Finite element analysis · Open scientific notes
Finite Element Theory and Computational Methods
Derivations, worked examples, and MATLAB programs
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A structured collection of notes on finite element formulations, nonlinear analysis, and computational methods.
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Contents
The material follows a numbered scientific sequence. New sections will be added in the same logical order.
Truss structures
Two-Dimensional Truss Structures
Three-Dimensional Truss Structures
2D state
Large Displacements of Beams
2D Beam Element, Linear Case
Beam Finite Element — Nonlinear Double-Clamped Beam
AvailableTotal Lagrangian Formulation (TL), 2D Isoparametric Beam Element
Updated Lagrangian Beam Formulation (UL)
2D Stress Analysis: Finite Element Types, Linear Case
- 9.1CST Finite ElementAvailable
- 9.2CST: Fast MATLAB SolverAvailable
- 9.3CST: Fast MATLAB Solver — Alternative VersionAvailable
- 9.4Isoparametric 4-node quadrilateral finite elementAvailable
- 9.5Isoparametric 4-node quadrilateral: fast solverAvailable
- 9.6Isoparametric 8-node quadrilateral finite elementAvailable
- 9.7Isoparametric 8-node quadrilateral: fast solverAvailable
Green–Lagrange Strain Tensor
Available“Exact” Strains
AvailableTotal Lagrangian Formulation (TL)
Euler–Almansi Strain Tensor
AvailableCauchy, First Piola–Kirchhoff, and Second Piola–Kirchhoff Stress Tensors
AvailableUpdated Lagrangian Formulation (UL)
- 15.1Plane Stress: Isoparametric 4-Node Quadrilateral Finite ElementAvailable
- 15.2Plane Stress: Isoparametric 4-Node Quadrilateral Finite Element — A Slightly Different ApproachAvailable
- 15.3Direct Evaluation of Euler–Almansi Strains and Comparison of UL ApproachesAvailable
- 15.4Direct Evaluation of Euler–Almansi Strains for the 8-Node Isoparametric Quadrilateral Finite ElementAvailable
- 15.5Updated Lagrangian Revisited: A Direct and Consistent Tangent Stiffness Matrix for the Q4 ElementNEWAvailable
- 15.6Updated Lagrangian Revisited: A Direct and Consistent Tangent Stiffness Matrix for the Q8 ElementNEWAvailable
- 15.7Comparison between TL and UL formulationsAvailable
Corotational formulation (CR)
Hencky and Biot strains
- 17.1Polar decomposition, Hencky strain and Biot strainAvailable
- 17.2Updated Lagrangian formulation: plane stress state, isoparametric 4-node quadrilateral, Hencky strainsAvailable
- 17.3UL formulation: plane stress, isoparametric 4-node quadrilateral, Hencky strain, improved approachAvailable
- 17.4UL formulation with multiplicative update of the deformation gradientAvailable
- 17.5Direct numerical evaluation of the strain–displacement and tangent stiffness matricesAvailable
- 17.6Vectorized double numerical differentiationAvailable
Hyperelastic Materials
- 18.1Hyperelastic materials, Mooney-Rivlin material model for nearly incompressible material, Part IAvailable
- 18.2Hyperelastic materials, Mooney-Rivlin material model for nearly incompressible materials, Part IIAvailable
- 18.3Mooney-Rivlin model for nearly incompressible materials, Selective Reduced IntegrationAvailable
Buckling of beam structures
AvailableBuckling, plane stress state
Available3D state
Finite rotations
Large displacements cantilever beam
3D beam elements. Linear case
Updated Lagrangian formulation, 3D isoparametric 2-node beam element
AvailableHexahedral isoparametric finite element. Linear case
- 25.1Hexahedral 8-node isoparametric finite element for linear analysisAvailable
- 25.2Fast program for the 8-node isoparametric hexahedral finite elementAvailable
- 25.3Hexahedral 20-node isoparametric finite element for linear analysisAvailable25.4Fast program for the 20-node isoparametric hexahedral finite elementAvailable
Total Lagrangian formulation (TL)
Updated Lagrangian formulation (UL)
Axisymmetric bodies
- 28.1Axisymmetric bodies. 4-node isoparametric finite element. Linear analysisPlanned
- 28.2Green–Lagrange strains in cylindrical coordinatesPlanned
...
Theory and implementation together
MATLAB programs in their mathematical context
Every program is presented beside the formulation it implements: inputs, governing equations, iteration strategy, numerical examples, and downloadable source files.
- Readable MATLAB source and direct downloads
- Explicit assumptions and notation
- Reproducible examples and comparison figures
- Tested MATLAB versions and expected compatibility are stated separately for each program package
% Newton–Raphson correction
dS = -K\F;
S = S + dS;
err = sqrt(dS'*dS/neq);
if err < tol
break
end% Moderately large displacement
ep = Nux*uel + 1/2*(Nvx*uel)^2;
ep1 = Nux' + (Nvx'*Nvx)*uel;
ep2 = Nvx'*Nvx;
ka = Nka*uel;
ka1 = Nka';% Load steps and equilibrium iterations
for ist = 1:nstep
F = ist/nstep*F0;
for iter = 1:itermax
[K,Psi] = stiff(S,F);
dS = -K\Psi;
S = S + dS;
end
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