| Matrix or Vector | Geometry | Shape Functions | Integration Points |
|---|---|---|---|
| Stiffness and Stress Stiffness Matrices; and Thermal Expansion Load Vector | Quad | Equation 11–122 and Equation 11–123 | 2 x 2 |
| Triangle | Equation 11–101 and Equation 11–102 | 1 | |
| Mass Matrix | Quad | Same as stiffness matrix | 2 x 2 |
| Triangle | 1 | ||
| Pressure Load Vector | Same as stiffness matrix, specialized to the face | 2 | |
| Thermal Conductivity Matrix and Heat Generation Load Vector | Quad | Equation 11–130 | 2 x 2 |
| Triangle | Equation 11–109 | 1 | |
| Specific Heat Matrix | Same as thermal conductivity matrix. | ||
| Convection Surface Matrix and Load Vector | Same as thermal conductivity matrix, specialized to the face | 2 | |
| Dielectric Permittivity and Electrical Conductivity Matrices; Charge Density, Joule Heating, and Peltier Heat Flux Load Vectors | Quad | Equation 11–131 | 2 x 2 |
| Triangle | Equation 11–110 | 1[a] | |
| Diffusivity Matrix and Diffusing Substance Generation Load Vector | Quad | Equation 11–132 | 2 x 2 |
| Triangle | Equation 11–111 | 1[a] | |
| Diffusion Damping Matrix | Same as diffusivity matrix. If KEYOPT(10)=1, matrix is diagonalized as described in Lumped Matrices | ||
| Diffusion Flux Load Vector | Same as diffusivity matrix, specialized to the face | 2 | |
| Thermoelastic Stiffness and Damping Matrices | Same as combination of stiffness and thermal conductivity matrices | ||
| Piezoelectric and Electroelastic Coupling Matrix | Same as combination of stiffness matrix and dielectric matrix | ||
| Seebeck Coefficient Coupling Matrix | Same as combination of electrical conductivity and thermal conductivity matrices | ||
| Surface Charge Density Load Vector | Same as dielectric matrix, specialized to the face | 2 | |
[a] Axisymmetric coupled-field analyses without structural degrees of freedom use a 3-point integration rule for all load vector and matrix calculations.
Structures describes the derivation of structural element matrices and load vectors as well as stress evaluations. General Element Formulations gives the general element formulations used by this element. Heat Flow describes the derivation of the thermal element matrices and load vectors as well as heat flux evaluations. Electromagnetics describes the derivation of the electric element matrices and load vectors, as well as electric field evaluations. Piezoelectrics discusses the piezoelectric capability used by the element. Piezoresistivity discusses the piezoresistive effect. Thermoelectrics discusses the thermoelectric effects. Thermoelasticity discusses the thermoelastic effects. Electroelasticity discusses the Maxwell stress electroelastic coupling. Thermoplasticity discusses the thermoplastic effect. Diffusion describes the derivation of diffusion matrices and load vectors. Structural-Diffusion Coupling discusses diffusion strain coupling.
This element uses the 
method (selective reduced integration technique for volumetric terms)
(Hughes([221]), Nagtegaal et al. ([222]))
If KEYOPT(6) = 2 or 3, the enhanced strain formulation from the works of Simo and Rifai([318]), Simo and Armero([319]), Simo et al.([320]), Andelfinger and Ramm([321]), and Nagtegaal and Fox([322]) is used. It introduces additional internal (user-inaccessible) structural degrees of freedom (DOFs) to prevent shear and volumetric locking (KEYOPT(6) = 2) or shear locking only (KEYOPT(6) = 3). These internal DOFs are created and condensed out at the element level.