Lecture Notes. — The George Washington University Press, 2010. — 101 p.These lecture notes are intended to supplement a one-semester graduate-level engineering course at The George Washington University in numerical methods for the solution of partial differential equations. Both finite difference and finite element methods are included.
The main prerequisite is a standard undergraduate calculus sequence including ordinary differential equations. In general, the mix of topics and level of presentation are aimed at upper-level undergraduates and first-year graduate students in mechanical, aerospace, and civil engineering.
Gordon Everstine.
Gaithersburg, Maryland.Numerical Solution of Ordinary Differential Equations.
Euler’s Method.
Truncation Error for Euler’s Method.
Runge-Kutta Methods.
Systems of Equations.
Finite Differences.
Boundary Value Problems.
Example.
Solving Tridiagonal Systems.
Shooting Methods.
Partial Differential Equations.
Classical Equations of Mathematical Physics.
Classification of Partial Differential Equations.
Transformation to Nondimensional Form.
Finite Difference Solution of Partial Differential Equations.
Parabolic Equations.
Explicit Finite Difference Method.
Crank-Nicolson Implicit Method.
Derivative Boundary Conditions.
Hyperbolic Equations.
The d’Alembert Solution of the Wave Equation.
Finite Differences.
Starting Procedure for Explicit Algorithm.
Nonreflecting Boundaries.
Elliptic Equations.
Derivative Boundary Conditions.
Direct Finite Element Analysis.
Linear Mass-Spring Systems.
Matrix Assembly.
Constraints.
Example and Summary.
Pin-Jointed Rod Element.
Pin-Jointed Frame Example.
Boundary Conditions by Matrix Partitioning.
Alternative Approach to Constraints.
Beams in Flexure.
Direct Approach to Continuum Problems.
Change of Basis.
Tensors.
Examples of Tensors.
Isotropic Tensors.
Calculus of Variations.
Example 1: The Shortest Distance Between Two Points.
Example 2: The Brachistochrone.
Constraint Conditions.
Example 3: A Constrained Minimization Problem.
Functions of Several Independent Variables.
Example 4: Poisson’s Equation.
Functions of Several Dependent Variables.
Variational Approach to the Finite Element Method.
Index Notation and Summation Convention.
Deriving Variational Principles.
Shape Functions.
Variational Approach.
Matrices for Linear Triangle.
Interpretation of Functional.
Stiffness in Elasticity in Terms of Shape Functions.
Element Compatibility.
Method of Weighted Residuals (Galerkin’s Method).
Potential Fluid Flow With Finite Elements.
Finite Element Model.
Application of Symmetry.
Free Surface Flows.
Use of Complex Numbers and Phasors in Wave Problems.
D Wave Maker.
Linear Triangle Matrices for 2-D Wave Maker Problem.
Mechanical Analogy for the Free Surface Problem.
Bibliography.
