Solver Reference

Each solver card shows its numerical scheme, AD strategy, and Tesseract schema; per-(solver, problem) exclusions and explained anomalies come from the problem configs and are listed alongside. Click Inputs / Outputs on any solver to expand its field tables. The ∂ column marks fields that support automatic differentiation (VJP/JVP).

Legend. Each card shows a Numerics: line — the discretization followed by the numerical method — and three badge categories: language (backend / runtime), AD: strategy (autodiff / adjoint / hybrid / forward-only), and GPU on GPU-capable solvers.

Navier–Stokes (Grid)

Exponax

Numerics: Spectral, ETDRK

jax AD: autodiff GPU

Upstream docs: https://fkoehler.site/exponax/

Path: mosaic/tesseracts/navier-stokes-grid/exponax

Image: exponax_navier_stokes_grid

Exponential time-differencing Runge–Kutta (ETDRK) spectral integrator for 3D Navier–Stokes, enforcing incompressibility to machine precision by construction. Gradients via JAX source-transformation AD

Inputs / Outputs

Inputs

Field	Type	∂	Description
drag	Array[(1,), Float32]	✓	Linear drag coefficient
order	int		ETDRK time integration order (0-4)
kolmogorov_forcing	bool		Enable Kolmogorov sinusoidal body forcing for sustained turbulence
injection_mode	int		Wavenumber at which energy is injected (Kolmogorov forcing)
injection_scale	Array[(1,), Float32]	✓	Amplitude of the Kolmogorov body forcing

Outputs: —

INS.jl

Numerics: Finite Difference, Projection, RK4

julia AD: autodiff

Upstream docs: https://agdestein.github.io/IncompressibleNavierStokes.jl/dev/

Path: mosaic/tesseracts/navier-stokes-grid/ins-jl

Image: ins_navier_stokes_grid

Julia finite-difference solver with spectral pressure projection and RK4 time integration; CPU-only. Differentiates through the time loop via Zygote.jl reverse-mode AD.

Inputs / Outputs

Inputs: —

Outputs: —

jax-cfd

Numerics: Finite Difference, Projection, semi-implicit

jax AD: autodiff GPU

Upstream docs: https://github.com/google/jax-cfd

Path: mosaic/tesseracts/navier-stokes-grid/jax-cfd

Image: jax_cfd_navier_stokes_grid

JAX-native incompressible Navier–Stokes solver on a staggered MAC grid with finite-difference advection and spectral FFT pressure projection. Gradients via JAX source-transformation AD through the time loop.

Inputs / Outputs

Inputs

Field	Type	∂	Description
density	Array[(1,), Float32]	✓	Density of the fluid
inner_steps	int		Solver sub-iterations per timestep (higher = more accurate pressure solve)

Outputs: —

OpenFOAM

Numerics: Finite Volume, PISO/PIMPLE, Euler

cpp AD: forward-only

Upstream docs: https://www.openfoam.com/documentation/overview

Path: mosaic/tesseracts/navier-stokes-grid/openfoam

Image: openfoam_navier_stokes_grid

OpenFOAM icoFoam PISO solver run as a forward-only reference baseline (C++ subprocess). No reverse-mode AD; used to anchor cross-solver comparisons.

Inputs / Outputs

Inputs: —

Outputs: —

PhiFlow

Numerics: Finite Difference, Semi-Lagrangian, Euler

jax AD: hybrid GPU

Upstream docs: https://tum-pbs.github.io/PhiFlow/

Path: mosaic/tesseracts/navier-stokes-grid/phiflow

Image: phiflow_navier_stokes_grid

Semi-Lagrangian advection with CG pressure projection, unconditionally stable at large dt. Gradients via JAX autodiff combined with analytic adjoint rules through the same simulation code.

Inputs / Outputs

Inputs: —

Outputs: —

PICT

Numerics: Finite Volume, PISO, BDF1

pytorch AD: autodiff GPU

Upstream docs: https://github.com/tum-pbs/PICT

Path: mosaic/tesseracts/navier-stokes-grid/pict

Image: pict_navier_stokes_grid

GPU-accelerated differentiable PISO solver built on PyTorch with custom CUDA kernels (PISOtorch). VJP w.r.t. v0, inflow_profile, and viscosity flows through the PISO time loop via PISOtorch_diff (tracks VELOCITY, BOUNDARY_VELOCITY, and VISCOSITY gradients). dt is a Python scalar consumed at construction time; zero gradients are returned for it.

Inputs / Outputs

Inputs: —

Outputs: —

Warp-NS

Numerics: Finite Difference, IPCS, SSP-RK3

warp AD: autodiff GPU

Upstream docs: https://github.com/NVIDIA/warp

Path: mosaic/tesseracts/navier-stokes-grid/warp-ns

Image: warp_ns_navier_stokes_grid

IPCS projection scheme implemented in NVIDIA Warp CUDA kernels with FFT pressure Poisson on periodic domains. Gradients via wp.Tape kernel-level VJP.

Inputs / Outputs

Inputs

Field	Type	∂	Description
num_iters_poisson	int		Minimum number of Jacobi iterations per 2-D streamfunction Poisson solve. The solver auto-scales to max(this value, min(4*N², 8000)) at runtime, so convergence is maintained across grid sizes N=16..128 without manual tuning. For N≥128 with production accuracy, a multigrid or FFT Poisson solver is recommended as Jacobi is capped at 8000 iterations.
num_iters_poisson_3d	int		Number of Jacobi iterations per 3-D pressure Poisson solve. 800 iterations is adequate for N≤32; increase for larger grids.

Outputs: —

XLB

Numerics: Lattice Boltzmann, Streaming

jax AD: autodiff GPU

Upstream docs: https://github.com/Autodesk/XLB

Path: mosaic/tesseracts/navier-stokes-grid/xlb

Image: xlb_navier_stokes_grid

JAX-accelerated lattice Boltzmann solver (D2Q9 / D3Q19) with BGK or KBC collision, running the streaming/collision loop on GPU. Gradients via source-transformation AD; recovers incompressible Navier–Stokes only to O(Ma²) via Chapman–Enskog expansion.

Inputs / Outputs

Inputs: —

Outputs: —

Structural mechanics

deal.II

Numerics: Finite Element, Direct (spsolve)

cpp AD: forward-only

Upstream docs: https://www.dealii.org/

Path: mosaic/tesseracts/structural-mesh/dealii-structural

Image: dealii_structural_mesh

deal.II Q1 trilinear hexahedral FEM linear-elasticity solver via C++ subprocess (UMFPACK direct). Forward-only — reference implementation for cross-framework validation.

Inputs / Outputs

Inputs

Field	Type	∂	Description
E_max	float		Young’s modulus of the fully solid material [MPa].
nu	float		Poisson’s ratio.
xmin	float		Void stiffness ratio (E_min = xmin * E_max).
penal	float		SIMP penalisation exponent (E(ρ) = E_min + (E_max−E_min)·ρ^penal).

Outputs: —

FEniCS

Numerics: Finite Element, GMRES+AMG

fenics AD: adjoint

Upstream docs: https://fenicsproject.org/

Path: mosaic/tesseracts/structural-mesh/fenics-structural

Image: fenics_structural_mesh

FEniCS/DOLFIN CG1 FEM linear-elasticity solver with SIMP penalisation. Gradient ∂C/∂ρ via dolfin-adjoint tape replay (discrete adjoint derived at the variational level via UFL).

Inputs / Outputs

Inputs

Field	Type	∂	Description
E_max	float		Young’s modulus of the fully solid material.
nu	float		Poisson’s ratio.
xmin	float		Void stiffness ratio (E_min = xmin * E_max).
penal	float		SIMP penalisation exponent (E(rho) = E_min + (E_max-E_min)*rho^penal).

Outputs: —

Firedrake

Numerics: Finite Element, GMRES+AMG

firedrake AD: adjoint

Upstream docs: https://www.firedrakeproject.org/

Path: mosaic/tesseracts/structural-mesh/firedrake-structural

Image: firedrake_structural_mesh

Firedrake CG1 FEM linear-elasticity solver with SIMP penalisation. Gradient ∂C/∂ρ via firedrake-adjoint (pyadjoint ReducedFunctional) — an independent tape-based discrete adjoint complement to FEniCS.

Inputs / Outputs

Inputs

Field	Type	∂	Description
E_max	float		Young’s modulus of the fully solid material.
nu	float		Poisson’s ratio.
xmin	float		Void stiffness ratio (E_min = xmin * E_max).
penal	float		SIMP penalisation exponent (E(rho) = E_min + (E_max-E_min)*rho^penal).

Outputs: —

JAX-FEM

Numerics: Finite Element, Direct (UMFPACK)

jax AD: hybrid GPU

Upstream docs: https://github.com/deepmodeling/jax-fem

Path: mosaic/tesseracts/structural-mesh/jax-fem

Image: jax_fem_structural_mesh

JAX-FEM HEX8 linear-elasticity solver with SIMP stiffness E(ρ) = E_min + (E_max − E_min)·ρ^penal. Gradient ∂C/∂ρ via JAX automatic differentiation through the linear FE solve (hybrid implicit-function-theorem adjoint).

Inputs / Outputs

Inputs: —

Outputs: —

TopOpt.jl

Numerics: Finite Element, Direct (CHOLMOD)

julia AD: adjoint

Upstream docs: https://github.com/JuliaTopOpt/TopOpt.jl

Path: mosaic/tesseracts/structural-mesh/topopt-jl

Image: topopt_jl_structural_grid

TopOpt.jl HEX8 FEM linear-elasticity solver with SIMP penalisation, Julia + Ferrite.jl backend. Gradient ∂C/∂ρ via analytical SIMP adjoint sensitivity computed alongside the forward solve.

Inputs / Outputs

Inputs

Field	Type	∂	Description
E	float		Young’s modulus of the solid material.
nu	float		Poisson’s ratio.
xmin	float		Minimum density (void stiffness) to prevent singular stiffness matrix.

Outputs: —

Heat transfer

deal.II

Numerics: Finite Element, Direct (spsolve)

cpp AD: forward-only

Upstream docs: https://www.dealii.org/

Path: mosaic/tesseracts/thermal-mesh/dealii-heat

Image: dealii_heat_thermal_mesh

deal.II Q1 trilinear hexahedral FEM heat-conduction solver with SIMP conductivity (C++ subprocess, UMFPACK direct). Forward-only — reference implementation for cross-framework validation.

Inputs / Outputs

Inputs

Field	Type	∂	Description
k_max	float		Maximum thermal conductivity (fully solid material).
p_exp	float		SIMP penalisation exponent p (k(ρ) = k_min + (k_max−k_min)·ρ^p).

Outputs: —

FEniCS

Numerics: Finite Element, GMRES+AMG

fenics AD: adjoint

Upstream docs: https://fenicsproject.org/

Path: mosaic/tesseracts/thermal-mesh/fenics-heat

Image: fenics_heat_thermal_mesh

FEniCS P1 FEM heat-conduction solver with SIMP penalisation. Gradients (∂C/∂ρ, ∂C/∂source) via dolfin-adjoint tape replay.

Inputs / Outputs

Inputs

Field	Type	∂	Description
k_max	float		Maximum thermal conductivity (fully solid/conducting material).
p_exp	float		SIMP penalisation exponent p (k(ρ) = k_min + (k_max−k_min)·ρ^p).

Outputs: —

Firedrake

Numerics: Finite Element, GMRES+AMG

firedrake AD: adjoint

Upstream docs: https://www.firedrakeproject.org/

Path: mosaic/tesseracts/thermal-mesh/firedrake-heat

Image: firedrake_heat_thermal_mesh

Firedrake P1 FEM heat-conduction solver with SIMP penalisation. Gradients (∂C/∂ρ, ∂C/∂source) via firedrake-adjoint, complementing the FEniCS reference.

Inputs / Outputs

Inputs

Field	Type	∂	Description
k_max	float		Maximum thermal conductivity (fully solid/conducting material).
p_exp	float		SIMP penalisation exponent p (k(ρ) = k_min + (k_max−k_min)·ρ^p).

Outputs: —

JAX-FEM

Numerics: Finite Element, Direct (UMFPACK)

jax AD: hybrid GPU

Upstream docs: https://github.com/deepmodeling/jax-fem

Path: mosaic/tesseracts/thermal-mesh/jax-fem

Image: jax_fem_thermal_mesh

JAX-FEM HEX8 steady-state heat-conduction solver with SIMP conductivity and optional volumetric source. Gradients ∂C/∂ρ and ∂C/∂source via JAX autodiff through the linear FE solve (hybrid implicit-function-theorem adjoint)

Inputs / Outputs

Inputs: —

Outputs: —

torch-fem

Numerics: Finite Element, Direct (spsolve)

pytorch AD: autodiff GPU

Upstream docs: https://github.com/meyer-nils/torch-fem

Path: mosaic/tesseracts/thermal-mesh/torch-fem-thermal

Image: torch_fem_thermal_mesh

torch-fem SolidHeat HEX8 heat-conduction solver with IsotropicConductivity3D SIMP material — the first PyTorch backend in thermal-mesh. Gradients via PyTorch autograd through the FE solve; GPU-native.

Inputs / Outputs

Inputs

Field	Type	∂	Description
k_max	float		Maximum thermal conductivity (fully solid material).
p_exp	float		SIMP penalisation exponent p (k(ρ) = k_min + (k_max−k_min)·ρ^p).

Outputs: —