Keywords: incompressible materials, standard displacement elasticity, meshless method, exponential basis function, fundamental solution, discrete transformation.

In this paper, the results of our latest investigations on the solution of elasticity and steady Navier-Stokes equations for fully incompressible materials are presented. The incompressible materials analysis is one of the interested research areas in the engineering mechanics and fluid dynamics. However, application of 'standard' displacement based (velocity) formulations to incompressible or nearly incompressible problems leads to indeterminacy of the mean stress or pressure which for compressible materials is related to the volumetric part of the strain via the bulk modulus [1].

In the method presented, the approximate solution of the standard form of elasticity and Navier-Stokes equations is expressed by a linear combination of exponential basis functions (EBFs) with complex-valued exponents multiplied by polynomials. These sets of functions are obtained by satisfying the equations exactly. The constant coefficients of the solution series are evaluated through a point collocation method on the domain boundaries via a complex discrete transformation technique. This method has been descriptively presented by Boroomand et al. [2] and applied to static and time harmonic elastic problems.

In contrast to other numerical methods based on displacement formulations (that fail when the material becomes fully incompressible), here we shall show that with the aid of EBFs, the standard equations can be solved even when Poisson's ratio becomes 0.5; and thus, it is not necessary to use methods with mixed formulations, iterative solutions and so on. To evaluate pressure we shall find appropriate basis functions by calculating the limit of the bulk modulus multiplied by volumetric strain when Poisson's ratio approaches to 0.5. The basis functions are then combined with other basis functions, associated with the deviatoric strains, to solve fully incompressible problems with general boundary conditions in a meshless style. This helps us to evaluate pressure without any oscillation.

The proposed numerical scheme is applied to the following four examples: (1) a square cavity, (2) a wave-shaped bottom cavity, (3) a circular cavity, and (4) a circular cavity with eccentric rotating cylinder. As compared to the analytical and other available numerical solutions, our numerical experiments demonstrate that the present method is capable of producing very accurate results using a few collocation points. The results promisingly show the capabilities of the proposed meshless method for being used as a unified formulation for solid and fluid phases in fluid-structure interaction problems with irregular geometries.

1

O.C. Zienkiewicz, R.C. Taylor, "The finite element method", 5th Edition, Vol. I, Butterworth-Heinemann, 2000.

2

B. Boroomand, S. Soghrati, B. Movahedian, "Exponential basis functions in solution of static and time harmonic elastic problems in a meshless style", International Journal for Numerical Methods in Engineering, 81, 971-1018, 2010. doi:10.1002/nme.2718

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