Justification of a certain algorithm for shape optimization in 3D elasticity

Abstract

The trabecular bone adapts its form to mechanical loads and is able to form structures that are both lightweight but very stiff. In this sense, it is a problem (for the Nature or living entities) similar to the structural optimization, especially the topology optimization. The structural optimization system developed by the authors previously was based directly on the bio-mechanical models. The paper aims to clarify the approach of using the bio-mechanical models of the trabecular bone remodelling phenomenon as a base for the structural optimization. The justified algorithm for shape optimization mimics the natural phenomenon to satisfy the condition of constant value of the strain energy density on the structural surface, issuing from the considerations in the area of mechanics. The main body of the paper contains the analysis of the stiffness optimization problem in the framework of speed method approach to shape optimization. The given numerical example (cantilever beam in bending) includes the influence of the structural surface curvature on the optimization procedure. The presented in the paper approach can be used as a method of structural optimization unrelated already to trabecular bone remodelling phenomenon.

Biomimetic Approach to Compliance Optimization and Multiple Load Cases

“…Relying on the approach presented in [12], using biomechanical models of the trabecular bone remodeling directly as a base for structural optimization, we will discuss a design with multiple load cases. In general, there is no solution to the problem of stiffest design (compliance minimization).…”

“…The assumption of constant strain energy density for the stiffest design [12] was justified in the same setting, but for a single load. The derivative of the Lagrange function gives (at the stationary point) the following condition for the stiffest design configuration:…”

“…To compute the strain energy density on the structural surface, the finite element method is used. (The details of the numerical environment can be found in [12].) For Fig.…”

“…The computations were run on 4 CPUs with the use of the cluster (16 GB RAM per node). The details of the mesh generation method can be found in [12]. Fig.…”

“…The most important theorem concerning the energy density on the structural surface states that for the stiffest design, the energy density must be constant along the designed shape. The new formulation of biomimetic optimization based on the trabecular bone phenomenon used in [12] was established with the use of the shape derivative concept. The mathematical analysis of the so-called speed method in shape optimization is performed, e.g., in [13].…”