A Tailored Strategy for PDE-Based Design of Hierarchically Structured Porous Catalysts

Abstract

Optimization problems involving the solution of partial differential equations (PDE) are often encountered in the context of optimal design, optimal control, and parameter estimation. Based on the reduced-gradient method, a general strategy is proposed to solve these problems by reusing existing software. As an illustration, this strategy was employed to solve a PDEbased optimization problem that arises from the optimal design of the network of pore channels in hierarchically structured porous catalysts. A Fortran implementation was developed by combining a gradient-based optimization package, NLPQL, a multigrid solver, MGD9V, and a limited amount of in-house coding. The value and gradient of the objective function are computed by solving the discretized PDE and another system of linear equations using MGD9V. These are subsequently fed into NLPQL to solve the optimization problem. The PDE was discretized in terms of a finite volume method on a matrix of computational cells. The number of the cells ranged from 129 £129 to 513 £513, and the number of the optimization variables ranged from 41 to 201. Numerical tests were carried out on a Dell laptop with a 2.16-GHz Intel Core2 Duo processor. The results show that the optimization typically converges in a limited number (i.e., 9‐48) of iterations. The CPU time is from 2.52 to 211.52 s. The PDE was solved 36‐201 times in each of the numerical tests. This study calls for the use of our strategy to solve PDE-based optimization problems.