Optimal shape design based on body-fitted grid generation.
Thesis DisciplineMechanical Engineering
Degree GrantorUniversity of Canterbury
Degree NameDoctor of Philosophy (PhD)
Shape optimization is an important step in many design processes. With the growing use of Computer Aided Engineering in the design chain, it has become very important to develop robust and efficient shape optimization algorithms. The field of Computer Aided Optimal Shape Design has grown substantially over the recent past. In the early days of its development, the method based on small shape perturbation to probe the parameter space and identify an optimal shape was routinely used. This method is nothing but an educated trial and error method. A key development in the pursuit of good shape optimization algorithms has been the advent of the adjoint method to compute the shape sensitivities more formally and efficiently. While undoubtedly, very attractive, this method relies on very sophisticated and advanced mathematical tools which are an impediment to its wider use in the engineering community. It that spirit, it is the purpose of this thesis to propose a new shape optimization algorithm based on more intuitive engineering principles and numerical procedures. In this thesis, the new shape optimization procedure which is proposed is based on the generation of a body-fitted mesh. This process maps the physical domain into a regular computational domain. Based on simple arguments relating to the use of the chain rule in the mapped domain, it is shown that an explicit expression for the shape sensitivity can be derived. This enables the computation of the shape sensitivity in one single solve, a performance analogous to the adjoint method, the current state-of-the art. The discretization is based on the Finite Difference method, a method chosen for its simplicity and ease of implementation. This algorithm is applied to the Laplace equation in the context of heat transfer problems and potential flows. The applicability of the proposed algorithm is demonstrated on a number of benchmark problems which clearly confirm the validity of the sensitivity analysis, the most important aspect of any shape optimization problem. This thesis also explores the relative merits of different minimization algorithms and proposes a technique to “fix” meshes when inverted element arises as part of the optimization process. While the problems treated are still elementary when compared to complex multiphysics engineering problems, the new methodology presented in this thesis could apply in principle to arbitrary Partial Differential Equations.