Field-Based Energy Formulation

• Field-based energy formulation is a mathematical framework that represents physical systems using energy densities, variational principles, and spatial-temporal fields.
• It underpins various disciplines such as continuum mechanics, electromagnetism, fracture, and fluid–structure interaction by enforcing conservation laws via Noether’s theorem.
• Structure-preserving discretization techniques are employed to maintain energy consistency and stability in the numerical simulation of complex, coupled multiphysics models.

A field-based energy formulation is a mathematical framework wherein the evolution, coupling, and conservation of physical systems are expressed in terms of spatially and temporally distributed fields and their associated energy functionals. This approach underpins a wide range of disciplines—including continuum mechanics, electromagnetism, fracture, thermodynamics, and coupled multiphysics systems—by capturing material behavior and interactions through energy densities, variational principles, and energy–momentum tensors. Its rigorous physical consistency is ensured by systematic adherence to variational theory, Noether’s theorem, conservative or dissipative Hamiltonian structure, and structure-preserving discretization.

1. Fundamental Principles and Variational Structure

The field-based energy formulation is grounded in the specification of energy functionals that depend on primary fields and their spatial derivatives. In continuum applications, the total energy $E$ is typically composed of kinetic energy $T$ and potential (stored) energy $\Psi$:

$$E({\rm fields}) = \int_\Omega \left[ T({\rm fields}, \partial_t {\rm fields}) + \Psi({\rm fields}, \nabla {\rm fields}, \ldots) \right]\,dV$$

Examples include displacement and damage variables in generalized continua (Abali, 2021), velocity and deformation gradient fields in fluid-structure interaction (Wang et al., 2020), phase and displacement fields in fracture (Paul et al., 2019, Fei et al., 2020, Lucarini et al., 2023), and electromagnetic potentials in electrodynamics (Westhoff, 12 Apr 2025, Crenshaw, 2017, Egger et al., 19 Jul 2025).

The governing equations are obtained by stationarity of the action under arbitrary admissible variations (Euler–Lagrange equations), yielding coupled PDEs for field evolution. Noether’s theorem and its extensions provide the formal link between invariance of the action and conservation laws, with energy-momentum tensors constructed as integrals over the fields and their canonical momenta (Abali, 2021, Canarutto, 2016, Öttinger, 2019).

2. Canonical Energy–Momentum Tensors and Conservation Laws

A central result of Noether-enhanced field-based energy formulation is the definition and local conservation of the energy–momentum tensor $T^{\mu\nu}$:

$$T^{\mu\nu} = \sum_{a} \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi^a)}\,\partial^\nu \phi^a - \eta^{\mu\nu} \mathcal{L}$$

$\partial_\mu T^{\mu\nu} = 0$

where $\mathcal{L}$ is the Lagrangian density and $\phi^a$ are the field variables. This tensor encodes energy density, flux, momentum density, and stress; its divergence-free property underpins local and global conservation laws (Öttinger, 2019, Canarutto, 2016). Extensions accommodate higher-gradient and dissipative systems, and coupling between subsystems emerges as energy–momentum exchange encoded in composite tensors (Abali, 2021, Lohmayer et al., 2024).

Covariant-differential formalism generalizes these constructions to gauge and curved geometries, in which momenta become vector-valued forms, and energy tensors are systematically identified as Noether currents of diffeomorphism invariance (Canarutto, 2016).

3. Specialized Formulations in Mechanics, Electrodynamics, and Fracture

3.1 Fluid–Structure Interaction

A one-field ALE approach (Wang et al., 2020) unifies velocity and deformation fields across fluid and solid domains, yielding a total energy functional containing fluid and solid kinetic energies plus solid potential energy. Conservative ALE mapping, exact quadrature, and appropriately chosen discrete variational forms guarantee unconditional energy stability—i.e., the discrete energy is non-increasing for any time-step, independent of mesh motion or coupling scheme.

3.2 Hyperelasticity (Mixed Field Formulations)

Mixed displacement–pressure formulations (Kadapa et al., 2020) leverage a split of the hyperelastic strain energy into volumetric and deviatoric parts, treating pressure as a fundamental field coupled via the energy functional. Linearization around current or prior states enables robust imposition of incompressibility and efficient Newton–Kantorovich solution of the weak form, with symmetric tangent matrices and reduced field variables compared to classical three-field formulations.

3.3 Fracture and Phase-Field Models

The field-based energy approach dominates modern computational fracture mechanics (Paul et al., 2019, Fei et al., 2020, Lucarini et al., 2023), utilizing phase-field variables to encode damage and crack growth. The total energy typically includes degraded bulk elastic energy, a regularized crack surface density, and (for multi-mode and fatigue phenomena) frictional dissipation, stored plastic work, and history-dependent driving forces. Coupled Euler–Lagrange equations for displacement and phase fields, discretized in suitable $C^1$ (or $C^0$) finite-element or FFT-based frameworks, are solved by monolithic or staggered schemes—ensuring energy consistency, irreversibility, and accurate resolution of diffusive versus localized crack evolution.

3.4 Electrodynamics and Multiphysics Coupling

Field-based energy formulations in electrodynamics utilize energy–momentum tensors expressed solely in terms of the primary fields (E, B), eschewing D and H as fundamental energy carriers. The universal energy exchange mechanism is $J_{\rm total}\cdot E$, and physically consistent energy balances, force densities, and dissipation emerge from this primitive (Westhoff, 12 Apr 2025). Lorentz covariance and conserved tensors underpin both vacuum and material media, and controversy over momentum density (Abraham–Minkowski) is resolved by expressing all energy and momentum in manifestly covariant, material-spacetime variables (Crenshaw, 2017). Port-Hamiltonian frameworks extend this structure to field–circuit coupled systems, with power-preserving interconnections and consistency across scales (Altmann et al., 17 Apr 2025).

4. Structure-Preserving Discretization and Computational Realizations

Energy-based modeling frameworks prioritize numerical discretizations that preserve the dissipation and conservation properties of the continuous models from PDE to algebraic equations (Rashid, 9 Dec 2025). Key principles include:

• Discrete gradients guaranteeing exact energy balance in time (Rashid, 9 Dec 2025).
• Choice of finite element spaces (e.g., $C^1$ LR-NURBS for shells and phase-fields; Nédélec edge elements for electromagnetic vector potentials) matched to variational and regularity requirements (Paul et al., 2019, Egger et al., 19 Jul 2025).
• Alternate minimization (block coordinate descent) and globally convergent Newton–Raphson for convex energy landscapes (Egger et al., 19 Jul 2025).
• FFT-based solvers employing staggered or monolithic updates for heterogeneous microstructures under cyclic loading (Lucarini et al., 2023).
• Port-Hamiltonian assembly of field and algebraic subsystems, ensuring passivity and correct power physical connections in field/circuit models (Altmann et al., 17 Apr 2025, Lohmayer et al., 2024).

Energy consistency, monotonicity (dissipation), and—where appropriate—entropy production are satisfied exactly at the discrete level. Rigorous a priori stability ensures robust computation for constrained, nonlinear, and multiphysics applications (Rashid, 9 Dec 2025).

5. Advanced Variational Extensions and Multiphysics Composition

The general field-based energy paradigm readily extends to complex systems involving higher-gradient elasticity, damage mechanics, internal variables, and modular coupling of subsystems. Extended Noether's formalism derives generalized energy–momentum tensors in strain gradient elasticity and damage (Abali, 2021). Compositional techniques—such as port-Hamiltonian networks and exergetic brackets—facilitate modular construction of multiphysics models (e.g., EMHD, MHD), preserving fundamental thermodynamic laws and ensuring non-negative entropy production (Lohmayer et al., 2024). Coupling is encoded via energy–momentum exchange or Dirac structure ports, maintaining invariant total energy and compliance with Onsager reciprocity.

6. Physical Consistency, Controversies, and Resolution

Field-based energy frameworks resolve several long-standing physical controversies and modeling ambiguities:

• In macroscopic electrodynamics, they clarify that D, H are mathematically auxiliary, and only E, B encode true field energy and momentum—ensuring correct energy storage and dissipation in stationary and moving media (Westhoff, 12 Apr 2025).
• By embedding continuum media into material-spacetime geometry and employing renormalized variables, they resolve Abraham–Minkowski momentum paradoxes, yielding unique, symmetric, and conserved energy–momentum tensors (Crenshaw, 2017).
• Complex multiphysics and constrained systems maintain physical consistency when field-based energy formulations and structure-preserving discretizations are employed, avoiding artificial energy creation or loss (Rashid, 9 Dec 2025).

7. Applications, Extensions, and Practical Implications

Field-based energy formulations are foundational in modern simulation codes for FSI, fracture, phase separation, electrodynamics in materials, and field/circuit interaction. They enable direct use of experimentally measured strengths and toughnesses, facilitate robust parameter calibration, and ensure physically interpretable, modular computational models. Ongoing developments extend the framework to high-index DAEs, nonlinear circuits, poroelasticity, and adaptive space-time computational physics.

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The modern field-based energy formulation, realized through variational, tensorial, and port-Hamiltonian formalisms, provides a unifying, physically consistent foundation for mathematical modeling, analysis, and simulation of coupled, nonlinear, and dissipative systems in engineering and the physical sciences (Wang et al., 2020, Abali, 2021, Westhoff, 12 Apr 2025, Lohmayer et al., 2024, Rashid, 9 Dec 2025, Paul et al., 2019, Egger et al., 19 Jul 2025).