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UBC Theses and Dissertations

A numerical shape optimization framework for generic problems solved on unstructured triangular meshes Gosselin, Serge


Until recently, experiments combined with trial and error have been the preferred methodology to refine designs in fields such as aerodynamics or heat transfer. Numerical shape optimization tools are becoming an important link in the design chain, accelerating this refinement process. Optimal design is a complex task requiring the integration of the many components necessary to perform accurate numerical simulations. In this thesis, a simplified shape optimization framework for generic applications is presented. The partial differential equations describing the physics of the optimization problems are solved on unstructured triangular meshes. The mesh generator guarantees the quality of the triangulation. The finite-volume method is used combined with an implicit Newton-Krylov GMRES solver for generic problems. The numerical and physical aspects of the problem are de-coupled inside the solver. This allows for simplified implementation of new physics package using interior and boundary fluxes. The shape to be optimized is defined with a set of control points. The movements of the boundary vertices are described by cubic spline interpolation. They are then propagated through the internal mesh by an explicit deformation law. Optimization constraints are enforced through a penalty formulation and the resulting unconstrained problem is solved using either the steepest descent method or a quasi-Newton method. Gradients of the objective function with respect to the shape are calculated using finite differences. The correct operation of the optimization framework is verified with validation problems. These allow the assessment of the performance of its different components. To indicate that a minimum has been attained, the gradient norm must be reduced by several orders of magnitude. Overall, the results show that the various components are properly integrated. The framework's range of applications is limited by the implicit solver, currently under development, and by finite-difference gradients. A discussion about necessary requirements to extend this framework to more practical applications is given.

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