Proposing a new meshless point interpolation method for solving engineering problems Sina Parand sinaparand@gmail.com Department of Civil Engineering Isfahan University of Technology, Isfahan 84156-83111, Iran Degree M.Sc Language: Farsi Supervisor: Prof. Bijan Boroomand, boromand@cc.iut.ac.ir Advisor: Prof. Mojtaba Azhari, mojtaba@cc.iut.ac.ir A new meshless method, in which essential boundary conditions are enforced easily, and does not have the limitations of previous meshless methods, has been proposed in this thesis. The proposed method has been developed in two different forms: The first form is for the cases that shape functions can have positive or negative values. The second form is for the situations that the non-negative property of shape functions is desirable for problem solution. For the case that negative values for the shape functions are permitted, an interpolation method has been proposed. The existence of interpolation property provides the ability to enforce essential boundary conditions directly and without any intricacy. For providing this property, the idea of “two-part shape functions” has been used. According to this idea, the final shape functions consist of two different parts. The first part provides the Kronecker delta property and the second one provides the necessary reproducing/consistency conditions. In this method, there is an ability to construct shape functions with arbitrary order of continuity and consistency. However, in this research, only first order reproducing/consistency conditions have been investigated. For the second form, i.e. when it is necessary to have non-negative shape functions, the suggested method has been proposed in three different types. In the first type, only weak Kronecker delta property has been provided on boundaries (whether convex or concave), and there is also the possibility of providing arbitrary order of continuity. In the second type, in addition to the weak Kronecker delta property, interpolation property has been provided on boundaries. However, in this form, the continuity order of the approximation, on boundary nodes, reduces to the zero-order. In the third form, the interpolation property has been provided in the whole domain, while the continuity order decreases to zero-order in all nodes. Laplace and elasticity equations have been solved in 2-D spaces to study the convergence behaviour and accuracy of the suggested method in different forms. In the case of using non-negative shape functions, because of the discontinuities in nodes, the problems have been solved only by using Galerkin weak form. When it is not necessary to have non-negative shape functions, and when arbitrary order of continuity is attainable, the problems are solved using both of Galerkin weak form and strong form collocation. The integrals, in the Galerkin weak form, have been evaluated by using a background mesh constructed by Delaunay triangulation algorithm. In the strong form, nodal points are used as the collocation points. By applications of regular and irregular discretizations, the effects of using irregular-node-distribution have been investigated. The results of the numerical solutions of the problems show the excellent performance and accuracy of the proposed method, especially in Galerkin weak form applications. Keywords Meshless Method, Point Interpolation, Partial Differential Equations, Galerkin Weak Form, Strong Form Collocation, Essential Boundary Conditio