Multiscale design of elastic solids with biomimetic cancellous bone cellular microstructures

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 distance-weighted 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.