Resolution

Settings of the solver

The BEMSolver class takes two (keyword-only) arguments at the time of its initialization:

from capytaine import BEMSolver
solver = BEMSolver(green_function=..., engine=...)

Let us discuss in more details these two objects.

Green function

A class used to evaluate the Green function, deriving from AbstractGreenFunction. In the present version, a single class is implemented, although it offers many parameters for customization:

Delhommeau (Default)

The method implemented in Nemoh (see [Del87] and [Del89]). See the documentation for details on the available options.

In Capytaine (and Nemoh), the integral of the wave term \(\mathcal{G}(r, z)\) (and its derivative \(\frac{\partial \mathcal{G}}{\partial r}\)) are approximated using surrogate models, which take the form of a tabulation of these function values for a grid of \((r, z)\), precomputed at the initialization of the program. A third-order Lagrange polynomial interpolation is employed to obtain the values between the precomputed values.

In version 1 of Capytaine (as in version 2 of Nemoh), the tabulation ranges of \(r\) and \(z\) are set as \([0, 100]\) with \(328\) discretization values and \([-16, 0]\) with \(46\) discretization values, respectively. In the new version, these can be user-defined with the following options:

import capytaine as cpt

# Legacy (versions 1.x)
gf = cpt.Delhommeau(tabulation_nr=324, tabulation_rmax=100,
                    tabulation_nz=46, tabulation_zmin=-16,
                    tabulation_nb_integration_points=251,
                    tabulation_grid_shape="legacy",
                    gf_singularities="high_freq")

# Default in Capytaine 2.1
gf = cpt.Delhommeau(tabulation_nr=676, tabulation_rmax=100,
                    tabulation_nz=372, tabulation_zmin=-251,
                    tabulation_nb_integration_points=1000,
                    tabulation_grid_shape="scaled_nemoh3",
                    gf_singularities="high_freq")

# Default in Capytaine 2.2
gf = cpt.Delhommeau(tabulation_nr=676, tabulation_rmax=100,
                    tabulation_nz=372, tabulation_zmin=-251,
                    tabulation_nb_integration_points=1000,
                    tabulation_grid_shape="scaled_nemoh3",
                    gf_singularities="low_freq")

In version 2.1, the default numbers of \(r\) and \(z\) values have been increased to \(676\) and \(372\), respectively. While the range of \(r\) is kept the same, the z range has been extended to \([-251, 0]\). The option tabulation_grid_shape is used to switched between the new distribution of points inspired by Nemoh version 3 or the "legacy" approach. The tabulation_nb_integration_points controls the accuracy of the precomputed tabulation points themselves.

In version 2.2, the way singularities are extracted of the infinite depth Green function to be integrated has changed. The "low_freq" variant is expected to be more accurate at low frequency and near the free surface. The former variant is still available by setting the gf_singularities parameter as in the above example.

The first time it is initialize with a given set of parameters, some tabulated data are precomputed and stored on disk. The default location is a os-dependant cache directory. The location at which the data is stored can be configured by passing tabulation_cache_dir to Delhommeau or by setting the environment variable CAPYTAINE_CACHE_DIR.

Advanced users can write their own class to evaluate the Green function. See the example in the Cookbook.

Engine

A class to build a interaction matrix, deriving from MatrixEngine. Two of them are available in the present version:

BasicMatrixEngine (Default)

A simple engine fairly similar to the one in Nemoh. It builds the full matrices with few optimizations. Only a reflection symmetry can be used to make the resolution faster.

The object can be initialized with the following options:

matrix_cache_size (Default: 1)

The solver keeps in memory the last interaction matrices that has been computed. This setting controls the number of old matrices that are saved. Setting it to 0 will reduce the RAM usage of the code but might increase the computation time.

linear_solver (Default: 'lu_decomposition')

This option is used to set the solver for linear systems that is used in the resolution of the BEM problem. Passing a string will make the code use one of the predefined solver. Three of them are available: 'direct' for a simple direct solver, 'lu_decomposition' for a faster direct solver with caching of the LU decomposition, or 'gmres' for an iterative solver.

A direct solver is used by default (since version 1.4) because it is more robust and the computation time is more predictable. Advanced users might want to change the solver to gmres, which is faster in many situations (and completely fails in other).

Alternatively, any function taking as arguments a matrix and a vector and returning a vector can be given to the solver:

import numpy as np

def my_linear_solver(A, b):
        """A dumb solver for testing."""
        return np.linalg.inv(A) @ b

my_bem_solver = cpt.BEMSolver(
   engine=BasicMatrixEngine(linear_solver=my_linear_solver)
   )

This option can be used for instance to apply a custom preconditioning to the iterative solver.

HierarchicalToeplitzMatrixEngine

Experimental engine using hierarchical structure in the mesh to build hierarchical influence matrices.

The object can be initialized with the following options:

matrix_cache_size (Default: 1)

Same as above.

ACA_distance and ACA_tol

Parameters of the Adaptive Cross Approximation (ACA) used to set the precision of the low-rank matrices.

Solving the problem

Once the solver has been initialized, it can be used to solve problems with the solve() method:

result = solver.solve(problem, keep_details=False, method='indirect')

The optional argument keep_details (default value: True) controls whether the source and potential distributions should be saved in the result object. These data are necessary for some post-processing such as the computation of the Kochin function or the reconstruction of the free surface elevation. However, when only the force on the body is of interest, they can be discarded to save space in memory.

The optional argument method (default value: indirect) controls the approach employed to solve for the potential velocity solutions. Two methods are implemented including 1) direct method (source-and-dipole formulation), and 2) indirect method (source formulation). The direct method appears to be slightly more accurate on some test cases but only allows for the computation of the forces on the floating body. Any other post-processing requires the indirect method.

A list of problems can be solved at once in an optimal order with:

list_of_results = solver.solve_all(list_of_problems, keep_details=False)

Parallelization

Capytaine includes two kinds of parallelization.

	joblib	OpenMP
Single resolution (solve)	✗	✓
Batch resolution (solve_all and fill_dataset)	✓ (if installed)	✓

Single problem with OpenMP

When solving a single problem, matrix constructions and linear algebra operations (using BLAS or MKL depending on your installation) can be parallelized by OpenMP. This feature is installed and on by default. The number of threads used can be controlled by the environment variable OMP_NUM_THREADS, as well as MKL_NUM_THREADS (for the linear algebra when using Intel’s MKL library usually distributed with conda). Note that the environment variable should be set before the start of the Python interpreter. Alternatively, if you’d like to change dynamically the number of threads, it can be done with the threadpoolctl library (see also GH 47).

Batch resolution with joblib

When solving several independent problems, they can be solved in parallel. This feature (new in version 1.4) requires the optional dependency joblib to be installed. The methods solve_all() and fill_dataset() take an optional keyword-argument n_jobs which control the number of jobs to run in parallel during the batch resolution. Since joblib may disturb user feedback (logging and error reporting), it is currently disabled by default.

When n_jobs=1 (the default) or joblib is not installed, no parallel batch resolution happens (although OpenMP parallelization might still be enabled).

When n_jobs=-1, all CPU cores are used (and joblib should automatically disable the OpenMP parallelization.)

The two parallelization layers (OpenMP and joblib) have different usage. If you have a relatively small mesh but study a large number of sea states, you should use the joblib parallelization. On the other hand, if your mesh is large or your available RAM is low, it might be beneficial to turn off the joblib parallelization and use only the OpenMP one.