A perimeter-constrained based heat conduction structure topology optimization method

Abstract

The application discloses a kind of based on perimeter constraint heat conduction structure topological optimization method.The method extracts structure boundary using two-step filtering projection, and uses structure boundary with certain width as structure perimeter, and constructs heat conduction structure topological optimization model based on perimeter constraint.The heat conduction structure topological optimization method based on perimeter constraint of the present application obtains an explicit generation method of variable-density topological optimization structure boundary by the way of modified two-step filtering projection, and solves the problem that heat conduction structure is complex after topological optimization in the prior art, difficult to process and manufacture or high processing cost.

Description

Technical Field

[0001] This invention relates to the field of thermal conductive structure design technology, and specifically to a topology optimization method for thermal conductive structures based on perimeter constraints. Background Technology

[0002] Thermal conductivity is a crucial performance indicator for products, and optimizing the heat dissipation performance of materials has become a common requirement for improving product quality. In aerospace, automotive, and microelectronic devices, components often require excellent thermal conductivity. When heat cannot be dissipated promptly, the operating temperature of the equipment will continuously rise, affecting its performance, reducing its lifespan, and in severe cases, causing damage. Therefore, improving the thermal conductivity of structures has become a major research focus in materials science.

[0003] Topology optimization is a simulation-driven structural design method. It offers significant advantages due to its high degree of design freedom, transforming the spatial arrangement of materials into an optimization problem that fully utilizes material properties. Currently, topology optimization is widely used in heat transfer. However, topology-optimized structures are complex, particularly in heat conduction, and the optimized results often contain many intractable, fine dendritic structures, posing challenges for traditional fabrication methods or high processing costs.

[0004] Regarding research on manufacturability, some scholars have proposed some topology optimization methods for the constraints of traditional processing technology. These methods have been applied in the topology optimization of mechanical structures, but they are difficult to meet the processing and manufacturing requirements of the topology optimization of heat transfer structures.

[0005] Although there are currently methods for generating coating structures based on SIMP, which can generate coating structures with controllable thickness on the surface of a substrate structure, and the coating structure of a certain width can be regarded as the perimeter of the structure, this method is mainly applied to the field of mechanical load-bearing structures and cannot be applied to the field of thermal conduction structures with complex optimization results.

[0006] Therefore, this invention develops a new design model for heat-conducting structures based on perimeter constraints. Summary of the Invention

[0007] To address the problems existing in the prior art, this invention provides a topology optimization method for thermally conductive structures based on perimeter constraints. By modifying the projection threshold and refining the structure perimeter, the structure perimeter is extracted using a SIMP-based coating structure generation method. The structure perimeter obtained by this method satisfies the requirements of the gradient method, resulting in an optimized structure that simultaneously considers manufacturing constraints and thermal conductivity.

[0008] The present invention provides a method for topology optimization of heat-conducting structures based on perimeter constraints, comprising the following steps: s1, extracting the structural boundary using a two-step filtered projection method, and using the structural boundary of a certain width as the structural perimeter; s2, applying constraints to the structural perimeter, incorporating the perimeter constraints into a traditional optimization model, constructing a topology optimization model for heat-conducting structures based on perimeter constraints, and selecting heat dissipation weakness as the optimization objective; s3, performing sensitivity analysis on the constraints and objective function, and solving the model using the MMA optimization algorithm to obtain the optimal material distribution structure.

[0009] Preferably, the two-step filtering projection method in s1 for extracting the structural boundary is as follows: First, a clear substrate structure is obtained through the first step of filtering and projection; a second step of filtering is introduced to smooth the boundary of the substrate structure; after the density gradient is calculated based on the smoothing, the density gradient norm with a value range of 0-1 is obtained through normalization; the gradient norm is used as the coating structure; finally, the gradient norm is projected a second time to obtain a clear structural boundary region; by changing the filtering radius of the second step of filtering, the coating structure thickness can be controlled, and a smaller projection threshold is selected in the second projection to add perimeter constraints.

[0010] Preferably, the method for adding perimeter constraints by selecting a smaller projection threshold in the second projection is as follows: Added coefficient Get new variables Right now

[0011]

[0012] in The density of the substrate structure after projection. Let the gradient norm be the one after projection. This refers to the refined gradient norm, which is also the final perimeter of the structure. This ensures the existence of the original structure within its specified region. Sometimes, The regions where the original structure does not exist, i.e. Sometimes, Therefore, Using this to represent material boundaries can avoid the problem of boundaries covering the original structural area and improve the accuracy of boundary calculations.

[0013] Preferably, in step s2, constraints are applied to the obtained material boundaries, and the SIMP method is incorporated to obtain a topology optimization model of the heat-conducting structure based on perimeter constraints:

[0014]

[0015]

[0016] in Here, is the divergence operator, k is the thermal conductivity, K is the global thermal conductivity matrix, T is the global temperature vector, P is the equivalent temperature load vector, Φ is the design variable, f(Φ) is the volume constraint, g(φ) is the perimeter constraint, c(Φ) is the objective function, and V is the volume fraction.

[0017] Preferably, the S3 sensitivity analysis specifically involves: first, performing sensitivity analysis on the filtering and projection functions, and then analyzing the sensitivity of the filtered density to the design variable. for:

[0018]

[0019]

[0020] T=∑∫NdΩ

[0021] Sensitivity of post-projection density to pre-projection density for:

[0022]

[0023] The sensitivity of the objective function to design variables can be obtained using the chain rule. for:

[0024]

[0025] Among them, the sensitivity of the objective function to the projected density for:

[0026]

[0027]

[0028] Sensitivity of design domain volume constraints to design variables for:

[0029]

[0030] The sensitivity of boundary constraints to design variables is:

[0031]

[0032]

[0033]

[0034] In the above sensitivity analysis, N is the shape function matrix. The density of the filtered material. Let R be the projected material density, R be the filter radius, β and η be the projection slope and projection threshold, respectively, and t be the projection threshold.e and k e These are the element temperature vector and the element thermal conductivity matrix, respectively, k min is a very small thermal conductivity value, considered as the thermal conductivity of the air-attacked area, and p is the penalty coefficient.

[0035] This invention provides a topology optimization method for heat-conducting structures based on perimeter constraints. By modifying the two-step filtering projection method, an explicit generation method for the boundary of a variable-density topology-optimized structure is obtained, which solves the problems of complex structure, difficulty in processing and manufacturing, or high processing cost of heat-conducting structures after topology optimization in the prior art. Attached Figure Description

[0036] Figure 1 This is a flowchart of the perimeter extraction process in the heat-conducting structure topology optimization method based on perimeter constraints of the present invention.

[0037] Figure 2 The boundary of the two-dimensional heat conduction structure of the present invention is extracted, wherein (a) is the material itself and (b) is the material boundary.

[0038] Figure 3 Figure (a) shows the boundary of the original structure, and Figure (b) shows the boundary of the structure after refinement, i.e., the perimeter.

[0039] Figure 4 This is a schematic diagram of the boundary conditions in this embodiment.

[0040] Figure 5 The results and temperature distribution diagram are obtained when optimizing the heat-conducting structure using the general SIMP method without applying perimeter constraints in Example 1.

[0041] Figure 6 The image shows the results and temperature distribution diagram obtained when optimizing the heat-conducting structure by applying perimeter constraints in Example 1.

[0042] Figure 7 This is a schematic diagram of the boundary conditions in Example 2.

[0043] Figure 8 The results and temperature distribution diagram are obtained when the heat-conducting structure is optimized using the general SIMP method without applying perimeter constraints in Example 2.

[0044] Figure 9 The image shows the results and temperature distribution diagram obtained when optimizing the heat-conducting structure by applying perimeter constraints in Example 2. Detailed Implementation

[0045] It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.

[0046] The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without creative effort are within the scope of protection of the present invention.

[0047] This invention first obtains a clear substrate structure through a first step of filtering and projection, namely, a substrate filtering projection method based on SIMP, calculates the material density gradient, and obtains the density gradient norm with a value range of [0,1] through normalization; the gradient norm is used as the coating structure, as detailed below.

[0048] The first step of this invention mainly studies the optimization problem of steady-state heat conduction. The principle is as follows: according to Fourier's law, steady-state heat conduction satisfies the following equation:

[0049]

[0050] in Let Q represent the divergence operator, k be the thermal conductivity of the material, T be the temperature, and Q be the internal heat source.

[0051] The equation can be solved under two types of boundary conditions:

[0052] Γ D :T=T0

[0053]

[0054] Γ D This is the Dirichlet boundary, where the temperature is a constant T0, Γ N This is the Newman boundary, on which a certain flux q0 can be applied. When q0 = 0, Γ N It is a thermally insulating boundary.

[0055] The finite element solution form of the above equation is:

[0056] KT = P

[0057] Where K is the overall thermal conductivity matrix, T is the overall temperature vector, and P is the equivalent temperature load vector, the expression for P is:

[0058] P=∫ Ω ρQN T dΩ+∫ s q f N T dA

[0059] Where ρ is the material density, Q is the internal heat source, and q f Let N be the boundary heat flux, and N be the shape function matrix.

[0060] With heat dissipation weakness as the objective and the volume fraction of thermally conductive material as a constraint, the following topology optimization model for the thermally conductive structure based on SIMP can be obtained:

[0061]

[0062]

[0063] Where Φ is the design variable, f(φ) is the volume constraint, and c(Φ) is the objective function. This paper adopts the commonly used heat dissipation weakness as the objective, namely:

[0064]

[0065] Its finite element discrete expression is:

[0066]

[0067] Where te and k0 are the unit temperature vector and the element thermal conductivity matrix, respectively. This represents the thermal conductivity of each element. This objective function measures the structure's heat dissipation capacity. Practical experience has shown that, under both boundary conditions, selecting either heat dissipation weakness or the average temperature of the design domain as the objective yields the same result.

[0068] Helmholtz equation filtering:

[0069] To prevent grid dependency and checkerboard patterns that often occur in topology optimization, it is necessary to filter design variables. Here, the Helmholtz equation is used for filtering:

[0070]

[0071] in R is the density after filtration, R is the filtration radius, and Φ is the original density.

[0072] Hyperbolic tangent projection:

[0073] While filtering avoids mesh dependency and checkerboard patterns, it introduces intermediate density elements at the edges of the optimized structure. These intermediate density elements lack practical physical meaning and cannot be processed. To obtain a clear 0-1 distribution, a hyperbolic tangent function is used to filter the density. Project:

[0074]

[0075] in Let denot be the density after projection, and β and η be the adjustment parameters of the projection function. The larger β is, the closer the projection function is to the step function.

[0076] SIMP material interpolation:

[0077] The thermal conductivity of the material is interpolated using the SIMP method:

[0078]

[0079] Where k is used as k SIMP Let represent the interpolated thermal conductivity of the material, k0 be the thermal conductivity value of the solid material, and k min ρ is a very small thermal conductivity value, which is regarded as the thermal conductivity value of the pore region to avoid numerical instability problems in the solution. ρ is a penalty coefficient, which can make the intermediate density further tend to 0 or 1.

[0080] The second step of the core content of this invention, filtering projection, specifically involves the following steps: A second-step filtering process is introduced to smooth the boundaries of the substrate structure; a density gradient norm within the range [0,1] is obtained through normalization; the gradient norm is used as the coating structure; and finally, a second projection is performed on the gradient norm to obtain a clear boundary region. The entire process is as follows: Figure 1 As shown. By changing the filtration radius of the second-step filtration, the coating thickness can be controlled. The larger the filtration radius of the second step, the larger the coating thickness. By reasonably selecting the coating thickness, the volume of the generated coating structure can be equivalent to the perimeter constraint of the structure. The boundary generated by the above method for the two-dimensional heat conduction structure is shown in the figure. Figure 2 As shown. Because a constraint needs to be imposed on the perimeter, using a general projection threshold in the projection step would result in the generation of grayscale cells. Therefore, a smaller projection threshold η = 0.01 is selected in the second projection. When the second projection threshold is small, the boundary region generated by the second projection will be wider, causing the boundary region to cover the original structure. This covering phenomenon is particularly obvious in areas with dense material distribution. This leads to an increase in the error of boundary area calculation, thus making it impossible to accurately apply boundary constraints. To address this problem, in Added coefficient Get new variables Right now

[0081]

[0082] in The density of the substrate structure after projection. Let the gradient norm be the one after projection. This refers to the refined gradient norm, which is also the final structure perimeter; this ensures that the original structure exists within the specified region. by Using this to represent material boundaries can avoid the problem of boundaries covering the original structural area (see...). Figure 3 This improves the accuracy of boundary calculations.

[0083] Topology optimization model for heat-conducting structures based on perimeter constraints:

[0084] The material boundary can be obtained using the above method. By applying constraints to it and incorporating the SIMP method, a topology optimization model of the heat-conducting structure based on perimeter constraints can be obtained.

[0085]

[0086]

[0087] Where k is the thermal conductivity, K is the overall thermal conductivity matrix, T is the overall temperature vector, P is the equivalent temperature load vector, Φ is the design variable, f(Φ) is the volume constraint, g(φ) is the perimeter constraint, c(Φ) is the objective function, and V is the volume fraction. The commonly used heat dissipation weakness is adopted as the objective.

[0088] Sensitivity analysis:

[0089] To solve the topology optimization model using the gradient descent method, it is necessary to first calculate the sensitivity of the objective function c(φ), volume constraint f(φ), and boundary constraint g(φ) to the design variable ρ, and then use the MMA algorithm to solve it.

[0090] First, sensitivity analysis is performed on the filtering and projection functions. The sensitivity of the density to the design variable after filtering is:

[0091]

[0092]

[0093] T=∑∫NdΩ

[0094] The sensitivity of the projected density to the unprojected density is:

[0095]

[0096] According to the chain rule, the sensitivity of the objective function to the design variables is:

[0097]

[0098] The sensitivity of the objective function to the projected density is:

[0099]

[0100]

[0101] The sensitivity of the design domain volume constraint to design variables is:

[0102]

[0103] The sensitivity of boundary constraints to design variables is:

[0104]

[0105]

[0106]

[0107] In the above sensitivity analysis The density of the filtered material. Let R be the projected material density, R be the filter radius, β and η be the projection slope and projection threshold, respectively, and t be the projection threshold. e and k e These are the element temperature vector and the element thermal conductivity matrix, respectively, k min is a very small thermal conductivity value, considered as the thermal conductivity of the air-attacked area, and p is the penalty coefficient.

[0108] An embodiment of the present invention provides a topology optimization method for heat-conducting structures based on perimeter constraints, see [link to relevant documentation]. Figure 1 In this embodiment, the design domain is a square with a size of 1×1 (m), the grid size is 0.01m, the thermal conductivity of the high thermal conductivity material is k=1W / (m·K), the thermal conductivity of the low thermal conductivity material is k0=0.001W / (m·K), the volume fraction of the material is set to 50% in the optimization, the two filtering radii are R1=R2=0.02m, the two projection slope values ​​β1 and β2 are both [8, 16, 32, 64, 128, 256], the initial iteration selects β=8, and then the β value is changed every 100 iterations. The first projection threshold is η1=0.5, and the second projection threshold is η2=0.01.

[0109] The boundary conditions of Example 1 are as follows: Figure 4 As shown, in this embodiment, a uniform heat source Q = 1W / m is set within the design domain. 2 A fixed temperature boundary with a length of 0.2m and a temperature of T=0℃ is set in the middle region of the bottom, while the remaining boundaries are thermally insulating boundaries. Without applying perimeter constraints, the results and temperature distribution obtained when optimizing the design using the general SIMP method are as follows: Figure 5 As shown, the structure exhibits numerous branches of varying sizes. The optimization results, perimeter distribution, and temperature distribution differ depending on the applied perimeter constraints. Figure 6As shown in Figures (a), (b), (c), and (d), the perimeter constraints are 80%, 70%, 60%, and 50%, respectively. When an 80% boundary constraint is applied, the number of fine branches in the optimized result decreases, and the high thermal conductivity material tends to concentrate in the larger branches. However, the fine branches do not completely disappear; most of the fine branch structures degenerate into needle-like structures. As the constraint strengthens, when the boundary constraint gradually decreases from 80% to 50%, the fine branches also gradually decrease, and the needle-like structures continue to degenerate and shorten. When the boundary constraint reaches 50%, the fine branches completely disappear. Compared with the optimization results obtained by traditional methods, the manufacturability of this result is significantly improved, and the manufacturing cost is greatly reduced. In addition, Table 1 shows the average temperature of the design domain for each optimization result. It can be seen that the average temperature increases slightly with the strengthening of the perimeter constraint. In practical applications, an appropriate perimeter constraint can be selected according to specific requirements to obtain an optimized structure that considers both manufacturing constraints and thermal conductivity.

[0110] Table 1. Average Temperature of Design Domain for Each Optimization Result

[0111] Design domain average temperature (K) 274.09 274.27 274.44 274.79 276.24

[0112] The boundary conditions of Example 2 are as follows Figure 7 As shown, in this embodiment, a uniform heat source Q = 2W / m is set within the design domain. 2 A uniform inward heat flux of q = 1 W / m is applied to the upper boundary to simulate heat transfer through the flow direction into the design domain. The bottom middle region is a fixed temperature boundary at T = 0℃, and the remaining boundaries are thermally insulating boundaries. First, the structure is optimized using the SIMP method without perimeter constraints, and the results are as follows... Figure 8 As shown. The optimization results, perimeter distribution, and temperature distribution are as follows when different perimeter constraints are applied. Figure 9 As shown in the figure, the perimeter constraints corresponding to Figures (a), (b), (c), and (d) are 80%, 70%, 60%, and 50%, respectively. Table 2 shows the design domain average temperature for each optimization result. It can also be seen that the perimeter constraint limits the generation of small branches in the structure, significantly improving the manufacturability of the structure.

[0113] Table 2 shows the average temperature of the design domain for each optimization result.

[0114] Design domain average temperature (K) 276.56 276.96 277.56 278.15 279.70

[0115] This invention provides an explicit generation method for variable density topology optimization structural boundaries by modifying the two-step filtering projection method. Based on the thermal conductivity structure topology optimization model and solution method with perimeter constraints, the average temperature of the results is compared through thermal conductivity structure optimization experiments under different perimeter ratio constraints, verifying the effectiveness of the proposed method. It also provides a variety of thermal conductivity structure topology optimization schemes that balance manufacturing constraints and thermal conductivity performance by selecting appropriate perimeter constraints according to actual needs.

[0116] The above are merely preferred embodiments of the present invention and do not limit the scope of the patent. Any equivalent structural or procedural transformations made based on the description and drawings of the present invention, or direct or indirect applications in other related technical fields, are similarly included within the scope of patent protection of the present invention.

Claims

1. A topology optimization method for heat-conducting structures based on perimeter constraints, characterized in that, Includes the following steps: s1, extract the structural boundary using a two-step filtered projection method, and use the structural boundary of a certain width as the structural perimeter; s2, imposing constraints on the structural perimeter, incorporating the perimeter constraint into the traditional optimization model, constructing a topology optimization model for the heat-conducting structure based on the perimeter constraint, and selecting heat dissipation weakness as the optimization objective; s3, performing sensitivity analysis on the constraints and objective function, and using the MMA optimization algorithm to solve the model to obtain the optimal material distribution structure; the two-step filtering projection method in s1 to extract the structural boundary specifically involves: firstly, obtaining a clear substrate structure through the first step of filtering and projection; then introducing a second step of filtering to smooth the boundary of the substrate structure, calculating the density gradient based on the smoothing, and obtaining the density gradient norm within the range of 0-1 through normalization, using the gradient norm as the coating structure, and finally performing a second projection on the gradient norm to obtain a clear structural boundary region. By changing the filtering radius of the second step of filtering, the coating structure thickness can be controlled, and a smaller projection threshold is selected in the second projection to add perimeter constraints. The smaller projection threshold is 0.

01. Pre-added coefficient ( ), to obtain new variables ,Right now in The density of the substrate structure after projection. Let be the norm of the projected gradient. This is the gradient norm after refinement, which is also the final structure perimeter.

2. The topology optimization method for heat-conducting structures based on perimeter constraints according to claim 1, characterized in that, The topology optimization model for the heat-conducting structure is as follows: ， in For divergence operators, Thermal conductivity, The overall thermal conductivity matrix is... For the overall temperature vector, Let f(Φ) be the equivalent temperature load vector, Φ be the design variable, and f(Φ) be the volume constraint. For perimeter constraints, Let be the objective function. This represents the volume fraction.

Images