Intro to Finite Elements

$$F_i = K_{ij}u_j$$

F and U are both vectors and K is a matrix.

Lecture 1: Linear Algebra and Notation

Finite Element Methods are suitable for nonsquare, or rectangular matrices which is a change from SM, which typically deals with square, and for that matter symmetric, matrices.

$$\boldsymbol{K} = \left[\begin{array}{cccc}k_{1,1} & k_{1,2} & \cdots & k_{1, m} \\k_{2,1} & k_{2,2} & \cdots & k_{2, m} \\\vdots & \vdots & \ddots & \vdots \\k_{n, 1} & k_{n, 2} & \cdots & k_{n, m}\end{array}\right]$$

Vectors are like matrices but has only one value in one of the 2 dimensions so the second index is omitted

$${\bf a} = \begin{bmatrix} a_1\\ a_2\\ a_3\\ \vdots\\ a_n \end{bmatrix}$$

Two special matrices, the zero matrix and the identity matrix. The zero matrix is the equivalent of identically zero in linear algebra. The identity matrix is a square matrix which has 1 along the main diagonal.

Lecture 2: Introduction to the Stiffness Displacement Method

Lecture 3: Finite Element System of Equations from Direct Stiffness

Lecture 4: Development of Displacement Based FEM in 1D, Formation of Stiffness Matrix

Lecture 5: Development of Displacement Based FEM in 2D, Constant Strain Triangle and Quadrilateral Elements

Lecture 6: Practical Considerations in FEM

Lecture 7: Convergence of FEM Results

Lecture 8: Higher Order Elements

Lecture 9: Isoparametric Formulation

Lecture 10: Numeric Integration in 2D

Lecture 11: Solution of Linear Algebraic Equations