Topology optimization of non-Fourier heat conduction problems considering global thermal dissipation energy minimization

Abstract

In this paper, a topology optimization model is proposed for non-Fourier heat conduction design. In this model, the finite element discretization scheme and the Wilson-θ time discretization method are combined to analyze non-Fourier transient heat conduction that is represented by the Cattaneo-Vernotte equation with a relaxation term. Based on the solid isotropic material with penalization (SIMP) interpolation model, the mathematical statement of the proposed optimization design is formulated by integrating the transient objective function over the time interval that considers thermal dissipation energy minimization. The adjoint variable method for sensitivity analysis and the method of moving asymptotes (MMA) for solving the optimization problem are discussed as well. Numerical examples illustrate the validity and applicability of the proposed non-Fourier heat conduction topology optimization by comparison with transient Fourier heat conduction design.

Appendix

The sensitivity analysis of the transient Fourier heat conduction problem is shown in this Appendix. For the two-dimensional (2D) temperature field T(t), the finite element model of the transient heat conduction problem based on Fourier’s law, given as

$$ \mathbf{C}\dot{\mathbf{T}}(t)+\mathbf{KT}(t)=\mathbf{F}(t)\kern2em t\in \left[0,{t}_n\right] $$

(33)

Here, we assume that the heat source and the initial temperature conditions are independent of the design variable, defined as

$$ \frac{\partial \mathbf{F}(t)}{\partial {x}_e}=0,\kern1em \frac{\partial \mathbf{T}(0)}{\partial {x}_e}=0\kern1em \mathrm{and}\kern1em \frac{\partial \dot{\mathbf{T}}(0)}{\partial {x}_e}=0 $$

(34)

For thermal dissipation energy problem

$$ f\left(\dot{\mathbf{T}}(t),\mathbf{T}(t),\mathbf{x}\right)=\frac{1}{2}{\mathbf{T}}^T\mathbf{KT} $$

(35)

According to the Lagrangian multiplier method, the augmented term of the equilibrium equation is expressed as

$$ \varphi ={\int}_0^{t_n}\frac{1}{2}{\mathbf{T}}^T\mathbf{KT} dt+{\int}_0^{t_n}{\boldsymbol{\upeta}}^T\left(\mathbf{C}\dot{\mathbf{T}}+\mathbf{KT}-\mathbf{F}\right) dt $$

(36)

The derivative of the function φ with respect to the design variable x_e is obtained as

$$ \frac{\partial \varphi }{\partial {x}_e}={\int}_0^{t_n}\left(\frac{1}{2}{\mathbf{T}}^T\frac{\partial \mathbf{K}}{\partial {x}_e}\mathbf{T}+{\mathbf{T}}^T\mathbf{K}\frac{\partial \mathbf{T}}{\partial {x}_e}\right) dt+{\int}_0^{t_n}\left[{\boldsymbol{\upeta}}^T\left(\frac{\partial \mathbf{C}}{\partial {x}_e}\dot{\mathbf{T}}+\mathbf{C}\frac{\partial \dot{\mathbf{T}}}{\partial {x}_e}+\frac{\partial \mathbf{K}}{\partial {x}_e}\mathbf{T}+\mathbf{K}\frac{\partial \mathbf{T}}{\partial {x}_e}\right)\right] dt $$

(37)

Given that ${\left.\boldsymbol{\upeta} \right|}_{t_n}=0$, the integral terms ${\int}_0^{t_n}{\boldsymbol{\upeta}}^T\mathbf{C}\frac{\partial \dot{\mathbf{T}}}{\partial {x}_e} dt$ are solved partial integration, transformed as

$$ {\int}_0^{t_n}{\boldsymbol{\upeta}}^T\mathbf{C}\frac{\partial \dot{\mathbf{T}}}{\partial {x}_e} dt=-{\int}_0^{t_n}{\dot{\boldsymbol{\upeta}}}^T\mathbf{C}\frac{\partial \mathbf{T}}{\partial {x}_e} dt $$

(38)

Then, the derivative of the function φ is reformulated by

$$ \frac{\partial \varphi }{\partial {x}_e}={\int}_0^{t_n}\left({\mathbf{T}}^T\mathbf{K}-{\dot{\boldsymbol{\upeta}}}^T\mathbf{C}+{\boldsymbol{\upeta}}^T\mathbf{K}\right)\frac{\partial \mathbf{T}}{\partial {x}_e} dt+{\int}_0^{t_n}\left({\boldsymbol{\upeta}}^T\left(\frac{\partial \mathbf{C}}{\partial {x}_e}\dot{\mathbf{T}}+\frac{\partial \mathbf{K}}{\partial {x}_e}\mathbf{T}\right)+\frac{1}{2}{\mathbf{T}}^T\frac{\partial \mathbf{K}}{\partial {x}_e}\mathbf{T}\right) dt $$

(39)

To eliminate the second term of its right-hand side $\frac{\partial \mathbf{T}}{\partial {x}_e}$and introduce the transformation u(s) = η(t_n − s), the Lagrangian multiplier η can be solved through the adjoint equation

$$ \mathbf{C}\dot{\mathbf{u}}+\mathbf{Ku}=-\mathbf{KT}\kern1.25em {\left.\mathbf{u}\right|}_0=0;{\left.\dot{\mathbf{u}}\right|}_0=0 $$

(40)

Then, the derivative $\frac{\partial \varphi }{\partial {x}_e}$ can be expressed as

$$ \frac{\partial \varphi }{\partial {x}_e}={\int}_0^{t_n}\left({\boldsymbol{\upeta}}^T\frac{\partial \mathbf{C}}{\partial {x}_e}\dot{\mathbf{T}}+{\boldsymbol{\upeta}}^T\frac{\partial \mathbf{K}}{\partial {x}_e}\mathbf{T}+\frac{1}{2}{\mathbf{T}}^T\frac{\partial \mathbf{K}}{\partial {x}_e}\mathbf{T}\right) dt $$

(41)