Deterministic Energy-Conserving Idempotent Multi-Physics Finite Element Solvers: Metric Acceleration Projection for Real-Time and Embedded Continuum Mechanics

Continuous multi-physics simulation—spanning unilateral contact elastodynamics, incompressible fluid flow (CFD), transient thermo-mechanics, and non-linear plasticity—presents substantial computational challenges for iterative numerical solvers. Commercial finite element (FEM) and multi-physics suites (e.g., Abaqus, ANSYS, LS-DYNA, OpenFOAM) typically rely on non-linear Newton-Raphson tangent iterations, penalty spring regularizations, and iterative Algebraic Multigrid (AMG) or Krylov pressure-Poisson solvers. These classical methods can encounter ill-conditioning, penalty-induced energy drift, time-step cutbacks upon contact chattering, and substantial iteration-dependent latency variations, creating severe bottlenecks for hard real-time execution in 1 kHz robotic Model Predictive Control (MPC) and surgical haptics. In this paper, we present IdemSolver, an in-situ deterministic energy-conserving multi-physics solver grounded in Carl Friedrich Gauss's Principle of Least Constraint (1829) and the Udwadia–Kalaba constrained motion formulation. We formulate physical constraints as single-pass orthogonal metric projection operators in Riemannian acceleration spaces, satisfying exact algebraic idempotence (Π² = Π). IdemSolver formulates four closed-form metric operators: (1) a Gauss-Delassus contact projector ΠM computing exact contact reaction impulses without penalty springs for localized contact patches (m ≤ 100); (2) a Hamiltonian energy hypersurface projector ΠH operating on nullspace velocities to preserve exact total mechanical energy (|ΔE| ≤ 10−7 J); (3) a discrete Hodge-Helmholtz projector Πdiv enforcing mass conservation (∇ · v ≤ 10−14 s−1) in a single pre-factored step without pressure Poisson loops; and (4) an in-situ radial yield surface projector ΠYield for associative non-linear elasto-plasticity without iterative root-finding. We conduct a comparative algorithmic evaluation against industry-standard FEM baselines (Newton-Raphson, Penalty Contact, and AMG-SIMPLE) as well as canonical impact and contact benchmarks. In embedded domains, IdemSolver achieves a deterministic mean step latency of 29.50 µs, a worst-case jitter of 59.80 µs, strictly 0.0 Bytes dynamic heap memory allocation, and guaranteed bounded mechanical energy stability. In non-linear impact simulation: on the canonical Taylor Bar impact benchmark, IdemSolver eliminates penalty eigenvalue inflation, enabling a 1.97× larger stable CFL time step (245.51 ns vs. 124.48 ns), running 1.8× faster than the explicit penalty baseline, and reducing penetration error to 0.001 mm (vs. 0.94 mm); on Hertzian contact, it attains equilibrium in a single pass (0.22 ms), eliminating non-linear tangent refactorization. IdemSolver thus provides a viable deterministic framework for embedded microsecond haptics/MPC and non-linear structural simulation.