Efficient topology optimization of thermo-elasticity problems using coupled field adjoint sensitivity analysis method
Introduction
A topology design optimization method helps designers to find a suitable material layout to achieve required performances. Ever since Bendsøe and Kikuchi [1] introduced a topology optimization method using a homogenization method, many topology optimization methods have been developed in many disciplines [2]. Since the topology optimization method involves many design variables, a gradient-based optimization method is generally preferred. Therefore, it is important that the sensitivity of performance measures with respect to design variables should be determined in a very efficient way. In the continuum DSA approach, the design sensitivity expressions are obtained by taking the first-order variation of the continuum variational equation [3]. In this paper, we will address the thermo-elasticity problems including both heat conduction and elasticity. To simplify the problem, a weakly coupled thermo-elasticity problem in steady state is considered. The design sensitivity expressions for heat conduction and elasticity problems are derived. The adjoint variable method is employed for the efficient computation of design sensitivity. In developing the adjoint DSA method for the thermo-elasticity problems, besides the adjoint equations for temperature and displacement fields, a coupled field adjoint equation in the temperature field is defined regarding the obtained adjoint displacement field as an adjoint load. We only need one more adjoint response in the coupled field instead of the whole field of the temperature sensitivity. The adjoint DSA method developed is then applied to the topology optimization of a thermo-elastic solid.
There are only a few literatures regarding topology optimization of thermal systems but quite a number of literatures on the shape DSA. Tortorelli et al. derived the shape design sensitivity for nonlinear transient heat conduction problems using a Lagrange multiplier method [4] and the adjoint method [5]. Yang [6] derived the shape design sensitivity expressions of thermo-elasticity problems by applying a material derivative approach to temperature and displacement fields. Sluzalec et al. [7] employed a Kirchhoff transformation to derive shape design sensitivity expressions for nonlinear heat conduction problems using the adjoint variable method. Bobaru et al. [8] employed an element-free Galerkin method in the DSA of thermo-elastic solids and applied it to thermal shape optimization problems. Jog performed a topology optimization for nonlinear thermo-elasticity with the perimeter method [9]. Li et al. [10] performed a discrete topology optimization using the ESO method.
The remainder of this paper is organized as follows: in Section 2, the governing equations for both heat conduction and elasticity problems are discussed. Weak formulations for the weakly coupled thermo- elasticity problems in steady state are derived. In Section 3, continuum-based DSA methods are formulated for the weakly coupled thermo-elasticity problems using the adjoint variable method in continuum form. In Section 4, a topology optimization method is formulated for the thermo-elasticity problem where the developed adjoint DSA method is applied. The penalization and parameterization methods of design variables are discussed. In Section 5, several numerical examples are presented to verify the accuracy of the proposed analytical DSA method compared with the finite difference sensitivity. Then, the efficiency of the developed method is discussed. The results of topology design optimization show very satisfactory results. Finally, concluding remarks are given in the last section.
Section snippets
Thermo-elasticity problems
Consider a body occupying an open domain in space that is bounded by a closed surface as shown in Fig. 1. Material properties are assumed to be linearly elastic and isotropic in domain . The body is subjected to the rate of internal heat generation and the following thermal boundary conditions: a prescribed temperature is imposed over the temperature boundary , a prescribed heat flux is applied to the flux boundary in the inward normal direction, and an ambient temperature
Design sensitivity equation
Consider a non-shape design variable vector that parameterizes the material distribution in heat conduction and elasticity problems. For the given design , Eqs. (21) and (25) can be written asandTaking the first-order variation of Eqs. (26) and (27), we have the following:andwhere the forms of
Formulation of topology design optimization
The objective of the topology optimization method is to find the optimal material distribution that maximizes the stiffness or minimizes the compliance of structural systems. In this paper, we discuss the topology design optimization method for coupled thermo-elasticity problems where the temperature and displacement fields are coupled in an identical domain. For the development of the topology optimization method, we assume the following:
- 1.
The design variables are associated with two material
Numerical examples
In the following numerical examples, for simplicity of problems, we consider two-dimensional problems in the absence of body force intensity and rate of heat generation . Example 1 Temperature sensitivity with respect to variation of thermal conductivity.
This example is intended to verify the DSA method in the temperature field only. Consider a square plate fixed along the bottom side as shown in Fig. 2.
The ambient temperature is 20 °C and the convection boundary conditions are imposed along the sides
Conclusions
In this paper, we have derived variational equations for the weakly coupled thermo-elasticity problems in steady state, where temperature and displacement fields are described in a common domain. Also, a topology design optimization method has been formulated by applying the adjoint DSA method. In developing the adjoint variable methods for the thermo-elasticity problems, in addition to the temperature and displacement adjoint equations, we defined a coupled field adjoint equation in the
Acknowledgements
This work was supported by the Advanced Ship Engineering Research Center of the Korea Science and Engineering Foundation under Grant No. R11-2002-104-03002-0 for the years 2003 and 2004. The support is gratefully acknowledged.
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