Finite Elements in Solid Mechanics - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 343

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××

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• • • •

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× ×

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• • • • • • × ×

W h h h Q h

Fig. 62. MITC7 Element (only tangential components are fixed at the points
marked with × )

We recall that elements in H 0 ( rot ,) have to have continuous tangential
components along the edges between the triangles. As with the Raviart-Thomas
element, it is easy to check that the tangential components of the vector expression
y
− x

are constant on every edge. Thus, the functions in h are linear on the edges,
and so are determined by the values at two points. In particular, these points can
be the sample points for a Gaussian quadrature formula which exactly integrates
quadratic polynomials. Thus (in agreement with Fig. 62) the six degrees of freedom
for functions in h are determined by the values on the sides. These six values
along with the two components at the midpoint of the triangle determine the eight
local degrees of freedom.
Therefore, the restriction operator R h described by (6.18) can be computed
from interpolation at the above six points on the sides along with two integrals
over the triangle. Thus, the values of R h η in a triangle depend only on the values
of η in the same triangle. This means that the system matrix can be assembled
locally, triangle by triangle. This would not have been the case if we had used the
L 2 -projector in place of R h .
For numerical results using this element, see Bathe, Brezzi, and Cho [1989]
and Bathe, Bucalem, and Brezzi [1991/92].

The Model without a Helmholtz Decomposition

The formulation (6.2) for the Mindlin plate is motivated by the similarity with the
mixed formulation of the Kirchhoff plate. Arnold and Brezzi [1993] developed
a clever modification which permits a development without using the Helmholtz
decomposition. There is a simple treatment in terms of the theory of saddle point
problems with penalty term developed in Ch. III, §4; cf. also Braess [1996]. A sim-
ilar modification can also be found in the treatment of shells by Pitkaranta [1992].