Structural and Multidisciplinary Optimization
| Hi-index | 0.00 |
Classical boundary variation techniques for performing shape optimization share some well-known difficulties, namely, maintaining a valid and good quality mesh that conforms to a dynamic, changing geometry and preserving the differentiability of the shape parameterization to accurately compute sensitivities of the cost functional and PDE with respect to design parameters. In the framework of a PDE-constrained optimization of the shape optimization problem, we overcome these difficulties by carrying out computations on a simply shaped (fictitious) domain in which the original shape is embedded. This fictitious domain is discretized using a uniform mesh which does not necessarily conform to the original shape. Nonconformity leads us to borrow the idea of distributed Lagrange multipliers from fictitious domain methods to approximately enforce boundary conditions via penalization. By freezing the mesh and evolving the shape on its vertices, we avoid remeshing and preserve the same linear algebra structures during the whole computation. This greatly simplifies implementation. As in standard level set methods, the oriented zero isocontour of a level set function is used in our formulation to represent the interface of a shape and to define its interior and exterior regions. Analytically, these regions are approximated by a continuous characteristic function that yields smooth derivatives necessary for our optimization algorithm. Unlike level set methods, we do not constrain the level set to a signed distance function nor develop the interface by solving the Hamilton-Jacobi equation. Rather, we interpret the level set values sampled on the mesh vertices as design variables which evolve, like the state and adjoint variables, to optimal values. Since there is an infinite number of implicit functions that define a single interface, our problem is ill-posed and therefore we regularize it to obtain a unique solution. A Newton-Krylov based optimization algorithm is employed to simultaneously solve the set of discretized nonlinear PDEs which correspond to the first order optimality conditions with respect to state, adjoint, and design variables. The presence of penalty parameters due to regularization and fictitious domain terms exacerbates the nonlinearity of the problem, thus requiring use of globalization methods. We resort to parameter continuation strategies in conjunction with an Armijo-type line search scheme to achieve robustness. Several shape optimization model problems are presented to demonstrate the effectiveness of our approach, including ones where the initial and final shape topologies differ significantly. Implementation is done using Sundance, a Sandia National Laboratories software package for rapid prototyping of PDE problems using finite element systems. We use its capabilities extensively to form and solve our Galerkin finite element systems.