Deformation models

Deformation models decide which shape changes matter. A single spring is a one-number approximation; a network of springs can represent a flexible rope, membrane, or coarse elastic object. The graph, rest geometry, mass distribution, constraints, and force law are the model—not the interpolated mesh animation.

Think like a programmer

Choose the smallest model that answers the question. Store stable node ids, masses, positions/velocities, edges, rest lengths, stiffness/damping, constraints, external loads, integration policy, and boundary conditions. Derive equal-and-opposite edge forces from a shared state snapshot, then integrate nodes synchronously. Keep mesh interpolation/rendering downstream of node state and log momentum, energy/transfer, constraint residual, minimum edge length, and time-step refinement.

Model checklist

Inputs: Nodes/masses, edge connectivity, rest lengths, force/damping law, fixed or constrained nodes, external loads, boundary contacts, integrator/step, and output mesh mapping.
State: Node positions/velocities, edge extensions/forces, constraint reactions, kinetic/elastic energy, internal-force sum, and diagnostics.
Rule: For each edge calculate extension along its current direction, add equal-and-opposite forces to endpoints, add constraints/external forces, then advance all free nodes together.
Output: Deformed node state, derived mesh, force/energy/constraint ledgers, and resolution/refinement result.
Check: Internal edge forces sum to zero; swapping endpoint storage gives identical physical forces; at rest geometry with zero velocity and no load, elastic force is zero; fixed nodes do not move but report reaction/transfer; refining segments changes the answer only under declared comparison geometry and tolerance.

\[\mathbf F_{ij}=k(\lVert\mathbf x_j-\mathbf x_i\rVert-L_{ij}^0)\frac{\mathbf x_j-\mathbf x_i}{\lVert\mathbf x_j-\mathbf x_i\rVert},\qquad\mathbf F_{ji}=-\mathbf F_{ij}\]

Spring force and stored energy

Change stiffness and extension. The spring force points back toward equilibrium, while the stored energy stays positive.

Stiffness: 40 N/mExtension: 0.35 m

Stiffness 40 N/m; extension 0.35 m; restoring force -14.00 N; elastic energy 2.45 J.

A spring network has resolution and modeling tradeoffs. More segments can resolve local curvature or deformation, but introduce more state, smaller stable time steps, and edge-direction artifacts. Compare models at matched physical length, mass density, boundary conditions, and load—not merely by node count. A mesh that visibly bends may still have unrealistic shear, area, or volume response.

Constraints are forces or impulses with a model role. A fixed node should not be silently overwritten every frame: record its reaction force and any energy/momentum exchange with the support. With only internal spring forces in isolation, net internal force must vanish; this pair-symmetry test detects update-order and edge-index bugs before a deformation image appears.

Try this experiment

Prediction: Adding more segments can represent bending better, but makes the update more expensive and potentially stiffer numerically.

Compare one spring with three springs in series at matched total rest length and mass. List the new node states, edge rest lengths, and constraints. Check internal-force sum and rest-state force. Apply the same end load, refine step size, and state whether endpoint displacement converged or merely changed because total stiffness was not calibrated.

Where this model breaks

A spring network may have grid-direction artifacts, poor volume preservation, unrealistic bending/shear, and stiffness that depends on discretization. Finite elements or continuum models are better when material fidelity matters. Collision/contact, self-intersection, plasticity, fracture, damping, and high stiffness can require specialized solvers; visual mesh smoothness does not validate force accuracy.

Summary

Model deformation with stateful nodes and force-generating connections. Preserve rest geometry, pair symmetry, constraints, and force/energy diagnostics; increase resolution only when a matched physical comparison shows it changes the answer needed.

Glossary

Rest length: unstrained length of a spring.
Node: simulated point carrying position and mass.
Damping: energy-loss term that reduces oscillation.
Constraint reaction: force required to enforce a fixed or prescribed condition.
Rest geometry: reference node/edge configuration used to compute deformation.

Self-check

What data defines a spring edge?
Why do internal forces come in pairs?
What diagnostic distinguishes a fixed-node constraint from silent position overwrite?
What trade-off comes with more nodes?

Sources

D. Baraff and A. Witkin, Large Steps in Cloth Simulation.
J. Baraff, An Introduction to Physically Based Modeling.

Model contract

Treat the lesson as a small function before treating it as a fact to memorize. Give every value a unit, keep only the state needed for the next step, and make the output easy to inspect.

\[\text{observable output} = f(\text{inputs},\,\text{state})\]

Inputs: Quantities you set or measure, with units and useful bounds.
State: Values the program must retain to reproduce the next result.
Rule: The relationship or update that turns inputs and state into a result.
Check: A known limit, unit check, invariant, or measured result that can expose a bad model.

Implement the idea as a model

For Deformation Models, write down the quantities you can control, the values your program must retain, and the result a reader could inspect. In Equilibrium and Elasticity, the useful program is not the drawing: it is the smallest explicit model that makes a prediction you can test.

Guided experiment

Prediction: changing one declared input while holding the others fixed should change only the outputs that the model connects to that input. Choose one input, predict the direction of change, then check a limiting case such as zero, a symmetric arrangement, or a familiar low-speed or small-change approximation.

Where this model breaks

This lesson is a teaching model, not a complete simulator. Before using it outside the stated question, check which interactions, scales, uncertainties, boundary conditions, and measurement limits it leaves out.

Summary

Treat Deformation Models as a contract: named inputs and units enter a rule, the rule produces an observable result, and a known limit or invariant checks whether the implementation deserves trust.

Glossary

Input: a measured value or chosen parameter supplied to a model.
State: the smallest set of values needed to continue or reproduce a model.
Validation: comparing an output with a known result, limit, invariant, or measurement.

Self-check

Which values are inputs, and which values must remain state?
What observable result would tell you the model is behaving as expected?
Which assumption would you test first before applying the model to a real system?

Model review: turn Deformation Models into a test

Represent flexible objects with stateful nodes and force-generating spring connections.

Name the inputs and units that the equilibrium and elasticity model needs.
Separate the state you must keep from values you can calculate when needed.
Write one rule that maps the current state and inputs to an observable result.
Choose a limiting case, unit check, invariant, or known result before trusting an output.
State one assumption you would change before using this simplified model for a real decision.

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