fourier

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fourier#

matinverse.fourier.Fourier(geo, boundary_conditions, thermal_conductivity=None, heat_capacity=None, conductance=None, heat_source=None, mode='linear', linear_solver='iterative', collapse_direct=False, compute_side_flux=None, batch_size=1, tol=1e-09, maxiter=5000, maxiter_nonlinear=100, scale_nonlinear=None, maxiter_transient=10000, saveat=None, DT=1.0, NT=1, X0=None)#

Differentiable Fourier solver based on the Finite-volume method.

Parameters:

  • geo (Union[Geometry2D, Geometry3D]) – The geometry object containing the mesh information.

  • boundary_conditions (BoundaryConditions) – The boundary conditions object

  • thermal_conductivity (Optional[Callable]) –

    A function that takes batch, space (as a function of volume index), temperature and time and returns the thermal conductivity tensor \([\mathrm{W\,m^{-1}\,K^{-1}}]\)

    Example:

    thermal_conductivity = lambda batch, space, temp, t: 100.*jnp.eye(geo.dim)

  • heat_capacity (Optional[Callable]) –

    A function that takes batch, space (as a function of volume index), temperature and time and returns the heat capacity \([\mathrm{J\,m^{-3}\,K^{-1}}]\). Example:

    heat_capacity = lambda batch, space, temp, t: 1e3

  • conductance (Optional[Callable]) –

    A function that takes the indices of the volumes adjacent to a given side and returns the batched conductance \([\mathrm{W\,m^{-2}\,K^{-1}}]\). If not provided, it is assumed to be large number so it won’t affect calculation.

    Example::

    conductance = lambda s1,s2: 1e3*jnp.ones(1) #assuming a single batch and uniform conductance

  • heat_source (Optional[Callable]) –

    A function that takes batch, space (as a function of volume index) and time and returns the heat source \([\mathrm{W\,m^{-3}}]\). If not provided, it is assumed to be zero.

    Example::

    heat_source = lambda b,s,t: 1e5

  • mode (str) – The mode of the solver. Can be ‘linear’, ‘nonlinear’ or ‘transient’.

  • linear_solver (str) – The linear solver to use. Can be ‘iterative’ or ‘direct’. If ‘iterative’, it uses the GMRES method. If ‘direct’, it uses sparse LU factorization.

  • collapse_direct (bool) – If True, it uses the same assembly matrix for all the batches in the direct solver. It can be used only for direct solvers and if no Robin BCs are used.

  • compute_side_flux (array) – A jnp.array containing the indices of the sides for which the flux will be computed. If not provided, no internal interfacial flux will be computed.

  • batch_size (int) – The number of batches to use. If > 1, the solver will be vectorized over the batches.

  • tol (float) – The tolerance for the linear solver.

  • maxiter (int) – The maximum number of iterations for the solver.

  • maxiter_nonlinear (int) – The maximum number of iterations for the nonlinear solver (default is 100). It is used only if mode='nonlinear'.

  • scale_nonlinear (float) – A scaling factor for the nonlinear solver. If not provided, it is set to \(10^{-4} dx^2\) It is used only if mode='nonlinear'.

  • maxiter_transient (int) – The maximum number of iterations for the transient solver (default is 10000). It is used only if mode='transient'.

  • saveat (array) – A jnp.array containing the time steps at which the solution will be saved. If not provided, it saves all time steps.

  • DT (float) – The time step for the transient solver.

  • NT (int) – The number of time steps for the transient solver.

  • X0 (array) – The initial guess for the temperature. It is expected to be of shape (batch_size, geo.nDOFs). If batch_size = 1 then it can be a 1D array. If not provided, it is assumed to be zero.

Returns:

(dict, dict) – A tuple containing the output dictionary and the statistics dictionary. The output dictionary contains the following fields:

  • ‘T’: The temperature field of shape (NT, batch_size, geo.nDOFs).

  • ‘J’: The heat flux field of shape (NT, batch_size, geo.nDOFs, geo.dim).

  • ‘kappa’: The thermal conductivity tensor field of shape (NT, batch_size, geo.nDOFs, geo.dim, geo.dim).

  • ‘kappa_effective’: The effective thermal conductivity field of shape (NT, batch_size, geo.nDOFs).

  • ‘P_boundary’: The boundary flux field of shape (NT, batch_size, geo.nBoundarySides).

  • ‘P_internal’: The internal flux field of shape (NT, batch_size, geo.nSides).

  • ‘T_boundary’: The temperature field at the boundaries of shape (NT, batch_size, geo.nBoundarySides).

The statistics dictionary contains the following fields:

  • ‘num_steps’: The number of steps taken by the nonlinear solver.

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