c The patch test can be used to verify the output of finite

(c) The patch test can be used to verify the output of finite element programs. Describe what the patch test is, how it is performed and why it works.

Solution

PATCH TEST :

In some cases considerable difficulty is experienced in finding displacement functions for an element which will automatically be continuous along the whole interface between adjacent elements. The discontinuity of displacement will cause infinite strains at the interfaces. However, if, in the limit, as the size of the subdivision decreases continuity is restored, then formulation will tend to the correct answer. This condition is always reached if (1) a constant strain condition automatically ensures displacement continuity. (2) the constant strain criterion is satisfied i.e., if nodal displacements are compatible with a constant strain condition such constant strain will in fact be obtainable. To test whether a non-conforming element will provide satisfactory result, a patch test is conducted. In this test, nodal displacements corresponding to any state of constant strain are imposed on an arbitrary patch of elements. If nodal equilibrium is simultaneously achieved without the imposition of external, nodal, forces and if a state of constant stress is obtained, then clearly no external work has been lost through inter-element discontinuity. Elements that pass such a patch test will converge.

Procedure:

A small number of elements are assembled into a patch.An element patch is the set of all elements attached to a patch node.

A patch of elements is subjected to the minimum displacement boundary conditions to eliminate all rigid body motions. Apply to boundary nodal points forces or displacements which should result in a state of constant stress within the assemblage. Nodes not on the boundary are neither loaded nor restrained. Compute the displacements of nodes which do not have a prescribed value.Compute the stresses and strains. The patch test is passed if the computed stresses and strains match the expected values to the limit of computer precision.Conforming elements ALWAYS pass the patch testNodes not on the boundary are neither loaded nor restrained.Since a patch may also consist of a single element, this test may be used to check the completeness of a single element.The number of constant stress states in a patch test depends on the actual number of constant stress states in the mathematical model