Source code for sfepy.terms.terms_elastic

import numpy as nm

from sfepy.linalg import dot_sequences
from sfepy.homogenization.utils import iter_sym
from sfepy.terms.terms import Term, terms
from sfepy.terms.terms_th import THTerm, ETHTerm

## expr = """
## e = 1/2 * (grad( vec( u ) ) + grad( vec( u ) ).T)
## D = map( D_sym )
## s = D * e
## div( s )
## """

## """
## e[i,j] = 1/2 * (der[j]( u[i] ) + der[i]( u[j] ))
## map =
## D[i,j,k,l]
## s[i,j] = D[i,j,k,l] * e[k,l]
## """

[docs]class LinearElasticTerm(Term): r""" General linear elasticity term, with :math:`D_{ijkl}` given in the usual matrix form exploiting symmetry: in 3D it is :math:`6\times6` with the indices ordered as :math:`[11, 22, 33, 12, 13, 23]`, in 2D it is :math:`3\times3` with the indices ordered as :math:`[11, 22, 12]`. Can be evaluated. Can use derivatives. :Definition: .. math:: \int_{\Omega} D_{ijkl}\ e_{ij}(\ul{v}) e_{kl}(\ul{u}) :Arguments 1: - material : :math:`D_{ijkl}` - virtual : :math:`\ul{v}` - state : :math:`\ul{u}` :Arguments 2: - material : :math:`D_{ijkl}` - parameter_1 : :math:`\ul{w}` - parameter_2 : :math:`\ul{u}` """ name = 'dw_lin_elastic' arg_types = (('material', 'virtual', 'state'), ('material', 'parameter_1', 'parameter_2')) arg_shapes = {'material' : 'S, S', 'virtual' : ('D', 'state'), 'state' : 'D', 'parameter_1' : 'D', 'parameter_2' : 'D'} modes = ('weak', 'eval') ## symbolic = {'expression': expr, ## 'map' : {'u' : 'state', 'D_sym' : 'material'}}
[docs] def get_fargs(self, mat, virtual, state, mode=None, term_mode=None, diff_var=None, **kwargs): vg, _ = self.get_mapping(state) if mode == 'weak': if diff_var is None: strain = self.get(state, 'cauchy_strain') fmode = 0 else: strain = nm.array([0], ndmin=4, dtype=nm.float64) fmode = 1 return 1.0, strain, mat, vg, fmode elif mode == 'eval': strain1 = self.get(virtual, 'cauchy_strain') strain2 = self.get(state, 'cauchy_strain') return 1.0, strain1, strain2, mat, vg else: raise ValueError('unsupported evaluation mode in %s! (%s)' % (self.name, mode))
[docs] def get_eval_shape(self, mat, virtual, state, mode=None, term_mode=None, diff_var=None, **kwargs): n_el, n_qp, dim, n_en, n_c = self.get_data_shape(state) return (n_el, 1, 1, 1), state.dtype
[docs] def set_arg_types(self): if self.mode == 'weak': self.function = terms.dw_lin_elastic else: self.function = terms.d_lin_elastic
[docs]class LinearElasticIsotropicTerm(LinearElasticTerm): r""" Isotropic linear elasticity term. :Definition: .. math:: \int_{\Omega} D_{ijkl}\ e_{ij}(\ul{v}) e_{kl}(\ul{u}) \mbox{ with } D_{ijkl} = \mu (\delta_{ik} \delta_{jl}+\delta_{il} \delta_{jk}) + \lambda \ \delta_{ij} \delta_{kl} :Arguments: - material_1 : :math:`\lambda` - material_2 : :math:`\mu` - virtual : :math:`\ul{v}` - state : :math:`\ul{u}` :Arguments 2: - material : :math:`D_{ijkl}` - parameter_1 : :math:`\ul{w}` - parameter_2 : :math:`\ul{u}` """ name = 'dw_lin_elastic_iso' arg_types = (('material_1', 'material_2', 'virtual', 'state'), ('material_1', 'material_2', 'parameter_1', 'parameter_2')) arg_shapes = {'material_1' : '1, 1', 'material_2' : '1, 1', 'virtual' : ('D', 'state'), 'state' : 'D', 'parameter_1' : 'D', 'parameter_2' : 'D'} geometries = ['2_3', '2_4', '3_4', '3_8']
[docs] def get_fargs(self, lam, mu, virtual, state, mode=None, term_mode=None, diff_var=None, **kwargs): from sfepy.mechanics.matcoefs import stiffness_from_lame mat = stiffness_from_lame(self.region.dim, lam, mu)[:, :, 0, 0, :, :] return LinearElasticTerm.get_fargs(self, mat, virtual, state, mode=mode, term_mode=term_mode, diff_var=diff_var, **kwargs)
[docs] def get_eval_shape(self, mat1, mat2, virtual, state, mode=None, term_mode=None, diff_var=None, **kwargs): return LinearElasticTerm.get_eval_shape(self, None, None, state)
[docs]class SDLinearElasticTerm(Term): r""" Sensitivity analysis of the linear elastic term. :Definition: .. math:: \int_{\Omega} \hat{D}_{ijkl}\ e_{ij}(\ul{v}) e_{kl}(\ul{u}) .. math:: \hat{D}_{ijkl} = D_{ijkl}(\nabla \cdot \ul{\Vcal}) - D_{ijkq}{\partial \Vcal_l \over \partial x_q} - D_{iqkl}{\partial \Vcal_j \over \partial x_q} :Arguments: - material : :math:`D_{ijkl}` - parameter_w : :math:`\ul{w}` - parameter_u : :math:`\ul{u}` - parameter_mesh_velocity : :math:`\ul{\Vcal}` """ name = 'd_sd_lin_elastic' arg_types = ('material', 'parameter_w', 'parameter_u', 'parameter_mesh_velocity') arg_shapes = {'material' : 'S, S', 'parameter_w' : 'D', 'parameter_u' : 'D', 'parameter_mesh_velocity' : 'D'} geometries = ['2_3', '2_4', '3_4', '3_8'] function = terms.d_sd_lin_elastic
[docs] def get_fargs(self, mat, par_w, par_u, par_mv, mode=None, term_mode=None, diff_var=None, **kwargs): vg, _ = self.get_mapping(par_u) grad_w = self.get(par_w, 'grad') grad_u = self.get(par_u, 'grad') grad_mv = self.get(par_mv, 'grad') return 1.0, grad_w, grad_u, grad_mv, mat, vg
[docs] def get_eval_shape(self, mat, par_w, par_u, par_mv, mode=None, term_mode=None, diff_var=None, **kwargs): n_el, n_qp, dim, n_en, n_c = self.get_data_shape(par_u) return (n_el, 1, 1, 1), par_u.dtype
[docs]class LinearElasticTHTerm(THTerm): r""" Fading memory linear elastic (viscous) term. Can use derivatives. :Definition: .. math:: \int_{\Omega} \left [\int_0^t \Hcal_{ijkl}(t-\tau)\,e_{kl}(\ul{u}(\tau)) \difd{\tau} \right]\,e_{ij}(\ul{v}) :Arguments: - ts : :class:`TimeStepper` instance - material : :math:`\Hcal_{ijkl}(\tau)` - virtual : :math:`\ul{v}` - state : :math:`\ul{u}` """ name = 'dw_lin_elastic_th' arg_types = ('ts', 'material', 'virtual', 'state') arg_shapes = {'material' : '.: N, S, S', 'virtual' : ('D', 'state'), 'state' : 'D'} function = staticmethod(terms.dw_lin_elastic)
[docs] def get_fargs(self, ts, mats, virtual, state, mode=None, term_mode=None, diff_var=None, **kwargs): vg, _ = self.get_mapping(state) n_el, n_qp, dim, n_en, n_c = self.get_data_shape(state) if mode == 'weak': if diff_var is None: def iter_kernel(): for ii, mat in enumerate(mats): strain = self.get(state, 'cauchy_strain', step=-ii) mat = nm.tile(mat, (n_el, n_qp, 1, 1)) yield ii, (ts.dt, strain, mat, vg, 0) fargs = iter_kernel else: strain = nm.array([0], ndmin=4, dtype=nm.float64) mat = nm.tile(mats[0], (n_el, n_qp, 1, 1)) fargs = ts.dt, strain, mat, vg, 1 return fargs else: raise ValueError('unsupported evaluation mode in %s! (%s)' % (self.name, mode))
[docs]class LinearElasticETHTerm(ETHTerm): r""" This term has the same definition as dw_lin_elastic_th, but assumes an exponential approximation of the convolution kernel resulting in much higher efficiency. Can use derivatives. :Definition: .. math:: \int_{\Omega} \left [\int_0^t \Hcal_{ijkl}(t-\tau)\,e_{kl}(\ul{u}(\tau)) \difd{\tau} \right]\,e_{ij}(\ul{v}) :Arguments: - ts : :class:`TimeStepper` instance - material_0 : :math:`\Hcal_{ijkl}(0)` - material_1 : :math:`\exp(-\lambda \Delta t)` (decay at :math:`t_1`) - virtual : :math:`\ul{v}` - state : :math:`\ul{u}` """ name = 'dw_lin_elastic_eth' arg_types = ('ts', 'material_0', 'material_1', 'virtual', 'state') arg_shapes = {'material_0' : 'S, S', 'material_1' : '1, 1', 'virtual' : ('D', 'state'), 'state' : 'D'} function = staticmethod(terms.dw_lin_elastic)
[docs] def get_fargs(self, ts, mat0, mat1, virtual, state, mode=None, term_mode=None, diff_var=None, **kwargs): vg, _, key = self.get_mapping(state, return_key=True) if diff_var is None: strain = self.get(state, 'cauchy_strain') key += tuple(self.arg_names[ii] for ii in [1, 2, 4]) data = self.get_eth_data(key, state, mat1, strain) fargs = (ts.dt, data.history + data.values, mat0, vg, 0) else: aux = nm.array([0], ndmin=4, dtype=nm.float64) fargs = (ts.dt, aux, mat0, vg, 1) return fargs
[docs]class LinearPrestressTerm(Term): r""" Linear prestress term, with the prestress :math:`\sigma_{ij}` given either in the usual vector form exploiting symmetry: in 3D it has 6 components with the indices ordered as :math:`[11, 22, 33, 12, 13, 23]`, in 2D it has 3 components with the indices ordered as :math:`[11, 22, 12]`, or in the matrix (possibly non-symmetric) form. Can be evaluated. :Definition: .. math:: \int_{\Omega} \sigma_{ij} e_{ij}(\ul{v}) :Arguments 1: - material : :math:`\sigma_{ij}` - virtual : :math:`\ul{v}` :Arguments 2: - material : :math:`\sigma_{ij}` - parameter : :math:`\ul{u}` """ name = 'dw_lin_prestress' arg_types = (('material', 'virtual'), ('material', 'parameter')) arg_shapes = [{'material' : 'S, 1', 'virtual' : ('D', None), 'parameter' : 'D'}, {'material' : 'D, D'}] modes = ('weak', 'eval')
[docs] def get_fargs(self, mat, virtual, mode=None, term_mode=None, diff_var=None, **kwargs): vg, _ = self.get_mapping(virtual) sh = mat.shape is_nonsym = sh[2] == sh[3] == vg.dim and not(vg.dim == 1) if is_nonsym: mat = mat.reshape(sh[:2] + (vg.dim**2, 1)) if mode == 'weak': return mat, vg else: if is_nonsym: strain = self.get(virtual, 'grad').transpose((0,1,3,2)) nel, nqp, nr, nc = strain.shape strain = strain.reshape((nel, nqp, nr*nc, 1)) else: strain = self.get(virtual, 'cauchy_strain') fmode = {'eval' : 0, 'el_avg' : 1, 'qp' : 2}.get(mode, 1) return strain, mat, vg, fmode
[docs] def get_eval_shape(self, mat, virtual, mode=None, term_mode=None, diff_var=None, **kwargs): n_el, n_qp, dim, n_en, n_c = self.get_data_shape(virtual) if mode != 'qp': n_qp = 1 return (n_el, n_qp, 1, 1), virtual.dtype
[docs] def d_lin_prestress(self, out, strain, mat, vg, fmode): aux = dot_sequences(mat, strain, mode='ATB') if fmode == 2: out[:] = aux status = 0 else: status = vg.integrate(out, aux, fmode) return status
[docs] def set_arg_types(self): if self.mode == 'weak': self.function = terms.dw_lin_prestress else: self.function = self.d_lin_prestress
[docs]class LinearStrainFiberTerm(Term): r""" Linear (pre)strain fiber term with the unit direction vector :math:`\ul{d}`. :Definition: .. math:: \int_{\Omega} D_{ijkl} e_{ij}(\ul{v}) \left(d_k d_l\right) :Arguments: - material_1 : :math:`D_{ijkl}` - material_2 : :math:`\ul{d}` - virtual : :math:`\ul{v}` """ name = 'dw_lin_strain_fib' arg_types = ('material_1', 'material_2', 'virtual') arg_shapes = {'material_1' : 'S, S', 'material_2' : 'D, 1', 'virtual' : ('D', None)} function = staticmethod(terms.dw_lin_strain_fib)
[docs] def get_fargs(self, mat1, mat2, virtual, mode=None, term_mode=None, diff_var=None, **kwargs): vg, _ = self.get_mapping(virtual) omega = nm.empty(mat1.shape[:3] + (1,), dtype=nm.float64) for ii, (ir, ic) in enumerate(iter_sym(mat2.shape[2])): omega[..., ii, 0] = mat2[..., ir, 0] * mat2[..., ic, 0] return mat1, omega, vg
[docs]class CauchyStrainTerm(Term): r""" Evaluate Cauchy strain tensor. It is given in the usual vector form exploiting symmetry: in 3D it has 6 components with the indices ordered as :math:`[11, 22, 33, 12, 13, 23]`, in 2D it has 3 components with the indices ordered as :math:`[11, 22, 12]`. The last three (non-diagonal) components are doubled so that it is energetically conjugate to the Cauchy stress tensor with the same storage. Supports 'eval', 'el_avg' and 'qp' evaluation modes. :Definition: .. math:: \int_{\Omega} \ull{e}(\ul{w}) .. math:: \mbox{vector for } K \from \Ical_h: \int_{T_K} \ull{e}(\ul{w}) / \int_{T_K} 1 .. math:: \ull{e}(\ul{w})|_{qp} :Arguments: - parameter : :math:`\ul{w}` """ name = 'ev_cauchy_strain' arg_types = ('parameter',) arg_shapes = {'parameter' : 'D'}
[docs] @staticmethod def function(out, strain, vg, fmode): if fmode == 2: out[:] = strain status = 0 else: status = terms.de_cauchy_strain(out, strain, vg, fmode) return status
[docs] def get_fargs(self, parameter, mode=None, term_mode=None, diff_var=None, **kwargs): vg, _ = self.get_mapping(parameter) strain = self.get(parameter, 'cauchy_strain') fmode = {'eval' : 0, 'el_avg' : 1, 'qp' : 2}.get(mode, 1) return strain, vg, fmode
[docs] def get_eval_shape(self, parameter, mode=None, term_mode=None, diff_var=None, **kwargs): n_el, n_qp, dim, n_en, n_c = self.get_data_shape(parameter) if mode != 'qp': n_qp = 1 return (n_el, n_qp, dim * (dim + 1) // 2, 1), parameter.dtype
[docs]class CauchyStrainSTerm(CauchyStrainTerm): r""" Evaluate Cauchy strain tensor on a surface region. See :class:`CauchyStrainTerm`. Supports 'eval', 'el_avg' and 'qp' evaluation modes. :Definition: .. math:: \int_{\Gamma} \ull{e}(\ul{w}) .. math:: \mbox{vector for } K \from \Ical_h: \int_{T_K} \ull{e}(\ul{w}) / \int_{T_K} 1 .. math:: \ull{e}(\ul{w})|_{qp} :Arguments: - parameter : :math:`\ul{w}` """ name = 'ev_cauchy_strain_s' arg_types = ('parameter',) integration = 'surface_extra'
[docs]class CauchyStressTerm(Term): r""" Evaluate Cauchy stress tensor. It is given in the usual vector form exploiting symmetry: in 3D it has 6 components with the indices ordered as :math:`[11, 22, 33, 12, 13, 23]`, in 2D it has 3 components with the indices ordered as :math:`[11, 22, 12]`. Supports 'eval', 'el_avg' and 'qp' evaluation modes. :Definition: .. math:: \int_{\Omega} D_{ijkl} e_{kl}(\ul{w}) .. math:: \mbox{vector for } K \from \Ical_h: \int_{T_K} D_{ijkl} e_{kl}(\ul{w}) / \int_{T_K} 1 .. math:: D_{ijkl} e_{kl}(\ul{w})|_{qp} :Arguments: - material : :math:`D_{ijkl}` - parameter : :math:`\ul{w}` """ name = 'ev_cauchy_stress' arg_types = ('material', 'parameter') arg_shapes = {'material' : 'S, S', 'parameter' : 'D'}
[docs] @staticmethod def function(out, coef, strain, mat, vg, fmode): if fmode == 2: out[:] = dot_sequences(mat, strain) status = 0 else: status = terms.de_cauchy_stress(out, strain, mat, vg, fmode) if coef is not None: out *= coef return status
[docs] def get_fargs(self, mat, parameter, mode=None, term_mode=None, diff_var=None, **kwargs): vg, _ = self.get_mapping(parameter) strain = self.get(parameter, 'cauchy_strain') fmode = {'eval' : 0, 'el_avg' : 1, 'qp' : 2}.get(mode, 1) return None, strain, mat, vg, fmode
[docs] def get_eval_shape(self, mat, parameter, mode=None, term_mode=None, diff_var=None, **kwargs): n_el, n_qp, dim, n_en, n_c = self.get_data_shape(parameter) if mode != 'qp': n_qp = 1 return (n_el, n_qp, dim * (dim + 1) // 2, 1), parameter.dtype
[docs]class CauchyStressTHTerm(CauchyStressTerm, THTerm): r""" Evaluate fading memory Cauchy stress tensor. It is given in the usual vector form exploiting symmetry: in 3D it has 6 components with the indices ordered as :math:`[11, 22, 33, 12, 13, 23]`, in 2D it has 3 components with the indices ordered as :math:`[11, 22, 12]`. Supports 'eval', 'el_avg' and 'qp' evaluation modes. :Definition: .. math:: \int_{\Omega} \int_0^t \Hcal_{ijkl}(t-\tau)\,e_{kl}(\ul{w}(\tau)) \difd{\tau} .. math:: \mbox{vector for } K \from \Ical_h: \int_{T_K} \int_0^t \Hcal_{ijkl}(t-\tau)\,e_{kl}(\ul{w}(\tau)) \difd{\tau} / \int_{T_K} 1 .. math:: \int_0^t \Hcal_{ijkl}(t-\tau)\,e_{kl}(\ul{w}(\tau)) \difd{\tau}|_{qp} :Arguments: - ts : :class:`TimeStepper` instance - material : :math:`\Hcal_{ijkl}(\tau)` - parameter : :math:`\ul{w}` """ name = 'ev_cauchy_stress_th' arg_types = ('ts', 'material', 'parameter') arg_shapes = {'material' : '.: N, S, S', 'parameter' : 'D'}
[docs] def get_fargs(self, ts, mats, state, mode=None, term_mode=None, diff_var=None, **kwargs): vg, _ = self.get_mapping(state) n_el, n_qp, dim, n_en, n_c = self.get_data_shape(state) fmode = {'eval' : 0, 'el_avg' : 1, 'qp' : 2}.get(mode, 1) def iter_kernel(): for ii, mat in enumerate(mats): strain = self.get(state, 'cauchy_strain', step=-ii) mat = nm.tile(mat, (n_el, n_qp, 1, 1)) yield ii, (ts.dt, strain, mat, vg, fmode) return iter_kernel
[docs] def get_eval_shape(self, ts, mats, parameter, mode=None, term_mode=None, diff_var=None, **kwargs): out = CauchyStressTerm.get_eval_shape(self, mats, parameter, mode, term_mode, diff_var, **kwargs) return out
[docs]class CauchyStressETHTerm(CauchyStressTerm, ETHTerm): r""" Evaluate fading memory Cauchy stress tensor. It is given in the usual vector form exploiting symmetry: in 3D it has 6 components with the indices ordered as :math:`[11, 22, 33, 12, 13, 23]`, in 2D it has 3 components with the indices ordered as :math:`[11, 22, 12]`. Assumes an exponential approximation of the convolution kernel resulting in much higher efficiency. Supports 'eval', 'el_avg' and 'qp' evaluation modes. :Definition: .. math:: \int_{\Omega} \int_0^t \Hcal_{ijkl}(t-\tau)\,e_{kl}(\ul{w}(\tau)) \difd{\tau} .. math:: \mbox{vector for } K \from \Ical_h: \int_{T_K} \int_0^t \Hcal_{ijkl}(t-\tau)\,e_{kl}(\ul{w}(\tau)) \difd{\tau} / \int_{T_K} 1 .. math:: \int_0^t \Hcal_{ijkl}(t-\tau)\,e_{kl}(\ul{w}(\tau)) \difd{\tau}|_{qp} :Arguments: - ts : :class:`TimeStepper` instance - material_0 : :math:`\Hcal_{ijkl}(0)` - material_1 : :math:`\exp(-\lambda \Delta t)` (decay at :math:`t_1`) - parameter : :math:`\ul{w}` """ name = 'ev_cauchy_stress_eth' arg_types = ('ts', 'material_0', 'material_1', 'parameter') arg_shapes = {'material_0' : 'S, S', 'material_1' : '1, 1', 'parameter' : 'D'}
[docs] def get_fargs(self, ts, mat0, mat1, state, mode=None, term_mode=None, diff_var=None, **kwargs): vg, _, key = self.get_mapping(state, return_key=True) strain = self.get(state, 'cauchy_strain') key += tuple(self.arg_names[1:]) data = self.get_eth_data(key, state, mat1, strain) fmode = {'eval' : 0, 'el_avg' : 1, 'qp' : 2}.get(mode, 1) return ts.dt, data.history + data.values, mat0, vg, fmode
[docs] def get_eval_shape(self, ts, mat0, mat1, parameter, mode=None, term_mode=None, diff_var=None, **kwargs): out = CauchyStressTerm.get_eval_shape(self, mat0, parameter, mode, term_mode, diff_var, **kwargs) return out
[docs]class NonsymElasticTerm(Term): r""" Elasticity term with non-symmetric gradient. The indices of matrix :math:`D_{ijkl}` are ordered as :math:`[11, 12, 13, 21, 22, 23, 31, 32, 33]` in 3D and as :math:`[11, 12, 21, 22]` in 2D. :Definition: .. math:: \int_{\Omega} \ull{D} \nabla\ul{u} : \nabla\ul{v} :Arguments 1: - material : :math:`\ull{D}` - virtual : :math:`\ul{v}` - state : :math:`\ul{u}` :Arguments 2: - material : :math:`\ull{D}` - parameter_1 : :math:`\ul{w}` - parameter_2 : :math:`\ul{u}` """ name = 'dw_nonsym_elastic' arg_types = (('material', 'virtual', 'state'), ('material', 'parameter_1', 'parameter_2')) arg_shapes = {'material' : 'D2, D2', 'virtual' : ('D', 'state'), 'state' : 'D', 'parameter_1' : 'D', 'parameter_2' : 'D'} modes = ('weak', 'eval') geometries = ['2_3', '2_4', '3_4', '3_8']
[docs] def get_fargs(self, mat, virtual, state, mode=None, term_mode=None, diff_var=None, **kwargs): vg, _ = self.get_mapping(state) if mode == 'weak': if diff_var is None: grad = self.get(state, 'grad').transpose((0,1,3,2)) nel, nqp, nr, nc = grad.shape grad = grad.reshape((nel,nqp,nr*nc,1)) fmode = 0 else: grad = nm.array([0], ndmin=4, dtype=nm.float64) fmode = 1 return grad, mat, vg, fmode elif mode == 'eval': grad1 = self.get(virtual, 'grad').transpose((0,1,3,2)) grad2 = self.get(state, 'grad').transpose((0,1,3,2)) nel, nqp, nr, nc = grad1.shape return 1.0,\ grad1.reshape((nel,nqp,nr*nc,1)),\ grad2.reshape((nel,nqp,nr*nc,1)),\ mat, vg else: raise ValueError('unsupported evaluation mode in %s! (%s)' % (self.name, mode))
[docs] def get_eval_shape(self, mat, virtual, state, mode=None, term_mode=None, diff_var=None, **kwargs): n_el, n_qp, dim, n_en, n_c = self.get_data_shape(state) return (n_el, 1, 1, 1), state.dtype
[docs] def set_arg_types(self): if self.mode == 'weak': self.function = terms.dw_nonsym_elastic else: self.function = terms.d_lin_elastic
def _build_wave_strain_op(vec, bf): dim = len(vec) if dim == 2: n0, n1 = vec nmat = nm.array([[n0, 0], [0, n1], [n1, n0]], dtype=nm.float64) else: n0, n1, n2 = vec nmat = nm.array([[n0, 0, 0], [0, n1, 0], [0, 0, n2], [n1, n0, 0], [n2, 0, n0], [0, n2, n1]], dtype=nm.float64) out = nm.einsum('ik,cqkj->cqij', nmat, bf) return out from sfepy.base.compat import block def _build_cauchy_strain_op(bfg): dim = bfg.shape[2] if dim == 2: g1, g2 = bfg[..., 0:1, :], bfg[..., 1:2, :] zz = nm.zeros_like(g1) out = block([[g1, zz], [zz, g2], [g2, g1]]) else: g1, g2, g3 = bfg[..., 0:1, :], bfg[..., 1:2, :], bfg[..., 2:3, :] zz = nm.zeros_like(g1) out = block([[g1, zz, zz], [zz, g2, zz], [zz, zz, g3], [g2, g1, zz], [g3, zz, g1], [zz, g3, g2]]) return out
[docs]class ElasticWaveTerm(Term): r""" Elastic dispersion term involving the wave strain :math:`g_{ij}`, :math:`g_{ij}(\ul{u}) = \frac{1}{2}(u_i \kappa_j + \kappa_i u_j)`, with the wave vector :math:`\ul{\kappa}`. :math:`D_{ijkl}` is given in the usual matrix form exploiting symmetry: in 3D it is :math:`6\times6` with the indices ordered as :math:`[11, 22, 33, 12, 13, 23]`, in 2D it is :math:`3\times3` with the indices ordered as :math:`[11, 22, 12]`. :Definition: .. math:: \int_{\Omega} D_{ijkl}\ g_{ij}(\ul{v}) g_{kl}(\ul{u}) :Arguments: - material_1 : :math:`D_{ijkl}` - material_2 : :math:`\ul{\kappa}` - virtual : :math:`\ul{v}` - state : :math:`\ul{u}` """ name = 'dw_elastic_wave' arg_types = ('material_1', 'material_2', 'virtual', 'state') arg_shapes = {'material_1' : 'S, S', 'material_2' : '.: D', 'virtual' : ('D', 'state'), 'state' : 'D'} geometries = ['2_3', '2_4', '3_4', '3_8']
[docs] @staticmethod def function(out, out_qp, geo, fmode): status = geo.integrate(out, out_qp) return status
[docs] def get_fargs(self, mat, kappa, virtual, state, mode=None, term_mode=None, diff_var=None, **kwargs): from sfepy.discrete.variables import create_adof_conn, expand_basis geo, _ = self.get_mapping(state) n_el, n_qp, dim, n_en, n_c = self.get_data_shape(virtual) ebf = expand_basis(geo.bf, dim) gmat = _build_wave_strain_op(kappa, ebf) if diff_var is None: econn = state.field.get_econn('volume', self.region) adc = create_adof_conn(nm.arange(state.n_dof, dtype=nm.int32), econn, n_c, 0) vals = state()[adc] # Same as nm.einsum('qij,cj->cqi', gmat[0], vals)[..., None] aux = dot_sequences(gmat, vals[:, None, :, None]) out_qp = dot_sequences(gmat, dot_sequences(mat, aux), 'ATB') fmode = 0 else: out_qp = dot_sequences(gmat, dot_sequences(mat, gmat), 'ATB') fmode = 1 return out_qp, geo, fmode
[docs]class ElasticWaveCauchyTerm(Term): r""" Elastic dispersion term involving the wave strain :math:`g_{ij}`, :math:`g_{ij}(\ul{u}) = \frac{1}{2}(u_i \kappa_j + \kappa_i u_j)`, with the wave vector :math:`\ul{\kappa}` and the elastic strain :math:`e_{ij}`. :math:`D_{ijkl}` is given in the usual matrix form exploiting symmetry: in 3D it is :math:`6\times6` with the indices ordered as :math:`[11, 22, 33, 12, 13, 23]`, in 2D it is :math:`3\times3` with the indices ordered as :math:`[11, 22, 12]`. :Definition: .. math:: \int_{\Omega} D_{ijkl}\ g_{ij}(\ul{v}) e_{kl}(\ul{u}) \;, \int_{\Omega} D_{ijkl}\ g_{ij}(\ul{u}) e_{kl}(\ul{v}) :Arguments 1: - material_1 : :math:`D_{ijkl}` - material_2 : :math:`\ul{\kappa}` - virtual : :math:`\ul{v}` - state : :math:`\ul{u}` :Arguments 2: - material_1 : :math:`D_{ijkl}` - material_2 : :math:`\ul{\kappa}` - state : :math:`\ul{u}` - virtual : :math:`\ul{v}` """ name = 'dw_elastic_wave_cauchy' arg_types = (('material_1', 'material_2', 'virtual', 'state'), ('material_1', 'material_2', 'state', 'virtual')) arg_shapes = {'material_1' : 'S, S', 'material_2' : '.: D', 'virtual' : ('D', 'state'), 'state' : 'D'} geometries = ['2_3', '2_4', '3_4', '3_8'] modes = ('ge', 'eg')
[docs] @staticmethod def function(out, out_qp, geo, fmode): status = geo.integrate(out, out_qp) return status
[docs] def get_fargs(self, mat, kappa, gvar, evar, mode=None, term_mode=None, diff_var=None, **kwargs): from sfepy.discrete.variables import create_adof_conn, expand_basis geo, _ = self.get_mapping(evar) n_el, n_qp, dim, n_en, n_c = self.get_data_shape(gvar) ebf = expand_basis(geo.bf, dim) gmat = _build_wave_strain_op(kappa, ebf) emat = _build_cauchy_strain_op(geo.bfg) if diff_var is None: avar = evar if self.mode == 'ge' else gvar econn = avar.field.get_econn('volume', self.region) adc = create_adof_conn(nm.arange(avar.n_dof, dtype=nm.int32), econn, n_c, 0) vals = avar()[adc] if self.mode == 'ge': # Same as aux = self.get(avar, 'cauchy_strain'), aux = dot_sequences(emat, vals[:, None, :, None]) out_qp = dot_sequences(gmat, dot_sequences(mat, aux), 'ATB') else: aux = dot_sequences(gmat, vals[:, None, :, None]) out_qp = dot_sequences(emat, dot_sequences(mat, aux), 'ATB') fmode = 0 else: if self.mode == 'ge': out_qp = dot_sequences(gmat, dot_sequences(mat, emat), 'ATB') else: out_qp = dot_sequences(emat, dot_sequences(mat, gmat), 'ATB') fmode = 1 return out_qp, geo, fmode