Isogeometric Topology Optimization of Multi-Material Structures under Thermal-Mechanical Loadings Using Neural Networks

Abstract

An isogeometric topology optimization (ITO) model for multi-material structures under thermal-mechanical loadings using neural networks is proposed. In the proposed model, a non-uniform rational B-spline (NURBS) function is employed for geometric description and analytical calculation, which realizes the unification of the geometry and computational models. Neural networks replace the optimization algorithms of traditional topology optimization to update the relative densities of multi-material structures. The weights and biases of neural networks are taken as design variables and updated by automatic differentiation without derivation of the sensitivity formula. In addition, the grid elements can be refined directly by increasing the number of refinement nodes, resulting in high-resolution optimal topology without extra computational costs. To obtain comprehensive performance from ITO for multi-material structures, a weighting coefficient is introduced to regulate the proportion between thermal compliance and compliance in the loss function. Some numerical examples are given and the validity is verified by performance analysis. The optimal topological structures obtained based on the proposed model exhibit both excellent heat dissipation and stiffness performance under thermal-mechanical loadings.

1. Introduction

Topology optimization, as one of the modern techniques of structural optimal design, can generate optimal configurations that satisfy specified loading conditions, performance, and constraints. Currently, the commonly used topology optimization methods are the homogenization method [1], level set method (LSM) [2], variable density method [3], bidirectional evolutionary structural optimization (BESO) method, moving morphable components (MMC) method [4], etc. In order to optimize the structure for increasing performance, the iterative process of topology optimization needs to be combined with numerical computational methods for structural performance analysis, such as the finite element method (FEM) [5], boundary element method (BEM) [6], meshless method (MM) [7], etc. These numerical methods utilize basis functions of unknown solution space to discrete geometrical domains, resulting in errors between the geometrical model and the computational model, and refined meshing or even re-meshing is required to improve the accuracy of the results, leading to an increase in computational costs. The isogeometric analysis (IGA) proposed by Hughes et al. [8] is an effective numerical computational method to accurately compute the unknown structural response. In IGA, non-uniform rational B-spline (NURBS) is applied for geometric models and numerical analysis. The use of NURBS ensures higher-order continuity between elements, computational efficiency, and better accuracy [9,10]. Seo et al. [11] successfully proposed an isogeometric topological optimization (ITO) by using a cropped surface analysis technique. However, most of the topology optimization methods require sensitivity analysis during the design process and usually have problems such as high computational costs and unstable values, which may make them challenging for multi-objective problems. So, the use of suitable optimization pathways, especially neural networks, has been a focus in the field of topology optimization.

Currently, a large number of neural network models have been used for topology optimization. Several processes of topology optimization can be improved or replaced by neural networks, such as structural response [12], objective function [13], sensitivity analysis, and post-processing [14]. Raissi et al. [15] developed a deep learning-based partial differential equation (PDE) solver, namely physical-informed neural networks (PINN), for both forward and inverse computational problems. Inspired by the recent success of PINN, physical-informed neural network topology optimization (PINNTO) has been developed as an alternative to classical topology optimization [16,17]. The design variables in traditional topology optimization have been replaced by parameters of neural networks. Jeong et al. [18] used PINN to improve its computational efficiency in the study of multi-loading problems and explored the effect of mesh size and neural network size on the optimal topology. Chandrasekhar et al. [19] employed neural networks to solve finite element equations during the process of topology optimization and used the activation function of the neural networks to replace the density field of the SIMP method [20]. What’s more, they added a Fourier mapping process to approximate the minimum or maximum structural scale and obtained a high resolution while maintaining computational efficiency [21]. PINNTO is different from the previous traditional data-driven methods because it does not rely on large datasets. The method utilizes automatic differentiation in neural networks with backpropagation to replace the traditional sensitivity analysis and train the neural network model toward better structural performance. In addition, most of the above studies are oriented to a single physical field, and the multi-objective topology optimization (MTO) under multi-loading conditions is rarely addressed.

In recent years, many scholars have investigated MTO considering both thermal and mechanical properties. Giraldo-Londoño et al. [22] used a density-based topology optimization formulation to design multi-material thermoelastic structures and support structures that facilitate additive manufacturing. Ooms et al. [23] developed a topology optimization to minimize structural compliance under thermal-mechanical loadings. Chen et al. [24] designed structures with lower thermal compliance and higher intrinsic frequency. The research mentioned above has greatly advanced the field of MTO with regard to thermal-mechanical loadings. However, there are usually problems of inconsistency between geometric and computational models for complex geometric problems and loss of accuracy of structural response, which can be effectively solved by using multi-objective isogeometric topological optimization (MITO). In addition, most of the structural topology optimization nowadays is usually necessary to derive the derivatives of the design variables in sensitivity analysis, while neural network-based topological optimization can decrease the costs during the pre-design process due to its strong learning and optimization capabilities. In this research, an isogeometric topology optimization model for multi-material structures under thermal-mechanical loadings using neural networks is proposed. The remainder of this article is structured as follows:

Section 2 introduces IGA for steady-state heat transfer and static mechanics structures. Section 3 describes the working principle of neural networks and the procedure of MITO based on neural networks. Section 4 presents numerical examples. Section 5 gives some conclusions.

2. Isogeometric Analysis

Isogeometric analysis (IGA), as an effective numerical method, uses non-uniform rational B-spline (NURBS) functions [8] as the shape functions of the computational model, which realizes the unification of the computational model and geometric model. It avoids errors introduced by segmented polynomial approximation of classical finite elements, and provides more accurate results.

The IGA theory involves three spaces shown in Figure 1: physical space, parameter space, and Gaussian space. The red section of Figure 1 represents an IGA element. The physical space is defined through spatial coordinates of the geometric model, while the parameter space consists of node vectors. Similar to an isoparametric element in finite element analysis, it is necessary to transform the data from parameter space to Gaussian space to facilitate integration operations.

Given a set of increasing sequences $\Xi =\left\{{\xi }_1, {\xi }_2,\cdots , {\xi }_{n+p+1}\right\}$, $\psi =\left\{{\eta }_1, {\eta }_2,\cdots , {\eta }_{m+q+1}\right\}$ in the parameter space. $p$ and $q$ denote the order of the NURBS basis functions in $\xi$ and $\eta$ directions, respectively. $n$ and $m$ denote the number of basis functions. The parametric element ${\widehat{\mathsf{\Omega }}}_e=\left[{\xi }_i,{\xi }_{i+1}\right]\times \left[{\eta }_i,{\eta }_{i+1}\right]$ in NURBS’s surface representation is mapped to the Gaussian space element $\tilde{\mathsf{\Omega }}=\left[-1, 1\right]$ by an isoparametric transformation with the expression. ${\xi }_{gs}$ and ${\eta }_{gs}$ are the coordinates in Gaussian space.

$$\left\{\begin{array}{l}\xi =\frac{{\xi }_{i+1}-{\xi }_i}{2}({\xi }_{gs}+1)\\ \eta =\frac{{\eta }_{i+1}-{\eta }_i}{2}({\eta }_{gs}+1)\end{array}\right.$$

(1)

Therefore, the integral operation of the physical element can be converted to Gaussian space through the spatial mapping as shown in Figure 1, and the corresponding conversion formula is

$${\displaystyle {\int }_{{\mathsf{\Omega }}_e}f(x,y)\mathrm{d}}{\mathsf{\Omega }}_e={\displaystyle {\int }_{{\widehat{\mathsf{\Omega }}}_e}f(x(\xi ),y(\eta ))\left|\begin{array}{ll}\frac{\partial x}{\partial \xi }& \frac{\partial y}{\partial \xi }\\ \frac{\partial x}{\partial \eta }& \frac{\partial \mathrm{y}}{\partial \eta }\end{array}\right|}\mathrm{d}{\widehat{\mathsf{\Omega }}}_e={\displaystyle {\int }_{\tilde{\mathsf{\Omega }}}f({\xi }_{gs},{\eta }_{gs})\left|\begin{array}{ll}\frac{\partial x}{\partial \xi }& \frac{\partial y}{\partial \xi }\\ \frac{\partial x}{\partial \eta }& \frac{\partial \mathrm{y}}{\partial \eta }\end{array}\right|}\left|\begin{array}{ll}\frac{\partial \xi }{\partial {\xi }_{gs}}& \frac{\partial \eta }{\partial {\xi }_{gs}}\\ \frac{\partial \xi }{\partial {\eta }_{gs}}& \frac{\partial \eta }{\partial {\eta }_{gs}}\end{array}\right|\mathrm{d}\tilde{\mathsf{\Omega }}$$

(2)

2.1. Isogeometric Analysis for Steady State Heat Transfer Structures

The governing equation of steady state heat transfer problems describes the equilibrium state of heat transfer within an object in the form of

$$k_{ij}\frac{{\partial }^{2}T}{\partial x_i\partial x_j}+\dot{Q}=0\hspace{1em}T(x,y)\in \mathsf{\Omega }$$

(3)

where $k_{ij}$ represents thermal conductivity, $T$ represents temperature at the IGA control point $(x,y)$ in physical space $\mathsf{\Omega }$, and $\dot{Q}$ represents heat source.

For the cases in this paper, the fixed temperature $\overline{T}$ on the Dirichlet boundary ${\Gamma }_1$, and density of the heat flow $\overline{q}$, and elemental external normal vector $n_j$ on the Neumann boundary ${\Gamma }_2$ are given to solve steady state heat transfer problems. The thermal boundary conditions are expressed as:

$$T(x, y)=\overline{T}\hspace{1em}\hspace{1em}(x, y)\in {\Gamma }_1$$

(4)

$$-k_{ij}\frac{\partial T}{\partial x_i}{n}_j=\overline{q}\hspace{1em}\hspace{1em}(x,y)\in {\Gamma }_2$$

(5)

Substituting Neumann boundary, Equation (3) can be converted to

$$\mathsf{\delta }{\mathsf{\Pi }}^{\prime }(T)={\displaystyle {\int }_{\mathsf{\Omega }}{k}_{ij}\frac{\partial T}{\partial x_i}\frac{\partial \mathsf{\delta }T}{\partial x_j}\mathrm{d}\mathsf{\Omega }}-{\displaystyle {\int }_{\mathsf{\Omega }}\dot{Q}\mathsf{\delta }T\mathrm{d}\mathsf{\Omega }}+{\displaystyle {\int }_{{\Gamma }_2}\overline{q}\mathsf{\delta }T\mathrm{d}\Gamma }$$

(6)

According to NURBS function, the interpolation yields the variants of the temperature $\mathsf{\delta }T$

$$\mathsf{\delta }T(\xi ,\eta )={\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1}^m{R}_{i,j}^{p,q}(\xi ,\eta )\mathsf{\delta }{T}_{i,j}}}$$

(7)

The IGA discrete governing equation for structural steady-state heat transfer problem can be derived as

$$K_{t}T=P$$

(8)

where $K_t$, $\mathbf{T}$, and $\mathbf{P}$ represent the IGA global thermal stiffness matrix, temperature vector, and thermal loading vector for multi-material structure, respectively. The IGA element thermal stiffness matrix and element thermal loading vector are

$$K_{te}^0={\displaystyle {\int }_{{\mathsf{\Omega }}_e}{B}_t^{\mathrm{T}}}\mathbf{k}{\mathbf{B}}_t\mathrm{d}{\mathsf{\Omega }}_{\mathrm{e}}$$

(9)

$$P_e={\displaystyle {\int }_{{\mathsf{\Omega }}_e}Q{R}_t^{\mathrm{T}}}\mathrm{d}{\mathsf{\Omega }}_{\mathrm{e}}-{\displaystyle {\int }_{{\Gamma }_2}\overline{q}{R}_t^{\mathrm{T}}}\mathrm{d}{\Gamma }_2$$

(10)

where ${\mathsf{\Omega }}_{\mathrm{e}}$ represents area of IGA element, $k$ represents thermal conductivity matrix, and $B_t$ represents IGA geometry matrix in the heat transfer problem

$${\mathbf{B}}_t={\left[\frac{\partial }{\partial x} \frac{\partial }{\partial y}\right]}^{\mathrm{T}}{\mathbf{R}}_t$$

(11)

where ${\mathbf{R}}_t$ represents matrix of IGA shape functions in the heat transfer problem

$${\mathbf{R}}_t=\left[R_1^{p,q}, R_2^{p,q}, R_3^{p,q},\cdots , R_{(p+1)\times (q+1)}^{p,q}\right]$$

(12)

where $R_i^{p,q}$ represents NURBS function of IGA control point.

2.2. Isogeometric Analysis for Static Mechanics Structures

In the 2-D linear elasticity problem, the equilibrium equation is

$$\nabla \cdot \mathsf{\sigma }+b=0\hspace{1em}\mathrm{in}\hspace{0.17em}\mathsf{\Omega }$$

(13)

where $\mathsf{\sigma }$ is stress vector and $b$ is the given body force in the design domain $\mathsf{\Omega }$. Displacement and stress boundary conditions are

$$U=g\hspace{1em}\hspace{1em}\mathrm{on} {\Gamma }_D$$

(14)

$$n\cdot \mathsf{\sigma }=t\hspace{1em}\hspace{1em}\mathrm{on} {\Gamma }_N$$

(15)

where the applied displacement and stress boundary conditions are denoted as $g$ and $t$. The displacement vector on displacement boundary ${\Gamma }_D$ is denoted as $U$ and the cosine of the outer normal direction of the force boundary ${\Gamma }_N$ is denoted as $n$. The weak form of equivalent integral of the 2-D linear elastic equilibrium equation with force boundary conditions is generally obtained by means of the virtual work principle:

$${\displaystyle {\int }_{\mathsf{\Omega }}\epsilon (U)\cdot \mathbf{D}\cdot }\epsilon (\mathsf{\delta }U)\mathrm{d}\mathsf{\Omega }={\displaystyle {\int }_{\mathsf{\Omega }}b\cdot }\mathsf{\delta }U\mathrm{d}\mathsf{\Omega }+{\displaystyle {\int }_{{\Gamma }_N}t\cdot }\mathsf{\delta }U\mathrm{d}\Gamma$$

(16)

According to the NURBS function, the interpolation yields the variants of the displacement $\mathsf{\delta }U$

$$\mathsf{\delta }U(\xi ,\eta )={\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1}^m{R}_{i,j}^{p,q}(\xi ,\eta )\mathsf{\delta }{U}_{i,j}}}$$

(17)

The IGA discrete governing equation for the 2-D elasticity problem can be obtained by coupling Equations (16) and (17):

$$K_{f}U=F$$

(18)

where $K_f$, $\mathbf{U}$, and $\mathbf{F}$ represent IGA global stiffness matrix, displacement vector and force loading vector for the multi-material structure, respectively. IGA element stiffness matrix and force loading are

$$K_{fe}^0={\displaystyle {\int }_{{\mathsf{\Omega }}_e}{B}_f^{\mathrm{T}}}\mathbf{D}{\mathbf{B}}_f\mathrm{d}{\mathsf{\Omega }}_{\mathrm{e}}$$

(19)

$$F_e={\displaystyle {\int }_{{\Omega }_e}{R}_f^{\mathrm{T}}}b\mathrm{d}{\Omega }_{\mathrm{e}}+{\displaystyle {\int }_{{\Gamma }_f}{R}_f^{\mathrm{T}}}t\mathrm{d}{\Gamma }_f$$

(20)

where $D$ is elasticity matrix, $B_f$ is IGA geometry matrix in the static mechanics problem;

$$B_f={\left[\begin{array}{ccc}\frac{\partial }{\partial x}& 0& \frac{\partial }{\partial y}\\ 0& \frac{\partial }{\partial y}& \frac{\partial }{\partial x}\end{array}\right]}^{\mathrm{T}}{R}_f$$

(21)

where ${\mathbf{R}}_f$ is matrix of IGA shape function in the mechanics problem

$$R_f=\left[\begin{array}{cccccccc}{R}_1^{p,q}& 0& R_2^{p,q}& 0& R_3^{p,q}& \cdots & R_{(p+1)\times (q+1)}^{p,q}& 0\\ 0& R_1^{p,q}& 0& R_2^{p,q}& 0& \cdots & 0& R_{(p+1)\times (q+1)}^{p,q}\end{array}\right]$$

(22)

3. MITO for Multi-Material Structure

3.1. Material Interpolation Model

The relative density of material-$I(I=1,2,3,\dots ,M)$ at the control point within an IGA element is interpolated according to NURBS function. The information of each IGA element is determined by its $(p+1)\times (q+1)$ element control points, and the IGA element density can be obtained by interpolating the relative densities of element control points and NURBS basis function:

$${\rho }_e^I={\displaystyle \sum _{i=1}^{(p+1)\times (q+1)}{R}_{e,i}^{p,q}{\rho }_{e,i}^I\left(\mathit{w}\right)}\hspace{1em}\hspace{1em}(e=1, 2, 3, \cdots , N_e)$$

(23)

where $R_{e,i}^{p,q}$ and ${\rho }_{e,i}^I$ represent NURBS function and relative density of material-$I$(MAT-$I$) at the $i$-th IGA control point of the $e$-th element, respectively. The value of relative density can be changed by the updating of the overall weights and bias $w$ of neural networks to enhance structural performance. In addition, the volume constraints of the material are set to achieve the lightweight design requirement in topology optimization. The volume fraction of the material is also affected by $w$ and fulfill $V^I=V_0^I$:

$$V^I={\displaystyle \sum _{e=1}^{N_e}\left({\displaystyle \sum _{i=1}^{(p+1)\times (q+1)}{R}_{e,i}^{p,q}{\rho }_{e,i}^I\left(\mathit{w}\right)}\right)A_e}$$

(24)

$$V_0^I=V_f^I{\displaystyle \sum _{e=1}^{N_e}{A}_e}$$

(25)

where $N_e$ represents the number of IGA elements. $A_e$ denotes the initial area of IGA element. $V^I$ denotes the volume of MAT-$I$. $V_0^I$ and $V_f^I$ denotes the constraints of volume and volume fraction, respectively. The thermal conductivity matrix expression can be formulated as

$$k_e={\displaystyle \sum _{I=1}^M{({\rho }_e^I)}^{\widehat{p}}{k}^I}$$

(26)

where $M$ represents the number of material types, $k^I$ represents the heat transfer coefficient matrix of MAT-$I$, and $\widehat{p}$ represents the material penalty factor. The IGA element thermal stiffness matrix of multi-material structure is transformed into:

$$K_{te}={\displaystyle {\int }_{{\mathsf{\Omega }}_e}{B}_t^{\mathrm{T}}}({\displaystyle \sum _{I=1}^M{({\rho }_e^I)}^{\widehat{p}}{k}^I})B_t\mathrm{d}{\mathsf{\Omega }}_e$$

(27)

The elastic matrix based on relative density ${\rho }_e^I$ of IGA element multi-material is as follows:

$$D_e={\displaystyle \sum _{I=1}^M{({\rho }_e^I)}^{\widehat{p}}{D}^I}$$

(28)

where $M$ is the number of material types and $D^I$ represents the elasticity matrix of MAT-$I$. The IGA element stiffness matrix is transformed into

$$K_{fe}={\displaystyle {\int }_{{\mathsf{\Omega }}_e}{B}_f^{\mathrm{T}}({\displaystyle \sum _{I=1}^M{({\rho }_e^I)}^{\widehat{p}}{D}^I})B_f\mathrm{d}{\mathsf{\Omega }}_e}$$

(29)

3.2. Neural Networks of MITO for Multi-Material Structures

The neural networks are constructed based on IGA theory with the objective of minimizing thermal compliance and compliance of multi-material structures simultaneously under thermal-mechanical loadings. Figure 2 shows the overview of MITO using neural networks.

As shown in Figure 2, IGA is used to discrete multi-material structure design domain and solve the steady state heat transfer and static mechanics governing equations, respectively. The relative densities of IGA control points are obtained by inputting IGA control point coordinates into the neural network model and updating the weights and biases of neurons through gradient computation and backpropagation, with scale filtering performed by Fourier projection. The high-resolution topology optimization results are obtained by inputting the coordinates of the high-density IGA control points into the optimized neural networks model.

In contrast to conventional ITO, the proposed model uses the overall weights and bias $w$ of neural networks rather than the densities of control points as design variables. As shown in Figure 3, the relative densities of multiple materials at the IGA control points are obtained by inputting the IGA control point coordinates $(x,y)$ into a fully-connected network. The network consists of several hidden layers, and the complexity and functionality of a neural network are inextricably linked to the number of hidden layers and neurons.

In order to convert the control point coordinates into relative densities and relate them to the neural network, $z_i^{\left[l\right]}=w_{ii}^{\left[l\right]}\widehat{x}+w_{ji}^{\left[l\right]}\widehat{y}+b_i^{\left[l\right]}$ are computed first, where $w_{ij}^{\left[l\right]}$ is the weight of the $i$-th neuron in the $l-1$ layer to the $j$-th neuron in the $l$ layer, and $b_j^{\left[l\right]}$ is the bias of the $j$-th neuron in the $l$ layer. To accelerate the training process of neural networks and improve the convergence and stability of the model, it is normalized by the BatchNorm function, and activated by the LeakyReLU function:

$${\widehat{z}}_i=\mathrm{LeakReLU}(z_i)$$

(30)

$$\mathrm{LeakReLU}(z_i)=\left\{\begin{array}{ll}{z}_i,& \mathrm{if} z_i\ge 0\\ \epsilon z_i, (\epsilon \sim {10}^{-2})& \mathrm{otherwise}\end{array}\right.$$

(31)

In the fully-connected network, the output of each layer is the input of the next layer. As to the output layer, the number of neurons is the same as the number of material types, which means that $i=I(I=1,2,3,\cdots ,M)$. In addition, the softMax function is set to scale relative density to $0<{\rho }_N^i<1$ ($N=1,2,3,\cdots ,N_c$, $N_c$ represents the number of IGA control points) and ensures the sum of relative densities of all types of materials equals 1 at each IGA control point:

$${\rho }_N^i=\frac{\exp \left({\widehat{z}}_i\right)}{{\displaystyle \sum _{I=1}^M\exp \left({\widehat{z}}_i\right)}}$$

(32)

The aforementioned neural network architecture ensures that the relative densities of the multi-material structure will undergo a change in accordance with alterations to the weights and bias of the neural networks. Moreover, the optimized neural network model can be stored for subsequent use; thereby, increasing the number of control points enhances the resolution of the final optimal structure without additional computational costs.

3.3. Loss Function and Sensitivity Analysis

Classical structural topology optimization is typically configured to solve the minimization problem with objectives such as thermal compliance, compliance, and so forth. In this work, neural networks are designed to replace classical optimization algorithms and perform topology optimization by minimizing the loss function with the Adam optimizer. Therefore, thermal compliance and compliance are incorporated into the loss function simultaneously, which means that a decrease in the loss function leads to an increase in heat dissipation and stiffness performance of the structure. The expressions of thermal compliance, compliance, and weighted function of IGA element are

$$S_e^0={T_e}^{\mathrm{T}}{K}_{te}^0{T}_e$$

(33)

$$C_e^0={U_e}^{\mathrm{T}}{K}_{fe}^0{U}_e$$

(34)

$$C{S}_e^0=\lambda S_e^0+(1-\lambda {)\mathrm{C}}_e^0$$

(35)

where $\lambda$ is the weighting coefficient that regulates the proportion of the two optimization objectives, and with the introduction of material penalty factor, the weighted function of IGA element is transformed into

$$C{S}_e=\lambda {T_e}^{\mathrm{T}}({\displaystyle \sum _{I=1}^M{({\rho }_e^I)}^{\widehat{p}}}{K}_{te}^0)T_e+(1-\lambda ){U_e}^{\mathrm{T}}({\displaystyle \sum _{I=1}^M}{({\rho }_e^I)}^{\widehat{p}}{K}_{fe}^0)U_e$$

(36)

The weighted objective function for multi-objective topology optimization is

$$CS={\displaystyle \sum _{e=1}^{N_e}C{S}_e}$$

(37)

In neural networks, the volumetric term is integrated into the loss function to satisfy the constraints on the volume of the material. As the loss function decreases, the volume of material approaches the corresponding constraint value. The loss function is constructed as

$$L\left(\mathit{w}\right)=\frac{CS\left(\mathit{w}\right)}{C{S}_0}+\alpha {{\displaystyle \sum _{I=1}^M\left(\frac{{\displaystyle \sum _{e=1}^{N\mathrm{e}}\left({\displaystyle \sum _{i=1}^{(p+1)\times (q+1)}{R}_{e,i}^{p,q}{\rho }_{e,i}^I\left(\mathit{w}\right)}\right)A_e^0}}{V_0^I}-1\right)}}^2$$

(38)

where $C{S}_0$ is the value of the initial weighted objective function of the system used for scaling and $\alpha$ is penalty factor. The sensitivity of loss function with respect to design variable $w_i$ is given by

$$\frac{\partial L}{\partial w_i}={\displaystyle \sum _{I=1}^M\frac{\partial L}{\partial {\rho }_{e,j}^I}\frac{\partial {\rho }_{e,j}^I}{\partial w_i}}$$

(39)

where $\frac{\partial {\rho }_{e,j}^I}{\partial w_i}$ can be solved automatically by the backpropagation of neural networks. $\frac{\partial L}{\partial {\rho }_{e,j}^I}$ involves both neural networks and IGA, and its expression is

$$\frac{\partial L}{\partial {\rho }_{e,j}^I}=\frac{1}{C{S}_0}\left[-\widehat{p}{{(\mathrm{CS}}_e^0)}^{\widehat{p}-1}\right]+\left[\frac{2\alpha }{V_0^I}\left(\frac{{\displaystyle \sum _{e=1}^{N\mathrm{e}}\left({\displaystyle \sum _{i=1}^{(p+1)\times (q+1)}{R}_{e,i}^{p,q}{\rho }_{e,i}^I}\right)A_e^0}}{V_0^I}-1\right){\displaystyle \sum _{e=1}^{N\mathrm{e}}\left({\displaystyle \sum _{i=1}^{(p+1)\times (q+1)}{R}_{e,i}^{p,q}}\right)A_e^0}\right]$$

(40)

Subsequently, the weights $\mathit{w}$ of the neural network are updated according to the backpropagation, and the penalty factors $\alpha$ and $\widehat{p}$ are updated during the iteration process. $\widehat{p}$ takes the form of gradual increment to avoid falling into a local minimum. The initial value of penalty factor $\alpha$ is taken as a small value and then gradually increases. The optimal structural performance is reached when the loss function is minimized.

3.4. Mathematical Model and Procedure of MITO

The mathematical model of MITO for multi-material structures under thermal-mechanical loadings using neural networks is established as

$$\left\{\begin{array}{ll}\underset{\mathit{w}}{\min }& CS(\mathit{w})=\lambda \frac{S\left(\mathit{w}\right)-S_{\min }}{S_{\max }-S_{\min }}+(1-\lambda )\frac{C\left(\mathit{w}\right)-C_{\min }}{C_{\max }-C_{\min }}\\ \mathrm{s}.\mathrm{t}.& K_t\left(\mathit{w}\right)T=P\\ & K_f\left(\mathit{w}\right)U=F\\ & {\displaystyle \sum _{e=1}^{N_e}\left({\displaystyle \sum _{i=1}^{(p+1)\times (q+1)}{R}_{e,i}^{p,q}{\rho }_{e,i}^I\left(\mathit{w}\right)}\right)A_e}=V_0^I\end{array}\right.$$

(41)

where $w$ is the overall weights and bias of neural network. $CS\left(\mathit{w}\right)$ is the weighted objective function. $S\left(\mathit{w}\right)$, $S_{\max }$, and $S_{\min }$ are the thermal compliance, maximum thermal compliance, and minimum thermal compliance, respectively. $C\left(\mathit{w}\right)$, $C_{\max }$, and $C_{\min }$ are the compliance, maximum compliance, and minimum compliance, respectively.

The procedure of MITO for multi-material structures under thermal-mechanical loadings using neural networks is depicted in Figure 4. The detailed steps are as follows:

• Input the size of neural networks (number of hidden layers and neurons) and multi-material properties (elastic modulus, Poisson’s ratio, thermal conductivity);
• Determine the initial design domain, volume constraints, weighting coefficients, maximum and minimum values of compliance, maximum and minimum values of thermal compliance, IGA parameter nodes, and control point coordinates for the multi-material structure;
• Set the number of IGA refinement nodes and perform IGA discretization of the multi-material structure;
• Construct loss function and initialize penalty factor, weights, and bias of neural networks;
• Input IGA control point coordinates to the neural networks and perform Fourier mapping;
• Output the relative densities of various materials at the IGA control points;
• Determinate the IGA overall force loading, boundary, and displacement conditions for the design domain;
• Assemble the IGA overall stiffness matrix, establish the IGA discrete governing equation for the 2-D elasticity problem, and solve the displacements at the IGA control points;
• Calculate compliance under the current material density distribution using IGA;
• Determinate the IGA overall thermal loading and thermal boundary conditions for the design domain;
• Assemble the IGA overall thermal stiffness matrix, establish the IGA discrete governing equation for the steady state heat transfer problem, and solve the temperature at the IGA control points;
• Calculate thermal compliance under the current structural density distribution using IGA;
• Establish the mathematical model for MITO of multi-material structures;
• Calculate the loss function and perform backpropagation of the neural networks;
• Update the material penalty factor and the volume penalty factor in the loss function;
• Judge the condition of iteration termination, if the condition is satisfied, get the optimized neural network model and go to the next step, if not, turn to step 5 and the aforementioned loop should be repeated until the condition is met;
• Set a higher number of IGA refinement nodes, input new IGA control point coordinates into the optimized neural network model, and output the high-resolution optimal topologies.

Figure 4. Procedure of MITO for multi-material structures under thermal-mechanical loadings using neural networks.

Figure 4. Procedure of MITO for multi-material structures under thermal-mechanical loadings using neural networks.

4. Numerical Examples

4.1. Michell Beam

As shown in Figure 5, the Michell beam is selected as the structural design domain, where temperature $T=50\hspace{0.17em}°\mathrm{C}$ is fixed at the upper midpoint, heat flux $\overline{q}=80\hspace{0.17em}{\mathrm{W}/\mathrm{m}}^2$ is applied at the lower edge, heat source $\dot{Q}=5\hspace{0.17em}{\mathrm{W}/\mathrm{m}}^3$ is applied at each of the lower endpoints, and force $F=1000\hspace{0.17em}\mathrm{N}$ in negative $y$-direction is applied at the lower midpoint. The parameters of the Michell beam with different numbers of material types are shown in

Table 1. Material parameters of the Michell beam with different numbers of material types.

Number of Material Types	1	2		3
Material	MAT-1	MAT-1	MAT-2	MAT-1	MAT-2	MAT-3
Color
Thermal conductivity	200	200	2 × 10−2	300	200	2 × 10−2
Elastic modulus	5 × 105	5 × 105	1 × 10−8	8 × 106	5 × 106	1 × 10−8
Poisson’s ratio	0.3	0.3	0.3	0.3	0.3	0.3
Volume fraction	0.4	0.5	0.5	0.25	0.15	0.6

.

Table 2. Optimal topology of the Michell beam with different weighting coefficients.

Number of Material Types	Method	Weighting Coefficient
		0	0.5	1
1	IGA- SIMP	S = 94.4 W, C = 43.7 $\mathrm{N}\cdot \mathrm{m}$	S = 74.1 W, C =45.5 $\mathrm{N}\cdot \mathrm{m}$	S = 62.3 W, C = 559.6 $\mathrm{N}\cdot \mathrm{m}$
	IGA- NN	S = 83.6 W, C = 41.0 $\mathrm{N}\cdot \mathrm{m}$	S = 69.4 W, C = 43.5 $\mathrm{N}\cdot \mathrm{m}$	S = 61.6 W, C = 493.1 $\mathrm{N}\cdot \mathrm{m}$
	IGA- NN- Fourier	S > 1000 W, C = 40.9 $\mathrm{N}\cdot \mathrm{m}$	S = 83.2 W, C = 41.2 $\mathrm{N}\cdot \mathrm{m}$	S = 61.8 W, C = 504.9 $\mathrm{N}\cdot \mathrm{m}$
2	IGA- SIMP	S = 75.2 W, C = 35.4 $\mathrm{N}\cdot \mathrm{m}$	S = 72.3 W, C = 44.1 $\mathrm{N}\cdot \mathrm{m}$	S = 54.6 W, C = 211.2 $\mathrm{N}\cdot \mathrm{m}$
	IGA- NN	S = 67.9 W, C = 34.4 $\mathrm{N}\cdot \mathrm{m}$	S = 62.0 W, C = 36.2 $\mathrm{N}\cdot \mathrm{m}$	S = 56.1 W, C = 186.7 $\mathrm{N}\cdot \mathrm{m}$
	IGA- NN- Fourier	S = 71.5 W, C = 34.4 $\mathrm{N}\cdot \mathrm{m}$	S = 61.1 W, C = 36.2 $\mathrm{N}\cdot \mathrm{m}$	S = 55.1 W, C = 184.3 $\mathrm{N}\cdot \mathrm{m}$
3	IGA- SIMP	S = 82.6 W, C = 36.6 $\mathrm{N}\cdot \mathrm{m}$	S = 81.0 W, C = 55.9 $\mathrm{N}\cdot \mathrm{m}$	S = 48.4 W, C = 222.7 $\mathrm{N}\cdot \mathrm{m}$
	IGA- NN	S > 1000 W, C = 31.2 $\mathrm{N}\cdot \mathrm{m}$	S = 56.3 W, C = 31.7 $\mathrm{N}\cdot \mathrm{m}$	S = 48.2 W, C = 280.3 $\mathrm{N}\cdot \mathrm{m}$
	IGA- NN- Fourier	S = 73.9 W, C = 30.8 $\mathrm{N}\cdot \mathrm{m}$	S = 57.3 W, C = 31.8 $\mathrm{N}\cdot \mathrm{m}$	S = 45.1 W, C = 160.1 $\mathrm{N}\cdot \mathrm{m}$

shows the optimal topologies of the Michell beam with different weighting coefficients. The topologies obtained based on the IGA-SIMP method have intermediate densities at the boundary, while the topologies obtained based on IGA and neural networks (IGA-NN) or IGA-NN via Fourier mapping (IGA-NN-Fourier) methods have almost no intermediate densities and exhibit smoother contours. This is because the neural networks model outputs the densities of control points refined by a large number of IGA refinement nodes without extra computational costs.

From Table 2, the topologies obtained based on the IGA-SIMP method have more truss structures, while the topologies obtained based on the IGA-NN method exhibit more concentrated material distribution. The IGA-NN-Fourier method performs a Fourier mapping on the coordinates of IGA control points based on the IGA-NN method, and more fine-branch structures appear in the obtained topologies. When weighting coefficient $\lambda =0$, the optimal topology shows a truss structure with better stiffness performance. While the optimal topology shows a tree root-like structure that extends downward to enhance heat transfer performance when $\lambda =1$. The material extends down from the center and the structure is formed by concentrating the material in the force conduction path when $\lambda =0.5$. The above phenomenon can be explained by the form in which the weighted objective function is composed. The part of the objective function that minimizes thermal compliance equals 0 when $\lambda =0$, and then the topology of the structure is completely in the direction of minimizing compliance. The topology of the structure is completely oriented to minimize thermal compliance when $\lambda =1$. Therefore, the optimal topology improves single structural performance when $\lambda =0$ or $\lambda =1$; however, it is unstable that the high values of structural thermal compliance or compliance occur since it is not optimized. When $\lambda =0.5$, the weighted objective function has both relevant terms for optimizing the heat transfer performance and stiffness performance, and the optimal topology obtained at this time has comprehensive performance.

The process of topology optimization for the Michell beam based on the IGA-NN-Fourier method is observed when $\lambda =0.5$. Figure 6 shows the iteration of the weighted objective function and volume fraction. Figure 7 shows topologies with different iteration numbers. From Figure 6, the weighted objective function has completed a sharp decrease within 10 iteration steps, and the volume fractions of MAT-1 and MAT-2 also reach the constraint condition quickly. The optimization has completed the general layout of the material at the 10th step of the iteration in Figure 7. The weighted objective function and volume fraction converge quickly and smoothly with few iterative steps, but the corresponding topology structure has a large number of intermediate densities. With the increase of iteration number, the penalty factor gradually increases, and the intermediate densities of topology are gradually reduced. The configuration of topology has nearly been determined when 50 iterations are completed, and the structure has a clear contour when the iteration number is 100.

Figure 8 shows the iterative process of thermal compliance and compliance with different methods. All three methods demonstrate rapid convergence with few iteration steps, and the convergence curves are generally smooth. The initial values of thermal compliance and compliance derived from all three methods are distinct, which is attributable to the fact that the initial material densities of the IGA-SIMP method are artificially set based on the volume fraction constraints. In contrast, the initial material densities of the proposed method are determined by the initial overall weights and biases of the neural networks, which are stochastic. Furthermore, the convergence values of thermal compliance and compliance obtained by the proposed method are lower than those obtained by the IGA-SIMP method. The classical method calculates the sensitivity of the objective function to the material density and adjusts the material density at each control point by means of specific algorithms. This enables the descending path of the objective function to be identified at the early stage of the iteration. However, it is challenging to further optimize the structure when the sensitivity values at a large number of control points exhibit minimal variation. While a neural network-based topology optimization is capable of optimizing the structure to a greater extent by continuously decreasing the loss function.

4.2. Reuleaux Triangular Structure

Figure 9 shows a Reuleaux triangular structure under thermal-mechanical loadings. The thermal and force boundary conditions and loads are applied as follows: temperature $T=50\hspace{0.17em}°\mathrm{C}$ is fixed at the center of Reuleaux triangular, heat flux $\overline{q}=200\hspace{0.17em}{\mathrm{W}/\mathrm{m}}^2$ is imposed on the outer edges of the three arcs, heat source $\dot{Q}=5\hspace{0.17em}{\mathrm{W}/\mathrm{m}}^3$ is applied to the two endpoints of the bottom edge, the displacements in the transverse and longitudinal directions are fixed in the middle part of the bottom edge, and the force loadings $F=1000\hspace{0.17em}\mathrm{N}$ are applied to each of the three endpoints. The paraments of elastic modulus, thermal conductivity, and constraint of volume fraction of Mat-1 (red), Mat-2 (black), and Mat-3 (white) are [6 × 10⁶, 3 × 10⁶, 1 × 10⁻⁸] Pa, [200, 100, 2 × 10⁻²] $\mathrm{W}/\left(\mathrm{m}\cdot \mathrm{K}\right)$ and [0.3, 0.3, 0.4], respectively.

Table 3. Optimal topological structures, temperature fields, and displacement fields with different weighting coefficients.

Weighting Coefficient $\mathit{\lambda }$	Optimal Topology	Temperature Field	Displacement Field
0.01	S = 92.5 W, C = 28.7 $\mathrm{N}\cdot \mathrm{m}$
0.25	S = 65.1 W, C = 29.7 $\mathrm{N}\cdot \mathrm{m}$
0.5	S = 60.2 W, C = 30.3 $\mathrm{N}\cdot \mathrm{m}$
0.75	S = 56.5 W, C = 36.2 $\mathrm{N}\cdot \mathrm{m}$
0.99	S = 52.9 W, C = 56.2 $\mathrm{N}\cdot \mathrm{m}$

shows the optimal topologies of a Reuleaux triangle structure with different weighting coefficients. Contradiction between objectives is the main difficulty in multi-objective problems. It means that there are counteracting effects between the objectives, making it difficult to optimize all the objectives simultaneously. In this case, with the increase of the weighting coefficient, the materials on the arc edges of the Reuleaux triangle become thicker, and the temperature field improves. This is due to the addition of the relevant term of thermal compliance in the weighted objective function, and the optimized distribution of material enhances structural heat transfer performance. In addition, as weighting coefficient increases, the topological results show a gradual decrease in the number of support branches favoring the force-bearing performance.

Figure 10 shows the iteration of the weighted objective function and volume fraction of each material when weighting coefficient $\lambda =0.5$. Figure 11 shows thermal compliance and compliance of the optimal topology obtained based on the IGA-NN method with different weighting coefficients. Figure 10 illustrates that the objective function undergoes a pronounced decline at the outset of the iteration, followed by a gradual increase, a subsequent gradual decline, and finally, a tendency towards stability. This is because the volume fraction of each material is lower than the constraint condition at the initial stage of optimization. The increase in the volume of MAT-1 (red) $V^1$ and MAT-2 (black) $V^2$, and the decrease in the volume of MAT-3 (white) $V^3$ reduce the value of the objective function in the loss function, resulting in a corresponding reduction in the value of the loss function. After the volume constraints of Mat-1 and Mat-2 are reached, $V^1$ and $V^2$ no longer rise. Mat-3 is void material, its volume $V^3$ keeps rising because the volume constraint of Mat-3 is not reached, then the change of the loss function depends on the variations in volume, and the weighted objective function increases with the rise of the volume of void material Mat-3. When the volume constraint of Mat-3 has been satisfied, the weighted objective function decreases slowly and finally converges.

From Figure 11, the thermal compliance of the optimal topology decreases as the weighting coefficient increases, which indicates that the structural heat dissipation performance is enhanced. While the compliance gradually increases, the displacement of the upper part of the structure increases, which means the stiffness performance of the structure is weakened. The above changes are directly related to the composition of the weighted objective function. As the weighting coefficient increases, the weight of the objective function for minimum thermal compliance in the weighted objective function increases, while the weight of the objective function for minimum compliance decreases, which is reflected in the configuration and structural performance of the optimal topologies. In order to achieve an optimal topology that exhibits both effective heat dissipation and stiffness performance, as illustrated in Table 3 and Figure 11, the recommended weighting coefficients are in a range from 0.25 to 0.75, at which point the two properties of the structure are satisfactorily compatible.

The parameters of volume fraction of materials are given as $V_f^3$ = 0.4, $V_f^1$ = [0.1, 0.2, 0.3, 0.4, 0.5], and $V_f^2$ = [0.5, 0.4, 0.3, 0.2, 0.1], respectively. Figure 12 shows the topological results obtained with different volume fractions of materials. Figure 13 shows structural thermal compliance and compliance with different volume fractions of materials.

From Figure 12, it can be observed that as the volume fraction of MAT-1 (red) increases, the red component of the optimal topology gradually extends upwards from the base of the Reuleaux triangle, ultimately becoming the predominant material in the topological structure. Figure 13 illustrates that the thermal compliance and compliance of the optimal topology decrease with the increase of the volume fraction of MAT-1. This indicates that the structural heat dissipation and stiffness performance have been enhanced because MAT-1 has the optimal elastic modulus and thermal conductivity among the three materials. The increase in the proportion of MAT-1 in the whole structure optimizes the comprehensive performance.

5. Conclusions

In this paper, a multi-objective isogeometric topology optimization model of multi-material structures under thermal-mechanical loading based on neural networks is proposed. In the proposed model, neural networks are formed to minimize thermal compliance and compliance of multi-material structures simultaneously. The weights and biases of neural networks are used as design variables, and a weighted thermal-mechanical objective function is incorporated into the loss function to obtain structures with comprehensive performance.

In comparison to the IGA-SIMP method, the optimal topologies obtained by the IGA-NN method exhibit higher resolution and smoother boundaries. The distribution of materials in the structures generated by the neural networks becomes more concentrated, and we can obtain results for more complex branching structures through Fourier mapping. The topology optimization based on neural networks is capable of optimizing the structure to a greater extent by continuously reducing a defined loss function.

The weighting coefficient directly determines the proportion of thermal compliance and compliance in the objective function. The stiffness performance of the structure is optimal when the weighting coefficient value is approximately 0, while the heat dissipation performance is optimal when the weighting coefficient value is approximately 1. The topologies with intermediate weighting coefficients have better compatibility in the structural performance.

The volume fraction of material directly affects the distribution of material in the design domain. Since the thermal conductivity and elastic modulus of each material are different, respectively, increasing the volume fraction of the materials with superior properties can improve the overall performance of the optimal topology.