CN114386299A - Topological optimization method and device based on PDE and storage medium - Google Patents

Abstract

The invention discloses a PDE-based topology optimization method, device and storage medium. The transmission method constructs the equivalent linear equation system of the Helmholtz type partial differential equation according to the discrete method of finite element analysis, calculates the node sensitivity filtering result of the equivalent linear equation system according to the multi-grid preprocessing conjugate gradient method, and filters the filtered results. The node sensitivity is mapped to the element sensitivity of the second structure topology optimization model to obtain a third structure topology optimization model; the design variables of the structure topology optimization model are updated according to the OC criterion; the finite element analysis and sensitivity filtering are repeated for the structure topology optimization model. and the operation process of updating the design variables until the structural topology optimization model satisfies the preset convergence condition, and then the final structural topology optimization model is output. The technical scheme of the invention reduces the time of topology optimization and improves the efficiency of topology optimization.

Description

A PDE-based topology optimization method, device and storage medium

technical field

The present invention relates to the technical field of topology optimization, in particular to a PDE-based topology optimization method, device and storage medium.

Background technique

Structural topology optimization refers to finding the optimal topology structure that satisfies the design constraints through a certain optimization method in a limited structural design space. Structural optimization design technology has spread in various fields of engineering practice, including aerospace, automobile manufacturing, shipbuilding industry, mobile communication, biological engineering and civil engineering, etc., all of which have been successfully applied. In the traditional structural topology optimization process, the sensitivity filtering and density filtering algorithms are time-consuming and computationally inefficient, making it an expensive operation to obtain other unit information from the neighborhood for large-scale numerical simulations in practical problems. The grid is divided more and more densely, and the calculation time will be longer and longer. When the traditional PDE filtering technology is applied to the structural topology optimization, with the increase of the grid size, the condition number of the linear equation system will be too large, that is, the model will be ill-conditioned. At the same time, when the direct method is used to solve the linear equation system It takes a long time, which makes the structure topology optimization time too long.

SUMMARY OF THE INVENTION

The invention provides a PDE-based topology optimization method, device and storage medium, which greatly reduces the topology optimization time and improves the topology optimization efficiency.

An embodiment of the present invention provides a PDE-based topology optimization method, comprising the following steps:

Initializing the parameters of the first structural topology optimization model, and defining the design area, constraint conditions and loads of the structural topology optimization model;

The finite element network is divided according to the design area of the first structural topology optimization model, and the corresponding finite element model is obtained, and the division of the finite element network is stopped until the finite element model meets the first preset condition; On this basis, the topology optimization model of SIMP flexible structure is established;

Finite element analysis: perform finite element analysis on the SIMP flexible structure topology optimization model, and calculate each element stiffness matrix and overall stiffness matrix of the SIMP flexible structure topology optimization model according to each constraint condition of the SIMP flexible structure topology optimization model , node displacement, flexibility and sensitivity to obtain the second structural topology optimization model;

Sensitivity filtering: Construct the equivalent linear equation system of Helmholtz-type partial differential equations according to the discrete method of finite element analysis, calculate the node sensitivity filtering results of the equivalent linear equation system according to the multi-grid preprocessing conjugate gradient method, and filter the filtered The node sensitivity is mapped to the element sensitivity of the second structure topology optimization model to obtain a third structure topology optimization model;

Update design variables: update the design variables of the third structural topology optimization model according to the OC criterion;

Repeat the operations of finite element analysis, sensitivity filtering and updating design variables for the third structural topology optimization model until the third structural topology optimization model satisfies the preset convergence condition, and output the third structural topology optimization Model.

Further, the SIMP flexible structure topology optimization model is expressed as:

F=KU

0<x _min ≤x _i ≤x _max ≤1

In the formula, the objective function C is the overall compliance of the topology structure, F is the force vector, U is the displacement array, K is the total stiffness matrix of the topology structure, x _i is the relative density of the element, u _i is the node displacement, k _i , k0 is the element stiffness matrix, V is the optimized structural volume, V ^* is the structural volume constraint, _V0 is the initial volume of the entire design domain, _f is the optimized volume ratio, _vi is the optimized element volume, x _min and x _max is the minimum limit value and maximum limit value of the relative density of cells, and N is the total number of discrete elements in the topology.

Further, after the filtered node sensitivity is mapped to the element sensitivity of the second structure topology optimization model to obtain the third structure topology optimization model, before updating the design variables, the third structure topology optimization model is updated according to the dichotomy method. , until the third structural topology optimization model satisfies the preset volume constraints.

Further, according to the discrete method of finite element analysis, an equivalent linear equation system of Helmholtz-type partial differential equations is constructed, specifically:

Taking the nodal sensitivity filter as the solution of the Helmholtz-type differential equation with Neumann boundary conditions, the Helmholtz-type differential equation is discretized by finite element to obtain its equivalent linear equation system.

Further, the node sensitivity filtering result of the equivalent linear equation system is calculated according to the multi-grid preprocessing conjugate gradient method, and the filtered node sensitivity is mapped to the unit sensitivity of the second structural topology optimization model to obtain the No. Three-structure topology optimization model, specifically:

According to the multigrid V-loop preprocessing method, the condition number of the coefficient matrix of the equivalent linear equation system is reduced to obtain the preprocessing linear equation system. In the preprocessing method, the multigrid is used for the method containing the preprocessing operator. algorithm solution;

The preprocessing linear equation system is solved according to the conjugate gradient algorithm, the node sensitivity filtering result of the preprocessing linear equation system is obtained, and the node sensitivity filtering result is mapped to the unit sensitivity filtering result of the second structural topology optimization model , the third structural topology optimization model is obtained.

Another embodiment of the present invention provides a PDE-based topology optimization device, including a topology model initialization module, a flexible topology establishment module, a finite element analysis module, a sensitivity filtering module, a design variable update module, and an optimization result detection module;

The topology model initialization module is used to initialize the parameters of the first structural topology optimization model, and define the design area, constraints and loads of the structural topology optimization model;

The establishing flexible topology module is used to divide the finite element network according to the design area of the first structural topology optimization model to obtain the corresponding finite element model, and stop dividing the finite element network until the finite element model meets the first preset condition ; On the basis of the finite element model, establish a SIMP flexible structure topology optimization model;

The finite element analysis module is used to perform finite element analysis on the SIMP flexible structure topology optimization model, and calculate the rigidity matrices of each element of the SIMP flexible structure topology optimization model according to each constraint condition of the SIMP flexible structure topology optimization model , the overall stiffness matrix, node displacement, flexibility and sensitivity to obtain the second structural topology optimization model;

The sensitivity filtering module is used to construct an equivalent linear equation system of Helmholtz type partial differential equations according to the discrete method of finite element analysis, and calculate the node sensitivity filtering result of the equivalent linear equation system according to the multigrid preprocessing conjugate gradient method, mapping the filtered node sensitivity to the element sensitivity of the second structure topology optimization model to obtain a third structure topology optimization model;

The updating design variable module is used for updating the design variables of the third structural topology optimization model according to the OC criterion;

The optimization result detection module is configured to repeat the operation process of the finite element analysis module, the sensitivity filtering module and the updating design variable module for the third structural topology optimization model until the third structural topology optimization model satisfies a preset convergence After the conditions are met, the third structural topology optimization model is output.

Further, the SIMP flexible structure topology optimization model is expressed as:

F=KU

0<x _min ≤x _i ≤x _max ≤1

In the formula, the objective function C is the overall compliance of the topology structure, F is the force vector, U is the displacement array, K is the total stiffness matrix of the topology structure, x _i is the relative density of the element, _ui is the node displacement, k _i , k0 is the element stiffness matrix, V is the optimized structural volume, V ^* is the structural volume constraint, _V0 is the initial volume of the entire design domain, _f is the optimized volume ratio, _vi is the optimized element volume, x _min and x _max is the minimum limit value and maximum limit value of the relative density of cells, and N is the total number of discrete elements in the topology.

Further, after the filtered node sensitivity is mapped to the element sensitivity of the second structure topology optimization model to obtain the third structure topology optimization model, before updating the design variables, the third structure topology optimization model is updated according to the dichotomy method. , until the third structural topology optimization model satisfies the preset volume constraints.

Further, an equivalent linear equation system of Helmholtz type partial differential equations is constructed according to the discrete method of finite element analysis, and the node sensitivity filtering result of the equivalent linear equation system is calculated according to the multi-grid preprocessing conjugate gradient method, and the filtered The node sensitivity is mapped to the element sensitivity of the second structure topology optimization model to obtain a third structure topology optimization model, which is specifically:

The nodal sensitivity filter is taken as the solution of the Helmholtz-type differential equation with Neumann boundary conditions, and its equivalent linear equation system is obtained by the finite element discretization of the Helmholtz-type differential equation;

According to the multigrid V-loop preprocessing method, the condition number of the coefficient matrix of the equivalent linear equation system is reduced to obtain the preprocessing linear equation system. In the preprocessing method, the multigrid is used for the method containing the preprocessing operator. algorithm solution;

The preprocessing linear equation system is solved according to the conjugate gradient algorithm, the node sensitivity filtering result of the preprocessing linear equation system is obtained, and the node sensitivity filtering result is mapped to the unit sensitivity filtering result of the second structural topology optimization model , the third structural topology optimization model is obtained.

Yet another embodiment of the present invention provides a readable storage medium, where the readable storage medium includes a stored computer program, and when the computer program is executed, controls a device where the readable storage medium is located to perform any one of the methods of the present invention The PDE-based topology optimization method described in this embodiment.

The embodiment of the present invention has the following beneficial effects:

The present invention provides a PDE-based topology optimization method, device and storage medium. The method obtains a finite element model by dividing the design area of the first structural topology optimization model into a finite element network, and then adds the finite element model to the finite element model. On the basis of , after establishing the SIMP flexible structure topology optimization model, the finite element analysis is performed on the SIMP flexible structure topology optimization model to obtain a second structure topology optimization model; for the second structure topology optimization model, the structure is discrete by finite element. The equivalent linear equations of the Helmholtz-type partial differential equations, and then the equivalent linear equations are solved according to the multigrid preprocessing conjugate gradient method, that is, the equivalent linear equations are calculated according to the multigrid preprocessing conjugate gradient method. The node sensitivity filtering result of the equation system is mapped to the element sensitivity filtering, and the optimized third structure topology optimization model is obtained. In this application, by performing multiple finite element analysis on the structural topology optimization model, constructing the equivalent linear equation system of the Helmholtz type partial differential equations discretized by the finite element for the structural topology optimization model, and calculating according to the conjugate gradient method of multi-grid preprocessing, etc. The node sensitivity filtering result of the valence linear equation system is obtained by mapping the element sensitivity filtering result, which greatly reduces the time of structural topology optimization and improves the efficiency of structural topology optimization.

Description of drawings

1 is a schematic flowchart of a PDE-based topology optimization method provided by an embodiment of the present invention;

2 is a schematic structural diagram of a PDE-based topology optimization device provided by an embodiment of the present invention;

FIG. 3 is a schematic diagram of the principle of multi-grid V-loop preprocessing of the PDE-based topology optimization method according to an embodiment of the present invention.

Detailed ways

The technical solutions in the present invention will be clearly and completely described below with reference to the accompanying drawings of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, but not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the protection scope of the present invention.

As shown in Figure 1, a PDE-based topology optimization method provided by an embodiment of the present invention includes the following steps:

Step S101: Initialize the parameters of the first structural topology optimization model, and define the design area, constraint conditions and loads of the structural topology optimization model.

Step S102: dividing a finite element network according to the design area of the first structural topology optimization model to obtain a corresponding finite element model, until the finite element model meets the first preset condition, then stop dividing the finite element network; Based on the meta-model, a topology optimization model of SIMP flexible structure is established.

As one of the embodiments, taking the minimum flexibility (that is, the maximum stiffness) of the first structural topology optimization model as the goal, the volume of the first structural topology optimization model being smaller than the preset threshold is a constraint condition, and the SIMP method is used as the interpolation method, and the 1 represents the solid part of the object, 0 represents the hollow part of the object, and the relative density feature of the unit is represented by the continuous function x _i ^p . The SIMP flexible structure topology optimization model is expressed as:

F=KU

0<x _min ≤x _i ≤x _max ≤1

In the formula, the objective function C is the overall compliance of the topology structure, F is the force vector, U is the displacement array, K is the total stiffness matrix of the topology structure, x _i is the relative density of the element, _ui is the node displacement, k _i , k0 is the element stiffness matrix, V is the optimized structural volume, V ^* is the structural volume constraint, _V0 is the initial volume of the entire design domain, _f is the optimized volume ratio, _vi is the optimized element volume, x _min and x _max is the minimum limit value and maximum limit value of the relative density of cells, and N is the total number of discrete elements in the topology.

The SIMP method is the power law, and its material interpolation model uses density as the design variable, and the relationship between the elastic modulus of the element material and the density satisfies the following formula:

Among them, E ₀ is the elastic modulus of solid isotropic material, and E _i is the elastic modulus after interpolation of element i.

Step S104: Finite element analysis: finite element analysis is performed on the SIMP flexible structure topology optimization model, and each element rigidity matrix of the SIMP flexible structure topology optimization model is calculated according to each constraint condition of the SIMP flexible structure topology optimization model, The overall stiffness matrix, node displacement, compliance and sensitivity are used to obtain the second structural topology optimization model.

Step S105: Sensitivity filtering: construct an equivalent linear equation system of Helmholtz-type partial differential equations according to the discrete method of finite element analysis, calculate the node sensitivity filtering result of the equivalent linear equation system according to the multi-grid preprocessing conjugate gradient method, and use The filtered node sensitivities are mapped to the element sensitivities of the second structure topology optimization model to obtain a third structure topology optimization model.

As one of the embodiments, step S105 includes the following sub-steps:

Sub-step S1051 : express the sensitivity filter as an implicit solution of the Helmholtz-type differential equation (2) with Neumann boundary conditions, and obtain the equivalent linear equation system (3) by finite element discretization of the Helmholtz-type differential equation (2).

x是单元相对密度，

和

是过滤前后的灵敏度，

是边界的法向量，K_F是标量问题的标准有限元刚度矩阵，T_F是将单元相对密度x映射到具有节点值的向量的矩阵，

是过滤区域内的节点表示。R_min是长度参数，用来确定过滤区域，它与标准的过滤半径r_min之间满足:in

x is the cell relative density,

and

is the sensitivity before and after filtering,

is the normal vector of the boundary, K _F is the standard finite element stiffness matrix for scalar problems, T _F is the matrix mapping the element relative density x to a vector with nodal values,

is the node representation within the filter area. R _min is a length parameter, used to determine the filter area, which satisfies the standard filter radius r _min :

Sub-step S1052: reduce the condition number of the coefficient matrix of the equivalent linear equation system (3) according to the multi-grid V-loop preprocessing method to obtain the preprocessing linear equation system (4), in the preprocessing method, The method of preprocessing operators is solved by a multigrid algorithm.

where M ^-1 is the preprocessing matrix

In this embodiment, the design area of the second structural topology optimization model is divided into nelx*nely unit small rectangles, then the dimension of the coefficient matrix K _F of the equivalent linear equation system of the Helmholtz equation discrete by the finite element The number is (nelx+1)(nely+1)*(nelx+1)(nely+1). With the continuous densification of the grid, the condition number of the coefficient matrix K _F also becomes larger, and sometimes it is even an ill-conditioned matrix. At this time, using the conventional solution method will greatly reduce the optimization efficiency of the model. Therefore, the embodiment of the present invention adopts the multi-grid V-cycle preprocessing method. Using multigrid as the preprocessing method refers to using the multigrid algorithm to solve the method containing the preprocessing operator in the preprocessing method.

Specifically, the inverse of the non-singular matrix M is multiplied on the left and right sides of the equation to obtain the pre-processed linear equation system (4). Then use the conjugate gradient algorithm to solve the preprocessing linear equations (4), which is the preprocessing conjugate gradient algorithm; in the traditional method, when using the preprocessing conjugate gradient algorithm, a specific preprocessing matrix M needs to be given, In this embodiment, the coefficient matrix is used as the preprocessing operator by using multiple grids where preprocessing is required in each conjugate gradient algorithm, that is, multigrid preprocessing.

The multi-grid preprocessing is specifically: first, using 2 to 3 smooth preprocessing techniques on the fine grid layer of the second structural topology optimization model to eliminate high-frequency components (the high-frequency components include wavelengths); The approximate solution of the grid equation is obtained; secondly, the limit operator is used to limit the error of the low-frequency component (the low-frequency component includes longer wavelength and low waveform frequency) to the coarser grid, so that the low frequency The component re-oscillates to get the high-frequency component; smooth it on the thicker grid, make the grid thicker layer by layer until the thickest layer, eliminate all errors, and get it on the thickest grid layer error exact solution;

Then use the interpolation operator to interpolate the exact solution to the fine grid layer step by step, and add the error exact solution and the original approximate solution to get the solution of the grid equation;

Finally, it is judged whether the accuracy requirements of the error solution and the limit of the maximum number of iterations are met. If so, the solution of the equation system is obtained. If it is not satisfied, continue to perform several iterations until the accuracy requirements of the error solution and the limit of the maximum number of iterations are satisfied, and finally the solution of the equation system is obtained.

As shown in FIG. 3 , the principle of the multi-grid V-loop preprocessing is as follows: using an iterative method to solve the linear equation system to obtain an approximate solution, and using the characteristics of different waveform frequencies of errors in different grids, the thinnest layer of grid The error on the grid is passed to the coarse grid through the limit operator, and the error is eliminated step by step until the error is completely eliminated. Then use interpolation to return the exact solution to the finest grid step by step to get the exact solution.

Therefore, when using the technical solution of the present invention to optimize the topology, the sensitivity filtering can be quickly completed, and even if the grid size increases and the grid division becomes denser, the optimization speed of the model will not be reduced, resulting in an ill-conditioned topology model.

Sub-step S1053: Solve the preprocessing linear equation system according to the conjugate gradient algorithm, map the node sensitivity filtering to the unit sensitivity filtering by mapping according to the node sensitivity filtering solution result of the preprocessing linear equation system, and perform the second structure The topology optimization model performs sensitivity filtering to obtain a third structural topology optimization model.

分别代表(4)式中的分量。As one of the embodiments, the process of solving the preprocessing linear equation system (4) according to the conjugate gradient algorithm x=mgcg(A, b, x ₀ ) is as follows, where: A=M ⁻¹ K _F , b= M ^-1 (T _F x),

respectively represent the components in (4).

enter:

Given a symmetric positive definite matrix A;

Given the right-hand term b;

Given an initial value of iteration x ₀ ∈R ⁿ ;

output:

the solution x of the system of preprocessed linear equations;

1 Calculate residuals on the finest mesh

2for i=1,2,...do;

3 Multigrid V loop to solve linear equations Az ₀ =r ₀ ;

4 initialize p ₀ =z ₀ ;

并根据多重网格V循环预处理方法求解z_i-1；5 Calculation step size

And solve zi _-1 according to the multigrid V cycle preprocessing method;

6 update the approximate solution x _i =x _i-1 +α _i p _i-1 ;

7. Update residual r _i =r _i-1 -α _i Ap _i-1 ;

8. According to the multigrid V, the linear equation system Az _i =r _i is solved cyclically;

9 Calculate the gradient direction coefficient

10 Set a new search direction p _i =z _i +β _i p _i-1 ;

11end;

12 until convergence;

As one of the embodiments: after step S105 and before step S106, update the Lagrangian multiplier of the third structural topology optimization model according to the dichotomy method until the third structural topology optimization model satisfies a preset volume Restrictions.

Step S106: Update design variables: update the design variables of the third structural topology optimization model according to the OC criterion.

Step S107: Repeat the operations of finite element analysis, sensitivity filtering and updating design variables for the third structural topology optimization model until the third structural topology optimization model satisfies a preset convergence condition, and output the third structural topology optimization model. Structural topology optimization model.

As an example, when the left side of the first structural topology optimization model is fixed, the lower side of the right side is a short cantilever beam, and is subjected to a vertical downward load F=1N. The design area is 480mm long and 160mm high, the volume ratio is 0.5, the Young's modulus of elasticity of the material is 1, and the Poisson's ratio is 0.3. During optimization, the penalty factor is 3, the moving limit move=0.2, and the filter radius is 3, 6, 9, 12, and 15 mm for experiments. It is found that under the condition of a certain filtering radius, the overall optimization time of the embodiment of the present invention is shorter than that of the traditional topology optimization solution, that is, the filtering efficiency during topology optimization can be improved by using the technical solution of the present invention. At the same time, under the condition of a certain filter radius, the flexibility value shows an increasing trend and the topology optimization time shows a decreasing trend with the increase of the filter radius in the embodiment of the present invention, that is, the accuracy of the embodiment of the present invention is relatively high, and the error is relatively high. Small.

As shown in FIG. 2 , another embodiment of the present invention provides a PDE-based topology optimization device, including a topology model initialization module, a flexible topology establishment module, a finite element analysis module, a sensitivity filtering module, a design variable update module, and an optimization result detection module ;

The topology model initialization module is used to initialize the parameters of the first structural topology optimization model, and define the design area, constraints and loads of the structural topology optimization model;

The establishing flexible topology module is used to divide the finite element network according to the design area of the first structural topology optimization model to obtain the corresponding finite element model, and stop dividing the finite element network until the finite element model meets the first preset condition ; On the basis of the finite element model, establish a SIMP flexible structure topology optimization model;

The finite element analysis module is used to perform finite element analysis on the SIMP flexible structure topology optimization model, and calculate the rigidity matrices of each element of the SIMP flexible structure topology optimization model according to each constraint condition of the SIMP flexible structure topology optimization model , the overall stiffness matrix, node displacement, flexibility and sensitivity to obtain the second structural topology optimization model;

The sensitivity filtering module is used to construct an equivalent linear equation system of Helmholtz type partial differential equations according to the discrete method of finite element analysis, and calculate the node sensitivity filtering result of the equivalent linear equation system according to the multigrid preprocessing conjugate gradient method, mapping the filtered node sensitivity to the element sensitivity of the second structure topology optimization model to obtain a third structure topology optimization model;

The updating design variable module is used for updating the design variables of the third structural topology optimization model according to the OC criterion;

The optimization result detection module is configured to repeat the operation process of the finite element analysis module, the sensitivity filtering module and the updating design variable module for the third structural topology optimization model until the third structural topology optimization model satisfies a preset convergence After the conditions are met, the third structural topology optimization model is output.

As one of the embodiments, the SIMP flexible structure topology optimization model is expressed as:

F=KU

0＜x _min ≤x _i ≤x _max ≤1

In the formula, the objective function C is the overall compliance of the topology structure, F is the force vector, U is the displacement array, K is the total stiffness matrix of the topology structure, x _i is the relative density of the element, _ui is the node displacement, k _i , k0 is the element stiffness matrix, V is the optimized structural volume, V ^* is the structural volume constraint, _V0 is the initial volume of the entire design domain, _f is the optimized volume ratio, _vi is the optimized element volume, x _min and x _max is the minimum limit value and maximum limit value of the relative density of cells, and N is the total number of discrete elements in the topology.

As one of the embodiments, after the filtered node sensitivity is mapped to the element sensitivity of the second structure topology optimization model to obtain a third structure topology optimization model, before updating design variables, the third structure topology optimization model is updated according to the dichotomy method. Lagrange multipliers of the structural topology optimization model until the third structural topology optimization model satisfies the preset volume constraint conditions.

As one of the embodiments, an equivalent linear equation system of Helmholtz-type partial differential equations is constructed according to the discrete method of finite element analysis, and the node sensitivity filtering result of the equivalent linear equation system is calculated according to the multigrid preprocessing conjugate gradient method, The filtered node sensitivity is mapped to the element sensitivity of the second structure topology optimization model to obtain a third structure topology optimization model, specifically:

The nodal sensitivity filter is taken as the solution of the Helmholtz-type differential equation with Neumann boundary conditions, and its equivalent linear equation system is obtained by the finite element discretization of the Helmholtz-type differential equation;

According to the multigrid V-loop preprocessing method, the condition number of the coefficient matrix of the equivalent linear equation system is reduced to obtain the preprocessing linear equation system. In the preprocessing method, the multigrid is used for the method containing the preprocessing operator. algorithm solution;

The preprocessing linear equation system is solved according to the conjugate gradient algorithm, the node sensitivity filtering result of the preprocessing linear equation system is obtained, and the node sensitivity filtering result is mapped to the unit sensitivity filtering result of the second structural topology optimization model , the third structural topology optimization model is obtained.

On the basis of the above method item embodiment, the present invention correspondingly provides the readable storage medium item embodiment;

Another embodiment of the present invention provides a readable storage medium, where the readable storage medium includes a stored computer program, and when the computer program is executed, controls the device where the readable storage medium is located to perform any one of the methods of the present invention. Item Method Item The PDE-based topology optimization method described in the embodiment.

Exemplarily, the computer program may be divided into one or more modules, and the one or more modules are stored in the memory and executed by the processor to accomplish the present invention. The one or more modules may be a series of computer program instruction segments capable of performing specific functions, and the instruction segments are used to describe the execution process of the computer program in the terminal device.

The terminal device may be a computing device such as a desktop computer, a notebook, a palmtop computer, and a cloud server. The terminal device may include, but is not limited to, a processor and a memory.

The processor may be a central processing unit (Central Processing Unit, CPU), or other general-purpose processors, digital signal processors (Digital Signal Processor, DSP), application specific integrated circuits (Application Specific Integrated Circuit, ASIC), off-the-shelf processors Programmable Gate Array (Field-Programmable Gate Array, FPGA) or other programmable logic devices, discrete gate or transistor logic devices, discrete hardware components, etc. The general-purpose processor can be a microprocessor or the processor can also be any conventional processor, etc. The processor is the control center of the terminal device, and uses various interfaces and lines to connect various parts of the entire terminal device.

The memory can be used to store the computer program and/or module, and the processor implements the terminal by running or executing the computer program and/or module stored in the memory and calling the data stored in the memory various functions of the device. The memory may mainly include a stored program area and a stored data area, wherein the stored program area may store an operating system, an application program required for at least one function (such as a sound playback function, an image playback function, etc.), etc.; the storage data area may store Data (such as audio data, phonebook, etc.) created according to the usage of the mobile phone, etc. In addition, the memory may include high-speed random access memory, and may also include non-volatile memory such as hard disk, internal memory, plug-in hard disk, Smart Media Card (SMC), Secure Digital (SD) card , a flash memory card (Flash Card), at least one magnetic disk storage device, flash memory device, or other volatile solid-state storage device.

Wherein, if the modules/units integrated in the terminal device are implemented in the form of software functional units and sold or used as independent products, they may be stored in a computer-readable storage medium (ie, the above-mentioned readable storage medium). Based on this understanding, the present invention can implement all or part of the processes in the methods of the above embodiments, and can also be completed by instructing relevant hardware through a computer program, and the computer program can be stored in a computer-readable storage medium. When the program is executed by the processor, the steps of the foregoing method embodiments can be implemented. Wherein, the computer program includes computer program code, and the computer program code may be in the form of source code, object code, executable file or some intermediate form, and the like. The computer-readable medium may include: any entity or device capable of carrying the computer program code, a recording medium, a U disk, a removable hard disk, a magnetic disk, an optical disk, a computer memory, a read-only memory (ROM, Read-Only Memory) , Random Access Memory (RAM, Random Access Memory), electric carrier signal, telecommunication signal and software distribution medium, etc.

It should be noted that the device embodiments described above are only schematic, wherein the units described as separate components may or may not be physically separated, and the components displayed as units may or may not be physical unit, that is, it can be located in one place, or it can be distributed over multiple network units. Some or all of the modules may be selected according to actual needs to achieve the purpose of the solution in this embodiment. In addition, in the drawings of the apparatus embodiments provided by the present invention, the connection relationship between the modules indicates that there is a communication connection between them, which may be specifically implemented as one or more communication buses or signal lines. Those of ordinary skill in the art can understand and implement it without creative effort. The above are the preferred embodiments of the present invention. It should be pointed out that for those skilled in the art, without departing from the principles of the present invention, several improvements and modifications can be made, and these improvements and modifications may also be regarded as It is the protection scope of the present invention. Those of ordinary skill in the art can understand that the realization of all or part of the processes in the above embodiments can be accomplished by instructing relevant hardware through a computer program, and the program can be stored in a computer-readable storage medium, and the program is During execution, the processes of the above-mentioned embodiments may be included. The storage medium may be a magnetic disk, an optical disk, a read-only memory (Read-Only Memory, ROM), or a random access memory (Random Access Memory, RAM) or the like.

Claims (10)

1. A topology optimization method based on PDE is characterized by comprising the following steps:

initializing parameters of a first structural topological optimization model, and defining a design area, constraint conditions and load of the structural topological optimization model;

dividing a finite element network aiming at the design area of the first structural topological optimization model to obtain a corresponding finite element model, and stopping dividing the finite element network until the finite element model meets a first preset condition; establishing an SIMP flexible structure topological optimization model on the basis of the finite element model;

finite element analysis: carrying out finite element analysis on the SIMP flexible structure topological optimization model, and calculating each unit rigidity matrix, total rigidity matrix, node displacement, flexibility and sensitivity of the SIMP flexible structure topological optimization model according to each constraint condition of the SIMP flexible structure topological optimization model to obtain a second structure topological optimization model;

and (3) sensitivity filtration: constructing an equivalent linear equation set of a Helmholtz type partial differential equation according to a finite element analysis discrete method, calculating a node sensitivity filtering result of the equivalent linear equation set according to a multiple grid preprocessing conjugate gradient method, and mapping the filtered node sensitivity to the unit sensitivity of the second structure topology optimization model to obtain a third structure topology optimization model;

updating design variables: updating the design variables of the third structural topology optimization model according to OC criteria;

and repeating the operation processes of finite element analysis, sensitivity filtering and design variable updating for the third structural topological optimization model until the third structural topological optimization model meets a preset convergence condition, and outputting the third structural topological optimization model.

2. The PDE-based topology optimization method of claim 1, wherein the SIMP flexible structure topology optimization model is represented as:

F＝KU

0<x_min≤x_i≤x_max≤1

in the formula, the target function C is the total flexibility of the topological structure, F is a force vector, U is a displacement array, K is a total rigidity matrix of the topological structure, and x_iIs the relative density of the unit, u_iIs node displacement, k_i,k₀Is a unit stiffness matrix, V is the optimized structure volume, V^*For structural volume constraints, V₀Initial volume for the entire design domain, f is the optimized volume ratio, v_iFor optimized cell volume, x_minAnd x_maxThe minimum limit and the maximum limit of the relative density of the units, and N is the total number of discrete units of the topological structure.

3. The PDE-based topology optimization method of claim 2, wherein after mapping the filtered node sensitivities to the unit sensitivities of the second structural topology optimization model to obtain a third structural topology optimization model, and before updating design variables, updating lagrangian multipliers of the third structural topology optimization model according to a bisection method until the third structural topology optimization model meets a preset volume constraint condition.

4. The PDE-based topology optimization method of claim 3, wherein an equivalent linear equation set of the Helmholtz-type partial differential equation is constructed according to a finite element analysis discretization method, specifically:

and filtering the node sensitivity to be used as a solution of a Helmholtz type differential equation with Neumann boundary conditions, and dispersing the Helmholtz type differential equation by finite elements to obtain an equivalent linear equation set of the Helmholtz type differential equation.

5. The PDE-based topology optimization method according to any one of claims 1 to 4, wherein a node sensitivity filtering result of the equivalent linear equation set is calculated according to a multiple mesh preprocessing conjugate gradient method, and the filtered node sensitivity is mapped to a unit sensitivity of the second structural topology optimization model to obtain a third structural topology optimization model, specifically:

reducing the condition number of a coefficient matrix of the equivalent linear equation set according to a multiple grid V circulation preprocessing method to obtain a preprocessing linear equation set, and solving the method containing a preprocessing operator in the preprocessing method by using a multiple grid algorithm;

and solving the preprocessed linear equation set according to a conjugate gradient algorithm to obtain a node sensitivity filtering result of the preprocessed linear equation set, and mapping the node sensitivity filtering result to a unit sensitivity filtering result of the second structural topological optimization model to obtain a third structural topological optimization model.

6. A topology optimization device based on PDE is characterized by comprising a topology model initialization module, a flexible topology establishment module, a finite element analysis module, a sensitivity filtering module, a design variable updating module and an optimization result detection module;

the topology model initialization module is used for initializing parameters of a first structural topology optimization model, and defining a design area, constraint conditions and load of the structural topology optimization model;

the flexible topology establishing module is used for dividing a finite element network aiming at the design region of the first structural topology optimization model to obtain a corresponding finite element model, and stopping dividing the finite element network until the finite element model meets a first preset condition; establishing an SIMP flexible structure topological optimization model on the basis of the finite element model;

the finite element analysis module is used for carrying out finite element analysis on the SIMP flexible structure topological optimization model, and calculating each unit rigidity matrix, the overall rigidity matrix, the node displacement, the flexibility and the sensitivity of the SIMP flexible structure topological optimization model according to each constraint condition of the SIMP flexible structure topological optimization model to obtain a second structure topological optimization model;

the sensitivity filtering module is used for constructing an equivalent linear equation set of a Helmholtz type partial differential equation according to a finite element analysis discrete method, calculating a node sensitivity filtering result of the equivalent linear equation set according to a multiple grid pretreatment conjugate gradient method, and mapping the filtered node sensitivity to the unit sensitivity of the second structure topology optimization model to obtain a third structure topology optimization model;

the design variable updating module is used for updating the design variables of the third structural topology optimization model according to OC criteria;

the optimization result detection module is used for repeating the operation processes of the finite element analysis module, the sensitivity filtering module and the design variable updating module aiming at the third structural topology optimization model until the third structural topology optimization model meets a preset convergence condition, and then outputting the third structural topology optimization model.

7. The PDE-based topology optimization device of claim 6, wherein the SIMP flexible structure topology optimization model is expressed as:

F＝KU

0<x_min≤x_i≤x_max≤1

in the formula, the target function C is the total flexibility of the topological structure, F is a force vector, U is a displacement array, K is a total rigidity matrix of the topological structure, and x_iIs the relative density of the unit, u_iIs node displacement, k_i,k₀Is a matrix of cell stiffness, V is the optimized structural volume, V^*For structural volume constraints, V₀Initial volume for the entire design domain, f is the optimized volume ratio, v_iFor optimized cell volume, x_minAnd x_maxThe minimum limit and the maximum limit of the relative density of the units, and N is the total number of discrete units of the topological structure.

8. The PDE-based topology optimization device of claim 7, wherein after mapping the filtered node sensitivities to the unit sensitivities of the second structure topology optimization model to obtain a third structure topology optimization model, and before updating design variables, the lagrangian multiplier of the third structure topology optimization model is updated according to a bisection method until the third structure topology optimization model meets a preset volume constraint condition.

9. The PDE-based topology optimization device according to any one of claims 6 to 8, wherein an equivalent linear equation set of the Helmholtz-type partial differential equation is constructed according to a finite element analysis discretization method, a node sensitivity filtering result of the equivalent linear equation set is calculated according to a multiple mesh preprocessing conjugate gradient method, and the filtered node sensitivity is mapped to a unit sensitivity of the second structural topology optimization model to obtain a third structural topology optimization model, specifically:

filtering the node sensitivity to be used as a solution of a Helmholtz type differential equation with Neumann boundary conditions, and dispersing the Helmholtz type differential equation by finite elements to obtain an equivalent linear equation set of the Helmholtz type differential equation;

reducing the condition number of a coefficient matrix of the equivalent linear equation set according to a multiple grid V circulation preprocessing method to obtain a preprocessing linear equation set, and solving the method containing a preprocessing operator in the preprocessing method by using a multiple grid algorithm;

and solving the preprocessed linear equation set according to a conjugate gradient algorithm to obtain a node sensitivity filtering result of the preprocessed linear equation set, and mapping the node sensitivity filtering result to a unit sensitivity filtering result of the second structural topological optimization model to obtain a third structural topological optimization model.

10. A readable storage medium, comprising a stored computer program which, when executed, controls an apparatus in which the readable storage medium is located to perform the method of PDE-based topology optimization as defined in any one of claims 1 to 5.

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