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Solution Procedures, Thermal Stress, and the Creep Route

Problem type is not a solver algorithm

Study answers what is being modeled: solid mechanics or heat transfer, static or transient, dimension, and kinematic assumptions. SolutionProcedure answers how it is solved:

Physical equation Standard route Explicit route
\(K u=F\) linear solve not applicable
\(R(u)=0\) incremental Newton not yet available
\(C\dot T+KT=Q\) implicit Euler not yet available
\(R(u,CE,t)=0\) backward-Euler creep + Newton not applicable
\(M\ddot u+C\dot u+Ku=F\) Newmark or generalized-\(\alpha\) central difference

This keeps second_order_dynamics from meaning “central difference” merely because that was the first implemented algorithm. It also keeps incrementation, Newton settings, time integration, and output cadence as separate decisions.

The implicit structural route currently assumes linear operators and time-invariant prescribed displacement supports. Newmark uses average acceleration by default. Generalized-\(\alpha\) derives \(\alpha_m,\alpha_f,\beta,\gamma\) from the spectral radius at infinite frequency and provides controllable high-frequency damping. Nonlinear implicit dynamics and moving supports require additional residual, linearization, and prescribed-kinematics work.

The useful first thermal-mechanical route

Many component analyses do not need a monolithic temperature-displacement solve. If temperature affects stress but deformation and dissipation do not meaningfully affect heat transfer, the transparent route is:

  1. solve \(\rho c_p \dot T-\nabla\cdot(k\nabla T)=Q\);
  2. retain the temperature history;
  3. evaluate \(\epsilon_{\mathrm{th}}=\alpha(T-T_{\mathrm{ref}})I\);
  4. solve mechanical equilibrium with the named equivalent operator \(F_{\mathrm{thermal}}\).

AgentFEM now stores \(E,\nu,\rho,\alpha,k,c_p,T_{\mathrm{ref}}\) in one thermoelastic property object, while heat and mechanics remain separate models sharing the mesh, material, and temperature field. Plane strain keeps the constrained out-of-plane thermal strain; plane stress uses the reduced constitutive relation.

This follows the same modeling distinction documented by Abaqus: fully coupled analysis is needed when stress and temperature mutually influence each other; sequential analysis is appropriate when the coupling is effectively one way. The current route does not claim monolithic coupling.

The first global creep state consumer

The local ArrheniusPowerLawCreep relation is normalized at a declared reference temperature:

\[ A(T)=A_{\mathrm{ref}} \exp\left[-\frac{Q}{R} \left(\frac{1}{T}-\frac{1}{T_{\mathrm{ref}}}\right)\right]. \]

The implementation uses the equivalent exponent form in code; all temperatures are absolute. The global three-dimensional Mises power-law route now accepts either an isothermal material or a scalar/finite-element temperature input for the normalized Arrhenius law. It provides:

  • committed/trial CE and CEEQ at every quadrature point;
  • a safeguarded backward-Euler local scalar solve over \(\Delta t\);
  • an analytical algorithmic consistent tangent consumed by global Newton;
  • atomic commit after acceptance and rollback after Newton, local, or maximum-creep-increment failure;
  • complete regional material assignment, fixed or automatic physical-time increments, dissipation history, and portable full-Step checkpoint/restart;
  • MPI-safe regional constitutive dispatch and a portable quadrature-state archive keyed by original physical cell, quadrature point, rule, material contract, and mesh fingerprint;
  • integration-point temperature consumption, TEMP output, increment-wise temperature evidence, and temperature identity in checkpoints;
  • a three-dimensional homogeneous stress-relaxation Golden contract.

This is implemented by reusing QuadratureTransaction; creep does not own a second private state store. Checkpoint recovery reconstructs accepted stress from displacement and committed CE without advancing a fictitious time increment. The route follows the same consistent-linearization principle that underpins robust Newton convergence in inelastic finite elements.

Arrhenius power-law creep is therefore a global consumer, while Sinh and Kachanov--Rabotnov remain material-point capabilities. A prescribed temperature field is not yet an automatic transfer of an entire transient thermal history, and it does not silently promote damage, mesh regularization, or structural rupture prediction.

Current boundary and next gates

Implemented now:

  • explicit central difference and implicit Newmark/generalized-\(\alpha\);
  • implicit-Euler heat transfer as a model-owned Step;
  • sequential isotropic thermal expansion as a visible vector operator;
  • normalized Arrhenius temperature dependence at material-point and global integration-point level;
  • global 3D J2 quadrature state with regional materials, analytical tangent, and portable full-Step restart;
  • cumulative J2 restart history, analytical uniaxial Golden verification, quadrature S/PE/PEEQ/MISES and nodal RF result fields, cyclic amplitude, physical-increment cutback, prescribed work, and internal-energy decomposition;
  • exact material-point Kachanov-Rabotnov creep-damage updates, Sinh creep, and modified-theta curve projection;
  • a sequential hot-wall FEM-to-creep assessment example with explicit calibration and maturity boundaries.
  • global 3D power-law creep with backward Euler, analytical tangent, shared transaction, automatic cutback, CE/CEEQ/S/MISES/RF/TEMP, prescribed Arrhenius temperature fields, dissipation history, regional materials, portable full-Step restart, MPI-portable quadrature state, and a relaxation Golden contract.

Next gates:

  1. finish field/energy/checkpoint products on top of the common complete execution-event trace, accepted-increment histories, status files, and result manifest across heat, implicit dynamics, and explicit dynamics;
  2. exercise the portable full-Step archive on larger partition changes and scheduled HPC restart campaigns;
  3. connect accepted transient heat-transfer histories to creep increments through an explicit field-transfer contract;
  4. add multi-element and external power-component benchmarks, then promote the experimental global Newton MPI path;
  5. introduce K-R/Liu--Murakami damage only with near-failure time control and a declared mesh-regularization policy;
  6. monolithic coupling only after a real case demonstrates two-way feedback.

References: