  We present a computational design methodology for topology optimization of
multi-material-based flexoelectric composites. The methodology extends our
recently proposed design methodology for a single flexoelectric material. We
adopt the multi-phase vector level set (LS) model which easily copes with
various numbers of phases, efficiently satisfies multiple constraints and
intrinsically avoids overlap or vacuum among different phases. We extend the
point wise density mapping technique for multi-material design and use the
B-spline elements to discretize the partial differential equations (PDEs) of
flexoelectricity. The dependence of the objective function on the design
variables is incorporated using the adjoint technique.The obtained design
sensitivities are used in the Hamilton Jacobi equation to update the LS
function. We provide numerical examples for two, three and four phase
flexoelectric composites to demonstrate the flexibility of the model as well as
the significant enhancement in electromechanical coupling coefficient that can
be obtained using multi-material topology optimization for flexoelectric
composites.
