Separate displacement and force tolerances, algebraic residual, discretization error and physical-model validity.
01 Model and scope
A consistent unit system makes every balance and tolerance dimensionally meaningful. Numerical checks answer different questions depending on whether they concern the assembled algebra, mesh approximation or a physical observable.
Route assumptions and calculation checks
- Use a declared consistent unit system and keep dimensional tolerances separate.
- Specify what is measured, its scale and the acceptance threshold.
- A small algebraic residual does not establish mesh convergence or physical validity.
02 Essential equations and symbols
Definitions and conventions
- L,F,T
- Dimensions of length, force and time.
- f_{ref},Q_{scale}
- Declared positive scales with the same units as the corresponding numerator.
- Dimensionless algebraic residual.
- Q_h,Q_{ref}
- Computed and reference values of the same observable.
03 Step by step derivation
Follow the assumptions, equations and discrete operations in order. The full explanation stays visible.
1 · Declare dimensions before choosing tolerances
Force and displacement tolerances have different dimensions. A gap threshold cannot serve as a contact-force threshold. A stiffness multiplies displacement into force, and the mass dimension follows from the dynamic balance.
2 · Check the discrete equations
Evaluate the residual on the appropriate unconstrained or projected coordinates. Constraint reactions generally make the unreduced residual nonzero on prescribed degrees of freedom. Declare the norm and force scale.
3 · Compare a defined observable
Displacement, frequency, energy and reaction are different observables. Match their location, component, units and normalization before computing an error. Near a zero reference value, use a justified absolute criterion or a separate positive scale.
4 · Keep the three questions separate
An algebraic residual concerns solution of the discrete equations. Refinement concerns the approximation of a specified model. Physical validation compares that model with relevant physical evidence. Passing one does not establish the others.
The displayed inequalities illustrate distinct acceptance questions. Their scales and thresholds must come from each case contract.
04 Algorithm and implementation
Follow the route sheets below to see how this shared mathematics is implemented.
05 Verification and limitations
FOUNDATION · explicit scales and observables
Each reference supports its own model, observable and analysis route. Read the original contract and limitations before transferring a conclusion.