The application of polynomial reproducing methods has been explored in the context of linear and non linear problems. Of specific interest is the application of a recently developed reproducing scheme, referred to as the error reproducing kernel method (ERKM), which uses non-uniform rational B-splines (NURBS) to construct the basis functions, an aspect that potentially helps bring in locall support, convex approximation and variation diminishing properties in the functional approximation. Polynomial reproducing methods have been applied to solve problems coming under the class of a simplified theory called Cosserat theory. Structures such as a rod which have special geometric properties can be modeled with the aid of such simplified theories. It has been observed that the application of mesh-free methods to solve the aforementioned problems has the advantage that large deformations and exact cross-sectional deformations in a rod could be captured exactly by modeling the rod just in one dimension without the problem of distortion of elements or element locking which would have had some effect if the problem were to be solved using mesh based methods. Polynomial reproducing methods have been applied to problems in fracture mechanics to study the propagation of crack in a structure. As it is often desirable to limit the use of the polynomial reproducing methods to some parts of the domain where their unique advantages such as fast convergence, good accuracy, smooth derivatives, and trivial adaptivity are beneficial, a coupling procedure has been adopted with the objective of using the advantages of both FEM and polynomial reproducing methods. Exploration of SMW (Sherman-Morrison-Woodbury) in the context of polynomial reproducing methods has been done which would assist in calculating the inverse of a perturbed matrix (stiffness matrix in our case). This would to a great extent reduce the cost of computation. In this thesis, as a first step attempts have been made to apply Mesh free cosserat theory to one dimensional problems. The idea was to bring out the advantages and limitations of mesh free cosserat theory and then extend it to 2D problems.

In the process of engineering design of structural shapes, the flat plate analysis results can be generalized to predict behaviors of complete structural shapes. In this case, the purpose of this project is to analyze a thin flat plate under conductive heat transfer and to simulate the temperature distribution, thermal stresses, total displacements, and buckling deformations. The current approach in these cases has been using the Finite Element Method (FEM), whose basis is the construction of a conforming mesh. In contrast, this project uses the mesh-free Scan Solve Method. This method eliminates the meshing limitation using a non-conforming mesh. I implemented this modeling process developing numerical algorithms and software tools to model thermally induced buckling. In addition, convergence analysis was achieved, and the results were compared with FEM. In conclusion, the results demonstrate that the method gives similar solutions to FEM in quality, but it is computationally less time consuming.

This thesis deals with smooth discretization schemes and inverse problems, the former used in efficient yet accurate numerical solutions to forward models required in turn to solve inverse problems. The aims of the thesis include, (i) development of a stabilization techniques for a class of forward problems plagued by unphysical oscillations in the response due to the presence of jumps/shocks/high gradients, (ii) development of a smooth hybrid discretization scheme that combines certain useful features of Finite Element (FE) and Mesh-Free (MF) methods and alleviates certain destabilizing factors encountered in the construction of shape functions using the polynomial reproduction method and, (iii) a first of its kind attempt at the joint inversion of both static and dynamic source parameters of the 2004 Sumatra-Andaman earthquake using tsunami sea level anomaly data. Following the introduction in Chapter 1 that motivates and puts in perspective the work done in later chapters, the main body of the thesis may be viewed as having two parts, viz., the first part constituting the development and use of smooth discretization schemes in the possible presence of destabilizing factors (Chapters 2 and 3) and the second part involving solution to the inverse problem of tsunami source recovery (Chapter 4). In the context of stability requirements in numerical solutions of practical forward problems, Chapter 2 develops a new stabilization scheme. It is based on a stochastic representation of the discretized field variables, with a view to reduce or even eliminate unphysical oscillations in the MF numerical simulations of systems developing shocks or exhibiting localized bands of extreme plastic deformation in the response. The origin of the stabilization scheme may be traced to nonlinear stochastic filtering and, consistent with a class of such filters, gain-based additive correction terms are applied to the simulated solution of the system, herein achieved through the Element-Free Galerkin (EFG) method, in order to impose a set of constraints that help arresting the spurious oscillations. The method is numerically illustrated through its application to a gradient plasticity model whose response is often characterized by a developing shear band as the external load is gradually increased. The potential of the method in stabilized yet accurate numerical simulations of such systems involving extreme gradient variations in the response is thus brought forth. Chapter 3 develops the MF-based discretization motif by balancing this with the widespread adoption of the FE method. Thus it concentrates on developing a 'hybrid' scheme that aims at the amelioration of certain destabilizing algorithmic issues arising from the necessary condition of moment matrix invertibility en route to the generation of smooth shape functions. It sets forth the hybrid discretization scheme utilizing bivariate simplex splines as kernels in a polynomial reproducing approach adopted over a conventional FE-like domain discretization based on Delaunay triangulation. Careful construction of the simplex spline knotset ensures the success of the polynomial reproduction procedure at all points in the domain of interest, a significant advancement over its precursor, the DMS-FEM. The shape functions in the proposed method inherit the global continuity ( C p 1 ) and local supports of the simplex splines of degree p . In the proposed scheme, the triangles comprising the domain discretization also serve as background cells for numerical integration which here are near-aligned to the supports of the shape functions (and their intersections), thus considerably ameliorating an oft-cited source of inaccuracy in the numerical integration of MF-based weak forms. Numerical experiments establish that the proposed method can work with lower order quadrature rules for accurate evaluation of integrals in the Galerkin weak form, a feature desiderated in solving nonlinear inverse problems that demand cost-effective solvers for the forward models. Numerical demonstrations of optimal convergence rates for a few test cases are given and the hybrid method is also implemented to compute crack-tip fields in a gradient-enhanced elasticity model. Chapter 4 attempts at the joint inversion of earthquake source parameters for the 2004 Sumatra-Andaman event from the tsunami sea level anomaly signals available from satellite altimetry. Usual inversion for earthquake source parameters incorporates subjective elements, e.g. a priori constraints, posing and parameterization, trial-and-error waveform fitting etc. Noisy and possibly insufficient data leads to stability and non-uniqueness issues in common deterministic inversions. A rational accounting of both issues favours a stochastic framework which is employed here, leading naturally to a quantification of the commonly overlooked aspects of uncertainty in the solution. Confluence of some features endows the satellite altimetry for the 2004 Sumatra-Andaman tsunami event with unprecedented value for the inversion of source parameters for the entire rupture duration. A nonlinear joint inversion of the slips, rupture velocities and rise times with minimal a priori constraints is undertaken. Large and hitherto unreported variances in the parameters despite a persistently good waveform fit suggest large propagation of uncertainties and hence the pressing need for better physical models to account for the defect dynamics and massive sediment piles. Chapter 5 concludes the work with pertinent comments on the results obtained and suggestions for future exploration of some of the schemes developed here.