## Mechanical conditions for stability and optimal convergence of mixed finite elements for linear plane elasticity.(English)Zbl 0729.73208

Summary: In order to develop an efficient and manageable tool for checking stability and optimal convergence of mixed finite elements (LBB- and ‘equilibrium’ condition), three mechanical conditions are stated. The first requires the continuity of the normal component of the stress tensor across interelement boundaries, the second forbids spurious modes on a two element patch, and the third is to avoid zero-energy-stresses on an element. The mathematical proof shows that the conditions are necessary and sufficient. Finally, the hybrid implementation of two plane mixed elements is carried out, and comparisons are made with two standard displacement elements. In particular, the mixed element with constant displacement shape functions (MMC) surpasses the linear displacement element by far and also the quadratic displacement element if the computational effort is compared.

### MSC:

 74S05 Finite element methods applied to problems in solid mechanics
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### References:

 [1] Herrmann, L.R., A bending analysis of plates, (), 577-602 [2] Prange, G., Das extremum der formänderungsarbeit, abhandlung zur erwerbung der venia legendi an der kgl., (1916), Technischen Hochschule zu Hannover [3] Hellinger, E., Die allgemeinen ansätze der mechanik der kontinua, (), 602-694, (Mechanik); Teilband 4 · JFM 45.1012.01 [4] Reissner, E., On a variational theorem in elasticity, J. math. phys., 29, 90-95, (1950) · Zbl 0039.40502 [5] Ladyszhenskaja, O.A., The mathematical theory of viscous incompressible flows, (1969), Gordon and Breach New York [6] Babus̆ka, I.; Aziz, A.K., Survey lectures on the mathematical foundations of the finite element method, (), 5-359 [7] Brezzi, F., On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, RAIRO serie rouge, 8, 129-151, (1974) · Zbl 0338.90047 [8] Oden, J.T.; Carey, G.F., () [9] Kikuchi, N.; Oden, J.T., Contact problems in elasticity: A study of variational inequalities and finite element methods, () · Zbl 0685.73002 [10] Raviart, P.A.; Thomas, J.M., A mixed finite element method for 2-nd order elliptic problems, () · Zbl 0362.65089 [11] Brezzi, F.; Douglas, J.; Marini, L.D., Two families of mixed finite elements for second order elliptic problems, Numer. math., 47, 217-235, (1985) · Zbl 0599.65072 [12] Johnson, C.; Mercier, B., Some equilibrium finite element methods for two-dimensional elasticity problems, Numer. math., 30, 103-116, (1978) · Zbl 0427.73072 [13] Arnold, D.N.; Brezzi, F.; Douglas, J., PEERS: A new mixed finite element for plane elasticity, Japan J. appl. math., 1, 347-367, (1984) · Zbl 0633.73074 [14] Fraeijs de Veubeke, B., Stress function approach, () · Zbl 0352.73061 [15] Stenberg, R., A family of mixed finite elements for the elasticity problem, Numer. math., 53, 513-538, (1988) · Zbl 0632.73063 [16] Punch, E.F.; Atluri, S.N., Development and testing of stable, invariant, isoparametric curvilinear 2- and 3-D hybrid-stress elements, Comput. methods appl. mech. engrg., 47, 331-356, (1984) · Zbl 0535.73057 [17] Xue, W.-M.; Karlovitz, L.A.; Atluri, S.N., On the existence and stability conditions for mixed-hybrid finite element solutions based on Reissner’s variational principle, Internat. J. solids and structures, 21, 97-116, (1985) · Zbl 0597.73072 [18] Xue, W.-M.; Atluri, S.N., Existence and stability, and discrete BB and rank conditions, for general mixed-hybrid finite elements in elasticity, (1986), Center for the Advancement of Computational Mechanics, School of Civil Engineering, Georgia Institute of Technology Atlanta · Zbl 0624.73094 [19] Zienkiewicz, O.C.; Qu, S.; Taylor, R.L.; Nakazawa, S., The patch test for mixed formulations, Internat. J. numer. methods engrg., 23, 1873-1883, (1986) · Zbl 0614.65115 [20] Pitkäranta, J., Local stability conditions for the babus̆ka method of Lagrange multipliers, Math. comp., 35, 1113-1129, (1980) · Zbl 0473.65072 [21] Pitkäranta, J.; Stenberg, R., Analysis of some mixed finite element methods for plane elasticity equations, Math. comp., 41, 399-423, (1983) · Zbl 0537.73057 [22] Stenberg, R., On the construction of optimal mixed finite element methods for the plane elasticity problem, Numer. math., 48, 447-462, (1986) · Zbl 0563.65072 [23] Olson, M.D., The mixed finite element method in elasticity and elastic contact problems, (), 19-49
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