Topological optimization design method for reliability of multiphase material structure

Abstract

The invention relates to the technical field of structural engineering, and discloses a multi-phase material structure reliability topological optimization design method, which comprises the following steps: S1, initializing, dividing units, and carrying out finite element initial setting; s2, setting an outer circulation ending condition, entering an outer circulation, and entering an alternating active phase algorithm circulation; s3, firstly, finite element analysis is carried out, and finite element analysis is carried out on the unit model to obtain overall flexibility; s4, carrying out sensitivity analysis; s5, entering sensitivity filtering calculation; s6, updating the optimization criterion density; s7, judging whether the itern is equal to the iterinmax or not; s8, judging whether b is equal to M or not; s9, judging whether a is equal to M-1 or not; and S10, judging whether the interout is equal to the inoutmax or not. Compared with traditional single-material topological optimization, the method has the advantages that a multi-material topological optimization result is provided to deal with the problem of a composite material; the alternative active phase algorithm is simple in thought, high in expandability and high in calculation efficiency; and a suppression function based on an exponential function is adopted, so that lower flexibility and a smaller gray scale unit proportion are obtained.

Description

Technical Field

[0001] The invention relates to the field of structural engineering design, and in particular to a reliability topological optimization design method for a multiphase material structure. Background Art

[0002] Structural optimization refers to the process of optimizing the performance of a structure by changing its material, size, distribution, and connection method through a certain optimization algorithm under a given design area, optimization target, and constraint conditions. It can improve product quality and performance, reduce production costs, and reduce energy consumption. It has been widely used in large-scale engineering machinery, agricultural production, chemical metallurgy, construction, transportation, and other fields. The three elements that constitute a structural optimization problem are: objective function, design variables, and constraints. The objective function is a direct indicator for determining structural performance, the design variables are the design parameters of the optimization process, and the constraints are restrictions on the design variables. Structural optimization can be roughly divided into three categories from the design level: Size Optimization, Shape Optimization, and Topology Optimization.

[0003] Size optimization refers to the optimization purpose achieved by changing the structural dimensions (such as the cross-sectional area of ​​the beam, the thickness of the plate, the diameter of the inner hole, etc.) without changing the structural shape and topological configuration. Its mathematical model is relatively simple, and the design variables are relatively single. It has been developed to be relatively mature, but lacks innovation.

[0004] Shape optimization refers to the process of changing the topological configuration of a structure (such as hole shape, node position, etc.) to find the optimal structure under given optimization target constraints, loads, and other conditions. The difficulty of shape optimization is that shape changes will cause changes in the finite element mesh, resulting in changes in a series of subsequent calculations, which will greatly increase the amount of calculations, including the sensitivity analysis of the objective function to the design variables. Therefore, its difficulty is slightly higher than size optimization, but its limitations are similar to those of size optimization.

[0005] Topological optimization refers to an optimization method that seeks the best material layout of a structure within a given design domain (such as whether there are holes in the structure, the location of the holes, the connection method of the components, etc.) under certain constraints. The difficulty of topological optimization is that there are many design forms that can meet certain requirements, and the topological form is generally difficult to quantitatively describe or parameterize. However, it has been proven that the structure after topological optimization is more reasonable.

[0006] These three optimizations correspond to the three stages of traditional design: first, conceptual design, topology optimization determines the optimal topology of the structure; then basic design, shape optimization determines the optimal shape of the structure; and finally, detailed design, size optimization determines the basic size of the structure. Obviously, topology optimization plays a core role here.

[0007] With the development of science and technology and humanities, the demand for lightweight and reliability in aerospace, automobile and other fields has become more and more diversified. Traditional single material topology optimization can no longer meet the needs. Multiphase material topology optimization provides a new technical means for such complex engineering problems. Multiphase material topology optimization refers to the method of combining materials with different elastic moduli during the optimization process so that the optimization result has more than one material. At present, it has also developed into one of the most popular research directions of topology optimization.

[0008] Existing literature 1, Zuo W, Saitou K. Multi-material topology optimization using ordered SIMP interpolation [J]. Structural and Multidisciplinary Optimization, 2017, 55: 477-491. provides a classic multi-phase material structure topology optimization method based on ordered SIMP interpolation. This optimization method provides the original alternating active phase algorithm. The defect of this method is that the boundary of the obtained optimized structure is relatively fuzzy, a large number of grayscale units are generated, the iteration speed is slow, and the iteration under multiple working conditions is not considered. Summary of the invention

[0009] In order to overcome or alleviate one or more of the above technical problems, the present invention aims to provide a reliability topology optimization method based on multi-phase material structures.

[0010] The present invention provides the following technical solutions:

[0011] A reliability topology optimization design method for a multiphase material structure comprises the following steps:

[0012] S1: Initialization, unit division, finite element initial setting, and topology optimization initial parameters, including material elastic modulus, Poisson's ratio, penalty coefficient, multiphase material volume fraction distribution, filter radius, and suppression factor;

[0013] S2: Set the end condition of the outer loop, enter the outer loop, enter the alternating active phase algorithm loop, and perform the inner loop in the order of a from 1 to p-1 and b from a+1 to p, where a is the first phase material phase of the two-phase loop, b is the second phase material phase of the two-phase loop, and p is the penalty coefficient;

[0014] S3: First, perform finite element analysis on the unit model to obtain the overall flexibility;

[0015] S4: Perform sensitivity analysis on the overall flexibility to obtain sensitivity;

[0016] S5: Enter sensitivity filtering calculation, perform sensitivity filtering calculation on the sensitivity, and obtain the filtered sensitivity;

[0017] S6: Optimize the density of the criterion, calculate the sensitivity after filtering to obtain the iteration factor, judge the iteration factor, and obtain the optimized unit density of each unit;

[0018] S7: Determine whether iterin and iterinmax are equal, iterin and iterinmax refer to the number of inner loop iterations and the maximum number of inner loop iterations respectively. If they are equal, proceed to S8; if not, return to S3;

[0019] S8: Determine whether b and M are equal, where M refers to the maximum number of material phases. If they are equal, proceed to S9; if they are not equal, b=b+1, and return to S2;

[0020] S9: Determine whether a and M-1 are equal, if they are equal, proceed to S10, otherwise a=a+1, return to S2;

[0021] S10: Determine the end condition of the outer loop, whether interout and ineroutmax are equal. interout and iteroutmax refer to the number of outer loop iterations and the maximum number of outer loop iterations respectively. If they are equal, draw the unit density image, obtain the topology optimization result, and end the outer loop. Otherwise, return to S1.

[0022] According to some embodiments, in step S3, the multiphase material topology optimization model takes the minimum structural flexibility as the objective function and the volume constraint as the constraint condition, which is described as:

[0023]

[0024] In the finite element analysis calculation, the element stiffness matrix K is:

[0025]

[0026] in, represents the relative volume fraction of the element of the mth material, Represents the stiffness matrix of the mth material, and substitutes it into the mathematical model to obtain:

[0027]

[0028] Where n is the number of units, V m is the volume of the mth material after optimization, v e represents the unit volume, f m represents the volume fraction of the mth material, u e is the unit node displacement, F is the force on the model, K is the overall stiffness matrix, C(x) is the overall flexibility of the model, and x is the volume fraction.

[0029] According to some embodiments, in step S6, in each iteration calculation, the volume constraints r of the active phase materials a and b are ab It can be calculated by the following formula:

[0030]

[0031] Where α m is the volume fraction of each unit of the m-phase material, because each unit has Therefore, if the density of phase a material is obtained, the density of phase b material is:

[0032] r b =r ab -r a

[0033] In a single optimization subproblem, the temporary upper limit of phase a is:

[0034] u a,temp =min(u a ,r ab )

[0035] The lower bound of the a-phase material remains unchanged, so the optimization subproblem of the two-phase material is abstractly expressed as:

[0036]

[0037] According to some embodiments, in step S5, the suppression function based on the exponential function is as follows:

[0038]

[0039] In the formula, δ(·) represents the suppression function in the design variable updating process, q is the suppression factor, is the unit optimization operator; therefore, the design variable update formula of the gray unit suppression method based on the optimization criterion method is changed to:

[0040]

[0041] When q=1, it is the original OC algorithm. As the q value gradually increases, the intermediate density units approach 0 or 1.

[0042] Compared with the prior art, the present invention has the following beneficial effects:

[0043] The reliability topology optimization method based on multi-phase material structure provided by the present invention, firstly, compared with the traditional single material topology optimization, the reliability topology optimization of multi-phase material structure provides the topology optimization results of multiple materials, which can deal with more complex engineering problems such as composite materials. Secondly, the alternating active phase algorithm adopted has the characteristics of simple ideas, strong scalability, and high computational efficiency. Finally, compared with the traditional OC algorithm, the suppression function based on the exponential function adopted by the present invention can obtain lower flexibility and a smaller grayscale unit ratio. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] Figure 1 A schematic diagram of an optimized structure with clear boundaries provided by an embodiment of the present invention.

[0045] Figure 2 This is a two-phase material structure distribution diagram provided by an embodiment of the present invention.

[0046] Figure 3 A sequence diagram for solving the four-phase material optimization sub-problems provided in an embodiment of the present invention.

[0047] Figure 4 A power function grayscale unit suppression function diagram provided by an embodiment of the present invention.

[0048] Figure 5 This is a graph of the exponential grayscale unit suppression function provided by an embodiment of the present invention.

[0049] Figure 6 A flow chart of multiphase material topology optimization provided by an embodiment of the present invention.

[0050] Figure 7 A schematic diagram of a planar MBB beam structure provided in an embodiment of the present invention.

[0051] Figure 8 Schematic diagram of MBB beam optimization results using different methods provided in embodiments of the present invention.

[0052] Fig. 9 Objective functions, i.e., flexibility value iteration curves, under three methods provided in the embodiments of the present invention.

[0053] Fig.10 Table 1 is a comparison table of topology optimization results under different algorithms provided in the embodiments of the present invention.

[0054] Fig.11 A schematic diagram of a planar cantilever beam structure provided in an embodiment of the present invention.

[0055] Fig.12 Schematic diagram of cantilever beam optimization results using different methods provided in embodiments of the present invention.

[0056] Fig.13 Iteration diagram of objective functions of different methods provided in embodiments of the present invention.

[0057] Fig.14 Table 2 is a comparison table of topology optimization results under different algorithms provided in the embodiments of the present invention.

[0058] Fig.15 A schematic diagram of a three-dimensional cantilever beam structure provided in an embodiment of the present invention.

[0059] Fig.16 Optimized structural diagrams of each single working condition of a three-dimensional cantilever beam provided in an embodiment of the present invention.

[0060] Fig.17 Optimization structure diagram of a three-dimensional cantilever beam at different iteration steps provided by an embodiment of the present invention.

[0061] Fig.18 Iteration diagram of objective functions for each single working condition of a three-dimensional cantilever beam structure provided in an embodiment of the present invention.

[0062] Fig.19 This is a diagram of the optimized structure of a three-dimensional cantilever beam under different combinations of weight coefficients for different working conditions provided in an embodiment of the present invention.

[0063] Fig. 20 A three-dimensional cantilever beam reliability topology optimization structure diagram under the working condition weight coefficient of (0.5, 0.5) provided in an embodiment of the present invention. DETAILED DESCRIPTION

[0064] The technical ideas of the present invention are as follows:

[0065] First, certain idealized premises and assumptions are put forward for each phase material. For example, during the optimization process, each material inside the discrete unit is isotropic, and the properties of each phase material in the unit change with the change of the relative density of the unit, and are exponentially related.

[0066] Secondly, the material design domain is divided into grid units, and multiphase materials are assigned to each unit so that the volume fractions of the multiphase materials satisfy that the sum is equal to 1. The alternating active phase algorithm proposed by Tavakoil and Mohseni in 2013 is used to perform topological optimization on the combination of different phase materials.

[0067] Finally, for the combination of different phase materials, the extended SIMP method is used to establish the material interpolation model for multiphase material topology optimization, and the volume fraction of each unit is iteratively calculated to finally obtain the reliability topology optimization result of the multiphase material structure. Figure 1 The multiphase material shown has clear boundaries and an optimized structure for reliability.

[0068] The present invention is described in detail below in conjunction with the embodiments and drawings, but it should be understood that the embodiments and drawings are only used to exemplify the present invention and do not constitute any limitation on the protection scope of the present invention. All reasonable changes and combinations within the scope of the inventive concept of the present invention fall within the protection scope of the present invention.

[0069] The present invention will be further described below in conjunction with the accompanying drawings.

[0070] Example 1

[0071] like Figure 6 This embodiment provides a reliability topology optimization design method for multiphase material structures, which is based on the alternating active phase algorithm. This multiphase material topology optimization includes two layers of internal and external cycles. The core is to allocate the volume fraction of each material so that the structure reaches the set minimum flexibility. The basic steps of the optimization design are:

[0072] S1: Initialization, unit division, initial finite element settings, and initial topology optimization parameters, including material elastic modulus, Poisson's ratio, penalty coefficient, multiphase material volume fraction distribution, filter radius, and suppression factor;

[0073] S2: Set the end condition of the outer loop, enter the outer loop, enter the alternating active phase algorithm loop, and perform the inner loop in the order of a from 1 to p-1 and b from a+1 to p, where a is the first phase material phase of the two-phase loop, b is the second phase material phase of the two-phase loop, and p is the penalty coefficient. The specific theory refers to the alternating active phase algorithm in the existing literature 1;

[0074] S3: First, perform finite element analysis on the unit model to obtain the overall flexibility;

[0075] S4: Perform sensitivity analysis on the overall flexibility to obtain sensitivity;

[0076] S5: Enter sensitivity filtering calculation, perform sensitivity filtering calculation on the sensitivity, and obtain the filtered sensitivity;

[0077] S6: Optimize the density of the criterion, calculate the sensitivity after filtering to obtain the iteration factor, judge the iteration factor, and obtain the optimized unit density of each unit;

[0078] S7: Determine whether iterin and iterinmax are equal, iterin and iterinmax refer to the number of inner loop iterations and the maximum number of inner loop iterations respectively. If they are equal, proceed to S8; if not, return to S3;

[0079] S8: Determine whether b and M are equal, where M refers to the maximum number of material phases. If they are equal, proceed to S9; if they are not equal, b=b+1, and return to S2;

[0080] S9: Determine whether a and M-1 are equal, if they are equal, proceed to S10, otherwise a=a+1, return to S2;

[0081] S10: Determine the end condition of the outer loop, whether interout and ineroutmax are equal. interout and interoutmax refer to the number of outer loop iterations and the maximum number of outer loop iterations respectively. If they are equal, draw the unit density image, obtain the topology optimization result, and end the outer loop. Otherwise, return to S1.

[0082] Among them, S1~S10 is the outer loop, which follows the alternating active phase algorithm; S3~7 is the inner loop, which follows the order of a from 1 to p-1 and b from a+1 to p.

[0083] The density update of S6 optimization criteria is specifically an intermediate density unit suppression method based on optimization criteria: the volume fraction in the a-phase material is updated by the optimization criteria. In this step, the intermediate density method based on the optimization criteria is applied. The specific theory is shown in formula 18. Note that the volume fraction of the a-phase material and r are obtained by subtracting the volume fraction of the a-phase material and b-phase material in this internal cycle from the total material volume fraction 1. Then, the volume fraction of the a-phase material is obtained by subtracting the volume fraction of the a-phase material from r. The specific theoretical part refers to the alternating active phase algorithm in the existing literature 1.

[0084] In the finite element analysis of step S3, this embodiment uses a simplified material interpolation model for programming and calculation. Tavakoil and Mohseni proposed a simplified material interpolation function as follows:

[0085]

[0086] Where m is the number of material phases, represents the elastic modulus of the m-phase material, M is the total number of phases in the material, and x em represents the volume fraction of the m-phase material in a unit, p is the penalty coefficient, and the sum of the relative volume fractions of each unit is equal to 1:

[0087]

[0088] Therefore, the material interpolation function for two-phase material topology optimization is simplified to:

[0089]

[0090] Schematic diagram as Figure 2 shown.

[0091] Step S3: The multiphase material topology optimization model takes the minimum structural flexibility as the objective function and the volume constraint as the constraint condition, which can be described as:

[0092]

[0093] According to the multiphase material interpolation formula described in the previous section, in the finite element analysis calculation, the unit stiffness matrix K is:

[0094]

[0095] in, represents the relative volume fraction of the element of the mth material, Represents the stiffness matrix of the mth material, and substitutes it into the above mathematical model:

[0096]

[0097] Where n is the number of units, V m is the volume of the mth material after optimization, v e represents the unit volume, f m represents the volume fraction of the mth material, u e is the displacement of the unit node, F is the force on the model, K is the overall stiffness matrix, C(x) is the overall flexibility of the model, and x is the volume fraction. In the optimization model, the first constraint is the static equilibrium equation, the second constraint is the volume constraint of each phase material, and the third constraint represents that the relative density of each phase material must comply with the upper and lower limit constraints.

[0098] In step 1, a variety of methods have emerged in the development of multiphase material topology optimization. Among them, the alternating active phase algorithm proposed by Tavakoil and Mohseni in 2013 stands out because of its simple idea, strong scalability and high computational efficiency. The multiphase material topology optimization model of this embodiment adopts this method, and its abstract expression calculation formula is given below.

[0099] First, within the design domain Ω, the material distribution of the m-th phase material is determined by its volume fraction α m (m=1,2,…,M). m The following relationship needs to be satisfied:

[0100] l m ≤α m ≤u m (8)

[0101]

[0102] In the formula, l m and u mRepresent the lower and upper bounds of the volume fraction, and their values ​​must be between 0 and 1. In addition, for each phase material, there is a global volume constraint to limit it, as shown below:

[0103]

[0104] In the formula, Λ m It is the volume constraint of each phase material set artificially. For the convenience of understanding and calculation, the vector field α={α1,α2,…,α M} is used to represent the set of design variables. In almost all topology optimization problems, material properties are domain functions of the volume fraction of each material phase in the optimization. The alternating active phase method uses the SIMP interpolation method to describe local material properties. These material parameters are given in advance, and in order to simplify the equation, both single-phase and multi-phase materials are represented by δ(α) to represent the material interpolation function. Partial differential equations (PDEs) are also an important component of topology optimization problems. Their solutions are represented by U(x) = μ(α(x)). The partial differential operator of the PDE constraint is represented by Therefore, the discrete mathematical model of the multiphase material topology optimization problem is:

[0105]

[0106] Where the objective function J(…) is the integral of α and U in the design domain Ω.

[0107] The subtlety of the alternating active phase algorithm lies in that it splits the multiphase material optimization problem into multiple two-phase material topology optimization sub-problems, which are implemented through loop nesting. Each outer loop requires solving M(M-1) / 2 sub-problems, and the two-phase material topology optimization structure in each sub-problem is calculated by the two-phase material topology optimization in the previous iteration. Figure 3 As shown in the figure, taking four-phase material as an example, the solution order of two-phase material sub-problems is given.

[0108] The subproblem in the alternating active phase algorithm of step S6 of this embodiment is solved based on the variable density method. The superscripts "a" and "b" are used to represent the two phase materials to be solved in the subproblem. Then, in the current iterative calculation, the M-2 phase material other than these two phases is fixed. Therefore, in each iterative calculation, the volume constraints r of the active phase materials a and b are ab It can be calculated by the following formula:

[0109]

[0110] Where α m is the volume fraction of each unit of the m-phase material, because each unit has Therefore, if the density of phase a material is obtained, the density of phase b material is:

[0111] r b =r ab -r a (13)

[0112] It can be seen from formula 14 that in a single optimization sub-problem, the temporary upper limit value of phase a is:

[0113] u a,temp =min(u a ,r ab ) (14)

[0114] The lower bound of the a-phase material remains unchanged, so the optimization subproblem of the two-phase material can be abstractly expressed as:

[0115]

[0116] As is known to all, most of the topology optimization methods based on variable density method have a common problem, that is, some intermediate density units, that is, gray units, will appear. This numerical instability phenomenon will greatly reduce the manufacturability of the optimized structure. Therefore, in the present invention, an intermediate density unit suppression method based on the optimization criterion method is used. This method is simple to implement and can always meet the constraints during the optimization process.

[0117] In step S5: Groenwold and Etman proposed in 2007 that when the optimization criterion method is used to update the design variables, a gray unit suppression function is applied to the design variables, thereby suppressing the gray units to a certain extent. This suppression function is expressed in the form of a power function. Zhang Yifei proposed a suppression function based on an exponential function, as follows:

[0118]

[0119] In the formula, δ(·) represents the inhibition function in the design variable updating process, q is the inhibition factor, is the unit optimization operator. Therefore, the design variable update formula of the gray unit suppression method based on the optimization criterion method can be changed to:

[0120]

[0121] From formula 18, we can see that when q = 1, it is the original OC algorithm. As the q value gradually increases, the intermediate density units are close to 0 or 1, such as Figure 4 and 5 shown.

[0122] Example 1

[0123] The first example considers Figure 7The MBB beam structure shown has an aspect ratio of 4:1 and a structural geometric size of 192mm×48mm. The structure is discretized into 192×48 units, the lower left corner of the structure constrains horizontal and vertical displacements, the lower right corner constrains vertical displacements, and the vertical concentrated force is applied at the midpoint of the upper side with a magnitude of 1N.

[0124] First, consider the optimization problem of two-phase solid materials. Red represents solid material 1, with the corresponding elastic modulus set to 2 and the volume fraction set to 0.4; blue represents solid material 2, with an elastic modulus of 1 and a volume fraction of 0.2; the porous material is white and colorless, with an elastic modulus set to 1e-9 and a volume fraction of 0.4.

[0125] The results of the classic alternating active phase algorithm in the existing literature, the grayscale unit suppression method based on the power function, and the grayscale unit suppression method based on the exponential function are analyzed and compared, and the most suitable method is selected. Among them, the initial value of the suppression factor q is set to 1, and its growth step is 0.01 per cycle. The optimization results are shown in Figure 8 As shown. Among them, Figure 8 .a is the classic algorithm in literature 1; Figure 8 .b is the grayscale unit suppression method based on power function; Figure 8 .c is the grayscale unit suppression method based on exponential function, Fig. 9 Iteration curves of the objective functions, i.e., flexibility values, under the three methods.

[0126] Fig.10 Table 1 lists the optimization result data under three different algorithm filtering with the same parameters, including the objective function value, number of iterations, and grayscale unit ratio.

[0127] according to Figure 8 As shown in the figure, the three algorithms can optimize the complete multi-phase material MBB beam structure, and the common point is that the solid material 1 with a larger elastic modulus is distributed around the beam, and the middle support part is filled with the solid material 2 with a smaller elastic modulus. The specific structural details are slightly different between the results of different methods, but in general, the materials with high stiffness are basically distributed on the main force transmission path of the structure, which verifies the effectiveness of this alternating active phase algorithm and proves that this algorithm can improve the utilization efficiency of materials while ensuring the overall performance of the structure. In addition, it can be clearly seen that compared with the original alternating active phase algorithm in the existing literature 1, the grayscale unit suppression method based on the power function and the grayscale unit suppression method based on the exponential function can obtain an optimized structure with clearer boundaries. This is from Fig.10 The grayscale unit ratio in Table 1 can be seen more intuitively.

[0128] from Fig. 9 You can see the iterative curves of the objective functions under the three methods, and then combine them Fig.10From the data in Table 1, it can be seen that the objective functions of the three methods gradually converge at about the tenth iteration, but the final objective function value of the original alternating active phase algorithm is the largest, the objective function value under the power function suppression method is second, and the exponential function method is the smallest. This means that the structural stiffness optimized by the original alternating active phase algorithm is the smallest, and the structural stiffness optimized by the grayscale unit suppression method based on the exponential function is the largest.

[0129] Example 2

[0130] The second example considers Fig.11 The plane cantilever beam structure shown has an aspect ratio of 2:1 and a structural geometric size of 96mm×48mm. The structure is discretized into 96×48 units, the left end of the beam is constrained by a fixed support, and a unit-sized concentrated force is applied to the lower right corner.

[0131] In this example, the three-phase solid material optimization problem is considered. The red color represents solid material 1, and the corresponding elastic modulus is set to 5 and the volume fraction is set to 0.2; the blue color represents solid material 2, the elastic modulus is 3, and the volume fraction is 0.1; the green color represents solid material 3, the elastic modulus is set to 1, and the volume fraction is set to 0.1; the hole material is white and colorless, the elastic modulus is set to 1e-9, and the volume fraction is 0.6. The other parameters such as the suppression factor and the cycle step size are the same as the first example. The results are as follows Fig.12 As shown, Fig.12 .a is the result of the classic method in literature 1, Fig.12 .b is the grayscale unit suppression method based on exponential function.

[0132] As in the previous example, first draw the objective function iteration curves of the two methods, and then list the objective function values, iteration times, and grayscale unit ratios of the optimization results of the two different algorithms. Fig.13 shown.

[0133] from Fig.14 It can be clearly seen from Table 2 that the grayscale unit suppression method based on the exponential function obtains a clearer optimized structure diagram, and its final objective function value is less than the objective function value obtained by the original alternating active phase method. In summary, both grayscale unit suppression methods can effectively suppress grayscale units, thereby obtaining a clearer structure. Although the grayscale unit suppression method based on the power function has the lowest grayscale unit ratio in the optimized structure, it sometimes has some small branch structures and large block structures, which is not as reasonable as the grayscale unit suppression method based on the exponential function, and the optimized structural stiffness is lower than that of the exponential function suppression method. Therefore, the multiphase material optimization model in the subsequent chapters uses the grayscale unit suppression method based on the exponential function to suppress the numerical instability of grayscale units.

[0134] Example 3

[0135] The last example also expands the 2D multiphase material topology optimization problem to a 3D multiphase material topology optimization problem. This code is modified based on the 2D multi-material code with reference to the classic 3D code for single-phase materials. Except for the finite element part, its core idea is exactly the same as the basic process.

[0136] Consider the three-dimensional cantilever beam structure shown in the figure. The structural geometric dimensions are 30cm×15cm×8cm. The left boundary of the structure is fully constrained. The working load F1 and working load F2 are as follows: Fig.15 As shown in the figure, the size is 1N. The structure is discretized into 3600 units, the penalty coefficient is set to 3, and the filter radius is set to 5.

[0137] This example considers a three-phase solid material topology optimization problem. Solid material 1 is red, with an elastic modulus of 5 and a volume fraction of 0.16; solid material 2 is blue, with an elastic modulus of 2 and a volume fraction of 0.16; solid material 3 is green, with an elastic modulus of 1 and a volume fraction of 0.16.

[0138] Fig.16 The optimized cantilever beam structure under a single working condition is shown. Fig.16 .a is the optimization result of working condition 1, Fig.16 .b is the optimization result of working condition 2. It can be seen from the figure that the structures under both working conditions are clear and reasonable. When working condition load F1 acts alone, the area around the load point is filled with red material with the largest elastic modulus, i.e. the largest stiffness, the part connected to the fixed end is filled with slightly smaller blue material, and the middle part is filled with green material with the smallest elastic modulus. The optimized structure of working condition 2 also follows this rule, but its geometric dimensions are different. This is because the maximum stress of the cantilever beam structure is concentrated at the load point and the fixed end contact point, so these main load-bearing parts should be filled with materials with large elastic modulus, which is consistent with the rules and results obtained from the two-dimensional cantilever beam example in Chapter 4. It also further confirms the effectiveness of this three-dimensional multi-material method.

[0139] The 10th, 20th, 50th, 100th, 150th, and 200th iterations are selected to further demonstrate their optimization processes. Fig.17 The structure obtained by single-condition topology optimization can be further used to obtain a multi-condition topology optimization structure. Here, two groups of condition weight coefficients are selected for calculation, and (ω1, ω2) are set to (0.2, 0.8) and (0.5, 0.5) respectively, as shown in Fig.19 As shown. Further select the second group of working condition weight coefficients and list the optimized structures with different β values, as shown in Fig.18 As shown. Fig. 20It can be seen that, similar to the two-dimensional example calculated above, the reliability topology optimization structure has basically unchanged configuration compared to the deterministic topology optimization structure, and for multiphase materials, its material distribution is basically unchanged. However, its geometric dimensions will change slightly. As the reliability index β increases, its structural length increases, its width and height decrease, and its volume fraction decreases.

[0140] The above embodiments are only preferred implementations of the present invention, and the protection scope of the present invention is not limited to the above embodiments. All technical solutions under the concept of the present invention belong to the protection scope of the present invention. It should be pointed out that for ordinary technicians in this technical field, improvements and modifications without departing from the principle of the present invention should also be regarded as the protection scope of the present invention.

Claims

1. A reliability topology optimization design method for a multiphase material structure, characterized by: The following steps are involved: S1: Initialization, unit division, finite element initial setting, and topology optimization initial parameters, including material elastic modulus, Poisson's ratio, penalty coefficient, multiphase material volume fraction distribution, filter radius, and suppression factor; S2: Set the end condition of the outer loop, enter the outer loop, enter the alternating active phase algorithm loop, and perform the inner loop in the order of a from 1 to p-1 and b from a+1 to p, where a is the first phase material phase of the two-phase loop, b is the second phase material phase of the two-phase loop, and p is the penalty coefficient; S3: First, perform finite element analysis on the unit model to obtain the overall flexibility; S4: Perform sensitivity analysis on the overall flexibility to obtain sensitivity; S5: Enter sensitivity filtering calculation, perform sensitivity filtering calculation on the sensitivity, and obtain the filtered sensitivity; S6: Optimize the density of the criterion, calculate the sensitivity after filtering to obtain the iteration factor, judge the iteration factor, and obtain the optimized unit density of each unit; S7: Determine whether iterin and iterinmax are equal, iterin and iterinmax refer to the number of inner loop iterations and the maximum number of inner loop iterations respectively. If they are equal, proceed to S8; if not, return to S3; S8: Determine whether b and M are equal, where M refers to the maximum number of material phases. If they are equal, proceed to S9; if they are not equal, b=b+1, and return to S2; S9: Determine whether a and M-1 are equal, if they are equal, proceed to S10, otherwise a=a+1, return to S2; S10: Determine the end condition of the outer loop, whether interout and ineroutmax are equal. interout and iteroutmax refer to the number of outer loop iterations and the maximum number of outer loop iterations respectively. If they are equal, draw the unit density image, obtain the topology optimization result, and end the outer loop. Otherwise, return to S1.

2. The reliability topology optimization design method for multiphase material structure according to claim 1, characterized in that: In step S3, the multiphase material topology optimization model takes the minimum structural flexibility as the objective function and the volume constraint as the constraint condition, which is described as: In the finite element analysis calculation, the element stiffness matrix K is: in, represents the relative volume fraction of the element of the mth material, Represents the stiffness matrix of the mth material, and substitutes it into the mathematical model to obtain: Where n is the number of units, V m is the volume of the mth material after optimization, v e represents the unit volume, f m represents the volume fraction of the mth material, u e is the unit node displacement, F is the force on the model, K is the overall stiffness matrix, C(x) is the overall flexibility of the model, and x is the volume fraction.

3. The reliability topology optimization design method of multiphase material structure according to claim 1 is characterized by: Step S6: In each iteration, the volume constraints r of active phase materials a and b are a b can be calculated by the following formula: Where α m is the volume fraction of each unit of the m-phase material, because each unit has Therefore, if the density of phase a material is obtained, the density of phase b material is: r b ＝r ab -r a In a single optimization subproblem, the temporary upper limit of phase a is: in a,temp =min(in a ,r ab ) The lower bound of the a-phase material remains unchanged, so the optimization subproblem of the two-phase material is abstractly expressed as:

4. The reliability topology optimization design method of multiphase material structure according to claim 1 is characterized by: In step S5, the suppression function based on the exponential function is as follows: In the formula, δ(·) represents the suppression function in the design variable updating process, q is the suppression factor, Optimize operators for cells; Therefore, the design variable update formula of the gray unit suppression method based on the optimization criterion method is changed to: When q=1, it is the original OC algorithm. As the q value gradually increases, the intermediate density units approach 0 or 1.

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