Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques

Cyril Touzé; Alessandra Vizzaccaro; Olivier Thomas

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Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques

Cyril Touzé, Alessandra Vizzaccaro, Olivier Thomas

arXiv:2107.05077v1Jul 11, 2021math.NAmath.DS

TL;DR

Geometric nonlinearities make large-amplitude structural vibrations difficult to reduce because all degrees of freedom can be nonlinearly coupled and dynamics can become complex. This review synthesizes nonlinear mappings and invariant-manifold methods, including finite-element adaptations, and concludes with applications, limitations, and future directions. It emphasizes that invariant manifolds provide theoretically grounded reduced domains, while practical methods must address computational cost, folding, velocity dependence, and nonsmooth dynamics.

Problem
Large-amplitude vibrations of thin structures produce distributed nonlinear coupling and complex dynamics, challenging efficient, predictive, simulation-free reduced-order modeling.
Method
The paper reviews nonlinear reduction methods based on invariant-manifold mappings, covering graph and normal-form approaches, parametrisation methods, finite-element procedures, stress manifolds, quadratic manifolds, and direct normal-form computation.
Results
The review identifies invariant manifolds as theoretically grounded reduced domains and surveys methods that compute ROM characteristics for geometrically nonlinear structures.
Takeaways & Limitations
Invariant-manifold theory reframes nonlinear reduction as computing a geometrically defined subset that encloses the relevant long-term dynamics.
Takeaways & Limitations
Invariant-manifold applications remain constrained by graph parametrisations that cannot pass folding points, velocity-independent quadratic manifolds, FE computational cost, and smoothness requirements for nonsmooth systems.

Abstract

from arXiv · show

This paper aims at reviewing nonlinear methods for model order reduction of structures with geometric nonlinearity, with a special emphasis on the techniques based on invariant manifold theory. Nonlinear methods differ from linear based techniques by their use of a nonlinear mapping instead of adding new vectors to enlarge the projection basis. Invariant manifolds have been first introduced in vibration theory within the context of nonlinear normal modes (NNMs) and have been initially computed from the modal basis, using either a graph representation or a normal form approach to compute mappings and reduced dynamics. These developments are first recalled following a historical perspective, where the main applications were first oriented toward structural models that can be expressed thanks to partial differential equations (PDE). They are then replaced in the more general context of the parametrisation of invariant manifold that allows unifying the approaches. Then the specific case of structures discretized with the finite element method is addressed. Implicit condensation, giving rise to a projection onto a stress manifold, and modal derivatives, used in the framework of the quadratic manifold, are first reviewed. Finally, recent developments allowing direct computation of reduced-order models (ROMs) relying on invariant manifolds theory are detailed. Applicative examples are shown and the extension of the methods to deal with further complications are reviewed. Finally, open problems and future directions are highlighted.

1 Introduction

The review addresses reduced-order modeling for large-amplitude vibrations of geometrically nonlinear structures, focusing on nonlinear mappings and invariant manifolds rather than enlarged linear bases.

Geometric nonlinearities arise in thin beams, plates, and shells during large-amplitude vibrations and couple all model degrees of freedom nonlinearly.
These structures exhibit instabilities, bifurcations, quasiperiodic and chaotic solutions, and even wave turbulence, complicating reduced-order modeling.
Linear reduction methods enlarge an orthogonal basis, but nonlinear couplings can require many basis vectors in finite-element models.
The reviewed nonlinear methods define variables through nonlinear mappings, representing reduced dynamics on curved manifolds in phase space.
Invariant-manifold techniques receive special attention, including adaptations for finite-element procedures and distinctions between intrusive and non-intrusive methods.

2 Framework

The framework covers analytical and finite-element formulations of geometrically nonlinear structures, then relates nonlinear dynamics and amplitude-dependent modal selection to ROM construction.

2.1 Equations of motion: Geometric nonlinearity exceeds small-motion assumptions while retaining a linear elastic constitutive law and polynomial dependence on displacements and velocities.
2.1.1 Analytical PDE models: Analytical PDE models describe thin structures using assumptions that expose membrane/bending or axial/longitudinal coupling as a source of geometric nonlinearity.
2.1.1 Analytical PDE models: Curved shells introduce linear and quadratic membrane/bending couplings through their non-flat geometry, with quadratic coupling potentially producing softening behavior.
2.1.2 Finite elements and space discretization: The geometrically exact finite-element formulation yields a generic polynomial stiffness operator with quadratic and cubic nonlinear terms after eliminating auxiliary strain and stress variables.
2.1.2 Finite elements and space discretization: Nonlinear coupling terms scale as N^4 in modal expansions, motivating careful treatment of the retained and eliminated coordinates.
2.3 Which ROM for which dynamics ?: Internal resonances make master-mode selection amplitude-dependent, so natural-frequency relationships alone may not identify all modes needed by a correct ROM.

3 Nonlinear methods and invariant manifolds

The paper introduces invariant-manifold-based nonlinear reduction through a geometric phase-space perspective, illustrated by a folded manifold associated with internal resonance.

3 Nonlinear methods and invariant manifolds: Invariant-manifold reduction is presented geometrically in phase space, where the reduced dynamics are represented on a lower-dimensional subset.
3 Nonlinear methods and invariant manifolds: The illustrated LSM is associated with the second-mode backbone curve and a 1:3 internal resonance between the second and fourth modes.
3 Nonlinear methods and invariant manifolds: The manifold is shown in the modal-coordinate subspace (x2, ˙x2, x4) through four continuation stages and exhibits apparent folding.

3.1 Invariant manifolds for dynamical systems

Invariant-manifold theory connects dynamical-systems geometry with nonlinear model reduction through graph and normal-form parametrisations, unified by an invariance equation.

3.1 Invariant manifolds for dynamical systems: Dynamical-systems theory organizes trajectories through invariant manifolds emerging from linearized eigendirections and governed by the eigenspectrum.
3.1 Invariant manifolds for dynamical systems: The parametrisation method maps original coordinates to lower-dimensional master coordinates and simultaneously defines the manifold and its reduced dynamics.
3.1 Invariant manifolds for dynamical systems: The invariance equation enforces that the nonlinear mapping remains invariant under the vector field and enables high-order expansions of the manifold and reduced dynamics.
3.1 Invariant manifolds for dynamical systems: Polynomial expansions convert the invariance equation into order-by-order co-homological equations for computing the manifold and reduced dynamics.
3.1 Invariant manifolds for dynamical systems: The graph style relates slave coordinates functionally to master coordinates, whereas the normal-form style introduces nonlinear coordinates and retains resonant reduced-dynamics terms.
3.1 Invariant manifolds for dynamical systems: In conservative systems, Lyapunov subcenter manifolds are two-dimensional manifolds filled with periodic orbits and provide the basis for nonlinear normal modes.

3.2 The graph style: nonlinear normal modes as invariant manifolds

The graph-style approach represents slave modal coordinates as functions of master displacements and velocities, then derives invariant-manifold geometry and reduced dynamics through invariance equations. Its versatility is balanced by assumptions that limit applicability near internal resonances, folding points, and very large finite-element models.

Graph-style formulation: The graph style uses functional relationships between slave and master coordinates, with both master displacements and velocities retained.The dependence on velocities reflects the two-dimensional modal eigenspace needed to represent oscillations as closed orbits.
Graph-style formulation: Differentiating the functional relationships and substituting them into the dynamics yields invariance PDEs describing the manifold geometry in phase space.For one master coordinate, the equations comprise 2N −2 PDEs and determine N −1 unknown slave-coordinate functions.
Single-master reduction: For a single master mode, the invariant-manifold expansion begins at second order because the manifold is tangent to its linear counterpart at the origin.Under conservative restoring forces, some polynomial coefficients vanish; damping or gyroscopic forces introduce additional terms.
Multi-master reduction: The single-master formulas require absence of a second-order internal resonance ωs = 2ωm, where strong modal coupling makes single-mode reduction unsuitable.The multi-mode extension introduces more master coordinates to address internal resonance and more complex nonlinear dynamics.
Single-master reduction: The single-master reduced dynamics retains self-quadratic and cubic contributions, including slave-mode effects gathered through sums over the slave modes.The displayed reduced model uses third-order truncation for both the coordinate relationship and reduced dynamics.
Applications and limitations: Graph-style invariant manifolds can be computed numerically through PDE solvers and have been applied to beams, frames, damping, forcing, and piecewise-linear systems.However, the graph assumption cannot represent folding points, and prior finite-element applications did not address models with millions of degrees of freedom.

3.3 Normal form approach

The normal form approach derives a nonlinear coordinate transformation, selects master normal coordinates, and reduces the dynamics to resonant terms. Compared with graph parametrisation, it gives the same second-order manifold geometry and predicts the same hardening or softening behavior, while supporting resonances, damping, forcing, and compact models.

3.3 Normal form approach: The normal form approach computes a complete nonlinear transformation before selecting a few resulting coordinates as masters for reduction.This allows reduced-order models with an arbitrary number of master modes, although the cited calculations were limited to third order.
3.3.1 Method and main results: The transformation is derived order by order by introducing an unknown nonlinear mapping and solving the associated homological equation.The method follows normal form theory while retaining oscillator-like mechanical equations in its real-valued formulation.
3.3.1 Method and main results: Velocities are included in the identity-tangent mapping, whose higher-order terms describe invariant-manifold curvature in phase space.The mapping introduces normal coordinates that are nonlinearly related to the original modal coordinates.
3.3.1 Method and main results: Selecting a few master normal coordinates and setting the remaining normal coordinates to zero parametrises the invariant manifold used for reduction.Internal resonances modify the retained normal-form terms by allowing additional resonant contributions.
3.3.1 Method and main results: The normal form retains only resonant monomials, eliminating quadratic terms in the nonresonant conservative case and collecting their effects in new fourth-order tensors.This produces a reduced equation containing the cubic terms associated with trivial resonances.
3.3.1 Method and main results: At second order, graph and normal-form parametrisations produce identical quadratic slave-mode geometry, while their reduced dynamics differs because the coordinates have different meanings.A nonlinear coordinate change makes the reduced equations equivalent, and both approaches predict the same hardening or softening behavior.
3.3.2 Applications: Parameter-dependent normal forms extend the method to modal damping, forcing, and frequency responses while preserving the conservative limit as damping vanishes.The reduced damping accounts for slave-mode damping coefficients, and damping can affect the type of nonlinearity.
3.3.2 Applications: Normal forms provide compact reduced dynamics containing the qualitative picture and bifurcations of the full system, supporting efficient ex-nihilo and identification models.Applications include structures with 1:1, 1:2, 1:3, 1:2:2:4, and related internal resonances.

3.4 Spectral submanifold

Spectral submanifolds address existence and uniqueness of invariant manifolds for dissipative systems, enabling reduced dynamics on selected modal directions. Their uniqueness depends on the spectral quotient and sufficiently high-order computation.

SSMs were introduced to clarify existence and uniqueness for dissipative systems and resolve terminology ambiguities surrounding nonlinear normal modes.
For a manifold associated with the first d modes, the spectral quotient compares the largest slave-mode damping rate with the smallest master-mode damping rate.
The existence theorem requires a sufficiently regular nonlinear vector field and non-resonance conditions on the eigenfrequencies.
A d-dimensional SSM is unique among invariant manifolds of smoothness σout + 1 sharing the same properties.
Increasing damping ratios with frequency can make the spectral quotient very large, so lower-order expansions may approximate infinitely many locally similar invariant manifolds.
The reviewed computational procedure was technically developed for a two-dimensional SSM, while extensions to multiple master modes require an additional calculation.
In polar coordinates, the reduced dynamics directly provides amplitude-frequency relationships without requiring time integration of the reduced model.
Applications include nonlinear Timoshenko and Rayleigh beams, backbone curves, frequency responses, experimental-data identification, and forced-response calculations.

4 ROMs for finite element problems

Finite-element reduction methods reviewed here range from non-intrusive static procedures and implicit condensation to quadratic manifolds and direct invariant-manifold computations. Their accuracy depends on capturing velocity dependence and respecting frequency-separation assumptions, while direct normal-form formulations extend simulation-free ROMs to internal resonances and large FE systems.

4.1 FE procedure and Stiffness Evaluation: FE codes often hide quadratic and cubic nonlinear terms, motivating non-intrusive static calculations that recover coefficients from accessible stiffness evaluations.These procedures exploit static FE computations because k(X) is available even when explicit nonlinear terms are not.
4.2 Implicit condensation and stress manifold: When the slave-to-master frequency ratio increases, ICE predictions approach the invariant-manifold geometry; near ρ = 2, strong 2:1 coupling makes single-mode reduction meaningless.The correction factor R tends to 1 as ρ increases, while the method predicts a 1% error at ρ = 11.7 and a 10% error at ρ = 4.15.
4.2 Implicit condensation and stress manifold: ICE constructs a stress manifold through static implicit condensation, but it is not invariant and omits velocity dependence.The resulting reduction approaches the invariant manifold only under a slow/fast separation between slave and master modes.
4.3 Modal derivatives and quadratic manifold: Quadratic manifolds built from modal derivatives can reproduce flat-beam responses, but curvature and incorrect self-quadratic treatment can cause departures from full-model results.The mapping is velocity-independent, so it is not an invariant manifold and requires a slow/fast assumption for accurate predictions.
4.3 Modal derivatives and quadratic manifold: Including velocity coordinates converts the quadratic-manifold mapping into a uniformly valid, simulation-free invariant-manifold formulation and bypasses the slow/fast restriction.The generalized formulation connects modal derivatives with normal-form theory and supports recursive higher-order developments in a non-intrusive FE setting.
4.4 Direct computation of normal form: Direct normal-form computation avoids expensive diagonalization in million-degree-of-freedom FE models and captures 1:2 and 1:3 internal-resonance responses with compact ROMs.Second-order mappings with third-order reduced dynamics already produce excellent results for numerous test cases up to comfortable vibration amplitudes.

5 Open problems and future directions

The review identifies unresolved questions about invariant-manifold ROM accuracy, parametrisation, and scope, while pointing to direct physical-space computation as a foundation for expansion.

Manifold geometry: The effects of invariant-manifold folding on dynamics, parametrisation, and ROM performance remain insufficiently understood, especially beyond apparent projection-induced loops.A clear folding without internal resonance would clarify differences between graph-style and normal-form parametrisations.
Accuracy and convergence: Invariant-manifold ROMs still lack simple a priori quality estimates comparable to singular-value estimates for linear POD reduction.The difficulty is tied to assessing manifold approximations and developing higher-order expansions.
Accuracy and convergence: Internal resonances between nonlinear frequencies at higher amplitudes remain an obstacle to blind convergence through increasing manifold order.The parametrisation method can inspect linear eigenfrequency relationships but cannot fully anticipate nonlinear-frequency resonances.
Future methodological development: Recent physical-space methods provide effective direct ROM computation, motivating extensions that enlarge their applicability and improve computation.The review connects this direction to attracting center sets containing long-term dynamical behavior.
Broader physical scope: Future applications include nonlinear damping, piezoelectric couplings and material nonlinearities, non-local nanostructure models, and nonsmooth contact or friction problems.Nonsmooth extensions are constrained by the smoothness assumptions underlying current invariant-manifold theorems.
Complex dynamics: Applying invariant-based ROMs to 10-20 master modes raises open questions about complex dynamics, chaotic vibrations, and wave turbulence.Most reported applications use only 1-3 master modes, leaving scalability in modal dimension an open direction.

6 Conclusion

The review presents nonlinear manifold-based ROM methods for geometrically nonlinear structures, emphasizing invariant manifolds and finite-element applications. It concludes that recent approaches can compute efficient, generally simulation-free ROMs directly from finite-element meshes while retaining broad applicability.

6 Conclusion: Nonlinear reduction uses a mapping to a curved manifold, embedding geometric complexity and potentially achieving greater accuracy with fewer master coordinates.This contrasts with linear methods that enlarge an orthogonal basis with additional vectors.
6 Conclusion: Invariant manifold methods receive special emphasis because dynamical-systems theory identifies subsets enclosing the long-term dynamics of mechanical systems.Their curvatures also encode non-resonant couplings, although computing them is more involved than linear reduction.
6 Conclusion: The review develops a unified parametrisation perspective, using LSM terminology for conservative systems and SSM terminology for damped systems, and gives special attention to finite-element structures.The presentation follows historical developments before addressing finite-element applications.
6 Conclusion: Finite-element methods such as STEP, implicit condensation, and modal derivatives offer ease, speed, efficiency, and non-intrusiveness but require extra assumptions and lack generality.They perform excellently when their assumptions, including slow/fast separation where needed, are satisfied.
6 Conclusion: Recent invariant-manifold ROMs can be computed directly from finite-element meshes, possibly non-intrusively, with generally simulation-free costs comparable to modal-derivative techniques.The review connects large finite-element models with lower-dimensional subsets containing important dynamics and identifies further generalization as an open direction.

Funding

The paper received no additional funding.

Funding: The work received no additional funding.

A Symmetry of the quadratic and cubic tensors

The appendix explains symmetry properties of the quadratic and cubic nonlinear tensors in finite-element and modal coordinates. These symmetries follow from index interchange, monomial commutativity, coefficient transformations, and the underlying potential energy.

A Symmetry of the quadratic and cubic tensors: Finite-element internal forces are represented using linear, quadratic, and cubic tensor coefficients in the model coordinates.The internal force vector is written as k(X) = KX + f_nl(X), with finite-element coordinates indexed through Einstein summation.
A Symmetry of the quadratic and cubic tensors: Finite-element strain and virtual-work expressions use discretized gradient operators built from spatial derivatives of shape functions.The matrices have rows corresponding to the six Voigt strain components.
A Symmetry of the quadratic and cubic tensors: The elasticity tensor symmetry yields a symmetric stiffness matrix and permits permutations of indices for cubic coefficients.For distinct indices, the cubic coefficients admit 4! = 24 index orderings.
A Symmetry of the quadratic and cubic tensors: Quadratic coefficients lack intrinsic index symmetry, but only paired sums matter because XiXj is unchanged when i and j are exchanged.An upper-triangular representation assigns nonzero coefficients only to increasing indices.
A Symmetry of the quadratic and cubic tensors: Redefining quadratic coefficients as symmetric parts permits index-order changes in the standard tensor form, and the same reasoning applies to modal coefficients.Modal coefficients inherit the corresponding symmetries through transformations involving eigenvector components.
A Symmetry of the quadratic and cubic tensors: Any permutation of modal tensor indices is allowed, and the full set of tensor symmetries follows from the existence of a quartic potential energy.The internal force components can be derived from elastic energy using Schwarz’s theorem.

B Classification of nonlinear terms

The appendix classifies nonlinear terms by their relationship to internal resonance and their effect on invariant subspaces. Resonant terms produce strong coupling and energy exchange, while non-resonant terms do not correspond to internal resonance.

B Classification of nonlinear terms: Geometrically nonlinear structures behave as large assemblies of nonlinearly coupled oscillators, producing many coupling terms.
B Classification of nonlinear terms: Invariant-breaking terms excite additional oscillators whenever the master coordinate xm is nonzero, breaking invariance of the corresponding linear eigensubspaces.These terms can be tracked directly in the equations defining invariant-manifold geometry.
B Classification of nonlinear terms: A nonlinear term can be interpreted as forcing an oscillator, so xm^2 contains frequency components 2ωm and 0.If ωp ≃ 2ωm, the forcing lies near oscillator p’s eigenfrequency and can produce a large response.
B Classification of nonlinear terms: The term g^p_m is resonant when ωp ≃ 2ωm and non-resonant when ωp ≠ 2ωm.The classification depends on whether the forcing frequency matches an internal eigenfrequency relationship.
B Classification of nonlinear terms: Trivially resonant monomials cannot be removed from the normal form, so the resulting normal form remains nonlinear even without other internal resonances.These monomials do not break invariance, consistent with dynamics represented in an invariant-based phase-space span.
B Classification of nonlinear terms: Non-resonant monomials are nonlinear terms not connected to internal resonance, whereas resonant monomials correspond to internal eigenfrequency relationships.
B Classification of nonlinear terms: Internal resonance creates strong coupling and energy exchange, leading to more complex dynamics and bifurcations in a larger phase space.Without internal resonance, the coupling is termed weak or non-resonant.

C Parametrisation of invariant manifold

The parametrisation method computes invariant manifolds and reduced dynamics order by order by solving homological equations for polynomial mappings and reduced vector fields. Graph and normal-form styles provide different choices for resolving these equations, while resonances constrain the reduction.

Order-by-order parametrisation: The unknown mapping W and reduced dynamics f are represented as homogeneous polynomial expansions and computed successively at each order.The mapping is n-dimensional and the reduced dynamics is expressed in the master coordinates.
Order-by-order parametrisation: The method derives order-k homological equations by substituting the manifold parametrisation into the invariance equation and matching terms of equal degree.The notation [.]_k selects kth-order terms, while W<k collects lower-order terms.
Normal and tangent equations: The normal co-homological equation is solved first for slave-coordinate corrections, followed by the tangent equation for master-coordinate corrections and reduced dynamics.The vector ξ is split into tangent and normal parts corresponding to master and slave coordinates.
Resonances: A cross resonance λ_i = mλ_L obstructs solving the normal equation, indicating strong master–slave coupling and motivating promotion of resonant slave modes to master coordinates.The obstruction arises when a slave eigenvalue matches a combination of master eigenvalues.
Resonances and parametrisation styles: An internal resonance λ_i = mλ_L among master eigenvalues obstructs linearisation of the reduced dynamics, and the resulting non-unique solutions yield graph and normal-form parametrisations.Graph style keeps higher-order master-coordinate corrections zero, whereas normal-form style retains resonant monomials and uses a full nonlinear mapping.

D Comparison of reduced dynamics

The graph-style and real normal-form formulations compute the same single-master invariant manifold but use different variables and reduced-dynamics representations. Under the stated modal assumptions, their equations are strictly equivalent through third order.

Equivalence: The comparison aims to establish equivalence between graph-style and real normal-form reduced dynamics for a single master mode.Both formulations theoretically compute the same invariant manifold, despite differing variables.
Formulation differences: The formulations differ through a quadratic term and a summation that excludes the master-mode term in one representation.The nonlinear relationship between modal and normal variables accounts for the differing forms.
Equivalence: Assuming harmonic lower-order master motion causes the quadratic terms to cancel when the modal-variable relationship is substituted.The assumptions include ˙R_m = iω_mR_m and ¨R_m = −ω_m^2R_m.
Equivalence: The two reduced-dynamics equations are strictly equivalent up to the third order.After cancellation, the cubic terms match the corresponding summation in the normal-form equation.

E Implicit static condensation

Implicit static condensation constructs a stress manifold by applying static forces along a selected master mode and measuring the resulting slave responses. The resulting condensed dynamics is equivalent to the full model with slave modal coordinates statically eliminated.

Stress-manifold construction: The method applies static forces aligned with a selected master eigenvector, computes finite-element displacements, and obtains modal coordinates for several force amplitudes.The forcing is fe = β_mMφ_m with β_m ∈ R.
Stress-manifold construction: Because the other oscillators receive zero direct forcing, the implicit function theorem yields a static nonlinear relationship between slave coordinates and the master coordinate.This relationship defines the slave coordinates as functions of the selected master coordinate.
Condensed dynamics: The reduced dynamics is obtained by substituting the slave-coordinate relationship into the master equation and fitting β_m(x_m) as a polynomial.Polynomial identification permits reduced-order dynamics at any chosen order.
Condensed dynamics: The condensed dynamics is equivalent to the full model with all slave modal coordinates statically condensed into the master dynamics.This equivalence is stated for the dynamics obtained from the static nonlinear relationship.
Approximation: The general explicit condensation solution is usually unavailable, so the slave functions are represented implicitly or approximated by polynomial expansions.The quadratic expansion is sufficient to derive third-order dynamics.
Stress-manifold geometry: Large invariant-breaking couplings create nonzero static slave responses, causing the manifold to depart from the linear eigensubspace and form the stress manifold.These couplings are identified in the nonlinear terms of the governing equations.