Marjan Šterk (2005) Mesh free methods on parallel computers. PhD thesis.
Numerical methods for the solution of partial differential equations without background mesh -- mesh free methods - constitute the subject of this dissertation. Particular emphasis is laid on their mathematical formulation, accuracy, time complexity, and applicability on parallel computers. Solving partial differential equations numerically involves their conversion into a system of linear or nonlinear equations, usually algebraic, which is then solved. The finite difference and finite element methods have traditionally been used to construct the algebraic system. The finite difference method uses a rectangular, usually equidistant, mesh of points. While it is simple and fast, and works well on domains with only vertical and horizontal boundaries, any other domain shape causes a significant loss of accuracy. In the finite element method, the domain is divided into a mesh of small elements, e.g. triangles or quadrilaterals in two-dimensional problems. The approximate solution is represented as a linear combination of base functions, which are defined on individual elements. Unfortunately, mesh construction is a complex task and is not fully automated, and thus requires human effort. The above problems can be overcome with mesh free methods, also known as meshless methods. Instead of the elements, an appropriate number of nodes are scattered throughout the domain and the base functions are defined on sets of nodes. Moreover, the solution accuracy does not depend as much on the distribution of nodes as it does in the finite element method. The different base function definition causes substantial differences in the mathematical formulation of the method as well as in the implementation on a computer. After the introductory part, this dissertation focuses on base functions constructed using the moving least squares approximation, which is the one most frequently used in mesh free methods. The steps involved in construction of the algebraic system are shown in detail for the three methods. The diffusion equation is used as a test case. A simple node generation algorithm is proposed for the mesh free method and two algorithms for determination of the support domain of a given point. The accuracy of each of the three methods for different numbers of nodes is analysed and comparisons are made. The analysis also includes accuracy optimization of the mesh free method by finding optimal parameter values. The superior accuracy of the mesh free method over the other two methods is shown for different domain shapes. A detailed description of the suggested implementation of the mesh free method is given. The key idea is the use of $k$-D trees for finding the nodes in the vicinity of a given point. Also, two numerical libraries are tested for the solution of small linear equation systems, which constitute a part of base function evaluation. Their performances are compared and the faster is chosen. The time and space complexity of the suggested implementation are analysed and compared to the finite difference and finite element methods. In the last part, the parallelization of the mesh free methods is discussed, again in comparison with the other two methods. Two domain decomposition methods are suggested, both of which guarantee ideal load balancing among the processors. In the one-dimensional decomposition, domain slices are assigned to each processor. The hierarchical distribution is similar to the $k$-D tree in the sense that it first divides the set of nodes into left and right halves, after which each is further divided into top and bottom, etc, until the number of subsets matches the number of processors. The initial, serial part of the program is somewhat longer with the hierarchical distribution. The speedups on a test cluster are measured and analysed for both decomposition methods, and the impacts of communication and computation speed are given. The above theoretical and practical analysis contributes some new knowledge in the field of mesh free methods, which is summarized in the Conclusions, together with possible directions for future research.
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