The shifted boundary method for embedded domain computations. Part I: Poisson and Stokes problems

Main, A.; Scovazzi, G.

doi:10.1016/j.jcp.2017.10.026

The shifted boundary method for embedded domain computations. Part I: Poisson and Stokes problems

Journal Article · Mon Oct 23 00:00:00 EDT 2017 · Journal of Computational Physics

DOI:https://doi.org/10.1016/j.jcp.2017.10.026· OSTI ID:1611740

Main, A. ^[1]; ^[2]

Duke Univ., Durham, NC (United States); DOE/OSTI
Duke Univ., Durham, NC (United States)

We propose a new finite element method for embedded domain computations, which falls in the category of surrogate/approximate boundary algorithms. The key feature of the proposed approach is the idea of shifting the location where boundary conditions are applied from the true to the surrogate boundary, and to appropriately modify the shifted boundary conditions, enforced weakly, in order to preserve optimal convergence rates of the numerical solution. This process yields a method which, in our view, is simple, efficient, and also robust, since it is not affected by the small-cut-cell problem. Although general in nature, here we apply this new concept to the Poisson and Stokes problems. We present in particular the full analysis of stability and convergence for the case of the Poisson operator, and numerical tests for both the Poisson and Stokes equations, for geometries of progressively higher complexity.

View Accepted Manuscript (DOE)

Research Organization:: Duke University, Durham, NC (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)

Grant/Contract Number:: SC0012169

OSTI ID:: 1611740

Journal Information:: Journal of Computational Physics, Journal Name: Journal of Computational Physics Journal Issue: C Vol. 372; ISSN 0021-9991

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

References (31)

Computational algorithms for tracking dynamic fluid-structure interfaces in embedded boundary methods: INTERFACE TRACKING IN EMBEDDED BOUNDARY METHODS Wang, K.; Grétarsson, J.; Main, A. International Journal for Numerical Methods in Fluids, Vol. 70, Issue 4 https://doi.org/10.1002/fld.3659	journal	April 2012
A fixed-grid b-spline finite element technique for fluid-structure interaction: FIXED-GRID b-SPLINE FINITE ELEMENT TECHNIQUE Rüberg, T.; Cirak, F. International Journal for Numerical Methods in Fluids, Vol. 74, Issue 9 https://doi.org/10.1002/fld.3864	journal	December 2013
A bubble-stabilized finite element method for Dirichlet constraints on embedded interfaces Mourad, Hashem M.; Dolbow, John; Harari, Isaac International Journal for Numerical Methods in Engineering, Vol. 69, Issue 4 https://doi.org/10.1002/nme.1788	journal	January 2006
An efficient finite element method for embedded interface problems Dolbow, John; Harari, Isaac International Journal for Numerical Methods in Engineering, Vol. 78, Issue 2 https://doi.org/10.1002/nme.2486	journal	April 2009
A face-oriented stabilized Nitsche-type extended variational multiscale method for incompressible two-phase flow: NITSCHE-TYPE EXTENDED VARIATIONAL MULTISCALE METHOD FOR TWO-PHASE FLOW Schott, B.; Rasthofer, U.; Gravemeier, V. International Journal for Numerical Methods in Engineering, Vol. 104, Issue 7 https://doi.org/10.1002/nme.4789	journal	October 2014
A Non-oscillatory Eulerian Approach to Interfaces in Multimaterial Flows (the Ghost Fluid Method) Fedkiw, Ronald P.; Aslam, Tariq; Merriman, Barry Journal of Computational Physics, Vol. 152, Issue 2 https://doi.org/10.1006/jcph.1999.6236	journal	July 1999
Computations of Compressible Multifluids Abgrall, Rémi; Karni, Smadar Journal of Computational Physics, Vol. 169, Issue 2 https://doi.org/10.1006/jcph.2000.6685	journal	May 2001
Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind Nitsche, J. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, Vol. 36, Issue 1 https://doi.org/10.1007/BF02995904	journal	July 1971
Finite cell method: h- and p-extension for embedded domain problems in solid mechanics Parvizian, Jamshid; Düster, Alexander; Rank, Ernst Computational Mechanics, Vol. 41, Issue 1 https://doi.org/10.1007/s00466-007-0173-y	journal	April 2007
Flow patterns around heart valves: A numerical method Peskin, Charles S. Journal of Computational Physics, Vol. 10, Issue 2 https://doi.org/10.1016/0021-9991(72)90065-4	journal	October 1972
A new finite element formulation for computational fluid dynamics: VII. The stokes problem with various well-posed boundary conditions: Symmetric formulations that converge for all velocity/pressure spaces Hughes, Thomas J. R.; Franca, Leopoldo P. Computer Methods in Applied Mechanics and Engineering, Vol. 65, Issue 1 https://doi.org/10.1016/0045-7825(87)90184-8	journal	November 1987
An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems Hansbo, Anita; Hansbo, Peter Computer Methods in Applied Mechanics and Engineering, Vol. 191, Issue 47-48 https://doi.org/10.1016/S0045-7825(02)00524-8	journal	November 2002
A finite element approach for the immersed boundary method Boffi, Daniele; Gastaldi, Lucia Computers & Structures, Vol. 81, Issue 8-11 https://doi.org/10.1016/S0045-7949(02)00404-2	journal	May 2003
The finite cell method for three-dimensional problems of solid mechanics Düster, A.; Parvizian, J.; Yang, Z. Computer Methods in Applied Mechanics and Engineering, Vol. 197, Issue 45-48 https://doi.org/10.1016/j.cma.2008.02.036	journal	August 2008
Subdivision-stabilised immersed b-spline finite elements for moving boundary flows Rüberg, T.; Cirak, F. Computer Methods in Applied Mechanics and Engineering, Vol. 209-212 https://doi.org/10.1016/j.cma.2011.10.007	journal	February 2012
A robust Nitsche’s formulation for interface problems Annavarapu, Chandrasekhar; Hautefeuille, Martin; Dolbow, John E. Computer Methods in Applied Mechanics and Engineering, Vol. 225-228 https://doi.org/10.1016/j.cma.2012.03.008	journal	June 2012
A new face-oriented stabilized XFEM approach for 2D and 3D incompressible Navier–Stokes equations Schott, B.; Wall, W. A. Computer Methods in Applied Mechanics and Engineering, Vol. 276 https://doi.org/10.1016/j.cma.2014.02.014	journal	July 2014
Ghost penalty Burman, Erik Comptes Rendus Mathematique, Vol. 348, Issue 21-22 https://doi.org/10.1016/j.crma.2010.10.006	journal	November 2010
A high-order immersed interface method for simulating unsteady incompressible flows on irregular domains Linnick, Mark N.; Fasel, Hermann F. Journal of Computational Physics, Vol. 204, Issue 1 https://doi.org/10.1016/j.jcp.2004.09.017	journal	March 2005
On the order of accuracy of the immersed boundary method: Higher order convergence rates for sufficiently smooth problems Griffith, Boyce E.; Peskin, Charles S. Journal of Computational Physics, Vol. 208, Issue 1 https://doi.org/10.1016/j.jcp.2005.02.011	journal	September 2005
An adaptive, formally second order accurate version of the immersed boundary method Griffith, Boyce E.; Hornung, Richard D.; McQueen, David M. Journal of Computational Physics, Vol. 223, Issue 1 https://doi.org/10.1016/j.jcp.2006.08.019	journal	April 2007
A higher-order generalized ghost fluid method for the poor for the three-dimensional two-phase flow computation of underwater implosions Farhat, Charbel; Rallu, Arthur; Shankaran, Sriram Journal of Computational Physics, Vol. 227, Issue 16 https://doi.org/10.1016/j.jcp.2008.04.032	journal	August 2008
Prediction of wall-pressure fluctuation in turbulent flows with an immersed boundary method Kang, Seongwon; Iaccarino, Gianluca; Ham, Frank Journal of Computational Physics, Vol. 228, Issue 18 https://doi.org/10.1016/j.jcp.2009.05.036	journal	October 2009
An enhanced FIVER method for multi-material flow problems with second-order convergence rate Main, Alex; Zeng, Xianyi; Avery, Philip Journal of Computational Physics, Vol. 329 https://doi.org/10.1016/j.jcp.2016.10.028	journal	January 2017
Fictitious domain methods using cut elements: III. A stabilized Nitsche method for Stokes’ problem Burman, Erik; Hansbo, Peter ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 48, Issue 3 https://doi.org/10.1051/m2an/2013123	journal	April 2014
Regularity estimates for elliptic boundary value problems in Besov spaces Bacuta, Constantin; Bramble, James H.; Xu, Jinchao Mathematics of Computation, Vol. 72, Issue 244 https://doi.org/10.1090/S0025-5718-02-01502-8	journal	December 2002
An Interior Penalty Finite Element Method with Discontinuous Elements Arnold, Douglas N. SIAM Journal on Numerical Analysis, Vol. 19, Issue 4 https://doi.org/10.1137/0719052	journal	August 1982
Weighted Extended B-Spline Approximation of Dirichlet Problems Höllig, Klaus; Reif, Ulrich; Wipper, Joachim SIAM Journal on Numerical Analysis, Vol. 39, Issue 2 https://doi.org/10.1137/S0036142900373208	journal	January 2001
Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems Arnold, Douglas N.; Brezzi, Franco; Cockburn, Bernardo SIAM Journal on Numerical Analysis, Vol. 39, Issue 5 https://doi.org/10.1137/S0036142901384162	journal	January 2002
Immersed Interface Methods for Stokes Flow with Elastic Boundaries or Surface Tension LeVeque, Randall J.; Li, Zhilin SIAM Journal on Scientific Computing, Vol. 18, Issue 3 https://doi.org/10.1137/S1064827595282532	journal	May 1997
Numerical Stability of the Finite Element Immersed Boundary Method Boffi, Daniele; Gastaldi, Lucia; Heltai, Luca Mathematical Models and Methods in Applied Sciences, Vol. 17, Issue 10 https://doi.org/10.1142/S0218202507002352	journal	October 2007

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Random geometries for optimal control PDE problems based on fictitious domain FEMS and cut elements Aretaki, Aikaterini; Karatzas, Efthymios N. arXiv https://doi.org/10.48550/arxiv.2003.00352	preprint	January 2020
Cut finite element error estimates for a class of nonlinear elliptic PDEs Katsouleas, Georgios; Karatzas, Efthymios N.; Travlopanos, Fotios arXiv https://doi.org/10.48550/arxiv.2003.06489	preprint	January 2020
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