Deep Energy Method for Large Deformation Analysis of Isotropic and Inhomogeneous Hyperelastic Ellipsoidal Pressurized Structures

Nasser Firouzi, Shaofan Li, Timon Rabczuk

ABSTRACT The accurate prediction of displacement and stress fields in pressure vessels is essential for the safe and reliable design of these structures, particularly when dealing with nonlinear behavior such as that of hyperelastic functionally graded materials (FGMs). Conventional finite element method (FEM) approaches, while robust, often require extensive meshing and may encounter challenges in graded or strongly nonlinear domains. In this work, we present a Deep Energy Method (DEM) framework, based on physics‐informed neural networks, to analyze the axisymmetric deformation of ellipsoidal vessels subjected to internal and external pressure loading. The formulation employs automatic differentiation and a hyperelastic constitutive model to evaluate the strain‐energy functional, with pressure boundary conditions incorporated through natural work terms. Both homogeneous and exponentially graded materials are studied, enabling direct assessment of how material gradation influences the structural response. Three neural network architectures with varying depth and width are examined to evaluate the effect of model complexity on accuracy and convergence. The DEM solution is obtained by minimizing the total potential energy, comprising the internal strain energy and pressure work, to train a neural network representation of the displacement field. Convergence analyses confirm stable optimization for all cases, with graded vessels showing fast and smooth loss reduction due to the stabilizing effect of material gradation. DEM predictions of displacement agree closely with FEM benchmarks in both expansion‐type (internal pressure) and contraction‐type (external pressure) scenarios, accurately capturing global deformation modes as well as local responses near the inner and outer boundaries. To further ensure numerical robustness, a Monte Carlo sampling strategy is also employed for a selected test case, yielding displacement and von Mises stress fields fully consistent with structured‐grid DEM and FEM solutions. von Mises stress fields derived from DEM match FEM references for both homogeneous and graded vessels. While stress concentrations appear near loaded boundaries in homogeneous materials, gradation leads to smoother stress transitions, and DEM successfully reproduces these patterns. Qualitative comparisons and magnitude‐wise agreement across all network architectures demonstrate the method's capability to reflect nonlinear constitutive behavior and spatial material variation. The findings highlight several advantages of DEM: it provides smooth mesh‐free solutions, naturally accommodates material gradation, and embeds the variational energy principle directly within the learning process. The method shows strong potential for extension to more complex geometries and nonlinear multiphysics problems, although sensitivity to optimization parameters and training cost remain important considerations. It also offers a powerful and reliable alternative to FEM for the nonlinear analysis of homogeneous and functionally graded pressure vessels. The method not only reproduces FEM accuracy but also provides additional flexibility and robustness across different network architectures and sampling schemes, positioning DEM as a promising tool for next‐generation computational mechanics.