In this article we construct the fundamental solutions for the wave equation arising in the de Sitter model of the universe. We use the fundamental solutions to represent solutions of the Cauchy problem and to prove the Lp − Lq-decay estimates for the solutions of the equation with and without a source term.

Most of reports deal with bounded simply-connected domains; only a few involve in exterior and annular problems (Chen et al. [3], Katsuroda[10] and Ushijima and Chibu [30]). For exterior problems of Laplace's equations, there exist two kinds of infinity conditions, (1) |u|≤C and (2) u=O( ln r), which must be complied with by the fundamental solutions chosen. For u=O(ln r), the traditional fundamental solutions can be used. However, for |u|≤C, new fundamental solutions are explored, with a brief error analysis. Numerical experiments are carried out to verify the theoretical analysis made. Numerical experiments are also provided for annular domains, to show that the method of fundamental solutions (MFS) is inferior to the method of particular solutions (MPS), in both accuracy and stability. MFS and MPS are classified into the Trefftz method (TM) using fundamental solutions (FS) and particular solutions (PS), respectively. The remarkable advantage of MFS over MPS is the uniform $ln|overline{PQ_i}|$, to lead to simple algorithms and programming, thus to save a great deal of human power. Hence, we may reach the engineering requirements by much less efforts and a little payment. Besides, the crack singularity in unbounded domain is also studied. A combination of both PS and FS is also employed, called combination of MFS. The numerical results of MPS and combination of MFS are coincident with each other. The study in this thesis may greatly extend the application of MFS from bounded simply-connected domains to other more complicated domains.

Convergence acceleration techniques for the iterative solution of system of equations arising in the discretisations of compressible flow problems governed by the steady state Euler or Navier-Stokes equations is considered. The system of PDE is discretised using a finite difference or finite volume method yielding a large sparse system of equations. A solution is computed by integrating the corresponding time dependent problem in time until steady state is reached. A convergence acceleration technique based on semicirculant approximations is applied. For scalar model problems, it is proved that the preconditioned coefficient matrix has a bounded spectrum well separated from the origin. A very simple time marching scheme such as the forward Euler method can be used, and the time step is not limited by a CFL-type criterion. Instead, the time step can asymptotically be chosen as a constant, independent of the number of grid points and the Reynolds number. Numerical experiments show that grid and parameter independent convergence is achieved also in more complicated problem settings. A comparison with a multigrid method shows that the semicirculant convergence acceleration technique is more efficient in terms of arithmetic complexity. Another convergence acceleration technique based on fundamental solutions is proposed. An algorithm based on Fourier technique is provided for the fast application. Scalar model problems are considered and a theory, where the preconditioner is represented as an integral operator is derived. Theory and numerical experiments show that for first order partial differential equations, grid independent convergence is achieved.

As the engineering profession moves from prescriptive or “deemed-to-satisfy” approaches towards design methodologies based on quantification of performance, sophisticated modelling tools are increasingly needed, especially when complex combinations of demand and capacity are encountered. Recourse is invariably made to advanced computational tools to provide high fidelity solutions to large and complex problems, such as the response of structural systems or components to thermomechanical actions. Software packages based on the finite element method are most commonly used for such analyses. There are some essential prerequisites to effective use of advanced computational software for complex nonlinear problems, which are often ignored, particularly in professional practice. These include a thorough understanding of the underlying mechanics of the problem under consideration; a good appreciation of the approximation methods for modelling the problem properly (e.g. the choice between elements, continuum or structural, low or high order interpolation, degree of mesh refinement necessary and so on); and perhaps most importantly ensuring that the software is reliable and is able to reproduce established fundamental solutions to an acceptable degree of accuracy. This thesis attempts to address most of these issues but focusses primarily on the last mentioned prerequisite and provides a range of novel and unprecedented fundamental solutions for beams, plates, and shallow shells subject to moderate or extreme thermomechanical loads such as those resulting from a fire. Geometric and material nonlinearities are included in the proposed formulations along with the most common idealised boundary conditions. Thermally induced deformations generate large displacements and require the solutions to account for geometric nonlinearity, while material nonlinearity arises from the degradation of the material at elevated temperatures. In the context of structural performance under extreme thermal action (such as fire), a finite element procedure is employed to analytically characterise generic temperature distributions through the thickness of a structural component arising from different types of fire exposure conditions including: a “short hot” fire leading to a high compartment temperature over a relatively short duration; and a “long cool” fire with lower compartment temperatures, but over a longer duration. Results have shown that despite the larger area under the long cool fire time-temperature curve, which traditionally represented the fire severity, the effect of the short hot fire on the nonlinear responses of beams, plates, and shallow shells is more pronounced. Also, the effect of temperature-dependent material properties is found to be more pronounced during the short hot fire rather than the long cool fire. Comparison studies have confirmed that while the current numerical and theoretical approaches for analysing of thin plates and shells are often computationally intensive, the proposed approach offers an adequate level of accuracy with a rapid convergence rate for such structures. The solutions developed can be used to: verify software used for modelling structural response to thermomechanical actions; help students and professionals appreciate the fundamental mechanics better; provide relatively quick solutions for component level analyses; and visualise internal load paths and stress trajectories in complex structural components such as composite shells that can help engineers develop deeper insights into the relevant mechanics. The formulations developed are versatile and can be used for other applications such as laminated composite or orthotropic shallow shells. A very significant by-product of developing such fundamental solutions is their potential use in the development of highly accurate hybrid elements for very efficient modelling of large problems. While this has not been fully developed and implemented in the current work, the requisite theoretical framework has been developed and reported in one of the appendices, which can be used to develop such elements and implement on an appropriate software platform.

This thesis consists of two parts. In the first part, to solve the boundary value problems of homogeneous equations, the fundamental solutions (FS) satisfying the homogeneous equations are chosen, and their linear combination is forced to satisfy the exterior and the interior boundary conditions. To avoid the logarithmic singularity, the source points of FS are located outside of the solution domain S. This method is called the method of fundamental solutions (MFS). The MFS was first used in Kupradze in 1963. Since then, there have appeared numerous reports of MFS for computation, but only a few for analysis. The part one of this thesis is to derive the eigenvalues for the Neumann and the Robin boundary conditions in the simple case, and to estimate the bounds of condition number for the mixed boundary conditions in some non-disk domains. The same exponential rates of Cond are obtained. And to report numerical results for two kinds of cases. (I) MFS for Motz's problem by adding singular functions. (II) MFS for Motz's problem by local refinements of collocation nodes. The values of traditional condition number are huge, and those of effective condition number are moderately large. However, the expansion coefficients obtained by MFS are scillatingly large, to cause another kind of instability: subtraction cancellation errors in the final harmonic solutions. Hence, for practical applications, the errors and the ill-conditioning must be balanced each other. To mitigate the ill-conditioning, it is suggested that the number of FS should not be large, and the distance between the source circle and the partial S should not be far, either. In the second part, to reduce the severe instability of MFS, the truncated singular value decomposition(TSVD) and Tikhonov regularization(TR) are employed. The computational formulas of the condition number and the effective condition number are derived, and their analysis is explored in detail. Besides, the error analysis of TSVD and TR is also made. Moreover, the combination of TSVD and TR is proposed and called the truncated Tikhonov regularization in this thesis, to better remove some effects of infinitesimal sigma_{min} and high frequency eigenvectors.

In this thesis, the two-dimensional initial and boundary value problems invol-ving convection and diffusion terms are solved using the boundary element method (BEM). The fundamental solution of steady magnetohydrodynamic (MHD) flow equations in the original coupled form which are convection-diffusion type is established in order to apply the BEM directly to these coupled equations with the most general form of wall conductivities. Thus, the solutions of MHD flow in rectangular ducts and in infinite regions with mixed boundary conditions are obtained for high values of Hartmann number, M. For the solution of transient convection-diffusion type equations the dual reciprocity boundary element method (DRBEM) in space is combined with the differential quadrature method (DQM) in time. The DRBEM is applied with the fundamental solution of Laplace equation treating all the other terms in the equation as nonhomogeneity. The use of DQM eliminates the need of iteration and very small time increments since it is unconditionally stable. Applications include unsteady MHD duct flow and elastodynamic problems. The transient Navier-Stokes equations which are nonlinear in nature are also solved with the DRBEM in space - DQM in time procedure iteratively in terms of stream function and vorticity. The procedure is applied to the lid-driven cavity flow for moderate values of Reynolds number. The natural convection cavity flow problem is also solved for high values of Rayleigh number when the energy equation is added.

In the thesis, the error and stability analysis is made for the 2D Helmholtz equation by the method of fundamental solutions (MFS) using both Bessel and Neumann functions. The bounds of errors in bounded simply-connected domains are derived, while the bounds of condition number are derived only for disk domains. The MFS using Bessel functions is more efficient than the MFS using Neumann functions. Interestingly, for the MFS using Bessel functions, the radius R of the source points is not necessarily larger than the maximal radius r_max of the solution domain. This is against the traditional condition: r_max < R for MFS. Numerical experiments are carried out to support the analysis and conclusions made.

In this thesis, the analysis of the method of fundamental solution(MFS) is expanded for biharmonic equations. The bounds of errors are derived for the traditional and the Almansi's approaches in bounded simply-connected domains. The exponential and the polynomial convergence rates are obtained from highly and finite smooth solutions, respectively. Also the bounds of condition number are derived for the disk domains, to show the exponential growth rates. The analysis in this thesis is the first time to provide the rigor analysis of the CTM for biharmonic equations, and the intrinsic nature of accuracy and stability is similar to that of Laplace's equation. Numerical experiment are carried out for both smooth and singularity problems. The numerical results coincide with the theoretical analysis made. When the particular solutions satisfying the biharmonic equation can be found, the method of particular solutions(MPS) is always superior to MFS, supported by numerical examples. However, if such singular particular solutions near the singular points can not be found, the local refinement of collocation nodes and the greedy adaptive techniques can be used. It seems that the greedy adaptive techniques may provide a better solution for singularity problems. Beside, the numerical solutions by Almansi's approaches are slightly better in accuracy and stability than those by the traditional FS. Hence, the MFS with Almansi's approaches is recommended, due to the simple analysis, which can be obtained directly from the analysis of MFS for Laplace's equation.

For the method of fundamental solutions(MFS), many reports deal with 2D problems. Since the MFS is more advantageous for 3D problems, this thesis is devoted to Laplace's equation in 3D problems. Since the fundamental solutions(FS) £X(x,y)=1/(4£k||x-y||), x,y∈R^3 are known, the location of source points is important in real computation. In this thesis, we choose a cylinder as the solution domain, and the source points on larger cylinders and spheres. Numerical results are reported, to draw some useful conclusions. The theoretical analysis will be explored in the future.

Size-dependent material responses observed at fine length scales are receiving growing attention due to the need in the modeling of very small sized mechanical structures. The conventional continuum theories do not suffice for accurate descriptions of the exact material behaviors in the fine-scale regime due to the lack of inherent material lengths. A number of new theories/models have been propounded so far to interpret such novel phenomena. In this dissertation a few enriched-continuum theories - the adhesive contact mechanics, surface elasticity and strain gradient elasticity - are employed to study the mechanical behaviors of a semi-infinite solid induced by the boundary forces. A unified treatment of axisymmetric adhesive contact problems is developed using the harmonic functions. The generalized solution applies to the adhesive contact problems involving an axisymmetric rigid punch of arbitrary shape and an adhesive interaction force distribution of any profile, and it links existing solutions/models for axisymmetric non-adhesive and adhesive contact problems like the Hertz solution, Sneddon's solution, the JKR model, the DMT model and the M-D model. The generalized Boussinesq and Flamant problems are examined in the context of the surface elasticity of Gurtin and Murdoch (1975, 1978), which treats the surface as a negligibly thin membrane with material properties differing from those of the bulk. Analytical solution is derived based on integral transforms and use of potential functions. The newly derived solution applies to the problems of an elastic half-space (half-plane as well) subjected to prescribed surface tractions with consideration of surface effects. The newly derived results exhibit substantial deviations from the classical predictions near the loading points and converge to the classical ones at a distance far away from those points. The size-dependency of material responses is clearly demonstrated and material hardening effects are predicted. The half-space contact problems are also studied using the simplified strain gradient elasticity theory which incorporates material microstructural effects. The solution is obtained by taking advantage of the displacement functions of Mindlin (1964) and integral transforms. Significant discrepancy between the current and the classical solutions is seen to exist in the immediate vicinity of the loading area. The discontinuity and singularity exist in classical solution are removed, and the stress and displacement components change smoothly through the solid body.