Method for applying periodic boundary conditions to non-matching grid representative volume elements and related apparatus

Abstract

The application discloses a non-matching mesh representative volume element periodic boundary condition application method and related equipment, and relates to the technical field of material simulation, and aims to solve the problems of low constraint precision and poor calculation stability in the prior art when a non-matching mesh representative volume element (RVE) is processed.

Description

Technical Field

[0001] This invention belongs to the field of multi-scale numerical simulation and computational homogenization technology of composite materials, specifically involving a method and related equipment for applying periodic boundary conditions to representative volume elements of non-matched meshes. Background Technology

[0002] The prediction of the macroscopic mechanical properties of composite materials typically relies on the accurate characterization of their microstructure. Multiscale computational methods often map the microscopic response to the macroscopic equivalent properties by establishing a representative volume element (RVE) model and applying periodic boundary conditions. Periodic boundary conditions require the displacement field on the relative boundaries of the RVE to satisfy a specific linear relationship, thereby simulating the continuity and periodicity of the material at the macroscopic scale. With the development of microstructure characterization techniques, methods based on experimental techniques such as X-ray computed tomography (CT) to reconstruct the true microstructure of materials and establish RVE models have gradually been applied. These RVE models can accurately reflect the complex geometric characteristics of materials, but their finite element meshes are usually irregular, non-periodic, and have mismatched boundary nodes.

[0003] For regular mesh models, the commonly used method for applying periodic boundaries is node pairing constraints. This method requires a strict one-to-one correspondence between mesh nodes on the relative periodic surface, achieving periodic displacement transfer by establishing linear constraint equations between node pairs. However, for non-matching mesh RVE models from sources such as CT reconstruction, the node pairing method is difficult to apply. For non-matching mesh cases, existing techniques mainly employ multi-point constraint (MPC) methods or weak constraint methods based on contact theory. Multi-point constraint methods establish linear constraint relationships between nodes on the periodic surface and use shape function interpolation to achieve displacement transfer between non-matching nodes. However, when the mesh heights on the periodic surfaces are mismatched, interpolation accuracy is difficult to guarantee, easily leading to constraint errors or numerical instability. While weak constraint methods based on contact or penalty functions can alleviate the node mismatch problem to some extent, their periodic displacement transfer depends on local contact relationships or the selection of penalty parameters, making it difficult to guarantee the overall consistency of the periodic constraints and exhibiting high parameter sensitivity.

[0004] In summary, the existing technologies mentioned above generally suffer from problems such as difficulty in guaranteeing constraint accuracy and insufficient computational stability when dealing with periodic boundary conditions in non-matched mesh RVEs. For complex geometric models obtained through methods such as CT reconstruction, how to stably and accurately apply periodic boundary conditions under non-matched mesh conditions has become an urgent technical problem to be solved in multi-scale numerical simulations. Summary of the Invention

[0005] To address the problems existing in the prior art, this invention provides a method and related equipment for applying periodic boundary conditions to representative volume elements of non-matched meshes. The purpose is to solve the common problems of difficulty in guaranteeing constraint accuracy and insufficient computational stability when dealing with periodic boundary conditions of non-matched mesh RVEs in the prior art, and to achieve stable and accurate application of periodic boundary conditions under non-matched mesh RVE conditions, so as to meet the needs of multi-scale numerical simulation.

[0006] To solve the above-mentioned technical problems, the present invention is achieved through the following technical solution: According to a first aspect of the present invention, a method for applying periodic boundary conditions to representative volume elements of a non-matching mesh is provided, comprising: Obtain a representative volume element finite element model of the material to be analyzed, wherein the representative volume element finite element model has a mismatched boundary mesh; Extract the boundary nodes located on each periodic surface of the representative volume element finite element model to obtain the coordinates of the periodic surface nodes; The coordinates of the periodic surface nodes are mapped to the parameter space to obtain the parameter coordinates of the boundary nodes; A virtual periodic shell with the same dimensions as the representative volume element finite element model is constructed, and the virtual periodic shell is divided into regular meshes to obtain the spatial coordinates of the reference nodes; Based on the parametric coordinates of the boundary nodes and the spatial coordinates of the reference nodes, a mapping relationship between the periodic surface nodes and the reference nodes on the virtual periodic shell is constructed through optimal transport mapping, resulting in a mapping matrix; Based on the mapping matrix, a periodic displacement constraint relationship is constructed, periodic constraint data is generated, and the periodic boundary conditions are applied.

[0007] In one possible implementation of the first aspect, the construction of a virtual periodic shell with dimensions consistent with the representative volume element finite element model, and the regular meshing of the virtual periodic shell to obtain the spatial coordinates of the reference nodes, specifically includes: Construct a hexahedral shell structure with dimensions consistent with the representative volume element finite element model; Set the material density of the hexahedral shell structure to zero; The hexahedral shell structure is divided into regular meshes so that the mesh nodes on the relative periodic surfaces of the hexahedral shell structure have a one-to-one correspondence, and the spatial coordinates of each mesh node are output as the spatial coordinates of the reference node.

[0008] In one possible implementation of the first aspect, mapping the coordinates of the periodic surface nodes to the parameter space to obtain the parameter coordinates of the boundary nodes specifically includes: Based on the coordinate extrema of the representative volume element finite element model in each direction, the parameter range of the periodic surface in the y and z directions is determined. The spatial coordinates of the boundary nodes on the periodic surface are converted into two-dimensional parametric coordinates using the following formula:

[0009]

[0010] In the formula, , These represent the maximum and minimum coordinate values ​​of the periodic surface in the y-direction; , These represent the maximum and minimum coordinate values ​​of the periodic surface in the z-direction; , For the first Spatial coordinate components of each boundary node; For the first The two-dimensional parameter coordinates of each boundary node after transformation; The transformed two-dimensional parameter coordinates are used as the parameter coordinates of the boundary nodes.

[0011] In one possible implementation of the first aspect, the step of constructing a mapping relationship between the periodic surface nodes and the reference nodes on the virtual periodic shell through optimal transport mapping based on the parametric coordinates of the boundary nodes and the spatial coordinates of the reference nodes, thereby obtaining a mapping matrix, specifically includes: Calculate the control area of ​​each boundary node in the parameter space, and use it as the node weight; Calculate the Euclidean distance between each boundary node in the parameter space; Based on the node weights and the Euclidean distance, construct and solve the following optimal transmission problem:

[0012] The constraints are:

[0013]

[0014]

[0015] in, To start from the first periodic surface The boundary node to the second periodic surface The mapping weights of each boundary node, the first periodic surface and the second periodic surface are a pair of opposite periodic surfaces; The Euclidean distance is given. For the first periodic surface The node weights of each boundary node; For the second periodic surface The node weights of each boundary node; Obtained from the solution This constitutes the mapping matrix.

[0016] In one possible implementation of the first aspect, the step of constructing a mapping relationship between the periodic surface nodes and the reference nodes on the virtual periodic shell through optimal transport mapping based on the parametric coordinates of the boundary nodes and the spatial coordinates of the reference nodes, thereby obtaining a mapping matrix, further includes: Obtain the strain energy density of the periodic surface nodes during the finite element solution process; Based on the mapping matrix and the strain energy density, an energy error term is constructed; By introducing the energy error term into the optimal transport problem, the following modified optimization model is constructed:

[0017]

[0018] The constraints are:

[0019]

[0020]

[0021] in, These are the corrected mapping weights; For the weighting factor; This refers to the energy error term; For the first periodic surface Strain energy density at each boundary node; For the second periodic surface Strain energy density at each boundary node; Obtained from the solution This forms the corrected mapping matrix.

[0022] In one possible implementation of the first aspect, the step of constructing periodic displacement constraint relationships and generating periodic constraint data based on the mapping matrix specifically includes: Obtain the macroscopic strain tensor and the dimensional parameters of the representative volume element finite element model; Based on the mapping matrix, the following mapping relationship is established between the boundary node displacements of the first periodic surface and the boundary node displacements of the second periodic surface:

[0023] Among them, the first periodic surface and the second periodic surface are a pair of opposing periodic surfaces. For the first periodic surface The displacement of each boundary node, For the second periodic surface The displacement of each boundary node, The mapping weights in the mapping matrix are, Let be the macroscopic strain tensor. The dimensional parameters of the representative volume element finite element model; The mapping relationship is used as the periodic displacement constraint relationship to generate the periodic constraint data.

[0024] In one possible implementation of the first aspect, after generating the periodic constraint data, the method further includes: Based on the aforementioned periodic displacement constraint relationship, the displacement error and stress error on the periodic surface are calculated, wherein the displacement error is:

[0025] The stress error is:

[0026] in, For the first periodic surface The displacement of each boundary node, For the second periodic surface The displacement of each boundary node, the first periodic surface and the second periodic surface are a pair of opposite periodic surfaces. The mapping weights in the mapping matrix are, For macroscopic strain tensor, The dimensional parameters of the representative volume element finite element model are... For the first periodic surface The stress tensor of each boundary node For the second periodic surface Stress tensor of each boundary node; A comprehensive error index is constructed based on the displacement error and the stress error:

[0027] in, As the displacement error weight, This represents the stress error weight; When the comprehensive error index exceeds a preset threshold, the mapping weights in the mapping matrix are adjusted using the following formula:

[0028] in, For the first Mapping weights for the next iteration For the first Mapping weights for the next iteration The learning rate; The mapping matrix is ​​updated according to the adjusted mapping weights, and the periodic displacement constraint relationship is reconstructed based on the updated mapping matrix until the comprehensive error index meets the requirements. According to a second aspect of the present invention, a computer device is provided, including a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor, when executing the computer program, implements the method for applying periodic boundary conditions to representative volume elements of a non-matching mesh.

[0029] According to a third aspect of the present invention, a computer-readable storage medium is provided, the computer-readable storage medium storing a computer program, which, when executed by a processor, implements the method for applying periodic boundary conditions to representative volume elements of a non-matching mesh.

[0030] According to a fourth aspect of the present invention, a computer program product is provided that, when executed by a processor, implements the method for applying periodic boundary conditions to representative volume elements of a non-matching mesh.

[0031] Compared with the prior art, the present invention has at least the following beneficial effects: Traditional methods for applying periodic boundary conditions typically require a strict one-to-one correspondence between mesh nodes on the relative periodic surface when dealing with mismatched meshes. This is often unsuitable for complex geometric models reconstructed through X-ray computed tomography. This invention extracts the coordinates of the periodic surface nodes, creating conditions for constraint construction that does not rely on direct node pairing, thus freeing the method from the limitations of mesh matching. Mapping the periodic surface node coordinates to the parameter space yields the parametric coordinates of the boundary nodes. Parametric processing transforms the boundary nodes in 3D space to a unified 2D parameter space, eliminating the influence of model geometry. This allows periodic surface nodes of different positions and sizes to be matched in the same coordinate system. The previously spatially dispersed boundary nodes are unified into a standardized parameter domain, providing a unified coordinate basis for establishing correspondences between nodes and avoiding matching difficulties caused by differences in model size or mesh density.

[0032] A virtual periodic shell with the same dimensions as the representative volume element is constructed and then regularly meshed to obtain the spatial coordinates of the reference nodes. This virtual periodic shell serves as a geometric reference system independent of the representative volume element. Its regular meshing ensures a strict one-to-one correspondence between the reference nodes on the relative periodic surfaces, providing an ideal standard template for applying periodic boundary conditions. Because the virtual shell has the same dimensions and spatial alignment as the representative volume element, the reference nodes accurately reflect the spatial position information of the periodic surfaces, thus establishing a spatial correspondence between mismatched boundary nodes and standard reference nodes.

[0033] Based on the parametric coordinates of boundary nodes and the spatial coordinates of reference nodes, an optimal transport mapping is used to construct the mapping relationship between periodic surface nodes and reference nodes on the virtual periodic shell, resulting in a mapping matrix. The optimal transport mapping can find the node matching scheme that minimizes the transport cost while satisfying mass conservation constraints, thereby establishing an optimal weight distribution relationship between mismatched mesh nodes. Compared to the traditional multi-point constraint method that relies on shape function interpolation, the optimal transport mapping can better handle extreme cases with unequal node numbers and mismatched positions, avoiding constraint errors caused by insufficient interpolation accuracy, while ensuring the mathematical optimality of the mapping relationship.

[0034] Periodic displacement constraints are constructed based on the mapping matrix, generating periodic constraint data and applying periodic boundary conditions. Since the mapping matrix itself establishes a correspondence between mismatched boundary nodes and regular reference nodes, the periodic displacement constraints constructed based on this matrix naturally inherit the mathematical properties of the optimal transport mapping, ensuring that displacement transfer between nodes on relatively periodic surfaces accurately satisfies the periodicity condition. The construction of the constraint relationships is entirely based on the mapping matrix, without relying on direct pairing between periodic surface nodes, thus completely eliminating the stringent requirements of mesh matching in traditional methods.

[0035] In summary, this invention achieves stable application of periodic boundary conditions under non-matched mesh conditions through the synergistic effect of parameterized mapping, virtual periodic shell construction, optimal transport mapping, and constraint construction based on the mapping matrix. This effectively solves the problems of high mesh matching requirements and difficulty in guaranteeing constraint accuracy in traditional methods. Attached Figure Description

[0036] To more clearly illustrate the technical solutions in the specific embodiments of the present invention, the drawings used in the description of the specific embodiments will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained from these drawings without creative effort.

[0037] Figure 1This is a schematic diagram of the overall process of applying periodic boundary conditions to a representative volume element of a non-matching mesh according to the present invention.

[0038] Figure 2 This is an RVE model, where the gray cube region represents the matrix and the yellow region represents the fibers. Figure 2 A three-dimensional Cartesian coordinate system was established, with the X direction representing the horizontal front-back direction, the Y direction representing the horizontal left-right direction, and the Z direction representing the vertical direction.

[0039] Figure 3 It is a virtual periodic shell structure.

[0040] Figure 4 The result of regular mesh partitioning for the virtual periodic shell.

[0041] Figure 5 This is a schematic diagram of the parametric mapping of RVE boundary nodes on the periodic reference plane.

[0042] Figure 6 The result is the displacement field contour plot of the RVE.

[0043] Figure 7 The result is the RVE stress field contour map.

[0044] Figure 8 This is a comparison between the RVE equivalent material stress-strain curve and the real material stress-strain curve. Detailed Implementation

[0045] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0046] like Figure 1 As shown, this invention provides a method for applying periodic boundary conditions to representative volume elements of a non-matching mesh, specifically including the following steps: S1. Obtain a representative volume element finite element model of the material to be analyzed, wherein the representative volume element finite element model has a non-matching boundary mesh.

[0047] For example, such as Figure 2As shown, the representative volumetric element model is a cubic structure, where the gray area represents the matrix and the yellow area represents the fibers. A three-dimensional Cartesian coordinate system is established in the model, with the X-direction representing the horizontal front-back direction, the Y-direction representing the horizontal left-right direction, and the Z-direction representing the vertical direction. This representative volumetric element finite element model was obtained through X-ray computed tomography (CT) reconstruction. Its boundary mesh exhibits mismatch characteristics, meaning that the number and positions of mesh nodes on the relative periodic surfaces are unequal.

[0048] S2. Extract the boundary nodes located on each periodic surface of the representative volume element finite element model to obtain the coordinates of the periodic surface nodes.

[0049] Specifically, from the obtained set of node coordinates of the representative volume element finite element model Extract the extreme values ​​in each direction. For the first Spatial coordinate vectors of boundary nodes , The total number of boundary nodes, and the coordinate vector of the boundary nodes. ,in The first The coordinate components of each boundary node in the X, Y, and Z directions are calculated. The maximum and minimum values ​​in each direction are calculated by traversing all boundary nodes. ，

[0050] ，

[0051] ，

[0052] Based on the above extreme values, a floating-point tolerance is set. (Typically, 1% to 5% of the mesh size is used) to identify the boundary nodes of periodic surfaces using the following criteria:

[0053] in The set of nodes of a periodic surface in the positive direction (e.g.) noodle), The set of nodes of the periodic surface in the negative direction (e.g.) Similarly, the sets of boundary nodes of the periodic surfaces in the Y and Z directions can be obtained, forming the set of node coordinates for each periodic surface.

[0054] S3. Map the coordinates of the periodic surface nodes to the parameter space to obtain the parameter coordinates of the boundary nodes.

[0055] By converting the coordinates of boundary nodes in three-dimensional space into two-dimensional parametric coordinates through parametric processing, the influence of geometric dimensions is eliminated, providing a unified coordinate basis for mapping construction.

[0056] S4. Construct a virtual periodic shell with the same dimensions as the representative volume element finite element model, and perform regular meshing on the virtual periodic shell to obtain the spatial coordinates of the reference nodes.

[0057] For example, a virtual cycle shell such as Figure 3 As shown, its dimensions are exactly the same as those of the representative volumetric element model. After regular meshing, the mesh nodes on the relative periodic surfaces have a one-to-one correspondence, such as... Figure 4 As shown.

[0058] S5. Based on the parameter coordinates of the boundary nodes and the spatial coordinates of the reference nodes, construct the mapping relationship between the periodic surface nodes and the reference nodes on the virtual periodic shell through optimal transmission mapping to obtain the mapping matrix.

[0059] The mapping matrix describes the correspondence between non-matching boundary nodes and rule reference nodes.

[0060] S6. Construct periodic displacement constraint relationships based on the mapping matrix, generate periodic constraint data, and complete the application of periodic boundary conditions.

[0061] By following the steps above, stable periodic boundary conditions can be established on a representative volumetric element model with a non-matching mesh, thereby meeting the requirements of multi-scale numerical simulation.

[0062] The method provided in this embodiment successfully establishes periodic boundary conditions under non-matched mesh conditions, solving the problem that the traditional node pairing method cannot be applied, and providing an accurate constraint basis for finite element solution.

[0063] In one possible implementation, a virtual periodic shell with the same dimensions as the representative volume element finite element model is constructed, and the virtual periodic shell is meshed regularly to obtain the spatial coordinates of the reference nodes, specifically including: First, a hexahedral shell structure with the same dimensions as the representative volume element finite element model is constructed. The geometric dimensions of this shell structure are exactly the same as those of the representative volume element model, that is, the six faces are aligned with the six periodic faces of the representative volume element finite element model.

[0064] Subsequently, the material density of the hexahedral shell structure is set to zero. That is, in the subsequent finite element solution, the shell structure does not contribute any stiffness, exists only as a geometric reference frame, and will not affect the mechanical response calculation results of the representative volume element model.

[0065] Next, the hexahedral shell structure is divided into regular meshes, ensuring a one-to-one correspondence between mesh nodes on relatively periodic faces. Specifically, a structured meshing method is used, with a mesh density set to [value missing]. ,in Indicates the number of divisions in the X direction. This represents the number of subdivisions in the Y direction. A denser mesh results in a smoother subsequent mapping. Structured meshing ensures that the number of nodes on two opposite periodic surfaces (left and right) is the same and their positions are symmetrical, the number of nodes on two opposite periodic surfaces (front and back) is the same and their positions are symmetrical, and the number of nodes on two opposite periodic surfaces (top and bottom) is the same and their positions are symmetrical.

[0066] After partitioning, output the set of shell mesh nodes. and periodic reference surface The set of shell mesh nodes is represented as follows: ,in Indicates the first There are 3 shell nodes, which serve as reference nodes for periodic constraints in subsequent steps. These nodes are regularly distributed and completely periodically symmetrical. The shell's six regular surfaces provide a standard periodic template, denoted as follows: , , , , , Each grid node on the reference plane has a one-to-one correspondence. Simultaneously, the spatial coordinates of each grid node are output as the spatial coordinates of the reference node.

[0067] By constructing a regular mesh shell with zero material density, this embodiment provides a standard reference system that does not participate in mechanical calculations, and provides a unified spatial coordinate reference for the mapping of mismatched mesh nodes to regular reference nodes.

[0068] In one possible implementation, mapping the coordinates of the periodic surface nodes to a parameter space to obtain the parameter coordinates of the boundary nodes specifically includes: First, based on the coordinate extrema of the representative volume element finite element model in each direction, the parameter ranges of the periodic surface in the Y and Z directions are determined. Specifically, the maximum coordinate value in the Y direction is obtained by traversing all nodes of the representative volume element finite element model. and minimum coordinate value and the maximum coordinate value in the Z direction. and minimum coordinate value These extreme values ​​determine the parameter range of the periodic surface.

[0069] Subsequently, the spatial coordinates of the boundary nodes on the periodic surface are converted into two-dimensional parametric coordinates using the following formula:

[0070]

[0071] In the formula, , These represent the maximum and minimum coordinate values ​​of the periodic surface in the y-direction; , These represent the maximum and minimum coordinate values ​​of the periodic surface in the z-direction; , For the first Spatial coordinate components of each boundary node; For the first The two-dimensional parameter coordinates of each boundary node after transformation.

[0072] Through the above transformation, for positively oriented periodic surfaces (such as...) (face), whose boundary nodes, after transformation, form the frontal parameter space set:

[0073] For negative direction periodic surfaces (such as...) (Surface), whose boundary nodes, after transformation, form a negative parameter space set:

[0074] Among them, the transformed two-dimensional parametric coordinates satisfy This achieves a normalized representation of boundary nodes in the parameter space.

[0075] The transformed 2D parametric coordinates are used as the parametric coordinates of the boundary node. Through the above parametric processing, the boundary nodes in 3D space are mapped to a unified 2D parametric space, eliminating the influence of model geometry and allowing the mapping to be constructed in a standardized coordinate space.

[0076] This implementation method unifies non-matching boundary nodes into a standard parametric coordinate system by normalizing parameter mapping, effectively solving the difficulty of directly pairing non-matching meshes.

[0077] In one possible implementation, based on the parametric coordinates of the boundary nodes and the spatial coordinates of the reference nodes, a mapping relationship between the periodic surface nodes and the reference nodes on the virtual periodic shell is constructed through optimal transport mapping, resulting in a mapping matrix, specifically including: After obtaining the parametric coordinates of the boundary nodes and the spatial coordinates of the reference nodes, the control area of ​​each boundary node in the parameter space is first calculated as the node weight. Node weight This reflects the area of ​​the region represented by the boundary node in the parameter space. It is calculated as follows: for each boundary node in the parameter space, the area corresponding to the boundary node is calculated based on the region enclosed by the boundary node and its neighboring nodes. ,Right now .

[0078] Calculate the Euclidean distance between each boundary node in the parameter space; Subsequently, the Euclidean distances of each boundary node in the parameter space are calculated. For the first periodic surface... The boundary node and the second periodic surface The boundary nodes and their Euclidean distances in the parameter space. Calculated using the following formula:

[0079] Based on the node weights and the Euclidean distance, construct and solve the following optimal transmission problem:

[0080] The constraints are:

[0081]

[0082]

[0083] in, To start from the first periodic surface The boundary node to the second periodic surface The mapping weights of each boundary node, the first periodic surface and the second periodic surface are a pair of opposite periodic surfaces; For the first periodic surface The node weights of each boundary node; For the second periodic surface The node weights of each boundary node.

[0084] The objective of solving this optimal transmission problem is to find the mapping scheme that minimizes the transmission cost while satisfying the mass conservation constraint. By solving this optimization problem, the mapping matrix between periodic surfaces can be obtained. The mapping matrix describes the weight distribution relationship between the nodes of the first periodic surface and the nodes of the second periodic surface. For example... Figure 5 As shown, this mapping matrix establishes a correspondence between non-matching boundary nodes and rule reference nodes.

[0085] This implementation establishes an optimal weight distribution relationship between non-matching grid nodes through optimal transmission mapping, realizing distributed mapping between nodes and avoiding the limitation of one-to-one matching of nodes in traditional methods.

[0086] In one possible implementation, based on the parametric coordinates of the boundary nodes and the spatial coordinates of the reference nodes, a mapping relationship between the periodic surface nodes and the reference nodes on the virtual periodic shell is constructed through optimal transport mapping to obtain a mapping matrix, and the process further includes the following steps: First, the strain energy density of the periodic surface nodes is obtained during the finite element solution process. Specifically, the strain energy density can be obtained through preliminary finite element analysis of a representative volume element model, or by gradually updating it during the iterative solution process. For the first periodic surface... The strain energy density of each boundary node is denoted as . ,in Boundary nodes The stress tensor, Boundary nodes The macroscopic strain tensor. The double dot product operation represents the tensor inner product. For the second periodic surface... The strain energy density of each boundary node is denoted as . Similarly, the definition is the same.

[0087] Subsequently, an energy error term is constructed based on the mapping matrix and the strain energy density. Specifically, the energy error term is defined as:

[0088] The summation sign must specify the range of summation: for the first periodic surface, , This represents the total number of boundary nodes on the first periodic surface; for the second periodic surface, , This represents the total number of boundary nodes on the second periodic surface. This error term measures the difference between the strain energy density of the boundary nodes on the first periodic surface and the weighted sum of the strain energy densities of the boundary nodes on the second periodic surface under the mapping relationship, reflecting the mechanical rationality of the mapping.

[0089] Next, the energy error term is introduced into the optimal transport problem to construct the following modified optimization model:

[0090] The constraints are:

[0091]

[0092]

[0093] in, These are the corrected mapping weights; This is a tradeoff coefficient, which takes the value of a positive real number and is used to control the balance between geometric matching and energy consistency. This refers to the energy error term; For the first periodic surface Strain energy density at each boundary node; For the second periodic surface Strain energy density at each boundary node.

[0094] Finally, the modified optimization model is solved, and the solution obtained is... This forms the corrected mapping matrix.

[0095] This embodiment introduces energy consistency constraints on the basis of geometric matching, so that the constructed mapping relationship not only satisfies the optimal matching in spatial position, but also takes into account the energy conservation of mechanical response, thereby improving the physical accuracy of periodic boundary conditions.

[0096] In one possible implementation, periodic displacement constraint relationships are constructed based on the mapping matrix to generate periodic constraint data, as detailed below: First, the macroscopic strain tensor and the dimensional parameters of the representative volume element finite element model are obtained. Macroscopic strain tensor The global strain applied to a representative volume element can be expressed as:

[0097] Dimensional parameters of a representative volume element finite element model The characteristic length of a representative volume element in the periodic direction is the side length for a cube model.

[0098] Subsequently, based on the mapping matrix, the following mapping relationship is established between the boundary node displacements of the first periodic surface and the boundary node displacements of the second periodic surface:

[0099] Among them, the first periodic surface and the second periodic surface are a pair of opposing periodic surfaces. For the first periodic surface The displacement of each boundary node, For the second periodic surface The displacement of each boundary node, The mapping weights in the mapping matrix are, Let be the macroscopic strain tensor. These are the dimensional parameters of the representative volume element finite element model.

[0100] This mapping relationship shows that the displacement of the boundary node of the first periodic surface is determined by the weighted sum of the displacements of the boundary node of the second periodic surface and the additional displacement caused by macroscopic strain.

[0101] Finally, the mapping relationship is used as the periodic displacement constraint relationship to generate the periodic constraint data, which can then be applied in the finite element analysis.

[0102] This implementation establishes the displacement transfer relationship between relative periodic surfaces through a mapping matrix, enabling the precise application of periodic boundary conditions under non-matched mesh conditions. At the same time, the introduction of macroscopic strain terms enables control over the overall deformation of representative volume elements.

[0103] In one possible approach, an adaptive error correction mechanism is introduced after generating the periodic constraint data, as follows: First, based on the periodic displacement constraint relationship, the displacement error and stress error on the periodic surface are calculated, wherein the displacement error is:

[0104] It should be understood that the displacement error measures the deviation between the actual displacement of the boundary node of the first periodic surface under the mapping relationship and the theoretical displacement calculated by the weighted sum of the displacements of the boundary node of the second periodic surface and the macroscopic strain.

[0105] The stress error is:

[0106] in, For the first periodic surface The displacement of each boundary node, For the second periodic surface The displacement of each boundary node, the first periodic surface and the second periodic surface are a pair of opposite periodic surfaces. The mapping weights in the mapping matrix are, For macroscopic strain tensor, The dimensional parameters of the representative volume element finite element model are... For the first periodic surface The stress tensor of each boundary node For the second periodic surface The stress tensor of each boundary node.

[0107] It should be understood that stress error measures the stress difference at corresponding nodes on a relative periodic surface.

[0108] After obtaining the displacement error and stress error, a comprehensive error index is constructed based on the displacement error and stress error:

[0109] in, As the displacement error weight, This represents the stress error weight; and All are positive real numbers, and can be selected according to the focus of the actual problem. For example, all can be set to 1.

[0110] When the comprehensive error index exceeds the preset threshold ,Right now When adjusting the mapping weights in the mapping matrix, the following formula is used:

[0111] in, For the first Mapping weights for the next iteration For the first Mapping weights for the next iteration The learning rate is a positive real number. The gradient of the comprehensive error index with respect to the mapped weights is used to guide the direction of iterative weight updates; the number of iteration steps. This is used to control the magnitude of each update until the overall error index meets the requirements.

[0112] The mapping matrix is ​​updated according to the adjusted mapping weights, and the periodic displacement constraint relationship is reconstructed according to the updated mapping matrix. The finite element solution is performed again and the error is evaluated. The above process is repeated until the comprehensive error index meets the requirements.

[0113] After completing the construction and adaptive correction of the periodic constraint data, the generated periodic constraint data is introduced into the finite element solution system. Specifically, constraint matrices are introduced into the finite element equations, forming the following system equations:

[0114] in, Here is the system stiffness matrix. This is the global node displacement vector. This represents the external load or equivalent load vector. By solving the system equations in conjunction with the constructed periodic displacement constraint relationship, periodic boundary conditions can be stably applied to the representative volume element model of the mismatched mesh, yielding the microscopic response results of the material, including the displacement field (e.g., ...). Figure 6 As shown), stress field (as shown) Figure 7 (as shown) and equivalent material stress-strain curves (as shown) Figure 8 (As shown).

[0115] This implementation introduces an adaptive error correction mechanism to iteratively optimize the mapping weights based on the displacement and stress errors actually generated by solving the periodic boundary conditions, thereby continuously improving the accuracy of the periodic constraints. Figure 6 and Figure 7 As shown, the displacement and stress field distributions obtained using this method are continuous and reasonable, with no obvious non-physical abrupt changes at the boundaries. Figure 8 As shown, the equivalent material stress-strain curve calculated by this method is in high agreement with the actual material stress-strain curve, verifying that the method is accurate and reliable when applying periodic boundary conditions under non-matched mesh conditions.

[0116] This invention establishes a unified, regularly distributed standard reference system for applying periodic boundary conditions by constructing a virtual periodic shell with dimensions consistent with representative volume elements and then performing regular meshing on this virtual periodic shell. Traditional methods, when dealing with mismatched meshes, often require approximations such as interpolation or contact because the nodes on the relative periodic surfaces cannot be directly paired, making it difficult to guarantee the accuracy of the constraints. This invention associates the mismatched boundary nodes with regular reference nodes through optimal transport mapping, essentially building a bridge between the mismatched meshes. This eliminates the dependence of applying periodic boundary conditions on the direct correspondence between the periodic surface nodes, thus freeing the traditional node pairing method from the strict requirements of mesh matching.

[0117] Optimal transport mapping is employed to construct the mapping relationship between boundary nodes and reference nodes. Under the premise of satisfying mass conservation constraints, the mapping scheme that minimizes transport costs is sought and applied to the construction of periodic boundary conditions. This enables the establishment of optimal weight allocation relationships among mismatched nodes, achieving distributed transport of node displacements. Compared to multi-point constraint methods that rely on shape function interpolation, optimal transport mapping better handles the situation of mesh height mismatch and avoids constraint errors caused by insufficient interpolation accuracy.

[0118] The virtual periodic shell exists only as a geometric reference frame, with its material density set to zero, and is not involved in the structural stiffness calculation. This design ensures that the introduction of the standard reference frame does not interfere with the mechanical response of the representative volume element itself, guaranteeing that the calculation results only reflect the mechanical behavior of the material itself.

[0119] The periodic displacement constraint relationship constructed by this method is directly based on the mapping matrix obtained from the optimal transport mapping. It expresses the displacement of the first periodic surface node as the sum of the weighted sum of the displacements of the second periodic surface node and the additional displacement caused by macroscopic strain. This constraint form fully embodies the mathematical essence of periodic boundary conditions, namely, the displacement difference on the relative boundaries is equal to the product of macroscopic strain and characteristic length. Introducing this constraint relationship into the finite element solution system enables accurate simulation of the continuity and periodicity of materials at the macroscopic scale under mismatched mesh conditions.

[0120] In another embodiment of the present invention, a computer device is provided, comprising a processor and a memory. The memory stores a computer program, which includes program instructions. The processor executes the program instructions stored in the computer storage medium. The processor may be a Central Processing Unit (CPU), or other general-purpose processors, digital signal processors (DSPs), application-specific integrated circuits (ASICs), field-programmable gate arrays (FPGAs), or other programmable logic devices, discrete gate or transistor logic devices, discrete hardware components, etc. It is the computing and control core of the terminal, suitable for implementing one or more instructions, specifically suitable for loading and executing one or more instructions in the computer storage medium to achieve a corresponding method flow or corresponding function. The processor described in this embodiment of the present invention can be used for the operation of a method for applying periodic boundary conditions to representative volume elements of a non-matched mesh.

[0121] In another embodiment of the present invention, a storage medium is provided, specifically a computer-readable storage medium (Memory), which is a memory device in a computer device used to store programs and data. It is understood that the computer-readable storage medium here can include both the built-in storage medium in the computer device and extended storage media supported by the computer device. The computer-readable storage medium provides storage space that stores the terminal's operating system. Furthermore, the storage space also stores one or more instructions suitable for loading and execution by a processor. These instructions can be one or more computer programs (including program code). It should be noted that the computer-readable storage medium here can be Random Access Memory (RAM) or non-volatile memory, such as at least one disk storage device. The processor can load and execute one or more instructions stored in the computer-readable storage medium to implement the corresponding steps of the method for applying periodic boundary conditions to representative volumetric cells of non-matching meshes in the above embodiments.

[0122] Those skilled in the art will understand that embodiments of the present invention can be provided as methods, systems, or computer program products. Therefore, the present invention can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, the present invention can take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, optical storage, etc.) containing computer-usable program code.

[0123] This invention is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the invention. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart illustrations and / or block diagrams. Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0124] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0125] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0126] This invention also provides a computer program product for executing the periodic boundary condition application method for representative volume elements of non-matched meshes described above. Since the computer program product provided by this invention and the periodic boundary condition application method for representative volume elements of non-matched meshes described above belong to the same inventive concept, the computer program product provided by this invention possesses all the advantages of the periodic boundary condition application method for representative volume elements of non-matched meshes described above. Therefore, the beneficial effects of the computer program product provided by this invention will not be elaborated upon here.

[0127] In this invention, the terms "one embodiment," "some embodiments," "example," "specific example," or "some examples," etc., refer to a specific feature, structure, material, or characteristic described in connection with that embodiment or example, which is included in at least one embodiment or example of the invention. In this specification, the illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples. Moreover, without contradiction, those skilled in the art can combine and integrate the different embodiments or examples described in this specification, as well as the features of different embodiments or examples.

[0128] Finally, it should be noted that the above-described embodiments are merely specific implementations of the present invention, used to illustrate the technical solutions of the present invention, and not to limit them. The scope of protection of the present invention is not limited thereto. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that any person skilled in the art can still modify or easily conceive of changes to the technical solutions described in the foregoing embodiments within the scope of the technology disclosed in the present invention, or make equivalent substitutions for some of the technical features; and these modifications, changes, or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention, and should all be covered within the scope of protection of the present invention.

Claims

1. A method for applying periodic boundary conditions to representative volume elements of a non-matching mesh, characterized in that, include: Obtain a representative volume element finite element model of the material to be analyzed, wherein the representative volume element finite element model has a mismatched boundary mesh; Extract the boundary nodes located on each periodic surface of the representative volume element finite element model to obtain the coordinates of the periodic surface nodes; The coordinates of the periodic surface nodes are mapped to the parameter space to obtain the parameter coordinates of the boundary nodes; A virtual periodic shell with the same dimensions as the representative volume element finite element model is constructed, and the virtual periodic shell is divided into regular meshes to obtain the spatial coordinates of the reference nodes; Based on the parametric coordinates of the boundary nodes and the spatial coordinates of the reference nodes, a mapping relationship between the periodic surface nodes and the reference nodes on the virtual periodic shell is constructed through optimal transport mapping, resulting in a mapping matrix; Based on the mapping matrix, a periodic displacement constraint relationship is constructed, periodic constraint data is generated, and the periodic boundary conditions are applied.

2. The method for applying periodic boundary conditions to representative volume elements of a non-matching mesh according to claim 1, characterized in that, The construction of a virtual periodic shell with dimensions identical to the representative volume element finite element model, and the generation of a regular mesh on the virtual periodic shell to obtain the spatial coordinates of the reference nodes, specifically includes: Construct a hexahedral shell structure with dimensions consistent with the representative volume element finite element model; Set the material density of the hexahedral shell structure to zero; The hexahedral shell structure is divided into regular meshes so that the mesh nodes on the relative periodic surfaces of the hexahedral shell structure have a one-to-one correspondence, and the spatial coordinates of each mesh node are output as the spatial coordinates of the reference node.

3. The method for applying periodic boundary conditions to representative volume elements of a non-matching mesh according to claim 1, characterized in that, The step of mapping the coordinates of the periodic surface nodes to the parameter space to obtain the parameter coordinates of the boundary nodes specifically includes: Based on the coordinate extrema of the representative volume element finite element model in each direction, the parameter range of the periodic surface in the y and z directions is determined. The spatial coordinates of the boundary nodes on the periodic surface are converted into two-dimensional parametric coordinates using the following formula: In the formula, , These represent the maximum and minimum coordinate values ​​of the periodic surface in the y-direction; , These represent the maximum and minimum coordinate values ​​of the periodic surface in the z-direction; , For the first Spatial coordinate components of each boundary node; For the first The two-dimensional parameter coordinates of each boundary node after transformation; The transformed two-dimensional parameter coordinates are used as the parameter coordinates of the boundary nodes.

4. The method for applying periodic boundary conditions to representative volume elements of a non-matching mesh according to claim 1, characterized in that, The mapping relationship between the periodic surface nodes and the reference nodes is constructed through optimal transport mapping based on the parameter coordinates of the boundary nodes and the spatial coordinates of the reference nodes, resulting in a mapping matrix. Specifically, this includes: Calculate the control area of ​​each boundary node in the parameter space, and use it as the node weight; Calculate the Euclidean distance between each boundary node in the parameter space; Based on the node weights and the Euclidean distance, construct and solve the following optimal transmission problem: The constraints are: in, To start from the first periodic surface The boundary node to the second periodic surface The mapping weights of each boundary node, the first periodic surface and the second periodic surface are a pair of opposite periodic surfaces; The Euclidean distance is given. For the first periodic surface The node weights of each boundary node; For the second periodic surface The node weights of each boundary node; Obtained from the solution This constitutes the mapping matrix.

5. The method for applying periodic boundary conditions to representative volume elements of a non-matching mesh according to claim 4, characterized in that, The method of constructing a mapping relationship between the periodic surface nodes and the reference nodes on the virtual periodic shell based on the parameter coordinates of the boundary nodes and the spatial coordinates of the reference nodes through optimal transport mapping to obtain a mapping matrix further includes: Obtain the strain energy density of the periodic surface nodes during the finite element solution process; Based on the mapping matrix and the strain energy density, an energy error term is constructed; By introducing the energy error term into the optimal transport problem, the following modified optimization model is constructed: The constraints are: in, These are the corrected mapping weights; For the weighting factor; This refers to the energy error term; For the first periodic surface Strain energy density at each boundary node; For the second periodic surface Strain energy density at each boundary node; Obtained from the solution This forms the corrected mapping matrix.

6. The method for applying periodic boundary conditions to representative volume elements of a non-matching mesh according to claim 1, characterized in that, The step of constructing periodic displacement constraint relationships based on the mapping matrix and generating periodic constraint data specifically includes: Obtain the macroscopic strain tensor and the dimensional parameters of the representative volume element finite element model; Based on the mapping matrix, the following mapping relationship is established between the boundary node displacements of the first periodic surface and the boundary node displacements of the second periodic surface: Among them, the first periodic surface and the second periodic surface are a pair of opposing periodic surfaces. For the first periodic surface The displacement of each boundary node, For the second periodic surface The displacement of each boundary node, The mapping weights in the mapping matrix are, Let be the macroscopic strain tensor. The dimensional parameters of the representative volume element finite element model; The mapping relationship is used as the periodic displacement constraint relationship to generate the periodic constraint data.

7. The method for applying periodic boundary conditions to representative volume elements of a non-matching mesh according to claim 1, characterized in that, After generating the periodic constraint data, the following is also included: Based on the aforementioned periodic displacement constraint relationship, the displacement error and stress error on the periodic surface are calculated, wherein the displacement error is: The stress error is: in, For the first periodic surface The displacement of each boundary node, For the second periodic surface The displacement of each boundary node, the first periodic surface and the second periodic surface are a pair of opposite periodic surfaces. The mapping weights in the mapping matrix are, For macroscopic strain tensor, The dimensional parameters of the representative volume element finite element model are... For the first periodic surface The stress tensor of each boundary node For the second periodic surface Stress tensor of each boundary node; A comprehensive error index is constructed based on the displacement error and the stress error: in, As the displacement error weight, This represents the stress error weight; When the comprehensive error index exceeds a preset threshold, the mapping weights in the mapping matrix are adjusted using the following formula: in, For the first Mapping weights for the next iteration For the first Mapping weights for the next iteration The learning rate; The mapping matrix is ​​updated according to the adjusted mapping weights, and the periodic displacement constraint relationship is reconstructed according to the updated mapping matrix until the comprehensive error index meets the requirements.

8. A computer device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that, When the processor executes the computer program, it implements the method for applying periodic boundary conditions to representative volume elements of non-matching meshes as described in any one of claims 1 to 7.

9. A computer-readable storage medium storing a computer program, characterized in that, When the computer program is executed by the processor, it implements the method for applying periodic boundary conditions to representative volume elements of non-matching meshes as described in any one of claims 1 to 7.

10. A computer program product, characterized in that, When executed by a processor, the computer program product implements the method for applying periodic boundary conditions to representative volume elements of a non-matching mesh as described in any one of claims 1 to 7.

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