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Change flex strain constraints for trilinear to a more stable formulation.

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This change improves trilinear flex elements by using reduced integration for volumetric quantities (strain trace and volume ratio) at the element center, while adding full integration for shear components at 8 Gauss points and removing the second strain invariant from the constraints, which is negligible for small strains. This reduced integration "B-bar" technique is standard in finite element analysis and prevents artificial stiffness that can occur when low-order elements are nearly incompressible. The constraint count per trilinear element changes from 24 to 26 compared to using invariants. For quadratic elements, the full 27 quadrature point are used resulting in 162 constraints.

PiperOrigin-RevId: 884570752
Change-Id: Ib74ece8f4712c2c81fbfd784524fcac52a2c80e5
This commit is contained in:
Alessio Quaglino
2026-03-16 12:04:46 -07:00
committed by Copybara-Service
parent a5098a068a
commit ac2730ab80
5 changed files with 371 additions and 230 deletions
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doc/XMLreference.rst
+7 -5
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@@ -4841,10 +4841,12 @@ degrees of freedom to avoid locking. It is only supported for dimension 2, i.e.,
:el-prefix:`equality/` |-| **flexstrain** |*|
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
This element constrains the strain invariants of a trilinear or quadratic flex to their initial values. Specifically, it
enforces that the trace and determinant of the deformation gradient remain constant, preserving volume and preventing
excessive stretching. This constraint type is only supported for dimension 3 trilinear flexes (i.e., volumetric
deformable bodies using trilinear interpolation). See `this model
This element constrains the strain of a trilinear or quadratic flex to its initial values. For trilinear elements,
a B-bar formulation is used to prevent volumetric locking: the trace of strain (I₁) and volume ratio (J-1 = det(F)-1)
are constrained at the element center, while the three off-diagonal shear components (E₁₂, E₁₃, E₂₃) are constrained
at each of the 8 Gauss points, giving 26 constraints per element. For quadratic elements, all 6 strain components
(3 invariants + 3 shear) are constrained at each of the 27 Gauss points, giving 162 constraints per element. This
constraint type is only supported for dimension 3 flexes with trilinear or quadratic interpolation. See `this model
<https://github.com/google-deepmind/mujoco/blob/main/model/flex/strain.xml>`__ for an example.
.. _equality-flexstrain-name:
@@ -4859,7 +4861,7 @@ deformable bodies using trilinear interpolation). See `this model
.. _equality-flexstrain-flex:
:at:`flex`: :at-val:`string, required`
Name of the flex whose strain invariants are being constrained.
Name of the flex whose strain is being constrained.
.. _equality-distance: