A shape optimization approach towards scalable algorithms allowing large deformations
Our project aims at finding optimal shapes inside a domain where a partial differential equations (PDEs) models a dynamical process.In particular, we focus on drag minimization of aerodynamical shapes in a flow tunnel where fluid dynamics are modelled by the Navier-Stokes equations.The main focus is in massively parallel scenarios, with dramatic number of degrees of freedom (DOF).We bring together the areas of parallel computing, shape optimization, and computational fluid dynamics to simulate phenomena in large, distributed-memory systems.
We use algorithms based on the solution of a nonlinear extension equation within the method of mappings, which links a boundary control variable to a deformation field. The focus is set on demonstrating how the selected extension allows for large deformations of the domain. We utilize the method of mappings and demonstrate how it is possible to preserve mesh quality throughout the optimization process. This is essential to solve optimization problems on complex shapes, which is mandatory for the application of our research in real-world applications.
In order to obtain solutions for the aforementioned complex shapes, it is necessary to use a high number of DOFs. This in turn requires scalable, parallel solution methods, of which the geometric multigrid is the foremost. This approach is proven to unfold optimal computational efficiency on hierarchically, distributed meshes. When it then comes to shape optimization, special attention has to be placed on the definition of the set of admissible shapes within a geometric multigrid framework.We therefore investigate several formulations of extension operators, which implicitly define the reachable shapes starting from a given reference configuration. We make use of the parallel simulation toolbox UG4, which is a C++ based framework for the solution of PDEs on massively parallel computer architectures.
Contact: Jose Alfonso Pinzon Escobar