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A Quadratic Manifold for Model Order Reduction of Nonlinear Structural Dynamics
Shobhit Jain, Paolo Tiso, Daniel J. Rixen, Johannes B. Rutzmoser
TL;DR
The paper addresses reduced-order modeling of geometrically nonlinear structural dynamics without relying on full nonlinear training simulations. It constructs a quadratic manifold from vibration modes and modal derivatives, then applies Galerkin projection on its configuration-dependent tangent space. In the tested flat-structure case, the quadratic manifold achieves accuracy comparable to linear manifolds with substantially fewer unknowns, while the authors limit its generality to systems where modal-derivative amplitudes can be quadratically enslaved.
Problem
Full-simulation-based reduction can be too costly, while vibration-mode bases miss nonlinear coupling and independent modal-derivative enrichment grows rapidly with the number of modes.
Method
The paper builds a quadratic displacement manifold from dominant vibration modes and modal derivatives or static modal derivatives, then projects the equations onto its configuration-dependent tangent space.
Results
In the flat-structure test, five-mode quadratic manifolds achieved the same accuracy as linear manifolds with one-quarter as many unknowns using static modal derivatives and six times fewer using modal derivatives.
Takeaways & Limitations
Quadratically enslaving modal-derivative amplitudes avoids the undesirable growth in unknowns with negligible tested accuracy loss.
Takeaways & Limitations
The quadratic enslavement is not expected to hold generally for arbitrary structural systems and is reported as especially suitable for slow dynamics dominated by a few separated modes.
Abstract
from arXiv · showhide
This paper describes the use of a quadratic manifold for the model order reduction of structural dynamics problems featuring geometric nonlinearities. The manifold is tangent to a subspace spanned by the most relevant vibration modes, and its curvature is provided by modal derivatives obtained by sensitivity analysis of the eigenvalue problem, or its static approximation, along the vibration modes. The construction of the quadratic manifold requires minimal computational effort once the vibration modes are known. The reduced order model is then obtained by Galerkin projection, where the configuration-dependent tangent space of the manifold is used to project the discretized equations of motion.
1 Introduction
Large nonlinear finite-element models are costly to simulate, while conventional reduced bases may miss geometric-nonlinearity coupling. The paper therefore develops a quadratic-manifold ROM using vibration modes and modal derivatives, with configuration-dependent tangent-space projection.
- Large finite-element models enable detailed structural representations but make routine exploration of loads, geometries, and materials prohibitively costly.
- Proper Orthogonal Decomposition can produce accurate Galerkin ROMs, but requires full high-fidelity solution snapshots and may be unsuitable for preliminary design.
- A basis containing only a few vibration modes performs poorly for geometric nonlinearities because it misses bending/torsion–stretching coupling.
- Modal derivatives enrich vibration-mode bases but grow quadratically with the number of vibration modes, reducing efficiency.
- The proposed quadratic manifold uses vibration modes and modal derivatives while projecting onto a configuration-dependent tangent space.
- The paper focuses on the reduction subspace and its curved-manifold generalization rather than hyper-reduction of nonlinear-term evaluation.
2 Model Order Reduction
The structural dynamics problem is reduced by approximating high-dimensional displacements with a low-dimensional nonlinear mapping and projecting the weak form onto admissible variations. Linear Galerkin projection is recovered when the mapping is a basis matrix.
- Finite-element discretization yields a high-dimensional second-order nonlinear dynamical system with mass, damping, internal-force, and external-load terms.
- Model reduction seeks a mapping from m reduced unknowns to the n-dimensional displacement space, with m ≪ n.
- Substituting the reduced mapping into the weak form restricts admissible variations to the reduced approximation.
- For a linear mapping Γ(q)=Vq, the formulation becomes Bubnov–Galerkin projection with reduced mass, damping, and stiffness matrices.
- The choice of basis or mapping controls reduced-solution accuracy, while the number of reduced unknowns controls computational speed-up.
3 Linear Manifold
Linear modal reduction is effective for small-displacement dynamics but becomes inadequate for geometric nonlinearities because truncated vibration modes omit important coupling. Modal derivatives add curvature information, although their number and computation can limit efficiency.
- POD bases require full nonlinear training runs and are typically suited only to the loading sets represented by their snapshots.
- A few low-frequency vibration modes can accurately approximate slowly varying linear responses and reduce the number of unknowns.
- As displacement grows beyond the linearization regime, vibration modes from the linearized model no longer adequately approximate the nonlinear response.
- Modal derivatives represent the sensitivity of one vibration mode to displacement in another modal direction and capture essential second-order nonlinearities.
- Static modal derivatives require one equilibrium-stiffness factorization, whereas dynamic modal derivatives require solving a high-dimensional system for each mode.
- Using all modal derivatives increases the basis size as O(m^2), motivating heuristic selection methods based on the applied loading.
- Modal-derivative ranking weights are not quantitative contribution measures and depend on vibration-mode normalization.
4 Quadratic Manifold
The quadratic manifold embeds modal-derivative curvature into a nonlinear displacement mapping whose tangent space changes with the modal coordinates. Galerkin projection on this configuration-dependent tangent space yields a reduced model with only the vibration-mode amplitudes as unknowns.
- 4 Quadratic Manifold: The quadratic-manifold approach treats modal derivatives as second-order components of an analytic manifold around the linear modal subspace.
- 4 Quadratic Manifold: The mapping uses q ∈ R^m, Φ ∈ R^(n×m), and a third-order tensor Ω to represent the quadratic displacement correction.
- 4 Quadratic Manifold: The manifold is tangent at equilibrium to the subspace spanned by the selected vibration modes.
- 4 Quadratic Manifold: The configuration-dependent tangent space incorporates modal-derivative corrections as the system departs from equilibrium.
- 4 Quadratic Manifold: Using symmetric or nonsymmetric modal derivatives changes the quadratic tensor construction, but only its symmetric part affects the mapping.
- 4 Quadratic Manifold: Velocity and acceleration mappings introduce a convective term quadratic in generalized velocities into the reduced equations.
- 4 Quadratic Manifold: The resulting reduced-order model is formulated in m unknowns and integrated with an implicit Newmark scheme and Newton–Raphson iterations.
- 4 Quadratic Manifold: Compared with a linear manifold containing independent modal-derivative amplitudes, the quadratic manifold preserves similar tested accuracy with fewer unknowns, but is never better in the Galerkin sense.
5 Applications and Results
The quadratic manifold reproduces nonlinear structural responses accurately while using substantially fewer reduced unknowns than linear-manifold formulations. Tests on a flat plate and a stiffened wing show that modal-derivative enrichment captures geometric nonlinearities, while POD with the same basis size can perform worse.
- Test setup: The study compares linear and quadratic manifolds, modal-derivative selection, and POD using nonlinear shell-structure simulations against full nonlinear solutions.Both models use triangular shell elements with six degrees of freedom per node, and accuracy is assessed with a global relative error measure.
- Model-I: Flat plate: The flat-plate load is chosen strongly enough that linear and nonlinear internal forces have the same order of magnitude, placing the response in the nonlinear regime.The plate is excited by a quasi-periodic pressure load with frequency selected as the first linearized-system eigenfrequency.
- Model-I: Flat plate: Using two bending VMs, the linear manifold requires modal-derivative enrichment because the VM-only basis misses bending–stretching coupling from geometric nonlinearities.The selected first and fifth modes reproduce the linear response, whereas the enriched reduced basis gives accurate nonlinear results.
- Model-I: Flat plate: The quadratic manifold uses two reduced unknowns instead of five for the all-MD linear manifold while reaching similar response accuracy.In this example, the quadratic manifold is constructed from the first and fifth VMs; the selected static and dynamic modal derivatives are identical.
- Model-II: Wing: For the wing model, five-mode quadratic manifolds achieve the same accuracy as linear manifolds with 20 unknowns for static modal derivatives and six times fewer unknowns for dynamic modal derivatives.Quadratic manifolds built with either static or dynamic modal derivatives show very similar accuracy.
- Comparison with POD: A five-vector POD basis performs worse than the quadratic manifold in the wing case, while adding more POD modes improves its performance.For the flat plate, five POD modes also do not provide comparable accuracy to the five-vector linear modal-derivative basis.
6 Conclusions
The quadratic manifold reduces thin-walled nonlinear structural dynamics by mapping vibration-mode amplitudes to full displacements with modal-derivative curvature, avoiding the unfavorable growth of explicit modal-derivative unknowns. Its accuracy is promising for suitable systems, but its scope and computational benefits remain subject to important limitations.
- Conclusions: The proposed quadratic mapping uses modal derivatives while avoiding the undesirable growth in unknowns associated with including those derivatives as independent basis vectors.The approach targets projection-based reduction for thin-walled structures with Von Kármán kinematics and does not require a full nonlinear solution run to construct the reduced model.
- Linear Manifold vs. Quadratic Manifold: Quadratic-manifold amplitudes enslave modal-derivative coordinates to vibration-mode amplitudes, whereas a linear manifold treats the corresponding derivative amplitudes as independent unknowns.The quadratic formulation is less general in this respect, although tested examples showed very similar solution accuracy between the two formulations.
- Speed: Reducing the quadratic-manifold system size does not directly translate into equivalent computational speed-up because nonlinear-term evaluation and projection remain bottlenecks.The authors identify hyper-reduction or tensor-based approaches as possible ways to address this issue, with further investigation ongoing.
- Limitations: The approach is presented as a local extension around equilibrium and is not specifically addressed to problems with local or global structural instabilities.The authors justify applicability to systems with mild nonlinearities and state that broader accuracy claims require more in-depth analytical research.
- Quadratic Manifold and Static Condensation: The quadratic-manifold reduced equations differ from statically condensed equations through additional inertial and damping contributions caused by nonlinear mapping and tangent-space projection.For the illustrative 2-DOF example, the reduction mappings are identical even though the resulting reduced-order models are not.