MECH345 Introduction to Finite Element Methods Chapter 1 Numerical Methods - Introduction

Numerical Methods - Introduction

Definition: - Methods that seek quantitative approximations to the solutions of mathematic problems. Applications: - structural and stress analysis - thermal analysis - dynamic analysis - acoustic analysis - manufacturing process modeling - fluid mechanics - ………….

Numerical Integration Calculate: Newton – Cotes integration Trapezoidal rule – 1 st order Newton-Cotes integration Trapezoidal rule – multiple application

Numerical Integration Calculate: Newton – Cotes integration Simpson 1/3 rule – 2 nd order Newton-Cotes integration

Numerical Integration Calculate: Gauss Quadrature Trapezoidal Rule : Gauss Quadrature : Chooseaccording to certain criteria

Numerical Integration Calculate: Gauss Quadrature 2pt Gauss Quadrature 3pt Gauss Quadrature Let:

Numerical Integration - Example Calculate: Trapezoidal rule Simpson 1/3 rule 2pt Gauss quadrature Exact solution

Linear System Solver Solve: Gauss Elimination: forward elimination + back substitution Example:

Linear System Solver Solve: Gauss Elimination: forward elimination Example:

Linear System Solver Solve: Gauss Elimination: back substitution Example: Computer sum

Linear System Solver Solve: Gauss Elimination: forward elimination + back substitution Pseudo code: Forward elimination: Back substitution: Do k = 1, n-1 Do i = k+1,n Do j = k+1, n Do ii = 1, n-1 i = n – ii sum = 0 Do j = i+1, n sum = sum +

Finite Difference Method Example 1: t u Find 0 1 Selection of :

Finite Difference Method Example 2: Find

Finite Difference Method Example 3: Find 0 1