A Constrained Variational Method for Two-Dimensional Shape Optimization

A constrained variational method is presented for the formulation and solution of a class of two-dimensional continuous shape optimization problems with equality constraints. Conceptually, the method casts the shape optimization problem as an analogous application of the principle of virtual work. It is postulated that the optimal shape is that equilibrium shape distinguished by the stationary value of the systems “effective” virtual work. The resulting formulation leads to a direct variational statement of the shape optimization problem, yielding the optimality criteria consisting of the Euler-Lagrange equations, constraints and boundary conditions. The Euler-Lagrange equations are linearized about the current shape and transformed into a set of Poisson’s equations. A direct boundary integral formulation is developed for the solution of Poisson’s equation that results in a continuous expression for the shape in terms of the Lagrange multipliers. The numerical solution procedure involves discretizing the shape into boundary and domain elements and using the direct boundary element method and the linearized constraint set to form a set of matrix equations. The solution to the set of matrix equations yields new estimates of the shape and the Lagrange multipliers. Convergence of the method is achieved when successive iterations of the shape and Lagrange multiplier estimates fail to improve by some prescribed limit. The classical problem of finding the curve with minimum perimeter and a prescribed enclosed area is used to illustrate the method.