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three-field-solid

Module: solid

Category: soliddomain

Type string: "three-field-solid"

Parameters

Name Description Default Units
laugon laugon false []
atol atol 0.01 []
minaug minaug 0 []
maxaug maxaug 0 []
mat_axis []

Description

The three-field-solid solid domain type implements a three-field formulation 1 for solving the finite element equations. This formulation is recommended for modeling (nearly-) incompressible materials using linear hexahedral and pentahedral elements. These elements tend to "lock" when modeling incompressible materials and the three-field formulation overcomes this issue. This formulation requires an uncoupled-elastic material to work.

In addition to the displacement, this formulation also solves for the pressure \(p\) and volume ratio \(J\). The pressure can be seen as a Lagrange multiplier and the volume constraint (i.e. \(J=1\)) is enforced using an augmented Lagrangian method. The bulk modulus \(k\) (defined on the material) is seen as a penalty factor for the constraint enforcement. Without augmentations, this amounts to a penalty method for enforcing the constraint. To enable augmentations, set laugon to 1 and specify a convergence tolerance using atol. The augmentation tolerance determines the convergence condition that is used for the augmented Lagrangian method: convergence is reached when the relative ratio of the Lagrange multiplier norm of the previous augmentation \(\left\Vert \lambda_{k}\right\Vert\) to the current one \(\left\Vert \lambda_{k+1}\right\Vert\) is less than the specified value:

\[ \left|\frac{\left\Vert \lambda_{k+1}\right\Vert -\left\Vert \lambda_{k}\right\Vert }{\left\Vert \lambda_{k+1}\right\Vert }\right|<\varepsilon \]

Thus, a value of 0.01 implies that the change in norm between the previous augmentation loop and the current loop is less than 1%.

In addition, you can control the minimum and maximum number of augmentations using minaug and maxaug.

Example:

<SolidDomain name="Part1" mat="Material1" type="three-field-solid">
    <laugon>1</laugon>
    <atol>0.01</atol>
    <minaug>0</minaug>
    <maxaug>0</maxaug>
</SolidDomain>

  1. Simo, J.C. and Taylor, R.L., "Quasi-incompressible finite elasticity in principal stretches: Continuum basis and numerical algorithms", Computer Methods in Applied Mechanics and Engineering 85 (1991), pp. 273-310. ↩