QF Solver
Get started
White box / Analysis methods / Composed nonlinear routes

Material + geometric nonlinear response

Composed formulation · assumptions, essential equations, full derivation, algorithm, code and verification references.

← All formulations and methodsComposed formulation
Browse another sheet Analysis methods / Composed nonlinear routes
01 Family
02 Chapter
03 Sheet
Analysis methodsComposed nonlinear routesSheet 2 / 2
Composed formulationOWNER-ACCEPTED BOUNDED STATIC EVIDENCE · selected combinations

Selected static TET4/TET10/HEX8/HEX20 cases compose material and geometric contributions. This is not a universal nonlinear route.

01 Model and scope

The residual combines internal forces from the constitutive state and the selected geometric formulation. Its tangent must differentiate the same route-specific contributions; contact, dynamics and other element/material compositions remain separate unless explicitly evidenced.

Read the kinematics, constitutive law and solution lifecycle together. A component's evidence does not qualify their combination.

Route assumptions and calculation checks

  1. Contact, dynamics, distributed execution, and external Code_Aster nonlinear correlation are outside the stated acceptance.
  2. Historical failures and limits remain part of the record.
  3. Each material and geometry subroute must match its declared state and strain assumptions.

Useful foundations and prerequisites

Strain, stress and rotation measuresSmall-strain J2 plasticityFull Newton with line search

02 Essential equations and symbols

R(u,α,S)=fext(α)−fintmat(u,S)−fintgeo(u)\mathbf R(\mathbf u,\alpha,\mathbf S)=\mathbf f_{ext}(\alpha)-\mathbf f_{int}^{mat}(\mathbf u,\mathbf S)-\mathbf f_{int}^{geo}(\mathbf u)KT=−∂R∂u=∂fintmat∂u+∂fintgeo∂u\mathbf K_T=-\frac{\partial\mathbf R}{\partial\mathbf u}=\frac{\partial\mathbf f_{int}^{mat}}{\partial\mathbf u}+\frac{\partial\mathbf f_{int}^{geo}}{\partial\mathbf u}

Definitions and conventions

fint,material
Internal force from the constitutive response.
fint,geometry
Internal force from the selected geometric formulation.
α\alpha{}
Applied load parameter.
KT=−∂R/∂u\mathbf K_T=-\partial\mathbf R/\partial\mathbf u
Negative derivative of the external-minus-internal residual for the declared route.
Material + geometric nonlinear response · method mapThe same named route must connect its kinematics, constitutive state, internal residual, tangent, and commit decision.
4 detailed steps

03 Step by step derivation

Follow the assumptions, equations and discrete operations in order. The full explanation stays visible.

1 · Identify the selected kinematics and material law

A combined route is not a single universal formulation. The kinematic measure may be corotational or total-Lagrangian; the constitutive update may be elastic, J2, or another declared law. They must use compatible stress and strain measures and the same committed state.

(E,K,M,Q,S)(\mathcal E,\mathcal K,\mathcal M,\mathcal Q,\mathcal S)

2 · Assemble internal force and consistent route tangent

Each integration point evaluates its current kinematics, material response, and internal-force contribution. The tangent must differentiate the same residual, including the geometric terms and algorithmic material tangent actually used by that route.

R(u,α,S)=fext(α)−fint,mat(u,S)−fint,geo(u)\mathbf R(\mathbf u,\alpha,\mathbf S)=\mathbf f_{ext}(\alpha)-\mathbf f_{int,mat}(\mathbf u,\mathbf S)-\mathbf f_{int,geo}(\mathbf u)KT=−∂R∂u\mathbf K_T=-\frac{\partial\mathbf R}{\partial\mathbf u}

3 · Iterate without advancing history prematurely

Newton corrections and line search operate on trial states derived from the last committed material history. Only after global equilibrium converges does the route commit state and advance the load parameter. Rejection must restore all state before a cutback.

KTΔu=R\mathbf K_T\Delta\mathbf u=\mathbf RSn+1=S∗if C=1\mathbf S_{n+1}=\mathbf S^*\quad\text{if }\mathcal C=1λn+1=λn+Δλ\lambda_{n+1}=\lambda_n+\Delta\lambda

4 · Treat route composition as the actual scope

The cumulative demonstration concerns selected static TET4/TET10/HEX8/HEX20 cases. Contact, dynamics, distributed execution, and external nonlinear correlation are outside its stated acceptance. Match the exact material, kinematic switch, load, element, and registry record before transferring any conclusion.

The 0.2.11 verification index records bounded cases and historical failures; it does not qualify every mathematically possible composition.

04 Algorithm and implementation

This sheet provides explanatory references rather than an inline source excerpt. Use the labeled sources to inspect the precise boundary.

05 Verification and limitations

OWNER-ACCEPTED BOUNDED STATIC EVIDENCE · selected combinations

Each reference supports its own model, observable and analysis route. Read the original contract and limitations before transferring a conclusion.

Cumulative nonlinear evidence
Capability matrix →Recorded cases and original sources →Run a maintained example →

06 Sources and related sheets

Cumulative nonlinear evidenceCurrent analysis map

Continue the reasoning

Corotational J2
Back to sheet contents ↑