import numpy as nm
from sfepy.terms.terms_multilinear import ETermBase
[docs]class MassTerm(ETermBase):
r"""
Mass term with lumping and RMM support [1]_.
The lumping parameter can be 'row_sum', 'hrz' or 'none' (default). It
applies for :math:`\beta` > 0:
- :math:`\beta` = 0 correponds to the consistent mass matrix :math:`M^C`;
- 0 < :math:`\beta` < 1 corresponds to the averaged mass matrix
:math:`M^A`.
- :math:`\beta` = 1 corresponds to the lumped mass matrix :math:`M^L`;
`term_mode` can be None (default), 'DPM' (diagonal projection matrix
:math:`A`), or 'RMM' (reciprocal mass matrix :math:`C`).
.. [1] González, J.A., Kolman, R., Cho, S.S., Felippa, C.A., Park, K.C.,
2018. Inverse mass matrix via the method of localized Lagrange
multipliers. International Journal for Numerical Methods in
Engineering 113, 277–295. https://doi.org/10.1002/nme.5613
:Definition:
.. math::
M^C = \int_{\cal{D}} \rho \ul{v} \cdot \ul{u} \\
M^L = \mathrm{lumping}(M^C) \\
M^A = (1 - \beta) M^C + \beta M^L \\
A = \sum_e A_e \\
C = \sum_e A_e^T (M_e^A)^{-1} A_e
:Arguments:
- material: :math:`\rho`
- material: lumping
- material: :math:`\beta`
- virtual/parameter_1: :math:`\ul{v}`
- state/parameter_2: :math:`\ul{u}`
"""
name = 'de_mass'
arg_types = ('material_rho', 'material_lumping', 'material_beta',
'virtual', 'state')
arg_shapes = {'material_rho' : '1, 1', 'material_lumping' : '.: str',
'material_beta' : '.: 1',
'virtual' : ('D', 'state'), 'state' : 'D'}
modes = ('weak', 'eval')
[docs] def get_function(self, rho, lumping, beta, virtual, state,
mode=None, term_mode=None, diff_var=None, **kwargs):
if ((term_mode is None)
and ((lumping == 'none') or (beta == 0.0))):
# Consistent mass matrix.
fun = self.make_function(
'00,i,i', rho, virtual, state, diff_var=diff_var,
)
return fun
_fun = self.make_function(
'00,i,i', rho, virtual, state, diff_var=state.name,
)
self.einfos[None] = self.einfos[state.name]
if (term_mode in ('RMM', 'DPM')) and (lumping == 'none'):
lumping = 'row_sum'
if (term_mode == 'DPM'):
beta = 1.0
def fun(out, *fargs):
# Compute M_C.
if diff_var is None:
sh = out.shape
_out = nm.zeros((sh[0], sh[1], sh[2], sh[2]), dtype=out.dtype)
else:
_out = out
status = _fun(_out, *fargs)
# Consistent element mass matrices M_C.
mc = _out if beta in (0.0, 1.0) else _out.copy()
# Diagonals of lumped element mass matrices M_L.
if lumping == 'row_sum':
mld = mc.sum(-1)
elif lumping == 'hrz':
n_bf = mc.shape[-1] // self.region.dim
det = fargs[2][0][0]
jrho = det * rho[..., 0, 0]
diag = nm.einsum('...ii->...i', mc)
intrho = jrho.sum(axis=1) # Element masses to preserve.
intdiag = diag[:, 0, :n_bf].sum(axis=1)
alpha = intrho / intdiag
mld = alpha[:, None, None] * diag
else:
raise ValueError('unknown lumping mode! (%s)' % lumping)
if beta == 1.0:
# M_A = M_L.
eye = nm.eye(mld.shape[-1])
_out[:] = nm.einsum('can,nm->canm', mld, eye)
elif beta > 0.0:
# M_A = (1 - beta) * M_C + beta * M_L.
_out[:] = (1.0 - beta) * mc
outd = nm.einsum('...ii->...i', _out) # View to diagonal.
outd += beta * mld
else:
# beta == 0.0, so M_A = M_C in RMM term_mode -> do nothing.
pass
if term_mode == 'RMM':
# out contains M_A, mld contains A_e diagonal.
ime = nm.linalg.inv(_out)
# ce = nm.einsum('cik,ckl,clj->cij', ae, ime, ae)
_out[:] = mld[..., None] * ime * mld[:, None, :]
if diff_var is None:
earg = self.einfos[None].eargs[2]
step_cache = earg.arg.evaluate_cache.setdefault('dofs', {})
cache = step_cache.setdefault(0, {})
dofs = earg.get_dofs(cache, step_cache, state.name + '.dofs')
dofs = dofs.reshape((sh[0], sh[2])) # Does a copy!
# Could be faster by exploiting that mld is a vector.
out[..., 0] = nm.einsum('caij,cj->cai', _out, dofs)
return status
return fun