×

zbMATH — the first resource for mathematics

A unified approach for embedded boundary conditions for fourth-order elliptic problems. (English) Zbl 1352.65507
Summary: An efficient procedure for embedding kinematic boundary conditions in the biharmonic equation, for problems such as the pure streamfunction formulation of the Navier-Stokes equations and thin plate bending, is based on a stabilized variational formulation, obtained by Nitsche’s approach for enforcing boundary constraints. The absence of kinematic admissibility constraints allows the use of non-conforming meshes with non-interpolatory approximations, thereby providing added flexibility in addressing the higher continuity requirements typical of these problems. Variationally conjugate pairs weakly enforce kinematic boundary conditions. The use of a scaling factor leads to a formulation with a single stabilization parameter. For plates, the enforcement of tangential derivatives of deflections obviates the need for pointwise enforcement of corner values in the presence of corners. The single stabilization parameter is determined from a local generalized eigenvalue problem, guaranteeing coercivity of the discrete bilinear form. The accuracy of the approach is verified by representative computations with bicubic B-splines, providing guidance to the determination of the scaling and exhibiting optimal rates of convergence and robust performance with respect to values of the stabilization parameter.

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J40 Boundary value problems for higher-order elliptic equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Landau, Fluid Mechanics. Translated from the Russian by J. B. Sykes and W. H. Reid. Course of Theoretical Physics 6 (1959)
[2] Kirchhoff, Über das Gleichgewicht und die Bewegung einer elastischen Scheibe, Journal für die Reine und Angewandte Mathematik 1850 (40) pp 51– (1850) · ERAM 040.1086cj
[3] Timoshenko, Theory of Plates and Shells, 2. ed. (1959)
[4] Clough RW Felippa CA A refined quadrilateral element for analysis of plate bending Proceedings of the Second Conference on Matrix Methods in Structural Mechanics AFFDL-TR-68-150 1968 399 440
[5] Clough RW Tocher JL Finite element stiffness matrices for analysis of plate bending Proceedings of the First Conference on Matrix Methods in Structural Mechanics AFFDL-TR-66-80 1966 515 546
[6] Bogner FK Fox RL Schmit LA The generation of interelement-compatible stiffness and mass matrices by the use of interpolation formulae Proceedings of the First Conference on Matrix Methods in Structural Mechanics AFFDL-TR-66-80 1966 397 443
[7] Hansbo, A discontinuous Galerkin method for the plate equation, Calcolo 39 (1) pp 41– (2002) · Zbl 1012.74066
[8] Engel, Continuous/discontinuous finite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity, Computer Methods in Applied Mechanics and Engineering 191 (34) pp 3669– (2002) · Zbl 1086.74038
[9] Wells, A C0 discontinuous Galerkin formulation for Kirchhoff plates, Computer Methods in Applied Mechanics and Engineering 196 (35-36) pp 3370– (2007) · Zbl 1173.74447
[10] Beirão da Veiga, family of C0 finite elements for Kirchhoff plates. I. Error analysis, SIAM Journal of Numerical Analysis 45 (5) pp 2047– (2007) · Zbl 1152.74043
[11] Hughes, Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Computer Methods in Applied Mechanics and Engineering 194 (39-41) pp 4135– (2005) · Zbl 1151.74419
[12] Kiendl, Isogeometric shell analysis with Kirchhoff-Love elements, Computer Methods in Applied Mechanics and Engineering 198 (49-52) pp 3902– (2009) · Zbl 1231.74422
[13] Echter, A hierarchic family of isogeometric shell finite elements, Computer Methods in Applied Mechanics and Engineering 254 pp 170– (2013) · Zbl 1297.74071
[14] Benson, Blended isogeometric shells, Computer Methods in Applied Mechanics and Engineering 255 pp 133– (2013) · Zbl 1297.74114
[15] Quarteroni, Primal hybrid finite element methods for 4th order elliptic equations, Calcolo 16 (1) pp 21– (1979) · Zbl 0435.65092
[16] Blum, On mixed finite element methods in plate bending analysis, Computational Mechanics 6 (3) pp 221– (1990) · Zbl 0736.73061
[17] Massimi, A discontinuous enrichment method for the efficient solution of plate vibration problems in the medium-frequency regime, International Journal for Numerical Methods in Engineering 84 (2) pp 127– (2010) · Zbl 1202.74201
[18] Dolbow, An efficient finite element method for embedded interface problems, International Journal for Numerical Methods in Engineering 78 (2) pp 229– (2009) · Zbl 1183.76803
[19] Gerstenberger, An embedded Dirichlet formulation for 3D continua, International Journal for Numerical Methods in Engineering 82 (5) pp 537– (2010) · Zbl 1188.74056
[20] Li, The Immersed Interface Method. Frontiers in Applied Mathematics 33 (2006) · Zbl 1122.65096
[21] Schillinger, An unfitted hp-adaptive finite element method based on hierarchical B-splines for interface problems of complex geometry, Computer Methods in Applied Mechanics and Engineering 200 (47-48) pp 3358– (2011) · Zbl 1230.74197
[22] Nitsche, Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilraume\"n, die keinen Randbedingungen unterworfen sind, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 36 (1) pp 9– (1971) · Zbl 0229.65079
[23] Stenberg, On some techniques for approximating boundary conditions in the finite element method, Journal of Computational and Applied Mathematics 63 (1-3) pp 139– (1995) · Zbl 0856.65130
[24] Barbosa, The finite element method with Lagrange multipliers on the boundary: circumventing the Babuskǎ-Brezzi condition, Computer Methods in Applied Mechanics and Engineering 85 (1) pp 109– (1991) · Zbl 0764.73077
[25] Harari, Finite element formulations for exterior problems: application to hybrid methods, non-reflecting boundary conditions, and infinite elements, International Journal for Numerical Methods in Engineering 40 (15) pp 2791– (1997) · Zbl 0890.76038
[26] Embar, Imposing Dirichlet boundary conditions with Nitsche’s method and spline-based finite elements, International Journal for Numerical Methods in Engineering 83 (7) pp 877– (2010) · Zbl 1197.74178
[27] Harari, Embedded kinematic boundary conditions for thin plate bending by Nitsche’s approach, International Journal for Numerical Methods in Engineering 99 (1) pp 99– (2012) · Zbl 1352.74162
[28] Höllig, Finite Element Methods with B-splines. Frontiers in Applied Mathematics 26 (2003) · Zbl 1020.65085
[29] Govindjee, Convergence of an efficient local least-squares fitting method for bases with compact support, Computer Methods in Applied Mechanics and Engineering 213-216 pp 84– (2012) · Zbl 1243.65028
[30] Harari, Consistent loading for thin plates, Journal of Mechanics of Materials and Structures 6 (5) pp 765– (2011)
[31] Cowper, The shear coefficient in Timoshenko’s beam theory, Transactions ASME Journal Applied Mechanics 33 (2) pp 335– (1966) · Zbl 0151.37901
[32] Prescott, Elastic waves and vibrations of thin rods, Philosophical Magazine (7) 33 (225) pp 703– (1942) · Zbl 0063.06338
[33] Harari, What are C and h?: inequalities for the analysis and design of finite element methods, Computer Methods in Applied Mechanics and Engineering 97 (2) pp 157– (1992) · Zbl 0764.73083
[34] Evans, Explicit trace inequalities for isogeometric analysis and parametric hexahedral finite elements, Numerische Mathematik 123 (2) pp 259– (2013) · Zbl 1259.65169
[35] Fix, An algorithm for the ill-conditioned generalized eigenvalue problem, SIAM Journal of Numerical Analysis 9 (1) pp 78– (1972) · Zbl 0252.65028
[36] Moler, An algorithm for generalized matrix eigenvalue problems, SIAM Journal of Numerical Analysis 10 (2) pp 241– (1973) · Zbl 0253.65019
[37] Peters, Ax = {\(\lambda\)}Bx and the generalized eigenproblem, SIAM Journal of Numerical Analysis 7 (4) pp 479– (1970) · Zbl 0276.15016
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.