Multiscale design of elastic solids with biomimetic cancellous bone cellular microstructures

Abstract

Natural (or biological) materials usually achieve outstanding mechanical performances. In particular, cancellous bone presents a high stiffness/strength to weight ratio, so its structure inspires the development of novel ultra-light cellular materials. A multiscale method for the design of elastic solids with a cancellous bone parameterized biomimetic microstructure is introduced in this work. The method combines a finite element model to evaluate the stiffness of the body at the macroscale with a gradient-based nonlinear constrained optimization solver to obtain the optimal values of the microparameters and microstructure orientation over the body domain. The most salient features of the implementation are an offline response surface methodology for the evaluation of the microstructure elastic tensor in terms of the microparameters, an adjoint method for the computation of the sensitivity of the macroscopic stiffness to the microparameters, a quasi-Newton approximation for the evaluation of the Hessian matrix of the nonlinear optimizer, and a distanceweighted filter of the microparameters to remediate checkerboard effects. The settings of the above features, the optimizer termination options, and the initial values of the microparameters are investigated for the best performance of the method. The effectiveness of the method is demonstrated for several examples, whose results are compared with the reference solutions calculated using a SIMP method. The method shows to be effective; it attains results coherent with SIMP approaches in terms of stiffness and spatial material distribution. The good performance of the multiscale method is attributed to the capability of the parameterized mimetic microstructure to attain bulk and shear moduli that are close to the Hashin-Shtrikman upper bounds over the complete solid volume fraction range.

Multiscale structural optimization with concurrent coupling between scales

“…This represents a significant reduction in the computational expense in comparison to contemporary sequential methods, which require fully populated databases to be assembled prior to the first optimization. In addition, the average number of parent microscale simulations required per optimization for the present framework is only 38,456, which is lower than the requirement to assemble the databases in contemporary frameworks which have 3 fewer design variables, for example (Imediegwu et al 2019) and (Colabella et al 2019) require 40,817 and 41,990 simulations respectively. It should be noted, however, that the reduction in computational cost is a function of the sampling plan used to populate the microstructural design space (full-factorial DOE) and may scale differently for alternative sampling plans.…”

“…The objective of this paper is to present a three-dimensional parameterization-based multiscale optimization framework, embedded with a novel concurrent coupling strategy between scales, and demonstrate its efficiency in comparison to sequential parameterization-based methods (Imediegwu et al 2019;Colabella et al 2019). Concurrent coupling ensures that only the microscale data required to evaluate the macroscale model during each iteration of optimization is collected and represents the principal novelty of this framework.…”

“…To demonstrate the efficacy of the concurrent multiscale optimization framework detailed within this paper, two classical problems are presented and solved. To highlight the efficiency, the total computational resource expended in the derivation of all optimized solutions presented within this paper is compared to the requirements of sequential parameterization-based methods (Imediegwu et al 2019;Colabella et al 2019). Objective values derived from optimized multiscale solutions are also compared to equivalent values derived through monoscale SIMP-based topology optimizations, to understand the relative merit of the present framework in comparison to the de facto standard method for structural optimization.…”

“…The number of levels afforded to each dimension provides sufficient coverage of the mechanical property space using the minimum number of nodes in each dimension. In combination, the DOE contains 153,664,000 unique permutations of the microscale parameters, which is several orders of magnitude larger than the size of the databases assembled within contemporary sequential frameworks (Imediegwu et al 2019;Colabella et al 2019).…”

“…Many multiscale optimization frameworks opt for a parameterization-based strategy, where the design variables controlling the microstructural geometry are optimized to fulfil functional objectives at the macroscale (Cheng et al 2018;Imediegwu et al 2019;Colabella et al 2019;Wang et al 2020;Thillaithevan et al 2020). Efficient parameterizations can provide clear accessibility to extremal point properties, as individual parameters can be configured to couple tightly with specific elements of the elasticity tensor, enabling straightforward control of point properties during optimization.…”