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Data-driven model order reduction for wave propagation in materials with damage and nonlinearities

This paper evaluates the performance of both intrusive projection-based (POD) and non-intrusive data-driven (DMD and OpInf) model order reduction methods for accelerating high-dimensional simulations of wave propagation in materials exhibiting nonlinearities and damage through three numerical examples.

Original authors: Saddam Hijazi, Nikiema Fulgence, Hannah Burmester, Natalie Rauter, Carmen Gräßle

Published 2026-03-24

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CC BY 4.0

Original authors: Saddam Hijazi, Nikiema Fulgence, Hannah Burmester, Natalie Rauter, Carmen Gräßle

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to predict how a ripple moves across a pond, but this pond is made of a complex, stretchy rubber sheet that has a few holes in it (damage) and behaves differently when you pull it hard (nonlinearity).

In the real world, engineers use massive supercomputers to simulate this. They break the rubber sheet into millions of tiny puzzle pieces (a method called Finite Element Method) to calculate exactly how every single piece moves. While accurate, this is like trying to count every grain of sand on a beach to predict the tide—it takes forever and uses up all your computer's memory.

The Goal of This Paper
The authors want to build a "shortcut" or a "cheat sheet" for these simulations. They want to create a Model Order Reduction (MOR). Think of this as taking a 4K movie of the ripple and compressing it into a high-quality GIF that still looks exactly the same but loads instantly.

They test three different ways to create this shortcut, ranging from "I know all the physics rules" to "I only have the video footage."

The Three "Shortcuts" Tested

1. The "Inside Job" (POD - Proper Orthogonal Decomposition)

The Analogy: Imagine you have the original blueprints of the building and the full engineering team. You ask the team, "What are the most important ways this building sways?" They show you a few key movements (like swaying left-right or twisting). You then build a tiny, simplified model that only moves in those specific ways.
How it works: This method requires full access to the computer code and the math equations. It looks at the data, finds the most common patterns, and builds a simplified version based on those patterns.
Result: It works very well, but it's like needing a master key to enter the building. If you don't have the source code (like in commercial software), you can't use this.

2. The "Black Box" Learner (DMD - Dynamic Mode Decomposition)

The Analogy: Imagine you don't have the blueprints, and you can't talk to the engineers. You only have a video camera recording the building swaying. You watch the video and say, "Okay, every time it moves left, it usually moves right next. I'll just guess the next move based on the last one."
How it works: This is purely data-driven. It looks at the snapshots of the simulation (the video frames) and tries to find a mathematical rule that connects one frame to the next. It doesn't care about the physics; it just cares about the pattern.
The Twist: The authors found that for complex, fast-moving waves, a standard "guess" wasn't good enough. They used a Multi-Resolution DMD (mrDMD).
- Analogy: Instead of watching the whole movie at once, mrDMD watches the movie in slow motion, then fast forward, then zooms in on specific scenes. It separates the slow, long-term swaying from the fast, chaotic vibrations. This worked much better for the tricky, nonlinear materials.

3. The "Detective" (Operator Inference - OpInf)

The Analogy: You are a detective trying to figure out the rules of a game by watching people play it. You see the players move, and you know some of the rules (like "there is a force pushing them"), but you don't know the exact formula for how they bounce. You try to reverse-engineer the formula that makes the players move the way they do.
How it works: This method sits in the middle. It knows the structure of the physics (e.g., "mass times acceleration equals force") but doesn't know the specific numbers. It uses the data to "infer" (guess) the missing numbers in the equation.
The Innovation: The authors created a new version of this detective work called Discrete OpInf with Re-projection.
- The Problem: Standard detective work sometimes gets the math slightly wrong, and the errors pile up over time (like a game of "Telephone" where the message gets garbled).
- The Fix: They used a technique called Re-projection. Imagine the detective makes a guess, runs the simulation, and then checks the result against the "real" video. If the guess is slightly off, they adjust the rules and try again. This "correction loop" made the shortcut incredibly accurate, almost as good as the slow, full simulation.

The Real-World Test: Finding Damage

The authors tested these shortcuts on three scenarios:

A simple wave: Like a ripple in a calm pond with a small rock (damage) in it.
A complex metal sheet: A fiber-metal laminate (like a high-tech airplane wing) with a crack.
A stretchy rubber sheet: A non-linear material that changes behavior when stretched, with a hidden flaw.

The Verdict:

Speed: The shortcuts were thousands of times faster than the original simulation.
Accuracy:
- For simple waves, all shortcuts worked well.
- For the complex, stretchy, damaged materials, the Multi-Resolution DMD and the Discrete OpInf with Re-projection were the winners. They could predict the waves and the damage effects with very high precision, even though they were running on a "cheat sheet."

Why Does This Matter?

In the real world, engineers need to check bridges, airplanes, and pipelines for damage right now. They can't wait hours for a supercomputer to simulate every possible crack.

This paper shows that we can build "smart, fast models" that learn from data. These models can run in real-time, allowing engineers to instantly detect where a crack is and how bad it is, potentially saving lives and saving millions in maintenance costs. It's like upgrading from a slow, manual map to a GPS that predicts traffic jams before they happen.

1. Problem Statement

The paper addresses the computational challenges associated with simulating wave propagation in materials exhibiting nonlinearities and damage. These simulations are critical for Structural Health Monitoring (SHM), where Guided Ultrasonic Waves (GUW) are used to detect and localize defects.

The Challenge: High-fidelity simulations using the Finite Element Method (FEM) result in high-dimensional systems of Ordinary Differential Equations (ODEs). Solving these repeatedly for parameter studies (e.g., varying damage location or severity) or inverse problems is computationally prohibitive due to time and memory constraints.
The Goal: To develop efficient Model Order Reduction (MOR) techniques that create low-dimensional surrogate models capable of accurately predicting wave dynamics and damage interactions while significantly reducing computational costs.
Specific Constraints: The study focuses on scenarios where the full-order model (FOM) code is a "black box" (e.g., commercial software like COMSOL), limiting access to system matrices. This necessitates non-intrusive methods that rely solely on solution data.

2. Methodology

The authors evaluate and compare three primary MOR approaches across three distinct numerical models:

A. Models Considered

Linear Wave Equation: A 2D scalar wave equation with parameterized damage (reduced wave speed).
Linear Mechanical System (GUW): A 2D fiber metal laminate (FML) model with damage (reduced Young's modulus), simulated via COMSOL.
Nonlinear Hyperelastic Model: An aluminum plate modeled with Neo-Hookean material behavior (large deformations), with and without damage.

B. Reduction Techniques

Intrusive (POD-Galerkin):
- Uses Proper Orthogonal Decomposition (POD) to generate a reduced basis from snapshot data.
- Projects the full-order equations onto this basis.
- Requirement: Full access to system matrices ( $M, K, B$ ). Used as a benchmark in the linear wave equation case.
Non-Intrusive: Dynamic Mode Decomposition (DMD):
- A purely data-driven method that approximates the system dynamics with a linear operator.
- Standard DMD: Assumes global linear dynamics.
- Multiresolution DMD (mrDMD): Decomposes the time domain hierarchically to capture localized, transient, and nonlinear features that standard DMD misses.
- Optimal DMD: Uses an optimization approach to compute initial mode amplitudes rather than simple projection.
Non-Intrusive: Operator Inference (OpInf):
- Infers reduced operators directly from data by solving a least-squares problem.
- Variants Tested:
  - Continuous Time OpInf: Requires time derivatives (approximated via finite differences). Includes "forces-informed" (constrained) and "unconstrained" formulations.
  - Discrete OpInf (dOpInf): Formulates the inference problem directly on the time-discretized dynamics (recurrence relations), avoiding the need for derivative estimation.
  - Re-projection Method: An extension where the FOM is solved iteratively using the reduced basis to generate "re-projected" trajectories. This eliminates non-Markovian closure errors, allowing OpInf to recover the intrusive reduced operators more accurately.
- Scaling: Data scaling is applied to handle numerical differences in magnitude between displacement and acceleration.

3. Key Contributions

Extension of OpInf to Second-Order Systems: The authors propose a discrete time OpInf formulation specifically for second-order mechanical systems ( $M\ddot{u} + Ku = f$ ), which avoids the numerical instability of estimating second derivatives from noisy data.
Re-projection for Second-Order Systems: They extend the re-projection method (previously used for first-order systems) to second-order mechanical systems. This allows the non-intrusive method to recover the accuracy of intrusive POD-Galerkin methods by generating trajectories that satisfy the reduced dynamics exactly.
Unconstrained OpInf Formulation: They propose an unconstrained formulation for forces-informed OpInf that preserves the structure (symmetry/positive definiteness) of mass and stiffness matrices while allowing the use of first-order optimizers.
Comprehensive Benchmarking: The study provides a rigorous comparison of intrusive vs. non-intrusive methods across linear, damaged, and highly nonlinear regimes.

4. Numerical Results

Case 1: Linear Wave Equation (Damage)

POD-ROM: Achieved good accuracy (error < 3% for $r=40$ ).
DMD: Standard DMD was less accurate than POD; Optimal DMD performed comparably to POD.
OpInf:
- Continuous OpInf: Struggled with accuracy due to derivative approximation errors.
- Discrete OpInf (dOpInf): Significantly outperformed continuous OpInf.
- dOpInf with Re-projection: Achieved the highest accuracy (error < 1% for $r \ge 60$ ), surpassing even the intrusive POD-ROM in some metrics. It successfully recovered the intrusive operators.

Case 2: Linear GUW in FML (Commercial Software/Black Box)

DMD vs. mrDMD: Standard DMD failed to capture the signal dynamics due to poor initial amplitude estimation. mrDMD was highly effective, isolating time-localized dynamics and achieving a sensor error of 0.137%.
OpInf: The unconstrained continuous OpInf provided qualitative accuracy but suffered from error accumulation over time. The choice of time integrator (Newmark parameters) significantly impacted OpInf-ROM accuracy.

Case 3: Nonlinear Hyperelastic Aluminum (with/without Damage)

Standard DMD: Failed completely (qualitatively and quantitatively) because it assumes linear dynamics and cannot capture the strong nonlinearities of the Neo-Hookean model.
mrDMD: Successfully captured the nonlinear transient behavior.
- Intact model error: 1.023%.
- Damaged model error: 4.388%.
Significance: This demonstrates that for nonlinear wave propagation, multiresolution analysis is essential, as global linear approximations (standard DMD) are insufficient.

5. Significance and Outlook

Practical Impact: The work demonstrates that non-intrusive, data-driven methods (specifically mrDMD and dOpInf with re-projection) can achieve accuracy comparable to intrusive methods even when the underlying physics code is inaccessible (e.g., commercial software).
SHM Application: These reduced models are computationally efficient enough to be used in inverse problems for damage identification (estimating location and severity), which requires thousands of forward simulations.
Future Work: The authors identify the need to extend these OpInf frameworks to parameterized systems (where damage parameters change the stiffness matrix). Since the reduced stiffness matrix is not parameter-separable in this black-box setting, future research will focus on hyper-reduction techniques or data-driven parameterization to decouple the reduced model from the high-dimensional full-order assembly.

Conclusion

The paper establishes that while intrusive POD remains a gold standard, discrete Operator Inference with re-projection and Multiresolution DMD are powerful, non-intrusive alternatives. They successfully handle the complexities of wave propagation in damaged and nonlinear materials, offering a viable pathway for real-time structural health monitoring where access to the full-order solver is restricted.