A direct approach to the finite element solution of elliptic optimal control problems

Abstract

A general framework is developed for the finite element solution of optimal control problems governed by elliptic nonlinear partial differential equations. Typical applications are steady-state problems in nonlinear continuum mechanics, where a certain property of the solution (a function of displacements, temperatures, etc.) is to be minimized by applying control loads. In contrast to existing formulations, which are based on the ''adjoint state,'' the present formulation is a direct one, which does not use adjoint variables. The formulation is presented first in a general nonlinear setting, then specialized to a case leading to a sequence of quadratic programming problems, and then specialized further to the unconstrained case. Linear governing partial differential equations are also considered as a special case in each of these categories.

Solution of static optimal control problems in non‐linear elasticity via quadratic programming

“…Recently, a new general framework has been developed by Givoli for the ÿnite element solution of optimal control problems governed by non-linear elliptic partial di erential equations [15]. In contrast to the ÿnite element schemes mentioned above, the approach in Reference [15] is a direct one, and does not involve adjoint variables.…”

“…In contrast to the ÿnite element schemes mentioned above, the approach in Reference [15] is a direct one, and does not involve adjoint variables. Computationally, this has the e ect of leading to a simpler formulation and reducing the number of variables by a factor of two.…”

“…In Reference [15] the numerical procedure has been developed, and the computational properties of the method have been discussed in detail. In addition, the method has been applied to a simple scalar one-dimensional model problem, but has not been tested in a more realistic setting.…”

“…In addition, the method has been applied to a simple scalar one-dimensional model problem, but has not been tested in a more realistic setting. In this paper we extend the formulation presented in Reference [15] from the scalar case to the vectorial elastic case, and apply the method to two-dimensional problems in non-linear elasticity. As examples, we consider the minimization of the lateral de ection of a membrane on a non-linear elastic foundation and the minimization of the in-plane large deformation of a hyperelastic plate.…”

“…As examples, we consider the minimization of the lateral de ection of a membrane on a non-linear elastic foundation and the minimization of the in-plane large deformation of a hyperelastic plate. In this context, we also discuss a practical uniqueness issue which has not been considered in Reference [15].…”