Energy-Based Modeling and Structure-Preserving Discretization of Physical Systems

This paper develops a comprehensive mathematical framework for energy-based modeling of physical systems, with particular emphasis on preserving fundamental structural properties throughout the modeling and discretization process. The approach provides systematic methods for handling challenging system classes including high-index differential-algebraic equations and nonlinear multiphysics problems. Theoretical foundations are established for regularizing constrained systems while maintaining physical consistency, analyzing stability properties, and constructing numerical discretizations that inherit the energy dissipation structure of the continuous models. The versatility and practical utility of the framework are demonstrated through applications across multiple domains including poroelastic media, nonlinear circuits, constrained mechanics, and phase-field models. The results ensure that essential physical properties such as energy balance and dissipation are maintained from the continuous formulation through to numerical implementation, providing robust foundations for computational physics and engineering applications.

💡 Research Summary

This paper presents a comprehensive and unified mathematical framework for the energy-based modeling and structure-preserving discretization of physical systems. At its core, the work addresses a critical challenge in computational physics and engineering: maintaining fundamental physical properties—specifically energy balance, dissipation, and interconnection structure—throughout the entire pipeline from continuous modeling to numerical implementation.

The foundation is a generalization of classical port-Hamiltonian (PH) system theory. The proposed framework introduces an extended state vector partitioned into energy variables (z1, z2) and algebraic variables (z3), described by a skew-symmetric interconnection matrix (J), a positive semi-definite dissipation matrix (R), and a Hamiltonian energy function (H). This formulation naturally encapsulates the energy dissipation inequality (dH/dt ≤ <y, u>) and is explicitly designed to handle challenging system classes, including high-index differential-algebraic equations (DAEs) and nonlinear multiphysics problems, more effectively than standard PH approaches.

The primary theoretical contributions are threefold. First, the paper introduces a structure-preserving regularization technique for high-index DAEs. By adding a small inertial term (ε dz3/dt) to the algebraic constraints, the index of the system is reduced to at most one, making it amenable to numerical simulation. Crucially, this regularization preserves a modified dissipation inequality, ensuring physical consistency is not sacrificed for numerical convenience, and the regularized system converges to the original one as ε → 0.

Second, the paper provides rigorous exponential stability analysis for the regularized systems under assumptions of strong convexity for the Hamiltonian and positive definiteness for the dissipation matrix. This offers a solid theoretical guarantee for the long-term behavior of the models.

Third, it details nonlinear structure-preserving discretization methods. For quadratic Hamiltonians, the implicit midpoint rule is shown to inherit the discrete dissipation inequality. For general nonlinear Hamiltonians, the discrete gradient method is employed, providing a powerful tool that exactly preserves the energy dissipation structure at the discrete level by design. The framework also proves to be closed under power-preserving interconnections, enabling modular modeling of complex systems.

The practical utility and versatility of the framework are demonstrated through detailed applications across four diverse domains: poroelastic media (coupling solid deformation and fluid flow), nonlinear electrical circuits (with components like diodes), constrained mechanical systems, and phase-field models. These examples validate that the abstract mathematical framework can be successfully applied to real-world, multi-physics problems.

In conclusion, this work bridges the gap between elegant mathematical theory and robust numerical practice. It provides systematic methods for transforming physical principles into computational models that are inherently stable and physically faithful, thereby establishing a robust foundation for high-fidelity simulation in science and engineering. The paper successfully argues that preserving geometric and energetic structure is not merely a theoretical nicety but a practical necessity for reliable long-term simulation.