Finite Element Method-Algebra Basics.pptx

FEM Algebra

1. ME- 6202 Finite Element Analysis Lect. # 2 Introduction to Matrix Algebra Dr. Nazeer Ahmad Anjum Mechanical Engineering Program University of Engineering Taxila

2. Introduction 3 9/25/2019 E ngineering P roblems are mathematical models of physical situations. Mathematical models are differential equations with a set of corresponding boundary and initial conditions . The differential equations are derived by applying the fundamental laws and principles of nature to a system or a control volume . These governing equations represent balance of Mass , Force , or Energy .

3. The analytical solutions are composed of two parts : a homogenous part and a particular part . In any given engineering problem, there are two sets of parameters that influence the way in which a system behaves. First , parameters that provide information regarding the natural behavior of a given system. These parameters include properties such as: modulus of elasticity , thermal conductivity , and viscosity Introduction 4 9/25/2019

4. Second, the parameters that produce disturbances in a system . These parameters include: External forces , moments , temperature difference across a medium, and pressure difference in fluid flow. Introduction 5 9/25/2019

5. A rectangular array of numbers (we will concentrate on real numbers). A n x m matrix has ‘n’ rows and ‘ m’ columns First column First row Second row Third row Second column Third column Fourth column Row number Column number What is a matrix ? 6 9/25/2019

6. Zero matrix: A matrix all of whose entries are zero Special matrix ? 7 9/25/2019 Identity matrix: A square matrix which has ‘1’ s on the diagonal and zeros everywhere else.

7. Equality of matrices Matrix Operations 8 9/25/2019 Addition of Matrices Properties of matrix addition: Matrix addition is commutative (order of addition does not matter) Matrix addition is associative Addition of the zero matrix Multiplication by a scalar Matrix Subtraction

8. Special operations Transpose Matrix Operations 9 9/25/2019 If A is a square matrix ( mxm ), it is called symmetric if Scalar (dot) product of two vectors

9. Properties Matrix multiplication Matrix Operations 10 9/25/2019 Matrix multiplication is non-commutative (order of addition does matter) 2 . Matrix multiplication is associative 3. Distributive law 4 . Multiplication by identity matrix 5 . Multiplication by zero matrix 6.

10. Matrix Operations 11 9/25/2019 Inverse of a matrix A matrix cannot have two or more inverses.

11. Matrix Operations 12 Methods of Solution of Simultaneous E quations This system is called a homogeneous system if all the elements of {F or P} are zero . This system has a trivial solution X l = X 2 = X 3 ••• = X n = 0. It is called a non-homogeneous system if at least one element of {F or P} is non-zero. The system is said to be consistent if [K] is a square, non-singular matrix. By Inversion of the Coefficient Matrix Method of Cofactors Gauss Jordan method Direct Methods Cramer's Rule Gauss Jordan method Gauss elimination method LU factorization method Doolittle method Crout's method Cholesky method F = Kx or P = kx

12. The TRACE & DETERMINANT are defined o nly if the matrix is square. Both quantities are single numbers, which are evaluated from the elements of the matrix and are therefore functions of the matrix elements. Product along red arrow minus product along Green arrow Trace and Determinant Matrix 13 For a 2x2 matrix: For a 1x1 matrix: The trace of the matrix A is denoted as tr (A) & is = where n is the order of A.   Determinant of an n x n matrix A is denoted as det A and is defined by the recurrence relation  

13. For ONLY a 3x3 matrix write down the first two columns after the third column Sum of products along red arrow minus sum of products along blue arrow This technique works only for 3x3 matrices Duplicate C olumn M ethod 14 9/25/2019

14. 32 3 -8 8 Sum of red terms = 0 + 32 + 3 = 35 Sum of blue terms = 0 – 8 + 8 = 0 Determinant of matrix A= det(A) = 35 – = 35 Example # 2 15 9/25/2019

15. Special case. If two rows or two columns are proportional (i.e. multiples of each other), then the determinant of the matrix is zero because rows 1 and 3 are proportional to each other If the determinant of a matrix is zero , it is called a singular matrix Finding determinant using inspection 16 9/25/2019

16. If A is a square matrix The minor , M ij , of entry a ij is the determinant of the submatrix that remains after the i th row and j th column are deleted from A. The cofactor of entry a ij is C ij =(-1) ( i+j ) M ij What is a cofactor? 17 9/25/2019 Cofactor method A matrix with elements that are the cofactors, term-by-term, of a given square matrix.

17. Sign of cofactor Find the minor and cofactor of a 33 Minor Cofactor What is a cofactor? 18 9/25/2019

18. Example # 3, Find the Cofactor matrix 19 9/25/2019 A= Solution The Cofactor matrix of A is =