PEM-UDE: Durability & Modeling Approach

• PEM-UDE method is a comprehensive computational strategy integrating coupled PDEs, finite element analysis, and machine learning for simulating durability and degradation in PEM systems.
• It couples electromechanical modeling with passive circuit optimization to predict vibration damping in smart structures and manage hydration-induced stresses in fuel cells.
• The approach combines physics-based simulations with hybrid PINNs to accelerate lifetime prognostics and deliver reliable diagnostics for PEM-based technologies.

The PEM-UDE method encompasses a set of computational and modeling strategies developed for the quantitative analysis, design, and ultimate durability estimation of proton exchange membrane (PEM) systems. Its applications span from the vibration damping of piezo-electromechanical (PEM) plates in smart structures to the prediction and management of mechanical degradation in PEM fuel cells and electrolyzers. The method integrates advanced numerical techniques, refined multiphysics models, and—more recently—physics-informed machine learning approaches, offering comprehensive tools for both performance optimization and lifetime prognostics.

1. Fundamental Concepts in PEM-UDE Modeling

At its core, the PEM-UDE method (Proton Exchange Membrane – Ultimate Durability Estimation or, in some contexts, Uncertain Degradation Estimation) addresses coupled physical phenomena in PEM-based devices. In smart structures, it enables the design of piezoelectromechanical plates where mechanical vibrations are converted and dissipated via tunable electric networks. In fuel cells and electrolyzers, it models hydration-induced mechanical stress and chemical membrane thinning, both of which lead to aging and failure. Across these domains, the PEM-UDE approach is distinguished by:

• Formulation and discretization of high-order coupled PDEs governing electromechanical and/or chemo-mechanical interactions.
• Calibration and synthesis of passive electric networks for damping or for lifetime extension.
• Modeling of cumulative damage in polymer membranes under cyclic environmental and operational stresses, utilizing both physically derived constitutive laws and modern hybrid learning frameworks.

2. Electromechanical Modeling and Numerical Strategies in PEM Plates

The canonical application of the PEM-UDE approach in smart structures is exemplified by the analysis and design of PEM plates equipped with distributed piezoelectric actuators networked via electric circuits. The vibrational and electromechanical behavior is governed by coupled partial differential equations:

$$\begin{align} □_4w + a\,\dot{w} + c\,\ddot{w} & = 0 \ □_2u + b\,\dot{u} + c_4w & = 0 \end{align}$$

where $w(\mathbf{r}, t)$ represents the mechanical displacement field and $u(\mathbf{r}, t)$ denotes the time-integral of voltage on the piezoelectric actuators. The spatial operators $□_4$ (biharmonic) and $□_2$ (Laplacian) are associated with the mechanical and electrical subsystems, respectively (1007.1581).

To resolve these high-order PDEs on arbitrarily shaped domains, a mixed non-conforming finite element method (FEM) is employed in space. Key computational aspects include:

• Derivation of a weak (variational) formulation coupling fourth-order mechanical plate theory and second-order membrane-like electrical behavior.
• Space discretization with a non-conforming triangular element that accommodates both subsystems on a unified mesh and satisfies the patch test, thus ensuring energy norm convergence.
• ODE system dimensionality reduction via projection onto a modal basis retaining dominant coupled modes, followed by time integration using a Runge–Kutta scheme.

This numerical strategy enables the design and simulation of complex PEM plates where analytical solutions are unattainable, while providing detailed insight into mode coupling, frequency matching, and damping efficiency.

3. Optimization of Passive Electrical Networks for Vibration Control

A defining element of the PEM-UDE method in smart structure applications is the systematic tuning of the interconnected passive electrical network to achieve optimal vibration damping. The design procedure entails:

• Synthesizing the passive electric network using lumped inductors ($L_N$), resistors ($R_N$), and inherent capacitance, with values selected such that the electrical system's dispersive properties and eigenfrequencies are commensurate with those of the target mechanical modes.
• Analytical tuning of $L_N$ to match the first electrical eigenfrequency to the controlled mechanical mode, feasible for canonical geometries such as the simply supported square plate.
• Optimization of $R_N$ to achieve the "critical" damping regime, wherein mechanically transduced energy converted to electrical form is dissipated prior to re-excitation of the mechanical subsystem. Sub-optimal resistance results in either persistent energy exchange (beating) or reduced coupling.

The evaluation of optimal parameters is embedded in the PEM-UDE numerical code, creating a feedback loop that maximizes overall damping effectiveness (1007.1581).

4. Predictive Durability Modeling Under Hydration Cycling

In the context of PEM fuel cells, the PEM-UDE methodology provides a mechanistic and quantitative framework for predicting membrane lifetime subjected to hydration/dehydration cycles. This framework comprises three primary components (1306.4575):

1. Fuel Cell Relative Humidity (RH) Distribution Model: Solves for the time- and position-dependent water content $\lambda(x, t)$ inside the membrane, governed by diffusion equations with boundary conditions tied to anode/cathode gas humidities.

   $$\frac{\partial \lambda}{\partial t} = \frac{\partial}{\partial x} \left[ D(\lambda)\frac{\partial \lambda}{\partial x} \right]$$

   with

   $$\lambda(x = 0, y, t) = \lambda_\mathrm{eq}(RH_c(y, t)), \quad \lambda(x = L_m, y, t) = \lambda_\mathrm{eq}(RH_a(y, t))$$

2. Hydration/Dehydration Induced Stress Model: Transforms water content fluctuations into in-plane stress via a linearized constitutive relation with viscoelastic correction using an extended Eyring model. For cyclic operation, the periodic stress regime is represented as:

   $$\sigma(x, t) = -E \left[\lambda(x, t) - \langle \lambda(x, t) \rangle \right]$$

3. Damage Accrual Model: Calculates the accumulation of irreversible plastic strain per cycle, triggered by nonlinear dependence on stress amplitude (modeled by a $\sinh$ law), and predicts failure once a critical threshold is reached. The number of cycles to failure is thus:

   $N_o = \frac{N_0}{\sinh(\sigma / \sigma_T)}$

The aggregate result is a robust, scenario-aware estimation of membrane lifetime as a function of RH cycling amplitude, spatial stress heterogeneity, and membrane mechanical properties.

5. Hybrid Physics-Informed Machine Learning Approaches

A recent extension of the PEM-UDE paradigm leverages Physics-Informed Neural Networks (PINNs) to model chemical degradation in PEM electrolyzers (2507.02887). The PINN framework integrates coupled ODEs representing both membrane thinning and cell voltage evolution directly into the neural network's loss function:

• Membrane Thinning Law:

$$\frac{d\, t_{\mathrm{mem}}}{dt} = -\frac{3.6 \cdot k_{5} \cdot C_{\mathrm{mem}} \cdot \mathrm{MM}_{\mathrm{F}} \cdot 3600}{0.82 \cdot \rho_{\mathrm{Naf}} \cdot 10^{6} \cdot c_{\mathrm{HO}} \cdot t_{\mathrm{mem}}}$$

• Cell Voltage Dynamics: An ODE coupling membrane thickness to the evolution of cell voltage, enforcing physically consistent prediction even with noisy or incomplete measurements.

This approach brings the following advantages:

• Parameter inference and system identification: PINNs can "learn" hidden physicochemical parameters (e.g., degradation rate constant $k_5$) from limited trajectory data while enforcing compliance with known physical laws.
• Superior generalizability and robustness: Models retain accuracy under sparse, noisy data and extrapolate reliably beyond the training window, outperforming purely black-box neural networks.
• Practical deployment: The integrated model underpins online calibration, diagnostics, and predictive maintenance in industrial PEM electrolyzer settings.

6. Practical Applications and Code Validation

The PEM-UDE method has been validated and benchmarked against test cases of increasing complexity:

• For PEM plates, comparison against semi-analytical modal solutions for simply supported square geometries demonstrates accuracy in capturing both mechanical and electrical modal shapes, eigenfrequencies, and energy exchange phenomena. In more complex geometries with clamped or grounded boundaries, the method elucidates the dependence of modal tuning and damping efficiency on network parameters and actuator placement (1007.1581).
• In fuel cell and electrolyzer systems, lifetimes predicted by the PEM-UDE models are consistent with experimental failure data. The flexible integration of new parameters (e.g., via PINNs) paves the way for scenario-specific prognostics over a broad range of operating and environmental conditions (1306.4575, 2507.02887).

7. Impact, Limitations, and Future Directions

The PEM-UDE method represents an overview of advanced modeling, optimization, and machine learning techniques for PEM-based system design and durability assessment. Its primary contributions are:

• Application of mixed non-conforming FEM and modal Runge–Kutta analysis to arbitrary PEM plate geometries for smart structural engineering.
• Mechanistically guided lifetime prediction under cyclic mechanical and chemical stress, with explicit connection to operating conditions and material properties.
• Integration of hybrid modeling frameworks, notably PINNs, for scalable, interpretable, and robust system diagnostics.

Principal limitations include dependence on reliable material property data and calibration of phenomenological parameters, as well as computational demands for fine discretizations or long-term prognostics. Future research directions include:

• Further incorporation of multi-scale and multi-physics effects (e.g., explicit Fenton-type reactions in membrane degradation).
• Expansion of data-driven components while safeguarding physical interpretability and cross-regime validity.
• Optimization of integrated network and material design for targeted performance metrics, extending applicability to new PEM technologies and operational frameworks (1007.1581, 1306.4575, 2507.02887).