Motivated by key advances in manufacturing techniques, the tailoring of materials with specific macroscopic properties has been the focus of active research in mechanical engineering and materials science over the past decade. The key challenge in this line of work is how to optimize the material microstructure to achieve a desired macroscopic constitutive response. The overwhelming majority of this type of inverse design work relies on topology optimization based, primarily, on linear theory.

In this work, we develop and implement a method to design particulate composites at the mesoscale using a shape optimization scheme to minimize or maximize a nonlinear cost function at the macroscale while satisfying a set of constraints associated, for example, with the volume fraction of inclusions or with the manufacturing technique. The optimization method relies on three key ‘modules’:​ multiscale modeling, sensitivity analysis, and optimization.

The multiscale modeling is based on a nonlinear finite element solver, which combines a classical homogenization scheme with a NURBS-based Interface-enriched Generalized Finite Element Method (NIGFEM) used to capture accurately and eciently the displacement field in a heterogeneous material with a finite element discretization that does not conform to the material interfaces. Damage evolution is captured using a three-parameter isotropic damage model able to simulate a wide range of failure responses.

The proposed gradient-based shape optimization scheme relies on the stationary nature of the non-conforming meshes used to discretize the periodic unit cell, thereby avoiding mesh distortion issues that plague conventional finite-element-based shape optimization studies. In the current approach, the finite element approximation space used in the NIGFEM is augmented with NURBS to allow for the accurate capture of the weak discontinuity present along complex, curvilinear material interfaces. NURBS are also used to parameterize the design geometry precisely and compactly by a small number of design variables.

To compute the derivatives of the cost and constraint functions with respect to the design variables, we also formulate an analytic nonlinear sensitivity, which is simplified by the fact that only the enrichment control points on material interfaces move, appear or disappear during the shape optimization process. The derivations uncover subtle but important new terms involved in the sensitivity of shape functions and their spatial derivatives. Our analytic nonlinear shape sensitivity avoids the technical diculties encountered in the finite difference or semi-analytical schemes when the boundary intersects an element very close to a node in a non-conforming mesh. In these situations, the boundary may move to another element during the design perturbation step, resulting in changes of the mesh topology, making the differentiation of the stiffness matrix and load vector problematic.

We apply the NIGFEM shape optimization scheme to several 2D and 3D structural problems including some benchmark and application examples to demonstrate the performance and accuracy of the method. Based on the multiscale approach, we also design the microstructure of a periodic particulate composite to optimize the volume fraction and distribution of the inclusions for a desired macroscopic nonlinear stress-strain curve.