Isogeometric Shape Optimization for Quasi-static and Transient Problems

Abstract

The recently developed isogeometric analysis (IGA) was aimed, from the start, at integrating computer aided design (CAD) and analysis. This synthesis of geometry and analysis has naturally led to renewed interest in developing structural shape optimization. The advantages of using isogeometric analysis in shape optimization are embodied in its ability to preserve exact CAD geometrical descriptions and its enhanced potential to perform shape sensitivity analysis. Recent contributions in shape optimization within IGA have been limited to static or steady-state loading conditions. The purpose of this work was to extend the isogeometric shape optimization and identification to quasi-static and transient problems. The normalization approaches for the search directions in isogeometric shape optimization scheme and the mean value property of B-spline basis were also studied. Shape sensitivity analysis plays a critical role in structural shape optimization. In this dissertation, an adjoint sensitivity analysis was performed for arbitrary objective functionals defined for quasi-static and transient problems at the continuous level. The sensitivity for quasi-static and transient problems are essentially different with each other since the transient case has a time-derivative term involved in the time interval. The transport relations considering discontinuities were studied and used to derive the continuous adjoint shape sensitivity. Consideration of the discontinuities enabled the shape sensitivity analysis to be applicable for the problems where discontinuities were involved in the objective functional and state equations. Then within the context of IGA, the continuous sensitivity was discretized to obtain the discrete design sensitivity with respect to the design discretization, which was used to find the search directions used to update the shape numerically. A interesting phenomenon in shape optimization is that the use of the search direction directly predicted from the discrete shape gradient makes the optimization history strongly dependent on the discretization. This discretization dependency can affect convergence and may lead the optimization process into a sub-optimal solution. The source of this discretization dependency was traced to the lack of consistency with the local steepest descent search direction in the continuous formulation. This inconsistency was analyzed using the shape variation equations and subsequently illustrated with a volume minimization problem. It was found that the inconsistency originates from the NURBS discretization which induces a discrete quadratic norm to represent the continuous Euclidean norm. To fix this inconsistency, a standard normalization approach, which is used to find the steepest descent direction for quadratic norm problems, was proposed to obtain a consistent discretization independent search direction. The standard approach requires solving a linear system of equations. Using the diagonally lumped mapping matrix (DLMM) and the partition of unity property of NURBS, two simpler normalization approaches, which do not require solving a linear system of equations, were proposed. The discretization-independence of the proposed approaches was verified with a benchmark problem. The superiority of the proposed search direction and its suitability for numerical implementation is illustrated with examples of shape optimization for mechanical and thermal problems. In the derivation of the simplified normalization approaches, the mean value property of B-spline basis function is proposed and proved using mathematical induction method. Using the normalization approaches, two frameworks to solve shape optimization and identification problems for quasi-static and transient process, respectively, were developed and implemented numerically within the context of isogeometric analysis. Generalized objective functionals were used to accommodate both structural shape optimization and identification problems in arbitrary forms. The methodology and its numerical implementation were tested using benchmark problems or passive control approaches with priori known solutions. For the quasi-static case, application problems were considered where an external load was allowed to move along the surface of a structure. The shape of the structure was modified to control the time-dependent displacement of the point where the load was applied according to a pre-specified target. For the transient case, the shape optimization and identification was performed for a plunger design under a transient heating process and a thermal protection layer design for a ballistic re-entry vehicle.