A tangent that is not positive definite is factored by LDLᵀ (Cholesky first, one symbolic
factorization) instead of cutting the increment back; each iteration records the tangent's
negative eigenvalues and the summary warns when a converged state keeps some. A model that can
move freely is still an error at its first tangent, located by the undeformed Cholesky factor;
a singular tangent later is a cutback. Nearly incompressible hybrid hex8, hex20 and tet10 at
ν = 0.4999 under prescribed displacement now converge to the closed form.
Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Temperatures now strain the material in nonlinear steps instead of being
refused. At finite strain the deformation gradient splits multiplicatively,
F = F_m F_theta with F_theta = diag(1 + alpha_i dT) at each point: the material
sees F_m and the loops run on the thermally expanded volume J_theta dV0 (mixed
elements project det F_m), so the tangent stays consistent. At small strain the
split is additive, and elements on their linear stiffness take the equivalent
nodal forces as a dead load. The step's temperatures are its temperature loads,
or the steady conduction of its temperature conditions and heat loads (one-way
coupling, as the linear coupled step); both at once are refused. They ramp with
the load factor from each material's reference temperature. Johnson-Cook
plasticity softens at the point's temperature. Verified: free expansion and a
fully constrained neo-Hookean block at 20 % thermal stretch, full and hybrid
(exact); a bar held at its ends heated past yield (exact); the linear limit and
the conducted temperatures against the linear solvers; tangents by central
differences with a temperature field.
Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm
Two more large-strain options on the isotropic constants, decoupled like
neo-Hookean with the volume kappa/2 (J - 1)^2: Mooney-Rivlin with C10 = mu/2 - C01
(0 <= C01 <= mu/2) and Yeoh with C10 = mu/2 and the higher terms C20, C30, so the
small-strain material is the elastic one everywhere. MaterialModel::Polynomial
gives the isochoric stress and tangent from Holzapfel's fictitious quantities
(eqs. 6.168, 6.193), consistent by central differences, equal to neo-Hookean
with C10 alone; hybrid elements take it. Abaqus writes *HYPERELASTIC,
MOONEY-RIVLIN and YEOH; LS-DYNA keeps refusing hyperelastic materials in
nonlinear steps by name. Verified in uniaxial and equibiaxial tension and pure
shear against the compressible closed form (exact).
Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm
The radial return of the explicit update is now J2::radial_return, called by
both solvers, with the consistent tangent J2::tangent (Simo & Hughes 1998,
Box 3.2). The new elastoplastic module gives the implicit material point:
Simo's multiplicative plasticity with the exponential return map in the
principal logarithmic strains (CMAME 99, 1992) and its spatial tangent through
the Daleckii-Krein divided differences of the logarithm, and the additive
radial return of small strain with kinematic hardening. MaterialModel::Plastic
carries it, decoupled (deviatoric return, Hencky volume) so hybrid elements
take it. The element loops of finite.rs gain small-strain kinematics for
inelastic materials and pass history through the mixed path. Every tangent is
checked against central differences, element by element, from a plastic
history.
Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm
solve.nonlinear follows a nonlinear static step in load increments: dead
loads and prescribed displacements ramp with the load factor, a pressure
follows the deformed faces (its load stiffness symmetrised so the
supernodal Cholesky stays), the consistent tangent is refactored every
iteration with an optional line search, and an increment converges on
force, displacement and energy together. Increments start from the
extrapolated last one, grow after fast convergence and are cut back on
inversion, a non-finite or growing residual, an indefinite tangent or too
many iterations; the step stops cleanly at the minimum increment, the
increment cap or the Newton iteration limit. Elements are assembled in
parallel in label order (same bits on any thread count). Results carry
Cauchy stress, logarithmic strain, the increment history and a displacement
frame per increment, which results.frame and the results panel show.
Trusses, shells, beams, plane stress, incompatible modes, temperature and
centrifugal loads are refused by name; couplings warn that they stay
linear.
Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01NmAttHhUZTokgPeEJia8sm
