Hybrid Chebyshev Polynomial Scheme for the Numerical Solution of Partial Differential Equations

In the numerical solution of partial differential equations (PDEs), it is common to find situations where the best choice is to use more than one method to arrive at an accurate solution. In this dissertation, hybrid Chebyshev polynomial scheme (HCPS) is proposed which is applied in two-step approach and one-step approach. In the two-step approach, first, Chebyshev polynomials are used to approximate a particular solution of a PDE. Chebyshev nodes which are the roots of Chebyshev polynomials are used in the polynomial interpolation due to its spectral convergence. Then, the resulting homogeneous equation is solved by boundary type methods including the method of fundamental solution (MFS) and the equilibrated collocation Trefftz method. However, this scheme can be applied to solve PDEs with constant coefficients only. So, for solving a wide variety of PDEs, one-step hybrid Chebyshev polynomial scheme is proposed. This approach combines two matrix systems of two-step approach into a single matrix system. The solution is approximated by the sum of particular solution and homogeneous solution. The Laplacian or biharmonic operator is kept on the left hand side and all the other terms are moved to the right hand side and treated as the forcing term. Various boundary value problems governed by the Poisson equation in two and three dimensions are considered for the numerical experiments. HCPS is also applied to solve an inhomogeneous Cauchy-Navier equations of elasticity in two dimensions. Numerical results show that HCPS is direct, easy to implement, and highly accurate

Local Radial Basis Function Methods for Solving Partial Differential Equations

Meshless methods are relatively new numerical methods which have gained popularity in computational and engineering sciences during the last two decades. This dissertation develops two new localized meshless methods for solving a variety partial differential equations. Recently, some localized meshless methods have been introduced in order to handle large-scale problems, or to avoid ill-conditioned problems involving global radial basis function approximations. This dissertation explains two new localized meshelss methods, each derived from the global Method of Approximate Particular Solutions (MAPS). One method, the Localized Method of Approximate Particular Solutions (LMAPS), is used for elliptic and parabolic partial differential equations (PDEs) using a global sparse linear system of equations. The second method, the Explicit Localized Method of Approximate Particular Solutions (ELMAPS), is constructed for solving parabolic types of partial differential equations by inverting a finite number of small linear systems. For both methods, the only information that is needed in constructing the approximating solution to PDEs, consists of the local nodes that fall within the domain of influence of the data. Since the methods are completely mesh free, they can be used for irregularly shaped domains. Both methods are tested and compared with existing global and local meshless methods. The results illustrate the accuracy and efficiency of our proposed methods

Localized Method of Approximate Particular Solutions for Solving Fourth-Order PDEs

In the past, dealing with fourth-order partial differential equations using the Local Method was not reliable due to difficulties in solving them directly. An approach such as splitting these equations into two Poisson differential equations was adopted to alleviate such challenges. However, this has a limitation since it is only applicable to Dirichlet and Laplace boundary conditions. In this paper, we solve fourth-order PDEs directly using the LMAPS. The improvement on the accuracy of this method was as a result of the proposed distribution of boundary conditions to alternating boundary points. And, also the use of suitable shape parameter; calculated using LOOCV(Leave-One-Out-Cross-Validation) Algorithm. The effectiveness of this Method was evident when we compared the results from two numerical examples

Meshfree and Particle Methods in Biomechanics: Prospects and Challenges

The use of meshfree and particle methods in the field of bioengineering and biomechanics has significantly increased. This may be attributed to their unique abilities to overcome most of the inherent limitations of mesh-based methods in dealing with problems involving large deformation and complex geometry that are common in bioengineering and computational biomechanics in particular. This review article is intended to identify, highlight and summarize research works on topics that are of substantial interest in the field of computational biomechanics in which meshfree or particle methods have been employed for analysis, simulation or/and modeling of biological systems such as soft matters, cells, biological soft and hard tissues and organs. We also anticipate that this review will serve as a useful resource and guide to researchers who intend to extend their work into these research areas. This review article includes 333 references

Adaptive meshless point collocation methods: investigation and application to geometrically non-linear solid mechanics

Conventional mesh-based methods for solid mechanics problems suffer from issues resulting from the use of a mesh, therefore, various meshless methods that can be grouped into those based on weak or strong forms of the underlying problem have been proposed to address these problems by using only points for discretisation. Compared to weak form meshless methods, strong form meshless methods have some attractive features because of the absence of any background mesh and avoidance of the need for numerical integration, making the implementation straightforward. The objective of this thesis is to develop a novel numerical method based on strong form point collocation methods for solving problems with geometric non-linearity including membrane problems. To address some issues in existing strong form meshless methods, the local maximum entropy point collocation method is developed, where the basis functions possess some advantages such as the weak Kronecker-Delta property on boundaries. r- and h-adaptive strategies are investigated in the proposed method and are further combined into a novel rh-adaptive approach, achieving the prescribed accuracy with the optimised locations and limited number of points. The proposed meshless method with h-adaptivity is then extended to solve geometrically non-linear problems described in a Total Lagrangian formulation, where h-adaptivity is again employed after the initial calculation to improve the accuracy of the solution effciently. This geometrically non-linear method is finally developed to analyse membrane problems, in which the out-of-plane deformation for membranes complicates the governing PDEs and the use of hyperelastic materials makes the computational modelling of membrane problems challenging. The Newton-Raphson arc-length method is adopted here to capture the snap-through behaviour in hyperelastic membrane problems. Several numerical examples are presented for each proposed algorithm to validate the proposed methodology and suggestions are made for future work leading on from the findings of this thesis

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described