Manchester Applied Mathematics and Numerical Analysis Seminars

Spring 1999

March 3, 1999, 2.30 pm

Lecture Theatre OF/B9 Oddfellows Hall (Material Science)

Finite Elements as ill-posed boundary value problems

Prof. Francis Molenkamp, Dept. of Civil Engineering, UMIST

In finite element analyses of static equilibrium of dilative geomaterials the phenomenon of locking causes the calculated displacement to vanish in the whole mesh. For regular plane meshes composed of 4-node quads, locking occurs because the stiffness of the bending modes, the so-called "hour-glass" modes, approach infinity.

The occurrence of spurious modes in such analyses causes the displacement field to become irrealistic as excessive internal deformation patterns are calculated.

To explain these numerical phenomena analytical parametric studies have been performed using for simplicity an equivalent elastic local continuum model instead of a more realistic local elasto-plastic model. The usage of this simplified approach will be justified. The observed numerical properties of the finite elements are investigated by considering the elements as boundary value problems with prescribed displacement (Dirichlet problem) or prescribed traction (Neumann problem).

For the Dirichlet problem the equivalent Poisson's ratio, p, for dilative materials enters the ill-posed parameter range 0.5 < p < 1. Nevertheless, partly unique solutions are shown to be obtained, explaining the locking of the 4-node quad. For the Neumann problem non-uniqueness can occur in this parameter range, explaining the observed spurious modes.

Assuming an increasing Poisson's ratio to represent approximately the increasing material dilation occurring during the loading process the limit of applicability of such finite elements is governed by the first occurrence of a locking or spurious eigen mode.

Do the requirements of uniqueness and avoidance of locking and spurious modes enforce the usage of non-local material models and corresponding finite elements?

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