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Preserving Lagrangian structure in nonlinear model reduction with application to structural dynamics

Kevin Carlberg, Ray Tuminaro, Paul Boggs

arXiv:1401.8044v1cs.CEcs.NAmath.NA

TL;DR

Large-scale mechanical simulations can be prohibitively costly, while common sampled equation-of-motion reductions can destroy Lagrangian structure. This work approximates Lagrangian ingredients before deriving reduced equations; on a nonlinear parameterized truss, it produced stable, often accurate responses, unlike frequently unstable alternatives.

  • Problem

    Sample-based complexity-reduction techniques can destroy Lagrangian structure in mechanical systems, while simulating large-scale systems can take weeks.

    Source: §1 · p. 1, §5.2 · p. 14

  • Method

    The method approximates Lagrangian ingredients and derives reduced equations from them, using RBS and matrix gappy POD to approximate parameterized reduced matrices while preserving symmetry and positive definiteness.

    Source: §1 · p. 2, §8 · p. 31

  • Results

    In parameterized nonlinear truss experiments, the structure-preserving method generated stable, often very accurate responses, while other techniques were often unstable and RBS and matrix gappy POD performed nearly alike.

    Source: §8 · p. 31

  • Takeaways & Limitations

    The truss experiments indicate that preserving Lagrangian structure can yield stable responses without the excessive sample indices that often left competing methods with little performance gain.

    Source: §8 · p. 31

    Robustness in the presence of strong nonlinearity remains an open issue identified for future work.

    Source: §8 · p. 31

Abstract

from arXiv · show

This work proposes a model-reduction methodology that preserves Lagrangian structure (equivalently Hamiltonian structure) and achieves computational efficiency in the presence of high-order nonlinearities and arbitrary parameter dependence. As such, the resulting reduced-order model retains key properties such as energy conservation and symplectic time-evolution maps. We focus on parameterized simple mechanical systems subjected to Rayleigh damping and external forces, and consider an application to nonlinear structural dynamics. To preserve structure, the method first approximates the system's `Lagrangian ingredients'---the Riemannian metric, the potential-energy function, the dissipation function, and the external force---and subsequently derives reduced-order equations of motion by applying the (forced) Euler--Lagrange equation with these quantities. From the algebraic perspective, key contributions include two efficient techniques for approximating parameterized reduced matrices while preserving symmetry and positive definiteness: matrix gappy POD and reduced-basis sparsification (RBS). Results for a parameterized truss-structure problem demonstrate the importance of preserving Lagrangian structure and illustrate the proposed method's merits: it reduces computation time while maintaining high accuracy and stability, in contrast to existing nonlinear model-reduction techniques that do not preserve structure.

1. Introduction.

Large-scale mechanical simulations can be too costly for time-critical applications, while existing efficient nonlinear reduction methods may lose Lagrangian structure. This work seeks parameterized reduced models that retain that structure by approximating its ingredients and deriving the reduced equations from them.

1. Introduction.

  • High-fidelity mechanical simulations can take weeks, limiting their use in time-critical applications that need accurate results quickly.Parameterized model reduction uses expensive offline training to support inexpensive online evaluations.Source: §1 · p. 1
  • Preserving Lagrangian structure retains properties including energy conservation, conserved quantities associated with symmetries, and symplectic time-evolution maps.Energy conservation applies in the absence of non-conservative forces.Source: §1 · pp. 1–2
  • Galerkin projection of the Euler–Lagrange equation preserves structure, but assembling its reduced equations scales with the high-fidelity model dimension and is efficient only in limited operator settings.The cited efficient cases require low-order polynomial state dependence and affine parameter dependence.Source: §1 · p. 2
  • Collocation, DEIM, and gappy POD can reduce nonlinear equation-assembly costs, but their approximations destroy Lagrangian structure in mechanical systems.These methods sample equations or nonlinear functions before Galerkin projection.Source: §1 · p. 2
  • The proposed method targets parameterized mechanical systems with Rayleigh damping and external forces by approximating Lagrangian ingredients and deriving motion equations with the Euler–Lagrange equation.The authors also state that the approach preserves Hamiltonian structure when formulated using Hamiltonian mechanics.Source: §1 · pp. 2–3

2. Preserving matrix symmetry and positive definiteness.

This section develops two ways to approximate parameterized reduced matrices at low online cost while retaining symmetry and positive definiteness. Reduced-basis sparsification and matrix gappy POD use different approximation constructions and have distinct exactness conditions.

2. Preserving matrix symmetry and positive definiteness.

  • The online problem is to approximate V^T A(µ⋆)V with a symmetric positive-definite matrix at cost independent of N, assuming only that individual entries of A(µ⋆) are inexpensive to compute.The matrix may be dense and parameterized without affine-dependence assumptions.Source: §2 · p. 3

2.1. Reduced-basis sparsification (RBS).

  • RBS replaces the dense reduced basis with a sparse basis supported on m sampled rows, preserving positive definiteness and enabling online computation from m² sampled matrix entries.The stated cost to compute the sparse basis is O(m²n + mn²).Source: §2.1 · p. 3

2.1.1. Exactness conditions.

  • RBS is exact under full sampling, and in broader cases exactness depends on whether sampled and reduced generalized eigenvalues interlace.For parameter-independent A, the text also gives exactness when m ≥ n.Source: §2.1.1 · pp. 4–5

2.2. Matrix gappy POD.

  • Matrix gappy POD approximates the reduced matrix as a combination of precomputed symmetric matrix-basis elements, with online coefficients chosen to preserve positive definiteness.The method modifies gappy POD on matrix data to include a positive-definiteness mechanism.Source: §2.2 · p. 5

2.2.1. Offline computation:

  • Offline, matrix gappy POD applies vectorized POD to matrix snapshots to form principal matrices that optimally represent the training snapshots in average Frobenius-norm projection error.The resulting basis matrices are symmetric because they are linear combinations of symmetric matrices.Source: §2.2.1 · pp. 5–6

2.2.2. Online computation: coefficients.

  • Online, the coefficients fit the sampled matrix entries through a constrained least-squares problem, whose positive-definiteness constraint preserves the reduced-matrix structure.The text notes that the constraints are typically inactive at the unconstrained solution.Source: §2.2.2 · pp. 6–7

2.2.3. Exactness conditions.

  • Matrix gappy POD is exact when sampled matrix data lie in the basis span and the sampled basis matrix has full column rank; full sampling ensures the rank condition.Source: §2.2.3 · p. 7

3. Preserving potential-energy structure.

The section approximates the nonlinear potential through a sparse, parameter-dependent reduced basis, preserving it as a scalar-valued function while reducing online costs. Its approach matches the reduced gradient locally, with solvability conditions and equilibrium centering shaping the method’s scope and efficiency.

3. Preserving potential-energy structure.

  • A sparse, parameter-dependent basis replaces the dense reduced basis in the potential approximation, which remains a parameterized scalar-valued function.The basis is computed online to approximate the reduced potential gradient.Source: §3 · p. 8
  • The gradient approximation matches a first-order expansion near the online reference configuration, assuming trajectories do not diverge greatly from that point.The authors motivate this local focus by the localized trajectories often found in mechanical oscillators, including structural-dynamics systems.Source: §3 · p. 8
  • With equilibrium as the reference, the low-order Taylor terms vanish and the reduced-Hessian equation can be solved exactly using m = n samples.This follows because the equilibrium gradient is zero, allowing Theorem 2.1 to apply.Source: §3 · pp. 8–9

3.1. Solvability of the two-term Taylor equation.

  • The two-term Taylor equation is not generally solvable; when the offset lies in the reduced subspace and has nonzero components, solvability requires full sampling, m = N.More broadly, solvability is conditional on the norm inequalities in (3.9), and large sampling may be needed when the offset lies primarily in the reduced subspace.Source: §3.1 · p. 10

3.2. Implementation and cost.

  • Under H-independence, evaluating the potential gradient requires only m components and can yield an N-independent operation count.Online steps 1–3 may still depend on N, but they depend only on the parameter instance and can be amortized across reduced configurations.Source: §3.2 · pp. 10–11
  • When H-independence fails, equilibrium centering can still give an N-independent operation count by sparsifying the nonlinear function’s argument.In the dense-Hessian example, the method computes only m components of the argument, regardless of Hessian sparsity.Source: §3.2 · p. 11

4. Lagrangian dynamics formulation.

The formulation describes parameterized simple mechanical systems using a configuration space, a Riemannian metric, and a potential-energy function. Rayleigh damping and external forces enter through a dissipation function and the forced Euler–Lagrange equation.

4. Lagrangian dynamics formulation.

  • The systems are parameterized simple mechanical systems, with a particular focus on finite-element structural-dynamics models.Parameters may describe variations in shape and material properties.Source: §4 · p. 11
  • The Lagrangian is kinetic energy minus potential energy, with kinetic energy defined through the metric induced by the symmetric positive-definite mass matrix.The configuration manifold is taken to be Q = R^N.Source: §4 · p. 11
  • Rayleigh viscous damping is represented by a positive-semidefinite dissipation function, while the external force follows the Lagrange–D’Alembert principle.The damping matrix combines the mass matrix and the potential Hessian at the parameterized equilibrium.Source: §4 · p. 11
  • Applying the forced Euler–Lagrange equation yields the equations of motion, including mass, damping, potential-gradient, and external-force terms.Conservative systems without external forces or damping conserve energy and symmetry-associated quantities, and have symplectic time-evolution maps.Source: §4 · p. 12
  • The reduced model aims to retain the configuration space, metric, potential, dissipation, and variationally derived external force so structure-preserving integration can retain characteristic properties.The first three define the simple mechanical system; dissipation and external force characterize its non-conservative forces.Source: §4 · p. 12

5. Existing model-reduction techniques.

The section contrasts structure-preserving Galerkin projection with complexity-reduction approaches that lower online costs but can destroy Lagrangian structure. Dense reduced bases and nonlinear terms also leave Galerkin models computationally expensive under general parameter dependence.

5. Existing model-reduction techniques.

  • Conventional reduction uses an offline training set to construct a low-dimensional configuration manifold, which is then used for online analyses at specified parameters.The reduced Euclidean configuration space has dimension n much smaller than N and may use a basis from POD or modal decomposition.Source: §5 · p. 12

5.1. Galerkin projection.

  • Galerkin projection on the Euler–Lagrange equation preserves Lagrangian structure, whereas projection on the first-order state-space form does not.The reduced model retains the original structure by defining reduced Lagrangian ingredients and deriving the equations of motion in the usual way.Source: §5.1 · p. 13

5.1.1. Computational bottleneck.

  • Despite its low-dimensional equations, Galerkin projection remains costly online because forming reduced operators with a dense basis requires operations that scale with N.For the reduced mass matrix, the stated products cost O(Nω), O(Nωn), and O(Nn^2), respectively.Source: §5.1.1 · pp. 13–14
  • With nonlinear potential energy, the potential gradient and its reduced projection must be evaluated for each reduced configuration, while the projected force is needed at each time instance.Affine parameter dependence can permit offline assembly and O(n^2) online products, but it is described as limiting and not generally applicable.Source: §5.1.1 · p. 14

5.2. Complexity reduction.

  • Complexity-reduction methods sample only selected entries before projection, but this strategy destroys Lagrangian structure for mechanical systems.The section examines collocation and DEIM/gappy POD as examples.Source: §5.2 · p. 14

5.2.1. Collocation.

  • Collocation approximates the reduced equations by projecting sampled full-order equations, but its reduced mass matrix is nonsymmetric and therefore does not define a metric.The reduced stiffness term is also nonsymmetric, so it cannot be the Hessian of a potential.Source: §5.2.1 · p. 14

5.2.2. DEIM/gappy POD.

  • DEIM and gappy POD approximate each nonlinear vector-valued term separately through least-squares regression or interpolation, often achieving higher accuracy than collocation.These methods still lose Lagrangian properties II–IV because the reduced mass, stiffness, and damping matrices are not symmetric; online savings require H-independence.Source: §5.2.2 · pp. 14–15

6. Efficient, structure-preserving model reduction.

The method approximates the Lagrangian ingredients of a Galerkin-projection reduced model, then derives its equations of motion using the forced Euler–Lagrange equation. Its ingredient-specific approximations aim to preserve structure while enabling efficient reduced simulations.

6. Efficient, structure-preserving model reduction.

  • The approach approximates the reduced model’s Lagrangian ingredients and derives its equations of motion by applying the forced Euler–Lagrange equation to those approximations.The ingredients include the metric, potential, dissipation, and external force.Source: §6 · p. 15

6.1. Riemannian-metric approximation ˜gr.

  • For the metric, reduced-basis sparsification and matrix gappy POD approximate the reduced matrix while preserving its symmetry and positive definiteness.The matrix represents the reduced Riemannian metric.Source: §6.1 · p. 16

6.2. Potential-energy-function approximation ˜Vr.

  • The potential approximation uses the reduced-potential method because only its gradient enters the equations of motion, with equilibrium as the reference configuration.The stated choice avoids limitations associated with other reference configurations.Source: §6.1 · p. 17, §6.2 · p. 17

6.3. Dissipation-function approximation Fr.

  • The dissipation approximation retains Rayleigh-damping form by combining the approximated mass matrix with the potential Hessian at equilibrium.The combination uses the Rayleigh damping coefficients.Source: §6.3 · p. 17

6.4. External-force approximation ˜fr.

  • The reduced external force is derived through the Lagrange–D’Alembert principle from an approximated force in the original coordinates, restricted to reduced configuration-space variations.Gappy POD can approximate the original-coordinate force; its error depends on sampling and how closely the force lies in the basis range.Source: §6.4 · p. 17

6.4.1. Exactness conditions.

  • For m < N, exactness is sufficient when a nonzero force lies in the force-basis range and the sampled basis has full column rank.With full sampling, the rank condition holds automatically, leaving the range condition.Source: §6.4.1 · p. 17

6.4.2. Implementation.

  • Procedure 4 gives offline and online steps for the force approximation; the subsequent evaluation tests practical numerical impact on a clamped–free truss.The paper frames that experiment as testing whether the theoretical structural advantage translates into improved numerical results.Source: §6.4.2 · p. 17, §7 · p. 17

7. Numerical experiments.

The numerical experiments compare reduced-order models for conservative and non-conservative truss dynamics with fixed and varying parameters, including a highly nonlinear case, assessing response accuracy, stability, and computational speedup. In the highly nonlinear case, the structure-preserving methods remain similar to one another but incur larger errors, while alternatives are often unstable or offer little speedup.

7. Numerical experiments.

  • The parameter-varying study used six Latin-hypercube training points and random online points; all ROMs shared a POD basis and snapshots covered only the first half of the time interval.Performance was evaluated using normalized time 1-norm error in the end-face bottom-left node’s y-displacement and simulation-time speedup against the full-order model.Source: §7 · p. 19

7.1. Conservative case.

  • With zero damping and no external forces, a timestep of 0.008 seconds gave a convergence rate of 1.98 and a Richardson-extrapolated error of 5.16 × 10^-7; Newton iterations averaged 3.15 per timestep.The conservative system is stiff, and Lagrangian-structure-preserving reduced models retain symplectic time-evolution maps under the implicit midpoint rule.Source: §7.1 · p. 20, Figure 7.2 · p. 20

7.1.1. Fixed parameters.

  • At 0.4% sampling, the structure-preserving ROM achieved 7.3% relative error and 207.0 speedup; gappy POD and collocation were unstable except full-sampling collocation.Galerkin had 5.42% relative error but only 1.69 speedup; sampling above 5% did not improve the proposed method’s accuracy and reduced speedup.Source: §7.1.1 · p. 21, Figure 7.4 · p. 22

7.1.2. Varying parameters.

  • At 4.9% sampling, method 2 achieved speedups of 15.9, 28.5, and 26.2 and relative errors of 11.6%, 13.0%, and 11.6%; gappy POD remained unstable and collocation stabilized only at full sampling.At 20% sampling, method 2’s relative errors were 1.51%, 5.83%, and 1.09%, with speedups of 4.84, 9.82, and 7.72.Source: §7.1.2 · p. 22, Figure 7.7 · p. 24

7.2. Non-conservative

  • With damping ζ = sin(5°), timestep 0.1 seconds gave a Richardson-extrapolated error of 1.07 × 10^-4; Newton iterations averaged 2.56 per timestep, indicating significant nonlinearity.All parameters were free to vary in this non-conservative case.Source: §7.2 · pp. 22–23

7.2.1. Fixed parameters.

  • At 2% sampling, the structure-preserving method achieved 6.28% error and 36.5 speedup; gappy POD was unstable below 20% sampling, while collocation stabilized only at full sampling.At 20% sampling, gappy POD achieved 1.53% error but only 4.1 speedup; the authors attribute its stabilization relative to the conservative case to less stiff dynamics with damping.Source: §7.2.1 · p. 25, Figure 7.9 · p. 25

7.2.2. Varying parameters.

  • At three predictive points, method 1 with 0.4% sampling achieved relative errors of 11.0%, 2.82%, and 10.3% and speedups of 73.3, 96.3, and 82.3; gappy POD and collocation yielded no performance improvement.Gappy POD stabilized only at 40% sampling, with speedup below 1; collocation stabilized at 80% sampling and also failed to improve performance.Source: §7.2.2 · p. 26

7.3. Effect of nonlinearity.

  • The stronger-nonlinearity study increased nominal forces by a factor of 2.5; responses had larger magnitudes than in the prior study, implying stronger geometric nonlinearity, and timestep 0.025 seconds gave error 3.62 × 10^-4.The study reused the parameter-varying experiment’s training and online points; the authors expected stronger nonlinearity to degrade the potential-energy approximation as configurations deviate further from equilibrium.Source: §7.3 · p. 26
  • The two structure-preserving techniques yielded very similar results, but errors were significantly larger than in the less nonlinear Section 7.2.2 experiments.The results indicate that increasing nonlinearity deleteriously affects method performance.Source: §7.3 · p. 27
  • Collocation was always unstable below 60% sampling and gappy POD below 80%; their best speedup was only 2.77.The best speedup was obtained by collocation at 60% sampling for online point 2.Source: §7.3 · pp. 27, 29
  • At 0.5% sampling, method 1’s relative errors were 21.3%, 11.1%, and 15.9%, with speedups of 116.4, 160, and 98.9 at the online points.The figure reports reduced-order model performance as a function of sampling percentage.Source: §7.3 · p. 27, Figure 7.12 · p. 28

8. Conclusions.

The conclusion presents structure-preserving reduction as an efficient approach for nonlinear simple mechanical systems, supported by truss experiments. It also identifies robustness, scale, sampling, and error-estimation work that remains.

8. Conclusions.

  • The method approximates quantities defining Lagrangian structure before deriving equations of motion, using RBS and matrix gappy POD to preserve reduced-matrix symmetry and positive definiteness.RBS was also used to preserve potential-energy structure.Source: §8 · pp. 29, 31
  • On the geometrically nonlinear parameterized truss, structure preservation yielded always-stable, often-accurate responses; competing methods were often unstable, and stability usually required too many samples for significant gains.RBS and matrix gappy POD produced nearly the same performance across the experiments.Source: §8 · p. 31
  • Future work targets stronger-nonlinearity robustness, truly large-scale applications, technique-specific sample selection, and rigorous error bounds and estimates.A non-local approximation of the potential-energy function is offered as one possible route to improved robustness.Source: §8 · p. 31

Appendix A. Hamiltonian dynamics.

The appendix recasts conservative simple mechanical systems in Hamiltonian terms and derives a reduced Galerkin model consistent with that formalism.

Appendix A. Hamiltonian dynamics.

  • For conservative systems, the mass matrix defines the parameterized Riemannian metric, kinetic energy is quadratic in velocity, and the Lagrangian is kinetic minus potential energy.The conjugate momenta are derived from the Lagrangian, and the Hamiltonian is defined through its Legendre transformation.Source: Appendix A · p. 31
  • The Hamiltonian is the total energy, and Hamilton’s equations yield motion equations equivalent to the conservative simple-system Lagrangian equations.Source: Appendix A · p. 31

A.1. Galerkin reduced-order model.

  • The reduced Galerkin model defines a reduced configuration space and Lagrangian, then obtains equations equivalent to the Lagrangian Galerkin equations and consistent with Hamiltonian structure.The displayed reduced equations are given in Eq. (A.18).Source: §A.1 · p. 32

A.2. Proposed structure-preserving methods.

The proposed reduced Hamiltonian models substitute approximated ingredients into the reduced Lagrangian and retain Hamiltonian structure. The appendix also describes POD basis construction and the matrix gappy POD optimization constraints.

A.2. Proposed structure-preserving methods.

  • The proposed methods replace the reduced Lagrangian with one using an SPD approximated mass matrix and approximated potential; Hamilton’s equations yield inexpensive equations retaining Hamiltonian structure.Source: §A.2 · pp. 32–33

Appendix B. Proper orthogonal decomposition.

  • The POD basis is computed from normalized snapshots by taking the left singular vectors of the snapshot matrix’s thin singular value decomposition.Source: Appendix B · p. 33

Appendix C. Solving the matrix gappy POD optimization problem.

  • The matrix gappy POD optimization reformulates constraints as reduced-matrix eigenvalues bounded below by a positive threshold; its constraint gradient assumes distinct eigenvalues.The unconstrained solution is often feasible, and the reported experiments did not require the cited treatment for repeated or nearby eigenvalues.Source: Appendix C · pp. 33–34

D.1. Proof of Theorem 2.1.

The proof uses a generalized eigenvalue-interlacing result to establish a matrix projection relation and an exactness condition for the parameterized matrix term.

D.1. Proof of Theorem 2.1.

  • For SPD matrices of dimensions m and n with m ≥ n, the generalized interlacing theorem guarantees an orthogonal U such that U^T B_hat(s) U = B_hat(r).Source: §D.1 · p. 35
  • The converse interlacing argument constructs the lower-dimensional matrix through successive dimension reductions, using intermediate eigenvalues that interlace the endpoint spectra.Source: §D.1 · p. 35
  • The exactness condition for A(µ) holds if and only if the eigenvalues of the general matrix pencil interlace those of the comparison matrix.The theorem assumes SPD matrices A1 and A2 and scalar functions h1 and h2 over the parameter domain.Source: §D.1 · pp. 35–36

D.2. Proof of Theorem 2.2.

The proof establishes exactness of the reduced matrix at μ⋆ by combining the stated condition with the preceding equation identities. Orthogonality satisfies the first part of (3.8), leaving solvability to be established.

D.2. Proof of Theorem 2.2.

  • The identities yield ξ_A(μ⋆) = z(μ⋆), and Eq. (2.6) then gives ˜A(μ⋆) = V^T A(μ⋆)V.This is the proof’s exactness result under condition 1.Source: §D.2 · pp. 36–37
  • Any orthogonal g_UV satisfies the first part of (3.8), so the remaining issue is solvability.Source: §D.2 · p. 37

D.3. Proof of Lemma 1.

The proof characterizes solvability of (3.8) by comparing the norms of the projected vectors. The condition is equality when m = n and a one-sided inequality when m > n.

D.3. Proof of Lemma 1.

  • For m > n, the rectangular orthogonal-basis mapping cannot increase the norm of P^T ˜c, which motivates the necessary norm inequality.The proof distinguishes vectors within the basis span, whose norm is preserved, from vectors only partially within it.Source: §D.3 · p. 37
  • When m = n, g_UV is square and full rank, so P^T ˜c must lie in its range and the norm condition must hold with equality.Source: §D.3 · p. 37
  • For m = n, solvability of (3.8) requires and is guaranteed by ||P^T ˜c||2 = ||˜V^T ˜c||2.The construction chooses orthogonal Q1 and Q2 so both transformed vectors equal the same multiple of e1.Source: §D.3 · p. 37
  • For m > n, solvability of (3.8) is equivalent to ||P^T ˜c||2 ≥ ||˜V^T ˜c||2.The proof constructs an orthogonal Q2 to match the projected vector’s norm, including when the inequality is strict.Source: §D.3 · pp. 37–38
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