Constitutive parameterized deep energy method for solid mechanics problems with random material parameters

Abstract

In practical structural design and solid mechanics simulations, material properties inherently exhibit random variations within bounded intervals. However, evaluating mechanical responses under continuous material uncertainty remains a persistent challenge. Traditional numerical approaches, such as the Finite Element Method (FEM), incur prohibitive computational costs as they require repeated mesh discretization and equation solving for every parametric realization. Similarly, data-driven surrogate models depend heavily on massive, high-fidelity datasets, while standard physics-informed frameworks (e.g., the Deep Energy Method) strictly demand complete retraining from scratch whenever material parameters change. To bridge this critical gap, we propose the Constitutive Parameterized Deep Energy Method (CPDEM). In this purely physics-driven framework, the strain energy density functional is reformulated by encoding a latent representation of stochastic constitutive parameters. By embedding material parameters directly into the neural network alongside spatial coordinates, CPDEM transforms conventional spatial collocation points into parameter-aware material points. Trained in an unsupervised manner via expected energy minimization over the parameter domain, the pre-trained model continuously learns the solution manifold. Consequently, it enables zero-shot, real-time inference of displacement fields for unknown material parameters without requiring any dataset generation or model retraining. The proposed method is rigorously validated across diverse benchmarks, including linear elasticity, finite-strain hyperelasticity, and complex highly nonlinear contact mechanics. To the best of our knowledge, CPDEM represents the first purely physics-driven deep learning paradigm capable of simultaneously and efficiently handling continuous multi-parameter variations in solid mechanics.

Figures (11)

• Figure 1: CPDEM architecture. The two encoders $g_{\theta_p}$ and $g_{\theta_c}$ are added to generate better representations for the material parameters and the collocation points coordinate. We also customize the manifold network $g_{\theta_g}$.
• Figure 2: The 1D bar example.
• Figure 3: CPDEM results for the 1D linear elastic bar with parametric Young's modulus $E$. Three representative moduli $E\in\{0.50,1.50,3.00\}$ are shown (top to bottom). For each $E$, the analytical displacement (left), the CPDEM prediction (middle), and the pointwise absolute error $|u_{\mathrm{pred}}-u_{\mathrm{ref}}|$ (right) are plotted along the axial coordinate; the relative $L^2$ error is reported in the title of the error panel.
• Figure 4: CPDEM parametric performance for the 1D linear elastic bar across the parameter range of $E$. Left: exact and predicted displacement $u(x)$ for several moduli (increasing $E$ reduces the maximum displacement). Middle: pointwise absolute error $|u_{\mathrm{pred}}-u_{\mathrm{ref}}|$ on a logarithmic scale. Right: relative $L^2$ error (%) versus $E$ for many sampled moduli in the trained range.
• Figure 5: The 2D bending beam example.