the material constant of each unit cell as the value of the parameter, a numerical value assigned to each matrix component of the reduced matrix is automatically calculated, and this numerical value is stored in each matrix of the memory. providing a simulation model creation method of heterogeneous material, characterized in Rukoto stored and held assigned to the address corresponding to the component.

The representative region of the heterogeneous material and the shape of the unit cell are, for example, a rectangular shape or a rectangular parallelepiped shape.
In the step of generating the reduced matrix that can be simulated, for example, when the shape of the representative region does not match the contour shape of the discretized model, the computer calculates a shape dimension in at least one direction of the discretized model. The processing is performed using a material constant determined for each unit cell of the discretization model by matching the contour shape of the discretization model with the shape of the representative region by scaling the discretization model by a constant multiplication. Run the routine.
In the step of generating a reduced matrix that can be simulated , the computer assigns one material phase to each unit cell of the discretized model, and processes the material constant value of the assigned material phase. It is preferable to give it to a routine for execution. In the step of generating a reduced matrix that can be simulated, the computer determines a material constant for each unit cell of the discretization model, and a unit of a plurality of material phases occupies a corresponding region of the unit cell. Preferably, the processing routine is executed by giving the value of the material constant of the material phase having the largest volume or area occupation ratio of the material phase in the corresponding region of the cell to the processing routine.

Further, in the step of generating a reduced matrix that can be simulated , the computer determines a material constant for each unit cell of the discretization model, and when a plurality of material phases occupy a corresponding region of the unit cell, It is also preferable that a weighted average value of material constants is obtained using the volume occupancy rate or area occupancy rate of the material phase in the corresponding region of the unit cell as a weighting factor, and this weighted average value is given to the processing routine for execution.

The discretization model is, for example , a finite element model in which the unit cell is a finite element.
Alternatively, the discrete model may it der having a discretized partial model by mesh-free method. In this case, the partial model preferably includes at least a model of an arrangement portion of the material phase having the lowest rigidity among the plurality of material phases.

  The representative region has a polygon or polyhedron shape having a corner portion in which a plurality of contour lines or contour surfaces are connected at an angle, and the discretized model is located at a position corresponding to the corner portion. The unit cell has discrete points, and the behavior of the discrete points is constrained when generating the reduced matrix.

The reduction matrix is, for example, a stiffness matrix in which each matrix component is digitized using the elastic modulus of the material phase as the parameter, and the physical quantity is a force, stress, or forced displacement applied to the heterogeneous material, The computer performs strain / stress analysis of the heterogeneous material by the simulation calculation. At that time, the maximum elastic modulus among the elastic moduli in the plurality of material phases is 100 times or more the minimum elastic modulus.
In the heterogeneous material, it is preferable that granular reinforcing materials such as carpon or silica are dispersedly arranged as a material phase. In the heterogeneous material, for example , a plurality of elastomers having different material characteristics are dispersedly arranged as a material phase.
A simulation model creation method for a heterogeneous material according to the present invention, and the computer uses the processing routine created by the simulation model creation method and the reduced matrix capable of simulation calculation, to the reduced matrix capable of simulation calculation. It is preferable to include the step of applying the physical quantity and performing the simulation calculation.
When the computer performs the simulation calculation by changing the volume occupancy or area occupancy of each material phase in the dispersed arrangement of the plurality of material phases or the representative region of the heterogeneous material by the computer, the computer Prior to the simulation calculation, the reduced matrix capable of simulation calculation is generated in advance using the processing routine without creating the entire matrix that can be simulated, and then the simulation calculation is performed by adding the physical quantity. Is preferably performed.

  In the present invention, a reduced matrix that eliminates the degrees of freedom of discrete points that are constrained by assigning periodic boundary conditions to the discretization model is generated as an operable matrix from the entire matrix that represents the discretization model of the heterogeneous material. A processing routine is prepared in advance. In this processing routine, a reduced matrix that can be calculated by inputting a material constant of a unit cell as a parameter is used. For this reason, a reduced matrix capable of calculating a simulation model of a representative region of a heterogeneous material can be created in a short time using the processing routine.

  Hereinafter, the method for creating a simulation model for a heterogeneous material according to the present invention will be described in detail based on a preferred embodiment shown in the accompanying drawings.

FIG. 1 is a block diagram functionally showing the configuration of a processing apparatus 10 that implements the method for creating a simulation model for a heterogeneous material of the present invention.
The processing device 10 includes an input operation system 12, a computer 14, and a display 16.
The input operation system 12 is a mouse or a keyboard, and is a device that inputs various types of information according to instructions from the operator.

The computer 14 includes a CPU 18 and a memory 20, and also includes a ROM (not shown). The computer 14 functionally forms a processing routine creation unit 22, a reduced matrix creation unit 24, and a simulation calculation unit 26 by executing computer software stored in a ROM or the like, and simulates the heterogeneous material of the present invention. This is the part that implements the model creation method.
The display 16 is a part that displays an input screen so that an operator can instruct using the input operation system 12 and displays a finite element model and a simulation calculation result to be described later.

The processing routine creation unit 22 is a part that creates a processing routine for creating a reduced matrix that can be simulated for analyzing bulk characteristics. The processing routine inputs a reduced matrix that eliminates the degrees of freedom of the nodes constrained by the periodic boundary conditions from the entire matrix representing the finite element model for simulation calculation, using the material constants of each finite element as parameters. This is a subprogram that is generated as a matrix that can be operated.
The processing routine creation unit 22 creates a finite element model that reproduces a representative region of a heterogeneous material by using a finite element (voxel), determines a periodic boundary condition to be given to the created finite element model, and further determines a node according to the periodic boundary condition. The above processing routine for generating a reduced matrix in which the degree of freedom is reduced is created. The created processing routine is stored and held in the memory 20.

  The reduction matrix creation unit 24 determines a material constant according to the material phase assigned to each finite element, calls a processing routine stored in the memory 20, and inputs a value of the determined material constant to the processing routine. In this way, numerical values are assigned to the matrix components of the reduced matrix to create an operable reduced matrix. In the reduced matrix, the value of each matrix component is address-managed and allocated and stored in the memory 20, and the numerical value is distributed to the memory of the corresponding matrix component by determining the material constant for each finite element. As a result, the reduced matrix becomes a matrix that can be simulated.

  The simulation calculation unit 26 performs a simulation calculation using the generated reduction matrix that can be calculated. For example, the reduced matrix is a stiffness matrix, and an external force or a forced displacement is applied to a predetermined position and a stress / strain analysis is performed to reproduce a deformation state or a maximum strain distribution is calculated. Alternatively, the reduced matrix is a heat conduction matrix, and a temperature distribution and a linear expansion coefficient are calculated by applying a temperature to a predetermined position and analyzing the heat conduction. Alternatively, by giving an object force distribution corresponding to the gradient of a predetermined physical quantity (displacement, temperature, concentration, etc.) as an external force, and further calculating a self-balancing equation considering the predetermined stress distribution, elastic characteristics, heat conduction characteristics, Thermoelastic characteristics, viscoelastic characteristics, and the like may be calculated.

A method for creating a simulation model of a heterogeneous material performed in such a processing apparatus 10 will be specifically described.
FIG. 2 is a flowchart showing an exemplary flow of a method for creating a simulation model of a heterogeneous material. FIG. 3 is a cross-sectional view of an example of a representative region of a heterogeneous material. This heterogeneous material includes a first polymer phase A (white region in FIG. 3), a second polymer phase B (black region in FIG. 3), and a granular filler phase C (circular gray in FIG. 3). Region) and filler-polymer boundary phase D (gray region in FIG. 3) surrounding the particle filler phase are heterogeneously distributed. That is, a plurality of elastomers having different material characteristics are dispersedly arranged as a material phase, and further, granular reinforcing materials are dispersedly arranged as a material phase. This inhomogeneous material has, for example, the maximum elastic modulus among the elastic moduli in a plurality of material phases is 100 times or more the minimum elastic modulus. The material phase constituting the heterogeneous material is not only a solid phase such as an elastomer or a filler, but may also be a void phase filled with gas or liquid.
A simulation model is created for such a heterogeneous material.

First, heterogeneous by arranging a plurality of voxels (unit cells) of the same rectangular parallelepiped (hexahedral) shape characterized by a plurality of discrete points adjacent to each other in three orthogonal directions. A finite element model (discretization model) of the representative region of the material is created (step S10).
The representative region of the heterogeneous material is, for example, a rectangular parallelepiped shape, and is divided into meshes in three directions orthogonal to each other, thereby continuously arranging cuboid-shaped finite elements, for example, 256 × 256 × 256 finite elements. Create a finite element model.
FIG. 4 shows an example of a finite element model composed of rectangular parallelepiped finite elements.

Next, the behavior of the node (discrete point) of the finite element located at one end of each arrangement direction of the finite element is smoothly connected to the behavior of the node (discrete point) located at the other end. Periodic boundary conditions are determined (step S20).
Next, by giving the defined periodic boundary condition to the finite element model, the node located at one end of the finite element model is restrained by the behavior of the node located at the other end. A processing routine for generating a reduced matrix in which the degrees of freedom of the nodes are eliminated from the entire matrix representing the finite element model is created (step S30).

Specifically, a reduced matrix that can be calculated is created by storing and holding the value of each matrix component at a predetermined address in the memory 20, and the processing routine stores each matrix component stored and held for each address. This is a part for calculating the value and designating the address of the memory 20 to be stored.
As described above, the reduced matrix is obtained by eliminating the degrees of freedom of nodes that are constrained by applying a periodic boundary condition to the whole matrix representing the finite element model. The reduced matrix is obtained by correcting the matrix components from the entire matrix formed by adding the matrix components of each finite element of the finite element model.

In this processing routine, the value given to the matrix component of the reduced matrix is set so that the material constant of each finite element can be input as a parameter, and the parameter value (material constant value) is input to reduce the value. A value assigned to each matrix component of the quantization matrix is automatically calculated, and this value is assigned to an address corresponding to each matrix component in the memory 20 and stored.
FIGS. 5A and 5B are examples showing a finite element model and matrix components of each finite element. Here, for the sake of simple explanation, a finite element model including finite elements A to D and nodes 1 to 9 will be described, and the matrix is assumed to be a stiffness matrix. Further, the degree of freedom of each node is assumed to be a one-degree-of-freedom system.

Each of the finite elements A to D in FIG. 5A is a stiffness matrix as shown in FIG.
For example, the 1 × 2 matrix component k1 ₁₂ of the finite element A represents the rigidity between the node 1 and the node 2 in the finite element A, and these matrix components are uniquely defined by defining the shape of the finite element. Determined. For example, by defining the material phase of each finite element, material constants such as Young's modulus, shear stiffness, and Poisson's ratio are determined, and the value of the matrix component of the stiffness matrix of each finite element is determined.
Such a stiffness matrix is added to the stiffness matrix in accordance with the finite element arrangement shown in FIG. 5A to become the following overall stiffness matrix M shown in the following [Equation 1].
Note that “Symm.” In FIG. 5B means that it has a symmetric matrix component centered on the diagonal component of the stiffness matrix. For example, the matrix component k2 ₂₂ having 2 rows and 3 columns is defined as the same as the matrix component having 3 rows and 2 columns.

Further, a non-zero matrix component of the entire stiffness matrix M is represented by K _ij (i, j = 1 to 8 natural numbers) and expressed as [Equation 2] below.

Then, a matrix equation is generated as shown in the following equation (1), with u _i (i = 1 to 9) as the degree of freedom of each node and F _i (i = 1 to 9) as an external force vector.

On the other hand, the periodic boundary condition in the finite element model shown in FIG. 5A is expressed as [Equation 4] below. The nodes 2 and 8 and the nodes 4 and 6 are corresponding nodes that are located at the ends of the finite element model and to which a periodic boundary condition should be given. Therefore, the degrees of freedom of the nodes 2 and 8 and the nodes 4 and 6 are constrained to the same value.
Further, since the nodes 1, 3, 7, and 9 are located at the corners of the boundary surface in the two directions of the finite element model, the nodes 1 and 3, the nodes 7 and 9, the nodes 1 and 7, and the nodes 3 and 9 Are constrained to the same value. The periodic boundary condition in which the behavior of the nodes located at the corners is smoothly connected means that the value of each node is constrained to 0.

Of these periodic boundary conditions, the periodic boundary condition (u ₁ = u ₃ = u ₇ = u ₉ = 0) of the nodes _{1, 3, 7, and 9} is given to the expression (1), so that the following expression (2) It becomes. Further, by assigning periodic boundary conditions of the nodes 2 and 8 and the nodes 4 and 6 to the equation (2) and eliminating the constraining nodes 8 and 6, the following equation (3) is obtained. The
When this expression (3) is expressed by each matrix component of the finite elements A to D, the following expression (4) is obtained.

The stiffness matrix on the left side of Equation (4) is a reduced stiffness matrix that is reduced by applying a periodic boundary condition to the overall stiffness matrix M. Since the matrix component of the reduced stiffness matrix is an addition sum of the matrix components of the predetermined stiffness matrix in each finite element A to D, the value of the matrix component of the stiffness matrix in each finite element A, that is, When the material constants of the finite elements A to D are determined and the values of the matrix components are known, the matrix components of the reduced matrix can be obtained.
The address of the memory 20 where the values of the matrix components of the reduced matrix represented by the above equation (4) are to be stored is set for each matrix component of the reduced stiffness matrix, and the processing routine determines the material constant of each finite element. Is input as a parameter, it is a subprogram that calculates the value of the matrix component of the stiffness matrix in each finite element, assigns the calculated value to the corresponding address in the matrix component of the reduced matrix, and stores it. is there. In this way, the value of the matrix component of each finite element is assigned to a predetermined address and added to the stored value and stored. In this way, the processing routine calculates the value of each matrix component of the reduced matrix shown on the left side in Equation (4), and creates a reduced matrix that can be calculated.

  For example, if the matrix components of the reduced matrix are a1 to a6 as shown in [Equation 8] below, addresses to be stored and held in the memory 20 are set as a1 to a6, and this address and the stiffness matrix of each finite element are set. The correspondence relationship with the matrix component is defined as [Equation 9] below.

By executing the processing routine, the value of each matrix component of the stiffness matrix of each finite element is assigned to the memory 20 in accordance with the correspondence relationship.
Processing routines are stored in memory. That is, once a processing routine is created in steps S10 to S30, this processing routine is called and used in subsequent steps S40 to S60.

Next, a material constant is set for each finite element (voxel) (step S40).
Specifically, a representative region of the heterogeneous material is cut out, and a material constant is determined for each finite element of the finite element model based on the dispersed arrangement of the material phase in this representative region.
FIG. 6 is a diagram showing an example of a finite element model created for the representative region shown in FIG.
As shown in the enlarged view on the right side in FIG. 6, each finite element unit is set to occupy one material phase.
When determining the material constant for each finite element of the finite element model, a plurality of material phases (gray phase and white phase in FIG. 7A) are provided in the corresponding region of the finite element as shown in FIG. Occupies the material constant value of the material phase having the largest volume or area occupancy of the material phase in the corresponding region of the finite element. In the region R in FIG. 7A, the gray phase occupancy is larger than the white phase occupancy, so the gray phase material constants are assigned to the finite elements in this region R, A constant is defined. When a plurality of material phases occupy one finite element region as shown in FIG. 7A, one material phase is determined in units of finite elements as shown in FIG. 7B.
The setting of the material constant is not limited to the above method. For example, the weighted average value of the material constant of each material phase is obtained using the volume occupancy or area occupancy of the material phase in the corresponding region of the finite element as a weighting factor, and the load average value is determined as the material constant in the finite element. Also good.

Next, by inputting the set material constant value as a parameter of the processing routine created in step 30 for each finite element, the processing routine, as described above, sets the matrix component of the matrix in each finite element. A value is calculated, and the calculated value is assigned to the address of the corresponding memory 20 in the reduced matrix component (step S50).
The reduced matrix created in this way is an operable matrix in which numerical values are given to the respective matrix components, and a periodic boundary condition is given. A predetermined external force is applied to the reduced matrix and a simulation calculation is performed (step S60). For example, when the reduction matrix is a stiffness matrix, the external force is a force or stress applied to the heterogeneous material, and when the reduction matrix is a heat conduction matrix, the external force is the amount of heat supplied to the heterogeneous material.

The simulation calculation is performed by calculating the behavior u _{i of} each node from the applied external force and the reduced matrix (step S60).
FIGS. 8A and 8B are diagrams showing results of creating a reduced stiffness matrix using the finite element model shown in FIG. 6 and performing a simulation calculation.
FIG. 8A is a deformation diagram showing a deformation state of the finite element model when a predetermined external force is applied, and FIG. 8B is a maximum main strain distribution at that time.
In FIG. 8A, white, gray, and black regions indicate the second polymer phase, the first polymer phase, the filler-polymer boundary phase, and the particulate filler phase in FIG. 3, respectively. The boundary surface of the finite element model is deformed by an external force.

  If the shape of the representative region cut out of the heterogeneous material does not match the contour shape of the finite element model, the finite element model is expanded and contracted by multiplying the finite element model by a constant size in at least one direction. The material constant in the finite element model may be set by matching the contour shape of the element model with the shape of the representative region.

The creation of a finite element model that reproduces the representative region of such a heterogeneous material can create a processing routine in advance and efficiently create a reduced matrix that can be simulated using this processing routine.
For example, in a finite element model with 256 × 256 × 1 three-dimensional finite element arrangement, a reduced matrix with periodic boundary conditions is created by setting the material constant value for each finite element. The processing time required for steps S40 and S50 to create a reduced matrix that can be calculated using this processing routine is 5 minutes, and this calculation processing required for the simulation calculation in step S60 The time is 14 minutes and the total processing time is 19 minutes. On the other hand, when a reduced matrix is created by assigning a periodic boundary condition after creating an entire matrix that can be operated as in the prior art, the preprocessing time required for creating the reduced matrix is 30 minutes. The processing time required for this calculation is 14 minutes, which is the same as the above processing time because the same reduced matrix is used, and the total processing time is 44 minutes.
In particular, when the simulation calculation is repeatedly performed by changing the dispersion arrangement of the material phase or changing the volume occupation ratio in the representative region of the material phase, it is necessary to repeatedly create an operable reduction matrix. Therefore, the reduced matrix can be created in a short time by using the processing routine many times.

  On the other hand, a penalty number K (a very large number compared to the matrix component) is used to create an entire matrix by adding an element matrix of finite elements according to an array of finite elements, and to give a periodic boundary condition to the entire matrix. Can be added to a predetermined matrix component to perform the simulation calculation. For example, when the nodes 4 and 6 of the entire matrix shown in [Equation 2] are constrained, a matrix of 4 rows, 4 columns, 4 rows, 6 columns, 6 rows, 4 columns, and 6 rows, 6 columns as shown in [Equation 10] below. The values of K, -K, K, and -K are added to the components, respectively.

However, in this case, since the degrees of freedom of nodes to be constrained are not deleted as in the above-described reduced matrix, this calculation is performed with the matrix size of the entire matrix. For this reason, the processing time of the calculation becomes 22 minutes to perform using a reduction matrix, increased compared to the total processing time 19 minutes prior to processing and the calculation in the present invention as described above, the total processing time 19 minutes Longer than that.
As described above, according to the present invention, the total processing time of the processing time for creating the reduced matrix and the processing time of the simulation calculation processing is shorter than that of the conventional method, and the simulation calculation can be performed efficiently.

  In the above embodiment, the representative region of the heterogeneous material is set as a three-dimensional region, and a three-dimensional finite element model in which a unit cell is formed using a finite element having a rectangular parallelepiped shape (including a cubic shape) is created. The cell is not limited to a rectangular parallelepiped shape, and a plurality of unit cells having the same shape such as a tetrahedron shape and a hexahedron shape are not arranged by joining the unit cells to each other and arranging them continuously in at least two directions. Any shape may be used as long as it occupies the space of the representative region of the homogeneous material without any gap. Further, the representative region of the heterogeneous material may be a two-dimensional planar region, and the finite element may be a rectangular shape, and a two-dimensional finite element model composed of a two-dimensional shape such as a regular triangle or a regular hexagon. May be created. In addition, a shape that occupies the region of the representative region of the heterogeneous material without gaps by joining a plurality of unit cell contours and arranging them continuously in at least two directions, such as a one-piece shape of a jigsaw puzzle Anything may be used.

Furthermore, the unit cell is not limited to a finite element. For example, the support range defined by the mesh-free method may be used as a unit cell, and modeling may be performed by the mesh-free method.
As shown in FIG. 9, the mesh-free method is within a distance ρ ₀ centering on the element point 50a of interest (black circle in FIG. 9) in a state where a plurality of element points (discrete points) are continuously arranged. In this range, the weight function w (x, y) is set by a spline function as shown in FIG. 9, for example, a quartic spline function. That is, a unit cell is set with each element point supporting a range within the distance ρ ₀ .

Using this weight function w (x, y), the position coordinates (x _i , y _i ) of each element point in the support around the element point of interest (i is an integer from 1 to n; n is in the support) The weighting coefficient is obtained from the total number of element points), and an interpolation function N (x, y) representing a physical quantity is set by the following equation (5) from the position coordinates of each element point and the weighting coefficient. Further, an approximate function φ ^h (x, y) of the physical quantity is determined from the physical quantity value φ _i (an integer from 1 to 1) at the element points in the support according to the following formula (6). This setting method is to determine the approximation function φ ^h (x, y) so that the error with respect to the value of the physical quantity φ _i is minimized within the support, that is, to be obtained by an approximation method using the moving least square method. It is. Details are described in “Computational Mechanics Handbook Vol. 1 Finite Element Method (Structure Editing)” (Japan Society of Mechanical Engineers, 1998, pp. 377 to 379).

  Represents an approximate function of the physical quantity in the stress field of one material phase of a heterogeneous material that deforms using this interpolation function, and this approximate function of the physical quantity is used in the finite element method formulated by the Galerkin method. By substituting into the governing equation of the stress field, a stiffness matrix corresponding to the stiffness matrix of each finite element in the finite element method can be created. Thus, the interpolation function created by the above equation (5) is determined by the position information of the element points in the support and the weighting coefficient determined by the weighting function w (x, y). The interpolation function varies depending on the point. This is different from the finite element method in which the interpolation function representing the physical quantity is uniquely determined according to the shape of the finite element when the model is set.

  As described above, in the model created by the mesh-free method, the interpolation function is determined by the position information of the element points in the support and the weighting coefficient determined by the weighting function w (x, y). Since the interpolation function changes depending on the point, it has a feature that it is difficult to diverge in the calculation of large deformation. From this, the part to be discretized by the mesh-free method includes at least the arrangement part of the material phase having the lowest rigidity among the plurality of material phases, and the remaining part is a model composed of finite elements. Also good. As a result, even if it is a heterogeneous material consisting of granular reinforcements such as carbon black and other elastomers that differ in rigidity by 100 times or more, simulation calculation can be performed without divergence against local large deformation of a weakly rigid material phase. It can be carried out.

  As mentioned above, although the simulation model creation method of the heterogeneous material of this invention was demonstrated in detail, this invention is not limited to the said embodiment, In the range which does not deviate from the main point of this invention, even if various improvement and a change are carried out. Of course it is good.

It is the block diagram which showed functionally the structure of the processing apparatus 10 which implements the simulation model preparation method of the heterogeneous material of this invention. It is a flowchart which shows the flow of an example of the simulation model creation method of the heterogeneous material of this invention. It is sectional drawing of an example of the representative area | region of a heterogeneous material. It is a figure which shows an example of the finite element model produced | generated in the simulation model creation method of the heterogeneous material of this invention. (A) And (b) is a figure explaining the finite element model and the matrix component of each finite element. It is a figure which shows an example of the finite element model produced with respect to the representative area | region shown in FIG. (A) And (b) is a figure explaining allocation of a material constant in the simulation model preparation method of the heterogeneous material of this invention. (A), (b) is a figure which shows the example of the result of having performed simulation calculation. It is a figure explaining the mesh free method used in the simulation model preparation method of the heterogeneous material of this invention.

Explanation of symbols

DESCRIPTION OF SYMBOLS 10 Processing apparatus 12 Input operation system 14 Computer 16 Display 18 CPU
20 Memory 22 Processing routine creation unit 24 Reduction matrix creation unit 26 Simulation operation unit

Claims (16)

A method for creating a heterogeneous material simulation model in which a simulation model of a heterogeneous material in which a plurality of material phases having different material properties are dispersedly arranged is created using a computer,
The computer is a discrete model of a representative area of the heterogeneous material by a unit cell of the same shape which shape is characterized by a plurality of discrete points, continuous plurality arranged along the two directions even without least The steps of creating
The computer smoothly connects the behavior of the discrete points of the unit cells located at one end of each arrangement direction of the unit cells in the discretization model to the behavior of the corresponding discrete points located at the other end. Defining a periodic boundary condition that constrains the behavior of the discrete points located at the one end to the behavior of the corresponding discrete points located at the other end; and
A reduced matrix in which the computer eliminates the degrees of freedom of the discrete points constrained by assigning the periodic boundary condition to the discretized model from the whole matrix representing the discretized model for performing a simulation operation. a step of previously created a process routine for generating a simulation operation can become a matrix by inputting a material constant of the unit cell as a parameter,
The computer stores the created processing routine in a memory; and
Based on the distributed material phase in the representative region, and the step material constants for each unit cell of the discrete model is that determined are entered into the computer from the input operation system in accordance with an instruction from the operator,
Matrix component wherein the computer invokes the routine that is stored and held in the memory, by executing the processing routine gives the value of the material constant defined in the processing routine, a corresponding of the reduction matrix giving numerical possess generating a simulation operation possible reduction matrix, to,
The reduced matrix that can be simulated is not created by the computer as a whole matrix that can be simulated, and the material constant of the unit cell is used as the parameter in the reduced matrix generated in the processing routine. It is generated in advance prior to the simulation calculation by being input, and then used for the simulation calculation performed by being given a predetermined physical quantity from the outside,
In the step of generating a reduced matrix that can be simulated, the material constant of each unit cell is input as the value of the parameter, so that a numerical value assigned to each matrix component of the reduced matrix is automatically calculated. is, this number is, the memory, the simulation model creation method of heterogeneous materials stored and held assigned to the address corresponding to each matrix element, characterized in Rukoto.   The method for creating a simulation model for a heterogeneous material according to claim 1, wherein the shape of the representative region of the heterogeneous material and the unit cell is a rectangular shape or a rectangular parallelepiped shape. In the step of generating a reduced matrix that can be simulated, if the shape of the representative region does not match the contour shape of the discretized model, the computer multiplies the shape size in at least one direction of the discretized model by a constant number. Then, by enlarging / reducing the discretization model, the processing routine is performed using material constants determined for each unit cell of the discretization model by matching the contour shape of the discretization model with the shape of the representative region. The method for creating a simulation model for a heterogeneous material according to claim 1 or 2 to be executed. The computer assigns one material phase to each unit cell of the discretization model and generates a material constant value of the assigned material phase in the processing routine in the step of generating a reduced matrix that can be simulated. The method for creating a simulation model for a heterogeneous material according to any one of claims 1 to 3, which is given and executed. In the step of generating a reduced matrix that can be simulated , the computer determines a material constant for each unit cell of the discretization model, and a unit of a plurality of material phases occupies a corresponding region of the unit cell. 5. The method for creating a heterogeneous material simulation model according to claim 4, wherein the processing routine is executed by giving a value of a material constant of a material phase having the largest volume occupancy ratio or area occupancy ratio of a material phase in a corresponding region of the cell to the processing routine. . In the step of generating a reduced matrix that can be simulated , the computer determines a material constant for each unit cell of the discretization model. When a plurality of material phases occupy a corresponding region of the unit cell, the unit cell The weighted average value of the material constant is obtained using the volume occupancy ratio or the area occupancy ratio of the material phase in the corresponding region of the material as a weighting factor, and the weighted average value is given to the processing routine to be executed. A method for creating a simulation model of a heterogeneous material as described in the paragraph.   The method of creating a heterogeneous material simulation model according to any one of claims 1 to 6, wherein the discretized model is a finite element model in which the unit cell is a finite element.   The method of creating a simulation model for a heterogeneous material according to claim 1, wherein the discretized model has a partial model discretized by a mesh-free method.   The method for creating a simulation model for a heterogeneous material according to claim 8, wherein the partial model includes at least a model of an arrangement portion of a material phase having the lowest rigidity among the plurality of material phases. The representative region has a polygonal or polyhedral shape having a corner portion where a plurality of contour lines or contour surfaces are connected at an angle,
The discretization model has discrete points of the unit cell at positions corresponding to the corners, and the behavior of the discrete points is constrained when generating the reduced matrix. A method for creating a simulation model of a heterogeneous material as described in the paragraph. The reduced matrix is a stiffness matrix in which each matrix component is digitized using the elastic modulus of the material phase as the parameter, and the physical quantity is a force, stress, or forced displacement applied to the heterogeneous material, and the computer The method for creating a simulation model for a heterogeneous material according to claim 1, wherein strain / stress analysis of the heterogeneous material is performed by the simulation calculation.   The method for creating a simulation model for a heterogeneous material according to claim 11, wherein the maximum elastic modulus among the elastic moduli in the plurality of material phases is 100 times or more the minimum elastic modulus.   The method for creating a simulation model for a heterogeneous material according to any one of claims 1 to 12, wherein the heterogeneous material includes granular reinforcing materials dispersedly arranged as a material phase. The non The homogeneous material, the simulation model creation method of heterogeneous material of any one of claims 1 to 13 in which a plurality of elastomer having different material properties are distributed as a material phase. A method for creating a simulation model of a heterogeneous material according to any one of claims 1 to 13,
The computer using the processing routine created by the simulation model creation method and the reduced matrix capable of simulation calculation, and performing the simulation calculation by assigning the physical quantity to the reduced matrix capable of simulation calculation; A method of calculating a simulation of a heterogeneous material, comprising: When performing the simulation calculation by changing the volume occupancy rate or area occupancy rate of each material phase in the dispersed arrangement of the plurality of material phases or the representative region of the heterogeneous material by the computer,
Prior to the simulation calculation, the computer generates the reduced matrix that can be simulated using the processing routine in advance without creating the entire matrix that can be simulated, and then assigns the physical quantity. The simulation calculation method for a heterogeneous material according to claim 15, wherein the simulation calculation is performed.

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