Finite element methods
Graduate · Engineering
Syllabus focus
Standard syllabus · STEM / applied
Pricing
Graduate-level rates are set on consultation. See the pricing page for K–12 and undergraduate rates.
Topics typically covered
Standard syllabus
Weak form and element formulations
- Calculus of variations and Euler-Lagrange equations
- Weighted residual methods: Galerkin approach
- Weak form derivation for Poisson and elasticity problems
- Shape functions for 1D and 2D elements
- Isoparametric mapping and numerical integration
- Element stiffness and mass matrix assembly
- Patch test and element completeness
- Locking phenomena: shear and volumetric
- Mixed formulations and reduced integration
- A priori and a posteriori error estimates (intro)
Solvers and nonlinear extensions
- Direct versus iterative linear solvers
- Preconditioning and sparse matrix storage
- Newton-Raphson for nonlinear equilibrium
- Material nonlinearities: plasticity intro
- Contact algorithms: penalty and Lagrange multipliers
- Large deformation and updated Lagrangian formulations
- Dynamic FE: Newmark time integration
- Eigenvalue solvers for modal analysis
- Adaptive mesh refinement strategies
- Parallel FE on HPC clusters
STEM / applied
Implementation and software development
- Building a 1D/2D FE code in MATLAB or Python
- Open-source frameworks: FEniCS, deal.II survey
- Pre- and post-processing pipelines
- Verification using manufactured solutions
- Validation against benchmark problems
- Coupled multiphysics FE (thermal-stress-fluid)
- Isogeometric analysis overview
- Reduced-order modeling from FE snapshots
- Industry ANSYS/Abaqus scripting and automation
- Thesis using custom or extended FE tools
Research applications
- Biomechanics FE: tissue and implant modeling
- Crashworthiness and impact simulation
- Geomechanics and coupled poroelastic FE
- Electromagnetics FE formulations (survey)
- Topology optimization with FE sensitivities
- Uncertainty quantification in FE predictions
- Digital twin calibration with sensor data
- Publication-quality FE result visualization
- Conference presentation of FE research
- Qualifying exam FEM preparation
Notes
Topics reflect common engineering syllabi at US colleges and universities. Exact order, depth, and applied emphasis vary by institution, department, and instructor.