Theory and Reference Manual

Theory & Reference Manual

Implementation-aligned derivations that connect equations, indexing rules, and constitutive updates to the exact array operations used in the source tree.

Core Mechanics

3 chapters

• 01 — Assembly & Global Indexing
• 02 — Boundary Conditions & Loads
• 03 — 1-D Bar Formulations

Element Families

4 chapters

• 04 — Triangular Elements (T3)
• 05 — Quadrilateral Elements (Q4)
• 06 — Tetrahedra & Hexahedra
• 07 — Constitutive Return Mapping

Dynamics, Periodicity & I/O

6 chapters

• 08 — Time Integration
• 09 — Modal & Periodic
• 10 — Mesh Import
• 11 — Nonlinear Solvers
• 12 — Plotting
• 13 — Legacy Wrappers

Reading order

Start with assembly (01-02), then the element families you use, then dynamics and I/O last.

• Core Assembly in Finite Element Analysis
  • Introduction
  • 1. Variational Foundation
  • 2. Assembly Identity and Derivation
  • Python implementation with numpy.ix_
  • Sparse and dense assembly in assmk
  • Complete example: 3-element manual assembly
• Boundary conditions and loads
  • 2.1 Applying external loads
  • 2.2 Dirichlet boundary conditions
  • 2.3 Multi-point constraints (Lagrange multipliers)
  • Summary
• 1D bar elements
  • 1. Continuum Mechanics Foundation
  • 2. Linear Bar Element
  • 3. Geometrically Nonlinear Formulation
  • 4. Mass Matrix
  • 5. Transformation to Global Coordinates
  • 6. Implementation Reference
• 2D Triangular Elements
  • Geometry and array layout
  • Shape functions and gradients
  • Strain-displacement matrix
  • Element stiffness derivation
  • Batched assembly in kt3e
  • Stress and flux recovery
  • Scalar potential version
  • Summary
  • Appendix: Constitutive Matrix Derivations
• 2D Isoparametric Quadrilateral Elements
  • Bilinear shape functions
  • Isoparametric mapping and the Jacobian
  • Strain-displacement matrix
  • Gauss-Legendre integration
  • Implementation in femlabpy
  • Scalar potential version
  • Summary
  • Appendix: Mathematical Foundations
• 3D Solid Elements: Tetrahedrons and Hexahedrons
  • Preliminaries: 3D Elasticity
  • 1. The 4-Node Tetrahedron Element (T4)
  • 2. The 8-Node Hexahedron Element (H8)
  • 3. Implementation Notes
• Constitutive Models and Plasticity
  • 1. Stress Invariants and Decomposition
  • 2. Von Mises Yield Criterion
  • 3. Return Mapping Algorithm
  • 4. Drucker-Prager Plasticity
  • 5. Material Row Layout
  • 6. Implementation Summary
• Structural Dynamics and Time Integration
  • 1. Mass and Damping Models
  • 2. Implicit Time Integration: Newmark-\(\beta\) Method
  • 3. Explicit Time Integration: Central Difference Method
  • 4. Dissipative Integration: HHT-\(\alpha\) Method
  • 5. Frequency Response Function (FRF)
  • Appendix: Mathematical Foundations
• Modal analysis and periodic boundaries
  • 1. Modal Analysis
  • 2. Periodic boundaries
  • 3. Volume averages and homogenization
  • 4. Practical workflow
• Mesh I/O and Gmsh Import
  • 1. The Core Data Structures
  • 2. Parsing the Mesh File
  • 3. Loader Functions
  • 4. Practical Extraction
• Nonlinear Solvers
  • 1. The Nonlinear Problem
  • 2. Newton-Raphson Method
  • 3. Load Control and Incremental Analysis
  • 4. Arc-Length Methods
  • 5. Tangent Stiffness Matrix
  • 6. Convergence Criteria
  • 7. Implementation in femlabpy
• Plotting and Postprocessing
  • 1. Stress Recovery Mathematics
  • 2. Reaction Forces
  • 3. Mesh and Field Plotting
  • Element Contours
  • Reactions
  • Typical Workflow
• Chapter 13: Legacy Wrappers
  • Why The Wrappers Exist
  • Shared Helpers
  • Benchmark Data Wrappers
  • Solver Wrappers
  • What The Wrappers Return
  • Reading Order