A semi‐Lagrangean time‐integration approach for extended finite element methods

Many computational problems incorporate discontinuities that evolve in time. The eXtendend Finite Element Method (XFEM) is able to represent discontinuities sharply on fixed arbitrary meshes, but numerical difficulties arise if these discontinuities move in time. We point out that this issue is crucial for interface problems with strongly discontinuous fields on fixed grids. A method using semi-Lagrangean techniques is proposed to adequately handle time integration based on finite difference schemes in the context of the XFEM. The basic idea is to adapt previous numerical solutions to the current interface position by tracking back virtual Lagrangean particles to their previous positions, where an appropriate solution can be extrapo- lated from a smooth field. Convergence properties of the proposed method in time and space are thoroughly studied for two one-dimensional model problems. Finally, the method is applied to the particularly challenging problem of premixed combustion, where the discontinuity appears at the flame front separating the burnt from the unburnt gases. A two-dimensional and a three-dimensional expanding flame demonstrates that the method is sufficiently accurate to retain the properties of the overall Nitsche-type formulation for interface problems with embedded strong discontinuities.

[1]  Yi Sun,et al.  Heterogeneous Multiscale Methods for Interface Tracking of Combustion Fronts , 2006, Multiscale Model. Simul..

[2]  Ted Belytschko,et al.  Elastic crack growth in finite elements with minimal remeshing , 1999 .

[3]  P. Minev,et al.  A finite element technique for multifluid incompressible flow using Eulerian grids , 2003 .

[4]  A. Staniforth,et al.  Semi-Lagrangian integration schemes for atmospheric models - A review , 1991 .

[5]  Isaac Harari,et al.  An efficient finite element method for embedded interface problems , 2009 .

[6]  Rainald Löhner,et al.  Adaptive Embedded/Immersed Unstructured Grid Techniques , 2007 .

[7]  Andreas Zilian,et al.  The enriched space–time finite element method (EST) for simultaneous solution of fluid–structure interaction , 2008 .

[8]  Douglas N. Arnold,et al.  Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems , 2001, SIAM J. Numer. Anal..

[9]  Zhaoyuan Wang,et al.  A sharp surface tension modeling method for two‐phase incompressible interfacial flows , 2009 .

[10]  Arnold Reusken,et al.  An extended pressure finite element space for two-phase incompressible flows with surface tension , 2007, J. Comput. Phys..

[11]  Ted Belytschko,et al.  Arbitrary discontinuities in space–time finite elements by level sets and X‐FEM , 2004 .

[12]  Guillaume Houzeaux,et al.  The fixed-mesh ALE approach for the numerical approximation of flows in moving domains , 2009, J. Comput. Phys..

[13]  J. Dolbow,et al.  Imposing Dirichlet boundary conditions with Nitsche's method and spline‐based finite elements , 2010 .

[14]  A. Huerta,et al.  Finite Element Methods for Flow Problems , 2003 .

[15]  J. Sethian,et al.  Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations , 1988 .

[16]  Gianluca Iaccarino,et al.  IMMERSED BOUNDARY METHODS , 2005 .

[17]  T. Belytschko,et al.  The extended finite element method (XFEM) for solidification problems , 2002 .

[18]  P. Hansbo,et al.  A Nitsche extended finite element method for incompressible elasticity with discontinuous modulus of elasticity , 2009 .

[19]  J. Nitsche.  Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind , 1971 .

[20]  Mark Sussman,et al.  A sharp interface method for incompressible two-phase flows , 2007, J. Comput. Phys..

[21]  Toward application of conformal decomposition finite elements to non‐colloidal particle suspensions , 2012 .

[22]  Brian T. Helenbrook,et al.  A Numerical Method for Solving Incompressible Flow Problems with a Surface of Discontinuity , 1999 .

[23]  T. Belytschko,et al.  An enriched finite element method and level sets for axisymmetric two‐phase flow with surface tension , 2003 .

[24]  Thomas-Peter Fries,et al.  The extended finite element method for two-phase and free-surface flows: A systematic study , 2011, J. Comput. Phys..

[25]  Volker Gravemeier,et al.  Numerical simulation of premixed combustion using an enriched finite element method , 2009, J. Comput. Phys..

[26]  Wolfgang A. Wall,et al.  An embedded Dirichlet formulation for 3D continua , 2010 .

[27]  David L. Chopp,et al.  A hybrid extended finite element/level set method for modeling phase transformations , 2002 .

[28]  A. Huerta,et al.  Finite Element Methods for Flow Problems , 2003 .

[29]  K. Hanjalic,et al.  Determination of the laminar burning velocity and the Markstein length of powder–air flames , 2002 .

[30]  P. Hansbo,et al.  An unfitted finite element method, based on Nitsche's method, for elliptic interface problems , 2002 .

[31]  A. Huerta,et al.  Arbitrary Lagrangian–Eulerian Methods , 2004 .

[32]  W. Wall,et al.  An eXtended Finite Element Method/Lagrange multiplier based approach for fluid-structure interaction , 2008 .

[33]  W. Wall,et al.  An extended residual-based variational multiscale method for two-phase flow including surface tension , 2011 .

[34]  T. Fries,et al.  On time integration in the XFEM , 2009 .

[35]  John E. Dolbow,et al.  A robust Nitsche’s formulation for interface problems , 2012 .

[36]  Ralf Hiptmair,et al.  Convergence of Lowest Order Semi-Lagrangian Schemes , 2012, Foundations of Computational Mathematics.

[37]  R. Fedkiw,et al.  A boundary condition capturing method for incompressible flame discontinuities , 2001 .

[38]  Paolo Zunino,et al.  Analysis of backward Euler/extended finite element discretization of parabolic problems with moving interfaces , 2013 .

[39]  T. Belytschko,et al.  The extended/generalized finite element method: An overview of the method and its applications , 2010 .

[40]  Wolfgang A. Wall,et al.  Quadrature schemes for arbitrary convex/concave volumes and integration of weak form in enriched partition of unity methods , 2013 .

[41]  Bernard J. Matkowsky,et al.  Flames as gasdynamic discontinuities , 1982, Journal of Fluid Mechanics.

[42]  Thomas J. R. Hughes,et al.  Finite element modeling of blood flow in arteries , 1998 .

[43]  Ronald Fedkiw,et al.  A Boundary Condition Capturing Method for Multiphase Incompressible Flow , 2000, J. Sci. Comput..

[44]  Kenneth E. Jansen,et al.  A stabilized finite element method for the incompressible Navier–Stokes equations using a hierarchical basis , 2001 .

[45]  S. Osher,et al.  A Non-oscillatory Eulerian Approach to Interfaces in Multimaterial Flows (the Ghost Fluid Method) , 1999 .