A theoretical and numerical analysis of a Dirichlet-Neumann domain decomposition method for diffusion problems in heterogeneous media

Problems with localized nonhomogeneous material properties present well-known challenges for numerical simulations. In particular, such problems may feature large differences in length scales, causing difficulties with meshing and preconditioning. These difficulties are increased if the region of localized dynamics changes in time. Overlapping domain decomposition methods, which split the problem at the continuous level, show promise due to their ease of implementation and computational efficiency. Accordingly, the present work aims to further develop the mathematical theory of such methods at both the continuous and discrete levels. For the continuous formulation of the problem, we provide a full convergence analysis. For the discrete problem, we show how the described method may be interpreted as a Gauss-Seidel scheme or as a Neumann series approximation, establishing a convergence criterion in terms of the spectral radius of the system. We then provide a spectral scaling argument and provide numerical evidence for its justification.

[1]  P. Michaleris,et al.  A Line Heat Input Model for Additive Manufacturing , 2015 .

[2]  Iain Todd,et al.  Design for additive manufacturing with site-specific properties in metals and alloys , 2017 .

[3]  F. Brezzi,et al.  A FAMILY OF MIMETIC FINITE DIFFERENCE METHODS ON POLYGONAL AND POLYHEDRAL MESHES , 2005 .

[4]  Luca F. Pavarino,et al.  Overlapping Schwarz Methods for Isogeometric Analysis , 2012, SIAM J. Numer. Anal..

[5]  Long Chen,et al.  An interface-fitted mesh generator and virtual element methods for elliptic interface problems , 2017, J. Comput. Phys..

[6]  L. Giraud,et al.  Algebraic Domain Decomposition Preconditioners , 2006 .

[7]  I. Babuska,et al.  The partition of unity finite element method: Basic theory and applications , 1996 .

[8]  Luca F. Pavarino,et al.  Positive definite balancing Neumann–Neumann preconditioners for nearly incompressible elasticity , 2006, Numerische Mathematik.

[9]  Wing Kam Liu,et al.  Mathematical foundations of the immersed finite element method , 2006 .

[10]  Alfio Quarteroni,et al.  Domain Decomposition Methods for Compressible Flows , 1999 .

[11]  Martin J. Gander,et al.  Comparison of the Dirichlet-Neumann and Optimal Schwarz Method on the Sphere , 2005 .

[12]  R. Glowinski,et al.  A fictitious domain method for Dirichlet problem and applications , 1994 .

[13]  Alessandro Reali,et al.  Suitably graded THB-spline refinement and coarsening: Towards an adaptive isogeometric analysis of additive manufacturing processes , 2018, Computer Methods in Applied Mechanics and Engineering.

[14]  Franco Brezzi,et al.  The Hitchhiker's Guide to the Virtual Element Method , 2014 .

[15]  E. Rank.  Adaptive remeshing and h-p domain decomposition , 1992 .

[16]  Valentino Pediroda,et al.  Fictitious Domain approach with hp-finite element approximation for incompressible fluid flow , 2009, J. Comput. Phys..

[17]  Barry Smith,et al.  Domain Decomposition Methods for Partial Differential Equations , 1997 .

[18]  Giancarlo Sangalli,et al.  The method of mothers for non-overlapping non-matching DDM , 2007, Numerische Mathematik.

[19]  T. Belytschko,et al.  A review of extended/generalized finite element methods for material modeling , 2009 .

[20]  Silvia Bertoluzza,et al.  A Fat boundary-type method for localized nonhomogeneous material problems , 2019, ArXiv.

[21]  Max D. Gunzburger,et al.  Solution of elliptic partial differential equations by an optimization-based domain decomposition method , 2000, Appl. Math. Comput..

[22]  A. Nassar,et al.  Physics-Based Multivariable Modeling and Feedback Linearization Control of Melt-Pool Geometry and Temperature in Directed Energy Deposition , 2017 .

[23]  Peter Hansbo,et al.  Hybridized CutFEM for Elliptic Interface Problems , 2018, SIAM J. Sci. Comput..

[24]  C. Peskin.  Numerical analysis of blood flow in the heart , 1977 .

[25]  R. Poprawe,et al.  Laser additive manufacturing of metallic components: materials, processes and mechanisms , 2012 .

[26]  Ernst Rank,et al.  Applying the hp–d version of the FEM to locally enhance dimensionally reduced models , 2007 .

[27]  K. Lipnikov,et al.  The nonconforming virtual element method , 2014, 1405.3741.

[28]  Luca F. Pavarino,et al.  Robust BDDC Preconditioners for Reissner-Mindlin Plate Bending Problems and MITC Elements , 2010, SIAM J. Numer. Anal..

[29]  Paul Steinmann,et al.  Thermomechanical finite element simulations of selective electron beam melting processes: performance considerations , 2014 .

[30]  Andrea Toselli,et al.  Domain decomposition methods : algorithms and theory , 2005 .

[31]  P. Angot,et al.  A Fictitious domain approach with spread interface for elliptic problems with general boundary conditions , 2007 .

[32]  Raducanu Razvan,et al.  MATHEMATICAL MODELS and METHODS in APPLIED SCIENCES , 2012 .

[33]  Ali Gökhan Demir,et al.  Selective laser melting finite element modeling: Validation with high-speed imaging and lack of fusion defects prediction , 2018, Materials & Design.

[34]  R. Codina,et al.  An adaptive Finite Element strategy for the numerical simulation of additive manufacturing processes , 2020, Additive Manufacturing.

[35]  Kenan Kergrene,et al.  Stable Generalized Finite Element Method and associated iterative schemes; application to interface problems , 2016, 1603.08571.

[36]  Janet Peterson,et al.  An optimization based domain decomposition method for partial differential equations , 1999 .

[37]  Robert H. Halstead,et al.  Matrix Computations , 2011, Encyclopedia of Parallel Computing.

[38]  P. Moireau,et al.  Sequential parameter estimation for fluid–structure problems: Application to hemodynamics , 2012, International journal for numerical methods in biomedical engineering.

[39]  Bertrand Maury,et al.  A Fat Boundary Method for the Poisson Problem in a Domain with Holes , 2002, J. Sci. Comput..

[40]  Ernst Rank,et al.  The hp‐d‐adaptive finite cell method for geometrically nonlinear problems of solid mechanics , 2012 .

[41]  Ernst Rank,et al.  The multi-level hp-method for three-dimensional problems: Dynamically changing high-order mesh refinement with arbitrary hanging nodes , 2016 .

[42]  Numerical solution of additive manufacturing problems using a two‐level method , 2021 .

[43]  B. Stucker,et al.  A Generalized Feed Forward Dynamic Adaptive Mesh Refinement and Derefinement Finite Element Framework for Metal Laser Sintering—Part I: Formulation and Algorithm Development , 2015 .

[44]  Nathalie Labonnote,et al.  Additive construction: State-of-the-art, challenges and opportunities , 2016 .

[45]  R. LeVeque,et al.  A comparison of the extended finite element method with the immersed interface method for elliptic equations with discontinuous coefficients and singular sources , 2006 .

[46]  Luca F. Pavarino,et al.  Isogeometric Schwarz preconditioners for linear elasticity systems , 2013 .

[47]  T. Zohdi,et al.  Numerical simulation of Laser Fusion Additive Manufacturing processes using the SPH method , 2018, Computer Methods in Applied Mechanics and Engineering.

[48]  Alexander Düster,et al.  Local enrichment of the finite cell method for problems with material interfaces , 2013 .

[49]  P. Michaleris,et al.  Numerical verification of an Octree mesh coarsening strategy for simulating additive manufacturing processes , 2019 .

[50]  R. Glowinski,et al.  A distributed Lagrange multiplier/fictitious domain method for particulate flows , 1999 .

[51]  Xiaoze Du,et al.  Finite element analysis of thermal behavior of metal powder during selective laser melting , 2016 .

[52]  Zi-Cai Li,et al.  Schwarz Alternating Method , 1998 .

[53]  K. Mills.  Recommended Values of Thermophysical Properties for Selected Commercial Alloys , 2001 .

[54]  Ernst Rank,et al.  A hierarchical computational model for moving thermal loads and phase changes with applications to selective laser melting , 2017, Comput. Math. Appl..

[55]  C. Kamath,et al.  Laser powder bed fusion additive manufacturing of metals; physics, computational, and materials challenges , 2015 .

[56]  Lourenço Beirão da Veiga,et al.  Curvilinear virtual elements for contact mechanics , 2020 .

[57]  Silvia Bertoluzza,et al.  Analysis of the fully discrete fat boundary method , 2011, Numerische Mathematik.

[58]  F. Auricchio,et al.  On a fictitious domain method with distributed Lagrange multiplier for interface problems , 2015 .

[59]  Brent Stucker,et al.  A Generalized Feed-Forward Dynamic Adaptive Mesh Refinement and Derefinement Finite-Element Framework for Metal Laser Sintering—Part II: Nonlinear Thermal Simulations and Validations , 2016 .