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EGN 5456 INTRO TO COMPUTATIONAL MECHANICS Fall 1996
Van Dommelen

GOALS
To familiarize students with basic properties of partial differential equations and their numerical discretizations.

PREREQUISITES
Some familiarity with complex Fourier series and integrals, fluid mechanics, partial differential equations.

INSTRUCTOR
Dr. Leon Van Dommelen (904) 487-6331
Office: T 3-4, 5-6 (before and after the ME seminar); W 10-11
dommelen@eng.famu.fsu.edu http://www.eng.famu.fsu.edu/ tex2html_wrap_inline136 dommelen

TIMES
Class: 12:55-1:45 MWF 223 CEB
Final: Monday 12/9/96 10:12 noon (FSU schedule)

TEXTBOOK
Selected notes will be provided.
  1.   Paul DuChateau and David W Zachmann Partial Differential Equations Schaum's Outline Series, McGraw-Hill (1986) ISBN 0-07-017897-6.
  2.   Bertil Gustafsson, Heinz-Otto Kreiss, and Joseph Oliger Time Dependent Problems and Difference Methods Wiley (1995) ISBN 0-471-50734-2. (Solid introduction to the fundamentals of finite difference methods. Will be used as a primary reference in this class.)

REFERENCES
Some recommended books:
  1.   Claudio Canuto, M. Yousuff Hussaini, A. Quarteroni, & T. A. Zang Spectral Methods in Fluid Mechanics Springer-Verlag (1987) ISBN 3-540-17371-4 (Berlin) 0-387-17371-4 (New York). (Standard reference on spectral methods.)
  2.   Germund Dahlquist and Åke Björck Numerical Methods Prentice Hall (1974) ISBN 0-13-627315-7. (Excellent introduction to basic numerical methods.)
  3.   Joel H. Ferziger Numerical Methods for Engineering Application Wiley (1981) ISBN 0-471-06336-3. (Stanford text.)
  4.   C.A.J. Fletcher Computational Techniques for Fluid Dynamics Vols. 1 and 2, Springer-Verlag (1988) ISBN 3-540-19466-5 (Berlin) 0-387-19466-5 (New York). (Good introduction to basic CFD.)
  5.   Wolfgang Hackbusch, Multi-Grid Methods and Application Springer-Verlag (1985) ISBN 0-387-12761-5.
  6.   Heinz Otto Kreiss Numerical Methods for Solving Time-Dependent problems for Partial Differential Equations Montreal: Les Presses de l'Universite de Montreal. Series: Seminaire de mathematiques superieures 65 (1978) ISBN 0840504306. (Read after Richtmyer & Morton.)
  7.   Leon Lapidus and George F. Pinder Numerical Solution of Partial Differential Equations in Science and Engineering Wiley-Interscience (1982) ISBN 0-471-09866-3. (Extensive reference on basic fundamentals.)
  8.   J.N. Reddy Functional Analysis and Variational Methods in Engineering McGraw-Hill (1986) ISBN 0-07-051348-1. (Read after Strang & Fix.)
  9.   Robert D. Richtmyer & K. W. Morton Difference Methods for Initial Value Problems Interscience, (1967) 2nd Ed. ISBN 0-470-72040-9. (Excellent but abstract introduction to the fundamentals of finite difference procedures.)
  10.   Gilbert Strang and George J. Fix An Analysis of the Finite Element Method Prentice-Hall, Englewood Cliffs, NJ (1973) ISBN 0-13-032946-0. (Excellent introduction to the fundamentals of the FE method.)
  11.   John C. Strikwerda Difference Schemes and Partial Differential Equations Wadsworth & Brooks/Cole, Belmont, CA (1989) ISBN 0-534-09984-X. (Solid introduction to the fundamentals of finite difference methods, but hard to read, many mistakes.)
  12.   Thomas A. Zang, C. L. Streett, & M. Y. Hussaini Spectral Methods for CFD ICASE Report No. 89-13, NASA CR 181803. (Similar to Canuto, but more concise, recent).

COURSE OUTLINE

Partial Differential Equations
As time and interest permits. Systems of first order equations arising in heat transfer, fluid and solid mechanics. Transformation of second order equations to first order systems. Quasi-linear and linear systems. Conservation laws. Shocks. Classification of partial differential equations and systems. Hyperbolic equations: characteristics, domains of influence and dependence, properly posed initial-boundary value problems. Fourier analysis. Parabolic equations, Fourier solutions, initial and boundary conditions. Elliptic equations, boundary conditions, Fourier solution, properly posed problems. [1,2,11,7]

Finite Difference Methods in One Dimension
As time and interest permits. Finite difference discretizations. Physical justifications. CFL condition, domain of dependence, maximum principles. Fourier solutions of difference equations. Taylor series expansions. Consistency, stability, dissipation and dispersion. Example problems from solid and fluid mechanics and heat transfer. Explicit and implicit schemes, upwinding, monotonicity preservation. [2,11,7,4]

Discretization Methods in Multiple Dimensions
As time and interest permits. General principles of curvi-linear grid generation. Finite difference discretizations on these grids. Finite volume discretizations. Galerkin and sub-domain finite element disretizations. [4]

Finite Element Methods in One Dimension
As time and interest permits. Weak formulation, Galerkin, collocation. Rayleigh-Ritz formulation. Finite elements, Lebesque integration, completeness, reduction of order. Energy/extremum principles. Convergence in natural and standard norms. [10,8].

Solution Methods in Multiple Dimensions
As time and interest permits. Discussion of direct and iterative solution procedures. Approximate factorization techniques, ADI and fractional step methods. Jacobi, Gauss-Seidel, SOR, multigrid, conjugate gradients, ILU methods. [11,7,3]

Spectral Methods
As time and interest permits. Introduction to the fast Fourier transform. Spectral accuracy. Spectral Galerkin, Tau, collocation. [1,12]

METHODS OF INSTRUCTION
Lectures, problem solving sessions, solution of selected problems on computers, examinations.

STUDENT EVALUATION
The course grade will be computed as: Students may not copy homework or tests or allow others to copy their homework or tests. Violations will result in reduced credit and a failing final grade.
Homework must be handed in at the start of the lecture at which it is due. It may not be handed in at the departmental office. Homework that is not received at the start of class on the due date listed above cannot be made up unless permission to hand in late has been given before the homework is due, or it was not humanly possible to ask for such permission before the class. If there is a chance you may be late in class, hand the homework in to the instructor the day before it is due. (Shove it under his door.)
Tests will be loosely based on the homework.
Immediately check the dates listed above for any conflicts.
Students are bound by the honor code of their university. It requires you to uphold academic integrity and combat academic dishonesty. Please see your student handbook.

COMPUTER USE
Requires knowledge of a major programming language such as Fortran, C, or Pascal.

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Author: Leon van Dommelen