| Modelling tools which are able to solve partial differential equations with increasing accuracy, complexity and ease of use are essential for engineers. Two main methods of solving these types of problems are the Finite Difference Method, and the Finite Element Method, both of which rely on a mesh to discretize the domain or solution space. These meshed methods are widely used and studied. However, they suffer from a variety of problems related to their construction and rigidity. A third type of solution, known as meshless or meshfree methods, are able to avoid meshing problems and are currently being heavily researched. In this thesis a promising type of meshless method, the Finite Cloud Method, is investigated and a `C' program implementing the method has been written. The method is applied to a range of problems including scalar and vectorial equations, coupled field and both time independent and time dependent solutions. In particular, the method is extended to include inhomogeneous domains (multiple materials) and spatially dependent material parameters. Physical situations addressed include: Heat Flow, Schrodinger's Equation, Maxwell's Equations and optical mode solving. Results for the various equation types have been very promising and in high agreement with both analytically and numerically solved solutions. As well, several improvements to the method have been developed and are detailed. The method is shown to be versatile, robust and highly accurate. |
