A reliability topology optimization design method for multiphase material structures

Abstract

This invention relates to the field of structural engineering technology and discloses a reliability topology optimization design method for multiphase material structures, comprising the following steps: S1: Initialization, element division, and finite element initial settings; S2: Setting the outer loop termination condition, entering the outer loop, and entering the alternating active phase algorithm loop; S3: First, performing finite element analysis on the element model to obtain the overall compliance; S4: Performing sensitivity analysis; S5: Entering sensitivity filtering calculation; S6: Optimization criterion density update; S7: Determining whether iterin and iterinmax are equal; S8: Determining whether b and M are equal; S9: Determining whether a and M-1 are equal; S10: Determining whether interout and ineroutmax are equal. Compared with traditional single-material topology optimization, this invention provides multi-material topology optimization results to address composite material problems; the alternating active phase algorithm is simple in concept, highly scalable, and computationally efficient; and the use of an exponential function-based suppression function achieves lower compliance and a smaller proportion of gray-scale elements.

Description

Technical Field

[0001] This invention relates to the field of structural engineering design, and in particular to a reliability topology optimization design method for multiphase material structures. Background Technology

[0002] Structural optimization refers to the process of optimizing a structure's performance by altering its materials, dimensions, distribution, and connection methods within a given design region, optimization objective, and constraints, using specific optimization algorithms. It can improve product quality and performance, reduce production costs, and lower energy consumption, and has been widely applied in large-scale engineering machinery, agricultural production, chemical metallurgy, construction, and transportation. The three essential elements of a structural optimization problem are: objective function, design variables, and constraints. The objective function is a direct indicator of structural performance, design variables are the design parameters used in the optimization process, and constraints impose restrictions on the design variables. From a design perspective, structural optimization can be broadly categorized into three types: Size Optimization, Shape Optimization, and Topology Optimization.

[0003] Dimensional optimization refers to achieving optimization by changing only the structural dimensions (such as the cross-sectional area of ​​a beam, the thickness of a plate, and the diameter of an internal hole) without altering the structural shape and topology. Its mathematical model is relatively simple, and the design variables are relatively singular. While it has matured considerably, it lacks innovation.

[0004] Shape optimization refers to finding the optimal structure under given optimization constraints and loads by changing the structural topology (e.g., hole shape, node location). The difficulty of shape optimization lies in the fact that shape changes alter the finite element mesh, requiring subsequent calculations to be modified accordingly, significantly increasing the computational load. Sensitivity analysis of the objective function to design variables is also challenging. Therefore, it is slightly more difficult than dimensional optimization, but its limitations are similar.

[0005] Topology optimization is an optimization method that seeks the optimal material layout (e.g., the presence and location of holes, and the connection methods of components) of a structure within a given design domain under certain constraints. The challenge of topology optimization lies in the fact that while there are many design forms that can meet certain requirements, topological forms are generally difficult to describe quantitatively or parameterize. However, it has been proven that topology-optimized structures are more rational.

[0006] These three optimization methods correspond to the three stages of traditional design: first, conceptual design, where topology optimization determines the optimal topology of the structure; then, basic design, where shape optimization determines the optimal shape of the structure; and finally, detailed design, where dimensional optimization determines the basic dimensions of the structure. Clearly, topology optimization plays a central role here.

[0007] With the development of science and technology and humanities, the demands for lightweighting and reliability in fields such as aerospace and automotive are becoming increasingly diversified. Traditional single-material topology optimization is no longer sufficient to meet these needs, and multiphase material topology optimization provides a new technical approach for such complex engineering problems. Multiphase material topology optimization refers to combining materials with different elastic moduli during the optimization process, thereby resulting in an optimization outcome that incorporates more than one material. Currently, it has also become one of the most popular research directions in topology optimization.

[0008] The first existing literature, Zuo W, Saitou K. Multi-material topology optimization using ordered SIMP interpolation[J]. Structural and Multidisciplinary Optimization, 2017, 55: 477-491, provides a classic method for topology optimization of multiphase material structures based on ordered SIMP interpolation. This optimization method provides the original alternating active phase algorithm. The drawbacks of this method are that the boundaries of the optimized structure are relatively blurry, a large number of gray 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 purpose of this invention is to provide a reliability topology optimization method based on multiphase material structures.

[0010] This invention provides the following technical solution:

[0011] A reliability topology optimization design method for multiphase material structures includes the following steps:

[0012] S1: Initialization, dividing the elements, performing finite element initial settings, setting topology optimization initial parameters, including material elastic modulus, Poisson's ratio, penalty coefficient, multiphase material volume fraction distribution, filtration radius, and inhibition factor;

[0013] S2: Set the outer loop termination condition, 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 cycle, b is the second phase material phase of the two-phase cycle, and p is the penalty coefficient.

[0014] S3: First, perform finite element analysis on the element model to obtain the overall compliance.

[0015] S4: Perform sensitivity analysis on the overall compliance to obtain the sensitivity.

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

[0017] S6: Optimize the density update criteria, calculate the iteration factor based on the filtered sensitivity, judge the iteration factor, and obtain the optimized unit density for each unit;

[0018] S7: Determine if iterin and iterinmax are equal. iterin and iterinmax refer to the number of iterations in the inner loop and the maximum number of iterations in the inner loop, respectively. If they are equal, proceed to S8; otherwise, return to S3.

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

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

[0021] S10: Determine the termination condition of the outer loop by checking if interout and iteroutmax are equal. interout and iteroutmax refer to the number of iterations and the maximum number of iterations of the outer loop, respectively. If they are equal, plot the cell density image, obtain the topology optimization result, and end the outer loop; otherwise, return to S1.

[0022] According to some implementation methods, in step S3, the multiphase material topology optimization model, with minimum structural compliance as the objective function and volume constraints as the constraint condition, is described as follows:

[0023]

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

[0025]

[0026] in, This represents the relative volume fraction of the m-th material per unit cell. Let m be the stiffness matrix of the m-th material. Substituting it into the mathematical model, we get:

[0027]

[0028] In the formula, n is the number of units, V m To optimize the volume of the m-th material, v e f represents the unit volume. m u represents the volume fraction of the m-th material. e Let F be the element nodal displacement, F be the force on the model, K be the overall stiffness matrix, C(x) be the overall flexibility of the model, and x be the volume fraction.

[0029] According to some implementation methods, in step S6, in each iteration calculation, the volume constraint r of the active phase materials a and b is... ab It can be calculated using the following formula:

[0030]

[0031] In the formula α m This represents the volume fraction of each unit cell in the m-phase material, because each unit cell has... Therefore, if the density value of phase a is determined, the density value of phase b is:

[0032] r b =r ab -r a

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

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

[0035] Since the lower bound of the a-phase material remains unchanged, the optimization subproblem of the two-phase material can be abstractly expressed as:

[0036]

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

[0038]

[0039] In the formula, δ(·) represents the inhibition function in the design variable update process, and q is the inhibition factor. For the element optimization operator; therefore, the design variable update formula for the gray-scale element suppression method based on the optimization criterion method is changed to:

[0040]

[0041] When q = 1, it is the original OC algorithm. As the value of q gradually increases, the intermediate density cells all move closer to 0 or 1.

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

[0043] The reliability topology optimization method for multiphase material structures provided by this invention offers several advantages. First, compared to traditional single-material topology optimization, multiphase material structure reliability topology optimization provides topology optimization results for multiple materials, enabling it to address more complex engineering problems such as composite materials. Second, the alternating active phase algorithm employed is characterized by its simplicity, strong scalability, and high computational efficiency. Finally, compared to traditional objective functions (OC), the suppression function based on an exponential function used in this invention achieves lower compliance and a smaller proportion of grayscale units. Attached Figure Description

[0044] Figure 1 This is a schematic diagram of an optimized structure with clear boundaries provided for an embodiment of the present invention.

[0045] Figure 2 The diagram shows the structural distribution of two-phase materials provided in an embodiment of the present invention.

[0046] Figure 3 The solution sequence diagram for the four-phase material optimization sub-problem provided in the embodiments of the present invention is shown.

[0047] Figure 4 The diagram shows the suppression function of the power function grayscale unit provided in the embodiment of the present invention.

[0048] Figure 5 The graph shows the suppression function of grayscale units for the exponential function provided in this embodiment of the invention.

[0049] Figure 6 A flowchart of multiphase material topology optimization provided for embodiments of the present invention.

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

[0051] Figure 8 This is a schematic diagram showing the MBB beam optimization results provided by different methods in the embodiments of the present invention.

[0052] Figure 9 The objective function, i.e., the compliance value, is shown as an iterative curve for the three methods provided in the embodiments of the present invention.

[0053] Figure 10 Table 1 compares the topology optimization results under different algorithms provided in the embodiments of the present invention.

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

[0055] Figure 12 This is a schematic diagram showing the optimization results of cantilever beams using different methods provided in the embodiments of the present invention.

[0056] Figure 13 Iterative graphs of the objective function for different methods provided in embodiments of the present invention.

[0057] Figure 14 Table 2 compares the topology optimization results under different algorithms provided in the embodiments of the present invention.

[0058] Figure 15 This is a schematic diagram of a three-dimensional cantilever beam structure provided for an embodiment of the present invention.

[0059] Figure 16 The following are optimized structural diagrams of a three-dimensional cantilever beam under various single working conditions provided in the embodiments of the present invention.

[0060] Figure 17 The diagram shows the optimized structure of a three-dimensional cantilever beam at different iteration steps, as provided in the embodiments of the present invention.

[0061] Figure 18 The objective function iteration diagram for each single working condition of the three-dimensional cantilever beam structure provided in the embodiments of the present invention is shown.

[0062] Figure 19 The diagram shows the optimized three-dimensional cantilever beam structure under different combinations of working condition weight coefficients, as provided in the embodiments of the present invention.

[0063] Figure 20 The three-dimensional cantilever beam reliability topology optimization structure diagram provided for embodiments of the present invention with a working condition weight coefficient of (0.5, 0.5). Detailed Implementation

[0064] The technical concept of this invention is as follows:

[0065] First, certain idealized premises and assumptions are proposed for each phase material. For example, in the optimization process, each material inside the discrete unit is isotropic, and the properties of each phase material in the unit change with the relative density of the unit in an exponential relationship.

[0066] Secondly, the material design domain is divided into grid cells, and multiphase materials are allocated to each cell so that the volume fraction of the multiphase materials satisfies the sum of 1. The alternating active phase algorithm proposed by Tavakoil and Mohseni in 2013 is used to perform topology optimization by combining different phase materials.

[0067] Finally, for combinations of different phase materials, an extended SIMP method is used to establish a material interpolation model for multiphase material topology optimization. The volume fraction of each element is iteratively calculated to obtain the reliability topology optimization results for the multiphase material structure. This allows for the establishment of... Figure 1 The multiphase material shown has clearly defined boundaries and an optimized structure with reliability.

[0068] The present invention will now be described in detail with reference to embodiments and accompanying drawings. However, it should be understood that the embodiments and drawings are for illustrative purposes only and do not constitute any limitation on the scope of protection of the present invention. All reasonable modifications and combinations included within the inventive spirit of the present invention fall within the scope of protection of the present invention.

[0069] The invention will now be further described with reference to the accompanying drawings.

[0070] Example 1

[0071] like Figure 6 This embodiment provides a reliability topology optimization design method for multiphase material structures, based on the alternating active phase algorithm. This multiphase material topology optimization includes two layers of loops, inner and outer. Its core lies in allocating the volume fraction of each material to achieve a predetermined minimum compliance of the structure. The basic steps of its optimization design are as follows:

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

[0073] S2: Set the outer loop termination condition, 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 cycle, b is the second phase material phase of the two-phase cycle, and p is the penalty coefficient. For specific theory, refer to the alternating active phase algorithm in the existing literature 1.

[0074] S3: First, perform finite element analysis on the element model to obtain the overall compliance.

[0075] S4: Perform sensitivity analysis on the overall compliance to obtain the sensitivity.

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

[0077] S6: Optimize the density update criteria, calculate the iteration factor based on the filtered sensitivity, judge the iteration factor, and obtain the optimized unit density for each unit;

[0078] S7: Determine if iterin and iterinmax are equal. iterin and iterinmax refer to the number of iterations in the inner loop and the maximum number of iterations in the inner loop, respectively. If they are equal, proceed to S8; otherwise, return to S3.

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

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

[0081] S10: Determine the termination condition of the outer loop by checking if interout and interoutmax are equal. interout and interoutmax refer to the number of iterations and the maximum number of iterations of the outer loop, respectively. If they are equal, plot the cell density image, obtain the topology optimization result, and end the outer loop; otherwise, return to S1.

[0082] S1 to S10 are the outer loops, which follow the alternating active phase algorithm; S3 to S7 are the inner loops, which are performed in the order of a from 1 to p-1 and b from a+1 to p.

[0083] The S6 optimization criterion density update specifically employs an intermediate density unit suppression method based on optimization criterion: The volume fraction of phase a material is updated using optimization criteria. This step applies an intermediate density method based on optimization criteria, as detailed in Formula 18. Note that the total material volume fraction 1 minus the volume fraction excluding phases a and b yields the volume fractions of phases a and b in this internal cycle, and r. Then, the volume fraction of phase b material is obtained by subtracting the volume fraction of phase a material from r. For the specific theoretical details, refer to the alternating active phase algorithm in 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] In the formula, m is the number of material phases. The modulus of elasticity of the m-phase material is represented by x, where M is the total number of phases in the material, and x is the modulus of elasticity of the m-phase material. em Let represent the volume fraction of phase m material within a single cell, p be the penalty coefficient, and the sum of the relative volume fractions of each cell equals 1.

[0087]

[0088] Therefore, the material interpolation function for topology optimization of two-phase materials simplifies to:

[0089]

[0090] Schematic diagram as follows Figure 2 As shown.

[0091] Step S3: The multiphase material topology optimization model, with minimum structural compliance as the objective function and volume constraints as the constraint conditions, is described as follows:

[0092]

[0093] Based on the multiphase material interpolation formula described in the previous section, the element stiffness matrix K in finite element analysis is:

[0094]

[0095] in, This represents the relative volume fraction of the m-th material per unit cell. Let m be the stiffness matrix of the m-th material. Substituting it into the mathematical model above, we get:

[0096]

[0097] In the formula, n is the number of units, V m To optimize the volume of the m-th material, v e f represents the unit volume. m u represents the volume fraction of the m-th material. e Let F be the element nodal displacement, K be the force on the model, C(x) be the overall stiffness matrix, C(x) be the overall flexibility of the model, and x be 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 upper and lower limit constraints.

[0098] In step 1, various methods for multiphase material topology optimization have emerged. Among them, the alternating active phase algorithm proposed by Tavakoil and Mohseni in 2013 stands out due to its simple approach, strong scalability, and high computational efficiency. The multiphase material topology optimization model in 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 The value is determined by (m = 1, 2, ..., M). α m The following relationship must be satisfied:

[0100] l m ≤α m ≤u m (8)

[0101]

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

[0103]

[0104] In the formula, Λ m These are artificially imposed volume constraints on the materials of each phase. For ease of understanding and calculation, a vector field α = {α1, α2, ..., α...} is used. M The set of design variables is represented by}. 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 to simplify the equations, the material interpolation function is represented by δ(α) for both single-phase and multi-phase materials. Partial differential equations (PDEs) are also an important component of topology optimization problems. Their solution is represented by U(x) = μ(α(x)). The partial differential operators constrained by PDEs are... Therefore, the discrete mathematical model for the topology optimization problem of multiphase materials is:

[0105]

[0106] In the formula, the objective function J(...) is the integral of α and U over the design domain Ω.

[0107] The brilliance of the alternating active phase algorithm lies in its decomposition of the multiphase material optimization problem into multiple two-phase material topology optimization subproblems, which are implemented through nested loops. Each outer loop requires solving M(M-1) / 2 subproblems, and the two-phase material topology optimization structure in each subproblem is obtained from the two-phase material topology optimization calculation in the previous iteration. For example... Figure 3 As shown, taking four-phase materials as an example, the solution order for subproblems of two-phase materials is given.

[0108] In this embodiment, the subproblem in the alternating active phase algorithm of step S6 is solved based on the variable density method. The superscripts "a" and "b" represent the two phase materials to be solved in the subproblem. Therefore, in each iteration, the M-2 phase material, excluding these two phases, remains constant. Thus, in each iteration, the volume constraint r of the active phase materials a and b... ab It can be calculated using the following formula:

[0109]

[0110] In the formula α m This represents the volume fraction of each unit cell in the m-phase material, because each unit cell has... Therefore, if the density value of phase a is determined, the density value of phase b is:

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

[0112] As shown in Equation 14, in a single optimization subproblem, the temporary upper bound of phase a is:

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

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

[0115]

[0116] As is well known, most topology optimization methods based on variable density methods share a common problem: the emergence of intermediate density units, also known as gray-scale units. This numerical instability significantly reduces the manufacturability of the optimized structure. Therefore, this invention employs an intermediate density unit suppression method based on optimization criteria. This method is simple to implement and consistently satisfies the constraints during the optimization process.

[0117] In step S5: Groenwold and Etman proposed in 2007 to apply a gray-scale unit suppression function to the design variables when updating design variables using the optimization criterion method, thereby suppressing the gray-scale 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 update process, and q is the inhibition factor. This is the element optimization operator. Therefore, the design variable update formula for the gray-scale element suppression method based on the optimization criterion method can be changed to:

[0120]

[0121] As shown in Formula 18, when q = 1, it represents the original OC algorithm. As the value of q gradually increases, the intermediate density cells all move closer to 0 or 1, such as... Figure 4 and 5 As shown.

[0122] Example 1

[0123] The first example considers, for instance... Figure 7The MBB beam structure shown has a length-to-width ratio of 4:1 and structural dimensions of 192mm × 48mm. The structure is discretized into 192 × 48 elements. The lower left corner of the structure is constrained for both horizontal and vertical displacement, while the lower right corner is constrained for vertical displacement. A concentrated vertical force of 1N is applied at the midpoint of the top edge.

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

[0125] The results of the classical alternating active phase algorithm, the gray-level unit suppression method based on the power function, and the gray-level unit suppression method based on the exponential function in existing literature 1 are analyzed and compared to select the most suitable method. The initial value of the suppression factor q is set to 1, and its growth step size is 0.01 per cycle. The optimization results are as follows: Figure 8 As shown. Among them, Figure 8 .a represents the classic algorithm from the literature; Figure 8 .b represents a gray-scale unit suppression method based on power functions; Figure 8 .c represents a gray-scale unit suppression method based on an exponential function. Figure 9 The graph shows the iterative curves of the objective function, i.e., the compliance value, under the three methods.

[0126] Figure 10 Table 1 lists the optimization results data for three different filtering algorithms under the same parameters, including the objective function value, number of iterations, and grayscale unit ratio.

[0127] according to Figure 8 As shown, all three algorithms can optimize complete multiphase material MBB beam structures. A common feature is that the solid material 1 with a higher elastic modulus is distributed around the perimeter of the beam, while the central support is filled with solid material 2 with a lower elastic modulus. The specific structural details vary slightly between the different methods, but overall, the materials with higher stiffness are generally distributed along the main force transmission paths of the structure. This verifies the effectiveness of the alternating active phase algorithm and proves that it can improve material utilization efficiency while ensuring the overall structural performance. Furthermore, it is quite evident that compared to the original alternating active phase algorithm in existing literature 1, both the power function-based gray-scale element suppression method and the exponential function-based gray-scale element suppression method can obtain optimized structures with clearer boundaries. This is evident from… Figure 10 The proportion of grayscale units in Table 1 makes the situation more intuitive.

[0128] from Figure 9 You can see the iterative curves of the objective function under the three methods, and then combine them with... Figure 10The data in Table 1 shows that the objective functions of the three methods gradually converge around the tenth iteration. However, the original alternating active phase algorithm has the largest final objective function value, followed by the power function suppression method, and the exponential function method has the smallest objective function value. This means that the original alternating active phase algorithm optimizes the smallest structural stiffness, while the gray-scale unit suppression method based on the exponential function optimizes the largest structural stiffness.

[0129] Example 2

[0130] The second example considers, for example Figure 11 The planar cantilever beam structure shown has a length-to-width ratio of 2:1 and structural dimensions of 96mm × 48mm. The structure is discretized into 96 × 48 elements. The left end of the beam is constrained by a fixed support, and a concentrated force of one unit magnitude is applied to the lower right corner.

[0131] In this example, a three-phase solid material optimization problem is considered. Red represents solid material 1, with an elastic modulus of 5 and a volume fraction of 0.2; blue represents solid material 2, with an elastic modulus of 3 and a volume fraction of 0.1; green represents solid material 3, with an elastic modulus of 1 and a volume fraction of 0.1; and the porous material is white (colorless), with an elastic modulus of 1e-9 and a volume fraction of 0.6. Other parameters, such as the suppression factor and cycle step size, are set the same as in the first example. The results are as follows: Figure 12 As shown, where Figure 12 .a represents the results from the classic method in the literature. Figure 12 .b represents the gray-scale unit suppression method based on the exponential function.

[0132] Similar to the previous example, first plot the iterative curves of the objective function under these two methods, and then list the objective function values, iteration counts, and grayscale unit ratios of the optimization results under these two different algorithms. For example... Figure 13 As shown.

[0133] from Figure 14 Table 2 clearly shows that the gray-scale element suppression method based on the exponential function yields a clearer optimized structure diagram, and its final objective function value is smaller than that obtained by the original alternating active phase method. In summary, both gray-scale element suppression methods effectively suppress gray-scale elements, resulting in clearer structures. Although the gray-scale element suppression method based on the power function has the lowest proportion of gray-scale elements in its optimized structure, it sometimes exhibits small branch structures and large block structures, making it less rational than the gray-scale element suppression method based on the exponential function. Furthermore, the optimized structure has lower stiffness than that of the exponential function suppression method. Therefore, the multiphase material optimization models in subsequent chapters will all adopt the gray-scale element suppression method based on the exponential function to suppress this numerical instability of gray-scale elements.

[0134] Example 3

[0135] The final example extends the two-dimensional multiphase material topology optimization problem to a three-dimensional multiphase material topology optimization problem. This code is a modification of the classic three-dimensional single-phase material code, based on the two-dimensional multiphase material code. Except for the finite element part, its core idea and basic process are completely consistent.

[0136] Consider the three-dimensional cantilever beam structure shown in the figure. The structural geometry is 30cm × 15cm × 8cm. The left boundary of the structure is fully constrained. The loads F1 and F2 are as follows: Figure 15 As shown, all elements are 1N in size. The structure is discretized into 3600 units, with a penalty coefficient of 3 and a filter radius of 5.

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

[0138] Figure 16 This demonstrates an optimized cantilever beam structure under a single load condition, in which... Figure 16 .a represents the optimization result for operating condition 1. Figure 16 Figure .b shows the optimization results for load case 2. As can be seen from the figure, the structures under both load cases are clear and reasonable. When load F1 acts alone, the area around the load application point is filled with the red material, which has the highest elastic modulus (i.e., the highest stiffness). The part connected to the fixed end is filled with a slightly smaller blue material, and the middle part is filled with the green material, which has the lowest elastic modulus. The optimized structure for load case 2 also follows this pattern, only with different geometric dimensions. This is because the maximum stress in the cantilever beam structure is concentrated at the load application point and the contact point at the fixed end. Therefore, these main load-bearing parts should be filled with materials with high elastic modulus. This is consistent with the pattern and results obtained from the two-dimensional cantilever beam example in Chapter 4. This further confirms the effectiveness of this three-dimensional multi-material method.

[0139] Six iterations—the 10th, 20th, 50th, 100th, 150th, and 200th—are selected to further demonstrate their optimization process. For example... Figure 17 As shown. The structure obtained from single-condition topology optimization can be further used to obtain a multi-condition topology optimization structure. Here, two sets of condition weight coefficient combinations are selected for calculation, with (ω1,ω2) set as (0.2,0.8) and (0.5,0.5) respectively, as shown. Figure 19 As shown. Further selecting the second set of working condition weight coefficients, the optimization structures for different β values ​​are listed, such as... Figure 18 As shown. From Figure 20As can be seen, similar to the two-dimensional examples calculated earlier, the configuration of the reliability-optimized topology structure remains basically unchanged compared to the deterministic topology-optimized structure, and for multiphase materials, the material distribution also remains 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 merely preferred embodiments of the present invention, and the scope of protection of the present invention is not limited to the above embodiments. All technical solutions falling within the scope of the present invention's concept are within the scope of protection of the present invention. It should be noted that for those skilled in the art, improvements and modifications made without departing from the principles of the present invention should also be considered within the scope of protection of the present invention.

Claims

1. A reliability topology optimization design method for multiphase material structures, characterized in that: The following steps are involved: S1: Initialization, dividing the elements, performing finite element initial settings, setting topology optimization initial parameters, including material elastic modulus, Poisson's ratio, penalty coefficient, multiphase material volume fraction distribution, filtration radius, and inhibition factor; S2: Set the outer loop termination condition, 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 cycle, b is the second phase material phase of the two-phase cycle, and p is the penalty coefficient. S3: First, perform finite element analysis on the element model to obtain the overall compliance. S4: Perform sensitivity analysis on the overall compliance to obtain the sensitivity. S5: Enter sensitivity filtering calculation, perform sensitivity filtering calculation on the sensitivity, and obtain the filtered sensitivity; S6: Optimize the density update criteria, calculate the iteration factor based on the filtered sensitivity, judge the iteration factor, and obtain the optimized unit density for each unit; S7: Determine if iterin and iterinmax are equal. iterin and iterinmax refer to the number of iterations in the inner loop and the maximum number of iterations in the inner loop, respectively. If they are equal, proceed to S8; otherwise, return to S3. S8: Determine if b and M are equal. M refers to the maximum number of material phases. If they are equal, proceed to S9. If they are not equal, then b = b + 1 and return to S2. S9: Determine if a and M-1 are equal. If they are equal, proceed to S10; otherwise, a = a + 1 and return to S2. S10: Determine the termination condition of the outer loop by checking if interout and iteroutmax are equal. interout and iteroutmax refer to the number of iterations and the maximum number of iterations of the outer loop, respectively. If they are equal, plot the cell 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 structures according to claim 1, characterized in that: In step S3, the multiphase material topology optimization model, with minimum structural compliance as the objective function and volume constraints as the constraint conditions, is described as follows: In finite element analysis, the element stiffness matrix K is: in, This represents the relative volume fraction of the m-th material per unit cell. Let m be the stiffness matrix of the m-th material. Substituting it into the mathematical model, we get: In the formula, n is the number of units, V m To optimize the volume of the m-th material, v e f represents the unit volume. m u represents the volume fraction of the m-th material. e Let F be the element nodal displacement, F be the force on the model, K be the overall stiffness matrix, C(x) be the overall flexibility of the model, and x be the volume fraction.

3. The reliability topology optimization design method for multiphase material structures according to claim 1, characterized in that: Step S6: In each iteration of the calculation, the volume constraint r of the active phase materials a and b is... a b can be calculated using the following formula: In the formula α m This represents the volume fraction of each unit cell in the m-phase material, because each unit cell has... Therefore, if the density value of phase a is determined, the density value of phase b is: r b ＝r ab -r a In a single optimization subproblem, the temporary upper bound of phase a is: in a,temp =min(in a ,r ab ) Since the lower bound of the a-phase material remains unchanged, the optimization subproblem of the two-phase material can be abstractly expressed as:

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

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