An isogeometric phase–field based shape and topology optimization for flexoelectric structures

Abstract

For considering topological changes in a CAD-based framework, an isogeometric phase-field based shape and topology optimization is developed. In this case, the diffuse interface, between a material and void phases, describes the boundaries of the geometry. The descent direction to minimize the objective function is found by using a bulk energy function, which is written in terms of the sensitivities of the objective function with respect to the design variables. The design variables are the local values of the phase-field variable. The evolution of the phase-field is determined by solving the time-dependent Allen–Cahn equation. The phase-field topology optimization approach is used to optimize flexoelectric structures, a problem for which we exploit the advantages of using NURBS to discretize the geometry and the solution fields. The C1 continuity required for the solution of the flexoelectric PDE is easily obtained using NURBS as basis functions. The adjoint method is used to compute the sensitivity analysis. The results of the optimal flexoelectric geometries are compared to the optimal geometries in case piezoelectric materials are used. Different geometries, boundary conditions and material configurations are considered to demonstrate the robustness of the method. Compared to pure piezoelectricity, the electric output of piezo-flexoelectric microbeams under bending is larger. In contrast, for a structure under compression, the minimized value of the objective function is larger for piezoelectricity. In general, the enhancement of the energy conversion factor is larger when flexoelectricity is considered together with piezoelectricity. The optimal results show the relevance of considering flexoelectricity at microscales (3∼5 μm).