dualmesh

A multiphysics framework for heat transfer, solid mechanics and fluid dynamics, with the dual mesh control domain method as an option.

dualmesh solves coupled boundary value problems: any number of fields (temperatures, displacements, velocities, or quantities of your own), each governed by a conservation law built from named terms that may depend on every field, solved together as one monolithic system by Newton's method with an exact Jacobian from automatic differentiation. The same problem description is discretised, by changing one keyword, with the Galerkin finite element method, the vertex-centred or cell-centred finite volume method, or the dual mesh control domain method (DMCDM) of J. N. Reddy. The core is C++17; everything is driven from Python.

The structure follows the MOOSE framework: physics is added as kernels, boundary conditions and materials, which are registered objects with validated, self-documenting parameters; meshes are generated or read from standard files; and the solvers are shared by every physics. See docs/scope.rst for what that covers and an honest comparison with MOOSE and COMSOL Multiphysics.

import dualmesh as dm

mesh = dm.generate_rectangle_mesh(
    x_min=0.0, x_max=0.1, y_min=0.0, y_max=0.05,
    num_x_elements=10, num_y_elements=5)

problem = dm.Problem(mesh, method="dmcdm")      # or method="fem"
problem.add_variable("temperature")
problem.add_kernel("HeatConduction", variable="temperature", thermal_conductivity=20.0)
problem.add_kernel("HeatSource", variable="temperature", heat_source=1.0e6)
problem.add_boundary_condition("DirichletBC", variable="temperature", boundary="left", value=40.0)
problem.add_boundary_condition("DirichletBC", variable="temperature", boundary="right", value=10.0)
problem.add_boundary_condition("ConvectiveHeatFluxBC", variable="temperature",
                               boundary="top", heat_transfer_coefficient=75.0)
problem.solve()

problem.sample("temperature", [[0.05, 0.0]])    # 83.142 (Reddy, Table 5.4.3)
problem.total_reaction("temperature", "left")   # heat flow through the left face
problem.write_vtu("bus_bar.vtu")

What the method is

The primal mesh is a mesh of finite elements and supplies the interpolation. The dual mesh is the set of node-centred control domains built from edge midpoints, face centroids and element centroids. The governing equation is satisfied in the integral sense over each control domain, with no weight function:

  -∮(∂CD) F·n dS + ∫(CD) S dV = 0

so the secondary variables (fluxes, forces, moments) appear naturally on the control-domain interfaces — the physical appeal of the finite volume method — while the primal interpolation removes the ad-hoc gradient reconstructions that the finite volume method needs. Every equation in the library is written in the canonical form -div F + S = 0, which is why the same kernels can be discretized either way.

Features

• Methods: Galerkin finite elements, vertex-centred and cell-centred finite volumes, and the dual mesh control domain method, selected by one argument. A method refuses, with the reason, an element or a physics it cannot treat correctly.
• Elements: Edge2, Edge3, Tri3, Tri6, Quad4, Quad8, Quad9, Tet4, Tet10, Hex8, Hex20, Hex27, Wedge6, Pyramid5, mixed freely in one mesh, on straight or distorted meshes, in 1D, 2D and 3D.
• Coordinate systems: Cartesian, axisymmetric (2πr), spherical (4πr²).
• Physics modules
  • heat transfer: conduction with temperature-dependent conductivity, volumetric heating, capacity, convection by a computed flow, convective, radiative and flux boundaries;
  • solid mechanics: linear elasticity in plane stress, plane strain, axisymmetric and three-dimensional form with thermal strain, tractions and pressures; and the structural members: mixed Euler–Bernoulli beams, displacement and mixed Timoshenko beams, axisymmetric circular plates and rectangular plates, functionally graded sections, von Kármán nonlinearity;
  • fluid dynamics: Stokes and Navier–Stokes flow by the penalty formulation with recovered pressure, and Boussinesq buoyancy;
  • framework: diffusion, anisotropic diffusion, reaction, advection, body force, time derivative, coupled force, Dirichlet/Neumann/Robin conditions, point sources, generic materials, and expressions such as "sin(pi*x)*exp(-t)" compiled to C++.
• Coupled problems: monolithic and fully coupled, verified on the natural convection benchmark of de Vahl Davis (Nusselt numbers within 0.2 %).
• Solvers: Newton's method with exact Jacobians from forward-mode automatic differentiation, direct (Picard) iteration with relaxation, load stepping, steady and transient (θ-method, adaptive step size) executioners. Linear systems are solved directly where that is cheap and by BiCGSTAB or GMRES with an ILU(0) preconditioner where it is not, chosen automatically.
• Parallel: threaded assembly, and an MPI solver with a two-level overlapping Schwarz preconditioner whose iteration count does not grow with the number of processes.
• Quadrature per kernel: Gauss rules, midpoint, trapezoid, Simpson, nodal lumping, interface and control-domain-trapezoid rules, and selective reduced integration for locking and penalty terms.
• Meshing: generators (line, rectangle, box, annulus, graded spacing), local and uniform refinement, and readers for Gmsh, Exodus, VTK, Abaqus and more through meshio.
• Verification: Reddy's book examples, analytical solutions, OpenFOAM cross-checks, and a method-of-manufactured-solutions study of the order of convergence of every method on every element type.
• Extensible from Python: kernels, boundary conditions and materials can be written in Python and still get exact derivatives.
• Input files: dualmesh run input.yaml, plus dualmesh list and dualmesh describe <type> for the object reference.

Verification

Every worked example of Reddy's book with published dual mesh results is reproduced by the test suite — one-dimensional and two-dimensional conduction, axisymmetric conduction, advection–diffusion at high Péclet number, nonlinear conduction, plane elasticity, pressurized cylinders, squeezed flow, the lid-driven cavity at Re = 0 and Re = 1000, functionally graded beams (linear and von Kármán), circular plates and rectangular plates. Natural convection is checked against de Vahl Davis (1983), and a manufactured-solution study checks the convergence order of every method and element type. See docs/verification.rst and tests/python. The suite also cross-checks against OpenFOAM (see verification/openfoam).

Installation

pip install .[all]           # from a clone
pytest                       # run the verification suite

Requirements: a C++17 compiler, CMake ≥ 3.18, Python ≥ 3.9. Eigen and pybind11 are found if installed and downloaded otherwise.

Documentation: https://dualmesh.readthedocs.io

Citing

If this software contributes to your work, please cite the method

J. N. Reddy, Computational Methods in Engineering: Finite Difference, Finite Volume, Finite Element, and Dual Mesh Control Domain Methods, CRC Press, 2024,

and the software itself; see CITATION.cff.

License

GNU Lesser General Public License, version 2.1 or later (the licence of the MOOSE framework, whose object model inspired this one). See LICENSE.