A separate isotropic research element splits deviatoric and volumetric stiffness integration. It is not a universal accuracy correction.
01 Model and scope
Selective reduced integration separates distortional and volumetric energy, integrating the former with the full rule and the latter at the centre. This can change volumetric locking behaviour, but the reduced term can also introduce rank or zero-energy problems; the card therefore stays research-only.
Route assumptions and calculation checks
- Deviatoric terms retain the eight-point rule; the volumetric term uses the element centre.
- The public HEX8 class and its maturity rows are unchanged.
- Rank, zero-energy modes, distortion sensitivity, and energy checks are required before any wider claim.
Useful foundations and prerequisites
02 Essential equations and symbols
Definitions and conventions
- Ddev, Dvol
- Deviatoric and volumetric constitutive parts.
- Bulk modulus in the volumetric contribution.
- Volumetric strain direction in the declared Voigt convention.
- Centre-point strain operator and Jacobian.
03 Step by step derivation
Follow the assumptions, equations and discrete operations in order. The full explanation stays visible.
1 · Split the isotropic constitutive energy
The research variant separates volumetric stiffness from the remainder of the isotropic 3D elastic matrix. m selects the normal-strain trace in the declared engineering-Voigt order. The bulk modulus-like factor grows as ν approaches 1/2, which is why full displacement integration can become too stiff in nearly incompressible response.
This is a selective integration research element and an isotropic-only implementation, not a replacement for the public fully integrated HEX8 formulation.
2 · Integrate distortional and volumetric terms differently
The deviatoric contribution is assembled with the standard eight HEX8 Gauss points. The volumetric contribution is evaluated at the element centre. The consistent mass is delegated to the standard HEX8 mass matrix.
The centre-point term has fewer constraints than the full volumetric rule. Whether that improves a case depends on mesh, distortion, constraints, and material; rank and zero-energy modes must be checked.
3 · Why the card remains research-only
A lower volumetric stiffness can reduce locking, but reduced integration can also produce spurious mechanisms or distortional sensitivity. Required evidence includes eigenvalue/rank checks, rigid and hourglass-mode search, patch/energy tests, distortion sweeps, convergence, and same-mesh comparisons against a named reference. The current public registry does not include this separate class; its availability in source is not a maturity claim.
04 Algorithm and implementation
Source excerpt · v0.2.11
The research class computes the deviatoric contribution at standard HEX8 points and the volumetric contribution once at the centre, then adds them.
Functions shown: volumetric_stiffness stiffness
def volumetric_stiffness(self, coords: np.ndarray) -> np.ndarray:
self.validate_geometry(coords)
_, volumetric = self._constitutive_split(self.material)
point = self.volumetric_integration_points[0]
b_matrix, determinant = self.b_matrix(coords, point)
stiffness = determinant * (b_matrix.T @ volumetric @ b_matrix)
return 0.5 * (stiffness + stiffness.T)
def stiffness(self, coords: np.ndarray) -> np.ndarray:
return self.deviatoric_stiffness(coords) + self.volumetric_stiffness(coords)05 Verification and limitations
RESEARCH_ONLY · not in the public element registry
Each reference supports its own model, observable and analysis route. Read the original contract and limitations before transferring a conclusion.