Efficient topology optimization of thermo-elasticity problems using coupled field adjoint sensitivity analysis method

doi:10.1016/j.finel.2005.05.003

Finite Elements in Analysis and Design

Volume 41, Issue 15, September 2005, Pages 1481-1495

https://doi.org/10.1016/j.finel.2005.05.003 Get rights and content

Abstract

We develop a unified and efficient adjoint design sensitivity analysis (DSA) method for weakly coupled thermo-elasticity problems. Design sensitivity expressions with respect to thermal conductivity and Young's modulus are derived. Besides the temperature and displacement adjoint equations, a coupled field adjoint equation is defined regarding the obtained adjoint displacement field as the adjoint load in the temperature field. Thus, the computing cost is significantly reduced compared to other sensitivity analysis methods. The developed DSA method is further extended to a topology design optimization method. For the topology design optimization, the design variables are parameterized using a bulk material density function. Numerical examples show that the DSA method developed is extremely efficient and the optimal topology varies significantly depending on the ratio of mechanical and thermal loadings.

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 $Ω$ in space that is bounded by a closed surface $Γ$ as shown in Fig. 1. Material properties are assumed to be linearly elastic and isotropic in domain $Ω$ . The body is subjected to the rate of internal heat generation $Q$ and the following thermal boundary conditions: a prescribed temperature $T_{0}$ is imposed over the temperature boundary $Γ_{T}^{0}$ , a prescribed heat flux $q$ is applied to the flux boundary $Γ_{T}^{1}$ in the inward normal direction, and an ambient temperature $T_{\infty}$

Design sensitivity equation

Consider a non-shape design variable vector $u$ that parameterizes the material distribution in heat conduction and elasticity problems. For the given design $u$ , Eqs. (21) and (25) can be written as $Find T \in Y such that A_{u} (T, \bar{T}) = L_{u} (\bar{T}) for all \bar{T} \in \bar{Y}$ and $Find z \in Z such that a_{u} (z, \bar{z}) = α_{u} (T, \bar{z}) + ℓ_{u} (\bar{z}) for all \bar{z} \in \bar{Z} .$ Taking the first-order variation of Eqs. (26) and (27), we have the following: $A_{u} (T^{'}, \bar{T}) = L_{δ u}^{'} (\bar{T}) - A_{δ u}^{'} (T, \bar{T}) for all \bar{T} \in \bar{Y}$ and $a_{u} (z^{'}, \bar{z}) = α_{δ u}^{'} (T, \bar{z}) + α_{u} (T^{'}, \bar{z}) + ℓ_{δ u}^{'} (\bar{z}) - a_{δ u}^{'} (z, \bar{z}) for all \bar{z} \in \bar{Z},$ where 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 $b$ and rate of heat generation $Q$ .

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.

References (10)

M.P. Bendsøe et al.
Generating optimal topologies in structural design using a homogenization method
Comput. Methods Appl. Mech. Eng.
(1988)
D.A. Tortorelli et al.
Design sensitivity analysis for nonlinear thermal systems
Comput. Methods Appl. Mech. Eng.
(1989)
R.J. Yang
Shape design sensitivity analysis of thermoelasticity problems
Comput. Methods Appl. Mech. Eng.
(1993)
A. Sluzalec et al.
Shape sensitivity analysis for nonlinear steady-state heat conduction problems
Int. J. Heat Mass Transfer
(1996)
C. Jog
Distributed-parameter optimization and topology design for nonlinear thermoelasticity
Comput. Methods Appl. Mech. Eng.
(1996)

There are more references available in the full text version of this article.

Cited by (55)

Multi–materials topology optimization using deep neural network for coupled thermo–mechanical problems
2024, Computers and Structures
In this paper, a density-based topology optimization method using neural networks is used to design domains composed of multi-materials under combined thermo–mechanical loading. The neural network produces the topology density field by minimizing a loss function defined for the problem. Length scale control is conducted using a Fourier space projection. JAX, a high-performance automatic differentiation (AD) Python library is used to build an end-to-end differentiable neural network. The model can handle the sensitivity analysis automatically using the backpropagation process (automatic differentiation of loss function) and the need for solving adjoint equations manual sensitivity analysis is removed. The optimization problem is solved by minimizing the loss function of the network and other optimization algorithms such as the Method of Moving Asymptotes (MMA) are not required. The performance of the method is demonstrated through several examples and compared with those obtained from the popular Solid Isotropic Material with Penalization (SIMP) method. The network is able to handle high-resolution re-sampling, resulting in a more refined and smooth variation of optimal topologies.
Multi-material topology optimization for maximizing structural stability under thermo-mechanical loading
2023, Computer Methods in Applied Mechanics and Engineering
Mechanical structures are often simultaneously subjected to thermal and mechanical loading, both of which can lead to buckling failure. Developing efficient structural forms with better capacity for stability is important to keep structures safe. This study aims to optimize structural buckling capacity by using a density-based topology optimization scheme. Instead of treating the mechanical and thermal loadings as a single coupled part in the linearized buckling analysis, the effects of mechanical and thermal loadings are decoupled, which allows to separately analyze and optimize buckling aspects induced by mechanical or thermal loading. Two optimization models based on the decoupled analysis models are developed to respectively maximize the critical load factor of buckling induced by mechanical loading under a specified thermal loading and buckling induced by thermal loading under a specified mechanical loading. Further, based on a three-phase material model, a multi-material topology optimization scheme is employed to optimize the buckling capacity of active structures made of structural and actuating materials and prestressed structures containing prestressed components. The actuation effects are mimicked by the thermal loading of active material. The sensitivities of the objective functions and constraints are derived through the adjoint technique, and the method of moving asymptotes (MMA) is employed to solve the topology optimization problems. Numerical examples are adopted to verify the effectiveness of the proposed approach.
Topology optimization of thermoelastic structures using MMV method
2022, Applied Mathematical Modelling
Citation Excerpt :
Sigmund and Torquato [11] used a three phase homogenization method to achieve the optimal distribution of each material phase with extremal thermal expansion coefficients. Based on the Solid Isotropic Material with Penalization (SIMP) method [12], Cho and Choi [13] developed an adjoint sensitivity analysis method of weakly-coupled thermoelastic topology optimization for finding the optimal structure in varying temperature field. Pedersen and Pedersen[14] suggested an alternate topology optimization of minimizing the von Mises stress instead of minimizing the traditional compliance in thermoelastic structures, and two sensitivities of von Mises stress are also derived by implementing the Rational Approximation of Material Properties (RAMP) and SIMP interpolation schemes under a uniform temperature elevation.
Thermoelastic problems refer to a coupled phenomenon existing in a wide range of practical applications where elastic responses are influenced by temperature variation. Meanwhile, Moving Morphable Void (MMV) approach receives extensive attentions owing to explicit topological geometric information. In MMV method, a closed B-spline curve defined by several design variables is adopted to describe void material in design domain. In present study, we intend to use the MMV method to topologically optimize the thermoelastic structures under uniform temperature increment and varying temperature field. The weak formulations of heat conduction and elasticity are combined to describe the thermoelastic problems. The optimization mathematical model with the aim of compliance minimization and volume constraint is constructed. The sensitivity of design variables is derived by material derivative method and adjoint sensitivity analysis method. The method of moving asymptote (MMA) algorithm is implemented to update the design variables for finding the best material distribution. Numerical examples are provided to demonstrate the effectiveness of the proposed method.
An all-movable rudder designed by thermo-elastic topology optimization and manufactured by additive manufacturing
2021, Computers and Structures
Citation Excerpt :
In 2008, Xia and Wang proposed to adopt the LSM to topology optimization for thermo-elastic structures [20] and since then several researchers have gone deeper in this field [21–24]. Under the frame of the density-based method, Cho and Choi developed an efficient adjoint sensitivity analysis method for the topology optimization of weakly coupled thermo-elastic problems [25]. Pedersen and Pedersen found compliance to be a questionable objective for such thermo-elastic design problems and proposed to achieve strength maximization by pursuing a uniform energy density[26].
In high-speed vehicles, rudders often endure both aerodynamic pressure and thermal loads. The innovative design of rudders is of great importance for the performance of the whole vehicle. In this work, thermo-elastic topology optimization is adopted to design a typical all-movable rudder structure. The compliance of the rudder skin is considered to be a new objective and the moment of inertia of the rudder is constrained during optimization to ensure its fast response to instructions of the control system. Then sensitivity analysis of the structural compliance and the moment of inertia is carried out. Optimization results show that thermal load has a great effect on the optimized configuration and minimizing the compliance of the rudder skin gives much better design than minimizing the global compliance. Subsequently, an engineering-oriented post-processing is conducted to make the optimized design suitable for additive manufacturing. An appropriate printing direction is selected based on the layout of the optimized ribs and certain ribs are reshaped with fillets to make the rudder free of internal support structures. Besides, according to a secondary topology optimization of the ribs and the stress distribution, a set of powder-discharge holes are properly opened on the ribs so that all cavities within the rudder are connected and the metal powder inside the rudder can be discharged with little effort after manufacturing. Finally, the optimized design is successfully printed using Selective Laser Melting, demonstrating the proposed post-processing is effective for additive manufacturing.
Topological optimization of thermoelastic composites with maximized stiffness and heat transfer
2019, Composites Part B: Engineering
Citation Excerpt :
Namely, the temperature field is considered first, and then it is matched with the field of deformations. In the mentioned cases, the design can be guided either by the mechanical target function [23] or by coupled mechanical and heat functions [24]. The qualitative seminal results in the problem of design of materials were obtained by Sigmund [15,16] who introduced the method of inverse optimization.
The topological optimization of composite structures is widely used while tailoring materials to achieve the required engineering physical properties. In this paper, the problem of topological optimization of the microstructure of a composite aimed at the construction of a material with most effective values of the bulk modulus of elasticity and thermal conductivity taking into account competing mechanical and thermal properties of the materials included in the composite is defined and solved. A two-phase composite consists of two base materials, one of which has a higher Young modulus but lower thermal conductivity, while the other has a lower Young modulus but higher thermal conductivity. A new class of problems for composites containing material pores or technological inclusions of different shapes is considered. Effective thermoelastic properties are obtained using the asymptotic homogenization method. A modified solid isotropic material with penalization (SIMP) model is used to regularize the problem. The problem of the isoperimetric constraints is solved by the method of moving asymptotes (MMA). The influence on optimal topology of the composite in the presence of two competing materials, and optimality criteria using linear weight functions are investigated. Pareto spaces that provide deep understanding of how these goals compete in achieving optimal topology are constructed.
Non-linear dynamics of size-dependent Euler–Bernoulli beams with topologically optimized microstructure and subjected to temperature field
2018, International Journal of Non-Linear Mechanics
This paper is devoted to the investigation of non-linear dynamics of non-homogeneous beams with a material optimally distributed along the height and length of the beam. The study was initiated by topological optimization for the given boundary and loading conditions, which yielded maximum stiffness of a beam microstructure. As a result, the beam with an optimized microstructure exhibiting non-homogeneity in two directions, i.e. along beam thickness and length, was obtained.
In the second step, a beam model was derived based on the kinematic Euler–Bernoulli hypotheses and the modified couple stress theory including the von Kármán geometric non-linearity and heat flow action obeying the Duhamel–Neumann law.
Both static and dynamic behaviour of the optimized (non-homogeneous) and homogeneous beams were studied for different values of the material length-dependent parameter and temperature. Differences and peculiarities in static and dynamic problems were illustrated and discussed. In particular, the influence of the scale size parameter on chaotic beam dynamics was investigated. Also, scenarios of transition into deterministic chaos were detected and analysed for both homogeneous and optimized beams.

View all citing articles on Scopus

View full text

Efficient topology optimization of thermo-elasticity problems using coupled field adjoint sensitivity analysis method

Abstract

Introduction

Section snippets

Thermo-elasticity problems

Design sensitivity equation

Formulation of topology design optimization

Numerical examples

Conclusions

Acknowledgements

Comput. Methods Appl. Mech. Eng.

Comput. Methods Appl. Mech. Eng.

Comput. Methods Appl. Mech. Eng.

Int. J. Heat Mass Transfer

Comput. Methods Appl. Mech. Eng.