multi_physics/biot_short_syntax.py

Description

Biot problem - deformable porous medium with a no-penetration boundary condition imposed in the weak sense on a boundary region, using the short syntax of keywords.

The Biot coefficient tensor \alpha_{ij} is non-symmetric. The mesh resolution can be changed by editing the shape variable.

This example demonstrates how to set up various linear solvers and preconditioners (see solvers dict):

  • ‘direct’ (a direct solver from SciPy), ‘iterative-s’ (an iterative solver from SciPy), ‘iterative-p’ (an iterative solver from PETSc) solvers can be used as the main linear solver.
  • ‘direct’, ‘cg-s’ (several iterations of CG from SciPy), ‘cg-p’ (several iterations of CG from PETSc), ‘pyamg’ (an algebraic multigrid solver) solvers can be used as preconditioners for the matrix blocks on the diagonal.

See setup_precond() and try to modify it.

The PETSc solvers can be configured also using command line options. For example, set 'ls' : 'iterative-p' in options, and run:

python simple.py examples/multi_physics/biot_short_syntax.py -ksp_monitor

or simply run:

python simple.py examples/multi_physics/biot_short_syntax.py -O "ls='iterative-p'"

to monitor the PETSc iterative solver convergence. It will diverge without preconditioning, see matvec_bj(), matvec_j() for further details.

The PETSc options can also be set in the solver configuration - try uncommenting the 'ksp_*' or 'pc_*' parameters in 'iterative-p'. Uncommenting all the lines leads to, among other things, using the GMRES method with no preconditioning and the condition number estimate computation. Compare the condition number estimates with and without a preconditioning (try, for example, using 'precond' : 'mg' or 'pc_type' : 'mg').

Find \ul{u}, p such that:

\int_{\Omega} D_{ijkl}\ e_{ij}(\ul{v}) e_{kl}(\ul{u})
- \int_{\Omega}  p\ \alpha_{ij} e_{ij}(\ul{v})
+ \int_{\Gamma_{TB}} \varepsilon (\ul{n} \cdot \ul{v}) (\ul{n} \cdot \ul{u})
= 0
\;, \quad \forall \ul{v} \;,

- \int_{\Omega} q\ \alpha_{ij} e_{ij}(\ul{u})
- \int_{\Omega} K_{ij} \nabla_i q \nabla_j p
= 0
\;, \quad \forall q \;,

where

D_{ijkl} = \mu (\delta_{ik} \delta_{jl}+\delta_{il} \delta_{jk}) +
\lambda \ \delta_{ij} \delta_{kl}
\;.

../../_images/multi_physics-biot_short_syntax.png

source code

r"""
Biot problem - deformable porous medium with a no-penetration boundary
condition imposed in the weak sense on a boundary region, using the short
syntax of keywords.

The Biot coefficient tensor :math:`\alpha_{ij}` is non-symmetric. The mesh
resolution can be changed by editing the `shape` variable.

This example demonstrates how to set up various linear solvers and
preconditioners (see `solvers` dict):

- `'direct'` (a direct solver from SciPy), `'iterative-s'` (an iterative solver
  from SciPy), `'iterative-p'` (an iterative solver from PETSc) solvers can be
  used as the main linear solver.
- `'direct'`, `'cg-s'` (several iterations of CG from SciPy), `'cg-p'` (several
  iterations of CG from PETSc), `'pyamg'` (an algebraic multigrid solver)
  solvers can be used as preconditioners for the matrix blocks on the diagonal.

See :func:`setup_precond()` and try to modify it.

The PETSc solvers can be configured also using command line options. For
example, set ``'ls' : 'iterative-p'`` in `options`, and run::

  python simple.py examples/multi_physics/biot_short_syntax.py -ksp_monitor

or simply run::

  python simple.py examples/multi_physics/biot_short_syntax.py -O "ls='iterative-p'"

to monitor the PETSc iterative solver convergence. It will diverge without
preconditioning, see :func:`matvec_bj()`, :func:`matvec_j()` for further
details.

The PETSc options can also be set in the solver configuration - try
uncommenting the ``'ksp_*'`` or ``'pc_*'`` parameters in ``'iterative-p'``.
Uncommenting all the lines leads to, among other things, using the GMRES method
with no preconditioning and the condition number estimate computation. Compare
the condition number estimates with and without a preconditioning (try, for
example, using ``'precond' : 'mg'`` or ``'pc_type' : 'mg'``).

Find :math:`\ul{u}`, :math:`p` such that:

.. math::
    \int_{\Omega} D_{ijkl}\ e_{ij}(\ul{v}) e_{kl}(\ul{u})
    - \int_{\Omega}  p\ \alpha_{ij} e_{ij}(\ul{v})
    + \int_{\Gamma_{TB}} \varepsilon (\ul{n} \cdot \ul{v}) (\ul{n} \cdot \ul{u})
    = 0
    \;, \quad \forall \ul{v} \;,

    - \int_{\Omega} q\ \alpha_{ij} e_{ij}(\ul{u})
    - \int_{\Omega} K_{ij} \nabla_i q \nabla_j p
    = 0
    \;, \quad \forall q \;,

where

.. math::
    D_{ijkl} = \mu (\delta_{ik} \delta_{jl}+\delta_{il} \delta_{jk}) +
    \lambda \ \delta_{ij} \delta_{kl}
    \;.
"""
from __future__ import absolute_import
import numpy as nm

from sfepy.base.base import Struct
from sfepy.mechanics.matcoefs import stiffness_from_lame
from sfepy.discrete.fem.meshio import UserMeshIO
from sfepy.mesh.mesh_generators import gen_block_mesh

def get_pars(ts, coor, mode, **kwargs):
    """
    Define the material parameters.
    """
    if mode == 'qp':
        n_nod, dim = coor.shape

        out = {}
        out['D'] = nm.tile(stiffness_from_lame(dim, lam=1.7, mu=0.3),
                           (coor.shape[0], 1, 1))

        alpha = [[0.132, 0.092],
                 [0.052, 0.132]]
        out['alpha'] = nm.tile(alpha, (coor.shape[0], 1, 1))

        perm = nm.eye(dim, dtype=nm.float64)
        out['K'] = nm.tile(perm, (coor.shape[0], 1, 1))

        out['np_eps'] = nm.tile(1e5, (coor.shape[0], 1, 1))

        return out

def post_process(out, pb, state, extend=False):
    """
    Compute derived quantities of interest..
    """
    from sfepy.base.base import Struct

    dvel = pb.evaluate('ev_diffusion_velocity.i.Omega(m.K, p)',
                       mode='el_avg')
    out['dvel'] = Struct(name='output_data',
                         mode='cell', data=dvel, dofs=None)

    stress = pb.evaluate('ev_cauchy_stress.i.Omega(m.D, u)',
                         mode='el_avg')
    out['cauchy_stress'] = Struct(name='output_data',
                                  mode='cell', data=stress, dofs=None)

    return out

# Mesh dimensions.
dims = [0.1, 0.1]

# Mesh resolution: increase to improve accuracy.
shape = [21, 21]

def mesh_hook(mesh, mode):
    """
    Generate the block mesh.
    """
    if mode == 'read':
        mesh = gen_block_mesh(dims, shape, [0, 0], name='user_block',
                              verbose=False)
        return mesh

    elif mode == 'write':
        pass

filename_mesh = UserMeshIO(mesh_hook)

materials = {
    'coef' : ({'val' : 1.0},),
}

regions = {
    'Omega' : 'all', # or 'cells of group 6'
    'GammaL' : ('vertices in (x < -0.0499)', 'facet'),
    'GammaR' : ('vertices in (x > 0.0499)', 'facet'),
    'GammaTB' : ('vertices of surface -s (r.GammaL +s r.GammaR)', 'facet')
}

fields = {
    'displacement': ('real', 'vector', 'Omega', 1),
    'pressure': ('real', 'scalar', 'Omega', 1),
}

variables = {
    'u'  : ('unknown field', 'displacement', 0),
    'v'  : ('test field',    'displacement', 'u'),
    'p'  : ('unknown field', 'pressure', 1),
    'q'  : ('test field',    'pressure', 'p'),
}

ebcs = {
    'inlet' : ('GammaL', {'p.0' : 1.0, 'u.all' : 0.0}),
    'outlet' : ('GammaR', {'p.0' : 0.0}),
}

integrals = {
    'i' : 2,
}

materials = {
    'm' : 'get_pars',
}

functions = {
    'get_pars' : (get_pars,),
}

equations = {
    'eq_1' :
    """+ dw_lin_elastic.i.Omega(m.D, v, u)
       - dw_biot.i.Omega(m.alpha, v, p)
       + dw_non_penetration_p.i.GammaTB(m.np_eps, v, u)
       = 0""",
    'eq_2' :
    """- dw_biot.i.Omega(m.alpha, u, q)
       - dw_diffusion.i.Omega(m.K, q, p)
       = 0""",
}

def setup_precond(mtx, problem):
    """
    Setup a preconditioner for `mtx`.
    """
    import scipy.sparse.linalg as spla
    from sfepy.solvers import Solver

    # Get active DOF indices for u, p.
    adi = problem.get_variables().adi
    iu = adi.indx['u']
    ip = adi.indx['p']

    # Get the diagonal blocks of the linear system matrix.
    K = mtx[iu, iu]
    M = mtx[ip, ip]

    # Create solvers for K, M blocks to be used in matvec_bj(). A different
    # solver for each block could be used.
    conf = problem.solver_confs['direct']
    # conf = problem.solver_confs['cg-s']
    # conf = problem.solver_confs['cg-p']
    # conf = problem.solver_confs['pyamg']
    ls1 = Solver.any_from_conf(conf, mtx=K, context=problem)
    ls2 = Solver.any_from_conf(conf, mtx=M, context=problem)

    def matvec_bj(vec):
        """
        The application of the Block Jacobi preconditioner.

        The exact version (as with the `'direct'` solver) can be obtained also
        by using the following PETSs command-line options, together with the
        `'iterative-p'` solver::

          -ksp_monitor -pc_type fieldsplit -pc_fieldsplit_type additive -fieldsplit_u_ksp_type preonly -fieldsplit_u_pc_type lu -fieldsplit_p_ksp_type preonly -fieldsplit_p_pc_type lu

        The inexact version (20 iterations of a CG solver for each block, as
        with the `'cg-s'` or `'cg-p'` solvers) can be obtained also by using
        the following PETSs command-line options, together with the
        `'iterative-p'` solver::

          -ksp_monitor -pc_type fieldsplit -pc_fieldsplit_type additive -fieldsplit_u_ksp_type cg -fieldsplit_u_pc_type none -fieldsplit_p_ksp_type cg -fieldsplit_p_pc_type none -fieldsplit_u_ksp_max_it 20 -fieldsplit_p_ksp_max_it 20
        """
        vu = ls1(vec[iu])
        vp = ls2(vec[ip])

        return nm.r_[vu, vp]

    def matvec_j(vec):
        """
        The application of the Jacobi (diagonal) preconditioner.

        The same effect can be obtained also by using the following PETSs
        command-line options, together with the `'iterative-p'` solver::

          -ksp_monitor -pc_type jacobi
        """
        D = mtx.diagonal()

        return vec / D

    # Create the preconditioner, using one of matvec_bj() or matvec_j().
    precond = Struct(name='precond', shape=mtx.shape, matvec=matvec_bj)
    precond = spla.aslinearoperator(precond)

    return precond

method = 'gmres'
i_max = 20
eps_r = 1e-8

solvers = {
    'direct' : ('ls.scipy_direct', {}),

    'iterative-s' : ('ls.scipy_iterative', {
        'method' : method,
        'i_max' : i_max,
        'eps_r' : eps_r,
        'setup_precond': setup_precond,
        'verbose' : 2,
    }),
    'cg-s' : ('ls.scipy_iterative', {
        'method' : 'cg',
        'i_max' : 20,
        'eps_r' : 1e-6,
        'verbose' : 0,
    }),

    'iterative-p' : ('ls.petsc', {
        'method' : method,
        'precond' : 'none',
        'i_max' : i_max,
        'eps_r' : eps_r,
        'verbose' : 2,
        # 'ksp_converged_reason' : None,
        # 'ksp_monitor_true_residual' : None,
        # 'ksp_monitor_singular_value' : None,
        # 'ksp_final_residual' : None,
        # 'ksp_type' : 'gmres', # Overrides `method`.
        # 'ksp_max_it' : 500,
        # 'ksp_gmres_restart' : 1000,
        # 'pc_type' : 'none', # Overrides `precond`.
    }),
    'cg-p' : ('ls.petsc', {
        'method' : 'cg',
        'precond' : 'none',
        'i_max' : 20,
        'eps_r' : 1e-6,
        'verbose' : 0,
    }),

    'pyamg' : ('ls.pyamg', {
        'method' : 'smoothed_aggregation_solver',
        'i_max' : 20,
        'eps_r' : 1e-6,
        'verbose' : 0,
    }),

    'newton' : ('nls.newton',
                {'i_max' : 1,
                 'eps_r' : 1e-6,
                 'eps_a' : 1.0,
    }),
}

options = {
    'nls' : 'newton',
    'ls' : 'iterative-s',

    'post_process_hook' : 'post_process',
}