Existence and stability, and discrete BB and rank conditions, for general mixed-hybrid finite elements in elasticity. (English) Zbl 0624.73094

Xue, W.-M.; Atluri, S. N.

Hybrid and mixed finite element methods, Winter Annu. Meet. ASME, Miami Beach/Fl. 1985, AMD-Vols. 73, 91-112 (1985).

[For the entire collection see Zbl 0623.00027.]
In this paper, all possible forms of mixed-hybrid finite element methods that are based on multi-field variational principles are examined as to the conditions for existence, stability, and uniqueness of their solutions. The reasons as to why certain “simplified hybrid-mixed methods” in general, and the so-called “simplified hybrid-displacement method” in particular (based on the so-called simplified variational principles), become unstable, are discussed. A comprehensive discussion of the “discrete” BB-conditions, and the rank conditions, of the matrices arising in mixed-hybrid methods, is given. Some recent studies aimed in the assurance of such rank conditions, and the related problem of the avoidance of spurious kinematic modes, are presented.

MSC:

74S05	Finite element methods applied to problems in solid mechanics
74S30	Other numerical methods in solid mechanics (MSC2010)
65K10	Numerical optimization and variational techniques

Keywords:

discrete BB-conditions; Hu-Washizu principle; discontinuous displacement fields; reciprocated traction fields; mixed-hybrid finite element methods; multi-field variational principles; existence; stability; uniqueness; simplified hybrid-displacement method; simplified variational principles; rank conditions

Citations:

Zbl 0623.00027