×

The stability of stars of simplicial hybrid equilibrium finite elements for solid mechanics. (English) Zbl 1352.74007

Summary: In this paper, we define the spurious kinematic modes of hybrid equilibrium 2D and 3D simplicial elements of general degree and present the results of studies on the stability of star patches of such elements. The approach used in these studies is based on first establishing the kinematic properties of a pair of elements that share an interface. This approach is introduced by its application to star patches of hybrid equilibrium triangular plate elements for modelling membrane and plate bending problems and then generalised to 3D continua. It is then shown how the existence of spurious kinematic modes depends on the topological and geometrical properties of a patch, as well as on the degree of the polynomial approximation functions of stress and displacement.

MSC:

74A05 Kinematics of deformation
74S05 Finite element methods applied to problems in solid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs

Software:

DLMF
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Veubeke, Upper and lower bounds in matrix structural analysis, Agardograph 72 pp 165– (1964) · Zbl 0131.22903
[2] Veubeke, Stress Analysis (1965)
[3] Sander, High Speed Computing of Elastic Structures pp 167– (1971)
[4] Kempeneers, Modèles équilibre pour l’analyse duale, Revue EuropÉenne des Éléments 12 (6) pp 737– (2003) · Zbl 1171.74444
[5] LadevÃĺze, Mastering Calculations in Linear and Nonlinear Mechanics (2005)
[6] Díez, Equilibrated patch recovery error estimates: simple and accurate upper bounds of the error, International Journal for Numerical Methods in Engineering 69 (10) pp 2075– (2007) · Zbl 1194.74382
[7] Wang, Stable linear traction-based equilibrium elements for elastostatics: Direct access to linear statically admissible stresses and quadratic kinematically admissible displacements for dual analysis, International Journal for Numerical Methods in Engineering 101 pp 887– (2015) · Zbl 1352.74445
[8] Heyman, The Stone Skeleton: Structural Engineering of Masonry Architecture (1995)
[9] Taylor, Structural design and the Eurocodes - a historical review, Structural Engineer 86 (20) pp 29– (2008)
[10] Weichert, Progress in the application of lower bound direct methods in structural design, International Journal of Applied Mechanics 2 (01) pp 145– (2010)
[11] Almeida, Alternative approach to the formulation of hybrid equilibrium finite elements, Computers & Structures 40 (4) pp 1043– (1991)
[12] Maunder, A general formulation of equilibrium macro-elements with control of spurious kinematic modes, International Journal for Numerical Methods in Engineering 39 (18) pp 3175– (1996) · Zbl 0878.73070
[13] Almeida, A set of hybrid equilibrium finite element models for the analysis of three-dimensional solids, International Journal for Numerical Methods in Engineering 39 (16) pp 2789– (1996) · Zbl 0873.73067
[14] Wang, A traction-based equilibrium finite element free from spurious kinematic modes for linear elasticity problems, International Journal for Numerical Methods in Engineering 99 pp 763– (2014) · Zbl 1352.74042
[15] Maunder, A triangular hybrid equilibrium plate element of general degree, International Journal for Numerical Methods in Engineering 63 (3) pp 315– (2005) · Zbl 1140.74555
[16] Maunder, The stability of stars of triangular equilibrium plate elements, International Journal for Numerical Methods in Engineering 77 (7) pp 922– (2009) · Zbl 1183.74294
[17] Almeida, A general degree hybrid equilibrium finite element for Kirchhoff plates, International Journal for Numerical Methods in Engineering 94 (4) pp 331– (2013) · Zbl 1352.74409
[18] Kempeneers, Pure equilibrium tetrahedral finite elements for global error estimation by dual analysis, International Journal for Numerical Methods in Engineering 81 (4) pp 513– (2010) · Zbl 1183.74284
[19] Pereira, Hybrid equilibrium hexahedral elements and super-elements, Communications in Numerical Methods in Engineering 24 pp 157– (2008) · Zbl 1132.74043
[20] Ramsay, Sub-modelling and boundary conditions with p-type hybrid-equilibrium plate-membrane elements, Finite Elements in Analysis and Design 43 (2) pp 155– (2006)
[21] Santos, A family of Piola-Kirchhoff hybrid stress finite elements for two-dimensional linear elasticity, Finite Elements in Analysis and Design 85 (0) pp 33– (2014)
[22] Veubeke, BM Fraeijs de Veubeke Memorial Volume of Selected Papers pp 569– (1980)
[23] NIST, Digital Library of Mathematical Functions (2014)
[24] Boffi, Mixed Finite Element Methods and Applications (2013) · Zbl 1277.65092
[25] Dong, A simple procedure to develop efficient & stable hybrid/mixed elements, and voronoi cell finite elements for macro-& micromechanics, Computers Materials and Continua 24 (1) pp 61– (2011)
[26] Cottrell, Isogeometric Analysis: Toward Integration of CAD and FEA (2009) · Zbl 1378.65009
[27] Munkres, Elements of Algebraic Topology (1984)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.