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An embedded mesh method using piecewise constant multipliers with stabilization: mathematical and numerical aspects. (English) Zbl 1354.65195
Summary: An embedded mesh method using piecewise constant multipliers originally proposed by the first author et al. [“An embedded mesh method in a multiple material ALE”, Comput. Methods Appl. Mech. Eng. 245–246, 273–289 (2012; doi:10.1016/j.cma.2012.07.014)] is analyzed here to determine effects of the pressure stabilization term and small cut cells. The approach is implemented for transient dynamics using the central difference scheme for the time discretization. It is shown that the resulting equations of motion are a stable linear system with a condition number independent of mesh size. Next, it is shown that the constraints and the stabilization terms can be recast as non-proportional damping such that the time integration of the scheme is provably stable with a critical time step computed from the undamped equations of motion. Effects of small cuts are discussed throughout the presentation. A mesh study is conducted to evaluate the effects of the stabilization on the discretization error and conditioning and is used to recommend an optimal value for stabilization scaling parameter. Several nonlinear problems are also analyzed and compared with comparable conforming mesh results. Finally, several demanding problems highlighting the robustness of the proposed approach are shown.

MSC:
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
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[1] Noh, Methods in Computational Physics pp 117– (1964)
[2] Peskin, Flow patters around heart valves: a numerical method, Journal of Computational Physics 10 pp 252– (1972) · Zbl 0244.92002 · doi:10.1016/0021-9991(72)90065-4
[3] Steger, A chimera grid scheme, Advances in Grid Generation, ASME FED-5 1 pp 59– (1983)
[4] Fedkiw, A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method, Journal of Computational Physics 152 pp 457– (1999) · Zbl 0957.76052 · doi:10.1006/jcph.1999.6236
[5] Wardlaw AB Luton JA Renzi JR Kiddy KC McKeown RM The Gemini coupled hydrocode Pomonkey, Maryland 2003
[6] Deiterding, A virtual test facility for the efficient simulation of solid material response under strong shock and detonation wave loading, Engineering with Computers 22 pp 325– (2006) · Zbl 05192772 · doi:10.1007/s00366-006-0043-9
[7] Wang, Algorithms for interface treatment and load computation in embedded boundary methods for fluid and fluid-structure interaction problems, International Journal for Numerical Methods in Fluids 67 pp 1175– (2011) · Zbl 1426.76436 · doi:10.1002/fld.2556
[8] Glowinski, A fictitious domain approach to the direct numerical simulation of incompressible viscous flow pas moving rigid bodies: application to particulate flow, Journal of Computational Physics 169 pp 363– (2001) · Zbl 1047.76097 · doi:10.1006/jcph.2000.6542
[9] BechĂ©t, A stable Lagrange multiplier space for stiff interface conditions within the extended finite element method, International Journal for Numerical Methods in Engineering 78 pp 931– (2009) · Zbl 1183.74259 · doi:10.1002/nme.2515
[10] Lew, A discontinuous-Galerkin based immersed boundary method, International Journal for Numerical Methods in Engineering 76 (4) pp 427– (2008) · Zbl 1195.76258 · doi:10.1002/nme.2312
[11] Sanders, An embedded mesh method for treating overlapping finite element meshes, International Journal for Numerical Methods in Engineering 91 pp 289– (2012) · Zbl 1246.74064 · doi:10.1002/nme.4265
[12] Gerstenberger, An embedded Dirichlet formulation for 3D continua, International Journal for Numerical Methods in Engineering 82 pp 537– (2010) · Zbl 1188.74056
[13] Hansbo, A finite element method on composite grids based on Nitsche’s method, ESAIM: Mathematical Modeling and Numerical Analysis 37 pp 209– (2003) · Zbl 1031.65128 · doi:10.1051/m2an:2003039
[14] Puso, An embedded mesh method in a multiple material ALE, Computer Methods in Applied Mechanics and Engineering 245 pp 273– (2012) · Zbl 1354.74294 · doi:10.1016/j.cma.2012.07.014
[15] Burman, Fictitious domain finite element methods using cut elements: I. a stabilized lagrange multiplier method, Computer Methods in Applied Mechanics and Engineering 199 pp 2680– (2010) · Zbl 1231.65207 · doi:10.1016/j.cma.2010.05.011
[16] Girault, Error analysis of a fictitious domain method applied to a Dirichlet problem, Japan Journal of Industrial and Applied Mathematics 12 pp 487– (1994) · Zbl 0843.65076 · doi:10.1007/BF03167240
[17] Annavarapu, Stable imposition of stiff constraints in explicit dynamics for embedded finite elements, International Journal for Numerical Methods in Engineering 92 pp 206– (2003) · Zbl 1352.74314 · doi:10.1002/nme.4343
[18] Wang, Fluid-structure interaction by the discontinuous-Galerkin method for large deformations, International Journal for Numerical Methods in Engineering 77 pp 30– (2009) · Zbl 1195.74204 · doi:10.1002/nme.2396
[19] Ern, Theory and Practice of Finite Elements (1992) · Zbl 1059.65103
[20] Wohlmuth, Discretization Methods and Iterative Solvers Based on Domain Decomposition (2001) · Zbl 0966.65097 · doi:10.1007/978-3-642-56767-4
[21] Carpenter, Lagrange constraints for transient finite element surface contact, International Journal for Numerical Methods in Engineering 32 pp 103– (1991) · Zbl 0763.73053 · doi:10.1002/nme.1620320107
[22] Hughes, Generalization of selective integration procedures to anisotropic and non-linear media, International Journal for Numerical Methods in Engineering 15 pp 1413– (1980) · Zbl 0437.73053 · doi:10.1002/nme.1620150914
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