Finite element method

Overview

Soft robots are continuous bodies, so their state is a field over the body rather than a finite set of joint angles 2112.03645 §II. The finite element method (FEM) approximates that field with a mesh of elements. For 3D bodies the static equilibrium is written as Q_int(q) = Q_ext(q), where q are the nodal unknowns, and transient problems add a dynamic ODE system 2112.03645 §IV.A. Nonlinearity can come from the geometry (large displacements) or from the material 2112.03645 §IV.A.

The shared core is the discretisation of the body and the balance between internal and external forces. Variants differ in the kind of body they discretise and in how they handle finite rotations.

FEM is also used outside robotics, as the reference solver for nonlinear continuum problems. In two-scale homogenisation (FE²) each macro integration point needs its own micro solve, which is expensive 2507.05291 §1. A learned model can replace those micro solves, but FE stays the reference against which it is measured 2507.05291 §4.3 2507.05291 §5.3.

Variants

3D continuum finite elements

The body is meshed in full three dimensions. The review lists this family under continuum mechanics and says it suits the design of new concepts because it covers a broad range of physics 2112.03645 tab. 1. Its main cost is that the model has many degrees of freedom, which the review judges too heavy for control 2112.03645 tab. 1 2112.03645 §IV.A. Card: Soft Robots Modeling.

Two formulations are named for capturing finite rotations: corotational and ANCF (absolute nodal coordinate formulation) 2112.03645 §IV.A. The review does not compare them.

For actuation, the review describes the SOFA handling of tendon-like actuators as a linearised constraint, with Lagrange multipliers λ and a three-step projection procedure 2112.03645 §IV.A.

Geometrically exact finite elements for rods (GE-FEM)

For slender, beam-like bodies the review places a geometrically exact finite element formulation in the directors-based family. It is an energetic Ritz method: it reduces the rod configuration to Lagrange ODEs [arxiv:2112.03645 §V.B]. The review names it as the main example of this reduction, next to modal Ritz and GVS 2112.03645 §V.B. It is treated on the Cosserat rod theory page, which covers the rod model it discretises.

FE as reference and data source for a learned surrogate

Here FE is not the model that runs in the loop. It produces the reference stresses: the paper's targets are nodal stresses averaged from FE solutions of periodic cells with a central hole 2507.05291 §4.1 [arxiv:2507.05291 §2.3]. A graph neural network is then trained to predict the stress at every node from the cell's mean stress 2507.05291 §3. The GNN is a learned surrogate, not a finite element solver; its architecture is covered on the graph neural network and physics-informed neural network pages.

The FE mesh also gives the discrete stress divergence used as a training penalty. Its operator is built from shape-function derivatives of the FE mesh 2507.05291 §2.3 2507.05291 eq. 31–35. The same operator can be used to check any stress field for local equilibrium, which is the residual the paper adds as a training penalty 2507.05291 §3.3.

Reported gains are author-reported and not like-for-like. On hyperelastic meshes the surrogate's inference is about 500 times faster than FE, but the timing compares GPU inference with CPU FE, excludes training, and uses no batching 2507.05291 §5.3.1. On linear meshes it is faster at every tested size, with no factor given 2507.05291 §4.3.1. Card: Physics-Informed Graph Neural Networks to Reconstruct Local Fields Considering Finite Strain Hyperelasticity.

When to use which

The review judges its options qualitatively. It gives no measured accuracy or run-time figures, so the rows about the review cite its statements, not benchmark results. The last row draws on the second card, whose figures are author-reported.

Implementation sketch

The review gives no algorithm and does not name the nonlinear solver to use. The sketch below is our paraphrase of the static case in [arxiv:2112.03645 §IV.A]. The choice of solver is left open here.

# Static equilibrium of a discretised body [arxiv:2112.03645 §IV.A]
# Input:  mesh of elements, initial nodal unknowns q0, external loads Q_ext
# Output: equilibrium nodal unknowns q with Q_int(q) = Q_ext

def static_equilibrium(mesh, q0, Q_ext):
    q = q0
    # residual: internal minus external generalised forces
    while not converged(q):
        # internal forces from the element strain energy (geometric or material)
        r = Q_int(mesh, q) - Q_ext
        # update q with a solver of your choice (not stated in the review)
        q = update(q, r)
    return q

Contradictions

• Discrete divergence of the FE reference. The linear-case divergence values differ between the paper's text and its table. Section 4.3 gives 4.56e-4 for the learned model and 9.43e-4 for FEM 2507.05291 §4.3. Table 1 gives 3.23e-4 and 9.18e-4 2507.05291 tab. 1. The paper does not explain the difference.
• Learned model below the FE reference on divergence. Both tables report a lower mean discrete divergence for the learned model than for FE: 3.23e-4 against 9.18e-4 in the linear case 2507.05291 tab. 1, and 1.56e-3 against 1.69e-3 in the hyperelastic case 2507.05291 tab. 2. The paper gives no explanation, so read this as a reported number, not as evidence that the learned stress is closer to equilibrium than FE.
• The review does not compare the corotational and ANCF formulations, nor does it report numbers for the cost of 3D FEM against the other families 2112.03645 §IV.A 2112.03645 tab. 1.

Papers

• Soft Robots Modeling: a Structured Overview: a review that sorts soft-robot models into four families and describes the finite element formulations for 3D continuum bodies and for rods (GE-FEM).
• Physics-Informed Graph Neural Networks to Reconstruct Local Fields Considering Finite Strain Hyperelasticity: a graph neural network that predicts per-node stress of a periodic micro-structure from its mean stress, trained against FE data with a discrete-divergence penalty.