Automated Computation of Autonomous Spectral Submanifolds for Nonlinear Modal Analysis

Sten Ponsioen; Tiemo Pedergnana; George Haller

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Automated Computation of Autonomous Spectral Submanifolds for Nonlinear Modal Analysis

Sten Ponsioen, Tiemo Pedergnana, George Haller

arXiv:1709.00886v3Sep 4, 2017math.DS

TL;DR

Nonlinear mechanical model reduction needs reliable invariant manifolds and efficient backbone-curve computation. The paper automates parameterization-based construction of two-dimensional SSMs with order selection by invariance error, achieving accurate backbone curves and substantial speed advantages over continuation.

Problem
Computing invariant manifolds tangent to modal subspaces in realistic nonlinear mechanical systems is challenging because prior invariance-equation approaches select among infinitely many solutions.
Method
The method constructs two-dimensional autonomous SSMs and their reduced dynamics with polynomial parameterizations, arbitrary approximation orders, and an a posteriori invariance-error estimate.
Results
The computed backbone curves agree closely with forced-response calculations, while the beam example produces curves in a fraction of the time required by COCO for one forcing amplitude.
Takeaways & Limitations
The automated SSM algorithm supports accurate reduced-order modeling and efficient backbone-curve computation for autonomous mechanical systems of arbitrary degrees of freedom.
Takeaways & Limitations
The current implementation is limited to two-dimensional SSMs in autonomous systems; higher-dimensional SSMs and forcing are left for future work.

Abstract

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We discuss an automated computational methodology for computing two-dimensional spectral submanifolds (SSMs) in autonomous nonlinear mechanical systems of arbitrary degrees of freedom. In our algorithm, SSMs, the smoothest nonlinear continuations of modal subspaces of the linearized system, are constructed up to arbitrary orders of accuracy, using the parameterization method. An advantage of this approach is that the construction of the SSMs does not break down when the SSM folds over its underlying spectral subspace. A further advantage is an automated a posteriori error estimation feature that enables a systematic increase in the orders of the SSM computation until the required accuracy is reached. We find that the present algorithm provides a major speed-up, relative to numerical continuation methods, in the computation of backbone curves, especially in higher-dimensional problems. We illustrate the accuracy and speed of the automated SSM algorithm on lower- and higher-dimensional mechanical systems.

1. Introduction

The paper situates spectral submanifolds within competing nonlinear normal-mode definitions and develops an automated parameterization-based method for computing two-dimensional SSMs and backbone curves.

1. Introduction: Rosenberg’s NNM is a synchronous periodic oscillation, whereas Shaw–Pierre’s NNM is an invariant manifold continuing a two-dimensional linear modal subspace.
1. Introduction: In non-conservative systems, periodic orbits are isolated while infinitely many invariant manifolds can remain tangent to each modal subspace.
1. Introduction: SSMs are the smoothest invariant manifolds tangent to modal subspaces, providing exactly invariant reduced-order models for nonlinear dynamics.They extend linear modal subspaces into nonlinear phase space.
1. Introduction: Backbone curves plot instantaneous vibration amplitude against instantaneous frequency and can be extracted from single-mode SSM dynamics under suitable non-resonance and damping conditions.
1. Introduction: The parameterization method computes unique SSM approximations to arbitrary orders and remains valid when an SSM folds over its underlying modal subspace.
1. Introduction: An a posteriori invariance error estimate lets users increase the approximation order until the computed SSM meets a required accuracy.

2. System set-up

The paper formulates autonomous nonlinear mechanical systems as first-order state-space dynamics, assuming positive-definite mass, symmetric damping and stiffness, and specified nonlinear smoothness.

2. System set-up: The mechanical model is M¨y + C ˙y + Ky + f(y, ˙y) = 0, with generalized positions y and nonlinear terms f.
2. System set-up: The mass matrix is symmetric positive definite, while damping and stiffness matrices are symmetric.
2. System set-up: The nonlinearities may be finitely differentiable, infinitely differentiable, or analytic through the allowed class C^r assumptions.
2. System set-up: Introducing x1 = y and x2 = ˙y transforms the second-order system into 2n first-order ordinary differential equations.
2. System set-up: The linearized first-order system has 2n eigenvalues, which are ordered according to their real parts.
2. System set-up: The analysis assumes a stable fixed point and a semisimple state matrix, enabling 2n linearly independent eigenvectors and associated real eigenspaces.

3. Spectral submanifolds for continuous mechanical systems

The paper defines SSMs as smoothest invariant manifolds tangent to selected modal subspaces and states their existence, uniqueness, parameterization, and reduced dynamics under spectral assumptions.

3. Spectral submanifolds for continuous mechanical systems: The selected modal subspace may represent a critically damped mode, two overdamped modes, or one underdamped mode.
3. Spectral submanifolds for continuous mechanical systems: An SSM is an invariant manifold tangent to a selected two-dimensional spectral subspace of the linearized system.
3. Spectral submanifolds for continuous mechanical systems: The outer spectral quotient compares transverse and in-manifold decay rates and determines the smoothness class in which the SSM is unique.
Theorem 1: Under the stated assumptions, a two-dimensional SSM exists, is tangent to the spectral subspace, and is unique among sufficiently smooth invariant manifolds tangent there.
Theorem 1: The SSM can be parameterized over an open subset of C2, with a polynomial reduced dynamics satisfying the invariance relationship.
Theorem 1: When inner resonances are absent, the reduced dynamics on the SSM can be linearized.

4. SSM computation

The computation approximates SSM embeddings and reduced dynamics with multivariate polynomials, solves order-by-order coefficient equations, and uses resonance structure to determine solvability and reduced-model terms.

4. SSM computation: The parameterization method expands the SSM, reduced dynamics, and nonlinearities as multivariate polynomial functions and solves the invariance equation at each order.
4.1. The Kronecker product: Kronecker products efficiently represent polynomial powers and their derivatives, while redundant tensor terms are constrained to encode the nonlinear vector field consistently.
4.2. The coefficient equations: At each order, known lower-order coefficients and nonlinear terms determine linear equations for the SSM and reduced-dynamics coefficients.
4.2.1. Partitioning the coefficient equations: Partitioning the coefficient equations separates modal and complementary blocks, whose eigenvalues generate the inner and outer non-resonance conditions.
4.2.1. Partitioning the coefficient equations: At an inner resonance, reduced-dynamics coefficients can cancel the resonant right-hand side and remove resonant terms from the SSM parameterization.
4.2.1. Partitioning the coefficient equations: An outer resonance leaves no corresponding freedom to modify the right-hand side and therefore causes the SSM construction to break down.

5. Reduced dynamics on the SSM

The reduced SSM dynamics are adapted to near-inner resonances through selective resonant-term removal and mixed parameterization, then expressed in amplitude-phase coordinates to construct backbone curves.

5.1. Near-inner-resonances: Near-inner resonances create small denominators that can reduce the Taylor series' convergence domain for the SSM parameterization.The resonance-closeness measure I quantifies proximity to resonances, and increasing δ allows weaker near-inner resonances to be included when invariance accuracy is unsatisfactory.
5.1. Near-inner-resonances: Resonant terms can be removed from the reduced dynamics by setting the corresponding cubic coefficients to -B_E, yielding a mixed parameterization style.The diagonal structure permits targeted removal of the resonant monomials rather than eliminating all terms at that order.
5.1. Near-inner-resonances: Higher-order near-inner resonances are handled by extending the resonance-closeness measure to selected integer pairs and incorporating nonzero coefficients at the corresponding orders.The relevant orders include pairs such as (2,1), (3,2), and (4,3), with coefficients associated with odd orders in the reduced dynamics.
5.2. Instantaneous amplitude and frequency: In amplitude-phase coordinates, the reduced dynamics assign each radius ρ an instantaneous frequency determined solely by ρ and a physically observable amplitude obtained from the SSM.The backbone curve is formed by mapping each fixed radius to its instantaneous amplitude-frequency point.
5.2. Instantaneous amplitude and frequency: The backbone curve consists of amplitude-frequency points computed from the parameterized SSM, while the continuous black comparison curve represents periodic orbits of a periodically forced system.

6. Invariance measure and order selection

The required SSM approximation order grows with the outer spectral quotient, motivating an invariance-error measure that guides automated order selection.

6. Invariance measure and order selection: A high outer spectral quotient can require a large Taylor-expansion order because transverse decay rates become much stronger than decay on the selected SSM.The unique SSM is approximated by expanding through order σout(E) + 1, and high-dimensional lightly damped systems can therefore demand high orders.
6. Invariance measure and order selection: The invariance error δinv compares full-system and reduced-system trajectories launched from a fixed-radius circle and integrated until they reach an inner radius.The measure averages the maximum Euclidean separation across trajectories and normalizes it by the distance from the origin to the initial circle.
6. Invariance measure and order selection: If δinv exceeds a prescribed bound, the SSM approximation order is increased until the computed manifold meets the required accuracy.

7. Applications

This section applies SSMtool to modified Shaw–Pierre systems and a discretized nonlinear Timoshenko beam, demonstrating its ability to compute reduced models that capture resonances, folds, forced responses, and high-dimensional beam dynamics. It also illustrates the method’s computational savings and the attraction of off-manifold trajectories to the computed SSM in a high-dimensional beam application.

7.1.1. Computing: The 15th-order approximations of W(E1) and W(E2) reduce invariance error enough to remain accurate through ρ0 = 0.35, corresponding to the reported physical displacement ranges.For W(E1), the maximum displacements are |x1| ≈0.66 m and |x2| ≈0.71 m; for W(E2), they are |x1| ≈0.73 m and |x2| ≈0.66 m.
7.1.2. Reduced Dynamics: Near-inner resonances introduce nonlinear terms into the reduced dynamics, whose O(15) backbone curves are compared with O(3) approximations for both in-phase and out-of-phase modes.The instantaneous frequencies depend only on ρ, and the O(15) curves are the red curves in figure 7a and figure 7b.
7.1.2. Reduced Dynamics: 3 minutes versus 8 minutes 6 seconds and 11 minutes 16 seconds: SSMtool’s O(15) backbone curves fit forced peak responses well while continuation computes single response curves.The SSM-derived curves are parameterized and can be evaluated at any required frequency, unlike the continuation timings reported for individual curves.
7.2. The modified Shaw–Pierre example: Outer resonances: Near outer resonances, the 15th-order parameterization captures a folded SSM that graph-based construction would miss, while trajectories initialized nearby converge toward W(E1).At k2 = 4 N m−1, construction breaks down because ΘC 3 becomes singular with a nonzero right-hand side; under the reported parameters, W(E1) remains an analytic SSM unique among all C4 invariant manifolds.
7.3. The discretized nonlinear Timoshenko beam: In the 32-dimensional beam model, the slowest eigenspace is about 50 times slower than the second slowest, making its two-dimensional SSM suitable for reduction.The slow eigenvalues are λ1,2 = −0.02286 ± 11.03i, so transverse trajectories decay rapidly while trajectories on the slow SSM remain active longer.
7.3. The discretized nonlinear Timoshenko beam: For the beam, the 4th-order W(E) is accurate through ρ0 = 1.5, while increasing to 10th order reduces invariance error by approximately three orders of magnitude.The tested radius corresponds to a maximum endpoint vertical displacement of 160 mm, and the 10th-order approximation is displayed in the 32-dimensional phase-space projections.
7.3. The discretized nonlinear Timoshenko beam: Near-inner-resonance conditions at orders O(|z|^i) for i = 3, 5, 7, 9 are satisfied within the spectral subspaces, enabling reduced dynamics on W(E).These reduced dynamics are obtained from SSMtool under the small negative real parts of λ1 and ¯λ1.
7.3. The discretized nonlinear Timoshenko beam: Under 4 minutes versus 4 hours 42 minutes 17 seconds, SSMtool computes the backbone curve far faster than COCO continuation for the 32-dimensional beam.The SSMtool curve is a tenth-order approximation computed to ρ = 1.3; COCO uses forcing amplitude A = 300 N.
7.3. The discretized nonlinear Timoshenko beam: Off-manifold trajectories converge toward the spectral submanifold W(E) because the slowest eigenspace is highly spectrally separated from the remaining eigenspaces.The reported trajectory comparison uses a full-system initial condition off the manifold and a reduced two-dimensional trajectory, with tend = 15 s.

8. Future work

The current SSMtool implementation computes two-dimensional spectral submanifolds in autonomous nonlinear mechanical systems of arbitrary finite dimension; future work targets higher-dimensional manifolds and forcing.

SSMtool currently computes two-dimensional spectral submanifolds for autonomous nonlinear mechanical systems with arbitrary degrees of freedom.
Future extensions will address higher-dimensional spectral submanifolds and possible forcing, supported by existing underlying theory.

9. Conclusions

The paper develops and tests an automated parameterization-method algorithm for constructing SSMs, reduced dynamics, and backbone curves at arbitrary orders. It handles folding and resonance-related structure, verifies backbone curves against continuation, and applies the method to a nonlinear Timoshenko beam with substantial computational savings.

The algorithm constructs SSMs, reduced dynamics, and backbone curves to any required precision for non-conservative systems of arbitrary finite dimension, subject to memory limits.The method is implemented in the MATLAB-based SSMtool package.
Parameterization-based embeddings remain valid when SSMs fold over their underlying spectral subspaces, unlike graph-based constructions.SSMtool detects near-outer and near-inner resonances; exact outer resonance causes construction breakdown, while near-inner resonances produce nonlinear reduced-dynamics terms.
For a two-degree-of-freedom non-conservative mechanical system, backbone curves were computed from SSMs up to 15th order and their accuracy was verified against periodically forced-system periodic orbits.
The method demonstrated SSM folding in a two-degree-of-freedom system by creating a near-outer resonance through parameter variation.
For a discretized nonlinear Timoshenko beam, the slow SSM produced a backbone curve closely agreeing with a COCO amplitude-frequency sweep and required a fraction of COCO's computation time.The spectral quotient indicates rapid decay transverse to the slow SSM, supporting its use for a two-dimensional reduced model.

Appendix A. Properties of the Kronecker product

Appendix A lists standard algebraic properties of the Kronecker product used in the paper's polynomial parameterization calculations.

The Kronecker product is associative for compatible matrices A, B, and C.
The Kronecker product is right-distributive.
The Kronecker product is left-distributive.
The product of two Kronecker products yields another Kronecker product for compatible matrices.

Appendix B. Equations of motion for the nonlinear Timoshenko beam

Appendix B derives the nonlinear Timoshenko-beam equations from Hamilton's principle, discretizes them with finite elements, and expresses the result as a nonlinear ODE system with mass, damping, stiffness, and nonlinear-force terms.

Hamilton's principle yields the beam's Euler–Lagrange equations and corresponding boundary conditions from a stationary action functional.
The beam model assumes plane stress or plane strain and neglects out-of-plane stress contributions because they enter at fourth order in the stiffest rotational degrees of freedom.External damping is modeled through velocity-proportional distributed forces and a force couple, producing damping proportional to the mass matrix.
The derivation uses kinematical and constitutive relations to express internal terms through the beam displacement field.
A finite-element discretization uses cubic, quadratic, and linear shape functions for u0, w, and φy, respectively, with a three-node beam element.The middle node is needed only to interpolate the transverse displacement, and matching orders are used to avoid shear and membrane locking.
The discretized beam dynamics take the form M¨y + C ˙y + Ky + f(y, ˙y) = 0, with y collecting the discretized degrees of freedom.M, C, and K are the mass, damping, and stiffness matrices, while f is the nonlinear force vector.
The nonlinear force components contain quadratic and cubic displacement terms together with velocity-dependent nonlinearities.The components are written using tensors D, G, H, and L under Einstein summation.

Appendix C. Multiple representations for the nonlinear coefficient matrices

The appendix explains how Kronecker-product representations create redundant monomial terms and how SSMtool resolves the resulting nonuniqueness in nonlinear coefficient matrices. It also enumerates unique monomials to clarify the representation.

A degree-i polynomial in 2n variables has S(2n, i) unique monomial terms, corresponding to multisets of cardinality i.
Unique monomial terms: For q ∈ C4 at degree two, the ten unique monomials are the multisets listed from {1,1} through {4,4}.
Multiple representations: The Kronecker product q ⊗q contains redundant cross terms, such as q2q1, so infinitely many coefficient matrices Gi can represent the same polynomial.
Multiple representations: SSMtool imposes independent constraints on redundant coefficients so each monomial maps to a unique matrix location and G_i q⊗i reproduces the nonlinear polynomial exactly.

Appendix D. Memory requirements for the coefficient matrices

The appendix quantifies memory growth in high-order SSM construction by estimating storage for dense coefficient matrices and illustrating the result on a two-degree-of-freedom system. Memory demand rises sharply with expansion order.

The most computationally demanding SSM-construction terms are the summations in equation (31) at order i.
The memory estimate assumes densely filled matrices with 8-byte doubles and depends on the system dimension n and expansion order i.
Example 4: Memory requirements for different orders: 2.0696 TB is required at order 17 versus 0.4846 TB at order 16 for the two-degree-of-freedom example.