Optimisation of a liquid-liquid extraction based sustainable water desalination process

In this paper, a waste heat liquid-liquid extraction method is considered for the seawater desalination. The mathematical model is a result of mass and energy balances, phase behaviour and salt distribution in the two phases. Because of the high non-linearity of the phase behaviour and the salt distribution these are being approximated by the use of piecewise linear approximations methods and the solutions are presented. The results indicate that the quality of solution is not affected by the use of the approximation

Canonical dual finite element method for solving nonconvex mechanics and topology optimization

Canonical duality theory (CDT) is a newly developed, potentially powerful methodological theory which can transfer general multi-scale nonconvex/discrete problems in Rn to a unified convex dual problem in continuous space Rm with m n and without a duality gap. The associated triality theory provides extremality criteria for both global and local optimal solutions, which can be used to develop powerful algorithms for solving general nonconvex variational problems. This thesis, first, presents a detailed study of large deformation problems in 2-D structural system. Based on the canonical duality theory, a canonical dual finite element method is applied to find a global minimization to the general nonconvex optimization problem using a new primal-dual semi-definite programming algorithm. Applications are illustrated by numerical examples with different structural designs and different external loads. Next, a new methodology and algorithm for solving post buckling problems of a large deformed elastic beam is investigated. The total potential energy of this beam is a nonconvex functional, which can be used to model both pre- and post-buckling phenomena. By using the canonical dual finite element method, a new primal-dual semi-definite programming algorithm is presented, which can be used to obtain all possible post-buckled solutions. In order to verify the triality theory, mixed meshes of different dual stress interpolation are applied to obtain the closed dimensions between discretized displacement and discretized stress. Applications are illustrated by several numerical examples with different boundary conditions. We find that the global minimum solution of the nonconvex potential leads to the unbuckled state, and both of these two solutions are numerically stable. However, the local minimum solution leads to an unstable buckled state, which is very sensitive to the external load, thickness of the beam, numerical precision, and the size of finite elements. Finally, a mathematically rigorous and computationally powerful method for solving 3-D topology optimization problems is demonstrated. This method is based on CDT developed by Gao in nonconvex mechanics and global optimization. It shows that the so-called NP-hard Knapsack problem in topology optimization can be solved deterministically in polynomial-time via a canonical penalty-duality (CPD) method to obtain precise global optimal 0-1 density distribution at each volume evolution. The relation between this CPD method and Gao's pure complementary energy principle is revealed for the first time. A CPD algorithm is proposed for 3-D topology optimization of linear elastic structures. Its novelty is demonstrated by benchmark problems. Results show that without using any artificial technique, the CPD method can provide mechanically sound optimal design, also it is much more powerful than the well-known BESO and SIMP methods. Finally, computational complexity and conceptual/mathematical mistakes in topology optimization modeling and popular methods are explicitly addressed.Doctor of Philosoph

Adaptive methods for linear dynamic systems in the frequency domain with application to global optimization

Designers often seek to improve their designs by considering several discrete modifications. These modifications may require changes in materials and geometry, as well as the addition or removal of individual components. In general, if the modifications are applied one at a time, none of them may sufficiently improve the performance. Also, the total number of modifications that may be included in the final design is often limited due to cost or other constraints. The designer must therefore determine the optimal combination of modifications in order to complete the design. While this design challenge arises fairly commonly in practice, very little research has studied it in its full generality. This work assumes that the mathematical description of the design and its modifications are frequency dependent matrices. Such matrices typically arise due to finite element analysis as well as other modeling techniques. Computing performance metrics related to steady-state forced response, also known as performing a frequency sweep, involves factorizing these matrices many times. Additionally, determining the globally optimum design in this case involves an exhaustive search of the combinations of modifications. These factors lead to prohibitively long run times particularly as the size of the system grows. The research presented here seeks to reduce these costs, making such a search feasible. Several innovative techniques have been developed and tested over the course of the research, focused in two primary areas: adaptive frequency sweeps and efficient combinatorial optimization. The frequency sweep methods rely on an adaptive bisection of the frequency range and either a subspace approximation based on implicit interpolatory model order reduction or an elementwise approximation using piecewise multi-point Padé interpolants. Additionally, a strategy for augmenting the adaptive methods with the system's modal information is presented. For combinatorial optimization, an approximation algorithm is developed that capitalizes on any presence of dynamic uncoupling between modifications. The net effect of this work is to allow designers and researchers to develop new dynamic systems and perform analyses faster and more efficiently than ever before

On the use of optimized cubic spline atomic form factor potentials for band structure calculations in layered semiconductor structures

The emperical pseudopotential method in the large basis approach was used to calculate the electronic bandstructures of bulk semiconductor materials and layered semiconductor heterostructures. The crucial continuous atomic form factor potentials needed to carry out such calculations were determined by using Levenberg-Marquardt optimization in order to obtain optimal cubic spline interpolations of the potentials. The optimized potentials were not constrained by any particular functional form (such as a linear combination of Gaussians) and had better convergence properties for the optimization. It was demonstrated that the results obtained in this work could potentially lead to better agreement between calculated and empirically determined band gaps via optimizationPhysicsM. Sc. (Physics

ANALYSIS OF ITERATIVE METHODS FOR THE SOLUTION OF BOUNDARY INTEGRAL EQUATIONS WITH APPLICATIONS TO THE HELMHOLTZ PROBLEM

This thesis is concerned with the numerical solution of boundary integral equations and the numerical analysis of iterative methods. In the first part, we assume the boundary to be smooth in order to work with compact operators; while in the second part we investigate the problem arising from allowing piecewise smooth boundaries. Although in principle most results of the thesis apply to general problems of reformulating boundary value problems as boundary integral equations and their subsequent numerical solutions, we consider the Helmholtz equation arising from acoustic problems as the main model problem. In Chapter 1, we present the background material of reformulation of Helmhoitz boundary value problems into boundary integral equations by either the indirect potential method or the direct method using integral formulae. The problem of ensuring unique solutions of integral equations for exterior problems is specifically discussed. In Chapter 2, we discuss the useful numerical techniques for solving second kind integral equations. In particular, we highlight the superconvergence properties of iterated projection methods and the important procedure of Nystrom interpolation. In Chapter 3, the multigrid type methods as applied to smooth boundary integral equations are studied. Using the residual correction principle, we are able to propose some robust iterative variants modifying the existing methods to seek efficient solutions. In Chapter 4, we concentrate on the conjugate gradient method and establish its fast convergence as applied to the linear systems arising from general boundary element equations. For boundary integral equalisations on smooth boundaries we have observed, as the underlying mesh sizes decrease, faster convergence of multigrid type methods and fixed step convergence of the conjugate gradient method. In the case of non-smooth integral boundaries, we first derive the singular forms of the solution of boundary integral solutions for Dirichlet problems and then discuss the numerical solution in Chapter 5. Iterative methods such as two grid methods and the conjugate gradient method are successfully implemented in Chapter 6 to solve the non-smooth integral equations. The study of two grid methods in a general setting and also much of the results on the conjugate gradient method are new. Chapters 3, 4 and 5 are partially based on publications [4], [5] and [35] respectively.Department of Mathematics and Statistics, Polytechnic South Wes