Internal Force Calculation

Internal forces are computed using the generalized relation:(1)

f_{i I}^{int} = \int_{Ω} σ_{i j} \frac{\partial Φ_{I}}{\partial x_{j}} d Ω

However, to increase the computational speed of the process, some simplifications are applied.

Reduced Integration Method

This is the default method for computing internal forces. A one point integration scheme with constant stress in the element is used. Due to the nature of the shape functions, the amount of computation can be substantially reduced:(2)

\frac{\partial Φ_{1}}{\partial x_{j}} = - \frac{\partial Φ_{7}}{\partial x_{j}}; \frac{\partial Φ_{2}}{\partial x_{j}} = - \frac{\partial Φ_{8}}{\partial x_{j}}; \frac{\partial Φ_{3}}{\partial x_{j}} = - \frac{\partial Φ_{5}}{\partial x_{j}}; \frac{\partial Φ_{4}}{\partial x_{j}} = - \frac{\partial Φ_{6}}{\partial x_{j}}

Hence, the value

\frac{\partial Φ_{I}}{\partial x_{j}}

is taken at the integration point and the internal force is computed using the relation:(3)

F_{i I} = σ_{i j} {(\frac{\partial Φ_{I}}{\partial x_{j}})}_{0} Ω

The force calculation is exact for the special case of the element being a parallelepiped.

Full Integration Method

The final approach that can be used is the full generalized formulation found in Equation 1. A classical eight point integration scheme, with non-constant stress, but constant pressure is used to avoid locking problems. This is computationally expensive, having eight deviatoric stress tensors, but will produce accurate results with no hourglass.

When assumed strains are used with full integration (HA8 element), the reduced integration of pressure is no more necessary, as the assumed strain is then a free locking problem.

ALE Improved Integration Method

This is an ALE method for computing internal forces (flag INTEG). A constant stress in the element is used.

The value $\int_{Ω} \frac{\partial Φ_{I}}{\partial x_{j}} d Ω$ is computed with Gauss points.