Solving linear systems in fluid dynamics P. Aaron Lott Applied Mathematics and Scientific Computation Program University of Maryland.

Slides:

Advertisements

Similar presentations

Stable Fluids A paper by Jos Stam.

Advertisements

My First Fluid Project Ryan Schmidt. Outline MAC Method How far did I get? What went wrong? Future Work.

03/29/2006, City Univ1 Iterative Methods with Inexact Preconditioners and Applications to Saddle-point Systems & Electromagnetic Maxwell Systems Jun Zou.

Solving Equations = 4x – 5(6x – 10) -132 = 4x – 30x = -26x = -26x 7 = x.

A New Domain Decomposition Technique for TeraGrid Simulations Leopold Grinberg 1 Brian Toonen 2 Nicholas Karonis 2,3 George Em Karniadakis 1 1 Brown University.

Multilevel Incomplete Factorizations for Non-Linear FE problems in Geomechanics DMMMSA – University of Padova Department of Mathematical Methods and Models.

Coupled Fluid-Structural Solver CFD incompressible flow solver has been coupled with a FEA code to analyze dynamic fluid-structure coupling phenomena CFD.

12/21/2001Numerical methods in continuum mechanics1 Continuum Mechanics On the scale of the object to be studied the density and other fluid properties.

MA5233: Computational Mathematics

Network and Grid Computing –Modeling, Algorithms, and Software Mo Mu Joint work with Xiao Hong Zhu, Falcon Siu.

Weak Formulation ( variational formulation)

Laminar Incompressible Flow over a Rotating Disk Numerical Analysis for Engineering John Virtue December 1999.

D. Zuzio* J-L. Estivalezes* *ONERA/DMAE, 2 av. Édouard Belin, Toulouse, France Simulation of a Rayleigh-Taylor instability with five levels of adaptive.

Numerical methods for PDEs PDEs are mathematical models for –Physical Phenomena Heat transfer Wave motion.

1 Parallel Simulations of Underground Flow in Porous and Fractured Media H. Mustapha 1,2, A. Beaudoin 1, J. Erhel 1 and J.R. De Dreuzy IRISA – INRIA.

ITERATIVE TECHNIQUES FOR SOLVING NON-LINEAR SYSTEMS (AND LINEAR SYSTEMS)

Lecture Objectives Review SIMPLE CFD Algorithm SIMPLE Semi-Implicit Method for Pressure-Linked Equations Define Residual and Relaxation.

Finite Mathematics Dr. Saeid Moloudzadeh Using Matrices to Solve Systems of Equations 1 Contents Algebra Review Functions and Linear Models.

MA5251: Spectral Methods & Applications

1 Discretization of Fluid Models (Navier Stokes) Dr. Farzad Ismail School of Aerospace and Mechanical Engineering Universiti Sains Malaysia Nibong Tebal.

ParCFD Parallel computation of pollutant dispersion in industrial sites Julien Montagnier Marc Buffat David Guibert.

CFD Lab - Department of Engineering - University of Liverpool Ken Badcock & Mark Woodgate Department of Engineering University of Liverpool Liverpool L69.

1 Discretization of Fluid Models (Navier Stokes) Dr. Farzad Ismail School of Aerospace and Mechanical Engineering Universiti Sains Malaysia Nibong Tebal.

Incompressible Flows Sauro Succi. Incompressible flows.

Computational Aspects of Multi-scale Modeling Ahmed Sameh, Ananth Grama Computing Research Institute Purdue University.

Lecture Objectives Review Define Residual and Relaxation SIMPLE CFD Algorithm SIMPLE Semi-Implicit Method for Pressure-Linked Equations.

Andrew McCracken Supervisor: Professor Ken Badcock

FALL 2015 Esra Sorgüven Öner

Discretization for PDEs Chunfang Chen,Danny Thorne Adam Zornes, Deng Li CS 521 Feb., 9,2006.

On the Use of Finite Difference Matrix-Vector Products in Newton-Krylov Solvers for Implicit Climate Dynamics with Spectral Elements ImpactObjectives 

Partial Derivatives Example: Find If solution: Partial Derivatives Example: Find If solution: gradient grad(u) = gradient.

An Overview of Meros Trilinos User’s Group Wednesday, November 2, 2005 Victoria Howle Computational Sciences and Mathematics Research Department (8962)

Consider Preconditioning – Basic Principles Basic Idea: is to use Krylov subspace method (CG, GMRES, MINRES …) on a modified system such as The matrix.

Lecture Objectives: Define 1) Reynolds stresses and

F. Fairag, H Tawfiq and M. Al-Shahrani Department of Math & Stat Department of Mathematics and Statistics, KFUPM. Nov 6, 2013 Preconditioning Technique.

A Preconditioned Domain Decomposition Algorithm for Contact Problem HARRACHOV 2007 A. Lotfi.

Quality of Service for Numerical Components Lori Freitag Diachin, Paul Hovland, Kate Keahey, Lois McInnes, Boyana Norris, Padma Raghavan.

Multipole-Based Preconditioners for Sparse Linear Systems. Ananth Grama Purdue University. Supported by the National Science Foundation.

Finding Rightmost Eigenvalues of Large, Sparse, Nonsymmetric Parameterized Eigenvalue Problems Minghao Wu AMSC Program Advisor: Dr. Howard.

An Introduction to Computational Fluids Dynamics Prapared by: Chudasama Gulambhai H ( ) Azhar Damani ( ) Dave Aman ( )

Differential Analysis. Continuity Equation Momentum Equation.

Model Anything. Quantity Conserved c  advect  diffuse S ConservationConstitutiveGoverning Mass, M  q -- M Momentum fluid, Mv -- F Momentum fluid.

Computational Fluid Dynamics Lecture II Numerical Methods and Criteria for CFD Dr. Ugur GUVEN Professor of Aerospace Engineering.

I- Computational Fluid Dynamics (CFD-I)

Matrix Addition and Scalar Multiplication

Matrix Multiplication

Cryogenic Flow in Corrugated Pipes

Objective Numerical methods SIMPLE CFD Algorithm Define Relaxation

Scientific Computing Lab

Scientific Computing Lab

Lecture Objectives: Review Explicit vs. Implicit

© Fluent Inc. 1/10/2018L1 Fluids Review TRN Solution Methods.

Meros: Software for Block Preconditioning the Navier-Stokes Equations

Robert Shuttleworth Applied Math & Scientific Computation (AMSC)

Objective Review Reynolds Navier Stokes Equations (RANS)

Objective Unsteady state Numerical methods Discretization

CFD – Fluid Dynamics Equations

Class # 27 ME363 Spring /23/2018.

Scientific Computing Lab

Analytical Tools in ME Course Objectives

Supported by the National Science Foundation.

Convergence in Numerical Science

QUANTIFYING THE UNCERTAINTY IN Wind Power Production Using Meteodyn WT

Objective Numerical methods Finite volume.

Finite Difference Method for Poisson Equation

 = N  N matrix multiplication N = 3 matrix N = 3 matrix N = 3 matrix

12. Navier-Stokes Applications

Ph.D. Thesis Numerical Solution of PDEs and Their Object-oriented Parallel Implementations Xing Cai October 26, 1998.

Solving a System of Linear Equations

Fig. 1 Steps in the PDE functional identification of nonlinear dynamics (PDE-FIND) algorithm, applied to infer the Navier-Stokes equations from data. Steps.

Presentation transcript:

Solving linear systems in fluid dynamics P. Aaron Lott Applied Mathematics and Scientific Computation Program University of Maryland

Non-zero structure of 2D Poisson Operator Using a Spectral Element Discretization

Incompressible Navier Stokes Equations

Discretized Steady Navier Stokes Each time step requires a Nonlinear Solve Each Nonlinear Solve requires a Linear solve Each Linear Solve can be expensive - Need efficient scalable solvers

Preconditioning

Preconditioner for Steady Navier Stokes Equations Choose P_F as an inexpensive approximation to F Choose P_S as an inexpensive approximation to the Schur complement of the system matrix

References High-Order Methods for Incompressible Fluid Flow. Deville Fischer and Mund Spectral/hp Element Methods for Computational Fluid Dynamics. Karniadakis and Sherwin Spectral Methods Fundamentals in Single Domains. Canuto Hussaini Quarteroni Zang Finite Element Methods and Fast Iterative Solvers with applications in incompressible fluid dynamics. Elman Silverster and Wathen Iterative Methods for Sparse Linear Systems. Saad

Opportunities/Resources Burgers Program in Fluid Dynamics Center for Scientific Computation and Mathematical Modeling AMSC Faculty Research Interests AMSC Wiki (CFD)