1. CHAP 0 MATHEMATICAL PRELIMINARY
FINITE ELEMENT ANALYSIS AND DESIGN
Nam-Ho Kim

2. MATHEMATICAL PRELIMINARY
Vector
a collection of scalars, defined using a bold typeface with braces
Matrix
a collection of vectors, defined using a bold typeface with brackets
dimension = N×K. When N = K, it is called a square matrix

3. MATRIX Transpose of a matrix: Change of row and column
Symmetric and Skew-symmetric matrices
Identity matrix

4. VECTOR-MATRIX CALCULUS
Addition
Scalar product between two vectors (must be the same dim)
Norm (Magnitude of a vector)

5. DETERMINANT Similar to the norm of a vector
Only defined for a square matrix
If a determinant is zero, the matrix is not invertible
A matrix is singular when its determinant is zero
For a 2x2 matrix:
For a 3x3 matrix

6. VECTOR-MATRIX CALCULUS cont.
Vector product
Scalar product  result = scalar
Vector product  result = vector

7. MATRIX-VECTOR MULTIPLICATION
Matrix  Vector = Vector
Vector  Matrix  Vector = Scalar

8. MATRIX-MATRIX MULTIPLICATION
Matrix  Matrix = Matrix
Inverse of a matrix:
A square matrix [A] is invertible, then
If a matrix is singular (|A| = 0), then the inverse does not exist

9. RULES OF MATRIX MULTIPLICATION
Associative rule:
Distributive rule:
Non-commutative:
Transpose of product:
Inverse of product:

10. MATRIX EQUATION N unknowns (x1, x2, …, xN) and N equations
unique solution if all equations are independent
Matrix form:
Solution: [A]–1 exists or [A] is not singular

11. EIGEN VALUE AND EIGEN VECTOR
Eigen value problem
: Eigen value
: Eigen vector
How to solve?
{x} = {0} is a solution (trivial solution)
In order to have non-trivial solution, the determinant must be zero.
Calculate from this equation and calculate from the eigen value problem

12. EIGEN VALUE AND EIGEN VECTOR
Characteristic equation
The textbook has a solution for [A]3x3 case
Eigen vectors
After solving for eigen values, substitute each of them to eigen problem
Since is singular, no unique solution exists
Practice example in the textbook

13. QUADRATIC FORM Quadratic form: quadratic function of all components
Matrix notation
Symmetric part is enough ([B] is not sym)

14. POSITIVE DEFINITE MATRIX
Positive semi-definite
Positive definiteness = each column of the matrix is linearly independent = the matrix is invertible = the matrix is not singular = the matrix equation has a unique solution.

15. MAXIMA & MINIMA OF FUNCTIONS
Single Variable f(x)
Taylor series expansion
In order for f to be extremum,
Condition for minima:
Condition for maxima:

16. MAXIMA & MINIMA OF FUNCTIONS cont.
Multi-Variable f(x)
Taylor series expansion
In order for f to be extremum,
Condition for minima: [H] is positive definite
Condition for maxima: [H] is negative definite
Hessian matrix Hij

17. MINIMUM PRINCIPLE Function in quadratic form Matrix equation
[A]: stiffness of the structure, {x}: displacement, {b}: applied force
Potential energy
structure is in equilibrium when F has a minimum value
Matrix equation
Solution of the matrix equation minimizes the quadratic form F.

18. Homework #1
5. For the two matrices [A] and [B] in Problem 2, answer the following questions.
(a) Evaluate the matrix–matrix multiplication [C] = [A][B].
(b) Evaluate the matrix–matrix multiplication [D] = [B][A].
7. Calculate the inverse of the matrix
9. Solve the following simultaneous system of equations using the matrix method:

19. Homework #1 11. Find the eigen values and eigen vectors
14. A function f(x1, x2) of two variables x1 and x2 is given by
Multiply the matrices and express f as a polynomial in x1 and x2.
Determine the extreme (maximum or minimum) value of the function and corresponding x1 and x2.
(c) Is this a maxima or minima?