ME 595M J.Murthy1 ME 595M: Computational Methods for Nanoscale Thermal Transport Lecture 9: Introduction to the Finite Volume Method for the Gray BTE J.

Slides:



Advertisements
Similar presentations
Fourier’s Law and the Heat Equation
Advertisements

2ª aula Evolution Equation. The Finite Volume Method.
Partial Differential Equations
Parallel Jacobi Algorithm Steven Dong Applied Mathematics.
Chapter 8 Elliptic Equation.
Finite Element Method (FEM) Different from the finite difference method (FDM) described earlier, the FEM introduces approximated solutions of the variables.
Lectures on CFD Fundamental Equations
Computational Methods in Physics PHYS 3437
Introduction to numerical simulation of fluid flows
Coupled Fluid-Structural Solver CFD incompressible flow solver has been coupled with a FEA code to analyze dynamic fluid-structure coupling phenomena CFD.
1 Lecture 24: Flux Limiters 2 Last Time… l Developed a set of limiter functions l Second order accurate.
ME 595M J.Murthy1 ME 595M: Computational Methods for Nanoscale Thermal Transport Lecture 10:Higher-Order BTE Models J. Murthy Purdue University.
1 Systems of Linear Equations Iterative Methods. 2 B. Iterative Methods 1.Jacobi method and Gauss Seidel 2.Relaxation method for iterative methods.
CHE/ME 109 Heat Transfer in Electronics LECTURE 12 – MULTI- DIMENSIONAL NUMERICAL MODELS.
PARTIAL DIFFERENTIAL EQUATIONS
Chapter 2 Section 2 Solving a System of Linear Equations II.
Basics of Finite Difference Methods
1 Lecture 11: Unsteady Conduction Error Analysis.
H=1 h=0 At an internal node vol. Balance gives vel of volume sides And continuity gives hence i.e., At each point in time we solve A steady state problem.
ME 595M J.Murthy1 ME 595M: Computational Methods for Nanoscale Thermal Transport Lecture 6: Introduction to the Phonon Boltzmann Transport Equation J.
Macquarie University The Heat Equation and Diffusion PHYS by Lesa Moore DEPARTMENT OF PHYSICS.
ME 595M J.Murthy1 ME 595M: Computational Methods for Nanoscale Thermal Transport Lecture 9: Boundary Conditions, BTE Code J. Murthy Purdue University.
ME 595M J.Murthy1 ME 595M: Computational Methods for Nanoscale Thermal Transport Lecture 11:Extensions and Modifications J. Murthy Purdue University.
ME 595M J.Murthy1 ME 595M: Computational Methods for Nanoscale Thermal Transport Lecture 11: Homework solution Improved numerical techniques J. Murthy.
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the.
1 Finite-Volume Formulation. 2 Review of the Integral Equation The integral equation for the conservation statement is: Equation applies for a control.
Scientific Computing Partial Differential Equations Poisson Equation Calculus of Variations.
Heat Transfer Modeling
© Fluent Inc. 9/5/2015L1 Fluids Review TRN Solution Methods.
Solving Scalar Linear Systems Iterative approach Lecture 15 MA/CS 471 Fall 2003.
Chapter 13 Multiple Integrals by Zhian Liang.
Lecture 16: Convection and Diffusion (Cont’d). Last Time … We l Looked at CDS/UDS schemes to unstructured meshes l Look at accuracy of CDS and UDS schemes.
Chapter 9: Differential Analysis of Fluid Flow SCHOOL OF BIOPROCESS ENGINEERING, UNIVERSITI MALAYSIA PERLIS.
02/16/05© 2005 University of Wisconsin Last Time Re-using paths –Irradiance Caching –Photon Mapping.
Pareto Linear Programming The Problem: P-opt Cx s.t Ax ≤ b x ≥ 0 where C is a kxn matrix so that Cx = (c (1) x, c (2) x,..., c (k) x) where c.
PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001S16-1MAR120, Section 16, December 2001 SECTION 16 HEAT TRANSFER ANALYSIS.
1 EEE 431 Computational Methods in Electrodynamics Lecture 4 By Dr. Rasime Uyguroglu
MEIAC 2001 COMET Laboratory for Computational Methods in Emerging Technologies Large-Scale Simulation of Ultra-Fast Laser Machining A preliminary outline.
7. Introduction to the numerical integration of PDE. As an example, we consider the following PDE with one variable; Finite difference method is one of.
Elliptic PDEs and the Finite Difference Method
© Fluent Inc. 11/24/2015J1 Fluids Review TRN Overview of CFD Solution Methodologies.
Radiosity Jian Huang, CS594, Fall 2002 This set of slides reference the text book and slides used at Ohio State.
Using the Segregated and Coupled Solvers
Jump to first page Chapter 3 Splines Definition (3.1) : Given a function f defined on [a, b] and a set of numbers, a = x 0 < x 1 < x 2 < ……. < x n = b,
1 Spring 2003 Prof. Tim Warburton MA557/MA578/CS557 Lecture 24.
J. Murthy Purdue University
FALL 2015 Esra Sorgüven Öner
Discretization Methods Chapter 2. Training Manual May 15, 2001 Inventory # Discretization Methods Topics Equations and The Goal Brief overview.
Discretization for PDEs Chunfang Chen,Danny Thorne Adam Zornes, Deng Li CS 521 Feb., 9,2006.
Lecture Objectives: Continue with linearization of radiation and convection Example problem Modeling steps.
Lecture Objectives: Review discretization methods for advection diffusion equation –Accuracy –Numerical Stability Unsteady-state CFD –Explicit vs. Implicit.
Lecture Objectives: Define 1) Reynolds stresses and
01/27/03© 2002 University of Wisconsin Last Time Radiometry A lot of confusion about Irradiance and BRDFs –Clarrified (I hope) today Radiance.
1 Spring 2003 Prof. Tim Warburton MA557/MA578/CS557 Lecture 25.
ERT 216 HEAT & MASS TRANSFER Sem 2/ Dr Akmal Hadi Ma’ Radzi School of Bioprocess Engineering University Malaysia Perlis.
Programming assignment # 3 Numerical Methods for PDEs Spring 2007 Jim E. Jones.
Lecture Objectives: - Numerics. Finite Volume Method - Conservation of  for the finite volume w e w e l h n s P E W xx xx xx - Finite volume.
Computational Fluid Dynamics Lecture II Numerical Methods and Criteria for CFD Dr. Ugur GUVEN Professor of Aerospace Engineering.
Example application – Finite Volume Discretization Numerical Methods for PDEs Spring 2007 Jim E. Jones.
J. Murthy Purdue University
© Fluent Inc. 1/10/2018L1 Fluids Review TRN Solution Methods.
Lecture 19 MA471 Fall 2003.
Convergence in Computational Science
Objective Review Reynolds Navier Stokes Equations (RANS)
Objective Numerical methods.
Finite Volume Method for Unsteady Flows
Objective Numerical methods Finite volume.
Sathish Vadhiyar Courtesy: Dr. David Walker, Cardiff University
SKTN 2393 Numerical Methods for Nuclear Engineers
Linear Algebra Lecture 16.
Presentation transcript:

ME 595M J.Murthy1 ME 595M: Computational Methods for Nanoscale Thermal Transport Lecture 9: Introduction to the Finite Volume Method for the Gray BTE J. Murthy Purdue University

ME 595M J.Murthy2 Gray Phonon BTE Recall gray phonon BTE: e ” is energy per unit volume per unit solid angle and depends on direction vector s. So there are as many pde’s as there are s directions. In each direction, e ” varies in space and time The e ” values in different directions are related to each other because of e 0 in the scattering term: Notice that This implies that there is no net energy source – scattering only shifts energy from one direction to another How would you add an net energy source to the gray BTE?

ME 595M J.Murthy3 Overview of Finite Volume Method Divide spatial domain into control volumes of extent  x  y Divide angular domain into control angles of extent  Divide time into steps of  t - but will only do steady here for simplicity Consider gray BTE in direction s. Integrate gray BTE over control volume and corresponding control angle. Get energy conservation statement for that direction for each spatial control volume Do the same for all directions. Solve each direction sequentially and iteratively Back out “temperature” from e 0 upon convergence using s

ME 595M J.Murthy4 Discretization Divide domain into rectangular control volumes of extent  x and  y. Assume 2D, so that depth into page (z) is one unit. Divide angular domain of 4  in N  xN  control angles per octant. Centroid of each control angle is (  i,  i ), extents are ( ,  ). For each control angle i: Important: The directions s are 3D even though we are considering 2D x y y x z s

ME 595M J.Murthy5 Discretization (cont’d) Control angle extent is In 2D, only directions in the “front” hemisphere are necessary. Thus  ranges from 0-  /2 and  =0-2  Thus, increase control angle extent to: Define for future use:

ME 595M J.Murthy6 Formula for S

ME 595M J.Murthy7 Spatial Discretization P E N W S yy xx e w n s e” stored at cell centroids

ME 595M J.Murthy8 Control Volume Balance Integrate governing equation over control volume and control angle: faces f n s w e P s

ME 595M J.Murthy9 Control Volume Balance (cont’d) Now look at RHS Collecting terms: Control volume balance says that net rate of energy entering the CV in direction s i must be balanced by net in- scattering to the direction i in the CV

ME 595M J.Murthy10 Upwinding e” is stored at cell centroids, but we need it on the CV faces Need to interpolate from cell centroid to face Can use a variety of schemes to perform interpolation  Central difference scheme Second-order accurate, but wiggles in spatial solution  Upwind difference scheme  Computationally convenient to write P E W e

ME 595M J.Murthy11 Discussion Upwinding, as shown, is only a first-order accurate scheme  Guaranteed smooth, bounded solutions  False diffusion In CFD, a variety of higher-order upwind-weighted schemes have been developed which typically involve other upwind points (P, W for face e) Will go with first-order upwind scheme for now. P E W e

ME 595M J.Murthy12 Discrete Equation Using upwinding and collecting terms, we obtain an algebraic equation: We obtain one such equation for each grid point P for each direction i The b term contains e 0 iP Once we have boundary conditions discretized, we can solve the set

ME 595M J.Murthy13 A Closer Look Consider a direction s i with s x >0, s y >0  Point p only connected to points south and west of it  Influence of other directions in b term  Influence of b term increases as acoustic thickness L/(v g  eff )increases  Diagonally dominant Other directions appear here

ME 595M J.Murthy14 Coefficient Structure P E N W S e w n s

ME 595M J.Murthy15 Discussion Prefer to solve iteratively and if possible, sequentially to keep memory requirements low For upwind scheme, diagonal dominance is guaranteed, making it possible to use iterative schemes Conservation of energy is guaranteed regardless of spatial and angular discretization  Confirm that sum of all scattering source terms at a point is zero regardless of discretization Any linear solver can be used – will use line-by-line tri- diagonal matrix algorithm (LBL-TDMA) for now.

ME 595M J.Murthy16 Overall Solution Algorithm 1. Initialize all e ” i values for all cell centroids and directions 2. Find e 0 P for each point P from current e ” values. 3. Start with direction i=1 4. For direction i:  Find discretized equations for direction i, assuming e 0 temporarily known  Solve for e ” i at all grid points using LBL-TDMA  Increment I as i=i+1 5. If (i.le.4*N  *N  ) go to 4 6. If (i>4*N  *N  ) check for convergence. If converged, stop. Else, go to 2.

ME 595M J.Murthy17 Conclusions In this lecture, we discretized the gray BTE. The discretization is guaranteed to give energy conservation regardless of the fineness of the spatial or angular discretization The discretization guarantees diagonal dominance and is hence suitable for iterative solvers such as the LBL TDMA. The next time, we will talk briefly of boundary conditions, and start looking at a finite volume code to solve the BTE.