Concurrent material and structure optimization of multiphase hierarchical systems within a continuum micromechanics framework

Abstract

We present a concurrent material and structure optimization framework for multiphase hierarchical systems that relies on homogenization estimates based on continuum micromechanics to account for material behavior across many different length scales. We show that the analytical nature of these estimates enables material optimization via a series of inexpensive “discretization-free” constraint optimization problems whose computational cost is independent of the number of hierarchical scales involved. To illustrate the strength of this unique property, we define new benchmark tests with several material scales that for the first time become computationally feasible via our framework. We also outline its potential in engineering applications by reproducing self-optimizing mechanisms in the natural hierarchical system of bamboo culm tissue.

Thermodynamically consistent concurrent material and structure optimization of elastoplastic multiphase hierarchical systems

“…We emphasize that in contrast to the equivalent strain energy maximization for concurrent optimization problems involving overall linear elastic multiphase hierarchical systems [15,46], finding the solution to the material optimization problem ( 45) is not straightforward. Both the macroscale plastic strain E p n+1 and the optimized microstructure mx,j n+1 are unknown.…”

“…where φC is the equivalent volume fraction of Material C computed as φC = (1 − φ x,j A,n+1 ) γ x,j C,n+1 . For a detailed derivation of the homogenized stiffness C(m x,j n+1 ), we refer interested readers to Appendix 1 in [46]. These estimates hold the following three properties that form the basis for further simplifications in the algorithmic procedure for the material optimization: Property 1: The microscale configuration mx,j n+1 corresponding to the maximum stiffness (maximum strain energy density) from ( 51) also maximizes the homogenized strength response.…”

“…Later, the optimized material orientation is updated for each load increment by aligning the material axis with the principal strain axes of the elastic part of the macroscale strain tensor E e n+1 . Furthermore, the continuum micromechanics-based estimates render the strain energy maximization (51) a straightforward constraint optimization problem that seeks the solution in the microscale design variable space using fast gradient-based optimization methods [66], which we detailed in our previous work [46].…”

“…1, the derivative of the homogenized stiffness C with respect to the element density ρ j in (62) can be evaluated by means of the chain rule. Following [46], the expression is…”

“…Continuum micromechanics provides a rigorous framework to derive analytical estimates of macroscale elastoplastic properties [37][38][39] and has been successfully applied to describe many multiphase hierarchical systems such as plant, wood, bone, and cementitious materials [40][41][42][43][44][45]. In the context of concurrent material and structure optimization, we recently integrated continuum micromechanics based homogenized estimates in end-compliance optimization problems, which rendered our framework computationally tractable for multiphase hierarchical systems with several material length scales [46].…”