Abstract
The variational approach to shape optimization in linearized elasticity is used in order to improve convergence of a known heuristic algorithm. The speed method of shape optimization is applied to obtain necessary optimality conditions for representative test examples. The algorithm originates from the biomimetic approach to compliance optimization. The trabecular bone adapts its form to mechanical loads and is able to form structures that are lightweight and very stiff at the same time. In this sense, it is a problem pertaining to both the nature or living entities which is similar to structural optimization, especially topology optimization. The paper presents the biomimetic approach, based on the trabecular bone remodeling phenomenon, with the aim of minimizing the compliance in multiple load cases. The method employed aims at minimizing the energy and combines structural evolution inspired by trabecular bone remodeling and the shape gradient framework, with strict analysis based on functionals in the 3-dimensional elasticity model. The method is enhanced to handle the problem of structural optimization under multiple loads. The new biomimetic approach does not require volume constraints. Instead of imposing volume constraints, shapes are parameterized by the assumed strain energy density on the structural surface. The stiffest design is obtained by adding or removing material on the structural surface in virtual space. Structural evolution is based on shape gradient approximation by the speed method, and it is separated from the finite element method of the model solution. Numerical examples confirm that the heuristic algorithm for structural optimization is efficient.
Highlights
The variational approach to shape and topology optimization can be developed, e.g., within the speed and the topological derivative methods
Instead of imposing a volume constraint, we parameterize shapes by the assumed energy density, which may be quite accurately predicted from yield criteria
Instead of imposing a volume constraint, we parameterize shapes by energy density, which may be quite accurately predicted from yield criteria
Summary
The variational approach to shape and topology optimization can be developed, e.g., within the speed and the topological derivative methods. Our aim in this paper is to improve the convergence of the biomimetic approach to compliance optimization and multiple load cases by an application of the speed method. This means that we introduce a variational problem for which the proposed optimality criterion of biomimetic approach becomes a necessary optimality condition. Other researchers suggested [8,9] that the trabecular tissue can adapt its structural form in reaction to external loads. In this sense, it is a problem which resembles structural optimization, especially topology optimization. The mathematical analysis of the so-called speed method in shape optimization is performed, e.g., in [13]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
