Generalized Midpoint Rule

Overview

By default the strain calculator evaluates the deformation gradient at the end of the time step (backward Euler). The alpha parameter on ComputeLagrangianStrain generalizes this to a generalized midpoint rule, sampling the deformation gradient – and therefore the stress and the internal force – at a weighted point within the step. It is an accuracy-versus-robustness choice for the time integration of the deformation gradient and rarely needs to be changed.

Formulation

The strain calculator forms the generalized-midpoint deformation gradient where and denote the previous and current steps and . The constitutive model is evaluated at , so the internal force in the balance of linear momentum is sampled at this weighted configuration and the displacement Jacobian carries the factor . The previous-step contribution is present only in a transient simulation; a Steady run takes .

Choosing alpha

• alpha = 1.0 (default): backward Euler. The deformation gradient is taken fully at the end of the step – first-order accurate over the step and the most robust choice.

• alpha = 0.5: the midpoint rule. Second-order accurate over the step, and matches the implicit integration used by Abaqus.

Intermediate values interpolate between the two; values below are not permitted. Set the parameter on the strain calculator:

[Materials]
  [strain]
    type = ComputeLagrangianStrain
    # ... displacements, base_name, etc.
    alpha = 0.5
  []
[]

The choice is independent of the kinematic approximation and the stabilization.