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Multiphysics Engine Roadmap

Overview

A new Engines/Multiphysics/ simulation engine enabling four student-friendly PDE simulations on simple geometries, designed for visualization in Unity or web platforms. Built on existing finite difference infrastructure with new engineering materials, Unity/web export, and optional ML/Monte Carlo integration following the GIS engine pattern.

Target complexity: ~20–25 files, ~3.5–5K LOC (comparable to Game/Quantum engines).

LBM Assessment

Lattice Boltzmann Methods were evaluated and rejected. FD (Grid2D, GridOperators, 4 time steppers) already covers all four simulation types. LBM would require ~1–2K LOC of new infrastructure (D2Q9 distributions, BGK collision, streaming) duplicating existing FD capability. LBM strengths (parallelism, complex boundaries) are unnecessary for simple student geometries.

Simulation Types

Simulation PDE Method Dimension
Heat plate ∂T/∂t = α∇²T FD (Grid2D + Laplacian2D + ITimeStepper) 2D
Pipe flow Navier-Stokes (simplified) FD (1D transient Hagen-Poiseuille) 1D (2D stretch)
Electric field ∇²φ = −ρ/ε FD (Poisson solver + Gradient2D) 2D
Beam stress EIu⁗ = q 1D FEM (Euler-Bernoulli) 1D

Phase 1 — Foundations & Shared Infrastructure

  • Add iterative Poisson solver (Gauss-Seidel) to GridOperators for ∇²φ = f on Grid2D with Dirichlet BC — GridOperators.SolvePoisson2D()
  • Add SolidExtensions to Physics/ — Hooke's law, second moment of area, Euler-Bernoulli beam equation, analytical deflections
  • Add minimal 1D FEM primitives in Numerics/Numerics/Numerics/FiniteElement/
    • IElement1D interface — shape functions, local stiffness, local load
    • BarElement (2-node, linear) — axial stress/strain
    • BeamElement (2-node, Hermite cubic) — Euler-Bernoulli bending
    • Assembler1D — local→global stiffness matrix assembly with Gaussian elimination solver
    • Mesh1D — 1D mesh from interval subdivision (nodes + elements)
  • Add EngineeringMaterial to Physics/Materials/Engineering/
    • EngineeringMaterial immutable struct: ThermalConductivity, SpecificHeat, Density, DynamicViscosity, ElectricPermittivity, YoungsModulus, PoissonsRatio
    • EngineeringLibrary with common materials: steel, aluminum, copper, water, air, concrete, glass
    • Extend Materials factory with Materials.Engineering("Steel") — add EngineeringMaterial? property to MaterialDescriptor

Phase 2 — Core Engine & Individual Solvers

  • Create engine scaffold in Engines/Multiphysics/ with fluent API (pattern from GIS RiskScenario)
    • SimulationType static entry point — SimulationType.Create(MultiphysicsType)
    • SimulationBuilder — fluent builder: .WithMaterial() → .WithGeometry() → .WithBoundary() → .WithInitialCondition() → .AddSource() → .Solve() / .Run()
    • MultiphysicsType enum: HeatPlate, PipeFlow, ElectricField, BeamStress
    • BeamSupport enum: Cantilever, SimplySupported, FixedFixed
    • SimulationResult — unified result type with 2D fields, 1D arrays, beam-specific outputs, E-field vectors
    • IMultiphysicsSolver — internal solver interface
  • HeatPlateSolver — 2D heat equation ∂T/∂t = α∇²T + source (Grid2D + Laplacian2D + forward Euler)
    • Material-driven α = k/(ρ·cp) from EngineeringMaterial
    • BCs: fixed temperature (Dirichlet edges)
    • Initial conditions: uniform or function-based
  • PipeFlowSolver — 1D transient Hagen-Poiseuille via FD (cylindrical Laplacian)
    • Material viscosity ν from EngineeringMaterial.KinematicViscosity
    • Symmetry BC at r=0, no-slip at r=R
    • Optional stretch: 2D lid-driven cavity (projection method + Poisson solver)
  • ElectricFieldSolver — 2D Poisson/Laplace ∇²φ = −ρ/ε
    • Poisson solver from Phase 1, E = −∇φ via GridOperators.Gradient2D
    • Conductor as BC (φ = V on edge cells), charge distributions as source
    • Material permittivity from EngineeringMaterial
  • BeamStressSolver — 1D Euler-Bernoulli via analytical SolidExtensions formulas
    • Support: cantilever, simply supported, fixed-fixed
    • Loads: point loads, distributed loads
    • Output: deflection curve, bending moment, shear force, stress distribution
    • EI from EngineeringMaterial.YoungsModulus + cross-section (rectangular, circular, or custom I)

Phase 3 — Export & Visualization Support

  • MultiphysicsBinaryExporter — Unity-compatible binary format (MPHY magic + header + float layers)
    • 2D timeline export: per-step float[Nx*Ny] layers
    • 1D beam export: positions + 4 layers (deflection, moment, shear, stress)
    • Reader: ReadHeader() and Read() for round-trip loading
  • MultiphysicsJsonExporter — JSON format for web visualization
    • SimulationResult direct export (all 4 simulation types)
    • SimulationTimeline export with per-step snapshots
    • FieldSnapshot single-frame export
    • BeamSnapshot export with all curves
    • ExportMetadata support (simulation, unit, density)
    • Save() to file for all types
  • Snapshot system
    • FieldSnapshot — time-stamped 2D scalar field with flat indexing, Min/Max, ToArray round-trip
    • BeamSnapshot — immutable 1D arrays (deflection, moment, shear, stress) with FromResult() factory
    • SimulationTimeline — ordered collection with FromResult(), InterpolateAt() linear blending

Phase 4 — ML & Monte Carlo Integration

  • MultiphysicsMonteCarloModel : IMonteCarloModel — runs solver N times with sampled parameters (depends on Phase 2; pattern from GIS PlumeMonteCarloModel)
    • ParameterVariation — ranges for material properties, BCs, loads
    • Output: scenario matrix, percentile maps
  • SurrogateTrainer — trains regression model (Linear/SVR/MLP) on MC scenario data (depends on MC model)
    • Predict simulation output without re-running solver
  • MultiphysicsClusterAnalyzer — wraps ClusteringExperiment on scenario matrix (parallel with surrogate; pattern from GIS ScenarioClusterAnalyzer)
    • Identifies representative scenarios from MC ensemble

Phase 5 — Tests & Documentation

  • Unit tests with analytical validation
    • Heat plate: steady-state convergence vs known Fourier solutions
    • Pipe flow: velocity profile vs analytical Poiseuille
    • Electric field: capacitor/line charge vs analytical E-field
    • Beam stress: cantilever deflection vs PL³/3EI
    • FEM primitives: stiffness matrix assembly, known beam cases (bar axial, cantilever, simply supported)
    • MC integration: scenario matrix dimensions, clustering produces valid results
    • Export: binary format header parsing, dimension checks
  • Update Engines/Multiphysics/README.md — add Multiphysics engine section
  • Update Numerics/README.md — add Poisson solver section
  • Update Physics/README.md — add EngineeringMaterial section

Key Files

Existing — to modify

  • Numerics/Numerics/Numerics/FiniteDifference/GridOperators.cs — add Poisson solver
  • Numerics/Numerics/Physics/Materials/Materials.cs — extend factory with Engineering path

Existing — to reuse as patterns

  • Numerics/Numerics/Engines/GIS/Simulation/PlumeMonteCarloModel.cs — MC integration
  • Numerics/Numerics/Engines/GIS/Analysis/ScenarioClusterAnalyzer.cs — clustering
  • Numerics/Numerics/Engines/GIS/Export/UnityBinaryExporter.cs — binary export format
  • Numerics/Numerics/Engines/Common/ISimulationEngine.cs — engine interface
  • Numerics/Numerics/Physics/HeatExtensions.cs — HeatEquationRate()
  • Numerics/Numerics/Physics/FluidExtensions.cs — NS residuals, pipe flow
  • Numerics/Numerics/Physics/ElectroMagneticFieldExtensions.cs — E-field calculations

New — to create

  • Numerics/Numerics/Numerics/FiniteElement/ — IElement1D, BarElement, BeamElement, Assembler1D, Mesh1D (~6 files)
  • Numerics/Numerics/Physics/Materials/Engineering/ — EngineeringMaterial, EngineeringLibrary (~2 files)
  • Numerics/Numerics/Engines/Multiphysics/ — engine, 4 solvers, config, export, snapshots, MC, ML (~15 files)

Decisions

  • FD over LBM — existing infrastructure sufficient; LBM complexity unjustified for simple geometries
  • 1D FEM primitives in Numerics — minimal (Element, Assembly, Mesh), beam solver logic in engine
  • Export like GIS — UnityBinaryExporter + JSON for web
  • Engineering materials — new struct + library extending existing MaterialDescriptor
  • Pipe flow scope — 1D Hagen-Poiseuille primary, 2D lid-driven cavity as stretch goal
  • Beam analysis — static only initially; dynamic vibration deferred
  • RL environment — deferred to follow-up roadmap