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Alejandro Ortiz-Bernardin's picture

Consistent and stable meshfree Galerkin methods using the virtual element decomposition

Paper Accepted for Publication in International Journal for Numerical Methods in Engineering

Consistent and stable meshfree Galerkin methods using the virtual element decomposition

A. Ortiz-Bernardin, A. Russo, N. Sukumar

 

Abstract

Alejandro Ortiz-Bernardin's picture

Linear smoothed polygonal and polyhedral finite elements

Paper Accepted for Publication in International Journal for Numerical Methods in Engineering

Linear smoothed polygonal and polyhedral finite elements

A. Francis, A. Ortiz-BernardinSPA. BordasS. Natarajan

ABSTRACT

N. Sukumar's picture

3. Meshfree methods and the patch test

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We are in the framework of small-strain two-dimensional linear elasticity without any body forces. Consider a domain that is discretized by a union of triangles and/or quadrilaterals (`patch of elements').  For C0 conforming approximations such as triangular/quadrilateral finite elements, the finite element approximation can exactly reproduce an arbitrary linear displacement field. Hence, if the exact solution is linear, then the finite element solution must match (within machine precision) the exact solution. In simple terms, passing the patch test for linear elasticity with standard conforming finite elements provides verification of one's implementation and is used to assess the same when new elements are proposed. For conforming elements, it is a sufficient condition for convergence (2nd order PDEs), and hence is the first problem that is solved when a new element/method is proposed. To carry out the patch test, the following steps are performed:

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