Modeling and simulation of keyhole-based welding as multi-domain problem using the extended finite element method

Abstract In this article, we demonstrate the flexibility of a multi-domain approach combined with the extended finite element method by addressing the modeling and simulation of keyhole-based welding. The welding process is modeled by the heat equation where the keyhole geometry and the interface separating molten and solid area are represented by two independent level set functions, separating the domain into three time-dependent subdomains. The keyhole shape is computed by an analytical approach based on the energy balance at the keyhole wall and its shape is assumed to be fixed. The solid-liquid interface is considered as free boundary whose evolution is described by the two-phase Stefan problem. The coupled problem including the two discontinuities is solved using a multi-domain XFEM implementation. The simulated results are shown together with experimental data on different welded materials.

[1]  Anders Logg,et al.  A compiler for variational forms , 2006, TOMS.

[2]  R. Nochetto,et al.  Theory of adaptive finite element methods: An introduction , 2009 .

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

[4]  Robert C. Kirby,et al.  Algorithm 839: FIAT, a new paradigm for computing finite element basis functions , 2004, TOMS.

[5]  Utkarsh Ayachit,et al.  The ParaView Guide: A Parallel Visualization Application , 2015 .

[6]  J. Goldak,et al.  A new finite element model for welding heat sources , 1984 .

[7]  Anders Logg,et al.  DOLFIN: Automated finite element computing , 2010, TOMS.

[8]  S. Osher,et al.  Regular Article: A PDE-Based Fast Local Level Set Method , 1999 .

[9]  Garth N. Wells,et al.  Automated Modelling of Evolving Discontinuities , 2009, Algorithms.

[11]  C. W. Hirt,et al.  An Arbitrary Lagrangian-Eulerian Computing Method for All Flow Speeds , 1997 .

[12]  Anders Logg,et al.  Unified form language: A domain-specific language for weak formulations of partial differential equations , 2012, TOMS.

[13]  Garth N. Wells,et al.  Optimizations for quadrature representations of finite element tensors through automated code generation , 2011, TOMS.

[14]  Min Li,et al.  Numerical Simulation of Stefan Problem Coupled with Mass Transport in a Binary System Through XFEM/Level Set Method , 2018, J. Sci. Comput..

[15]  J. C. Jaeger,et al.  Conduction of Heat in Solids , 1952 .

[16]  Anders Logg,et al.  Unified framework for finite element assembly , 2009, Int. J. Comput. Sci. Eng..

[17]  J A Sethian,et al.  A fast marching level set method for monotonically advancing fronts. , 1996, Proceedings of the National Academy of Sciences of the United States of America.

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

[19]  Roland Herzog,et al.  Implementation of an X-FEM Solver for the Classical Two-Phase Stefan Problem , 2012, J. Sci. Comput..

[20]  Anders Logg,et al.  Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book , 2012 .

[21]  L. I. Rubinshteĭn.  The Stefan Problem , 1971 .

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

[23]  Komkamol Chongbunwatana,et al.  Simulation of vapour keyhole and weld pool dynamics during laser beam welding , 2014, Prod. Eng..

[24]  A. Kaplan.  A MODEL OF DEEP PENETRATION LASER WELDING BASED ON CALCULATION OF THE KEYHOLE PROFILE , 1994 .

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

[26]  C. J. Smithells,et al.  Smithells metals reference book , 1949 .

[27]  Baskar Ganapathysubramanian,et al.  Modelling dendritic solidification with melt convection using the extended finite element method , 2006, J. Comput. Phys..

[28]  G. C. Danielson,et al.  Thermal Properties of Armco Iron , 1967 .

[29]  A. Reusken,et al.  Numerical Methods for Two-phase Incompressible Flows , 2011 .

[30]  John E. Dolbow,et al.  Solving thermal and phase change problems with the eXtended finite element method , 2002 .

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

[32]  Thomas-Peter Fries,et al.  Higher‐order XFEM for curved strong and weak discontinuities , 2009 .

[33]  E. Bänsch,et al.  An ALE finite element method for a coupled Stefan problem and Navier–Stokes equations with free capillary surface , 2013 .