To investigate the lateral load distribution of each pile in a pile group, the bending moment distribution along the pile and the lateral group displacements with respect to pile location in the group, pile spacing, pile diameter and soil stiffness three dimensional finite element analysis were performed on 4x4 pile groups in clay. Different Elatic Modulus values, pile spacings, pile diameters and lateral load levels used in this study. In the analysis PLAXIS 3D Foundation geotechnical finite element package was used. It is found that, lateral load distribution among the piles was mainly a function of row location in the group independent from pile spacing. For a given load the leading row piles carried the greatest load. However, the trailing row piles carried almost the same loads. For a given load, bending moment values of the leading row piles were greater than the trailing row piles. On the other hand, as the spacing increased group displacements and individual pile loads decreased under the same applied load. However, this behavior was seen more clearly in the first and the second row piles. For the third and the fourth row piles, pile spacing became a less significant factor affecting the load distribution. It is also found that, pile diameter and soil stiffness are not significant factors on lateral load distribution as row location and pile spacing.

Conduction based compact thermal modeling of DC/DC converters, which are electronic components commonly used in military applications, are investigated. Three carefully designed numerical case studies are carried out at component, board and system levels using ICEPAK software. Experiments are conducted to gather temperature data that can be used to study compact thermal models (CTMs) with different levels of simplification. In the first (component level) problem a series of conduction based CTMs are generated and used to study the thermal behavior of a Thin-Shrink Small Outline Package (TSSOP) type DC/DC converter under free convection conditions. In the second (board level) case study, CTM alternatives are produced and investigated for module type DC/DC converter components using a printed circuit board (PCB) of an electro-optic system. In the last case study, performance of the CTM alternatives generated for the first case are assessed at the system level using them on a PCB placed inside a realistic avionic box. v Detailed comparison of accuracy of simulations obtained using CTMs with various levels of simplification is made based on experimentally obtained temperature data. Effects of grid size and quality, choice of turbulence modeling and space discretization schemes on numerical solutions are discussed in detail. It is seen that simulations provide results that are in agreement with measurements when appropriate CTMs are used. It is also showed that remarkable reductions in modeling and simulation times can be achieved by the use of CTMs, especially in system level analysis.

In this thesis, the two-dimensional initial and boundary value problems governed by unsteady partial differential equations are solved by making use of boundary element techniques. The boundary element method (BEM) with time-dependent fundamental solution is presented as an efficient procedure for the solution of diffusion, wave and convection-diffusion equations. It interpenetrates the equations in such a way that the boundary solution is advanced to all time levels, simultaneously. The solution at a required interior point can then be obtained by using the computed boundary solution. Then, the coupled system of nonlinear reaction-diffusion equations and the magnetohydrodynamic (MHD) flow equations in a duct are solved by using the time-domain BEM. The numerical approach is based on the iteration between the equations of the system. The advantage of time-domain BEM are still made use of utilizing large time increments. Mainly, MHD flow equations in a duct having variable wall conductivities are solved successfully for large values of Hartmann number. Variable conductivity on the walls produces coupled boundary conditions which causes difficulties in numerical treatment of the problem by the usual BEM. Thus, a new time-domain BEM approach is derived in order to solve these equations as a whole despite the coupled boundary conditions, which is one of the main contributions of this thesis. Further, the full MHD equations in stream function-vorticity-magnetic induction-current density form are solved. The dual reciprocity boundary element method (DRBEM), producing only boundary integrals, is used due to the nonlinear convection terms in the equations. In addition, the missing boundary conditions for vorticity and current density are derived with the help of coordinate functions in DRBEM. The resulting ordinary differential equations are discretized in time by using unconditionally stable Gear&#039 / s scheme so that large time increments can be used. The Navier-Stokes equations are solved in a square cavity up to Reynolds number 2000. Then, the solution of full MHD flow in a lid-driven cavity and a backward facing step is obtained for different values of Reynolds, magnetic Reynolds and Hartmann numbers. The solution procedure is quite efficient to capture the well known characteristics of MHD flow.

In statistics, regression analysis is a technique, used to understand and model the relationship between a dependent variable and one or more independent variables. Multiple Adaptive Regression Spline (MARS) is a form of regression analysis. It is a non-parametric regression technique and can be seen as an extension of linear models that automatically models non-linearities and interactions. MARS is very important in both classification and regression, with an increasing number of applications in many areas of science, economy and technology. In our study, we analyzed Generalized Partial Linear Models (GPLMs), which are particular semiparametric models. GPLMs separate input variables into two parts and additively integrates classical linear models with nonlinear model part. In order to smooth this nonparametric part, we use Conic Multiple Adaptive Regression Spline (CMARS), which is a modified form of MARS. MARS is very benefical for high dimensional problems and does not require any particular class of relationship between the regressor variables and outcome variable of interest. This technique offers a great advantage for fitting nonlinear multivariate functions. Also, the contribution of the basis functions can be estimated by MARS, so that both the additive and interaction effects of the regressors are allowed to determine the dependent variable. There are two steps in the MARS algorithm: the forward and backward stepwise algorithms. In the first step, the model is constructed by adding basis functions until a maximum level of complexity is reached. Conversely, in the second step, the backward stepwise algorithm reduces the complexity by throwing the least significant basis functions from the model. In this thesis, we suggest not using backward stepwise algorithm, instead, we employ a Penalized Residual Sum of Squares (PRSS). We construct PRSS for MARS as a Tikhonov Regularization Problem. We treat this problem using continuous optimization techniques which we consider to become an important complementary technology and alternative to the concept of the backward stepwise algorithm. Especially, we apply the elegant framework of Conic Quadratic Programming (CQP) an area of convex optimization that is very well-structured, hereby, resembling linear programming and, therefore, permitting the use of interior point methods. At the end of this study, we compare CQP with Tikhonov Regularization problem for two different data sets, which are with and without interaction effects. Moreover, by using two another data sets, we make a comparison between CMARS and two other classification methods which are Infinite Kernel Learning (IKL) and Tikhonov Regularization whose results are obtained from the thesis, which is on progress.

Desirability Functions continue to attract attention of scientists and researchers working in the area of multi-response optimization. There are many versions of such functions, differing mainly in formulations of individual and overall desirability functions. Derringer and Suich&rsquo / s desirability functions being used throughout this thesis are still the most preferred ones in practice and many other versions are derived from these. On the other hand, they have a drawback of containing nondifferentiable points and, hence, being nonsmooth. Current approaches to their optimization, which are based on derivative-free search techniques and modification of the functions by higher-degree polynomials, need to be diversified considering opportunities offered by modern nonlinear (global) optimization techniques and related softwares. A first motivation of this work is to develop a new efficient solution strategy for the maximization of overall desirability functions which comes out to be a nonsmooth composite constrained optimization problem by nonsmooth optimization methods. We observe that individual desirability functions used in practical computations are of mintype, a subclass of continuous selection functions. To reveal the mechanism that gives rise to a variation in the piecewise structure of desirability functions used in practice, we concentrate on a component-wise and generically piecewise min-type functions and, later on, max-type functions. It is our second motivation to analyze the structural and topological properties of desirability functions via piecewise max-type functions. In this thesis, we introduce adjusted desirability functions based on a reformulation of the individual desirability functions by a binary integer variable in order to deal with their piecewise definition. We define a constraint on the binary variable to obtain a continuous optimization problem of a nonlinear objective function including nondifferentiable points with the constraints of bounds for factors and responses. After describing the adjusted desirability functions on two well-known problems from the literature, we implement modified subgradient algorithm (MSG) in GAMS incorporating to CONOPT solver of GAMS software for solving the corresponding optimization problems. Moreover, BARON solver of GAMS is used to solve these optimization problems including adjusted desirability functions. Numerical applications with BARON show that this is a more efficient alternative solution strategy than the current desirability maximization approaches. We apply negative logarithm to the desirability functions and consider the properties of the resulting functions when they include more than one nondifferentiable point. With this approach we reveal the structure of the functions and employ the piecewise max-type functions as generalized desirability functions (GDFs). We introduce a suitable finite partitioning procedure of the individual functions over their compact and connected interval that yield our so-called GDFs. Hence, we construct GDFs with piecewise max-type functions which have efficient structural and topological properties. We present the structural stability, optimality and constraint qualification properties of GDFs using that of max-type functions. As a by-product of our GDF study, we develop a new method called two-stage (bilevel) approach for multi-objective optimization problems, based on a separation of the parameters: in y-space (optimization) and in x-space (representation). This approach is about calculating the factor variables corresponding to the ideal solutions of each individual functions in y, and then finding a set of compromised solutions in x by considering the convex hull of the ideal factors. This is an early attempt of a new multi-objective optimization method. Our first results show that global optimum of the overall problem may not be an element of the set of compromised solution. The overall problem in both x and y is extended to a new refined (disjunctive) generalized semi-infinite problem, herewith analyzing the stability and robustness properties of the objective function. In this course, we introduce the so-called robust optimization of desirability functions for the cases when response models contain uncertainty. Throughout this thesis, we give several modifications and extensions of the optimization problem of overall desirability functions.

In this thesis we present a numerical algorithm to solve the singular Inverse Sturm-Liouville problems with symmetric potential functions. The singularity, which comes from the unbounded domain of the problem, is treated by considering the limiting case of the associated problem on the symmetric finite interval. In contrast to regular problems which are considered on a finite interval the singular inverse problem has an ill-conditioned structure despite of the limiting treatment. We use the regularization techniques to overcome the ill-posedness difficulty. Moreover, since the problem is nonlinear the iterative solution procedures are needed. Direct computation of the eigenvalues in iterative solution is handled via psoudespectral methods. The numerical examples of the considered problem are given to illustrate the accuracy and convergence behaviour.

Medical image segmentation is a challenging problem, and is studied widely. In this thesis, the main goal is to develop automatic segmentation techniques of human mimic muscles and to compare them with ground truth data in order to determine the method that provides best segmentation results. The segmentation methods are based on Bayesian with Markov Random Field (MRF) and Level Set (Active Contour) models. Proposed segmentation methods are multi step processes including preprocess, main muscle segmentation step and post process, and are applied on three types of data: Magnetic Resonance Imaging (MRI) data, Computerized Tomography (CT) data and unified data, in which case, information coming from both modalities are utilized. The methods are applied both in three dimensions (3D) and two dimensions (2D) data cases. A simulation data and two patient data are utilized for tests. The patient data results are compared statistically with ground truth data which was labeled by an expert radiologist.

Optimal control of PDEs has a crucial place in many parts of sciences and industry. Over the last decade, there have been a great deal in, especially, control problems of elliptic problems. Optimal control problems of Burgers equation that is as a simplifed model for turbulence and in shock waves were recently investigated both theoretically and numerically. In this thesis, we analyze the space-time simultaneous discretization of control problem for Burgers equation. In literature, there have been two approaches for discretization of optimization problems: optimize-then-discretize and discretize-then-optimize. In the first part, we follow optimize-then-discretize appoproach. It is shown that both distributed and boundary time dependent control problem can be transformed into an elliptic pde. Numerical results obtained with adaptive and non-adaptive elliptic solvers of COMSOL Multiphysics are presented for both the unconstrained and the control constrained cases. As for second part, we consider discretize-then-optimize approach. Discrete adjoint concept is covered. Optimality conditions, KKT-system, lead to a saadle point problem. We investigate the numerical treatment for the obtained saddle point system. Both direct solvers and iterative methods are considered. For iterative mehods, preconditioners are needed. The structures of preconditioners for both distributed and boundary control problems are covered. Additionally, an a priori error analysis for the distributed control problem is given. We present the numerical results at the end of each chapter.

It is obvious that differential equations can be used to model real-life problems. Although it is possible to obtain analytical solutions of some of them, it is in general difficult to find closed form solutions of differential equations. Finding thus approximate solutions has been the subject of many researchers from different areas. In this thesis, we propose a new approach to Variational Iteration Method (VIM) to obtain the solutions of systems of first-order differential equations. The main contribution of the thesis to VIM is that proposed approach uses restricted variations only for the nonlinear terms and builds up a matrix-valued Lagrange multiplier that leads to the extension of the method (EVIM). Close relation between the matrix-valued Lagrange multipliers and fundamental solutions of the differential equations highlights the relation between the extended version of the variational iteration method and the classical variation of parameters formula. It has been proved that the exact solution of the initial value problems for (nonhomogenous) linear differential equations can be obtained by such a generalisation using only a single variational step. Since higher-order equations can be reduced to first-order systems, the proposed approach is capable of solving such equations too / indeed, without such a reduction, variational iteration method is also extended to higher-order scalar equations. Further, the close connection with the associated first-order systems is presented. Such extension of the method to higher-order equations is then applied to solve boundary value problems: linear and nonlinear ones. Although the corresponding Lagrange multiplier resembles the Green&rsquo / s function, without the need of the latter, the extended approach to the variational iteration method is systematically applied to solve boundary value problems, surely in the nonlinear case as well. In order to show the applicability of the method, we have applied the EVIM to various real-life problems: the classical Sturm-Liouville eigenvalue problems, Brusselator reaction-diffusion, and chemical master equations. Results show that the method is simple, but powerful and effective.

In this study, the application of a Monte Carlo method (MCM) for radiative heat transfer in three-dimensional rectangular enclosures was investigated. The study covers the development of the method from simple surface exchange problems to enclosure problems containing absorbing, emitting and isotropically/anisotropically scattering medium. The accuracy of the MCM was first evaluated by applying the method to cubical enclosure problems. The first one of the cubical enclosure problems was prediction of radiative heat flux vector in a cubical enclosure containing purely, isotropically and anisotropically scattering medium with non-symmetric boundary conditions. Then, the prediction of radiative heat flux vector in an enclosure containing absorbing, emitting, isotropically and anisotropically scattering medium with symmetric boundary conditions was evaluated. The predicted solutions were compared with the solutions of method of lines solution (MOL) of discrete ordinates method (DOM). The method was then applied to predict the incident heat fluxes on the freeboard walls of a bubbling fluidized bed combustor, and the solutions were compared with those of MOL of DOM and experimental measurements. Comparisons show that MCM provides accurate and computationally efficient solutions for modelling of radiative heat transfer in 3-D rectangular enclosures containing absorbing, emitting and scattering media with isotropic and anisotropic scattering properties.