# zbMATH — the first resource for mathematics

An embedded Dirichlet formulation for 3D continua. (English) Zbl 1188.74056
Summary: This paper presents a new approach for imposing Dirichlet conditions weakly on non-fitting finite element meshes. Such conditions, also called embedded Dirichlet conditions, are typically, but not exclusively, encountered when prescribing Dirichlet conditions in the context of the eXtended finite element method (XFEM). The method’s key idea is the use of an additional stress field as the constraining Lagrange multiplier function. The resulting mixed/hybrid formulation is applicable to 1D, 2D and 3D problems. The method does not require stabilization for the Lagrange multiplier unknowns and allows the complete condensation of these unknowns on the element level. Furthermore, only non-zero diagonal-terms are present in the tangent stiffness, which allows the straightforward application of state-of-the-art iterative solvers, like algebraic multigrid (AMG) techniques. Within this paper, the method is applied to the linear momentum equation of an elastic continuum and to the transient, incompressible Navier-Stokes equations. Steady and unsteady benchmark computations show excellent agreement with reference values. The general formulation presented in this paper can also be applied to other continuous field problems.

##### MSC:
 74S05 Finite element methods applied to problems in solid mechanics 74B05 Classical linear elasticity
ML
Full Text:
##### References:
 [1] Wall, FluidâStructure Interaction Modelling Simulation, Optimisation (2006) [2] Gerstenberger, An extended finite element method/Lagrange multiplier based approach for fluidâstructure interaction, Computer Methods in Applied Mechanics and Engineering 197 (19â20) pp 1699– (2008) · Zbl 1194.76117 [3] Legay, An EulerianâLagrangian method for fluidâstructure interaction based on level sets, Computer Methods in Applied Mechanics and Engineering 195 (17â18) pp 2070– (2006) [4] Belytschko, Elastic crack growth in finite elements with minimal remeshing, International Journal for Numerical Methods in Engineering 45 (5) pp 601– (1999) · Zbl 0943.74061 [5] MoÃ“s, A finite element method for crack growth without remeshing, International Journal for Numerical Methods in Engineering 46 (1) pp 131– (1999) · Zbl 0955.74066 [6] Sukumar, Modeling holes and inclusions by level sets in the extended finite-element method, Computer Methods in Applied Mechanics and Engineering 190 (46â47) pp 6183– (2001) · Zbl 1029.74049 [7] Ji, On strategies for enforcing interfacial constraints and evaluating jump conditions with the extended finite element method, International Journal for Numerical Methods in Engineering 61 (14) pp 2508– (2004) · Zbl 1075.74651 [8] MoÃ“s, Imposing Dirichlet boundary conditions in the extended finite element method, International Journal for Numerical Methods in Engineering 67 (12) pp 1641– (2006) · Zbl 1113.74072 [9] Zilian, The enriched spaceâtime finite element method (EST) for simultaneous solution of fluidâstructure interaction, International Journal for Numerical Methods in Engineering 75 (3) pp 305– (2008) · Zbl 1195.74212 [10] Dolbow, Residual-free bubbles for embedded Dirichlet problems, Computer Methods in Applied Mechanics and Engineering 197 pp 3751– (2008) · Zbl 1197.65180 [11] Mourad, A bubble-stabilized finite element method for Dirichlet constraints on embedded interfaces, International Journal for Numerical Methods in Engineering 69 (4) pp 772– (2007) · Zbl 1194.65136 [12] Sanders, On methods for stabilizing constraints over enriched interfaces in elasticity, International Journal for Numerical Methods in Engineering 78 (9) pp 1009– (2009) · Zbl 1183.74313 [13] 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 [14] FernÃ!ndez-MÃ©ndez, Imposing essential boundary conditions in mesh-free methods, Computer Methods in Applied Mechanics and Engineering 193 (12â14) pp 1257– (2004) · Zbl 1060.74665 [15] Hansbo, An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems, Computer Methods in Applied Mechanics and Engineering 191 (47â48) pp 5537– (2002) · Zbl 1035.65125 [16] Hansbo, A finite element method for the simulation of strong and weak discontinuities in solid mechanics, Computer Methods in Applied Mechanics and Engineering 193 (33â35) pp 3523– (2004) · Zbl 1068.74076 [17] Mergheim, A hybrid discontinuous Galerkin/interface method for the computational modelling of failure, Communications in Numerical Methods in Engineering 20 (7) pp 511– (2004) · Zbl 1302.74166 [18] Nitsche, Ãber ein Variationsprinzip zur LÃ§sung von Dirichlet-Problemen bei Verwendung von TeilrÃ\currencyumen, die keinen Randbedingungen unterworfen sind, Abhandlungen aus dem Mathematisches Seminar der UniversitÃ\currencyt 36 pp 9– (1971) [19] Hemker, Discontinuous Galerkin discretization with embedded boundary conditions, Computational Methods in Applied Mathematics 3 pp 135– (2003) · Zbl 1035.65128 · doi:10.2478/cmam-2003-0010 [20] Lew, A discontinuous-Galerkin-based immersed boundary method, International Journal for Numerical Methods in Engineering 76 (4) pp 427– (2008) · Zbl 1195.76258 [21] Gee M, Sala M, Siefert C, Hu J, Tuminaro R. Ml 5.0 smoothed aggregation user’s guide. SAND2006-2649, 2006. [22] Sala, A new PetrovâGalerkin smoothed aggregation preconditioner for nonsymmetric linear systems, SIAM Journal on Scientific Computing 31 pp 143– (2008) · Zbl 1183.76673 [23] Gerstenberger A, Mayer UM, Wall WA. A 3d XFEM/Lagrange multiplier based approach for fluid structure interaction. Eighth World Congress on Computational Mechanics/Fifth European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS 2008, Venice, July 2008. [24] Zilian, A localized mixed-hybrid method for imposing interfacial constraints in the extended finite element method (XFEM), International Journal for Numerical Methods in Engineering (2009) · Zbl 1171.74456 [25] Daux, Arbitrary cracks and holes with the extended finite element method, International Journal for Numerical Methods in Engineering 48 (12) pp 1741– (2000) · Zbl 0989.74066 [26] Legrain, Stability of incompressible formulations enriched with x-FEM, Computer Methods in Applied Mechanics and Engineering 197 (21â24) pp 1835– (2008) · Zbl 1194.74426 [27] Behr, Stabilized finite element methods for the velocityâpressureâstress formulation of incompressible flows, Computer Methods in Applied Mechanics and Engineering 104 (1) pp 31– (1993) [28] Brooks, Streamline upwind/PetrovâGalerkin formulations for convection dominated flows with particular emphasis on the incompressible NavierâStokes equations, Computer Methods in Applied Mechanics and Engineering 32 (1â3) pp 199– (1982) · Zbl 0497.76041 [29] Hughes, Encyclopedia of Computational Mechanics 3 (2004) [30] Gresho, Incompressible Flow and the Finite Element Method, Volume 2: Isothermal Laminar Flow (2000) · Zbl 0988.76005 [31] Engels, Numerical Quadrature and Cubature (1980) [32] Peano, GaussâLobatto integration of high precision tetrahedral elements, International Journal for Numerical Methods in Engineering 18 (2) pp 311– (1982) · Zbl 0473.73083 [33] Mayer, Interface handling for three-dimensional higher-order XFEM computations in fluidâstructure interaction, International Journal for Numerical Methods in Engineering 554 pp 4545– (2009) · Zbl 1171.74447 [34] Hamel, SpiralfÃ§rmige Bewegungen zÃ\currencyher FlÃ$$\tfrac14$$ssigkeiten, Jahresbericht der Deutschen Mathematiker-Vereinigung 25 pp 34– (1916) [35] Jeffery, The two-dimensional steady motion of a viscous fluid, Philosophical Magazine 29 pp 455– (1915) · JFM 45.1088.01 · doi:10.1080/14786440408635327 [36] Rosenhead, The steady two-dimensional radial flow of viscous fluid between two inclined plane walls, Proceedings of the Royal Society of London, Series A, Mathematical and Physical Sciences 175 (963) pp 436– (1940) · JFM 66.1385.02 [37] Corless, ISSAC ’07: Proceedings of the 2007 International Symposium on Symbolic and Algebraic Computation (2007) [38] Bagheri, Implementing, using high-order finite element methods, Finite Elements in Analysis and Design 16 (3â4) pp 175– (1994) · Zbl 0812.76042 [39] SchÃ\currencyfer, Benchmark Computations of Laminar Flow Around a Cylinder pp 547– (1996) [40] Gerstenberger, Enhancement of fixed-grid methods towards complex fluidâstructure interaction applications, International Journal for Numerical Methods in Fluids 57 (9) pp 1227– (2008) · Zbl 1338.74038 [41] Wall, Fluidâstructure interaction approaches on fixed grids based on two different domain decomposition ideas, International Journal of Computational Fluid Dynamics 22 (6) pp 411– (2008) · Zbl 1184.76732
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.