Development of a Maxwell Equation Solver for Application to Two Fluid Plasma Models. C. Aberle, A. Hakim, and U. Shumlak
|
|
- Ashlee Norman
- 5 years ago
- Views:
Transcription
1 Development of a Maxwell Equation Solver for Application to Two Fluid Plasma Models C. Aberle, A. Hakim, and U. Shumlak Aerospace and Astronautics University of Washington, Seattle American Physical Society - Division of Computational Physics August 2002 San Diego, California For a copy of this poster, visit
2 Abstract A Maxwell Equation Solver is developed for use in two fluid plasma solvers. The application requires that the solver be adaptive to the geometry, handle discrete currents and charge distributions, and capture dynamic behavior. Two approaches, Finite Volume Time Domain (FVTD) and Mixed potentials, are investigated, compared, and results are presented. The FVTD scheme, Weighted Essentially Non-Oscillatory (WENO) is applied to Cartesian and triangular meshes. Solvers using Cartesian meshes perform adequately without currents, but they fail to accurately model situations with discrete currents. Solvers using triangular meshes successfully overcome this problem by reducing cross-term dissipation. The mixed potential solver is second order accurate in space and time. It uses quadratic upwind for the spatial fluxes and a central differencing for the time. Successful results are obtained in simulations with discrete currents.
3 Motivation Many current plasma simulation codes are based on the magnetohydrodynamic (MHD) model whose derivation involves several assumptions that severely limit its applicability, particularly for Hall effect physics. The two-fluid plasma model only assumes local thermodynamic equilibrium for each species and, therefore, more accurately models the appropriate physical processes. Accurate electromagnetic (EM) field solvers are needed to properly account for the interactions between the fields. The solver must be capable of handling propagation of EM waves in all directions; and must be able to account for charges and currents as both continuous and discrete sources. In this work, we develop and compare two different approaches to developing an EM field solver. The first approach solves the Maxwell equations using Finite Volume Time Domain (FVTD) methods and the second approach uses a mixed potential formulation with a quadratic upwind differencing scheme.
4 Maxwell Equations These equations are solved in two dimensions. The transverse electric mode subset of these equations are used.
5 Conservative Form of Equations Using a non-dimensionalization to set, Maxwell's equations are written in conservative form: where
6 Potential Formulation where with and connected by the Lorentz gauge condition
7 Potential Formulation These can be transformed from the physical space to the grid space using transformation formula for the Laplacian for each and where is the metric tensor, and is divergence operator in space.
8 Discretization Schemes To solve the Maxwell Equations, they must be discretized. Two approaches are used, Finite Volume Time Domain (FVTD) and potential formulation. The FVTD approach divides the domain into cells, and tracks the movement of quantities between the cells. This approach is based upon the methods of computational fluid dynamics (CFD) of the past several decades. Shang calls these 'characteristic methods'. The potential formulation uses a finite difference scheme to discretize the potential form of the Maxwell equations. The spatial derivatives are discretized using Quadratic Upwind Scheme (QUICK).
9 Finite Volume Time Domain (FVTD) For FVTD, integrating the conservative form of Maxwell equations over a cell, we get where is the cell area and is the length of the cell edge. is a numerical flux, the movement of the conserved variables between cells in the domain. Each FVTD scheme calculates and uses this value to update. FVTD is tested on two different types of grids, one with rectangular shaped cells, and the other with triangular shaped cells.
10 Upwind with Finite Volume The first order upwind finite volume scheme calculates the numerical flux using the quantities of the nearest cell upwind of the cell edge. For Maxwell equations, where these and are to lie in the x direction. Temporal discretization is accomplished using a simple first order Euler time integration.
11 Weighted Essentially Non- Oscillatory (WENO) Scheme WENO schemes are a variation on Essentially Non-Oscillatory (ENO) schemes. WENO and ENO schemes were developed by Harten, Shu, and others over the past decade. ENO schemes interpolate the flux from neighboring cells. An ENO scheme chooses those neighboring cells that have the smoothest solution. To form an ENO scheme, stencils are made of the cells neighboring the cell edge. For example, to interpolate on to the cell edge as shown, three stencils would be created for a 2 nd order solver. The first stencil would use cell 1 and 2. The second stencil would use cell 1 and 3. The third stencil would use cell 1 and 4. For rectangular grid cells, these stencils are used on the axis normal to the cell edge.
12 WENO WENO schemes recognize that for smooth solutions, all the cells can be safely used for interpolation. Therefore, instead of selecting the interpolation of only a single stencil, WENO performs a weighted average on the interpolations of every stencil. The stencils are weighted in such a way that a really smooth solution will be using a higher order interpolation consisting of all the cells. Using WENO schemes with first order Euler time is unstable. Time integration methods of higher order are needed. Using WENO on a rectangular grid uses an explicit Runge Kutta method of 3 rd order. Using WENO on a rectangular grid requires the use of an implicit time method. The implicit time method uses a residual iterative technique for each time step. By applying conjugate gradient repeatedly, the residual is driven towards zero for the time step.
13 Potential Formulation: Temporal Discretization To advance the solution in time, the temporal derivatives are discretized using central differencing scheme:
14 Potential Formulation: Spatial Discretization The spatial derivative are discretized by fitting a quadratic between two upwind points and one downwind point where are constants determined from the grid spacing and the values of at the upwind and downwind points. In practice it is simplest to explicitly determine from values at the neighboring nodes and then use them in calculating the needed derivatives. For uniform rectangular grids, the differencing formula reduces to
15 Sine Wave This simulation tests the propagation of an initial sine wave across the domain with periodic boundary conditions. The simulation runs for one time unit, at which time, the solution should look exactly like the original waveform. This gives a good measure of dissipation Standing wave in 1-D: QUICK scheme Solid lines shows the exact solution while dotted lines shows the solution obtained with QUICK scheme. Ex This uses 100 points while the FVTD used 20 points x
16 Sine Wave This shows the results of upwind on a rectangular mesh. As is shown, increasing the resolution of the mesh decreases the error, as anticipated. This shows the results of using 2 nd order WENO on a triangular mesh. As is shown, increasing the resolution of the mesh decreases the error, as shown. Note, the accuracy for the 20x20 is much better than the rectangular upwind.
17 Square Wire The schemes must be able to handle discrete currents within the domain. Therefore, a square wire is simulated and results are compared. This square wire is centered in the domain with absorbing boundary conditions. The simulation runs until steady state is achieved The resulting magnetic field is compared against the analytic magnetostatic solution. This analytic solution is determined by applying the Biot Savart law to this square wire. The size of the domain is ten times in length and a hundred time in area with respect to the wire Magnetic field around square Wire Solid lines shows the exact solution while dotted lines shows the solution obtained with QUICK scheme. Radial magnetic field Radial distance from wire center
18 Square Wire, Success & Failure Triangular Grid Rectangular Grid By looking at these plots, the upwind on the rectangular grid is an obvious failure. The upwind on the triangular grid is generating a reasonable solution. Therefore, the upwind on the rectangular grid is unable to a solution that is qualitatively reasonable.
19 Square Wire, FVTD Triangular Grid Rectangular Grid These plots illustrate the accuracy of the solvers. These plots are taken along the x=y diagonal of the domain. The upwind on the triangular grid performs quite, whereas the upwind on the rectangular grid reiterates its failures.
20 Analysis of Rectangular vs Triangular Grids The behavior of FVTD on rectangular and triangular grids is analyzed using the Von Neumann method. This type of analysis looks at the behavior of Fourier components to the solution. While Von Neumann is typically used to determine stability, the method obtains information about dissipation. The dissipation of the schemes is crucial to explain their behavior in the simulation of the square wire. Each Fourier component is designated as where the range is from 0 to +1. Each component is a separate 2D wave mode that has the spatial form The Von Neumann method calculates the Gain G that each mode experiences for each time step. If G >1, then the scheme is unstable. If G =1, then there is no dissipation. If G <1, then the scheme is dissipative. This method is applied to both the rectangular and triangular upwind FVTD schemes.
21 Rectangular vs Triangular Grids for the Scalar Wave Equation Triangular Grid Rectangular Grid This shows the dissipation behavior of the simple scalar wave propagation in the +xy diagonal direction. As can be seen by both mesh plots, the dissipation is acceptable and both grids would perform adequately. Given the simplicity of a rectangular grid, it is more favorable for the scalar equation.
22 Rectangular vs Triangular Grids for the Transverse Electric Mode (TEM) Triangular Grid Rectangular Grid These plots show the dissipation for the electric field component of TEM. Yikes, the dissipative behavior of the rectangular grid for the TEM is vastly different than for the scalar wave equation. It is this massive dissipation that accounts for the inability for the rectangular upwind to propagate along a diagonal. The triangular upwind is capable of diagonal propagation and this is justified by this dissipation plot.
23 Summary Electromagnetic field solvers for use in plasma simulations are developed in this work. Two different approaches are taken: a Finite Volume Time Domain (FVTD) approach and an approach using the mixed potential formulation. The FVTD method gives accurate results for simple problems. For problems requiring sources, FVTD on rectangular grids fails while it succeeds on triangular grids. The mixed potential formulation using quadratic upwinding gives accurate results on arbitrary two dimensional grids for the test problems considered. It is seen that the solver is able to account for discrete sources well. Although preliminary results look promising, much work needs to be done to test the efficacy of this approach in plasma simulations.
24 Summary The two methods developed give comparable results, at least for the test problems considered. The FVTD method is more flexible as it can deal with arbitrary grids, while the potential solver needs a transformation from physical to logical space. Further, as the potential solver uses finite differences, it may be difficult to handle shocks using it. Also, the conversion to E and B fields from the potentials involves a loss in accuracy.
CS205b/CME306. Lecture 9
CS205b/CME306 Lecture 9 1 Convection Supplementary Reading: Osher and Fedkiw, Sections 3.3 and 3.5; Leveque, Sections 6.7, 8.3, 10.2, 10.4. For a reference on Newton polynomial interpolation via divided
More informationThe WENO Method in the Context of Earlier Methods To approximate, in a physically correct way, [3] the solution to a conservation law of the form u t
An implicit WENO scheme for steady-state computation of scalar hyperbolic equations Sigal Gottlieb Mathematics Department University of Massachusetts at Dartmouth 85 Old Westport Road North Dartmouth,
More informationMid-Year Report. Discontinuous Galerkin Euler Equation Solver. Friday, December 14, Andrey Andreyev. Advisor: Dr.
Mid-Year Report Discontinuous Galerkin Euler Equation Solver Friday, December 14, 2012 Andrey Andreyev Advisor: Dr. James Baeder Abstract: The focus of this effort is to produce a two dimensional inviscid,
More informationFinal Report. Discontinuous Galerkin Compressible Euler Equation Solver. May 14, Andrey Andreyev. Adviser: Dr. James Baeder
Final Report Discontinuous Galerkin Compressible Euler Equation Solver May 14, 2013 Andrey Andreyev Adviser: Dr. James Baeder Abstract: In this work a Discontinuous Galerkin Method is developed for compressible
More information1.2 Numerical Solutions of Flow Problems
1.2 Numerical Solutions of Flow Problems DIFFERENTIAL EQUATIONS OF MOTION FOR A SIMPLIFIED FLOW PROBLEM Continuity equation for incompressible flow: 0 Momentum (Navier-Stokes) equations for a Newtonian
More informationStudies of the Continuous and Discrete Adjoint Approaches to Viscous Automatic Aerodynamic Shape Optimization
Studies of the Continuous and Discrete Adjoint Approaches to Viscous Automatic Aerodynamic Shape Optimization Siva Nadarajah Antony Jameson Stanford University 15th AIAA Computational Fluid Dynamics Conference
More informationA-posteriori Diffusion Analysis of Numerical Schemes in Wavenumber Domain
2th Annual CFD Symposium, August 9-1, 218, Bangalore A-posteriori Diffusion Analysis of Numerical Schemes in Wavenumber Domain S. M. Joshi & A. Chatterjee Department of Aerospace Engineering Indian Institute
More informationNIA CFD Seminar, October 4, 2011 Hyperbolic Seminar, NASA Langley, October 17, 2011
NIA CFD Seminar, October 4, 2011 Hyperbolic Seminar, NASA Langley, October 17, 2011 First-Order Hyperbolic System Method If you have a CFD book for hyperbolic problems, you have a CFD book for all problems.
More informationBACK AND FORTH ERROR COMPENSATION AND CORRECTION METHODS FOR REMOVING ERRORS INDUCED BY UNEVEN GRADIENTS OF THE LEVEL SET FUNCTION
BACK AND FORTH ERROR COMPENSATION AND CORRECTION METHODS FOR REMOVING ERRORS INDUCED BY UNEVEN GRADIENTS OF THE LEVEL SET FUNCTION TODD F. DUPONT AND YINGJIE LIU Abstract. We propose a method that significantly
More informationA Toolbox of Level Set Methods
A Toolbox of Level Set Methods Ian Mitchell Department of Computer Science University of British Columbia http://www.cs.ubc.ca/~mitchell mitchell@cs.ubc.ca research supported by the Natural Science and
More informationDebojyoti Ghosh. Adviser: Dr. James Baeder Alfred Gessow Rotorcraft Center Department of Aerospace Engineering
Debojyoti Ghosh Adviser: Dr. James Baeder Alfred Gessow Rotorcraft Center Department of Aerospace Engineering To study the Dynamic Stalling of rotor blade cross-sections Unsteady Aerodynamics: Time varying
More informationThis is an author-deposited version published in: Eprints ID: 4362
This is an author-deposited version published in: http://oatao.univ-toulouse.fr/ Eprints ID: 4362 To cite this document: CHIKHAOUI Oussama, GRESSIER Jérémie, GRONDIN Gilles. Assessment of the Spectral
More informationHigh-Order Finite Difference Schemes for computational MHD
High-Order Finite Difference Schemes for computational MHD A. Mignone 1, P. Tzeferacos 1 and G. Bodo 2 [1] Dipartimento di Fisica Generale, Turin University, ITALY [2] INAF Astronomic Observatory of Turin,,
More informationPartial Differential Equations
Simulation in Computer Graphics Partial Differential Equations Matthias Teschner Computer Science Department University of Freiburg Motivation various dynamic effects and physical processes are described
More informationFaculty of Mechanical and Manufacturing Engineering, University Tun Hussein Onn Malaysia (UTHM), Parit Raja, Batu Pahat, Johor, Malaysia
Applied Mechanics and Materials Vol. 393 (2013) pp 305-310 (2013) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/amm.393.305 The Implementation of Cell-Centred Finite Volume Method
More informationComputer Project 3. AA Computational Fluid Dyanmics University of Washington. Mishaal Aleem March 17, 2015
Computer Project 3 AA 543 - Computational Fluid Dyanmics University of Washington Mishaal Aleem March 17, 2015 Contents Introduction........................................... 1 3.1 Grid Generator.......................................
More informationAdaptive Mesh Astrophysical Fluid Simulations on GPU. San Jose 10/2/2009 Peng Wang, NVIDIA
Adaptive Mesh Astrophysical Fluid Simulations on GPU San Jose 10/2/2009 Peng Wang, NVIDIA Overview Astrophysical motivation & the Enzo code Finite volume method and adaptive mesh refinement (AMR) CUDA
More informationFluent User Services Center
Solver Settings 5-1 Using the Solver Setting Solver Parameters Convergence Definition Monitoring Stability Accelerating Convergence Accuracy Grid Independence Adaption Appendix: Background Finite Volume
More informationOn the order of accuracy and numerical performance of two classes of finite volume WENO schemes
On the order of accuracy and numerical performance of two classes of finite volume WENO schemes Rui Zhang, Mengping Zhang and Chi-Wang Shu November 29, 29 Abstract In this paper we consider two commonly
More informationNumerical Methods for (Time-Dependent) HJ PDEs
Numerical Methods for (Time-Dependent) HJ PDEs Ian Mitchell Department of Computer Science The University of British Columbia research supported by National Science and Engineering Research Council of
More informationUnstructured Mesh Generation for Implicit Moving Geometries and Level Set Applications
Unstructured Mesh Generation for Implicit Moving Geometries and Level Set Applications Per-Olof Persson (persson@mit.edu) Department of Mathematics Massachusetts Institute of Technology http://www.mit.edu/
More informationA methodology for the rigorous verification of plasma simulation codes
A methodology for the rigorous verification of plasma simulation codes Fabio Riva P. Ricci, C. Beadle, F.D. Halpern, S. Jolliet, J. Loizu, J. Morales, A. Mosetto, P. Paruta, C. Wersal École Polytechnique
More informationVon Neumann Analysis for Higher Order Methods
1. Introduction Von Neumann Analysis for Higher Order Methods Von Neumann analysis is a widely used method to study how an initial wave is propagated with certain numerical schemes for a linear wave equation
More informationDevelopment of an Integrated Computational Simulation Method for Fluid Driven Structure Movement and Acoustics
Development of an Integrated Computational Simulation Method for Fluid Driven Structure Movement and Acoustics I. Pantle Fachgebiet Strömungsmaschinen Karlsruher Institut für Technologie KIT Motivation
More informationNew Very High-Order Upwind Multilayer Compact Schemes with Spectral-Like Resolution for Flow Simulations
New Very High-Order Upwind Multilayer Compact Schemes with Spectral-Lie Resolution for Flow Simulations Zeyu Bai and Xiaolin Zhong University of California, Los Angeles, CA, 995, USA Hypersonic boundary
More informationTechniques for Using the Method of Manufactured Solutions for Verification and Uncertainty Quantification of CFD Simulations Having Discontinuities
Techniques for Using the Method of Manufactured Solutions for Verification and Uncertainty Quantification of CFD Simulations Having Discontinuities Ben Grier Clemson University Richard Figliola, Larry
More informationThird Order WENO Scheme on Three Dimensional Tetrahedral Meshes
COMMUNICATIONS IN COMPUTATIONAL PHYSICS Vol. 5, No. 2-4, pp. 86-848 Commun. Comput. Phys. February 29 Third Order WENO Scheme on Three Dimensional Tetrahedral Meshes Yong-Tao Zhang 1, and Chi-Wang Shu
More informationNumerical Analysis of Shock Tube Problem by using TVD and ACM Schemes
Numerical Analysis of Shock Tube Problem by using TVD and Schemes Dr. Mukkarum Husain, Dr. M. Nauman Qureshi, Syed Zaid Hasany IST Karachi, Email: mrmukkarum@yahoo.com Abstract Computational Fluid Dynamics
More informationHigh-Order Numerical Algorithms for Steady and Unsteady Simulation of Viscous Compressible Flow with Shocks (Grant FA )
High-Order Numerical Algorithms for Steady and Unsteady Simulation of Viscous Compressible Flow with Shocks (Grant FA9550-07-0195) Sachin Premasuthan, Kui Ou, Patrice Castonguay, Lala Li, Yves Allaneau,
More informationHigh-Order CENO Reconstruction Scheme For Three-Dimensional Unstructured Mesh
High-Order CENO Reconstruction Scheme For Three-Dimensional Unstructured Mesh by Al-Amin Aziz A thesis submitted in conformity with the requirements for the degree of Masters of Applied Science Graduate
More informationOutline. Level Set Methods. For Inverse Obstacle Problems 4. Introduction. Introduction. Martin Burger
For Inverse Obstacle Problems Martin Burger Outline Introduction Optimal Geometries Inverse Obstacle Problems & Shape Optimization Sensitivity Analysis based on Gradient Flows Numerical Methods University
More informationComputational Astrophysics 5 Higher-order and AMR schemes
Computational Astrophysics 5 Higher-order and AMR schemes Oscar Agertz Outline - The Godunov Method - Second-order scheme with MUSCL - Slope limiters and TVD schemes - Characteristics tracing and 2D slopes.
More informationHigh-order, conservative, finite difference schemes for computational MHD
High-order, conservative, finite difference schemes for computational MHD A. Mignone 1, P. Tzeferacos 1 and G. Bodo 2 [1] Dipartimento di Fisica Generale, Turin University, ITALY [2] INAF Astronomic Observatory
More informationA Hybrid Magnetic Field Solver Using a Combined Finite Element/Boundary Element Field Solver
A Hybrid Magnetic Field Solver Using a Combined Finite Element/Boundary Element Field Solver Abstract - The dominant method to solve magnetic field problems is the finite element method. It has been used
More informationMulti-Mesh CFD. Chris Roy Chip Jackson (1 st year PhD student) Aerospace and Ocean Engineering Department Virginia Tech
Multi-Mesh CFD Chris Roy Chip Jackson (1 st year PhD student) Aerospace and Ocean Engineering Department Virginia Tech cjroy@vt.edu May 21, 2014 CCAS Program Review, Columbus, OH 1 Motivation Automated
More informationFAST ALGORITHMS FOR CALCULATIONS OF VISCOUS INCOMPRESSIBLE FLOWS USING THE ARTIFICIAL COMPRESSIBILITY METHOD
TASK QUARTERLY 12 No 3, 273 287 FAST ALGORITHMS FOR CALCULATIONS OF VISCOUS INCOMPRESSIBLE FLOWS USING THE ARTIFICIAL COMPRESSIBILITY METHOD ZBIGNIEW KOSMA Institute of Applied Mechanics, Technical University
More informationA Higher-Order Accurate Unstructured Finite Volume Newton-Krylov Algorithm for Inviscid Compressible Flows
A Higher-Order Accurate Unstructured Finite Volume Newton-Krylov Algorithm for Inviscid Compressible Flows by AMIR NEJAT B.Sc. (Aerospace Engineering), AmirKabir University of Technology, 1996 M.Sc. (Aerospace
More informationLagrangian methods and Smoothed Particle Hydrodynamics (SPH) Computation in Astrophysics Seminar (Spring 2006) L. J. Dursi
Lagrangian methods and Smoothed Particle Hydrodynamics (SPH) Eulerian Grid Methods The methods covered so far in this course use an Eulerian grid: Prescribed coordinates In `lab frame' Fluid elements flow
More informationOn the high order FV schemes for compressible flows
Applied and Computational Mechanics 1 (2007) 453-460 On the high order FV schemes for compressible flows J. Fürst a, a Faculty of Mechanical Engineering, CTU in Prague, Karlovo nám. 13, 121 35 Praha, Czech
More informationOn the Construction, Comparison, and Local Characteristic Decomposition for High-Order Central WENO Schemes
Journal of Computational Physics 8, 87 09 (00) doi:0.006/jcph.00.79 On the Construction, Comparison, and Local Characteristic Decomposition for High-Order Central WENO Schemes Jianxian Qiu, and Chi-Wang
More informationRecent developments for the multigrid scheme of the DLR TAU-Code
www.dlr.de Chart 1 > 21st NIA CFD Seminar > Axel Schwöppe Recent development s for the multigrid scheme of the DLR TAU-Code > Apr 11, 2013 Recent developments for the multigrid scheme of the DLR TAU-Code
More informationNumerical Methods for Hyperbolic and Kinetic Equations
Numerical Methods for Hyperbolic and Kinetic Equations Organizer: G. Puppo Phenomena characterized by conservation (or balance laws) of physical quantities are modelled by hyperbolic and kinetic equations.
More informationHigh Order Weighted Essentially Non-Oscillatory Schemes for Convection. Dominated Problems. Chi-Wang Shu 1
High Order Weighted Essentially Non-Oscillatory Schemes for Convection Dominated Problems Chi-Wang Shu Division of Applied Mathematics, Brown University, Providence, Rhode Island 09 ABSTRACT High order
More informationMass-Spring Systems. Last Time?
Mass-Spring Systems Last Time? Implicit Surfaces & Marching Cubes/Tetras Collision Detection & Conservative Bounding Regions Spatial Acceleration Data Structures Octree, k-d tree, BSF tree 1 Today Particle
More informationOn the thickness of discontinuities computed by THINC and RK schemes
The 9th Computational Fluid Dynamics Symposium B7- On the thickness of discontinuities computed by THINC and RK schemes Taku Nonomura, ISAS, JAXA, Sagamihara, Kanagawa, Japan, E-mail:nonomura@flab.isas.jaxa.jp
More informationNUMERICAL VISCOSITY. Convergent Science White Paper. COPYRIGHT 2017 CONVERGENT SCIENCE. All rights reserved.
Convergent Science White Paper COPYRIGHT 2017 CONVERGENT SCIENCE. All rights reserved. This document contains information that is proprietary to Convergent Science. Public dissemination of this document
More informationA DETAILED COMPARISON OF WENO AND SFV HIGH-ORDER METHODS FOR INVISCID AERODYNAMIC APPLICATIONS
Blucher Mechanical Engineering Proceedings May 2014, vol. 1, num. 1 www.proceedings.blucher.com.br/evento/10wccm A DETAILED COMPARISON OF WENO AND SFV HIGH-ORDER METHODS FOR INVISCID AERODYNAMIC APPLICATIONS
More informationMiddle School Math Course 3
Middle School Math Course 3 Correlation of the ALEKS course Middle School Math Course 3 to the Texas Essential Knowledge and Skills (TEKS) for Mathematics Grade 8 (2012) (1) Mathematical process standards.
More informationApplication of Finite Volume Method for Structural Analysis
Application of Finite Volume Method for Structural Analysis Saeed-Reza Sabbagh-Yazdi and Milad Bayatlou Associate Professor, Civil Engineering Department of KNToosi University of Technology, PostGraduate
More informationModule 1: Introduction to Finite Difference Method and Fundamentals of CFD Lecture 13: The Lecture deals with:
The Lecture deals with: Some more Suggestions for Improvement of Discretization Schemes Some Non-Trivial Problems with Discretized Equations file:///d /chitra/nptel_phase2/mechanical/cfd/lecture13/13_1.htm[6/20/2012
More informationTutorial 1. Introduction to Using FLUENT: Fluid Flow and Heat Transfer in a Mixing Elbow
Tutorial 1. Introduction to Using FLUENT: Fluid Flow and Heat Transfer in a Mixing Elbow Introduction This tutorial illustrates the setup and solution of the two-dimensional turbulent fluid flow and heat
More informationThe Finite Element Method
The Finite Element Method A Practical Course G. R. Liu and S. S. Quek Chapter 1: Computational modeling An overview 1 CONTENTS INTRODUCTION PHYSICAL PROBLEMS IN ENGINEERING COMPUTATIONAL MODELLING USING
More informationFOURTH ORDER COMPACT FORMULATION OF STEADY NAVIER-STOKES EQUATIONS ON NON-UNIFORM GRIDS
International Journal of Mechanical Engineering and Technology (IJMET Volume 9 Issue 10 October 2018 pp. 179 189 Article ID: IJMET_09_10_11 Available online at http://www.iaeme.com/ijmet/issues.asp?jtypeijmet&vtype9&itype10
More informationCompressible Flow in a Nozzle
SPC 407 Supersonic & Hypersonic Fluid Dynamics Ansys Fluent Tutorial 1 Compressible Flow in a Nozzle Ahmed M Nagib Elmekawy, PhD, P.E. Problem Specification Consider air flowing at high-speed through a
More informationConservative high order semi-lagrangian finite difference WENO methods for advection in incompressible flow. Abstract
Conservative high order semi-lagrangian finite difference WENO methods for advection in incompressible flow Jing-Mei Qiu 1 and Chi-Wang Shu Abstract In this paper, we propose a semi-lagrangian finite difference
More informationMESHLESS SOLUTION OF INCOMPRESSIBLE FLOW OVER BACKWARD-FACING STEP
Vol. 12, Issue 1/2016, 63-68 DOI: 10.1515/cee-2016-0009 MESHLESS SOLUTION OF INCOMPRESSIBLE FLOW OVER BACKWARD-FACING STEP Juraj MUŽÍK 1,* 1 Department of Geotechnics, Faculty of Civil Engineering, University
More informationPartition Design and Optimization for High-Order Spectral Volume Schemes on Tetrahedral Grids
48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition 4-7 January 200, Orlando, Florida AIAA 200-720 Partition Design and Optimization for High-Order Spectral Volume
More informationPost Processing, Visualization, and Sample Output
Chapter 7 Post Processing, Visualization, and Sample Output Upon successful execution of an ADCIRC run, a number of output files will be created. Specifically which files are created depends upon how the
More informationObservations on the fifth-order WENO method with non-uniform meshes
Observations on the fifth-order WENO method with non-uniform meshes Rong Wang, a, Hui Feng, b, Raymond J. Spiteri a,, a Department of Computer Science, University of Saskatchewan, Saskatoon, SK, S7N 5C9,
More informationWeno Scheme for Transport Equation on Unstructured Grids with a DDFV Approach
Weno Scheme for Transport Equation on Unstructured Grids with a DDFV Approach Florence Hubert and Rémi Tesson Abstract In this paper we develop a DDFV approach for WENO scheme on unstructred grids for
More informationAvailable online at ScienceDirect. Procedia Engineering 99 (2015 )
Available online at www.sciencedirect.com ScienceDirect Procedia Engineering 99 (2015 ) 575 580 APISAT2014, 2014 Asia-Pacific International Symposium on Aerospace Technology, APISAT2014 A 3D Anisotropic
More informationAn Efficient, Geometric Multigrid Solver for the Anisotropic Diffusion Equation in Two and Three Dimensions
1 n Efficient, Geometric Multigrid Solver for the nisotropic Diffusion Equation in Two and Three Dimensions Tolga Tasdizen, Ross Whitaker UUSCI-2004-002 Scientific Computing and Imaging Institute University
More informationGENUINELY MULTIDIMENSIONAL NON-DISSIPATIVE FINITE VOLUME SCHEMES FOR TRANSPORT
GENUINELY MULTIDIMENSIONAL NON-DISSIPATIVE FINITE VOLUME SCHEMES FOR TRANSPORT BRUNO DESPRÉS, FRÉDÉRIC LAGOUTIÈRE Commissariat à l Énergie Atomique, Bruyères-le-Châtel. despres@cmpax.polytechnique.fr e-mail:
More informationParallel Algorithms: Adaptive Mesh Refinement (AMR) method and its implementation
Parallel Algorithms: Adaptive Mesh Refinement (AMR) method and its implementation Massimiliano Guarrasi m.guarrasi@cineca.it Super Computing Applications and Innovation Department AMR - Introduction Solving
More informationStudy on the Numerical Accuracy for the CFD
Study on the Numerical Accuracy for the CFD T.Yamanashi 1, H.Uchida, and M.Morita 1 Department of Mathematics, Master s Research Course of Faculty of Science, Tokyo University of Science,1-3 Kagurazaka,
More informationSemester Final Report
CSUMS SemesterFinalReport InLaTex AnnKimball 5/20/2009 ThisreportisageneralsummaryoftheaccumulationofknowledgethatIhavegatheredthroughoutthis semester. I was able to get a birds eye view of many different
More informationAccuracy Analysis of Charged Particle Trajectory CAE Software
www.integratedsoft.com Accuracy Analysis of Charged Particle Trajectory CAE Software Content Executive Summary... 3 Overview of Charged Particle Beam Analysis... 3 Types of Field Distribution... 4 Simulating
More informationAdvective and conservative semi-lagrangian schemes on uniform and non-uniform grids
Advective and conservative semi-lagrangian schemes on uniform and non-uniform grids M. Mehrenberger Université de Strasbourg and Max-Planck Institut für Plasmaphysik 5 September 2013 M. Mehrenberger (UDS
More informationA fully implicit Navier-Stokes algorithm for unstructured grids incorporating a two-equation turbulence model
Copyright 1996, American Institute of Aeronautics and Astronautics, Inc. AIAA Meeting Papers on Disc, January 1996 A9618376, AIAA Paper 96-0414 A fully implicit Navier-Stokes algorithm for unstructured
More informationOutline. Visualization Discretization Sampling Quantization Representation Continuous Discrete. Noise
Fundamentals Data Outline Visualization Discretization Sampling Quantization Representation Continuous Discrete Noise 2 Data Data : Function dependent on one or more variables. Example Audio (1D) - depends
More informationModeling Unsteady Compressible Flow
Tutorial 4. Modeling Unsteady Compressible Flow Introduction In this tutorial, FLUENT s density-based implicit solver is used to predict the timedependent flow through a two-dimensional nozzle. As an initial
More informationThe Immersed Interface Method
The Immersed Interface Method Numerical Solutions of PDEs Involving Interfaces and Irregular Domains Zhiiin Li Kazufumi Ito North Carolina State University Raleigh, North Carolina Society for Industrial
More informationParameterization of Meshes
2-Manifold Parameterization of Meshes What makes for a smooth manifold? locally looks like Euclidian space collection of charts mutually compatible on their overlaps form an atlas Parameterizations are
More informationThe Development of a Navier-Stokes Flow Solver with Preconditioning Method on Unstructured Grids
Proceedings of the International MultiConference of Engineers and Computer Scientists 213 Vol II, IMECS 213, March 13-15, 213, Hong Kong The Development of a Navier-Stokes Flow Solver with Preconditioning
More information1.7.1 Laplacian Smoothing
1.7.1 Laplacian Smoothing 320491: Advanced Graphics - Chapter 1 434 Theory Minimize energy functional total curvature estimate by polynomial-fitting non-linear (very slow!) 320491: Advanced Graphics -
More informationA Parallel Explicit/Implicit Time Stepping Scheme on Block-Adaptive Grids
A Parallel Explicit/Implicit Time Stepping Scheme on Block-Adaptive Grids Gábor Tóth, Darren L. De Zeeuw, Tamas I. Gombosi, Kenneth G. Powell Center for Space Environment Modeling, University of Michigan,
More informationLimiters for Unstructured Higher-Order Accurate Solutions of the Euler Equations
Limiters for Unstructured Higher-Order Accurate Solutions of the Euler Equations Krzysztof Michalak and Carl Ollivier-Gooch Advanced Numerical Simulation Laboratory University of British Columbia Higher-order
More informationNonoscillatory Central Schemes on Unstructured Triangulations for Hyperbolic Systems of Conservation Laws
Nonoscillatory Central Schemes on Unstructured Triangulations for Hyperbolic Systems of Conservation Laws Ivan Christov Bojan Popov Department of Mathematics, Texas A&M University, College Station, Texas
More informationSkåne University Hospital Lund, Lund, Sweden 2 Deparment of Numerical Analysis, Centre for Mathematical Sciences, Lund University, Lund, Sweden
Volume Tracking: A New Method for Visualization of Intracardiac Blood Flow from Three-Dimensional, Time-Resolved, Three-Component Magnetic Resonance Velocity Mapping Appendix: Theory and Numerical Implementation
More informationMATHEMATICAL ANALYSIS, MODELING AND OPTIMIZATION OF COMPLEX HEAT TRANSFER PROCESSES
MATHEMATICAL ANALYSIS, MODELING AND OPTIMIZATION OF COMPLEX HEAT TRANSFER PROCESSES Goals of research Dr. Uldis Raitums, Dr. Kārlis Birģelis To develop and investigate mathematical properties of algorithms
More informationAnalysis of the Adjoint Euler Equations as used for Gradient-based Aerodynamic Shape Optimization
Analysis of the Adjoint Euler Equations as used for Gradient-based Aerodynamic Shape Optimization Final Presentation Dylan Jude Graduate Research Assistant University of Maryland AMSC 663/664 May 4, 2017
More informationMetafor FE Software. 2. Operator split. 4. Rezoning methods 5. Contact with friction
ALE simulations ua sus using Metafor eao 1. Introduction 2. Operator split 3. Convection schemes 4. Rezoning methods 5. Contact with friction 1 Introduction EULERIAN FORMALISM Undistorted mesh Ideal for
More informationTheoretical Background for OpenLSTO v0.1: Open Source Level Set Topology Optimization. M2DO Lab 1,2. 1 Cardiff University
Theoretical Background for OpenLSTO v0.1: Open Source Level Set Topology Optimization M2DO Lab 1,2 1 Cardiff University 2 University of California, San Diego November 2017 A brief description of theory
More informationMATLAB. Advanced Mathematics and Mechanics Applications Using. Third Edition. David Halpern University of Alabama CHAPMAN & HALL/CRC
Advanced Mathematics and Mechanics Applications Using MATLAB Third Edition Howard B. Wilson University of Alabama Louis H. Turcotte Rose-Hulman Institute of Technology David Halpern University of Alabama
More informationA Hybrid Cartesian Grid and Gridless Method for Compressible Flows
rd AIAA Aerospace Sciences Meeting and Exhibit,, January 5, Reno, Nevada A Hybrid Cartesian Grid and Gridless Method for Compressible Flows Hong Luo and Joseph D. Baum Science Applications International
More informationThe Level Set Method. Lecture Notes, MIT J / 2.097J / 6.339J Numerical Methods for Partial Differential Equations
The Level Set Method Lecture Notes, MIT 16.920J / 2.097J / 6.339J Numerical Methods for Partial Differential Equations Per-Olof Persson persson@mit.edu March 7, 2005 1 Evolving Curves and Surfaces Evolving
More informationA new multidimensional-type reconstruction and limiting procedure for unstructured (cell-centered) FVs solving hyperbolic conservation laws
HYP 2012, Padova A new multidimensional-type reconstruction and limiting procedure for unstructured (cell-centered) FVs solving hyperbolic conservation laws Argiris I. Delis & Ioannis K. Nikolos (TUC)
More informationContents. I The Basic Framework for Stationary Problems 1
page v Preface xiii I The Basic Framework for Stationary Problems 1 1 Some model PDEs 3 1.1 Laplace s equation; elliptic BVPs... 3 1.1.1 Physical experiments modeled by Laplace s equation... 5 1.2 Other
More informationHigh Order Schemes for CFD: A Review. Juan Cheng 1. Institute of Applied Physics and Computational Mathematics, Beijing , China.
High Order Schemes for CFD: A Review Juan Cheng 1 Institute of Applied Physics and Computational Mathematics, Beijing 100088, China and Chi-Wang Shu 2 Division of Applied Mathematics, Brown University,
More informationIntroduction to Computational Fluid Dynamics Mech 122 D. Fabris, K. Lynch, D. Rich
Introduction to Computational Fluid Dynamics Mech 122 D. Fabris, K. Lynch, D. Rich 1 Computational Fluid dynamics Computational fluid dynamics (CFD) is the analysis of systems involving fluid flow, heat
More informationIntroduction to C omputational F luid Dynamics. D. Murrin
Introduction to C omputational F luid Dynamics D. Murrin Computational fluid dynamics (CFD) is the science of predicting fluid flow, heat transfer, mass transfer, chemical reactions, and related phenomena
More informationOverview of Traditional Surface Tracking Methods
Liquid Simulation With Mesh-Based Surface Tracking Overview of Traditional Surface Tracking Methods Matthias Müller Introduction Research lead of NVIDIA PhysX team PhysX GPU acc. Game physics engine www.nvidia.com\physx
More informationA Central Compact-Reconstruction WENO Method for Hyperbolic Conservation Laws
AIAA SciTech Forum 8 January 08, Kissimmee, Florida 08 AIAA Aerospace Sciences Meeting 0.54/6.08-0067 A Central Compact-Reconstruction WENO Method for Hyperbolic Conservation Laws Kilian Cooley and Dr.
More informationHigh Order Fixed-Point Sweeping WENO Methods for Steady State of Hyperbolic Conservation Laws and Its Convergence Study
Commun. Comput. Phys. doi:.48/cicp.375.6a Vol., No. 4, pp. 835-869 October 6 High Order Fixed-Point Sweeping WENO Methods for Steady State of Hyperbolic Conservation Laws and Its Convergence Study Liang
More informationA HYBRID SEMI-PRIMITIVE SHOCK CAPTURING SCHEME FOR CONSERVATION LAWS
Eighth Mississippi State - UAB Conference on Differential Equations and Computational Simulations. Electronic Journal of Differential Equations, Conf. 9 (), pp. 65 73. ISSN: 7-669. URL: http://ejde.math.tstate.edu
More informationCFD MODELING FOR PNEUMATIC CONVEYING
CFD MODELING FOR PNEUMATIC CONVEYING Arvind Kumar 1, D.R. Kaushal 2, Navneet Kumar 3 1 Associate Professor YMCAUST, Faridabad 2 Associate Professor, IIT, Delhi 3 Research Scholar IIT, Delhi e-mail: arvindeem@yahoo.co.in
More informationLevel set methods Formulation of Interface Propagation Boundary Value PDE Initial Value PDE Motion in an externally generated velocity field
Level Set Methods Overview Level set methods Formulation of Interface Propagation Boundary Value PDE Initial Value PDE Motion in an externally generated velocity field Convection Upwind ddifferencingi
More informationRevision of the SolidWorks Variable Pressure Simulation Tutorial J.E. Akin, Rice University, Mechanical Engineering. Introduction
Revision of the SolidWorks Variable Pressure Simulation Tutorial J.E. Akin, Rice University, Mechanical Engineering Introduction A SolidWorks simulation tutorial is just intended to illustrate where to
More informationIMA Preprint Series # 2033
FOURTH ORDER PARTIAL DIFFERENTIAL EQUATIONS ON GENERAL GEOMETRIES By John B. Greer Andrea L. Bertozzi and Guillermo Sapiro IMA Preprint Series # 2033 ( March 2005 ) INSTITUTE FOR MATHEMATICS AND ITS APPLICATIONS
More informationThe Total Variation Approach to Approximate Hyperbolic Wave Equations
2 CUMMING The Total Variation Approach to Approximate Hyperbolic Wave Equations Christine Cumming Faculty Sponsor: Dr. Robert H. Hoar, Department of Math ABSTRACT From ultrasound for medical purposes to
More information