Shape optimization for 2D diffusive scalar transport

Abstract

A new theoretical formulation is presented for the shape optimization problem associated with maximizing or minimizing the diffusive scalar transport from a two-dimensional body. In particular, we consider the diffusive transport of heat from an isothermal body into a medium with constant temperature at the far-field. The formulation also applies to mass and momentum transport. The diffusion problem, which is governed by the Laplace equation, is addressed using conformal mapping techniques where the two-dimensional domain is mapped onto a simpler domain where an analytical solution can be readily obtained. The objective function of the optimization problem is the length of the object in the transformed domain and the variables of the optimization are the parameters of the Schwarz-Christoffel transformation. The length of the object in the transformed domain is related to the scalar displacement, which corresponds to a far-field temperature drop or rise (slip velocity in case of momentum transport), that depends on the shape of the body and it quantifies the enhancement or reduction in transport rate. The mathematical formulation is validated by addressing two fundamental shape optimization problems associated with maximizing or minimizing the transport rate (drag in case of momentum transport) from a two-dimensional body of unit span: i) for a given surface area to obtain the shape that maximizes the transport rate from a body, ii) for a given volume to obtain the shape that minimizes the transport rate from a body. For both cases we compute numerically that the cylinder is the optimal shape. The versatility of the formulation is further demonstrated by including constraints with respect to the length of the body.