This course provides a comprehensive introduction to the finite element method (FEM) with emphasis on structural mechanics.

1. Direct Stiffness Method (DSM)

Element equations for bars and frames, global assembly procedures, imposition of boundary conditions, solution of the linear system, and post-processing of displacements and forces. Algorithmic structure and basic implementation concepts are introduced.

2. Constraint treatment in the DSM

Master-slave techniques, penalty method, and Lagrange multiplier adjunction. Multi-freedom constraints. Numerical properties and practical implications are discussed.

3. Variational foundations of FEM

Strong form of a boundary value problem. Principle of minimum total potential energy, method of weighted residuals, and their equivalence. Derivation of weak formulations.

4. Beam elements

Variational formulation of the plane beam element. Discretization with Hermite functions.

5. Continuum elements and element technology

Kinematics, strain-stress relations, and constitutive equations of plane stress linear elasticity. 3-node triangular elements (CST). 4-node quadrilateral elements, isoparametric mapping, tensor-product basis functions, and numerical integration concepts. Consistent treatment of Neumann boundary conditions. Stress recovery and post-processing.

6. Convergence and accuracy

Fundamental notions of completeness, consistency, and stability. Error analysis, convergence, optimal rates. Influence of mesh refinement and interpolation order.

7. Hands-on computational practice

Guided implementation of selected formulations in MATLAB.

Felippa, C.A., Introduction to Finite Element Methods. Lecture Notes, CU Boulder.