Keywords: shear-locking, finite element and meshfree coupling, matching field approach.

It is well-known in numerical analysis that in the thin-beam limit of shear-deformable Timoshenko beams, stress oscillations and a reduction in the optimal convergence rate, a phenomenon called shear-locking occurs due to inconsistencies in the approximations of the transverse displacement and rotation fields. In order to eliminate shear-locking, the so-called matching field strategy can be adopted by using meshfree methods in which the approximation functions of the rotation field can be obtained from the derivatives of the approximation functions of the transverse displacement field [1,2]. However, using finite element modelling for some portions of the structure, where finite element analysis is accurate, is computationally less expensive and thus coupling of mesh-free and finite element methods may be advantageous in some cases.

In order to benefit from the advantages of both methods, the continuous blending method was introduced [3], which provides the possibility of using the mesh-free and finite element method jointly by mixing the mesh-free and finite element approximations in the interface regions and using meshfree and finite element methods alone in separate regions of the computational domain. The continuous blending method ensures continuity and consistency for the mixed approximations in the interface regions and produces shape functions that satisfy the Kronecker delta property at the required nodes, so that finite elements can be connected or the boundary conditions can be imposed directly at these nodes.

On the other hand, when the matching field approach is used, the approximation functions of the rotation field are constrained to be identical to the derivatives of the approximation functions of the transverse displacement field, and so the continuous blending method has to be modified in order to be able to satisfy this constraint condition in the interface regions. Recently, a meshfree and finite element coupling technique that satisfies the constraint conditions of the matching field approach was developed [4]. This technique is adopted herein for the locking-free analysis of shear deformable beams. For both transverse displacement and rotation fields, the developed technique produces approximation functions that satisfy the Kronecker delta property at the required nodes of the meshfree region, so that finite elements can be connected or the boundary conditions can be imposed to these nodes directly when the matching field approach is adopted.

1

B.M. Donning, W.K. Liu, "Meshless methods for shear-deformable beams and plates", Computer Methods in Applied Mechanics and Engineering, 152, 47-71, 1998. doi:10.1016/S0045-7825(97)00181-3

2

W. Kanok-Nukulchai, W. Barry, K. Saran-Yasoontorn, P.H. Bouillard, "On elimination of shear locking in the element-free Galerkin method", International Journal for Numerical Methods in Engineering, 52, 705-725, 2001. doi:10.1002/nme.223

3

A. Huerta, S. Fernandez-Mendez, "Enrichment and coupling of the finite element and the meshless methods", International Journal for Numerical Methods in Engineering, 48, 1615-1636, 2000. doi:10.1002/1097-0207(20000820)48:11<1615::AID-NME883>3.0.CO;2-S

4

R.E. Erkmen, M.A. Bradford, "Coupling of finite element and meshfree methods for locking-free analysis of shear-deformable beams and plates", Engineering Computations, 2011. doi:10.1108/02644401111179009(accepted)

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