Exponential basis functions in solution of problems with fully incompressible materials: A mesh-free method

In this paper exponential basis functions (EBFs) satisfying the governing equations of elastic problems with incompressible materials are introduced. Due to similarity between elasticity problems and steady state fluid problems the bases found for the former problems are used for latter problems. We discuss on using single field form known as displacement/velocity based formulation and also on using a two-field form known as u-p formulation. In the first formulation we find the pressure bases through performing a limit analysis using a fictitious bulk modulus while in the second formulation the bases are found directly by considering the pressure as a separate variable. In the second formulation we directly apply the condition of incompressibility. It is shown that both formulations yield identical bases meaning that the first one may be used in a standard approach. However, it is also shown that when the incompressibility condition is applied by a Laplacian of pressure in the second formulation, some additional spurious EBFs may be obtained. Having defined appropriate bases, we follow the solution strategy recently introduced by the authors for other engineering problems. Some well-known benchmark problems are solved to show the capabilities of the method.