A solution to linear elasticity using locally supported RBF collocation in a generalised finite-difference mode

• Published in: Engineering analysis with boundary elements 2013-01, Vol.37 (1), p.32-41
• Main Authors: Stevens, D., Power, H., Cliffe, K.A.
• Format: Article
• Language: English
• Subjects: Boundaries; Collocation; Displacement; Finite difference; Finite difference method; Linear elasticity; Mathematical analysis; Mathematical models; Meshless; Operators; Partial differential equations; RBF
• ISSN: 0955-7997; 1873-197X
• DOI: 10.1016/j.enganabound.2012.08.005

This work presents a novel meshless numerical approach for the solution of linear elasticity problems, using locally supported RBF collocation. The Kansa (unsymmetric) RBF collocation method is used to form local collocation systems, which enforce the PDE governing and boundary operators. With the displacement values acting as the unknowns in the system, a sparse global system is formed. This global matrix is formed in a manner analogous to a finite difference method, with the displacement values at each internal node defined in terms of the displacements at other nodes within the local stencil. In contrast to traditional finite difference methods, here the RBF collocation assumes the role traditionally played by polynomial interpolants. The RBF collocation does itself satisfy the governing PDE operator at some collocation points, and therefore allows for a significantly more accurate reconstruction than is found from simple polynomial interpolants. In addition, the boundary operators (for applied displacement and applied surface traction) are enforced directly within the local RBF collocation systems, rather than being enforced at the global matrix. In contrast to traditional finite difference methods based on polynomial interpolation, the RBF collocation does not require a regular arrangement of nodes. Therefore, the proposed numerical method is directly applicable to unstructured datasets.