This work is devoted to the theoretical and numerical aspects of shape optimization. The first part (chapter I to IV) deals with optimization problems under convexity constraint or constant width constraint. We give several new results related to Newton's problem and Meissner's conjecture. The second part (chapter V to VII) deals with the numerical study of shape optimization problems where many shapes or phases are involved. Some new numerical methods are introduced to study optimal configurations of famous problems : Kelvin's problem and Caffarelli's conjecture. The last part (chapter VIII and IX) is devoted to optimal transportation problems and irrigation problems. More precisely, we introduce a general framework, where different kind of cost functions are allowed. This seems relevant in some problems presenting congestion effects as for instance traffic on a highway, crowds moving in domains with obstacles. In the last chapter we give preliminary results related to the numerical approximation of optimal irrigation networks.

Thesis (M.S.)--Auburn University, 2008. / Abstract. Vita. Includes bibliographical references (p. 65-68).

The absorption of the energy in the form of heat from electromagnetic radiation is strongly dependent on the shape of the surface. Also, the transfer of this generated thermal energy is dependent on the surface area of the object in contact with the surrounding medium. Here in this thesis, we present a structural optimization method for metal nanostructures based on the shape dependency of their electromagnetic heat dissipation and thermodynamic transfer to the surroundings. We have used a parallel genetic algorithm (GA) in conjunction with a coupled electromagnetic (FDTD) and thermodynamic modeling of the metallic nanostructures for the optimization. The optimized nano-structure demonstrates significant improvement in electromagnetic heating in the spectral window of optimization as well as expedited cooling properties. The symmetry of the structures which is inherent in the design procedure makes them independent of the polarization at normal incidence and insensitive to the azimuthal direction of incidence.

The absorption of the energy in the form of heat from electromagnetic radiation is strongly dependent on the shape of the surface. Also, the transfer of this generated thermal energy is dependent on the surface area of the object in contact with the surrounding medium. Here in this thesis, we present a structural optimization method for metal nanostructures based on the shape dependency of their electromagnetic heat dissipation and thermodynamic transfer to the surroundings. We have used a parallel genetic algorithm (GA) in conjunction with a coupled electromagnetic (FDTD) and thermodynamic modeling of the metallic nanostructures for the optimization. The optimized nano-structure demonstrates significant improvement in electromagnetic heating in the spectral window of optimization as well as expedited cooling properties. The symmetry of the structures which is inherent in the design procedure makes them independent of the polarization at normal incidence and insensitive to the azimuthal direction of incidence.

Wafer probing is a testing process to inspect semiconductor wafers before packaging for defects by checking the electrical conductivity via physical contact between the wafers and the probe card. During the contact process, the shape of the probe needle and the mounting configuration onto the probe card have large influences on the stresses and contact force that the probe needles experience. In this paper, static performance of a vertical-type probe needle integrated with floating mount technology is analyzed with a nonlinear finite element analysis. The comparison between fixed mount and floating mount technologies is a part of the analyses. The geometry of a vertical probe needle is optimized to minimize the stress that occurs during the overdrive process, while maintaining adequate force for proper contact with the wafer. Effects of major overall dimensions of probe needle on the maximum stress and contact force is analyzed first, and then curvature of the probe needle body is optimized by employing a constrained minimization function, fmincon, in MATLAB. The maximum stress in the vertical probe pin at 125 ??m overdrive is effectively reduced from 1339 MPa to 972 MPa by applying floating mount technology over the fixed mount, and further reduced to 666 MPa by applying the optimization scheme. The final optimized design induced the contact force of 5.217 gf, which is in the range of the required contact force of 5 to 8 gf. Fatigue life increased from 19,219 cycles to 108,129 by applying floating mount over fixed mount, and further increased to 830,596 for the optimized design.

Numerical noise is a prevalent concern in many practical optimization problems. Convergence of gradient based optimization algorithms in the presence of numerical noise is not always assured. One way to improve optimization algorithm performance in the presence of numerical noise is to adjust the method of gradient computation. This study investigates the use of Continuous Sensitivity Equation (CSE) gradient approximations in the context of numerical noise and optimization. Three problems are considered: a problem with a system of ODE constraints, a single parameter flow problem constrained by the Navier-Stokes equations, and a multiple parameter flow problem constrained by the Navier-Stokes equations. All three problems use adaptive methods in the simulation of the constraint and are numerically noisy. Gradients for each problem are computed with both CSE and finite difference methods. The gradients are analyzed and compared. The two flow problems are optimized with a trust region optimization algorithm using both sets of gradient calculations. Optimization results are also compared, and the CSE gradient approximation yields impressive results for these examples. / Master of Science

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Adapting pipes are a significant part of any pipe-line network and they are the sources of substantial hydraulic losses. They are designed for a manufacturing simplicity, regardless of flow. This paper concerns with lowering of hydraulic losses of adapting pipes by means of the shape optimization. Several methods of a mathematical optimization are tested and due to the complexity of the task and the need of the computational distribution among several computers, the gradient based algorithm is used. These methods loop together with a CFD software then automatically explore the design space. Several optimizations of diffusers with different opening angles, shape parameterizations and boundary conditions are made for the better insight on hydraulic losses. In three chapters there is description of development of parametric description of bend by means of Bezier surfaces. At the end optimum shape is found with hydraulic losses were decreased about 22%, which was not validated by experiment. In the final chapter is application of developed software on the shape optimization of Kaplan draft tube.

An evolutionary geometry parametrization is established to represent aerodynamic configurations. This geometry parametrization technique is constructed by integrating the classical B-spline formulation with the knot insertion algorithm. It is capable of inserting control points to a given parametrization without modifying its geometry. Taking advantage of this technique, a shape design problem can be solved as a sequence of optimizations from the basic parametrization to more refined parametrizations. Owing to the nature of the B-spline formulation, feasible parametrization refinements are not unique; guidelines based on sensitivity analysis and geometry constraints are developed to assist the automation of the proposed optimization sequence. Test cases involving airfoil optimization and induced drag minimization are solved adopting this method. Its effectiveness is demonstrated through comparisons with optimizations using uniform refined parametrizations.

An evolutionary geometry parametrization is established to represent aerodynamic configurations. This geometry parametrization technique is constructed by integrating the classical B-spline formulation with the knot insertion algorithm. It is capable of inserting control points to a given parametrization without modifying its geometry. Taking advantage of this technique, a shape design problem can be solved as a sequence of optimizations from the basic parametrization to more refined parametrizations. Owing to the nature of the B-spline formulation, feasible parametrization refinements are not unique; guidelines based on sensitivity analysis and geometry constraints are developed to assist the automation of the proposed optimization sequence. Test cases involving airfoil optimization and induced drag minimization are solved adopting this method. Its effectiveness is demonstrated through comparisons with optimizations using uniform refined parametrizations.