Multi-objective isogeometric multi-scale topology optimization method for high stiffness and high thermal conductivity

Abstract

The present invention belongs to the field of structural optimization, and discloses a multi-objective isogeometric multi-scale topology optimization method for high stiffness and high thermal conductivity, comprising the following steps: (1) calculating the equivalent elastic tensor and the equivalent thermal conductivity tensor of each gradient lattice sample of the porous structure to be optimized; (2) using the Kriging meta-model to fit the equivalent density and the corresponding equivalent elastic tensor and equivalent thermal conductivity tensor of all gradient lattice samples to construct a surrogate model, and further predicting the equivalent elastic tensor and equivalent thermal conductivity tensor of any equivalent density gradient lattice of the porous structure to be optimized; (3) constructing a multi-objective isogeometric multi-scale topology optimization model with the objective of minimizing both the static compliance and the heat dissipation compliance, and further optimizing the equivalent density of the gradient lattice in all cells within the macroscopic design domain of the porous structure to be optimized, so as to obtain an optimized three-dimensional porous structure. The present invention can simultaneously consider the stiffness performance and the heat dissipation performance of the porous structure.

Description

Technical Field

[0001] The present invention belongs to the technical field related to structural optimization, and more specifically, relates to a multi-objective isogeometric multi-scale topology optimization method for high stiffness and high thermal conductivity. Background Art

[0002] Porous structures are ubiquitous in nature, such as bones, honeycombs, sponges, etc., and have excellent properties such as super light weight, impact energy absorption, heat insulation and heat prevention, and can be flexibly designed according to the needs of the application field. The multi-scale topology optimization method is an intelligent design method for automatically finding the optimal material distribution. Due to its wide design freedom, it can design novel porous structures beyond people's intuition and experience. In engineering applications, porous structures often need to have good load-bearing capacity and heat dissipation capacity at the same time. Therefore, it is necessary to improve the current multi-scale topology optimization method so that it can design porous structures with high stiffness and high thermal conductivity.

[0003] Regarding the multi-scale topology optimization method for high stiffness and high thermal conductivity, relevant technical personnel in this field have conducted some research. For example, in Document 1: "A. Ali Musaddiq, S. Masatoshi. Toward multiphysics multiscale concurrent topology optimization for lightweight structures with high heat conductivity and high stiffness using MATLAB[J]. Structural and Multidisciplinary Optimization, 2022, 65(7).", the topological configuration of the microstructure and its distribution in the porous structure are optimized simultaneously, and multi-objective functions are used to consider the stiffness and thermal conductivity of the porous structure at the same time. In this method, macro and micro optimizations are carried out simultaneously, and the computational efficiency is very low. Moreover, this method considers the case of periodic and uniform arrangement of microstructures in the porous structure, with limited design freedom. At the same time, this method only studies two-dimensional problems and does not study three-dimensional problems. Regarding the multi-scale topology optimization method based on functionally graded lattices, relevant technical personnel in this field have also conducted some research. For example, in Document 2: "X. Liu, L. Gao, M. Xiao, et al. Kriging-assisted design of functionally graded cellular structures with smoothly-varying lattice unit cells[J]. Computer Methods in Applied Mechanics and Engineering, 2022, 390.", a multi-scale topology optimization method for functionally graded lattices is realized based on the Kriging surrogate model. This method considers the point-by-point gradient change of the lattice, improves the design freedom while reducing the computational cost. However, this method is only limited to optimizing the stiffness performance of the porous structure and does not further optimize its heat conduction performance.

[0004] Therefore, effectively releasing the design space, designing a three-dimensional porous structure with a gradually changing distribution of gradient lattices, and simultaneously improving the load-bearing performance and heat dissipation performance of the porous structure with relatively low computational cost and high computational accuracy are hot research issues to be solved currently. Summary of the Invention

[0005] In view of the above deficiencies or improvement requirements of the prior art, the present invention provides a multi-objective isogeometric multi-scale topology optimization method for high stiffness and high thermal conductivity. The method uses a level set function to construct a prototype lattice unit cell of a triply periodic minimal surface lattice, and interpolates the level set function of the prototype lattice unit cell through shape interpolation technology to obtain a series of gradient lattices. The equivalent elastic tensor and equivalent thermal conductivity tensor of the gradient lattice are obtained by using the homogenization method based on isogeometric analysis, and these data are input into the Kriging meta-model to construct a surrogate model that can predict the equivalent elastic tensor and equivalent thermal conductivity tensor of any equivalent density gradient lattice. The sub-objectives of minimizing the static compliance and minimizing the heat dissipation compliance are coupled in a normalized linear weighted manner to construct a multi-objective function, and then a multi-objective isogeometric multi-scale topology optimization model for high stiffness and high thermal conductivity is used to optimize the distribution of the gradient lattice in the macroscopic design domain. Finally, the corresponding gradient lattice configurations are filled into the macroscopic design domain one by one to obtain the final porous structure, while improving the load-bearing capacity and heat dissipation capacity of the porous structure, thus realizing the topology optimization process.

[0006] To achieve the above object, according to one aspect of the present invention, there is provided a multi-objective isogeometric multi-scale topology optimization method for high stiffness and high thermal conductivity, which mainly includes the following steps:

[0007] (1) Calculate the equivalent elastic tensor and equivalent thermal conductivity tensor of each gradient lattice sample of the porous structure to be optimized by using the homogenization method based on isogeometric analysis;

[0008] (2) Use the Kriging meta-model to fit the equivalent density and the corresponding equivalent elastic tensor and equivalent thermal conductivity tensor of all gradient lattice samples to construct a surrogate model, and use this surrogate model to predict the equivalent elastic tensor and equivalent thermal conductivity tensor of any equivalent density gradient lattice of the porous structure to be optimized;

[0009] (3) Based on the equivalent elastic tensor and equivalent thermal conductivity tensor of the lattice obtained by this surrogate model, construct a multi-objective isogeometric multi-scale topology optimization model with the goal of simultaneously minimizing the static compliance and the heat dissipation compliance, and use this multi-objective isogeometric multi-scale topology optimization model to optimize the equivalent density of the gradient lattice in all cells in the macroscopic design domain of the porous structure to be optimized, thereby obtaining an optimized three-dimensional porous structure; the mathematical expression of this multi-objective isogeometric multi-scale topology optimization model is:

[0010]

[0011] In the formula, is the initial density at N c macro control points in the isogeometric grid, that is, the design variable; is the multi-objective function; and Sub-objective functions for static compliance and heat dissipation compliance respectively, and w st and w th are the weights corresponding to the two, and w st + w th = 1; and are the static compliance and thermal compliance of the initial structure of the topology optimization respectively; and are the optimal solutions considering each sub-objective separately; K st is the total stiffness matrix, U is the total displacement field, and F is the applied force load matrix; K th is the total heat conduction matrix, T is the total temperature field, and P is the applied heat load matrix; represents the volume constraint of the structure, V M is the set maximum volume ratio, ν M is the element volume ratio, that is, the volume ratio of the gradient lattice, and Ω M is the total macroscopic design domain, is the minimum value of the design variable, and χ c is and the corresponding NURBS basis function R i The design variable field formed by the linear combination.

[0012] Furthermore,

[0013] and The expressions of are:

[0014]

[0015] Among them, U e is the element displacement, T e is the element temperature, and are the element stiffness matrix and the element heat conduction matrix respectively.

[0016] Furthermore, and The expressions of are:

[0017]

[0018] Among them, D e is the element elastic tensor, B st is the strain-displacement matrix calculated through the partial derivatives of the NURBS basis function; κ e is the element thermal conductivity tensor, B th is the temperature gradient-temperature matrix calculated through the partial derivatives of the NURBS basis function; ω i 、ω jand ω k are the weights of the 3×3×3 Gaussian integration points in each element in the three parametric directions respectively; J1 is the Jacobian matrix mapping from the parametric domain to the physical domain, and J2 is the Jacobian matrix mapping from the parent domain to the parametric domain.

[0019] Furthermore, D e and κ e are both obtained by interpolation using the SIMP method in isogeometric topology optimization, and the corresponding expressions are:

[0020]

[0021] where and are the macroscopic element constitutive elastic tensor and the heat conduction tensor respectively.

[0022] Furthermore, and are equivalent to the gradient lattice equivalent elastic tensor and heat conduction tensor calculated by the homogenization method based on isogeometric analysis, and are also equivalent to the gradient lattice equivalent elastic tensor and heat conduction tensor predicted by the Kriging surrogate model, that is:

[0023]

[0024] where and are the gradient lattice equivalent elastic tensor and heat conduction tensor predicted by the Kriging surrogate model respectively.

[0025] Furthermore, the method for updating the design variables is the optimization criterion method based on sensitivity information, and the corresponding formula is:

[0026]

[0027] where is the control point density at the (k + 1)-th iteration, is the control point density at the k-th iteration, τ and η are the step size limit and the damping coefficient respectively, ρ min and ρ max are the specified minimum density and maximum density respectively, is the update factor.

[0028] Furthermore, the prototype lattice unit cell is a triply periodic minimal surface lattice, the number of gradient lattice samples constructed based on the prototype lattice unit cell is 50, and the equivalent density distribution of the gradient lattice samples is evenly spaced, and the equivalent density value range of the gradient lattice samples is [0.12, 1].

[0029] Furthermore, based on the equivalent density values of all the gradient dot lattices optimized, the gradient dot lattice configurations corresponding to each equivalent density value are obtained based on the level set function and shape interpolation, and then the obtained gradient dot lattice configurations are filled into each unit correspondingly to obtain the final three-dimensional porous structure.

[0030] Furthermore, the prototype lattice unit cell of the triply periodic minimal surface lattice of the porous structure to be optimized is constructed using the level set function, and then the level set function of the prototype lattice unit cell is interpolated by shape interpolation to obtain the gradient dot lattice; the sub-goals of minimizing the static compliance and minimizing the heat dissipation compliance are coupled in a normalized linear weighted manner to construct a multi-objective function.

[0031] Furthermore, based on the homogenization method of isogeometric analysis, the equivalent elastic tensor and equivalent thermal conductivity tensor of the gradient dot lattice samples are obtained, and the corresponding formulas are:

[0032]

[0033] where u is the displacement field, |Ω m | is the volume of the gradient dot lattice sample, D pq is the constitutive matrix of the gradient dot lattice, is the applied multi-directional unit test strain field, ε p (u i ) is the unknown strain field, t is the temperature field, is the applied test unit temperature gradient field, κ is the heat conduction tensor, g p (t i ) is the unknown temperature gradient field.

[0034] Generally speaking, compared with the prior art by the above technical solutions conceived by the present invention, the multi-objective isogeometric multi-scale topology optimization method for high stiffness and high thermal conductivity provided by the present invention mainly has the following beneficial effects:

[0035] 1. The present invention constructs a multi-objective function in the form of normalized linear weighting, which can simultaneously consider the stiffness performance and heat dissipation performance of the porous structure, and can effectively improve the bearing capacity and heat dissipation capacity of the porous structure at the same time.

[0036] 2. The present invention uses the Kriging surrogate model to replace the homogenization method to predict the equivalent elastic tensor and thermal conductivity tensor of the gradient dot lattice with any equivalent density, and at the same time considers the case of the gradient gradual change distribution of the lattice in the porous structure. Therefore, while reducing the calculation cost, it can effectively release the design space.

[0037] 3. The method provided by the present invention adopts the isogeometric analysis method, which ensures the unity of the CAD model and the CAE model during the optimization process, eliminates the geometric approximation error existing in the conversion process between the CAD model and the CAE model, and improves the calculation accuracy during the optimization process. Description of the Drawings

[0038] Figure 1 is a schematic flow chart of a multi-objective isogeometric multi-scale topology optimization method for high stiffness and high thermal conductivity provided by the present invention;

[0039] Figure 2 is a schematic configuration diagram of the triply periodic minimal surface lattice constructed by the present invention;

[0040] Figure 3 is a relationship curve diagram between the equivalent elastic modulus and the equivalent density obtained by the present invention;

[0041] Figure 4 is a relationship curve diagram between the equivalent thermal conductivity and the equivalent density obtained by the present invention;

[0042] Figure 5 is a schematic diagram of the porous structure design domain and boundary conditions constructed by the present invention;

[0043] Figure 6 is constructed by the present invention Figure 5 optimization results under the multi-objective function composed of different weights of the porous structure therein;

[0044] Figure 7 is constructed by the present invention Figure 5 schematic diagram of the iteration curve of the optimization process of the porous structure when the weights of the two sub-objective functions both take 0.5 therein:

[0045] Figure 8 is constructed by the present invention Figure 5 schematic diagram of the lattice filling after the optimization of the porous structure when the weights of the two sub-objective functions both take 0.5 therein. Detailed Embodiment

[0046] In order to make the objectives, technical solutions and advantages of the present invention clearer, the present invention will be further described in detail below with reference to the drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below can be combined with each other as long as they do not conflict with each other.

[0047] The present invention provides a multi-objective isogeometric multi-scale topology optimization method for high stiffness and high thermal conductivity, and the method mainly includes the following steps:

[0048] Step 1: Calculate the equivalent elastic tensor and equivalent thermal conductivity tensor of each gradient lattice sample of the porous structure to be optimized using the homogenization method based on isogeometric analysis.

[0049] Use the level set function to construct the prototype lattice unit cell of the triply periodic minimal surface lattice of the porous structure to be optimized, and then interpolate the level set function of the prototype lattice unit cell through shape interpolation technology to obtain a series of gradient lattices. Use the homogenization method based on isogeometric analysis to obtain the equivalent elastic tensor and equivalent thermal conductivity tensor of the gradient lattice.

[0050] The equivalent elastic tensor and equivalent thermal conductivity tensor of the gradient lattice samples are obtained by the homogenization method based on isogeometric analysis, and the corresponding formulas are:

[0051]

[0052] where \(u\) is the displacement field, \(|\Omega|\) is the volume of the gradient lattice sample, \(D\) is the gradient lattice constitutive matrix, \(\overline{\varepsilon}\) is the applied multi-directional unit test strain field, \(\varepsilon(u)\) is the unknown strain field, \(t\) is the temperature field, \(\overline{\theta}\) is the applied test unit temperature gradient field, \(\kappa\) is the heat conduction tensor, and \(g(t)\) is the unknown temperature gradient field. m | is the volume of the gradient lattice sample, D pq is the gradient lattice constitutive matrix, is the applied multi-directional unit test strain field, ε p (u i ) is the unknown strain field, t is the temperature field, is the applied test unit temperature gradient field, κ is the heat conduction tensor, g p (t i ) is the unknown temperature gradient field.

[0053] ε p (u i ) and g p (t i ) are calculated through the linear elasticity equation, and the corresponding expressions are:

[0054]

[0055] where \(\delta u\) is the virtual displacement in the gradient lattice, represents the total displacement space, \(\delta t\) is the virtual temperature field in the gradient lattice, is the total temperature space.

[0056] Step 2: Use the Kriging meta-model to fit the equivalent density and the corresponding equivalent elastic tensor and equivalent thermal conductivity tensor of all gradient lattice samples to construct a surrogate model, and use this surrogate model to predict the equivalent elastic tensor and equivalent thermal conductivity tensor of any equivalent density gradient lattice of the porous structure to be optimized.

[0057] Step 3: Based on the equivalent elastic tensor and equivalent heat conduction tensor of the dot matrix obtained by the surrogate model, construct a multi-objective isogeometric multi-scale topology optimization model with the goal of minimizing both the static compliance and the heat dissipation compliance, and use this multi-objective isogeometric multi-scale topology optimization model to optimize the equivalent density of the gradient dot matrix in all cells within the macroscopic design domain of the porous structure to be optimized, thereby obtaining an optimized three-dimensional porous structure; the mathematical expression of this multi-objective isogeometric multi-scale topology optimization model is:

[0058]

[0059] In the formula, is the initial density at N c macroscopic control points in the isogeometric grid, that is, the design variable; is the multi-objective function; and are the sub-objective functions of the static compliance and the heat dissipation compliance respectively, and w st and w th are the weights corresponding to the two, and w st +w th = 1; and are the static compliance and the heat compliance of the initial structure of the topology optimization respectively; and are the optimal solutions considering each sub-objective separately; K st is the total stiffness matrix, U is the total displacement field, and F is the applied force load matrix; K th is the total heat conduction matrix, T is the total temperature field, and P is the applied heat load matrix; represents the volume constraint of the structure, V M is the set maximum volume ratio, ν M is the unit volume ratio, that is, the volume ratio of the gradient dot matrix, Ω M is the total macroscopic design domain, is the minimum value of the design variable, χ c is and the corresponding NURBS basis function R i the design variable field formed by the linear combination.

[0060] χ c is and the corresponding NURBS basis function R i the design variable field formed by the linear combination, and the corresponding expression is:

[0061]

[0062] and The expressions of are:

[0063]

[0064] Among them, U e is the element displacement, T e is the element temperature, and are the element stiffness matrix and the element heat conduction matrix respectively.

[0065] and The expressions of

[0066]

[0067] are as follows: Among them, D e is the element elastic tensor, B st is the strain-displacement matrix calculated by the partial derivatives of the NURBS basis functions; κ e is the element heat conduction tensor, B th is the temperature gradient-temperature matrix calculated by the partial derivatives of the NURBS basis functions; ω i 、ω j and ω k are the weights of the 3×3×3 Gauss integration points in each element in the three parameter directions respectively; J1 is the Jacobian matrix mapping from the parameter domain to the physical domain, and J2 is the Jacobian matrix mapping from the parent domain to the parameter domain.

[0068] D e and κ e are both obtained by interpolation using the SIMP method in isogeometric topology optimization, and the corresponding expressions are:

[0069]

[0070] Among them, and are the macroscopic element constitutive elastic tensor and the heat conduction tensor respectively.

[0071] and are equivalent to the gradient lattice equivalent elastic tensor and the heat conduction tensor calculated by the homogenization method based on isogeometric analysis, and are also equivalent to the gradient lattice equivalent elastic tensor and the heat conduction tensor predicted by the Kriging surrogate model, that is:

[0072]

[0073] Among them, and are the gradient lattice equivalent elastic tensor and the heat conduction tensor predicted by the Kriging surrogate model respectively.

[0074] The method for updating design variables is the optimization criterion method based on sensitivity information, specifically as follows:

[0075]

[0076] Among them, is the control point density at the (k + 1)-th iteration, is the control point density at the k-th iteration, τ and η are the step size limit and damping coefficient respectively, and ρ min and ρ max are the specified minimum density and maximum density respectively, is the update factor.

[0077] The prototype lattice unit cell is a triply periodic minimal surface lattice. The number of gradient lattice samples constructed according to the prototype lattice unit cell is 50, and the equivalent density distribution of the gradient lattice samples is evenly spaced. The equivalent density value range of the gradient lattice samples is [0.12, 1].

[0078] In this embodiment, isogeometric analysis is introduced to construct the design domain and mesh division, and a multi-objective function is constructed by the normalized linear weighting method. According to the equivalent density values of the gradient lattices in all elements obtained by optimization, the specific gradient lattice configurations corresponding to each equivalent density value are obtained based on the level set function and shape interpolation technology, and then filled into each element correspondingly to obtain the final three-dimensional porous structure.

[0079] The following takes specific embodiments to further elaborate on the present invention in detail.

[0080] Embodiment 1

[0081] In this embodiment, the design domain, load and boundary conditions of the porous structure to be optimized are as Figure 5 shown, which is a cuboid with L = 60 mm, W = 16 mm, and H = 60 mm; the left side of the cuboid is completely fixed, and a vertical downward force F = 1000 N is applied at the center point of the bottom edge of the right side surface. A uniform thermal load with a power of 0.1 W is applied throughout the cuboid, and the temperature T 0 is fixed on the four sides parallel to the y-axis as the heat dissipation side. Here, aluminum alloy is selected as the structural material, and its Young's modulus is: the thermal conductivity is 120 W / (m·K), and the Poisson's ratio is 0.3. The entire design domain is divided into 30×8×30 mesh elements, the volume constraint is set to 0.5, and in order to reduce the number of iteration steps, the initial design variables are uniformly set to 0.5.

[0082] As Figure 1 shown, a multi-objective isogeometric multi-scale topology optimization method provided by Embodiment 1 of the present invention includes the following steps:

[0083] Step 1: Describe the topological configuration of the triply periodic minimal surface lattice with an equivalent density of 0.12 using a level set function. Take it as the prototype lattice unit cell. Apply shape interpolation technology to this prototype lattice unit cell to obtain a series of gradient lattice samples with similar configurations. Use the homogenization method based on isogeometric analysis to obtain the equivalent elastic tensors and heat conduction tensors of all gradient lattice samples.

[0084] Specifically, it includes the following sub-steps:

[0085] (1.1) Describe the topological configuration of the triply periodic minimal surface prototype lattice unit cell using a level set function. The specific formula is:

[0086]

[0087] where \(x\) represents the global coordinates of any point within the region \(H\), and \(H\) is a specified design region. denotes the lattice structure boundary, and \(\Omega\) s is the region occupied by the lattice. Therefore, the part where \(\varphi\) s (x) is greater than 0 represents the solid of the lattice, the part where \(\varphi\) s (x) equals 0 represents the lattice structure boundary, while the part where \(\varphi\) s (x) is less than 0 represents the holes.

[0088] The explicit level set function of the triply periodic minimal surface lattice is:

[0089] \(\varphi\) SGSH (x) = (sin(2πx)cos(2πy) + sin(2πy)cos(2πz) + sin(2πz)cos(2πx)) 2 -t 2

[0090] where \(x\), \(y\), and \(z\) are the three-dimensional coordinates of a point in Euclidean space, and \(t\) is a variable that determines the equivalent density of the SGSH lattice.

[0091] (1.2) Use shape interpolation technology on the triply periodic minimal surface prototype lattice unit cell to obtain 50 gradient lattice samples with equivalent densities uniformly distributed in the interval [0.12, 1]. The formula for shape interpolation technology is:

[0092]

[0093] where \(\varphi\) e (x) is the new level set function with adjusted height obtained through shape interpolation technology, used to represent the newly obtained gradient lattice. \(\varphi\) pro (x) is the level set function of the prototype lattice unit cell. All other gradient lattices are obtained by adjusting the cross-section height of \(\varphi\) pro (x). is the interpolation coefficient matrix. By using the bisection method to determine the value of can obtain the gradient lattice level set function φ with any volume fraction e (x).

[0094] (1.3) The homogenization method based on isogeometric analysis is used to calculate the equivalent elastic modulus and thermal conductivity tensor of all gradient lattice samples. The corresponding formulas are:

[0095]

[0096] Among them, u is the displacement field, |Ω m | is the volume of the gradient lattice sample, D pq is the gradient lattice constitutive matrix, is the applied multi-directional unit test strain field, ε p (u i ) is the unknown strain field, t is the temperature field, is the applied test unit temperature gradient field, κ is the thermal conductivity tensor, g p (t i ) is the unknown temperature gradient field.

[0097] ε p (u i ) and g p (t i ) are solved by the linear elastic equation. The corresponding formulas are:

[0098]

[0099] Among them, δu is the virtual displacement in the gradient lattice, represents the total displacement space, δt is the virtual temperature field in the gradient lattice, is the total temperature space.

[0100] Step 2: Input the equivalent density and the corresponding equivalent elastic tensor and thermal conductivity tensor of 50 gradient lattice samples into the Kriging meta-model for fitting to construct a surrogate model that can predict the equivalent elastic tensor and thermal conductivity tensor of any equivalent density gradient lattice, so that the equivalent elastic tensor and thermal conductivity tensor of the gradient lattice with any equivalent density can be directly obtained, greatly improving the calculation efficiency.

[0101] Step 3: Introduce isogeometric analysis to construct the design domain and mesh generation, construct the multi-objective function by means of normalized linear weighting, and then construct a multi-objective isogeometric multi-scale topology optimization model based on the equivalent elastic tensor and equivalent heat conduction tensor of the lattice obtained by the surrogate model. Couple the two sub-objectives of minimizing the static compliance and minimizing the heat dissipation compliance and consider them simultaneously. Use the optimization criterion method to optimize the equivalent density of the gradient lattice in all elements, so that the porous result has good load-bearing capacity and heat dissipation capacity at the same time.

[0102] Specifically, it includes the following sub-steps:

[0103] (3.1) The mathematical expression of the multi-objective isogeometric multi-scale topology optimization model for high stiffness and high heat conduction is:

[0104]

[0105] Among them, is the initial density at N c macro control points in the isogeometric mesh, that is, the design variable, is the multi-objective function, and are the sub-objective functions of static compliance and thermal compliance respectively, while w st and w th are the weights corresponding to the two, and w st + w th = 1; and are the static compliance and thermal compliance of the initial structure of the topology optimization respectively, and are the optimal solutions considering each sub-objective separately, K st is the total stiffness matrix, U is the total displacement field, F is the applied force load matrix, K th is the total heat conduction matrix, T is the total temperature field, P is the applied heat load matrix, represents the volume constraint of the structure, V M is the specified maximum volume ratio, ν M is the unit volume ratio, that is, the volume ratio of the gradient lattice, Ω M is the total macroscopic design domain, is the minimum value of the design variable, χ c is and the corresponding NURBS basis function R i The design variable field formed by the linear combination is expressed as:

[0106]

[0107] Specifically, and The expressions of are:

[0108]

[0109] Among them, U e is the element displacement, T e is the element temperature, and are the element stiffness matrix and the element heat conduction matrix respectively, and the corresponding expressions are:

[0110]

[0111] Among them, D e is the element elastic tensor, B st is the strain-displacement matrix calculated through the partial derivatives of the NURBS basis functions, κ e is the element heat conduction tensor, B th is the temperature gradient-temperature matrix calculated through the partial derivatives of the NURBS basis functions, ω i , ω j and ω k are the weights of the 3×3×3 Gaussian integration points in each element in the three parameter directions respectively. J1 is the Jacobian matrix mapping from the parameter domain to the physical domain, and J2 is the Jacobian matrix mapping from the parent domain to the parameter domain.

[0112] Specifically, both D e and κ e are obtained by interpolation using the SIMP method in isogeometric topology optimization, and the corresponding formulas are:

[0113]

[0114] Among them, and are the macroscopic element constitutive elastic tensor and heat conduction tensor respectively; based on the homogenization method, and are equivalent to the gradient lattice equivalent elastic tensor and heat conduction tensor calculated by the homogenization method, and are also equivalent to the gradient lattice equivalent elastic tensor and heat conduction tensor predicted by the Kriging surrogate model, that is:

[0115]

[0116] Among them, and are the gradient lattice equivalent elastic tensor and heat conduction tensor predicted by the Kriging surrogate model respectively.

[0117] (3.2) Calculate the sensitivities of the multi-objective function and the volume constraint condition to the design variables. The sensitivity expression of the multi-objective function to the design variables is:

[0118]

[0119] Among them, and The expressions of are:

[0120]

[0121] Furthermore, the sensitivity calculation formula of the volume constraint condition with respect to the design variable is:

[0122]

[0123] When updating the design variable, filtering is performed by using the mean of the adjacent control point densities to replace the current control point density, so as to avoid numerical instability phenomena such as checkerboards and grid dependencies. Here, the filtering radius is taken as 1.5.

[0124] (3.3) Substitute the multi-objective function and the sensitivity of the volume constraint condition with respect to the design variable obtained in step (3.2) into the optimization criterion method to update the design variable. Among them, the gradient-based heuristic criterion method is used to update the design variable, and the corresponding formula is:

[0125]

[0126] Among them, is the control point density at the (k + 1)-th iteration, is the control point density at the k-th iteration, τ and η are the step size limit and damping coefficient respectively, ρ min and ρ max are the minimum density and maximum density respectively, and here they are taken as 0.1 and 1 respectively, is the update factor, and the corresponding formula is:

[0127]

[0128] Among them, is the multi-objective function with respect to the design variable ρ i The first-order derivative of, is the first-order derivative of the volume constraint G with respect to ρ i The first-order derivative of.

[0129] (3.4) Construct the convergence condition according to the maximum change value of the design variable between two iterations. If the maximum change value of the design variable between two iterations is less than 1% or the number of iteration steps reaches 180 steps, stop the iteration and output the equivalent density of the gradient lattice in all elements of the design domain. If the convergence condition is not satisfied, return to step (3.1) to continue updating the design variable.

[0130] Step 4: Based on the equivalent density of the optimized gradient lattice points in all cells output after meeting the convergence condition in Step 3, the gradient lattice configuration corresponding to the equivalent density of each cell is obtained by using the level set function and shape interpolation technology, and is filled into the corresponding cell respectively, obtaining the final three-dimensional porous structure, thus realizing the topology optimization process.

[0131] Please refer to Figures 2 to 8 , and the following uses the design of a three-dimensional square plate to further illustrate the present invention.

[0132] As Figure 2 shown is a schematic diagram of the topological configuration of a triply periodic minimal surface lattice. As Figure 3 and Figure 4 shown are schematic diagrams of the Kriging surrogate models constructed according to the equivalent density, equivalent elastic tensor, and equivalent heat conduction tensor corresponding to the gradient lattice samples, where aluminum alloy is selected as the structural material, with a Young's modulus of 70 GPa, a heat conduction coefficient of 120 W / (m·K), and a Poisson's ratio of 0.3. It can be seen that the equivalent densities of the gradient lattice samples are uniformly arranged in the interval [0.12, 1].

[0133] As Figure 5 shown is a square plate design domain with L = 60 mm, W = 16 mm, and H = 60 mm. The left side of the cuboid is completely fixed, and a vertical downward force F = 1000 N is applied at the center point of the bottom edge of the right side. And a uniform heat load with a power of 0.1 W is applied throughout the cuboid, and the temperature T 0 is fixed on the four sides parallel to the y-axis as the heat dissipation sides, and the volume constraint is set to 0.5.

[0134] As Figure 6 shown are the optimization results under multi-objective functions composed of different weights, where the abscissa is the static compliance and the ordinate is the thermal compliance. It can be seen that these optimization results are all Pareto optimal solutions under different weight values, together constituting the Pareto optimal front, clearly showing the trade-off between the two sub-objectives of thermal compliance and static compliance. From the performance parameters, it can be found that the larger the w st , the smaller the static compliance of the optimization result, and the larger the w th , the smaller the thermal compliance of the optimization result. From the gradient lattice density distribution diagram of the optimization results, it can be observed that as w th decreases and w st increases, the density distribution of the optimization results gradually changes from the structure with the best heat dissipation ability to the structure with the best load-bearing ability.

[0135] As Figure 7 shown is when w st = 0.5, w thThe optimized iteration curve when =0.5. The iteration process is relatively stable, and both the static compliance and the heat dissipation compliance decrease simultaneously. As Figure 8 shown in w st =0.5, w th =0.5 is the optimized structure after gradient lattice filling. It can be seen from the figure that a multi-objective isogeometric multi-scale topology optimization method for high stiffness and high thermal conductivity provided by the present invention, compared with the traditional method, simultaneously considers the stiffness performance and the thermal conductivity of the porous structure, has a lower calculation cost, better connectivity between gradient lattices, and ensures the unity of the CAD model, the CAE model and the TO model during the optimization process by introducing isogeometry, improves the calculation accuracy, greatly expands the design space of the porous structure, and can effectively improve the bearing capacity and the heat dissipation capacity of the porous structure at the same time.

[0136] It is easy for those skilled in the art to understand that the above are only the preferred embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims

1. A multi-objective isogeometric multi-scale topology optimization method for high stiffness and high thermal conductivity, characterized in that, The method includes the following steps: (1) Calculate the equivalent elastic tensor and equivalent thermal conductivity tensor of each gradient lattice sample of the porous structure to be optimized by using the homogenization method based on isogeometric analysis; (2) Fit the equivalent density and the corresponding equivalent elastic tensor and equivalent thermal conductivity tensor of all gradient lattice samples by using the Kriging meta-model to construct a surrogate model, and use this surrogate model to predict the equivalent elastic tensor and equivalent thermal conductivity tensor of any equivalent density gradient lattice of the porous structure to be optimized; (3) Construct a multi-objective isogeometric multi-scale topology optimization model with the goal of minimizing both the static compliance and the heat dissipation compliance based on the equivalent elastic tensor and equivalent thermal conductivity tensor of the lattice obtained by this surrogate model, and use this multi-objective isogeometric multi-scale topology optimization model to optimize the equivalent density of the gradient lattice in all cells within the macroscopic design domain of the porous structure to be optimized, and then obtain the optimized three-dimensional porous structure; the mathematical expression of this multi-objective isogeometric multi-scale topology optimization model is: In the formula, is the initial density at N c macro control points in the isogeometric grid, that is, the design variable; is the multi-objective function; and are the sub-objective functions of static compliance and heat dissipation compliance respectively, while w st and w th are the weights corresponding to the two, and w st + w th = 1; and are the static compliance and thermal compliance of the initial structure of the topology optimization respectively; and are the optimal solutions considering each sub-objective separately; K st is the total stiffness matrix, U is the total displacement field, and F is the applied force load matrix; K th is the total heat conduction matrix, T is the total temperature field, and P is the applied heat load matrix; represents the volume constraint of the structure, V M is the set maximum volume ratio, ν M is the element volume ratio, that is, the volume ratio of the gradient lattice, and Ω M is the total macro design domain, is the minimum value of the design variable, is and the corresponding NURBS basis function R i The design variable field formed by the linear combination.

2. The multi-objective isogeometric multi-scale topology optimization method for high stiffness and high thermal conductivity according to claim 1, characterized in that: and The expression is: Among them, U e is the element displacement, T e is the element temperature, and are the element stiffness matrix and the element heat conduction matrix, respectively.

3. The multi-objective isogeometric multi-scale topology optimization method for high stiffness and high thermal conductivity according to claim 2, characterized in that: and The expression for: where D e is the element elastic tensor, B st is the strain-displacement matrix calculated from the partial derivatives of the NURBS basis functions; κ e is the element heat conduction tensor, B th is the temperature gradient-temperature matrix calculated from the partial derivatives of the NURBS basis functions; ω i , ω j and ω k are the weights of the 3×3×3 Gaussian integration points in each element in the three parametric directions respectively; J1 is the Jacobian matrix mapping from the parametric domain to the physical domain, and J2 is the Jacobian matrix mapping from the parent domain to the parametric domain.

4. The multi-objective isogeometric multi-scale topology optimization method for high stiffness and high thermal conductivity according to claim 3, characterized in that: D e and κ e are both obtained by interpolation using the SIMP method in isogeometric topology optimization, and the corresponding expressions are: Among them, D 0 (ρ i c ) and κ 0 (ρ i c ) are the macroscopic unit constitutive elastic tensor and the heat conduction tensor, respectively.

5. The multi-objective isogeometric multi-scale topology optimization method for high stiffness and high thermal conductivity according to claim 4, characterized in that: and equivalent to the equivalent elastic tensor and heat conduction tensor of the gradient lattice calculated by the homogenization method based on isogeometric analysis, and also equivalent to the equivalent elastic tensor and heat conduction tensor of the gradient lattice predicted by the Kriging surrogate model, that is: Among them, and are the equivalent elastic tensor and heat conduction tensor of the gradient lattice predicted by the Kriging surrogate model, respectively.

6. The multi-objective isogeometric multi-scale topology optimization method for high stiffness and high thermal conductivity as claimed in claim 1, characterized in that: The method for updating the design variables is the optimization criterion method based on sensitivity information, and the corresponding formula is: Among them, is the control point density of the (k + 1)-th iteration, is the control point density of the k-th iteration, τ and η are the step size limit and the damping coefficient respectively, ρ min and ρ max are the specified minimum density and maximum density respectively, is the update factor.

7. The multi-objective isogeometric multi-scale topology optimization method for high stiffness and high thermal conductivity according to any one of claims 1-6, characterized in that: The prototype lattice unit cell is a triply periodic minimal surface lattice, the number of gradient lattice samples constructed according to the prototype lattice unit cell is 50, and the equivalent density distribution of the gradient lattice samples is evenly spaced, and the value range of the equivalent density of the gradient lattice samples is [0.12, 1].

8. The multi-objective isogeometric multi-scale topology optimization method for high stiffness and high thermal conductivity according to any one of claims 1-6, characterized in that: Based on the equivalent density values of the gradient lattice in all cells obtained by optimization, obtain the gradient lattice configuration corresponding to each equivalent density value based on the level set function and shape interpolation, and then fill the obtained gradient lattice configuration into each cell correspondingly to obtain the final three-dimensional porous structure.

9. The multi-objective isogeometric multi-scale topology optimization method for high stiffness and high thermal conductivity according to any one of claims 1-6, characterized in that: Use the level set function to construct the prototype lattice unit cell of the triply periodic minimal surface lattice of the porous structure to be optimized, and then interpolate the level set function of the prototype lattice unit cell by shape interpolation to obtain the gradient lattice; Couple the sub-goals of minimizing the static compliance and minimizing the heat dissipation compliance in a normalized linear weighted manner to construct a multi-objective function.

10. The multi-objective isogeometric multi-scale topology optimization method for high stiffness and high thermal conductivity according to any one of claims 1-6, characterized in that: Obtain the equivalent elastic tensor and equivalent thermal conductivity tensor of the gradient lattice sample by the homogenization method based on isogeometric analysis, and the corresponding formula is: where u is the displacement field, |Ω m | is the volume of the gradient lattice sample, D pq is the gradient lattice constitutive matrix, is the applied multi-direction unit test strain field, ε p (u i ) is the unknown strain field, t is the temperature field, is the applied test unit temperature gradient field, κ is the heat conduction tensor, g p (t i ) is the unknown temperature gradient field.

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