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# Theory

The finite element method is basically concerned with the determination of field variables. The most important ones are the stress and strain fields. As basic measure of strain in CalculiX the Lagrangian strain tensor E is used for elastic media, the Eulerian strain tensor e is used for deformation plasticity and the deviatoric elastic left Cauchy-Green tensor is used for incremental plasticity. The Lagrangian strain satisfies ([18]):

| (1)

where are the displacement components in the material frame of reference and repeated indices imply summation over the appropriate range. In a linear analysis, this reduces to the familiar form:

| (2)

The Eulerian strain satisfies ([18]):

| (3)

where are the displacements components in the spatial frame of reference.

Finally, the deviatoric elastic left Cauchy-Green tensor is defined by ([56]):

| (4)

where is the elastic Jacobian and is the elastic deformation gradient. The above formulas apply for Cartesian coordinate systems.

The stress measure consistent with the Lagrangian strain is the second Piola-Kirchhoff stress S. This stress, which is internally used in CalculiX for all applications (the so-called total Lagrangian approach, see [9]), can be transformed into the first Piola-Kirchhoff stress P (the so-called engineering stress, a non-symmetric tensor) and into the Cauchy stress t (true stress). All CalculiX input (e.g. distributed loading) and output is in terms of true stress. In a tensile test on a specimen with length L the three stress measures are related by:

| (5)

where is the engineering strain defined by

| (6)

Subsections

- Node Types
- Element Types
  - Eight-node brick element (C3D8 and F3D8)
  - C3D8R and F3D8R
  - Incompatible mode eight-node brick element (C3D8I)
  - Twenty-node brick element (C3D20 and F3D20)
  - C3D20R and F3D20R
  - C3D20RI
  - Four-node tetrahedral element (C3D4 and F3D4)
  - Ten-node tetrahedral element (C3D10 and F3D10)
  - Six-node wedge element (C3D6 and F3D6)
  - Fifteen-node wedge element (C3D15 and F3D15)
  - Three-node shell element (S3)
  - Four-node shell element (S4 and S4R)
  - Six-node shell element (S6)
  - Eight-node shell element (S8 and S8R)
  - Three-node plane stress element (CPS3)
  - Four-node plane stress element (CPS4 and CPS4R)
  - Six-node plane stress element (CPS6)
  - Eight-node plane stress element (CPS8 and CPS8R)
  - Three-node plane strain element (CPE3)
  - Four-node plane strain element (CPE4 and CPE4R)
  - Six-node plane strain element (CPE6)
  - Eight-node plane strain element (CPE8 and CPE8R)
  - Three-node axisymmetric element (CAX3)
  - Four-node axisymmetric element (CAX4 and CAX4R)
  - Six-node axisymmetric element (CAX6)
  - Eight-node axisymmetric element (CAX8 and CAX8R)
  - Two-node beam element (B31 and B31R)
  - Three-node beam element (B32 and B32R)
  - Three-node network element (D)
  - Two-node unidirectional gap element (GAPUNI)
  - Two-node 3-dimensional dashpot (DASHPOTA)
  - Two-node 3-dimensional spring (SPRINGA)
  - One-node coupling element (DCOUP3D)
- Fluid Section Types: Gases
  - Orifice
  - Bleed Tapping
  - Preswirl Nozzle
  - Straight and Stepped Labyrinth
  - Characteristic
  - Carbon Seal
  - Gas Pipe
  - Gas Pipe (Fanno)
  - Restrictor, Long Orifice
  - Restrictor, Enlargement
  - Restrictor, Contraction
  - Restrictor, Bend
  - Restrictor, Wall Orifice
  - Restrictor, Entrance
  - Restrictor, Exit
  - Restrictor, User
  - Branch, Joint
  - Branch, Split
  - Cross, Split
  - Vortex
  - Möhring
  - Change absolute/relative system
- Fluid Section Types: Liquids
  - Pipe, Manning
  - Pipe, White-Colebrook
  - Pipe, Sudden Enlargement
  - Pipe, Sudden Contraction
  - Pipe, Entrance
  - Pipe, Diaphragm
  - Pipe, Bend
  - Pipe, Gate Valve
  - Pump
- Fluid Section Types: Open Channels
  - Straight Channel
  - Sluice Gate
  - Sluice Opening
  - Weir Crest
  - Weir slope
  - Discontinuous Slope
  - Discontinuous Opening
  - Reservoir
  - Contraction
  - Enlargement
  - Drop
  - Step
- Boundary conditions
  - Single point constraints (SPC)
  - Multiple point constraints (MPC)
  - Contact
- Materials
  - Linear elastic materials
  - Ideal gas for quasi-static calculations
  - Hyperelastic and hyperfoam materials
  - Deformation plasticity
  - Incremental (visco)plasticity
  - Tension-only and compression-only materials.
  - Fiber reinforced materials.
  - The Cailletaud single crystal model.
  - Elastic anisotropy with isotropic viscoplasticity.
  - User materials
- Types of analysis
  - Static analysis
  - Frequency analysis
  - Complex frequency analysis
  - Buckling analysis
  - Modal dynamic analysis
  - Steady state dynamics
  - Direct integration dynamic analysis
  - Heat transfer
  - Acoustics
  - Shallow water motion
  - Hydrodynamic lubrication
  - Irrotational incompressible inviscid flow
  - Electrostatics
  - Stationary groundwater flow
  - Diffusion mass transfer in a stationary medium
  - Aerodynamic Networks
  - Hydraulic Networks
  - Turbulent Flow in Open Channels
  - Three-dimensional Navier-Stokes Calculations
- Convergence criteria
- Loading
  - Point loads
  - Facial distributed loading
  - Centrifugal distributed loading
  - Gravity distributed loading
  - Temperature loading in a mechanical analysis
  - Initial(residual) stresses
  - Concentrated heat flux
  - Distributed heat flux
  - Convective heat flux
  - Radiative heat flux
- Error estimators
  - Zienkiewicz-Zhu error estimator
  - Extrapolation error estimator
- Output variables

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