Abstract
A coupled topology and domain shape optimization framework is presented that is based on incorporating the shape design variables of the design domain in the Solid Isotropic Material with Penalization topology optimization method. The shape and topology design variables are incrementally updated in a sequential fashion, using a staggered numerical update scheme. Non-Uniform Rational B-Splines are employed to parameterize the shape of the design domain. This not only guarantees a highly accurate description of the shape boundaries by means of smooth basis functions with compact support, but also enables an efficient control of the design domain with only a few control points. Furthermore, the optimization process is performed in a computationally efficient way by applying a gradient-based optimization algorithm, for which the sensitivities can be computed in closed form. The usefulness of the coupled optimization approach is demonstrated by analyzing several benchmark problems that are subjected to different types of initial conditions and domain bounds. The variation in simulation results denotes that a careful construction of the initial design domain is necessary and meaningful.
Highlights
Topology optimization is a mathematical method that optimally places the material within a given design domain for the specific loading and boundary conditions applied
The shape and topology design variables are incrementally updated in a sequential fashion, using a so-called staggered numerical update scheme
In order to solve an optimization problem with the finite element method, the design domain needs to be meshed by finite elements. This is done by creating a fixed auxiliary mesh through defining the finite element node locations in the parametric domain, see Zhang et al (2010), and subsequently mapping these node locations to the actual physical design domain using Eq (18), see Fig. 2. This enables the construction of an explicit relationship between the shape design variables and the Finite Element Method (FEM) mesh, so that the FEM mesh is automatically adapted for changes of the structural boundary caused by shape optimization
Summary
Topology optimization is a mathematical method that optimally places the material within a given design domain for the specific loading and boundary conditions applied. Due to the nature of the SIMP method, the possible solutions of the density distribution are bounded by the specific size and shape of the design domain chosen This limitation can be reduced by choosing the design domain as large as possible, this may lead to relatively large computational times. A first possible approach is based on the so-called design space optimization method (Kim and Kwak 2002; Jang and Kwak 2008), which adjusts the design domain boundary during topology optimization by adding new design elements. The FEM formulation forms the basis for the computation of the sensitivities of the structural compliance to the shape and topology design variables, which is presented in Sect.
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