A multi-material stress-constrained continuum step length topology optimization method

Abstract

The present application relates to the technical field of structural design and optimization, in particular to a continuity step length topology optimization method based on multi-material stress constraint, which comprises parameter initialization, material attribute interpolation and finite element calculation, interpolation calculation of each unit stress constraint, stress constraint relaxation criterion construction, incremental Lagrange stress constraint integration to form an optimization problem and calculate the sensitivity of design variables, moving progressive line method to update multiple design variables, iterative optimization and step length parameter update until the convergence condition is met. The present application is applicable to more material types and different stress criterion combinations, can meet the continuity step length minimum value and control the step length change rate in the specified iteration round, the output structure is reasonable and reasonable, the boundary is clear and the structure is novel, the design variable change in the iteration process is stable, and the abnormal numerical oscillation problem is solved.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of structural design and optimization, in particular to a continuity step topology optimization method based on multi-material stress constraint. BACKGROUND

[0002] With the rapid development of computer technology and finite element theory, computer-aided design has become an indispensable tool in the design of the construction industry. Among them, topology optimization as a powerful structural design automation tool, its goal is to find the optimal material distribution under the given design domain, load, boundary conditions and constraints, to achieve specific performance goals. With the development of advanced manufacturing technology (such as additive manufacturing), it is possible to design complex structures containing multiple materials with different properties (such as high strength, light weight, toughness, thermal conductivity, etc.), which has spawned a hot wave of multi-material topology optimization. Multi-material structures can make full use of the characteristics of different materials (such as high strength, light weight, toughness, thermal conductivity, etc.), and design innovative components with performance far superior to single-material structures. Expanding topology optimization to the multi-material field and considering actual engineering constraints, especially stress constraints, will involve different stress criteria for different materials. Traditional stress-constrained topology optimization problems mostly use single von-Mises stress solution, which is obviously biased with different material strength criteria, and the abnormal numerical oscillation problem in stress optimization problems needs to be improved. SUMMARY

[0003] The purpose of the present application is to provide a solution that not only solves the multi-material and multi-stress criterion constraint problem in the topology optimization process, but also effectively controls the numerical oscillation problem in the optimization process.

[0004] In order to achieve the above purpose, the present application provides a continuity step topology optimization method based on multi-material stress constraint, comprising:

[0005] Establish a finite element model according to the actual engineering, and initialize parameters;

[0006] Establish a physical field for multi-material design variables, and the elastic modulus of a single material is interpolated to form an interpolated penalty form of elastic modulus that can represent the intermediate density value of multiple materials;

[0007] Different stress criteria exist for different materials to describe material stress behavior, including Von-Mises and Drucker-Prager two strength criteria, and element stress interpolation is performed to represent the intermediate density stress value;

[0008] Based on the singularity problem of stress constraint, a stress constraint function of stress constraint relaxation criterion is constructed;

[0009] The relative density method in the SIMP method is taken as a physical model, the density of the design domain design variable is taken as an objective function, the stress of each unit in the design domain satisfies a set value as a constraint, an optimization criterion is set to solve a nonlinear optimization problem, and optimization iteration is performed; a design variable optimization allowed step length which changes in the form of a continuous function with iteration rounds is constructed, and the optimization iteration obtains a structure which satisfies the design domain stress intensity and volume minimization, and makes the optimization process design variable convergence process smooth.

[0010] Further, the elastic modulus in the form of interpolation penalty representing the intermediate density value of multiple materials is expressed by interpolation of multiple materials, including and two design variables representing materials; for representing the proportion of the corresponding unit entity material, when 1 indicates that the unit is an entity, and 0 indicates that the unit is an empty unit; representing the proportion of the unit material, when 1 indicates that the unit is filled with material one, and 0 indicates that the unit is filled with material two.

[0011] Further, the interpolation expression of the elastic modulus is:

[0012] ;

[0013] wherein, represents the elastic modulus, and are the Young's modulus of material one and material two respectively, is a constant for preventing numerical singularity, and are SIMP penalty parameters respectively.

[0014] Further, material one and material two adopt Von-Mises and Drucker-Prager two strength criteria respectively, and the expression is:

[0015] ;

[0016] ;

[0017] wherein, represents the strength criterion of material one, represents the strength criterion of material two, represents the entity stress vector, is the strength limit of von-Mises material, and are parameters under the Drucker-Prager strength criterion respectively, and:

[0018] ;

[0019] ;

[0020] and are the uniaxial compressive and tensile strength limits of the Drucker-Prager strength criterion material, respectively; is the deviatoric second invariant:

[0021] ;

[0022] where, is the stress vector of each element , is the auxiliary matrix for calculating the deviatoric second invariant, is a constant for avoiding numerical instability problem caused by negative number in the calculation of the square root.

[0023] Further, a SIMP form of the strength interpolation function is used to interpolate the stress of the elements, so as to consider the different strength criterion functions of the two materials and obtain an interpolated strength criterion:

[0024] ;

[0025] where, is the interpolated strength criterion considering the two materials and the two strength criteria, is the bulk stress vector of the material , is the constitutive matrix of the material , is the strain-displacement conversion matrix of the point , is the local displacement vector of the point , is the penalty term of the stress interpolation.

[0026] Further, the relaxation criterion is introduced to the constraint function and expressed as:

[0027] ;

[0028] where, represents the constraint function, is the global vector representation of , , is the global displacement vector, is the relaxation criterion, is the stress interpolation function, and the cubic polynomial can accelerate the convergence of the optimization iteration process, represents the total number of elements after meshing.

[0029] Further, the constraint function is used to constrain the objective function of optimization:

[0030]

[0031] wherein, is the volume fraction in the design domain, is the volume of element , is the global stiffness matrix derived from the interpolated elastic modulus, is the global load vector.

[0032] Further, the unconstrained incremental Lagrange method is used to solve the nonlinear optimization problem, the objective function of the incremental Lagrange method is composed of the penalty term of the original objective function and the constraint function, to integrate the objective function and the constraint function to form an unconstrained sub-optimization problem, the unconstrained optimization sub-problem is solved by the moving asymptote method in the gradient processing method, the moving asymptote is dynamically adjusted to construct the conservative convex approximation model of the objective function and the constraint function under the current design variable, and the new design variable is obtained by solving the model.

[0033] Further, the function form of the decay step control method is used in the iterative optimization, to control the allowable step to reach the predetermined conservative step in the specified round according to the calculation form, while ensuring the continuity of the step change.

[0034] The above scheme of the present application has the following beneficial effects:

[0035] The continuity step topology optimization method based on multi-material stress constraint provided by the present application obtains the multi-material attribute parameters based on the design variables in the interpolation form, expresses the multiple strength criteria of different materials through a reasonable interpolation function expression, forms the constraints of the optimization problem by relaxing the criteria, takes the volume minimization in the design domain as the objective function, integrates the objective function and the constraint function to form a series of unconstrained optimization problems by the incremental Lagrange method, solves the series of nonlinear optimization problems by the moving asymptote method of the continuity step, is suitable for more material types and different stress criterion combinations, can meet the minimum value of the continuity step in the specified iteration round and control the step change rate, the output structure is reasonable and novel, the design variable change in the iteration process is stable, and the abnormal numerical shock problem is solved.

[0036] Other beneficial effects of the present application will be described in detail in the subsequent specific embodiment part. BRIEF DESCRIPTION OF DRAWINGS

[0037] Figure 1 is the flowchart of the present application;​​

[0038] Figure 2 Design domain plot for case 1 of the present invention;

[0039] Figure 3 Optimization result plot for case 1 of the present invention, where (a) is the optimization result, (b) is the stress distribution of the optimization result;

[0040] Figure 4 Plot of the change of the average value of the design variables and the allowed step size during the convergence iteration process for case 1 of the present invention.

[0041] Figure 5 Design domain plot for case 2 of the present invention;

[0042] Figure 6 Optimization result plot for case 2 of the present invention, where (a) is the optimization result, (b) is the stress distribution of the optimization result;

[0043] Figure 7 Plot of the change of the average value of the design variables and the allowed step size during the convergence iteration process for case 2 of the present invention. DETAILED DESCRIPTION

[0044] Other advantages and embodiments of the disclosure will be disclosed in the following specific examples. Those of ordinary skill in the art can easily understand other advantages and benefits of the disclosure from the content disclosed in the specification. Obviously, the described embodiments are only a part of the embodiments of the disclosure, not all. The disclosure can also be implemented or applied by other different specific embodiments, and the details in the specification can be modified or changed based on different views and applications without departing from the spirit of the disclosure. It should be noted that the following embodiments and features in the embodiments can be combined with each other without conflict. Based on the embodiments in the disclosure, all other embodiments obtained by those of ordinary skill in the art without creative labor are within the scope of protection of the disclosure.

[0045] It should be noted that various aspects of the embodiments described below are within the scope of the appended claims. It should be apparent that the aspects described herein can be embodied in a wide variety of forms and that any specific structure and / or function described herein is merely illustrative. Based on the teachings provided herein one skilled in the art will appreciate that one or more aspects described herein can be used alone or in any combination of the described aspects. For example, devices and / or methods described herein can be used in combination with each other and / or can be used in combination with one or more other structures and / or functionalities described or not described herein.

[0046] It is also necessary to note that the drawings provided in the following embodiments only illustrate the basic concept of the present disclosure in a schematic manner, and only show the components related to the present disclosure in the drawings, not drawn according to the number, shape and size of the components in actual implementation, and the shape, number and ratio of each component in actual implementation can be a random change, and the component layout pattern can be more complex. In addition, in the following description, specific details are provided to facilitate a thorough understanding of the examples. However, those skilled in the art will understand that the aspects can be practiced without these specific details.

[0047] As shown in Figure 1 , the embodiment of the present application provides a multi-material stress constraint based continuous step length topology optimization method, which specifically comprises the following steps:

[0048] S1, according to the engineering practice, a finite element model is established, and the parameter initialization is carried out. The finite element model can be in two or three dimensions, and the method provided in the embodiment can be used for topology optimization of two or three dimensional finite element model.

[0049] S2, based on the physical model of SIMP (Solid Isotropic Material with Penalization) method, a physical field expressed by multi-material design variables is established, and the elastic modulus of a single material is interpolated to form an interpolated elastic modulus in the form of penalty, which can represent the intermediate density value of multiple materials.

[0050] In the embodiment, the expression of the elastic modulus is formed by multi-material interpolation, which mainly contains the materials represented by two design variables. and Among them, is used to represent the proportion of the unit solid material, which is mainly embodied as 1 when the unit is solid and 0 when the unit is empty; represents the proportion of the unit material, which is mainly embodied as 1 when the unit is filled with material 1 and 0 when the unit is filled with material 2. The interpolation expression of the elastic modulus is specifically:

[0051] (1)

[0052] Among them, represents the elastic modulus, and are the Young's modulus of material 1 and material 2 respectively, is a small constant used to prevent numerical singularity, and These are the SIMP penalty parameters, both of which are set to 3 in this embodiment. The elastic modulus is constructed using equation (1), making the expression easy to understand and effectively penalizing intermediate density units, thus making the optimization result close to the 0-1 distribution.

[0053] S3 uses different stress criteria to describe the stress behavior of different materials, including the Von-Mises and Drucker-Prager strength criteria, and performs element stress interpolation to represent the intermediate density stress value.

[0054] In this embodiment, we take the Von-Mises and Drucker-Prager strength criteria for materials 1 and 2 respectively as examples for illustration. Their expressions can be written as:

[0055] (2)

[0056] (3)

[0057] in, This indicates the strength criterion for material 1. This indicates the strength criterion for material 2. Represents the stress vector of the solid. It is the strength limit of von Mises materials. and These are the parameters under the Drucker-Prager strength criterion, and:

[0058] (4)

[0059] (5)

[0060] in, and These are the uniaxial compressive and tensile strength limits of materials according to the Drucker-Prager strength criterion. The second-order invariant of the deviatoric stress is expressed as follows:

[0061] (6)

[0062] in, Each unit The stress vector, It is an auxiliary matrix for calculating the second-order invariants of deviatoric stress. It is a small constant set to avoid numerical instability caused by negative square roots when calculating strength criteria.

[0063] Therefore, when different strength criteria exist within a multi-material stress-constrained topology optimization framework, a SIMP-type strength interpolation function can be used to simultaneously consider the different strength criterion functions of two materials, thus deriving the interpolated strength criterion:

[0064] (7)

[0065] in, To simultaneously consider two materials and two strength criteria, an interpolated strength criterion, It is a material The entity stress vector, It is a material constitutive matrix, It is a point The strain-displacement transformation matrix, It is a point The local displacement vector, This is the penalty term for stress interpolation; in this embodiment, a linear term is used, so it is set to 1.

[0066] S4. Based on the singularity problem of stress constraints, a stress constraint function for stress constraint relaxation criteria is constructed to alleviate the problem of abrupt stress increase at singularities.

[0067] It should be noted that, in this embodiment, when using the physical model of the SIMP method, all structural units are assumed to be composed of materials with a relative density between 0 and 1. and These are two design variables used to characterize the multiphase material within the design domain. Due to the singularity problem of stress constraints—that is, high stress concentration still exists in low-density regions—a relaxation criterion needs to be introduced into the constraint function, specifically expressed as:

[0068] (8)

[0069] in, Represents the constraint function. yes , The global vector representation, It is a global displacement vector. That is, the relaxation principle, when When the value is 0, this relaxation criterion can effectively make the element stress tend to 0, thereby achieving the effect of relaxing the element stress. This is the stress interpolation function, which includes a cubic polynomial to accelerate the convergence of the optimization iteration process. This represents the total number of cells after meshing.

[0070] At the same time, the objective function for optimization is constrained by the constraint function, namely:

[0071] (9)

[0072] where, is the volume fraction in the design domain, is the volume of the element is the global stiffness matrix derived from the interpolated elastic modulus, is the global load vector.

[0073] S5, the relative density method in the SIMP method is taken as the physical model, the density of the design variable in the design domain is taken as the objective function, the stress of each element in the design domain needs to meet the set value as the constraint, and the optimization criterion is used to solve the nonlinear optimization problem.

[0074] In this embodiment, the unconstrained incremental Lagrangian method is adopted. In the normalized incremental Lagrangian method, the initial problem is divided into a series of non-constrained optimization sub-problems and finally converges to the solution of the original problem. The objective function of the incremental Lagrangian method is composed of the original objective function and the penalty term of the constraint function, wherein the first The objective function of the kth iteration can be expressed as:

[0075] (10)

[0076] where

[0077] (11)

[0078] is the penalty term in the objective function, the normalization is formed by the number of elements N, is the virtual value of the design variable in the kth iteration, is the equivalent constraint form, written as:

[0079] (12)

[0080] where is the Lagrange multiplier estimate, is the quadratic penalty parameter, and its update in the iteration is written as:

[0081] (13)

[0082] where, is the update parameter, for example, it can be taken as 1.05, is the numerical upper limit to prevent numerical instability. The Lagrange multiplier update can be expressed as:

[0083] (14)

[0084] ​In this embodiment, this series of unconstrained optimization sub-problems are further solved by the method of moving asymptotes (MMA) in the gradient-based approach, which is a sequential convex approximation optimization algorithm based on gradient information. It dynamically adjusts the moving asymptotes to construct a conservative convex approximation model of the objective function and constraint functions under the current design variables, and solves the model to obtain new design variables, solving the unconstrained sub-problems generated by the incremental Lagrange method in each iteration k.

[0085] Specifically, in solving each round of incremental Lagrange sub-problems, the following approximate convex sub-problems are used to replace the original non-convex sub-problems:

[0086] (15)

[0087] Where, because the optimization process is to optimize two design variables synchronously, their expressions are the same, so the two design variables and are expressed as and respectively take and , and the parameters and are obtained by and , where is the allowed step size of each iteration. In addition:

[0088] (16)

[0089] (17)

[0090] (18)

[0091] Where , , are the derivatives of the unconstrained problem with respect to the design variables at the point.

[0092] The upper and lower asymptotes in the approximate convex sub-problems are expressed as:

[0093] (19)

[0094] When , the upper and lower asymptotes , are:

[0095] ​​​​​ (20)

[0096] When and , ; when there is:

[0097] (21)

[0098] The final updated design variable is:

[0099] (22)

[0100] wherein

[0101] (23)

[0102] It should be noted that after solving the nonlinear optimization problem in S5, there will still be different degrees of numerical oscillation phenomenon during optimization, therefore, in this embodiment, the design variable optimization allowed step size which changes continuously with the iteration round is also constructed. Considering that in the early stage of optimization, the variable range of the design variable is large, i.e. the allowed degree of numerical oscillation is high, after the optimization is stable, large numerical oscillation should be avoided, i.e. the allowed degree of numerical oscillation is low. Based on this, in this embodiment, a function form decay step size control method is adopted to control the allowed step size to reach the predetermined conservative step size in the specified round according to the calculation form, while ensuring the continuity of the step size change, and the expression is:

[0103] (24)

[0104] (25)

[0105] wherein, is the allowed step size in each iteration round, is the initial setting step size, is an update parameter, is the minimum value of the step size in the continuous iteration, and the update iteration parameter is used to ensure continuous change in rounds of iteration and reach the minimum value of the allowed step size in rounds. In addition, a number of additional stable iteration rounds are further added to verify that the numerical oscillation in the structure is indeed controlled by the effective iteration round k. After updating the design variable in each round, the step size parameter is updated.

[0106] It can be understood that two design variables are repeated S2-S5 every time they are updated until the iteration round reaches the upper limit value, wherein the first The step size is changed continuously in each iteration. It is worth noting that the incremental Lagrange method is performed 5 times in each iteration, i.e., the total number of incremental Lagrange optimization is 5 times the number of iterations. In addition, based on practical experience, the initial step size can be increased to 0.5, and the minimum allowable step size is set to 0.02 in the specified number of iterations.

[0107] By using the method provided in the embodiment, based on the material properties of a given structure, the structure optimization region, the constraint condition, the load type, the load size, the allowable stress of the material, the continuity step size parameter and the like, a structure satisfying the design domain stress intensity and volume minimization can be obtained, and the convergence process of the design variable in the optimization process is stable. The calculated structure is analyzed and evaluated for discreteness and stress. Generally speaking, under the same material, different design domains can be optimized to obtain an optimization result with uniform stress distribution, clear topology contour and novel structure design, and the iterative process of the design variable is stable and there is no abnormal numerical oscillation problem.

[0108] The effect of the scheme is further illustrated by specific cases as follows. Case 1 is as shown in Figure 2 , which is a multi-material different stress criterion optimization analysis case, and the design domain is a pillow beam. Figure 2 The size in is 100 mm, the load is 60 N, which is applied to the design domain by a solid rectangular material, and the upper end and the lower end of the pillow beam are fixed supports. The optimization material selection is two kinds, in which the Poisson's ratio of material 1 and material 2 is 0.3, and the elastic modulus is 10 and 6 respectively, the strength criterion of material 1 is Von-Mises stress criterion, the strength limit , and the strength criterion of material 2 is Drucker-Prager stress criterion, and the uniaxial tensile and compressive limit is . Figure 3 The optimization result of example 1 is shown in, in which (a) is the optimization result, (b) is the stress distribution of the optimization result, and in the figure represents the material proportion of the optimization result, represents the gray unit percentage in the optimization result, it represents the total iteration rounds, represents the maximum value of the stress in the design domain of the optimization result. Figure 4 The change curves of the average value of the design variable and the allowable step size (Allowable Move) in the convergence iteration (Iteration) process of example 1 are shown in.

[0109] Case 2 is as shown in Figure 5 , which is a multi-material different stress criterion optimization analysis case, and the design domain is an L-shaped beam. Figure 5 The size in is 200 mm, the load 80 N, applied to the design domain by solid material, the upper end of the L-shaped beam is fixed support. The optimization material selection is two kinds, the elastic modulus of material 1 and material 2 is the same and is set to 10, the Poisson's ratio is 0.3, the strength criterion of material 1 is Von-Mises stress criterion, the strength limit , the strength criterion of material 2 is Drucker-Prager stress criterion, the uniaxial tensile and compressive limit is . Figure 6 The optimization results of example 2 are shown, wherein (a) is the optimization result, (b) is the stress distribution of the optimization result, the material proportion of the optimization result is represented by , the gray unit percentage in the optimization result is represented by , it represents the total iteration rounds, , and the maximum value of stress in the design domain of the optimization result is represented by Figure 7 The curves of the average value of the design variable and the allowable move in the convergence iteration process of example 2 are shown.

[0110] According to the above, the multi-material stress constraint-based continuity step topology optimization method provided in the embodiment obtains the multi-material attribute parameters based on the design variables through the interpolation form, expresses the strength criteria of different materials through a reasonable interpolation function expression, forms the constraints of the optimization problem by relaxing the criteria, takes the volume minimization in the design domain as the objective function, integrates the objective function and the constraint function by the incremental Lagrange method to form a series of unconstrained optimization problems, solves the series of nonlinear optimization problems by the continuity step moving asymptote method, is suitable for more material types and different stress criterion combinations, can meet the requirements of reaching the minimum value of the continuity step and controlling the change rate of the step in the specified iteration rounds, and finally, the case 1 and the case 2 verify that the stress of the output structure is uniform and reasonable, the boundary is clear, the structure is novel, the design variable changes stably in the iteration process, and the abnormal numerical oscillation problem is solved.

[0111] Based on the same inventive concept, the embodiment further provides a device, which comprises at least one processor and a memory connected with the at least one processor in communication. The memory stores instructions executable by the at least one processor, and the instructions are executed by the at least one processor to enable the at least one processor to execute the algorithm part of the foregoing multi-material stress constraint-based continuity compensation topology optimization method.

[0112] Based on the same inventive concept, the embodiment further provides a computer readable storage medium, which stores a computer program, and the program is executed by a processor to implement the algorithm part of the foregoing multi-material stress constraint-based continuity compensation topology optimization method.

[0113] ​The computer readable medium includes, but is not limited to, any type of disk including floppy disks, optical disks, CD-ROMs, and magnetic-optical disks, ROMs, RAMs, EPROMs (Erasable Programmable Read-Only Memory), EEPROMs, flash cards, magnetic cards, or optical cards. That is, the computer readable medium includes any medium that is capable of storing or carrying information that can be read by the device.

[0114] The apparatus, computer readable storage medium, and so on provided by the embodiments have the same inventive concept and the same beneficial effects as the foregoing method, and will not be described here.

[0115] The technical features of the above embodiments can be combined in any manner. To make the description concise, not all possible combinations of the technical features in the above embodiments are described, but it should be considered that any combination of the technical features is within the scope of the present disclosure as long as there is no contradiction.

[0116] The above embodiments only express several implementation manners of the present application, and the description is specific and detailed, but it should not be understood as a limitation on the scope of the application. It should be pointed out that for those skilled in the art, some modifications and improvements can be made without departing from the concept of the present application, and these are within the protection scope of the present application. Therefore, the protection scope of the present application should be subject to the appended claims.

Claims

1. A continuous step-size topology optimization method based on multi-material stress constraints, characterized in that, include: A finite element model is established based on the actual engineering situation, and parameters are initialized. A physical field expressing multiple material design variables is established. The elastic modulus of a single material is interpolated to form an interpolated penalized form of elastic modulus that can represent the intermediate density values ​​of multiple materials. Different stress criteria exist for different materials to describe material stress behavior, including two strength criteria: Von-Mises and Drucker-Prager. Element stress interpolation is performed to represent intermediate density stress values. Based on the singularity problem of stress constraints, a stress constraint function for stress constraint relaxation criteria is constructed. Using the relative density method in the SIMP method as the physical model, the design variable density of the design domain as the objective function, the stress of each element in the design domain satisfies the set value as a constraint, and the optimization criteria are set to solve the nonlinear optimization problem and perform optimization iteration. The allowable step size for design variable optimization is constructed to vary with the continuity function form of the iteration rounds. The optimization iteration obtains a structure that satisfies the minimum stress intensity and volume of the design domain, and makes the convergence process of the design variables in the optimization process smooth. Here, the elastic modulus, representing the interpolation penalty form of the intermediate density values ​​of multiple materials, is expressed by interpolation of multiple materials, including... and The materials represented by the two design variables; Used to represent the material percentage of the corresponding unit entity, when A value of 1 indicates that the cell is a solid, while a value of 0 indicates that the cell is an empty cell. This indicates the proportion of the material phase in this unit. A value of 1 indicates that the entire unit is filled with material 1, and a value of 0 indicates that the entire unit is filled with material 2. The interpolation expression for the elastic modulus is: ； in, Indicates the elastic modulus. and These are the Young's moduli of Material 1 and Material 2, respectively. It is a constant used to prevent numerical singularity. and These are the SIMP penalty parameters.

2. The continuous step-size topology optimization method based on multi-material stress constraints according to claim 1, characterized in that, Material 1 and Material 2 are respectively subjected to the Von-Mises and Drucker-Prager strength criteria, with the following expressions: ； ； in, This indicates the strength criterion for material one. This indicates the strength criterion for material two. Represents the stress vector of the solid. It is the strength limit of von Mises materials. and These are the parameters under the Drucker-Prager strength criterion, and: ； ； and These are the uniaxial compressive and tensile strength limits of materials according to the Drucker-Prager strength criterion. For the second-order invariant of deviatoric stress: ； in, Each unit The stress vector, It is an auxiliary matrix for calculating the second-order invariants of deviatoric stress. It is a constant set to avoid numerical instability caused by negative square roots when calculating strength criteria.

3. The continuous step-size topology optimization method based on multi-material stress constraints according to claim 2, characterized in that, When performing element stress interpolation, a SIMP-form strength interpolation function is used to simultaneously consider the different strength criterion functions of the two materials, resulting in the interpolated strength criterion: ； in, To simultaneously consider two materials and two strength criteria, an interpolated strength criterion, It is a material The entity stress vector, For materials constitutive matrix, It is a point The strain-displacement transformation matrix, It is a point The local displacement vector, It is a penalty term for stress interpolation.

4. The continuous step-size topology optimization method based on multi-material stress constraints according to claim 3, characterized in that, The relaxation criterion introduced for the constraint function is expressed as follows: ； in, Represents the constraint function. yes , The global vector representation, It is a global displacement vector. To relax the guidelines, As the stress interpolation function, the cubic polynomial can accelerate the convergence of the optimization iteration process. This represents the total number of cells after meshing.

5. The continuous step-size topology optimization method based on multi-material stress constraints according to claim 4, characterized in that, Constraint functions are used to constrain the objective function of optimization: ； in, It is the volume fraction within the design domain. It is a unit volume, It is a global stiffness matrix obtained using interpolated elastic modulus. It is the global load vector.

6. The continuous step-size topology optimization method based on multi-material stress constraints according to claim 1, characterized in that, When solving nonlinear optimization problems, the unconstrained incremental Lagrangian method is adopted. The objective function of the incremental Lagrangian method consists of the original objective function and the penalty term of the constraint function. The objective function and the constraint function are integrated to form an unconstrained sub-optimization problem. The unconstrained optimization sub-problem is solved by the moving asymptote method in the gradient processing method. By dynamically adjusting the moving asymptote, a conservative convex approximation model of the objective function and constraint function under the current design variables is constructed, and the new design variables are obtained by solving the model.

7. The continuous step-size topology optimization method based on multi-material stress constraints according to claim 1, characterized in that, During iterative optimization, a decay step size control method in the form of a function is adopted to control the allowable step size to reach the predetermined conservative step size in the specified number of rounds, while ensuring the continuity of step size changes.

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