MECH300H Introduction to Finite Element Methods Lecture 2 Review

Matrix Algebra Row and column vectors Addition and Subtraction – must have the same dimensions Multiplication – with scalar, with vector, with matrix Transposition – Differentiation and Integration

Matrix Algebra Determinant of a Matrix: Matrix inversion - Important Matrices diagonal matrix identity matrix zero matrix eye matrix

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: Gaussian Quadrature Trapezoidal Rule : Gaussian Quadrature : Chooseaccording to certain criteria

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

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

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

Linear System Solver Solve: Gaussian 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 +