Coupling SUNDIALS with OpenFOAM 6 th OpenFOAM Workshop. E. David Huckaby June 14, 2011
|
|
|
- Hubert Johnson
- 5 years ago
- Views:
Transcription
1 Coupling SUNDIALS with OpenFOAM 6 th OpenFOAM Worshop E. David Hucaby June 14, 2011 EDH-OFW2011
2 Outline Motivation Mathematics Basic Design Eamples Food Web Steady & Unsteady Pitz-Daily 2
3 Motivation Why couple SUNDIALS and OpenFOAM? Improve solution convergence over current segregated solution algorithms SUNDIALS vs. alternatives Comprehensive (write own solver Lightweight Trilinos, PetSC User Familiarity (via. Cantera vs. NITSOL [1] Luo, Baum, Lohner, 1998, JCP 46, [2] Pawlowsi, Shadid, Simonis, Waler, 2006, SIAM Review 48-2, [3] Pernice, Tocci, SIAM J. Sci. Comp. 23-2, [4] Evans, Knoll, Pernice, 2006, JCP 219, [5] Vui, Saghir, Boerstoel, 2000, IJ Num. Meth. Fluids 33, [6] Bocelie, Tang, Denison, Wang, Cremer, Chen, 2003, Worshop on Solution Methods for Large Scale Non-Linear Problems [7] Pernice, Zhou, Waler, 1997, University of Utah Center for High Performance Computing 3
4 SUNDIALS Suite of solvers CVODE ode CVODES add sensitivity analysis KINSOL nonlinear algebraic equations IDA differential algebraic equations (DAE IDAS add sensitivity analysis User supplies the physics function, F F dy = dt F( y = F( y, t 0 ( y, y, t = 0 Features Serial & Parallel, Eact & Ineact Newton Solvers Hindmarsh, Brown, Grant, Lee, Serban, Shumaer, Woodward 4
5 5 Nonlinear Iteration At each time step a DAE or ODE system form nonlinear algebraic system Newton Solve Until Approimate Newton (Picard, SIMPLE, Ineact Newton F J s F s J = + = = +1 ( ( ( ( appr F s J = ( ( ( F F s J η < + rel abs F F ε < ε + + ( ( 0 1 Knoll and Keys, 2006, JCP 193, Collier, Hindmarsh, Serban, Woodward,KINSOL v.2.6
6 Ineact Newton J ( s + F( 2 < η F( Solve for s using Jacobian Free Krylov Iteration Method only requires matri-vector product Method Krylov Variable None s F( Objective Function Js MV Product F( + ε s F( ε Right z = P s F( Left s P -1 F( Fied Point s G( = F( - ( P J ( P z 1 F( + ε ( P z F( ε P J s s J G 1 F( + ε s P F( ε G( + ε s G( ε 6 Knoll and Keys, 2006, JCP 193,
7 Design - Interface to SUNDIALS GOAL reduce eplicit dependencies in two pacages SUNDIALS Design - Objected Oriented C Classes: KinMEM, IDAMEM, NVector,.. Member functions, KinSet (KinMEM,. Interface (KINSOL, IDA similar to Cantera s interface to CVODE(S Interface (.H is not dependent on SUNDIALS Dependency is in the implementation (.C Static call bac functions Classes: Kinsol, KinsolUser, Ida, IdaUser physics module which inherits KinsolUser or IdaUser 7
8 Design Physics Module Physics Modules (OpenFOAM Turn solvers into objects Class contains fields AND equations Similar to turbulence and radiation models No dependence on SUNDIALS SUNDIALS/OpenFOAM Physics Module Inherits from KinsolUser or IDAUser Container for OpenFOAM solver object FoodWeb, Incompressible Flow, 8
9 Outline Motivation Mathematics Basic Design Eamples Food Web Steady & Unsteady Pitz-Daily 9
10 Food Web Model Reaction-Diffusion Equation m i c t i = ( d c + f (, c c i : M prey & M predators i i i Model Parameters f = i (, c ci bi g( + a i j g( = 1+ α y + β + y ij c ( sin(4π sin(4π j Steady (KINSOL M = 3 c prey ( = 0 = 1 c (, t = 0 = 3 10 pred d pred = 1 d prey = 0.5 α = 1 β = 0 Unsteady (IDA M = 1 c c prey pred (, t = 0 = 10 + i 5 (, t = 0 = 10 prey ( 16(1 y(1 y d pred = 1 d prey = 0.5 α = 50 β = a ij = A = 1 A 0 E E m b prey 0 A E E E = 10 prey = 1 4 = 1 G G A 0 G G 0 A G = 5 10 m b pred prey = 4 0 = 1 10 Brown, 1986, SIAM J. Applied Math, v.46, no. 3 Collier and Serban (KINSOL, Hindmarsh, Serban, Collier (IDA
11 Food Web Physics Class Calculate increment from the residual OF does time integration (KINSOL void PredatorPrey::calcIncrement(u[].. tmp<fvscalarmatri> CEqn( M_[i]*(fvm::ddt( dc_ - fvc::ddt(dc_ - fvm::laplacian(diff_[i],dc_,"laplacian(diff,conc" + fvm::sp( (rateaa_[i]*freqoverconc*conc_[i],dc_ -Resid ; u P 1 u Time integration outside of OF void PredatorPrey::calcIncrement(Rz[], z[], alpha.. tmp<fvscalarmatri> CEqn( fvm::sp( timecoeff*scalar(alpha*m_[i], dc_ - fvm::laplacian(diff_[i],dc_,"laplacian(diff,conc" + fvm::sp( (rateaa_[i]*freqoverconc*conc_[i],dc_ - Resid ; z ( P ss + Pα t 1 R z 11
12 Prey Number Density for Kinsol Run problem using tol abs = tol rel =10-5 on a 2020 mesh Kinsol C Kinsol C++ Segregated OpenFOAM Kinsol OpenFOAM Reaction Laplacian + Reaction Diagonal Lower Left Upper Right Prey Predator Prey Predator
13 Kinsol-Open Solutions Food for prey poison for predator Prey/m 3 g ( =1+ y Residuals ( d c + f (, c i i i Pred/m 3 13
14 Prey Number Density for IDA Run problem using tol abs = tol rel =10-5 on a 2020 mesh Time Differences due to: IDA-C Finite Difference space+diag Eact Newton IDA-C++ Finite Difference In-Eact Newton round-off error FV (cell vs. FD (point Newton methods Preconditioner OpenFOAM Finite Volume In-Eact Newton 14
15 Prey Number Density Green - OpenFOAM (cell Red -IDA-C++ (point Blue IDA-C++ (point->cell 15
16 Final Prey Number Density IDA - C++ IDA- OpenFOAM 16
17 Incompressible Flow Simple class for (steady & unsteady incompressible flow based on SIMPLE algorithm Residual & Left - Preconditioner for Ida Residual & Right - Preconditioner for Kinsol Fied-Point Residual Preconditioning for Kinsol Necessarily have a standard SIMPLE solver Advantageous as solver compleity increases <?>SimpleFoam solver Provides interfaces between simple class and SUNDIALS pacages 17
18 Laminar Pitz Daily Tested on Pitz-Daily with ν = 0.01 m/s 2 Kinsol Right Precond Kinsol Fied Point Picard SIMPLE Time(s 129 s 138 s 198 s 146 s Machine zero Machine zero 1000 iters 1000 iters Mass L2 1.1 e e-11 2e Mass abs-ma 8.9 e e e Mom. L2 2.1 e e Mom. abs-ma 9.0 e e
19 Kinsol-Right Laminar Pitz-Daily +1 in Kinsol-Fied Point Picard-SIMPLE -1 in 0 m/s 5 m/s Velocity Profiles are nearly identical 12 m/s 19
20 Turbulent Pitz-Daily Solver Approach 1. nonlinear iteration on velocity & pressure followed by 2. N SIMPLE sweeps on U, p & M 1 turbulence corrections 3. M 2 turbulence corrections System is much less robust Selection of input parameters loose tolerance at the beginning tighten toward convergence step-size Turbulence model Reynolds stress seemed to wor better Need Globalization Methods 20
21 Results Fied Point Right Pre. Not quite converged.. 21
22 Future Wor Testing and eploration of eisting software interfaces Solution Strategies Improve variable and residual scaling Ida with incompressible flow Include turbulence models in nonlinear iteration Code clean up generalization, templates, better names, comments,.. Etend to other physical systems Additional Capabilities Sensitivity Analysis IDAS, CVodeS,.. Alternative Pacages - NITSOL, Trillinos (NOX,.. 22
Numerical Methods for Large-Scale Nonlinear Equations
Slide 1 Numerical Methods for Large-Scale Nonlinear Equations Homer Walker MA 512 April 28, 2005 Inexact Newton and Newton Krylov Methods a. Newton-iterative and inexact Newton methods. Slide 2 i. Formulation
Preconditioning the Newton-Krylov method for reactive transport
Preconditioning the Newton-Krylov method for reactive transport Michel Kern, L. Amir, A. Taakili INRIA Paris-Rocquencourt Maison de la Simulation MoMaS workshop Reactive Transport Modeling in the Geological
Preconditioners for the incompressible Navier Stokes equations
Preconditioners for the incompressible Navier Stokes equations C. Vuik M. ur Rehman A. Segal Delft Institute of Applied Mathematics, TU Delft, The Netherlands SIAM Conference on Computational Science and
OpenFOAM selected solver
OpenFOAM selected solver Roberto Pieri - SCS Italy 16-18 June 2014 Introduction to Navier-Stokes equations and RANS Turbulence modelling Numeric discretization Navier-Stokes equations Convective term {}}{
Coupling transport with geochemistry Applications to CO2 sequestration
Coupling transport with geochemistry Applications to CO2 sequestration Michel Kern with L. Amir, B. Gueslin, A. Taakili INRIA Paris Rocquencourt 3ème Conférence Internationale de la Société Marocaine de
Newton s Method and Efficient, Robust Variants
Newton s Method and Efficient, Robust Variants Philipp Birken University of Kassel (SFB/TRR 30) Soon: University of Lund October 7th 2013 Efficient solution of large systems of non-linear PDEs in science
Towards Accommodating Comprehensive Chemical Reaction Mechanisms in Practical Internal Combustion Engine Simulations
Paper # 0701C-0326 Topic: Internal Combustion and gas turbine engines 8 th U. S. National Combustion Meeting Organized by the Western States Section of the Combustion Institute and hosted by the University
Solving Large Nonlinear Sparse Systems
Solving Large Nonlinear Sparse Systems Fred W. Wubs and Jonas Thies Computational Mechanics & Numerical Mathematics University of Groningen, the Netherlands f.w.wubs@rug.nl Centre for Interdisciplinary
Coupled calculations in OpenFOAM -
Coupled calculations in OpenFOAM - Multiphysics handling, structures and solvers, Gothenburg Region OpenFOAM User Group Meeting Klas Jareteg Chalmers University of Technology November 14, 2012 Outline
Fast Iterative Solution of Saddle Point Problems
Michele Benzi Department of Mathematics and Computer Science Emory University Atlanta, GA Acknowledgments NSF (Computational Mathematics) Maxim Olshanskii (Mech-Math, Moscow State U.) Zhen Wang (PhD student,
Lecture 8: Fast Linear Solvers (Part 7)
Lecture 8: Fast Linear Solvers (Part 7) 1 Modified Gram-Schmidt Process with Reorthogonalization Test Reorthogonalization If Av k 2 + δ v k+1 2 = Av k 2 to working precision. δ = 10 3 2 Householder Arnoldi
A high order adaptive finite element method for solving nonlinear hyperbolic conservation laws
A high order adaptive finite element method for solving nonlinear hyperbolic conservation laws Zhengfu Xu, Jinchao Xu and Chi-Wang Shu 0th April 010 Abstract In this note, we apply the h-adaptive streamline
A Robust Preconditioned Iterative Method for the Navier-Stokes Equations with High Reynolds Numbers
Applied and Computational Mathematics 2017; 6(4): 202-207 http://www.sciencepublishinggroup.com/j/acm doi: 10.11648/j.acm.20170604.18 ISSN: 2328-5605 (Print); ISSN: 2328-5613 (Online) A Robust Preconditioned
Fast solvers for steady incompressible flow
ICFD 25 p.1/21 Fast solvers for steady incompressible flow Andy Wathen Oxford University wathen@comlab.ox.ac.uk http://web.comlab.ox.ac.uk/~wathen/ Joint work with: Howard Elman (University of Maryland,
Implicit solvers for large-scale nonlinear problems
Journal of Physics: Conference Series Implicit solvers for large-scale nonlinear problems To cite this article: David E Keyes et al 2006 J. Phys.: Conf. Ser. 46 433 View the article online for updates
ODE solvers in Julia. Gabriel Ingesson. October 2, 2015
ODE solvers in Julia Gabriel Ingesson October 2, 2015 Motivation General: Numerical methods for solving ODE s is important for system simulations. Simulation is important for controller design. Personal:
TAU Solver Improvement [Implicit methods]
![TAU Solver Improvement [Implicit methods]](/thumbs/95/123936756.jpg "TAU Solver Improvement [Implicit methods]")
TAU Solver Improvement [Implicit methods] Richard Dwight Megadesign 23-24 May 2007 Folie 1 > Vortrag > Autor Outline Motivation (convergence acceleration to steady state, fast unsteady) Implicit methods
A Review of Solution Techniques for Unsteady Incompressible Flow
Zeist 2009 p. 1/57 A Review of Solution Techniques for Unsteady Incompressible Flow David Silvester School of Mathematics University of Manchester Zeist 2009 p. 2/57 PDEs Review : 1966 1999 Update : 2000
Algorithms for Nonlinear and Transient Problems
Algorithms for Nonlinear and Transient Problems David E. Keyes Department of Applied Physics & Applied Mathematics Columbia University Institute for Scientific Computing Research Lawrence Livermore National
Energy-Conserving Numerical Simulations of Electron Holes in Two-Species Plasmas
Energy-Conserving Numerical Simulations of Electron Holes in Two-Species Plasmas Yingda Cheng Andrew J. Christlieb Xinghui Zhong March 18, 2014 Abstract In this paper, we apply our recently developed energy-conserving
IMEX Methods in SUNDIALS: The ARKode Solver
IMEX Methods in SUNDIALS: The ARKode Solver Daniel R. Reynolds reynolds@smu.edu Department of Mathematics Southern Methodist University BOUT++ 2013 Workshop September 4, 2013 Outline 1 Motivation 2 ARK
Adaptive preconditioners for nonlinear systems of equations
Adaptive preconditioners for nonlinear systems of equations L. Loghin D. Ruiz A. Touhami CERFACS Technical Report TR/PA/04/42 Also ENSEEIHT-IRIT Technical Report RT/TLSE/04/02 Abstract The use of preconditioned
Optimizing Runge-Kutta smoothers for unsteady flow problems
Optimizing Runge-Kutta smoothers for unsteady flow problems Philipp Birken 1 November 24, 2011 1 Institute of Mathematics, University of Kassel, Heinrich-Plett-Str. 40, D-34132 Kassel, Germany. email:
Progress in Parallel Implicit Methods For Tokamak Edge Plasma Modeling
Progress in Parallel Implicit Methods For Tokamak Edge Plasma Modeling Michael McCourt 1,2,Lois Curfman McInnes 1 Hong Zhang 1,Ben Dudson 3,Sean Farley 1,4 Tom Rognlien 5, Maxim Umansky 5 Argonne National
Efficient Augmented Lagrangian-type Preconditioning for the Oseen Problem using Grad-Div Stabilization
Efficient Augmented Lagrangian-type Preconditioning for the Oseen Problem using Grad-Div Stabilization Timo Heister, Texas A&M University 2013-02-28 SIAM CSE 2 Setting Stationary, incompressible flow problems
1. Introduction. Consider the Navier Stokes equations ηu t ν 2 u + (u grad) u + grad p = f div u = 0 (1.1)
University of Maryland Department of Computer Science TR-C4930 University of Maryland Institute for Advanced Computer Studies TR-009-0 BOUNDARY CONDITIONS IN APPROXIMATE COMMUTATOR PRECONDITIONERS FOR
Parallel Algorithms for Solution of Large Sparse Linear Systems with Applications
Parallel Algorithms for Solution of Large Sparse Linear Systems with Applications Murat Manguoğlu Department of Computer Engineering Middle East Technical University, Ankara, Turkey Prace workshop: HPC
Multi-Point Special Boundary-Value Problems and Applications to Fluid Flow Through Porous Media
Multi-Point Special Boundary-Value Problems and Applications to Fluid Flow Through Porous Media Mohamed A. Hajji Abstract In this paper we propose a numerical scheme based on finite differences for the
Space-time XFEM for two-phase mass transport
Space-time XFEM for two-phase mass transport Space-time XFEM for two-phase mass transport Christoph Lehrenfeld joint work with Arnold Reusken EFEF, Prague, June 5-6th 2015 Christoph Lehrenfeld EFEF, Prague,
Large Scale Fluid-Structure Interaction by coupling OpenFOAM with external codes
Large Scale Fluid-Structure Interaction by coupling OpenFOAM with external codes Thomas Gallinger * Alexander Kupzok Roland Wüchner Kai-Uwe Bletzinger Lehrstuhl für Statik Technische Universität München
Reactive transport in porous media: formulations, non-linear solvers and preconditioners
Reactive transport in porous media: formulations, non-linear solvers and preconditioners Michel Kern with L. Amir, B. Gueslin, A. Taakili Institut National de Recherche en Informatique et Automatique High
Fast Solvers for Unsteady Incompressible Flow
ICFD25 Workshop p. 1/39 Fast Solvers for Unsteady Incompressible Flow David Silvester University of Manchester http://www.maths.manchester.ac.uk/~djs/ ICFD25 Workshop p. 2/39 Outline Semi-Discrete Navier-Stokes
CLASS NOTES Computational Methods for Engineering Applications I Spring 2015
CLASS NOTES Computational Methods for Engineering Applications I Spring 2015 Petros Koumoutsakos Gerardo Tauriello (Last update: July 2, 2015) IMPORTANT DISCLAIMERS 1. REFERENCES: Much of the material
Linear Solvers. Andrew Hazel
Linear Solvers Andrew Hazel Introduction Thus far we have talked about the formulation and discretisation of physical problems...... and stopped when we got to a discrete linear system of equations. Introduction
Sensitivity Analysis of Differential-Algebraic Equations and Partial Differential Equations
Sensitivity Analysis of Differential-Algebraic Equations and Partial Differential Equations Linda Petzold Department of Computer Science University of California Santa Barbara, California 93106 Shengtai
Elmer. Introduction into Elmer multiphysics. Thomas Zwinger. CSC Tieteen tietotekniikan keskus Oy CSC IT Center for Science Ltd.
Elmer Introduction into Elmer multiphysics FEM package Thomas Zwinger CSC Tieteen tietotekniikan keskus Oy CSC IT Center for Science Ltd. Contents Elmer Background History users, community contacts and
Reduced System Computing and MMOs. Warren Weckesser. Department of Mathematics Colgate University
Reduced System Computing and MMOs Warren Weckesser Department of Mathematics Colgate University Acknowledgments NSF Grant DMS-54468: RUI: Reduced System Computing for Singularly Perturbed Differential
Implicit Solution of Viscous Aerodynamic Flows using the Discontinuous Galerkin Method
Implicit Solution of Viscous Aerodynamic Flows using the Discontinuous Galerkin Method Per-Olof Persson and Jaime Peraire Massachusetts Institute of Technology 7th World Congress on Computational Mechanics
ETNA Kent State University
Electronic Transactions on Numerical Analysis. Volume 35, pp. 57-80, 009. Copyright 009,. ISSN 1068-9613. ETNA BOUNDARY CONDITIONS IN APPROXIMATE COMMUTATOR PRECONDITIONERS FOR THE NAVIER-STOKES EQUATIONS
ADAPTIVE ACCURACY CONTROL OF NONLINEAR NEWTON-KRYLOV METHODS FOR MULTISCALE INTEGRATED HYDROLOGIC MODELS
XIX International Conference on Water Resources CMWR 2012 University of Illinois at Urbana-Champaign June 17-22,2012 ADAPTIVE ACCURACY CONTROL OF NONLINEAR NEWTON-KRYLOV METHODS FOR MULTISCALE INTEGRATED
A Taxonomy and Comparison of Parallel Block Multi-level Preconditioners for the Incompressible Navier Stokes Equations
SANDIA REPORT SAND2007-2761 Unlimited Release Printed May 2007 A Taxonomy and Comparison of Parallel Block Multi-level Preconditioners for the Incompressible Navier Stokes Equations Howard Elman, Victoria
THE solution of the absolute value equation (AVE) of
The nonlinear HSS-like iterative method for absolute value equations Mu-Zheng Zhu Member, IAENG, and Ya-E Qi arxiv:1403.7013v4 [math.na] 2 Jan 2018 Abstract Salkuyeh proposed the Picard-HSS iteration method
PRECONDITIONING TECHNIQUES FOR NEWTON S METHOD FOR THE INCOMPRESSIBLE NAVIER STOKES EQUATIONS
BIT Numerical Mathematics 43: 961 974, 2003. 2003 Kluwer Academic Publishers. Printed in the Netherlands. 961 PRECONDITIONING TECHNIQUES FOR NEWTON S METHOD FOR THE INCOMPRESSIBLE NAVIER STOKES EQUATIONS
A monolithic FEM solver for fluid structure
A monolithic FEM solver for fluid structure interaction p. 1/1 A monolithic FEM solver for fluid structure interaction Stefan Turek, Jaroslav Hron jaroslav.hron@mathematik.uni-dortmund.de Department of
Newton-Multigrid Least-Squares FEM for S-V-P Formulation of the Navier-Stokes Equations
Newton-Multigrid Least-Squares FEM for S-V-P Formulation of the Navier-Stokes Equations A. Ouazzi, M. Nickaeen, S. Turek, and M. Waseem Institut für Angewandte Mathematik, LSIII, TU Dortmund, Vogelpothsweg
OpenFOAM SIMULATION OF THE FLOW IN THE HÖLLEFORSEN DRAFT TUBE MODEL
Turbine-99 III Proceedings of the third IAHR/ERCOFTAC workshop on draft tube flow 8-9 December 2005, Porjus, Sweden Paper No. 8 OpenFOAM SIMULATION OF THE FLOW IN THE HÖLLEFORSEN DRAFT TUBE MODEL Nilsson
Scalable Non-Linear Compact Schemes
Scalable Non-Linear Compact Schemes Debojyoti Ghosh Emil M. Constantinescu Jed Brown Mathematics Computer Science Argonne National Laboratory International Conference on Spectral and High Order Methods
Solving Symmetric Indefinite Systems with Symmetric Positive Definite Preconditioners
Solving Symmetric Indefinite Systems with Symmetric Positive Definite Preconditioners Eugene Vecharynski 1 Andrew Knyazev 2 1 Department of Computer Science and Engineering University of Minnesota 2 Department
THE SEPARATRIX FOR A SECOND ORDER ORDINARY DIFFERENTIAL EQUATION OR A 2 2 SYSTEM OF FIRST ORDER ODE WHICH ALLOWS A PHASE PLANE QUANTITATIVE ANALYSIS
THE SEPARATRIX FOR A SECOND ORDER ORDINARY DIFFERENTIAL EQUATION OR A SYSTEM OF FIRST ORDER ODE WHICH ALLOWS A PHASE PLANE QUANTITATIVE ANALYSIS Maria P. Skhosana and Stephan V. Joubert, Tshwane University
Finding Rightmost Eigenvalues of Large, Sparse, Nonsymmetric Parameterized Eigenvalue Problems
Finding Rightmost Eigenvalues of Large, Sparse, Nonsymmetric Parameterized Eigenvalue Problems AMSC 663-664 Final Report Minghao Wu AMSC Program mwu@math.umd.edu Dr. Howard Elman Department of Computer
Implementation of an Alternate Chemistry Library into OpenFOAM TM
of an Alternate Chemistry Library into OpenFOAM TM Bernhard F.W. Gschaider 1 Markus Rehm 2 Peter Seifert 2 Bernd Meyer 2 1 ICE Strömungsforschung Leoben, Austria 2 Institute of Energy Process Enineering
Unsteady Incompressible Flow Simulation Using Galerkin Finite Elements with Spatial/Temporal Adaptation
Unsteady Incompressible Flow Simulation Using Galerkin Finite Elements with Spatial/Temporal Adaptation Mohamed S. Ebeida Carnegie Mellon University, Pittsburgh, PA 15213 Roger L. Davis and Roland W. Freund
Introduction to Compact Dynamical Modeling. II.1 Steady State Simulation. Luca Daniel Massachusetts Institute of Technology. dx dt.
Course Outline NS & NIH Introduction to Compact Dynamical Modeling II. Steady State Simulation uca Daniel Massachusetts Institute o Technology Quic Snea Preview I. ssembling Models rom Physical Problems
An Accelerated Block-Parallel Newton Method via Overlapped Partitioning
An Accelerated Block-Parallel Newton Method via Overlapped Partitioning Yurong Chen Lab. of Parallel Computing, Institute of Software, CAS (http://www.rdcps.ac.cn/~ychen/english.htm) Summary. This paper
C. Vuik 1 S. van Veldhuizen 1 C.R. Kleijn 2
On iterative solvers combined with projected Newton methods for reacting flow problems C. Vuik 1 S. van Veldhuizen 1 C.R. Kleijn 2 1 Delft University of Technology J.M. Burgerscentrum 2 Delft University
DELFT UNIVERSITY OF TECHNOLOGY
DELFT UNIVERSITY OF TECHNOLOGY REPORT 13-10 Comparison of some preconditioners for the incompressible Navier-Stokes equations X. He and C. Vuik ISSN 1389-6520 Reports of the Delft Institute of Applied
Computation for the Backward Facing Step Test Case with an Open Source Code
Computation for the Backward Facing Step Test Case with an Open Source Code G.B. Deng Equipe de Modélisation Numérique Laboratoire de Mécanique des Fluides Ecole Centrale de Nantes 1 Rue de la Noë, 44321
An Efficient Low Memory Implicit DG Algorithm for Time Dependent Problems
An Efficient Low Memory Implicit DG Algorithm for Time Dependent Problems P.-O. Persson and J. Peraire Massachusetts Institute of Technology 2006 AIAA Aerospace Sciences Meeting, Reno, Nevada January 9,
Научный потенциал регионов на службу модернизации. Астрахань: АИСИ, с.
MODELING OF FLOWS IN PIPING TREES USING PROJECTION METHODS В.В. Войков, Астраханский инженерно-строительный институт, г. Астрахань, Россия Jason Mayes, Mihir Sen University of Notre Dame, Indiana, USA
Nonlinear iterative solvers for unsteady Navier-Stokes equations
Proceedings of Symposia in Applied Mathematics Nonlinear iterative solvers for unsteady Navier-Stokes equations Philipp Birken and Antony Jameson This paper is dedicated to Gene Golub. Abstract. The application
PALADINS: Scalable Time-Adaptive Algebraic Splitting and Preconditioners for the Navier-Stokes Equations
2013 SIAM Conference On Computational Science and Engineering Boston, 27 th February 2013 PALADINS: Scalable Time-Adaptive Algebraic Splitting and Preconditioners for the Navier-Stokes Equations U. Villa,
Chapter 6 Laminar External Flow
Chapter 6 aminar Eternal Flow Contents 1 Thermal Boundary ayer 1 2 Scale analysis 2 2.1 Case 1: δ t > δ (Thermal B.. is larger than the velocity B..) 3 2.2 Case 2: δ t < δ (Thermal B.. is smaller than
Semi-implicit Krylov Deferred Correction Methods for Ordinary Differential Equations
Semi-implicit Krylov Deferred Correction Methods for Ordinary Differential Equations Sunyoung Bu University of North Carolina Department of Mathematics CB # 325, Chapel Hill USA agatha@email.unc.edu Jingfang
A finite element solver for ice sheet dynamics to be integrated with MPAS
A finite element solver for ice sheet dynamics to be integrated with MPAS Mauro Perego in collaboration with FSU, ORNL, LANL, Sandia February 6, CESM LIWG Meeting, Boulder (CO), 202 Outline Introduction
In particular, if the initial condition is positive, then the entropy solution should satisfy the following positivity-preserving property
1 3 4 5 6 7 8 9 1 11 1 13 14 15 16 17 18 19 1 3 4 IMPLICIT POSITIVITY-PRESERVING HIGH ORDER DISCONTINUOUS GALERKIN METHODS FOR CONSERVATION LAWS TONG QIN AND CHI-WANG SHU Abstract. Positivity-preserving
Efficient FEM-multigrid solver for granular material
Efficient FEM-multigrid solver for granular material S. Mandal, A. Ouazzi, S. Turek Chair for Applied Mathematics and Numerics (LSIII), TU Dortmund STW user committee meeting Enschede, 25th September,
Shifted Laplace and related preconditioning for the Helmholtz equation
Shifted Laplace and related preconditioning for the Helmholtz equation Ivan Graham and Euan Spence (Bath, UK) Collaborations with: Paul Childs (Schlumberger Gould Research), Martin Gander (Geneva) Douglas
Time-adaptive methods for the incompressible Navier-Stokes equations
Time-adaptive methods for the incompressible Navier-Stokes equations Joachim Rang, Thorsten Grahs, Justin Wiegmann, 29.09.2016 Contents Introduction Diagonally implicit Runge Kutta methods Results with
CHAPTER 4 BOUNDARY LAYER FLOW APPLICATION TO EXTERNAL FLOW
CHAPTER 4 BOUNDARY LAYER FLOW APPLICATION TO EXTERNAL FLOW 4.1 Introduction Boundary layer concept (Prandtl 1904): Eliminate selected terms in the governing equations Two key questions (1) What are the
Preliminary Results of GRAPES Helmholtz solver using GCR and PETSc tools
Preliminary Results of GRAPES Helmholtz solver using GCR and PETSc tools Xiangjun Wu (1),Lilun Zhang (2),Junqiang Song (2) and Dehui Chen (1) (1) Center for Numerical Weather Prediction, CMA (2) School
Chapter 6. Nonlinear Equations. 6.1 The Problem of Nonlinear Root-finding. 6.2 Rate of Convergence
Chapter 6 Nonlinear Equations 6. The Problem of Nonlinear Root-finding In this module we consider the problem of using numerical techniques to find the roots of nonlinear equations, f () =. Initially we
Application of Dual Time Stepping to Fully Implicit Runge Kutta Schemes for Unsteady Flow Calculations
Application of Dual Time Stepping to Fully Implicit Runge Kutta Schemes for Unsteady Flow Calculations Antony Jameson Department of Aeronautics and Astronautics, Stanford University, Stanford, CA, 94305
Computational Methods
1 / 55 Computational Methods Copyright c 2018 by Nob Hill Publishing, LLC We summarize many of the fundamental computational procedures required in reactor analysis and design. We illustrate these procedures
2 integration methods estimate and control the temporal error through adapting both the temporal step size and the order of the temporal approximation
1 Higher order, locally conservative temporal integration methods for modeling multiphase flow in porous media C.T. Miller aλ, M.W. Farthing ay, C.E. Kees a z, and C.T. Kelley bx a Center for the Advanced
1. Introduction. Let the nonlinear set of equations arising in nonlinear transient elasticity be given by. r(u) = 0, (1.1)
PSEUDO-TRANSIENT CONTINUATION FOR NONLINEAR TRANSIENT ELASTICITY MICHAEL W. GEE, C. T. KELLEY, AND R. B. LEHOUCQ Abstract. We show how the method of pseudo-transient continuation can be applied to improve
The solution of the discretized incompressible Navier-Stokes equations with iterative methods
The solution of the discretized incompressible Navier-Stokes equations with iterative methods Report 93-54 C. Vuik Technische Universiteit Delft Delft University of Technology Faculteit der Technische
HIGHER-ORDER LINEARLY IMPLICIT ONE-STEP METHODS FOR THREE-DIMENSIONAL INCOMPRESSIBLE NAVIER-STOKES EQUATIONS
STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume LIII, Number 1, March 2008 HIGHER-ORDER LINEARLY IMPLICIT ONE-STEP METHODS FOR THREE-DIMENSIONAL INCOMPRESSIBLE NAVIER-STOKES EQUATIONS IOAN TELEAGA AND JENS
A high-order discontinuous Galerkin solver for 3D aerodynamic turbulent flows
A high-order discontinuous Galerkin solver for 3D aerodynamic turbulent flows F. Bassi, A. Crivellini, D. A. Di Pietro, S. Rebay Dipartimento di Ingegneria Industriale, Università di Bergamo CERMICS-ENPC
A Review of Preconditioning Techniques for Steady Incompressible Flow
Zeist 2009 p. 1/43 A Review of Preconditioning Techniques for Steady Incompressible Flow David Silvester School of Mathematics University of Manchester Zeist 2009 p. 2/43 PDEs Review : 1984 2005 Update
Far Field Noise Minimization Using an Adjoint Approach
Far Field Noise Minimization Using an Adjoint Approach Markus P. Rumpfkeil and David W. Zingg University of Toronto Institute for Aerospace Studies 4925 Dufferin Street, Toronto, Ontario, M3H 5T6, Canada
Nonlinear Preconditioning in PETSc
Nonlinear Preconditioning in PETSc Matthew Knepley PETSc Team Computation Institute University of Chicago Challenges in 21st Century Experimental Mathematical Computation ICERM, Providence, RI July 22,
CHAPTER 7 NUMERICAL MODELLING OF A SPIRAL HEAT EXCHANGER USING CFD TECHNIQUE
CHAPTER 7 NUMERICAL MODELLING OF A SPIRAL HEAT EXCHANGER USING CFD TECHNIQUE In this chapter, the governing equations for the proposed numerical model with discretisation methods are presented. Spiral
Linear Algebraic Equations
Linear Algebraic Equations Linear Equations: a + a + a + a +... + a = c 11 1 12 2 13 3 14 4 1n n 1 a + a + a + a +... + a = c 21 2 2 23 3 24 4 2n n 2 a + a + a + a +... + a = c 31 1 32 2 33 3 34 4 3n n
A SPARSE APPROXIMATE INVERSE PRECONDITIONER FOR NONSYMMETRIC LINEAR SYSTEMS
INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING, SERIES B Volume 5, Number 1-2, Pages 21 30 c 2014 Institute for Scientific Computing and Information A SPARSE APPROXIMATE INVERSE PRECONDITIONER
2 CAI, KEYES AND MARCINKOWSKI proportional to the relative nonlinearity of the function; i.e., as the relative nonlinearity increases the domain of co
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS Int. J. Numer. Meth. Fluids 2002; 00:1 6 [Version: 2000/07/27 v1.0] Nonlinear Additive Schwarz Preconditioners and Application in Computational Fluid
Structure preserving preconditioner for the incompressible Navier-Stokes equations
Structure preserving preconditioner for the incompressible Navier-Stokes equations Fred W. Wubs and Jonas Thies Computational Mechanics & Numerical Mathematics University of Groningen, the Netherlands
Compressible Navier-Stokes (Euler) Solver based on Deal.II Library
Compressible Navier-Stokes (Euler) Solver based on Deal.II Library Lei Qiao Northwestern Polytechnical University Xi an, China Texas A&M University College Station, Texas Fifth deal.ii Users and Developers
M.A. Botchev. September 5, 2014
Rome-Moscow school of Matrix Methods and Applied Linear Algebra 2014 A short introduction to Krylov subspaces for linear systems, matrix functions and inexact Newton methods. Plan and exercises. M.A. Botchev
= V I = Bus Admittance Matrix. Chapter 6: Power Flow. Constructing Ybus. Example. Network Solution. Triangular factorization. Let
Chapter 6: Power Flow Network Matrices Network Solutions Newton-Raphson Method Fast Decoupled Method Bus Admittance Matri Let I = vector of currents injected into nodes V = vector of node voltages Y bus
Code Verification of Multiphase Flow with MFIX
Code Verification of Multiphase Flow with MFIX A N I R U D D H A C H O U D H A R Y ( A N I R U D D @ V T. E D U ), C H R I S T O P H E R J. ROY ( C J R O Y @ V T. E D U ), ( V I R G I N I A T E C H ) 23
2 In this paper we model saturation with the van Geneuchten formula, [2], S a (ψ) = ( S r + (1 Sr) [1+(ffjψj) n ] m ; ψ<0 1; ψ 0 : (2) In (2), S r is
![2 In this paper we model saturation with the van Geneuchten formula, [2], S a (ψ) = ( S r + (1 Sr) [1+(ffjψj) n ] m ; ψ<0 1; ψ 0 : (2) In (2), S r is](/thumbs/92/109106657.jpg "2 In this paper we model saturation with the van Geneuchten formula, [2], S a (ψ) = ( S r + (1 Sr) [1+(ffjψj) n ] m ; ψ<0 1; ψ 0 : (2) In (2), S r is")
1 Nonsmooth Nonlinearities and Temporal Integration of Richards' Equation K. R. Kavanagh aλ, C. T. Kelley ay, R. C. Berger b,j.p. Hallberg b, and Stacy E. Howington b a Department of Mathematics and Center
Accelerating incompressible fluid flow simulations on hybrid CPU/GPU systems
Accelerating incompressible fluid flow simulations on hybrid CPU/GPU systems Yushan Wang 1, Marc Baboulin 1,2, Karl Rupp 3,4, Yann Fraigneau 1,5, Olivier Le Maître 1,5 1 Université Paris-Sud, France 2
Draft Tube calculations using OpenFOAM-1.5dev and validation with FLINDT data
6th OpenFOAM Workshop, June 13-16 2011, PennState University, USA Draft Tube calculations using OpenFOAM-1.5dev and validation with FLINDT data C. Devals, Y. Zhang and F.Guibault École Polytechnique de
FVM for Fluid-Structure Interaction with Large Structural Displacements
FVM for Fluid-Structure Interaction with Large Structural Displacements Željko Tuković and Hrvoje Jasak Zeljko.Tukovic@fsb.hr, h.jasak@wikki.co.uk Faculty of Mechanical Engineering and Naval Architecture
Math 19, Homework-1 Solutions
SSEA Summer 207 Math 9, Homework- Solutions. Consider the graph of function f shown below. Find the following its or eplain why they do not eist: (a) t 2 f(t). = 0. (b) t f(t). =. (c) t 0 f(t). (d) Does
1. Introduction. The problem of interest is a system of nonlinear equations
SIAM J. NUMER. ANAL. Vol. 46, No. 4, pp. 2112 2132 c 2008 Society for Industrial and Applied Mathematics INEXACT NEWTON DOGLEG METHODS ROGER P. PAWLOWSKI, JOSEPH P. SIMONIS, HOMER F. WALKER, AND JOHN N.
Lecture 1: Course Introduction.
Lecture : Course Introduction. What is the Finite Element Method (FEM)? a numerical method for solving problems of engineering and mathematical physics. (Logan Pg. #). In MECH 40 we are concerned with
Termination criteria for inexact fixed point methods
Termination criteria for inexact fixed point methods Philipp Birken 1 October 1, 2013 1 Institute of Mathematics, University of Kassel, Heinrich-Plett-Str. 40, D-34132 Kassel, Germany Department of Mathematics/Computer
Third-Order Active-Flux Scheme for Advection Diffusion: Hyperbolic Diffusion, Boundary Condition, and Newton Solver
Preprint accepted for publication in Computers and Fluids, 16 http://d.doi.org/1.116/j.compfluid.15.1. Third-Order Active-Flu Scheme for Advection Diffusion: Hyperbolic Diffusion, Boundary Condition, and
Multigrid absolute value preconditioning
Multigrid absolute value preconditioning Eugene Vecharynski 1 Andrew Knyazev 2 (speaker) 1 Department of Computer Science and Engineering University of Minnesota 2 Department of Mathematical and Statistical