Topology Optimization of Coupled Multi-Physics Problems
Topology optimization provides a promising approach to systematically design multi-physics problems, such as thermo-mechanical, electro-static, and fluid-structure interaction problems. This class of design problems is often dominated by nonlinear phenome
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Department of Aerospace Engineering Sciences, University of Colorado Boulder, Boulder, CO, USA
Abstract. Topology optimization provides a promising approach to systematically design multi-physics problems, such as thermomechanical, electro-static, and fluid-structure interaction problems. This class of design problems is often dominated by nonlinear phenomena and is not well suited for intuitive design strategies. In this lecture we will discuss applications of topology optimization methods to coupled multi-physics problems, emphasizing the differences between volumetric and interface coupling in the context of topology optimization. Focusing on density methods, topology optimization of piezo-electric devices and fluid-structure interaction problems will be studied.
1
Introduction
The performance of many engineering systems often depends on multiple physical phenomena belonging to different engineering disciplines, such as solid mechanics, fluid mechanics, and heat transfer. Topology optimization methods have been developed and applied to problems that are dominated G. Rozvany, T. Lewiński (Eds.), Topology Optimization in Structural and Continuum Mechanics, CISM International Centre for Mechanical Sciences, DOI 10.1007/978-3-7091-1643-2_18, © CISM, Udine 2014
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K. Maute
by a single phenomenon, such as elastic deformations in structural mechanics and the flow of liquids and gases in fluid mechanics. The reader is referred to the textbook by Bendsøe and Sigmund (2003) for an overview of topology optimization. In this lecture we will discuss topology optimization methods for coupled problems where the interaction of multiple physical phenomena needs to be accounted for. Such problems are often labeled “multi-physics” and this terminology will be used subsequently. Before discussing individual applications, first we will examine the basic modes in which different physical fields interact and how this interaction is modeled in the context of topology optimization. 1.1
Volumetric and Surface Coupling
Let us assume two scalar physical fields, u and v, which are governed by two partial differential equations, here linear diffusion equations. For the sake of simplicity, we study these fields for steady state conditions, i.e. they do not vary with time. First we consider the case where both fields are defined over the same domain Ω. ∂ u j + q u = 0 in Ω ∂xi i ∂ v j + q v = 0 in Ω ∂xi i
jiu ni = Qu
on Γqu
u=u ˆ
on Γu ,
(1)
jiv ni = Qv
on Γqv
v = vˆ on Γv ,
(2)
with ji denotes the diffusive fluxes, q are external volumetric fluxes, and Q are external surface fluxes. The fields are coupled via the constitutive laws and/or the external volumetric fluxes: jiu = jiu (uj , vj ) q u = q u (uj , vj )
and and
jiv = jiv (uj , vj ) , v
v
q = q (vj , uj ) .
(3) (4)
In both cases, the fields are coupled in each material point or a subset of points within the volume of the body. Examples for this type of coupling include Joule heating for electro-thermal problems, thermal expansion in thermo-elastic problems, and piezo-elec
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