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Finite Element Methods

From transcript: 20ME630 Finite Element Methods (E)

Cheat sheet

Printable study sheet: formulas → definitions → topic notes → traps.

Formulas

Weak form / virtual work → element equations k e u e = f e

Assemble K U = F ; apply BCs → solve reduced system

Linear tetra/brick/shell/beam element choices by geometry & stress state

Strain–displacement ε = B u ; stress σ = E ε (elasticity matrix)

Convergence: refine mesh (h) or raise order (p); energy norm

Modal: det(K − ω² M)=0 ; transient M ü + C u̇ + K u = F(t)

Error indicators / adaptive meshing at gradients & singularities

Weak form → [K]{u} = {F}

Element stiffness (bar): k = AE/L [ 1 −1 ; −1 1 ]

Shape functions Ni interpolate u(x) = Σ Ni ui

Assembly: map local DOF → global; apply BCs

von Mises: σv = √(σ1²+σ2²+σ3² − σ1σ2 − σ2σ3 − σ3σ1)

Definitions

Discretization: Replace continuum with finite elements & nodes
DOF: Unknowns at nodes — translations/rotations
Shape functions N: Interpolate field inside element from nodal values
Stiffness matrix K: Maps nodal displacements to nodal forces
Mesh quality: Aspect ratio, skew, Jacobian — affects accuracy
Singularity: Stress→∞ at sharp reentrant corners — interpret carefully
Verification vs validation: Code/math correct vs model matches reality
Linear vs nonlinear: Geometry/material/contact nonlinearity
Convergence: Refine mesh / raise order until results stabilize
h- vs p-refinement: Smaller elements vs higher polynomial order

Topic-wise short notes

Study these first — one block per syllabus topic. Then read the deep notes below.

FEM formulation basics

Strong form PDE → multiply by weight → integrate by parts → weak form.
Galerkin: weight = shape function family.
Isoparametric mapping: parent element → physical via Jacobian.
2D: plane stress vs plane strain vs axisymmetric assumptions.

FEA workflow

Geometry cleanup → material → mesh → BCs/loads → solve → postprocess.
Symmetry planes cut cost; check against hand calc on simple cases.
Contact, bolts, welds idealized — state assumptions in reports.
Fatigue/FEA: use principal/von Mises histories with Kt awareness.

Element selection cheat

Thin structures → shells; long slender → beams; 3D stress → solids.
Incompressible materials / plasticity need special elements.
Contact: friction, separation; convergence harder — stabilize carefully.

Exam traps & quick notes

Garbage in → garbage out: BCs and loads dominate accuracy.
Check reaction equilibrium and mesh quality (aspect, Jacobian).
Project: Ansys on Aero structures / BAJA chassis — validate vs hand calc.
Garbage BCs → garbage results; check reaction sum = applied load.
Singularities: report nearby stress or use fillets in real design.
Units consistency: N–mm–MPa or N–m–Pa — never mix.
Hourglass/locking: choose element tech (reduced integration, etc.).
Always mesh convergence study before claiming a number.

Comprehensive notes

FEM formulation basics

Potential energy minimization, isoparametric elements, numerical integration (Gauss).

External: NPTEL FEM ↗ Ansys student resources ↗

FEA workflow

Geometry cleanup, contacts, singularity awareness at sharp corners, linear vs nonlinear.

External: NAFEMS primers ↗

Weak form to [K]{u}={F}

Shape functions, element stiffness, assembly, BCs. Bar/truss/beam/2D continuum progression. Numerical integration.

External: NPTEL ↗ Engineering Toolbox ↗

FEA practice & trust

Mesh quality, contacts, singularities, linear vs nonlinear, convergence. Always sanity-check with hand calc. BAJA/Aero structures in Ansys.

External: NPTEL ↗ Engineering Toolbox ↗

Interview Q&A for this subject

Q: Why refine the mesh?
A: Reduce discretization error until KPI converges — but fix BCs first.

External: ASQ / quality ↗ MIT OCW ↗