Source-linked AI summary
A stabilized, nodal strain finite element method for small and large deformations: formulation, analysis, and impact on remeshing strategies
Nestor Rossi, Ignacio Romero
TL;DR
Large-deformation remeshing can diffuse material history and reduce simulation accuracy. This paper develops variationally formulated, stabilized nodal-strain finite elements for small and large strains. Its tests show reduced internal-variable diffusion, supporting their use in finite-strain simulations with large distortions.
Problem
In large-deformation simulations, remeshing addresses mesh distortion but can diffuse localized internal-variable data and compromise the accuracy of history-dependent analyses.
Source: §1 · pp. 1–2Method
The paper formulates nodal-strain methods as mixed finite elements on primal and dual meshes, then eliminates dual-mesh fields to obtain assumed-strain formulations.
Source: §6 · p. 27Results
Tests show nodal-strain methods are less afflicted by internal-variable diffusion and, with their robustness regarding locking, are well-suited to finite-strain simulations with large distortions.
Source: §6 · p. 28Takeaways & Limitations
The nodal-strain formulation’s reduced information diffusion supports its use in finite-strain simulations with large distortions.
Source: §6 · p. 28For heterogeneous materials, the dual cells are assumed homogeneous, so nodes cannot lie on material interfaces and the method cannot reproduce the exact displacement solution in the described problem.
Source: §2.3.1 · p. 10
Abstract
from arXiv · showhide
We present in this article the discretization of small and finite strain mechanics using variationally consistent, stabilized, nodal strain finite element formulations. We prove that these methods derive from a mixed variational principle that uses different meshes for the primal and dual variables. We demonstrate that, for both mechanical problems, nodal strain finite elements can be written as pure primal formulations with assumed strain operators. The stabilization terms are shown to be necessary and a new, general class of stabilizing functions is proposed for the finite strain regime. The ensuing formulations can be interpreted as pure particle methods because all kinematic and material information is stored at the nodes. This point of view explains the favorable properties of nodal strain methods when employed in problems that require frequent remeshing and involve inelastic materials. In these cases, the diffusion of internal variables is kept to a minimum. The numerical examples presented validate the claims.
1 Introduction
Large-deformation remeshing can distort meshes and diffuse material history, threatening accuracy in history-dependent simulations. This work extends variationally grounded nodal-strain methods to finite strains and investigates their remeshing behavior.
1 Introduction
- Remeshing is used to address severe mesh distortion in large-deformation Lagrangian analyses, but transferring internal variables can diffuse localized material data and reduce accuracy.Standard formulations may require history remapping even when nodal positions are unchanged, because integration-point locations generally change with connectivity.Source: §1 · pp. 1–2
- Existing nodal-strain stabilization strategies often lack mathematical analysis, while the variational derivation previously described was limited to small-strain linear elasticity.The paper also notes recognized low-energy modes and loss of stability under mesh refinement in some formulations.Source: §1 · p. 3
- The work extends a variationally derived nodal-strain framework to large-strain kinematics and constitutive models with internal variables.It represents strain and history variables as piecewise constants on nodal patches and introduces a general structure for variational stabilization.Source: §1 · p. 3
- The method has mixed finite-element and particle-like interpretations, with material information associated with nodes that are preserved when connectivity changes.The paper investigates whether this nodal structure reduces information loss from remeshing while retaining variationally consistent stabilization.Source: §1 · p. 3
- The numerical examples show the proposed method is robust for finite-strain formulations and reduces artificial diffusion from remeshing.The introduction presents these outcomes as validation of the method’s claims.Source: §1 · p. 4
2 Mixed formulation and nodal strain approximation
The nodal strain method is derived from a mixed variational formulation on primal and dual meshes, then reduced to a displacement-only assumed-strain formulation. Stabilization is needed for stable results in the coarse-mesh static example, while dual-cell material storage gives the method a particle interpretation.
2.1 Mixed variational formulation
- The nodal strain formulation starts from a Hu-Washizu saddle-point problem on primal and dual meshes, with the dual mesh associated to nodes.In the continuous problem the auxiliary energy has no effect, but its discrete counterpart can stabilize the formulation.Source: §2.1 · p. 5, Figure 1 · p. 5, §2.2 · p. 6
2.2 Finite element discretization
- The discrete projection links the symmetric displacement gradient to strain values that are constant on each dual cell.With the selected spaces, the mixed problem reduces to a displacement-only formulation with an assumed strain operator.Source: §2.2 · pp. 6, 8
- Because dual cells correspond to nodes, strain, stress, and material history can be associated with nodes, giving the method a particle-method interpretation.The elastic auxiliary material used for stabilization does not alter the constitutive response or add material history.Source: §2.2 · p. 8
2.3 On the need for stabilization and Lamichhane’s results
- Lamichhane proved coercivity and O(h) H1 displacement convergence for linear elasticity on tetrahedra with the stated stabilizing material parameters.The coercivity constant is independent of λ/µ, explaining the method’s locking-free behaviour in that setting.Source: §2.3 · pp. 8–9
2.3.1 Numerical example
- The example’s mesh sequence places the material interface inside primal elements because dual cells are assumed homogeneous and nodes cannot lie on the interface.A single primal element shape function therefore cannot reproduce the exact displacement field’s change across materials.Source: §2.3.1 · p. 10
- Without stabilization, displacement gradients do not converge in the H1-norm, while stabilization restores convergence; unstabilized displacement and strain fields oscillate.The oscillations occur even though displacement and strain converge in the L2-norm.Source: §2.3.1 · p. 10, Figure 5 · p. 11
- In the two-material bar example, unstabilized solutions show unrealistic oscillations in static analysis with coarse meshes.The example also shows failure to converge in the expected norm without stabilization.Source: §2.3.1 · p. 10
3 Extension to large-strain kinematics
The section extends the Hu–Washizu nodal-strain framework to finite-strain kinematics, including a general variationally consistent stabilizer and an assumed-strain operator updated for changes of reference configuration. Its example shows that avoiding oscillations alone is insufficient: the stabilizer must also prevent primal-mesh element inversion.
3 Extension to large-strain kinematics
- The section generalizes the Hu–Washizu functional to finite-strain kinematics and discusses continuum and discrete formulations, stabilization, reference changes, and stabilizer choice.The introduced functional spaces are not claimed to suffice for proving existence of a solution.Source: §3 · p. 11, §3.1 · p. 11
3.2 The stabilization function
- The finite-strain stabilizer is a Bregman divergence of a smooth frame-indifferent function, which ensures objectivity and satisfies the stated discrete-variation requirements.The divergence is the difference between the function evaluated at the deformation gradient and its linearization around the independent field.Source: §3.2 · pp. 13–14
3.3 Finite element discretization
- The discrete formulation uses primal and dual meshes, with the assumed deformation gradient interpreted as a modified gradient operator on the discrete deformation.The derivation identifies compatible stabilization with Bregman divergences of frame-indifferent functions.Source: §3.3 · pp. 14–15
3.4 The projection operator in a change of reference configuration
- After remeshing, the assumed-gradient operator is updated using the last remeshed configuration, while the weak form remains computed in the initial configuration.The intermediate configuration is the configuration at the last remeshing.Source: §3.4 · pp. 15–16
3.5 On the choice of the stabilizing function ˆg
- The stabilizer's domain controls its behavior under severe distortion; growth to infinity as det G approaches zero from above makes the primal-mesh term a barrier against degeneration and inversion.The dual-mesh deformation gradient having positive determinant does not guarantee the same for its primal-mesh counterpart.Source: §3.5 · pp. 16–17
3.5.1 Numerical example
- The stabilizing function may be chosen independently of the Saint Venant–Kirchhoff material model while retaining variational consistency.The example compares two stabilizing functions with the unstabilized formulation.Source: §3.5.1 · pp. 17–18
- The barrier-capable stabilizer removes oscillations and primal-mesh element inversion, unlike the quadratic choice, despite using a smaller stabilization coefficient.The quadratic stabilizer removes oscillations but still permits primal-mesh inversion; the barrier-capable choice achieves both goals.Source: §3.5.1 · pp. 17–18
4 Application to history-dependent problems with remeshing
Section 4 applies the nodal formulation to history-dependent problems: material history remains at nodes during connectivity-only remeshing, while the dual mesh and equilibrium must still be updated. It introduces finite-strain plasticity and an entropy-based indicator for assessing diffusion of internal variables.
4.1 Finite-strain elastoplastic constitutive model
- The formulation uses classical finite-strain J2 plasticity, with internal variables updated once per dual cell from nodal averaged deformation gradients.The material state includes the deformation gradient and elastic and hardening variables; the implementation uses MUESLI.Source: §4.1 · pp. 18, 20
4.2 Remeshing
- Connectivity-only remeshing avoids remapping nodal material history, but changes the dual mesh and requires rebalancing; the primal-mesh deformation gradient still needs remapping.The examples restrict remeshing to node reconnection through flip operations.Source: §4.2 · pp. 20–21
4.3 Measuring diffusion
- The indicator measures how diffused, or close to homogeneous, a field is; it takes values only in (0, 1], where values close to 0 indicate a very localized field and feq = 1 represents the homogeneous case.Source: §4.3 · p. 22
- For equivalent plastic strain, the indicator represents an equivalent plastified volume fraction; a more localized plastic-strain distribution yields a lower value.A distribution homogeneous on a subdomain has an indicator equal to that subdomain’s volume fraction.Source: §4.3 · pp. 22–23
5 Numerical examples
The examples compare standard FEM and nodal strain under cyclic large distortions that require remeshing. Across the cylinder, block, and tube cases, remeshing produces larger indicator jumps for standard FEM, while the tube case shows why jumps matter relative to the field’s level.
5 Numerical examples
- The study uses quasi-static cyclic large-distortion examples and compares remeshed with unremeshed runs for standard FEM and the nodal-strain approach.The standard-FEM baseline uses a displacement-pressure mixed P1/P1 formulation with projection stabilization and remaps internal variables by element-shape-function projection.Source: §5 · p. 23
5.1 Torsion of a cylinder
- In cylinder torsion, remeshing causes diffusion-indicator jumps in both methods, but the jumps are clearly larger for standard FEM and accumulate over successive remeshes.The meaningful comparison is the remeshing-induced jumps, not absolute indicator levels across methods, which use different discrete representations.Source: §5.1 · pp. 24–25, Figure 12 · p. 25
5.2 Distortion of a block
- In the block distortion case, standard FEM shows noticeable jumps at remeshing, whereas the nodal-strain curve is virtually continuous at the same scale.The standard-FEM jumps occur during indicator growth, decay, or near local extrema, linking them to remeshing rather than loading stage.Source: §5.2 · pp. 25–26, Figure 14 · p. 26
5.3 Rotation of a tube
- In tube rotation, plastic strain remains localized and the indicator stays low, making standard-FEM remeshing jumps a much larger fraction of the indicator itself.The indicator’s equivalent-volume interpretation connects those relative increments to plastic slip spreading over a significant body volume.Source: §5.3 · p. 27
6 Conclusions
The conclusions frame nodal strain as a mixed-mesh method reducible to a single-field assumed-strain formulation, with stabilization functions proposed for finite strains. Its nodal material representation avoids history transfer during remeshing and reduces internal-variable diffusion in the reported tests.
6 Conclusions
- The nodal strain formulation is a mixed method on primal and dual meshes that can eliminate dual-mesh fields into a single-field assumed-strain method.The conclusions characterize the resulting method as locking-free.Source: §6 · p. 27
- For finite strains, the proposed general class of stabilizing functions preserves variational consistency and objectivity while allowing relatively simple implementation.The numerical examples confirm the selected stabilization choices.Source: §6 · p. 28
- Because material points coincide with mesh nodes, the method avoids projecting material history when remeshing adds no new nodes, and the examples show reduced internal-variable diffusion.If remeshing introduces new nodes, internal variables must be interpolated to them.Source: §6 · p. 28
A Appendix: remeshing performance in numerical experiments
The appendix describes flip-based remeshing operations and their quality controls. It notes that flip-only strategies do not guarantee every mesh-quality tolerance, while the whole-domain projection makes diffusion relatively insensitive to the resulting mesh configuration.
A Appendix: remeshing performance in numerical experiments
- The algorithm supports four interior tetrahedral flip types and one boundary flip type, and reports flip counts and changes in total element count.Quality checks use minimum dihedral angle and meanratio thresholds; boundary flips also face a maximum patch-volume-change criterion.Source: A Appendix: remeshing performance in… · pp. 28–29
- Flip-only remeshing prevents mesh quality from worsening but does not guarantee that all prescribed quality tolerances will be met.Because solution projection covers the whole domain, diffusion is relatively insensitive to the new mesh configuration, whether produced by flips or a more sophisticated strategy.Source: A Appendix: remeshing performance in… · p. 29