Solution Structures for Imposing Boundary Conditions in Mesh Free Analysis of Heat Conduction Problems

One of the main advantages of meshless methods is that it eliminates the mesh generation, but it is still necessary to place nodes with controlled spacing variation on the boundary and within the domain. However, due to lack of connectivity between nodes it is more difficult to interpolate the field variables and impose boundary conditions. In this paper, a mesh free method is presented for analysis using a structured grid that does not conform to the geometry of the domain. The geometry of the domain is independent of the structured grid and is represented using implicit equations. The implicit equations of the boundaries can be used to construct solution structures that satisfy boundary conditions exactly even though the nodes of the grid are not on the boundaries of the domain. The solution structures are constructed using the implicit equations of the boundary together with a piece-wise interpolation over the structured grid. The implicit equations are also used to construct step function of solid such that its value is equal to unity inside the solid and zero outside. The step function of the solid is used for volume integrations needed for the analysis. The traditional weak form for Poisson’s equation is modified by using this solution structure to eliminate the surface integration terms. The accuracy and implementation of the present mesh free method is illustrated for two-dimensional heat conduction problems governed by Poisson’s equation. Satisfactory results are obtained when compared with analytical results and results from commercial finite element software.