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A directly differentiated form of boundary integral equation with respect to geometric design variables is used to calculate shape design sensitivities. This allows the coupling of an optimizing technique and a boundary element elastic stress analyser to form an optimum shape design algorithm in two dimensions. Hermitian cubic spline functions used to represent boundary shapes offer considerable advantages in fitting a wide range of curves, and in the automatic remeshing process. They also reduce the need for optimization constraints to avoid impractical designs during the optimization procedure. The boundary element method offers advantages over the finite element method, and is applicable to a wide range of shape optimization problems.

SUMMARYA boundary element method for steady two-dimensional low-to-moderate-Reynolds number flows of incompressible fluids, using primitive variables, is presented. The velocity gradients in the Navier -Stokes equations are evaluated using the alternatives of upwind and central finite difference approximations, and derivatives of finite element shape functions. A direct iterative scheme is used to cope with the non-linear character of the integral equations. In order to achieve convergence, an underrelaxation technique is employed at relatively high Reynolds numbers. Driven cavity flow in a square domain is considered to validate the proposed method by comparison with other published data.