High-order Gaussian quadrature in X-FEM with the Lagrange-multiplier for fluid–structure coupling. (English) Zbl 1427.76145
Summary: This paper presents an alternative choice for the construction of integration cells and an additional Lagrange-multiplier mesh in computations using the eXtended finite element method (X-FEM) with the Lagrange-multiplier. The proposed computational target is flow simulations with the Dirichlet boundary conditions on a non-boundary-fitted mesh. The methods studied in this paper make use of a high-order Gaussian quadrature to relax the complexity in implementing the X-FEM with the Lagrange-multiplier. The first of these is the straightforward employment of the Gaussian quadrature as a numerical integration scheme for enriched fluid elements. The second makes use of free Lagrange-multipliers that are not placed at crossing points between fluid elements and the Dirichlet boundary. This method considers the scalability of fluid–structure and fluid–thin object interactions. A unique benchmark problem in which a computational fluid domain is partitioned into two physical domains with the Lagrange-multipliers is introduced to validate the proposed method. Numerical results show that this method is applicable in practice with acceptable numerical accuracy.
MSC:
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 65M50 | Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs |
| 74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |
Keywords:
X-FEM; discontinuity; Dirichlet boundary condition; weak imposition; Lagrange-multiplier; numerical integration; fluid; structure interactionsReferences:
| [1] | Tezduyar, Space-time finite element techniques for computation of fluid-structure interactions, Computer Methods in Applied Mechanics and Engineering 195 (17-18) pp 2002– (2006) |
| [2] | Tezduyar, Interface-tracking and interface-capturing techniques for finite element computation of moving boundaries and interfaces, Computer Methods in Applied Mechanics and Engineering 195 pp 2983– (2006) · Zbl 1176.76076 |
| [3] | Hughes, Lagrangian-Eulerian finite element formulation for incompressible viscous flows, Computer Methods in Applied Mechanics and Engineering 29 pp 329– (1981) · Zbl 0482.76039 |
| [4] | Belytschko, Finite element method with user-controlled meshes for fluid-structure interactions, Computer Methods in Applied Mechanics and Engineering 33 pp 689– (1982) |
| [5] | Singh, Coupled shell-fluid interaction problems with degenerate shell and three-dimensional fluid elements, Computers and Structures 38 (5) pp 515– (1991) · Zbl 0825.73502 |
| [6] | Nitikitpaiboon, An arbitrary Lagrangian-Eulerian velocity potential formulation for fluid-structure interaction, Computers and Structures 47 (4) pp 871– (1993) · Zbl 0800.73296 |
| [7] | Tezduyar, A new strategy for finite element computations involving moving boundaries and interfaces-the deforming-spatial-domain/space-time procedure: I. The concept and the preliminary numerical tests, Computer Methods in Applied Mechanics and Engineering 94 pp 339– (1992) · Zbl 0745.76044 |
| [8] | Tezduyar, A new strategy for finite element computations involving moving boundaries and interfaces-the deforming-spatial-domain/space-time procedure: II. Computations of free-surface flows, two-liquid flows, and flows with drifting cylinders, Computer Methods in Applied Mechanics and Engineering 94 pp 353– (1992) · Zbl 0745.76045 |
| [9] | Tezduyar, Computation of moving boundaries and interfaces and stabilization parameters, International Journal for Numerical Methods in Fluids 43 pp 555– (2003) · Zbl 1201.76123 |
| [10] | Zhang, Analysis of fluid-structure interaction problems with structural buckling and large domain changes by ALE finite element method, Computer Methods in Applied Mechanics and Engineering 190 pp 6341– (2001) · Zbl 1015.74064 |
| [11] | Kuhl, An arbitrary Lagrangian Eulerian finite-element approach for fluid-structure interaction phenomena, International Journal for Numerical Methods in Engineering 57 pp 117– (2003) · Zbl 1062.74617 |
| [12] | Watanabe, Multiphysics simulation of left ventricular filling dynamics using fluid-structure interaction finite element method, Biophysical Journal 87 (3) pp 2074– (2004) |
| [13] | Sawada, Fluid-structure interaction analysis of the two-dimensional flag-in-wind problem by an interface-tracking ALE finite element method, Computers and Fluids 36 pp 136– (2007) · Zbl 1181.76099 |
| [14] | Peskin, Flow patterns around heart valves: a numerical method, Journal of Computational Physics 10 pp 252– (1972) · Zbl 0244.92002 |
| [15] | Peskin, The immersed boundary method, Acta Numerica 11 pp 479– (2002) · Zbl 1123.74309 |
| [16] | Zhu, Simulation of a flapping flexible filament in a flowing soap film by the immersed boundary method, Journal of Computational Physics 179 pp 452– (2002) · Zbl 1130.76406 |
| [17] | Boffi, A finite element approach to the immersed boundary method, Computers and Structures 81 pp 491– (2003) |
| [18] | Wang, Extended immersed boundary method using FEM and RKPM, Computer Methods in Applied Mechanics and Engineering 193 pp 1305– (2004) · Zbl 1060.74676 |
| [19] | Zhang, Immersed finite element method, Computer Methods in Applied Mechanics and Engineering 193 pp 2051– (2004) · Zbl 1067.76576 |
| [20] | Wang, The immersed/fictitious element method for fluid-structure interaction: volumetric consistency, compressibility and thin members, International Journal for Numerical Methods in Engineering 74 pp 32– (2008) · Zbl 1159.74437 |
| [21] | Glowinski, Distributed Lagrange multiplier methods for incompressible viscous flow around moving rigid bodies, Computer Methods in Applied Mechanics and Engineering 151 pp 181– (1998) · Zbl 0916.76052 |
| [22] | Glowinski, A fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies: application to particulate flow, Journal of Computational Physics 169 pp 363– (2001) · Zbl 1047.76097 |
| [23] | Hart, A two-dimensional fluid-structure interaction model of the aortic valve, Journal of Biomechanics 33 pp 1079– (2000) |
| [24] | Yu, A DLM/FD method for fluid/flexible-body interactions, Journal of Computational Physics 207 (1) pp 1– (2005) |
| [25] | Li, Immersed interface methods for moving interface problems, Numerical Algorithms 14 (4) pp 269– (1997) · Zbl 0886.65096 |
| [26] | Li, An overview of the immersed interface method and its applications, Taiwanese Journal of Mathematics 7 (1) pp 1– (2003) · Zbl 1028.65108 |
| [27] | Wagner, The extended finite element method for rigid particles in Stokes flow, International Journal for Numerical Methods in Engineering 51 pp 293– (2001) · Zbl 0998.76054 |
| [28] | Legay, An Eulerian-Lagrangian method for fluid-structure interaction based on level sets, Computer Methods in Applied Mechanics and Engineering 195 pp 2070– (2006) · Zbl 1119.74053 |
| [29] | Zilian, The enriched space-time finite element method (EST) for simultaneous solution of fluid-structure interaction, International Journal for Numerical Methods in Engineering 75 pp 305– (2008) · Zbl 1195.74212 |
| [30] | Gerstenberger, An extended finite element method/Lagrange multiplier based approach for fluid-structure interaction, Computer Methods in Applied Mechanics and Engineering 197 pp 1699– (2008) · Zbl 1194.76117 |
| [31] | Mayer, Interface handling for three-dimensional higher-order XFEM-computations in fluid-structure interaction, International Journal for Numerical Methods in Engineering 79 (7) pp 846– (2009) · Zbl 1171.74447 |
| [32] | Wang, Fluid-structure interaction by the discontinuous-Galerkin method for large deformations, International Journal for Numerical Methods in Engineering 77 (1) pp 30– (2009) · Zbl 1195.74204 |
| [33] | Tezduyar, The multi-domain method for computation of the aerodynamics of a parachute crossing the far wake of an aircraft, Computer Methods in Applied Mechanics and Engineering 191 pp 705– (2001) · Zbl 1113.76406 |
| [34] | Tezduyar, Finite element methods for flow problems with moving boundaries and interfaces, Archives of Computational Methods in Engineering 8 pp 83– (2001) |
| [35] | Loon, A combined fictitious domain/adaptive meshing method for fluid-structure interaction in heart valves, International Journal for Numerical Methods in Engineering 46 pp 533– (2004) · Zbl 1060.76582 |
| [36] | Shi, Distributed Lagrange multiplier/fictitious domain method in the framework of lattice Boltzmann method for fluid-structure interactions, Journal of Computational Physics 206 (1) pp 81– (2005) · Zbl 1087.76543 |
| [37] | Akin, Computation of flow problems with the mixed interface-tracking/interface-capturing technique (MITICT), Computers and Fluids 36 pp 2– (2007) · Zbl 1181.76105 |
| [38] | Cruchaga, A numerical model based on the mixed interface-tracking/interface-capturing technique (MITICT) for flows with fluid-solid and fluid-fluid interfaces, International Journal for Numerical Methods in Fluids 54 pp 1021– (2007) · Zbl 1128.76033 |
| [39] | Wall, Fluid-structure interaction approaches on fixed grids based on two different decomposition ideas, International Journal of Computational Fluid Dynamics 22 (6) pp 411– (2008) · Zbl 1184.76732 |
| [40] | Gerstenberger, Enhancement of fixed-grid methods towards complex fluid-structure interaction applications, International Journal for Numerical Methods in Fluids 57 (9) pp 1227– (2008) · Zbl 1338.74038 |
| [41] | Tezduyar, Parallel finite-element computation of 3D flows, Computer 26 pp 27– (1993) |
| [42] | Johnson, Mesh update strategies in parallel finite element computations of flow problems with moving boundaries and interfaces, Computer Methods in Applied Mechanics and Engineering 119 pp 73– (1994) · Zbl 0848.76036 |
| [43] | Johnson, Advanced mesh generation and update methods for 3D flow simulations, Computational Mechanics 23 (2) pp 130– (1999) · Zbl 0949.76049 |
| [44] | Stein, Mesh moving techniques for fluid-structure interactions with large displacements, Journal of Applied Mechanics 70 pp 58– (2003) · Zbl 1110.74689 |
| [45] | Moës, 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 |
| [46] | Barbosa, The finite element method with Lagrange multipliers on the boundary: circumventing the Babuska-Brezzi condition, Computer Methods in Applied Mechanics and Engineering 85 (1) pp 109– (1991) · Zbl 0764.73077 |
| [47] | Flemisch, Stable Lagrange multipliers for quadrilateral meshes of curved interfaces in 3D, Computer Methods in Applied Mechanics and Engineering 196 (8) pp 1589– (2007) · Zbl 1173.74416 |
| [48] | Cho, Boundary locking induced by penalty enforcement of essential boundary conditions in mesh-free methods, Computer Methods in Applied Mechanics and Engineering 197 (13-16) pp 1167– (2008) |
| [49] | Moës, A finite element method for crack growth without remeshing, International Journal for Numerical Methods in Engineering 46 (1) pp 131– (1999) · Zbl 0955.74066 |
| [50] | Belytschko, Arbitrary discontinuities in finite elements, International Journal for Numerical Methods in Engineering 50 (4) pp 993– (2001) · Zbl 0981.74062 |
| [51] | Sukumar, Modeling holes and inclusions by level sets in the extended finite-element method, Computer Methods in Applied Mechanics and Engineering 190 pp 6183– (2001) · Zbl 1029.74049 |
| [52] | Wagner, Particulate flow simulations using lubrication theory solution enrichment, International Journal for Numerical Methods in Engineering 56 pp 1261– (2003) · Zbl 1032.76037 |
| [53] | Moës, A computational approach to handle complex microstructure geometries, Computer Methods in Applied Mechanics and Engineering 192 pp 3163– (2003) · Zbl 1054.74056 |
| [54] | 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 |
| [55] | Chessa, Arbitrary discontinuities in space-time finite elements by level sets and X-FEM, International Journal for Numerical Methods in Engineering 61 pp 2595– (2004) · Zbl 1077.76039 |
| [56] | Sawada, Extended-FEM for the solid-fluid mixture two-scale problems with BCC and FCC microstructures, Interaction and Multiscale Mechanics: An International Journal 2 (1) pp 45– (2009) · doi:10.12989/imm.2009.2.1.045 |
| [57] | Brooks, Streamline upwind/Petrov-Galerkin formulation for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Computer Methods in Applied Mechanics and Engineering 32 pp 199– (1982) · Zbl 0497.76041 |
| [58] | Tezduyar, Stabilized finite element formulations for incompressible flow computations, Advances in Applied Mechanics 28 pp 1– (1992) · Zbl 0747.76069 |
| [59] | Tezduyar, Incompressible flow computations with stabilized bilinear and linear equal-order-interpolation velocity-pressure elements, Computer Methods in Applied Mechanics and Engineering 95 pp 221– (1992) · Zbl 0756.76048 |
| [60] | Liu, Generalized multiple scale reproducing kernel particle methods, Computer Methods in Applied Mechanics and Engineering 136 pp 91– (1996) · Zbl 0896.76069 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
