Sandwich structures are especially interesting when multiple functionalities (such as stiffness and thermal insulation) are required. Properties of these structures are strongly dependent on the general geometry of the sandwich, but also on the detailed patterns of matter partitioning within the core. Therefore it seems possible to tailor the core pattern in order to obtain the desired properties. But multi-functional specifications and the infinite number of possible shapes, leads to non-trivial selection and/or optimization problems. In this context of "material by design", we propose a numerical approach, based on structural optimization techniques, to find the core pattern that leads to the best performances for a given set of conflicting specifications. The distribution of matter is defined thanks to a level-set function, and the convergence toward the optimized pattern is performed through the evolution of this function on a fixed grid. It is shown that the solutions of optimization are strongly dependent on the formulation of the problem, which have to be chosen with respect to the physics. A first application of this approach is presented for the design of sandwich core materials, in order to obtain the best compromise between flexural stiffness and relative density. The influence of both the initialization (starting geometry) and the formulation of the optimization problem are detailed.