Unstructured mesh topology optimization method and system based on algebraic multigrid

Abstract

The present application belongs to the technical field of mesh topology optimization, in order to solve the problem of geometric multigrid algorithm in topology optimization problem solving is limited, put forward based on algebraic multigrid unstructured mesh topology optimization method and system, calculate the element stiffness matrix of each finite element unit, and accumulate to get the overall stiffness matrix;Adopt algebraic multigrid algorithm to the overall stiffness matrix is coarsened, construct the multilevel grid structure containing multilayer interpolation operator and coarse grid operator;Using conjugate gradient method in multilevel grid system under the processing of linear elastic problem under different frequency error component, through multiple iterations to get the displacement vector that meets the requirements of residual norm;According to the displacement vector, the flexibility of the structure, the derivative of the flexibility to the density, the volume constraint function, the derivative of the volume constraint function to the density are calculated, and the optimization algorithm is used to update the density;Repeat the above steps until the structure meets the design requirements;The present application makes the optimization problem not limited by geometric model.

Description

Technical Field

[0001] This invention belongs to the technical field of mesh topology optimization, and particularly relates to a method and system for unstructured mesh topology optimization based on algebraic multigrids. Background Technology

[0002] The statements in this section are merely background information related to the present invention and do not necessarily constitute prior art.

[0003] Structural optimization design methods include three types: size optimization, shape optimization, and topology optimization. Compared to size and shape optimization, topology optimization is widely used in various practical engineering fields due to its flexible design methods and independence from the initial configuration. Density-based topology optimization is currently the mainstream method. Its core idea is to assign density to the mesh of discrete geometric entities, obtain the equivalent elastic modulus of each mesh through density interpolation, and then obtain the equivalent element stiffness matrix and the overall stiffness matrix under density-based topology optimization.

[0004] To obtain a more refined optimized structure, the number of discrete grids must be increased (millions, tens of millions, or even hundreds of millions of grids). This brings two computational problems: (1) The sharp increase in the number of grids directly leads to a surge in the dimension of the matrix of the equation to be solved, resulting in a huge memory consumption when using the direct method; (2) The density-based topology optimization will generate a large number of gray units in the early and middle stages of optimization, and a large number of 0-1 units in the later stages of optimization. These units, through density interpolation, will severely degrade the condition number of the matrix, causing the relatively memory-efficient iterative algorithm to converge poorly or even fail to converge. Currently, in density-based topology optimization, the main method to solve this problem is to improve the condition number of the matrix to be solved by using a geometric multigrid algorithm, thereby improving the convergence performance of the traditional iterative algorithm.

[0005] While geometric multigrid algorithms can effectively accelerate the solution efficiency of topology optimization problems, they have strict restrictions on the type of mesh. Geometric multigrid algorithms require that the mesh of the discrete solid domain must be a geometrically regular mesh, such as a hexahedral mesh. This mesh restriction makes geometric multigrid algorithms unable to handle complex design domains and hollow models well. However, actual engineering models are usually quite complex and cannot be discretized using only regular hexahedral meshes, which greatly limits the practical application of geometric multigrid algorithms. Summary of the Invention

[0006] To overcome the shortcomings of the prior art, this invention provides an unstructured mesh topology optimization method and system based on algebraic multigrids. It combines the algebraic multigrid algorithm with the conjugate gradient method to handle error components of different frequencies in linear elastic problems at multiple mesh levels. This solves the problem that traditional iterative algorithms suffer from poor convergence or even failure to converge when dealing with topology optimization problems due to ill-conditioned overall stiffness matrix.

[0007] To achieve the above objectives, the present invention adopts the following technical solution: In a first aspect, the present invention provides an unstructured mesh topology optimization method based on algebraic multigrids, comprising: The element stiffness matrix of each element is calculated based on the finite element information of the geometric model to be optimized, and then summed to obtain the overall stiffness matrix. An algebraic multigrid algorithm is used to coarsen the overall stiffness matrix and construct a multi-level grid structure containing multi-level interpolation operators and coarse grid operators. Based on a multi-level mesh structure, the conjugate gradient method is used to handle error components of different frequencies in linear elastic problems, and the displacement vector that meets the residual norm requirement is obtained through multiple iterations. Based on the calculated displacement vector, the structural flexibility, the derivative of flexibility with respect to density, the volume constraint function, and the derivative of volume constraint function with respect to density are calculated. The density is then updated using an optimization algorithm to obtain the structure after one optimization. Repeat the above steps of calculating the unit stiffness matrix and the overall stiffness matrix to update the density until the structure meets the design requirements.

[0008] Secondly, the present invention provides an unstructured mesh topology optimization system based on algebraic multigrids, comprising: The calculation module is configured to: calculate the element stiffness matrix of each element based on the finite element information of the geometric model to be optimized, and sum them to obtain the overall stiffness matrix; The coarsening module is configured to: use an algebraic multigrid algorithm to coarse the overall stiffness matrix and construct a multi-level mesh structure containing multi-level interpolation operators and coarse mesh operators; The residual module is configured to: process the error components of different frequencies of linear elastic problems based on a multi-level mesh structure using the conjugate gradient method, and obtain the displacement vector that meets the residual norm requirement through multiple iterations; The density update module is configured to: calculate the structural compliance, the derivative of compliance with density, the volume constraint function, and the derivative of the volume constraint function with density based on the calculated displacement vector; update the density using an optimization algorithm to obtain the structure after one optimization. The iterative module is configured to repeat the steps of calculating the unit stiffness matrix and the overall stiffness matrix to update the density until the structure meets the design requirements.

[0009] Thirdly, the present invention provides an electronic device including a memory and a processor, and computer instructions stored in the memory and running on the processor, wherein the computer instructions, when executed by the processor, perform the method described in the first aspect.

[0010] Fourthly, the present invention provides a computer-readable storage medium for storing computer instructions, which, when executed by a processor, perform the method described in the first aspect.

[0011] Fifthly, the present invention provides a computer program product, including a computer program that, when executed by a processor, implements the method described in the first aspect.

[0012] The above one or more technical solutions have the following beneficial effects: In this invention, an algebraic multigrid algorithm is used to coarsen the overall stiffness matrix, constructing multiple grid levels of varying coarseness. This is combined with the conjugate gradient method to handle error components of different frequencies in linear elastic problems. Based on the displacement vector, the structural compliance, the derivative of compliance with density, the volume constraint function, and the derivative of the volume constraint function with density are calculated. An optimization algorithm is then used to update the density, obtaining a structure after one optimization. The steps of calculating the unit stiffness matrix and the overall stiffness matrix, up to updating the density, are repeated until the structure meets the design requirements. This invention solves the problem of poor convergence or even non-convergence caused by ill-conditioned overall stiffness matrix in traditional iterative algorithms when dealing with topology optimization problems. It makes the optimization problem unrestricted by geometric models, broadening its practical application scope.

[0013] Advantages of additional aspects of the invention will be set forth in part in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention. Attached Figure Description

[0014] The accompanying drawings, which form part of this invention, are used to provide a further understanding of the invention. The illustrative embodiments of the invention and their descriptions are used to explain the invention and do not constitute an improper limitation of the invention.

[0015] Figure 1 This is a flowchart of the unstructured mesh topology optimization method based on algebraic multigrid in Embodiment 1 of the present invention; Figure 2 This is a three-dimensional model diagram of the hollow semicircular ring in Embodiment 1 of the present invention; Figure 3 This is a diagram of the hollow semi-circular torus tetrahedral mesh model in Embodiment 1 of the present invention; Figure 4 This is a schematic diagram of the load constraint of the hollow semicircular ring in Embodiment 1 of the present invention; Figure 5The optimized configuration of the hollow semi-circular ring in Embodiment 1 of the present invention has a density greater than 0.5. Figure 6 This is a three-dimensional model diagram of the cantilever beam in Embodiment 1 of the present invention; Figure 7 This is a hexahedral mesh model of the cantilever beam in Embodiment 1 of the present invention; Figure 8 This is a schematic diagram of the load constraint of the cantilever beam in Embodiment 1 of the present invention; Figure 9 The optimized configuration of the cantilever beam in Embodiment 1 of the present invention has a density greater than 0.5 after optimization. Detailed Implementation It should be noted that the following detailed descriptions are exemplary and intended to provide further illustration of the invention. Unless otherwise specified, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention pertains.

[0016] It should be noted that the terminology used herein is for the purpose of describing particular implementations only and is not intended to limit the exemplary implementations of the present invention.

[0017] Where there is no conflict, the embodiments and features in the embodiments of the present invention can be combined with each other.

[0018] Example 1 This embodiment discloses an unstructured mesh topology optimization method based on algebraic multigrids, including: The element stiffness matrix of each element is calculated based on the finite element information of the geometric model to be optimized, and then summed to obtain the overall stiffness matrix. An algebraic multigrid algorithm is used to coarsen the overall stiffness matrix, and inter-grid interpolation operators and coarse grid operators are constructed under the multi-grid structure. Based on a multi-level mesh structure, the conjugate gradient method is used to process error components of different frequencies in linear elastic problems under a multi-level mesh system. Through multiple iterations, a displacement vector that meets the residual norm requirement is obtained. Based on the displacement vector obtained from the above solution, combined with the element stiffness matrix, global stiffness matrix, element degrees of freedom, and actual element volume, the structural flexibility, the derivative of flexibility with respect to density, the volume constraint function, and the derivative of the volume constraint function with respect to density are calculated. The density is filtered using a density filtering method and subjected to Heaviside projection. The density is updated using the moving asymptote optimization algorithm to obtain the optimized structure. Repeat the above steps of calculating the unit stiffness matrix and the overall stiffness matrix to update the density until the structure meets the design requirements.

[0019] This embodiment reads the geometric model, discretizes it using an irregular mesh, calculates the element stiffness matrix and the global stiffness matrix, and uses algebraic multigrid processing to obtain multi-level interpolation operators and coarse-grid operators for the global stiffness matrix. It combines multi-level meshing, smoothing operators, and the conjugate gradient method to handle error components of different frequencies in linear elastic problems at the multi-level mesh hierarchy. This solves the problem of poor convergence or even non-convergence caused by the ill-conditioned global stiffness matrix K in traditional iterative algorithms when dealing with topology optimization problems. This embodiment differs from traditional geometric multigrid algorithms, making the optimization problem unrestricted by the geometric model and broadening its practical application scope.

[0020] The algebraic multigrid algorithm and conjugate gradient algorithm used in this embodiment have excellent parallel characteristics on computer systems. Combined with multi-CPU parallel computing, the computational efficiency is greatly improved to meet the needs of practical applications such as large-scale topology optimization. In addition, the calculation of the element stiffness matrix and the large number of large-scale matrix-matrix multiplications and matrix-vector multiplications involved in the optimization process can be performed in parallel on multiple CPUs, further improving computational efficiency.

[0021] The following section uses a geometric model—a hollow semi-circular ring structure—as an example to illustrate the unstructured mesh topology optimization method based on algebraic multigrid proposed in this embodiment: The hollow semi-circular ring has the following geometric dimensions: an outer diameter of 30mm, an inner diameter of 10mm, and a scanning radius of 100mm for its horizontal cross-section. The load is applied vertically downwards to one end of the horizontal cross-section, while the other end has a fixed horizontal cross-section. The entire geometric region is the design domain. Figure 2 As shown. The hollow semi-circular torus tetrahedral mesh model is as follows. Figure 3 As shown, the load constraint of the hollow semicircular ring is as follows: Figure 4 As shown, the optimized configuration of the hollow semi-circular ring exhibits a density greater than 0.5. Figure 5 As shown.

[0022] Specifically, the geometric model of the hollow semi-circular ring structure was imported into Abaqus. Material definition, load constraint application, and tetrahedral mesh generation were then performed. A static analysis was completed in Abaqus, and the accompanying finite element file—a ".inp" file—was saved. The tetrahedral mesh generation was done using Abaqus' automatic tetrahedral mesh generation, with a mesh size of 2 mm, a total of 592,121 meshes, 108,156 nodes, and 324,468 degrees of freedom.

[0023] The .inp file contains the finite element information of the optimization model. This information is extracted and used to complete the calculation of the stiffness matrix of the finite element elements, the accumulation of the overall stiffness matrix, and the extraction of information such as loads, constraints, and materials in subsequent topology optimization.

[0024] The optimization objective is to minimize the structural flexibility under static conditions. The volume fraction and initial element density are determined, and the element stiffness matrix and global stiffness matrix of all elements are calculated.

[0025] Specifically, based on the .inp file, information such as mesh number, node number, node coordinates, and material elastic modulus is read to calculate the stiffness matrix Ke of each element. Then, the overall stiffness matrix K is obtained by accumulating the elements according to the node correspondence.

[0026] Tetrahedral mesh element stiffness matrix The calculation formula is:

[0027] Where B is the strain matrix, D is the constitutive matrix, and J is the Jacobian matrix. The strain matrix B and the Jacobian matrix J are calculated based on the mesh node information in the inp file. The constitutive matrix D of the element is determined by the elastic modulus E and Poisson's ratio ν of the material actually used in the structure. The elastic modulus of the material is also directly controlled by the design variable density.

[0028] The physical density is obtained by mapping the element density to Heaviside after density filtering. Based on the improved SIMP interpolation formula, the equivalent elastic modulus of the element is obtained. The element stiffness matrix is ​​expressed as a function of the physical density. The function. Where, E e Through unit physical density The calculated "equivalent elastic modulus" of the element; It is the minimum elastic modulus of the material. This is to prevent the matrix from becoming singular and unsolvable when the element density is 0 and the element equivalent elastic modulus is 0; This is the density penalty coefficient. This refers to the elastic modulus of the material itself.

[0029] The algebraic multigrid algorithm is used to coarsen the overall stiffness matrix, generate algebraic multigrid multilayer grid operators and interpolation operators, and form multiple relatively small linear systems.

[0030] Specifically, first, a coarsening algorithm is selected, such as the classic RS coarsening algorithm, CLJP coarsening algorithm, Falgout coarsening algorithm, HMIS coarsening algorithm, etc. The coarsening algorithm performs coarsening based on the formula for judging the strength of influence.

[0031]

[0032] The above formula is the formula for judging the strength of influence, where... The coarsening algorithm uses the connection strength threshold as a threshold. The values ​​are used to compare the coefficients in the overall stiffness matrix K. This allows us to determine the relative strength of influence between matrix coefficients, thereby guiding the coarsening algorithm to generate multigrid systems. This represents the value in the i-th row and j-th column of the matrix.

[0033] Overall stiffness matrix coefficients The size is directly determined by the topological density of the connected units. In the later stages of optimization, the structural density gradient will evolve towards 0-1, producing distinct void regions and solid regions. The algebraic multigrid coarsening algorithm, based on the formula for judging the strength of influence, calculates the relationship between the coefficients of the overall stiffness matrix at each step of the optimization process, and adaptively coarses the mesh along the material distribution.

[0034] Choose a smoothing algorithm such as weighted Jacobi smoothing, Gauss-Seidel smoothing, or Chebyshev smoothing.

[0035] Weighted Jacobian smoothing formula:

[0036] in, Here, represents the weighting coefficients; U and L are the negative values ​​of the upper and lower triangular matrices of the global stiffness matrix K; and D is the diagonal matrix of the global stiffness matrix K. This is the solution from the previous iteration. Let b be the smoothed solution, and b be the residual vector of the multigrid system.

[0037] Choose an interpolation algorithm, such as the classic interpolation algorithm, the direct interpolation algorithm, or the extended interpolation algorithm.

[0038]

[0039] in, The coefficients of the global stiffness matrix are represented; e represents the error component of the iterative solution to the linear elasticity problem. The set of grid points that have a weak influence on a certain grid point i; The set of coarse grid points that strongly influence a certain grid point i; This is the set of fine grid points that strongly influence a certain grid point i.

[0040] The above formula is a classic interpolation formula that uses the strength of the relationship between nodes to define the interpolation method, enabling the values ​​on a few nodes to be interpolated to obtain values ​​on more nodes. By combining the algebraic multigrid algorithm and the conjugate gradient method, the residual vector obtained by the conjugate gradient method is processed in a multi-level system, and multiple iterations are used to obtain the displacement vector under the allowed residual norm.

[0041] In this embodiment, the algebraic multigrid system, combined with the conjugate gradient method, processes the residuals and updates the solution vector. The specific process is as follows: Input: Global stiffness matrix K, initial guesses of displacement vector U and load vector F, maximum number of iterations M.

[0042] Output: The calculated displacement vector.

[0043] Step 1: Calculate the initial residual r under the initial conjectured solution U.

[0044] Step 2: Use the initial residual r as input to the algebraic multigrid algorithm, preprocess the initial residual r, and output the processed residual r.

[0045] Step 3: The residual r after processing by the algebraic multigrid algorithm is used as input to the conjugate gradient method for subsequent calculation of step size, etc., and the displacement vector U is updated. After updating the displacement vector U, a new residual r is obtained.

[0046] Step 4: Apply algebraic multigrid processing to the new residual r and update the residual r again.

[0047] Step 5: Update the search direction of the conjugate gradient method using the updated residual r, so that the conjugate gradient method can update the displacement vector U next time.

[0048] Step 6: Determine if the current residual is less than the specified value. If it is, the current optimization step is complete. Otherwise, return to the conjugate gradient method in step 3 to update U once more. Then, continue to use the algebraic multigrid algorithm to process the residual until the residual is less than the specified value and then stop iterating.

[0049] This embodiment uses the HMIS coarsening algorithm and the extended interpolation algorithm. In the initial optimization stage, the connection strength threshold is set to 0.5. As topology optimization progresses, the number of gray and blank elements gradually increases. To prevent excessive deterioration of the stiffness matrix condition number, the connection strength threshold is appropriately reduced to 0.2-0.3. Although this operation increases the construction time of the algebraic multigrid levels, it improves the preprocessing performance of the algebraic multigrid algorithm and prevents the conjugate gradient method from failing to converge. In actual testing, as topology optimization progresses and a large number of gray and blank elements appear, appropriately reducing the connection strength threshold reduces the number of iterations of the conjugate gradient algorithm, thus improving overall computational efficiency. When the structural topology tends to stabilize, the solution changes between the previous and next steps are relatively small, and the conjugate gradient algorithm converges without a large number of iterations. The connection strength threshold is then restored to 0.5 to balance overall computational efficiency.

[0050] Based on the calculated displacement vector, the structural flexibility, the derivative of flexibility with respect to density, the volume constraint function, and the derivative of volume constraint function with respect to density are calculated. The density is updated using the MMA optimization algorithm to obtain a new optimized structure. The element stiffness matrix and the global stiffness matrix are recalculated, and a new algebraic multigrid hierarchy is reconstructed to solve the displacement vector for the next optimization step until the structure meets the design requirements.

[0051] Specifically, after the algebraic multigrid preprocessing conjugate gradient method calculation is completed, the compliance of the objective function is calculated based on the calculated displacement vector U:

[0052] in, Let be the objective function. This is a density vector containing the physical density of all elements. The element node displacement vector; The global stiffness matrix is ​​assembled from the local stiffness matrices of all elements; N is the total number of elements in the finite element model; e is the index number of the specific element currently being calculated. This is the element stiffness matrix under unit elastic modulus.

[0053] By employing density filtering and Heaviside projection, and according to the chain rule, the sensitivity of the objective function compliance and volume constraint function to the design variable density is calculated:

[0054]

[0055] In the above formula, V is the volume constraint function and C is the objective function. The intermediate density after filtration. For intermediate density The physical density after Heaviside projection. The design variables are updated using the MMA (Moving Asymptote Method) algorithm, and the final design structure is obtained through repeated iterations. This embodiment mainly targets the topology optimization problem of complex geometric models under irregular mesh discretization, but it also has universality and is applicable to common simple geometric models, such as common cantilever beam models. Simple geometric models are usually discretized using regular quadrilateral or hexahedral meshes. In this case, geometric multigrids can be well adapted. Geometric multigrids can directly coarsen based on the actual mesh to generate multiple mesh levels of different coarseness. Combined with iterative algorithms, it can handle error components of different frequencies in linear elasticity problems. Geometric multigrids are limited or even unusable when dealing with more complex models. The direct reason is that actual complex models cannot be discretized using only regular meshes and must be discretized using irregular meshes such as tetrahedrons. The algebraic multigrid used in this invention is mainly for complex model problems, but it can also handle simple geometric models that geometric multigrids excel at. This is because, regardless of the model, the algebraic multigrid algorithm in this embodiment does not deal with the mesh itself, but with the finite element stiffness equations.

[0056] like Figure 1 As shown, the specific process of the topology optimization method is as follows: Step 1: Create a 3D model and define the constraint and load areas; Step 2: Divide the model using solid meshes, which can be hexahedral, tetrahedral, or a mixture of both, and extract the mesh node information; Step 3: Calculate the element stiffness matrix Ke and the global stiffness matrix K, and extract the load and constraint vectors; Step 4: Process the global stiffness matrix K using algebraic multigrid to generate a multi-level mesh; Step 5: Use CG as the basic iterative algorithm and process the residuals after calculation using a multi-level mesh system; Step 6: Determine if the residuals have converged. If the residuals have converged, proceed to Step 7; otherwise, return to Step 5 to continue iterative calculation. Step 7: Use the calculated displacement vector to complete the topology optimization work, such as sensitivity calculation, filtering, density update, etc. Step 8: Optimize whether to continue. If not, proceed to Step 9. If yes, return to Step 3 and recalculate. Step 9: Output the final optimized result image.

[0057] The purpose of this embodiment is to illustrate that unstructured mesh topology optimization systems based on algebraic multigrids are also applicable to simple geometric models such as cantilever beams. The geometric model of a cantilever beam is as follows: Figure 6 As shown, the geometric dimension is 100. 100 200mm, discretized using a regular hexahedral mesh, the mesh model is as follows: Figure 7 As shown, the mesh size is 2mm, the total number of nodes is 262,701, and the total number of degrees of freedom is 788,103. The load and constraint conditions are as follows: Figure 8 As shown.

[0058] The core difference in unstructured mesh topology optimization systems based on algebraic multigrids when handling simple geometric models discretized from regular meshes lies in the finite element method of computation; otherwise, the entire process is essentially the same. For models discretized from different meshes, the formulas for calculating the element stiffness matrix are formally identical and not fundamentally different. After obtaining the corresponding .inp file, the element stiffness matrix and the global stiffness matrix can be calculated based on the node coordinates and coordinate order of the elements. Subsequent topology optimization and the algebraic multigrid coarsening combined with the conjugate gradient algorithm for solution are consistent. Figure 9 The figure shown is a simple static topology optimization result of a cantilever beam model, showing only elements with a density greater than 0.5.

[0059] Example 2 This embodiment provides an unstructured mesh topology optimization system based on algebraic multigrids, including: The calculation module is configured to: calculate the element stiffness matrix of each element based on the finite element information of the geometric model to be optimized, and sum them to obtain the overall stiffness matrix; The coarsening module is configured to: use an algebraic multigrid algorithm to coarse the overall stiffness matrix and construct a multi-level mesh structure containing multi-level interpolation operators and coarse mesh operators; The residual module is configured to: process the error components of different frequencies of linear elastic problems based on a multi-level mesh structure using the conjugate gradient method, and obtain the displacement vector that meets the residual norm requirement through multiple iterations; The density update module is configured to: calculate the structural compliance, the derivative of compliance with density, the volume constraint function, and the derivative of the volume constraint function with density based on the calculated displacement vector; update the density using an optimization algorithm to obtain the structure after one optimization. The iterative module is configured to repeat the steps of calculating the unit stiffness matrix and the overall stiffness matrix to update the density until the structure meets the design requirements.

[0060] In further embodiments, the following is also provided: An electronic device includes a memory and a processor, as well as computer instructions stored in the memory and running on the processor. When executed by the processor, the computer instructions perform the method described in Embodiment 1. For brevity, further details are omitted here.

[0061] It should be understood that in this embodiment, the processor can be a central processing unit (CPU), or it can be other general-purpose processors, digital signal processors (DSPs), application-specific integrated circuits (ASICs), programmable gate arrays (FPGAs), or other programmable logic devices, discrete gate or transistor logic devices, discrete hardware components, etc. The general-purpose processor can be a microprocessor or any conventional processor, etc.

[0062] Memory may include read-only memory and random access memory, and provides instructions and data to the processor. A portion of memory may also include non-volatile random access memory. For example, memory may also store information about the device type.

[0063] A computer-readable storage medium for storing computer instructions, which, when executed by a processor, perform the method described in Embodiment 1.

[0064] The method in Embodiment 1 can be directly implemented by a hardware processor, or implemented by a combination of hardware and software modules within the processor. The software modules can reside in readily available storage media in the art, such as random access memory, flash memory, read-only memory, programmable read-only memory, electrically erasable programmable memory, or registers. This storage medium is located in memory; the processor reads information from the memory and, in conjunction with its hardware, completes the steps of the above method. To avoid repetition, a detailed description is not provided here.

[0065] A computer program product includes a computer program that, when executed by a processor, implements the method described in Embodiment 1.

[0066] The present invention also provides at least one computer program product tangibly stored on a non-transitory computer-readable storage medium. The computer program product includes computer-executable instructions, such as instructions included in program modules, which execute in a device on a target real or virtual processor to perform the processes / methods described above. Typically, program modules include routines, programs, libraries, objects, classes, components, data structures, etc., that perform specific tasks or implement specific abstract data types. In various embodiments, the functionality of program modules can be combined or divided among program modules as needed. The machine-executable instructions for the program modules can execute within a local or distributed device. In a distributed device, the program modules can reside in both local and remote storage media.

[0067] The computer program code used to implement the methods of the present invention may be written in one or more programming languages. This computer program code may be provided to a processor of a general-purpose computer, a special-purpose computer, or other programmable data processing device, such that when executed by the computer or other programmable data processing device, the program code causes the functions / operations specified in the flowcharts and / or block diagrams to be implemented. The program code may be executed entirely on a computer, partially on a computer, as a stand-alone software package, partially on a computer and partially on a remote computer, or entirely on a remote computer or server.

[0068] In the context of this invention, computer program code or related data may be carried by any suitable carrier to enable a device, apparatus, or processor to perform the various processes and operations described above. Examples of carriers include signals, computer-readable media, and the like. Examples of signals may include electrical, optical, radio, sound, or other forms of propagation signals, such as carrier waves, infrared signals, etc.

[0069] Those skilled in the art will recognize that the units and algorithm steps described in conjunction with the embodiments herein can be implemented in electronic hardware or a combination of computer software and electronic hardware. Whether these functions are implemented in hardware or software depends on the specific application and design constraints of the technical solution. Those skilled in the art can use different methods to implement the described functions for each specific application, but such implementation should not be considered beyond the scope of this application.

[0070] While the specific embodiments of the present invention have been described above in conjunction with the accompanying drawings, this is not intended to limit the scope of protection of the present invention. Those skilled in the art should understand that various modifications or variations that can be made by those skilled in the art without creative effort based on the technical solutions of the present invention are still within the scope of protection of the present invention.

Claims

1. An unstructured mesh topology optimization method based on algebraic multigrids, characterized in that, include: The element stiffness matrix of each element is calculated based on the finite element information of the geometric model to be optimized, and then summed to obtain the overall stiffness matrix. An algebraic multigrid algorithm is used to coarsen the overall stiffness matrix and construct a multi-level grid structure containing multi-level interpolation operators and coarse grid operators. Based on a multi-level mesh structure, the conjugate gradient method is used to handle error components of different frequencies in linear elastic problems, and the displacement vector that meets the residual norm requirement is obtained through multiple iterations. Based on the calculated displacement vector, the structural flexibility, the derivative of flexibility with respect to density, the volume constraint function, and the derivative of volume constraint function with respect to density are calculated. The density is then updated using an optimization algorithm to obtain the structure after one optimization. Repeat the above steps of calculating the unit stiffness matrix and the overall stiffness matrix to update the density until the structure meets the design requirements.

2. The unstructured mesh topology optimization method based on algebraic multigrid as described in claim 1, characterized in that, The acquisition of finite element information for the geometric model to be optimized is specifically as follows: after performing unstructured mesh generation on the geometric model, applying loads and fixed constraints, and completing static simulation analysis, finite element information is obtained; wherein, the finite element information includes mesh node coordinate information, number of meshes, mesh type, mesh number, material properties, and load and constraint application node location information.

3. The unstructured mesh topology optimization method based on algebraic multigrid as described in claim 1, characterized in that, An algebraic multigrid algorithm is used to coarsen the overall stiffness matrix, constructing a multi-level mesh structure containing multiple interpolation operators and coarse mesh operators, specifically: By comparing the coefficients of the finite element stiffness matrix based on the connection strength threshold, the relationship between the strength and weakness of nodes can be obtained. The coarsening algorithm selects coarse grid points from the stiffness matrix coefficients based on the strong and weak influence relationships between them, and combines them with the interpolation algorithm to form a multi-level grid system.

4. The unstructured mesh topology optimization method based on algebraic multigrid as described in claim 1, characterized in that, Based on a multi-level mesh structure, the conjugate gradient method is used to handle error components of different frequencies in linear elastic problems. Through multiple iterations, a displacement vector satisfying the residual norm requirement is obtained, specifically: The residuals processed by the algebraic multigrid algorithm are used as inputs to the conjugate gradient method to calculate the step size and update the displacement vector, thus obtaining a new residual. The new residuals obtained by the conjugate gradient method are processed using algebraic multigrid to update the residuals again; The search direction and displacement vector of the conjugate gradient method are updated using the updated residuals. The iteration continues until the residual is less than the specified value, at which point the iteration stops.

5. The unstructured mesh topology optimization method based on algebraic multigrid as described in claim 1, characterized in that, Density filtering and Heaviside projection are used, and the sensitivity of the objective function and volume constraint function to the design variable density is calculated according to the chain rule.

6. The unstructured mesh topology optimization method based on algebraic multigrid as described in claim 1, characterized in that, The element stiffness matrix is ​​calculated based on the coordinate information of each element node in the global coordinate system, the element and node numbering order, the material elastic modulus, and Poisson's ratio.

7. An unstructured mesh topology optimization system based on algebraic multigrids, characterized in that, include: The calculation module is configured to: calculate the element stiffness matrix of each element based on the finite element information of the geometric model to be optimized, and sum them to obtain the overall stiffness matrix; The coarsening module is configured to: use an algebraic multigrid algorithm to coarse the overall stiffness matrix and construct a multi-level mesh structure containing multi-level interpolation operators and coarse mesh operators; The residual module is configured to: process the error components of different frequencies of linear elastic problems based on a multi-level mesh structure using the conjugate gradient method, and obtain the displacement vector that meets the residual norm requirement through multiple iterations; The density update module is configured to: calculate the structural compliance, the derivative of compliance with density, the volume constraint function, and the derivative of the volume constraint function with density based on the calculated displacement vector; update the density using an optimization algorithm to obtain the structure after one optimization. The iteration module is configured to repeat the steps of calculating the unit stiffness matrix and the overall stiffness matrix to update the density until the structure meets the design requirements.

8. An electronic device, characterized in that, It includes a memory and a processor, as well as computer instructions stored in the memory and running on the processor, which, when executed by the processor, perform the method according to any one of claims 1-6.

9. A computer-readable storage medium, characterized in that, Used to store computer instructions, which, when executed by a processor, perform the method described in any one of claims 1-6.

10. A computer program product, characterized in that, Includes a computer program, which, when executed by a processor, implements the method described in any one of claims 1-6.

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