The DUNE Framework: Basic Concepts and Recent Developments

This paper presents the basic concepts and the module structure of the Distributed and Unified Numerics Environment and reflects on recent developments and general changes that happened since the release of the first Dune version in 2007 and the main papers describing that state [1, 2]. This discussion is accompanied with a description of various advanced features, such as coupling of domains and cut cells, grid modifications such as adaptation and moving domains, high order discretizations and node level performance, non-smooth multigrid methods, and multiscale methods. A brief discussion on current and future development directions of the framework concludes the paper.

[1]  P. Hansbo,et al.  A finite element method for domain decomposition with non-matching grids , 2003 .

[2]  C. M. Elliott,et al.  Computation of geometric partial differential equations and mean curvature flow , 2005, Acta Numerica.

[3]  Panayot S. Vassilevski,et al.  Almost optimal interior penalty discontinuous approximations of symmetric elliptic problems on non-matching grids , 2003, Numerische Mathematik.

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

[5]  王东东,et al.  Computer Methods in Applied Mechanics and Engineering , 2004 .

[6]  Andreas Dedner,et al.  A generic grid interface for parallel and adaptive scientific computing. Part I: abstract framework , 2008, Computing.

[7]  T. Hughes Multiscale phenomena: Green's functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods , 1995 .

[8]  Luciano Teresi,et al.  Navier-Stokes equations in a moving domain , 2006 .

[9]  KretzMatthias,et al.  Vc: A C++ library for explicit vectorization , 2012 .

[10]  Peter Bastian,et al.  An HPC Perspective on Generative Programming , 2019, 2019 IEEE/ACM 14th International Workshop on Software Engineering for Science (SE4Science).

[11]  Stefan Turek,et al.  Advances Concerning Multiscale Methods and Uncertainty Quantification in EXA-DUNE , 2016, Software for Exascale Computing.

[12]  P. Alam,et al.  R , 1823, The Herodotus Encyclopedia.

[13]  Markus Blatt A Parallel Algebraic Multigrid Method for Elliptic Problems with Highly Discontinuous Coefficients , 2010 .

[14]  Thomas Y. Hou,et al.  A Multiscale Finite Element Method for Elliptic Problems in Composite Materials and Porous Media , 1997 .

[15]  Daniel Kressner,et al.  Numerical Mathematics and Advanced Applications - ENUMATH 2013 - Proceedings of ENUMATH 2013, the 10th European Conference on Numerical Mathematics and Advanced Applications, Lausanne, Switzerland, August 2013 , 2015, ENUMATH.

[16]  René Milk,et al.  DUNE-multiscale (LSSC '15) , 2015 .

[17]  Martin Jakob A New Cement to Glue Nonconforming Grids with Robin Interface Conditions : The Finite Element Case , 2016 .

[18]  Benedikt Oswald,et al.  Dune-CurvilinearGrid: Parallel Dune Grid Manager for Unstructured Tetrahedral Curvilinear Meshes , 2016, ArXiv.

[19]  Martin J. Gander,et al.  Algorithm 932: PANG: Software for nonmatching grid projections in 2D and 3D with linear complexity , 2013, TOMS.

[20]  Peter Bastian,et al.  The Iterative Solver Template Library , 2006, PARA.

[21]  Nicolai Schoch,et al.  HiFlow3 – Technical Report on Release 2.0 , 2017 .

[22]  Christoph Gersbacher The Dune-PrismGrid Module , 2012 .

[23]  Ben Schweizer,et al.  An Adaptive Multiscale Finite Element Method , 2014, Multiscale Model. Simul..

[24]  René Milk,et al.  Extending DUNE: The dune-xt modules , 2016, ArXiv.

[25]  Yalchin Efendiev,et al.  Generalized multiscale finite element methods (GMsFEM) , 2013, J. Comput. Phys..

[26]  T. Hughes,et al.  The variational multiscale method—a paradigm for computational mechanics , 1998 .

[27]  Benjamin S. Kirk,et al.  Library for Parallel Adaptive Mesh Refinement / Coarsening Simulations , 2006 .

[28]  Steffen Müthing,et al.  The interface for functions in the dune-functions module , 2015, ArXiv.

[29]  René Milk,et al.  dune-gdt (LSSC '15) , 2015 .

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

[31]  Wim Vanroose,et al.  Hiding global synchronization latency in the preconditioned Conjugate Gradient algorithm , 2014, Parallel Comput..

[32]  Thomas Hérault,et al.  An evaluation of User-Level Failure Mitigation support in MPI , 2012, Computing.

[33]  Tom Henderson,et al.  A general method for modeling on irregular grids , 2011, Int. J. High Perform. Comput. Appl..

[34]  Ulrike Meier Yang,et al.  On the use of relaxation parameters in hybrid smoothers , 2004, Numer. Linear Algebra Appl..

[35]  Volker Lindenstruth,et al.  Vc: A C++ library for explicit vectorization , 2012, Softw. Pract. Exp..

[36]  E Weinan,et al.  The Heterognous Multiscale Methods , 2003 .

[37]  Steffen Müthing,et al.  Automatic Code Generation for High-performance Discontinuous Galerkin Methods on Modern Architectures , 2018, ACM Trans. Math. Softw..

[38]  M. Larson,et al.  Adaptive Variational Multiscale Methods Based on A Posteriori Error Estimation: Duality Techniques for Elliptic Problems , 2005 .

[39]  Andrew T. T. McRae,et al.  A structure-exploiting numbering algorithm for finite elements on extruded meshes, and its performance evaluation in Firedrake , 2016 .

[40]  Yalchin Efendiev,et al.  An Adaptive Generalized Multiscale Discontinuous Galerkin Method for High-Contrast Flow Problems , 2018, Multiscale Model. Simul..

[41]  Mario Ohlberger Error control based model reduction for multiscale problems , 2015 .

[42]  Lisandro Dalcin,et al.  Parallel distributed computing using Python , 2011 .

[43]  Robert Klöfkorn,et al.  hp–Adaptive Discontinuous Galerkin Methods for Porous Media Flow , 2017 .

[44]  Mario Ohlberger,et al.  True Error Control for the Localized Reduced Basis Method for Parabolic Problems , 2016, 1606.09216.

[45]  G. G. Stokes "J." , 1890, The New Yale Book of Quotations.

[46]  Martin J. Gander,et al.  A new cement to glue nonconforming grids with Robin interface conditions: the finite element case. Domain decomposition methods in science and engineering , 2005 .

[47]  Peter Bastian,et al.  On the generic parallelisation of iterative solvers for the finite element method , 2008, Int. J. Comput. Sci. Eng..

[48]  Oliver Sander,et al.  Solving primal plasticity increment problems in the time of a single predictor–corrector iteration , 2017, Computational Mechanics.

[49]  Andreas Dedner,et al.  On discontinuous Galerkin approach for atmospheric flow in the mesoscale with and without moisture , 2014 .

[50]  P. Henning,et al.  A localized orthogonal decomposition method for semi-linear elliptic problems , 2012, 1211.3551.

[51]  Robert Scheichl,et al.  Algebraic multigrid for discontinuous Galerkin discretizations of heterogeneous elliptic problems , 2012, Numer. Linear Algebra Appl..

[52]  D. Barkley A model for fast computer simulation of waves in excitable media , 1991 .

[53]  Steffen Müthing,et al.  Function space bases in the dune-functions module , 2018, ArXiv.

[54]  Oliver Sander,et al.  Truncated Nonsmooth Newton Multigrid Methods for Block-Separable Minimization Problems , 2017 .

[55]  E Weinan,et al.  The heterogeneous multiscale method* , 2012, Acta Numerica.

[56]  Howard C. Elman,et al.  Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics , 2014 .

[57]  Ulrich Rüde,et al.  A Massively Parallel Multigrid Method for Finite Elements , 2006, Computing in Science & Engineering.

[58]  Timothy A. Davis,et al.  Algorithm 832: UMFPACK V4.3---an unsymmetric-pattern multifrontal method , 2004, TOMS.

[59]  Oliver Sander,et al.  The dune-subgrid module and some applications , 2009, Computing.

[60]  Andreas Dedner,et al.  Python Framework for HP Adaptive Discontinuous Galerkin Method for Two Phase Flow in Porous Media , 2018, Applied Mathematical Modelling.

[61]  Matthias Kretz,et al.  Extending C++ for explicit data-parallel programming via SIMD vector types , 2015 .

[62]  Chunxiao Wu,et al.  Multigrid Methods for Obstacle Problems , 2013 .

[63]  Steffen Müthing,et al.  High-performance Implementation of Matrix-free High-order Discontinuous Galerkin Methods , 2017, ArXiv.

[64]  Michael J. Holst,et al.  The Implementation of the Colored Abstract Simplicial Complex and Its Application to Mesh Generation , 2019, ACM Trans. Math. Softw..

[65]  Robert Klöfkorn,et al.  Distributed Newest Vertex Bisection , 2016, J. Parallel Distributed Comput..

[66]  Andreas Dedner,et al.  The Distributed and Unified Numerics Environment,Version 2.4 , 2016 .

[67]  Gabriel Wittum,et al.  UG 4: A novel flexible software system for simulating PDE based models on high performance computers , 2013, Comput. Vis. Sci..

[68]  Bernard Haasdonk,et al.  THE LOCALIZED REDUCED BASIS MULTISCALE METHOD , 2015 .

[69]  Fred P. Brooks,et al.  The Mythical Man-Month , 1975, Reliable Software.

[70]  J. Henson,et al.  Plasticity , 2010, Neurology.

[71]  Peter Hansbo,et al.  CutFEM: Discretizing geometry and partial differential equations , 2015 .

[72]  J. Oden,et al.  A Posteriori Error Estimation in Finite Element Analysis , 2000 .

[73]  Oliver Sander,et al.  Infrastructure for the Coupling of Dune Grids , 2010 .

[74]  Robert Klöfkorn,et al.  Efficient matrix-free implementation of discontinuous Galerkin methods for compressible flow problems , 2015 .

[75]  Carsten Burstedde,et al.  p4est: Scalable Algorithms for Parallel Adaptive Mesh Refinement on Forests of Octrees , 2011, SIAM J. Sci. Comput..

[76]  Christian Engwer,et al.  Dune-UDG: A Cut-Cell Framework for Unfitted Discontinuous Galerkin Methods , 2012 .

[77]  Robert Klöfkorn,et al.  A Weak Compatibility Condition for Newest Vertex Bisection in any dimension , 2017, SIAM J. Sci. Comput..

[78]  Gebräuchliche Fertigarzneimittel,et al.  V , 1893, Therapielexikon Neurologie.

[79]  Steffen Müthing,et al.  A stable and high-order accurate discontinuous Galerkin based splitting method for the incompressible Navier-Stokes equations , 2016, J. Comput. Phys..

[80]  Mario Ohlberger,et al.  A Posteriori Error Estimates for the Heterogeneous Multiscale Finite Element Method for Elliptic Homogenization Problems , 2005, Multiscale Model. Simul..

[81]  H. Rentz-Reichert,et al.  UG – A flexible software toolbox for solving partial differential equations , 1997 .

[82]  Christian Wieners,et al.  Numerical approximation of incremental infinitesimal gradient plasticity , 2009 .

[83]  Yalchin Efendiev,et al.  An adaptive GMsFEM for high-contrast flow problems , 2013, J. Comput. Phys..

[84]  C. Engwer,et al.  An unfitted finite element method using discontinuous Galerkin , 2009 .

[85]  Christophe Geuzaine,et al.  Gmsh: A 3‐D finite element mesh generator with built‐in pre‐ and post‐processing facilities , 2009 .

[86]  Andreas Dedner,et al.  Efficient multigrid preconditioners for atmospheric flow simulations at high aspect ratio , 2014, 1408.2981.

[87]  Mario Ohlberger,et al.  Error Control for the Localized Reduced Basis Multiscale Method with Adaptive On-Line Enrichment , 2015, SIAM J. Sci. Comput..

[88]  Gerhard Wellein,et al.  A Unified Sparse Matrix Data Format for Efficient General Sparse Matrix-Vector Multiplication on Modern Processors with Wide SIMD Units , 2013, SIAM J. Sci. Comput..

[89]  P. Henning,et al.  Efficient implementation of the localized orthogonal decomposition method , 2016, Computer Methods in Applied Mechanics and Engineering.

[90]  Mark S. Shephard,et al.  Generic components for petascale adaptive unstructured mesh-based simulations , 2012, Engineering with Computers.

[91]  Assyr Abdulle,et al.  On A Priori Error Analysis of Fully Discrete Heterogeneous Multiscale FEM , 2005, Multiscale Model. Simul..

[92]  P. Alam ‘S’ , 2021, Composites Engineering: An A–Z Guide.

[93]  Andreas Dedner,et al.  A generic grid interface for parallel and adaptive scientific computing. Part II: implementation and tests in DUNE , 2008, Computing.

[94]  Andreas Dedner,et al.  The DUNE-ALUGrid Module , 2014, ArXiv.

[95]  Steffen Müthing,et al.  Matrix-free multigrid block-preconditioners for higher order Discontinuous Galerkin discretisations , 2018, J. Comput. Phys..

[96]  Stefan Turek,et al.  Hardware-Based Efficiency Advances in the EXA-DUNE Project , 2016, Software for Exascale Computing.

[97]  Frederick P. Brooks,et al.  The Mythical Man-Month: Essays on Softw , 1978 .

[98]  Andreas Dedner,et al.  Continuous Finite-Elements on Non-Conforming Grids Using Discontinuous Galerkin Stabilization , 2014 .

[99]  Robert Klöfkorn,et al.  Performance Pitfalls in the Dune Grid Interface , 2012 .

[100]  Dominic Kempf,et al.  System testing in scientific numerical software frameworks using the example of DUNE , 2017 .

[101]  Rainer Helmig,et al.  A New Simulation Framework for Soil–Root Interaction, Evaporation, Root Growth, and Solute Transport , 2018, Vadose Zone Journal.

[102]  Dominik Göddeke,et al.  A High-Level C++ Approach to Manage Local Errors, Asynchrony and Faults in an MPI Application , 2018, 2018 26th Euromicro International Conference on Parallel, Distributed and Network-based Processing (PDP).

[103]  Tsuyoshi Murata,et al.  {m , 1934, ACML.

[104]  Andreas Dedner,et al.  The Dune Python Module , 2018, ArXiv.

[105]  Daniel Peterseim,et al.  Localization of elliptic multiscale problems , 2011, Math. Comput..

[106]  Andrew T. T. McRae,et al.  Firedrake: automating the finite element method by composing abstractions , 2015, ACM Trans. Math. Softw..

[107]  Katharina Kormann,et al.  Fast Matrix-Free Evaluation of Discontinuous Galerkin Finite Element Operators , 2017, ACM Trans. Math. Softw..

[108]  Knut-Andreas Lie,et al.  The Open Porous Media Flow Reservoir Simulator , 2019, Comput. Math. Appl..

[109]  S. Orszag Spectral methods for problems in complex geometries , 1980 .

[110]  Andreas Dedner,et al.  The Distributed and Unified Numerics Environment (DUNE) , 2006 .

[111]  Kunibert G. Siebert,et al.  Design of Adaptive Finite Element Software - The Finite Element Toolbox ALBERTA , 2005, Lecture Notes in Computational Science and Engineering.

[112]  Robert Klöfkorn,et al.  Comparison of linear reconstructions for second-order finite volume schemes on polyhedral grids , 2017, Computational Geosciences.

[113]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[114]  David Wells,et al.  The deal.II library, Version 9.1 , 2019, J. Num. Math..

[115]  Ralf Kornhuber,et al.  Numerical simulation of coarsening in binary solder alloys , 2014 .

[116]  Andreas Klöckner,et al.  Loo.py: transformation-based code generation for GPUs and CPUs , 2014, ARRAY@PLDI.

[117]  Frédéric Hecht,et al.  New development in freefem++ , 2012, J. Num. Math..

[118]  Danna Zhou,et al.  d. , 1840, Microbial pathogenesis.

[119]  P. Alam ‘A’ , 2021, Composites Engineering: An A–Z Guide.

[120]  Christian Engwer,et al.  Geometric Reconstruction of Implicitly Defined Surfaces and Domains with Topological Guarantees , 2017, ACM Trans. Math. Softw..

[121]  Steffen Müthing,et al.  A flexible framework for multi physics and multi domain PDE simulations , 2015 .

[122]  Steffen Müthing,et al.  Concepts for Flexible Parallel Multi-domain Simulations , 2016 .

[123]  Oliver Sander,et al.  The Dune FoamGrid implementation for surface and network grids , 2015, ArXiv.

[124]  P. Alam,et al.  H , 1887, High Explosives, Propellants, Pyrotechnics.

[125]  Santiago Badia,et al.  FEMPAR: An Object-Oriented Parallel Finite Element Framework , 2017, Archives of Computational Methods in Engineering.

[126]  René Milk,et al.  DUNE-mlmc (SPPEXA AnPleMeet '16) , 2015 .

[127]  Axel Voigt,et al.  AMDiS: adaptive multidimensional simulations , 2007 .

[128]  Robert Klöfkorn,et al.  Parallel Adaptive Simulation of PEM Fuel Cells , 2008 .

[129]  Martin Beck,et al.  DuMux 3 - an open-source simulator for solving flow and transport problems in porous media with a focus on model coupling , 2019, Computers & Mathematics with Applications.

[130]  Sebastian Götschel,et al.  Solving Optimal Control Problems with the Kaskade 7 Finite Element Toolbox , 2012 .

[131]  Christian Engwer,et al.  Scalable Hybrid Parallelization Strategies for the DUNE Grid Interface , 2013, ENUMATH.

[132]  Steffen Müthing,et al.  Integrating Multi-threading and Accelerators into DUNE-ISTL , 2013, ENUMATH.

[133]  Aiqing Zhang,et al.  JAUMIN: a programming framework for large-scale numerical simulation on unstructured meshes , 2019, CCF Trans. High Perform. Comput..

[134]  E. Rank,et al.  A multiscale finite-element method , 1997 .

[135]  Bernard Haasdonk,et al.  A Software Framework for Reduced Basis Methods Using Dune-RB and RBmatlab , 2012 .

[136]  Anthony T. Patera,et al.  Domain Decomposition by the Mortar Element Method , 1993 .

[137]  Martin Kronbichler,et al.  Scalability of high-performance PDE solvers , 2020, Int. J. High Perform. Comput. Appl..

[138]  Andreas Dedner,et al.  A Generic Stabilization Approach for Higher Order Discontinuous Galerkin Methods for Convection Dominated Problems , 2011, J. Sci. Comput..

[139]  Thomas Hérault,et al.  Failure Detection and Propagation in HPC systems , 2016, SC16: International Conference for High Performance Computing, Networking, Storage and Analysis.

[140]  Stefan Turek,et al.  EXA-DUNE: Flexible PDE Solvers, Numerical Methods and Applications , 2014, Euro-Par Workshops.