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PDE-constrained optimization for virtual sensing in structural dynamics: Full-field displacement and force recovery from sparse sensors

This paper presents a GPU-accelerated PDE-constrained optimization framework that jointly recovers full-field displacement and force distributions from sparse sensor measurements with significantly higher accuracy than modal expansion methods, as demonstrated on increasingly complex structural examples ranging from a cantilever plate to a nuclear reactor pressure vessel.

Original authors: Minjae Kim, Jaehwan Jeong, Jaemin Kim

Published 2026-06-30

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Minjae Kim, Jaehwan Jeong, Jaemin Kim

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 have a giant, complex machine, like a nuclear reactor or a massive bridge. You want to know exactly how every single part of it is shaking and where the invisible forces are pushing or pulling on it. But you can't put sensors everywhere; you only have a few tiny microphones (accelerometers) stuck to the outside.

This paper presents a new "super-smart" way to figure out the whole story from just those few tiny clues. The authors call this Virtual Sensing.

Here is the simple breakdown of how they did it and why it's better than the old way.

The Old Way: Guessing with a "Short List" (Modal Expansion)

Think of the old method like trying to describe a complex song by only humming the first few notes.

How it works: Engineers usually assume the machine vibrates in a few simple, pre-defined patterns (like the main notes of a song). They listen to the few sensors, try to match those notes, and then guess the rest of the song based on that short list.
The Problem: If the machine does something complicated (like a high-pitched squeal near a clamp), the "short list" doesn't have those notes. The guess is wrong. Also, if they try to figure out where the force is coming from, they just do a math trick to reverse-engineer it, which often leads to "ghost forces" appearing in places where nothing is actually touching the machine.

The New Way: The "Physics Detective" (PDE-Constrained Optimization)

The authors propose a new method that acts like a strict detective who never breaks the laws of physics.

The Setup: Instead of guessing a short list of notes, they use a full, detailed digital twin of the machine (a computer model with thousands of tiny pieces).
The Rule: They tell the computer: "You must find the invisible forces that, when applied to this digital model, make the few sensors on the outside shake exactly like the real sensors do."
The Magic: The computer doesn't just guess. It solves a massive puzzle where the "laws of physics" (the equations that govern how steel bends and shakes) are the rules. It forces the solution to be physically possible at every single point, not just where the sensors are.

The Creative Analogy: The Dark Room and the Flashlight

Imagine a dark room (the machine) where you can't see anything, but you have a few tiny mirrors (sensors) reflecting a little bit of light.

The Old Method: You try to guess the shape of the room by only looking at the reflections in those few mirrors. You might get the general shape right, but you'll miss the corners and details. If you try to guess where the light bulb is, you might think it's in the middle of the room when it's actually in the corner.
The New Method: You have a perfect 3D blueprint of the room. You ask the computer: "If I put a light bulb here, would the mirrors reflect what I see? What if I put it there?" The computer runs millions of simulations in a split second, using the laws of light (physics) to ensure that the reflections must match your observation. It finds the exact spot of the light bulb and the exact shape of the room, even in the dark corners where you have no mirrors.

Why This Paper is a Big Deal

The authors tested this on three things: a flat metal plate, a bent pipe, and a massive nuclear reactor pressure vessel.

It's More Accurate: In every test, their new method was much better at predicting how the whole machine moves. For the flat plate, it was 17 times more accurate than the old method.
It Finds the "Ghost" Forces: The old method often invented fake forces in the middle of the machine. The new method correctly identified that the force was only on the specific edge or inner surface where it actually was.
It's Fast Enough to Use: Solving these massive puzzles usually takes forever. But the authors used a special trick (using a powerful graphics card, or GPU, like the ones in gaming computers) to speed it up by 64 times. This makes it possible to do this "virtual sensing" almost in real-time.

The Bottom Line

This paper shows that by treating the problem as a strict physics puzzle rather than a simple pattern-matching game, we can "see" the entire inside of a machine and find invisible forces just by listening to a few sensors on the outside. It's like having X-ray vision for structural health, but it's done entirely with math and computers.

Technical Summary: PDE-Constrained Optimization for Virtual Sensing in Structural Dynamics

Problem Statement
Virtual sensing—the recovery of full-field structural responses (displacement, strain, acceleration) and applied forces from sparse sensor measurements—is a critical capability for structural health monitoring (SHM), fatigue assessment, and damage detection. While the structural dynamics community has widely adopted modal expansion for this purpose, the method suffers from fundamental limitations. Modal expansion represents displacement as a linear combination of a truncated set of dominant eigenmodes ( $N_d$ ), where $N_d$ is bounded by the number of sensors ( $N_s$ ). This approach introduces modal truncation error, particularly in thin structures where high-frequency modes are significant. Furthermore, modal expansion is purely kinematic; it fits sensor data to mode shapes without enforcing the governing partial differential equation (PDE). Consequently, it cannot propagate information to unobserved degrees of freedom (DOFs) via physical laws, leading to inaccurate reconstructions away from sensor locations. Additionally, force recovery via modal expansion (back-calculation) is unstable, as the dynamic stiffness operator amplifies truncation errors and fails to localize forces to specific boundaries.

Methodology
This study formulates virtual sensing as a PDE-constrained optimization (PDE-CO) problem. Unlike modal expansion, which reduces the problem to a low-dimensional modal space, PDE-CO operates on the full-order finite element (FE) system.

Formulation:
- Governing Equation: The linear elastodynamic equation under harmonic excitation serves as an equality constraint: $D_{ff}(\omega)\hat{U}_f = \hat{F}_f$ , where $D_{ff} = K_{ff} - \omega^2 M_{ff}$ is the dynamic stiffness matrix, $\hat{U}_f$ is the free-DOF displacement, and $\hat{F}_f$ is the external force.
- Optimization Variable: The unknown force distribution is parameterized as $\hat{F}_f = P\theta$ , where $\theta$ is the vector of force amplitudes at candidate loaded DOFs and $P$ is a Boolean selection matrix.
- Objective Function: The goal is to minimize the misfit between predicted and observed sensor accelerations, regularized by a Tikhonov term:
  $\min_{\theta} J(\theta) = \| -\omega^2 S_f D_{ff}^{-1} P \theta - \hat{a}_{obs} \|^2 + \alpha \|\theta\|^2$
  Here, $S_f$ is the sensor selection matrix, and $\alpha$ is the regularization weight.
Solution Strategy:
- Reduced-Space Approach: The displacement $\hat{U}_f$ is eliminated by solving the PDE constraint for each trial $\theta$ , reducing the problem to an unconstrained optimization over $\theta$ .
- Gradient Computation: Instead of deriving adjoint equations manually, the framework utilizes reverse-mode automatic differentiation (AD) via the JAX library. This automatically computes the gradient $\nabla_\theta J$ by differentiating through the linear solver ( $D_{ff}^{-1}$ ), effectively performing the adjoint solve ( $D_{ff}^\top \lambda = w$ ) implicitly.
- Optimization Algorithm: The L-BFGS quasi-Newton method iteratively updates $\theta$ to minimize the objective. Each iteration requires one forward solve and one adjoint solve (both reusing the LU factorization of $D_{ff}$ ).
- Implementation Pipeline: The framework couples an offline stage (FEniCSx on CPU) for FE assembly and matrix generation with an online stage (JAX on GPU) for differentiable optimization and reconstruction.

Key Contributions

Novel Framework: This is the first application of PDE-CO to recover full-field displacement and force from sparse vibration sensors in structural dynamics, moving beyond the limitations of modal truncation.
Joint Recovery: The method simultaneously recovers the full-field displacement and the applied force distribution within a single optimization loop, ensuring the solution is physically consistent with the governing PDE at every iteration.
Differentiable Simulation: The integration of FEniCSx and JAX enables efficient gradient computation via AD, eliminating the need for manual adjoint derivation and allowing the framework to extend naturally to nonlinear problems.
GPU Acceleration: The implementation leverages GPU acceleration (NVIDIA RTX 5070 Ti) to handle the dense linear solves required by the full-order FE system, achieving significant speedups over CPU-based approaches.

Numerical Results
The framework was validated on three benchmark problems of increasing complexity: a cantilever plate, a 90° elbow pipe, and a reactor pressure vessel (RPV). In all cases, PDE-CO outperformed modal expansion in displacement recovery accuracy and provided superior force localization.

Case	Geometry/Load	Sensors	Modal Expansion Error ( $\epsilon_u$ )	PDE-CO Error ( $\epsilon_u$ )	Improvement
Cantilever Plate	Thin plate, tip load	4	7.69%	0.45%	17×
Elbow Pipe	Curved pipe, tip load	8	1.11%	0.28%	4.0×
RPV	Nuclear vessel, inner surface load	36	1.86%	0.885%	2.1×

Displacement Accuracy: PDE-CO consistently achieved sub-percent accuracy. The improvement was most dramatic for the thin plate, where modal truncation severely degraded the baseline method.
Force Recovery: Unlike modal expansion, which produced spurious, non-localized force estimates, PDE-CO correctly identified the loading regions (e.g., the free-end face for the plate/pipe and the inner surface for the RPV).
Computational Performance: While PDE-CO is computationally more intensive than modal expansion, GPU acceleration provided speedups of up to 64.8× for the RPV case (reducing per-iteration solve time from ~7.6s to ~0.15s), making large-scale optimization tractable.

Significance and Claims
The authors claim that PDE-CO offers a robust alternative to modal expansion by eliminating modal truncation error and enforcing physical consistency through the governing PDE. The method's ability to propagate sparse sensor information to the entire domain via the dynamic stiffness operator allows for accurate reconstruction even in regions far from sensors.

Crucially, the paper highlights that force identification is a distinctive advantage of PDE-CO. While modal expansion can only derive force estimates via unstable back-calculation, PDE-CO treats the force as the primary optimization variable, jointly optimizing it with the displacement. This capability is particularly significant for structural health monitoring scenarios involving inaccessible surfaces (e.g., the inner wall of a pressure vessel), where direct measurement is impossible.

The study concludes that while modal expansion remains a strong baseline for thick-walled structures with favorable sensor-to-mode ratios, PDE-CO provides superior accuracy and unique force localization capabilities, especially for thin structures or complex loading conditions. The framework's reliance on differentiable simulation and GPU acceleration positions it as a viable path toward real-time, physics-based digital twins for structural dynamics.