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Diamond elements: a finite element/discrete-mechanics approximation scheme with guaranteed optimal convergence in incompressible elasticity. (English) Zbl 1194.74406
Summary: We present a finite element discretization scheme for the compressible and incompressible elasticity problems that possess the following properties: (i) the discretization scheme is defined on a triangulation of the domain; (ii) the discretization scheme is defined – and is identical – in all spatial dimensions; (iii) the displacement field converges optimally with mesh refinement; and (iv) the inf-sup condition is automatically satisfied. The discretization scheme is motivated both by considerations of topology and analysis, and it consists of the combination of a certain mesh pattern and a choice of interpolation that guarantees optimal convergence of displacements and pressures. Rigorous proofs of the satisfaction of the inf-sup condition are presented for the problem of linearized incompressible elasticity. We additionally show that the discretization schemes can be given a compelling interpretation in terms of discrete differential operators. In particular, we develop a discrete analogue of the classical tensor differential complex in terms of which the discrete and continuous boundary-value problems are formally identical. We also present numerical tests that demonstrate the dimension-independent scope of the scheme and its good performance in problems of finite elasticity.

MSC:
74S05 Finite element methods applied to problems in solid mechanics
74B15 Equations linearized about a deformed state (small deformations superposed on large)
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