Source code for sfepy.terms.terms_elastic

import numpy as nm

from sfepy.base.base import use_method_with_name, assert_
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 check_shapes(self, mat, virtual, state): n_el, n_qp, dim, n_en, n_c = self.get_data_shape(state) sym = (dim + 1) * dim / 2 assert_(mat.shape == (n_el, n_qp, sym, sym))
[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 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'} function = terms.d_lin_elastic @staticmethod
[docs] def op_dv(vgrad): nel, nlev, dim, _ = vgrad.shape sd = nm.zeros((nel, nlev, dim**2, dim**2), dtype=vgrad.dtype) if dim == 1: sd[...] = vgrad[:,:] elif dim == 2: sd[:,:,0:2,0:2] = vgrad[:,:] sd[:,:,2:4,2:4] = vgrad[:,:] elif dim == 3: sd[:,:,0:3,0:3] = vgrad[:,:] sd[:,:,3:6,3:6] = vgrad[:,:] sd[:,:,6:9,6:9] = vgrad[:,:] else: exit('not yet implemented!') return sd
[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').transpose((0,1,3,2)) grad_u = self.get(par_u, 'grad').transpose((0,1,3,2)) nel, nqp, nr, nc = grad_u.shape strain_w = grad_w.reshape((nel, nqp, nr * nc, 1)) strain_u = grad_u.reshape((nel, nqp, nr * nc, 1)) mat_map = {1: nm.array([0]), 3: nm.array([0, 2, 2, 1]), 6: nm.array([0, 3, 4, 3, 1, 5, 4, 5, 2])} mmap = mat_map[mat.shape[-1]] mat_ns = mat[nm.ix_(nm.arange(nel), nm.arange(nqp), mmap, mmap)] div_mv = self.get(par_mv, 'div') grad_mv = self.get(par_mv, 'grad') opd_mv = self.op_dv(grad_mv) aux = dot_sequences(mat_ns, opd_mv) mat_mv = mat_ns * div_mv - (aux + aux.transpose((0,1,3,2))) return 1.0, strain_w, strain_u, mat_mv, 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 LinearElasticIsotropicTerm(Term): 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}` """ name = 'dw_lin_elastic_iso' arg_types = ('material_1', 'material_2', 'virtual', 'state') arg_shapes = {'material_1' : '1, 1', 'material_2' : '1, 1', 'virtual' : ('D', 'state'), 'state' : 'D'} function = staticmethod(terms.dw_lin_elastic_iso)
[docs] def check_shapes(self, lam, mu, virtual, state): n_el, n_qp, dim, n_en, n_c = self.get_data_shape(state) assert_(lam.shape == (n_el, n_qp, 1, 1)) assert_(mu.shape == (n_el, n_qp, 1, 1))
[docs] def get_fargs(self, lam, mu, 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 strain, lam, mu, vg, fmode else: raise ValueError('unsupported evaluation mode in %s! (%s)' % (self.name, mode))
[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') 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') 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 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]`. 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'} modes = ('weak', 'eval')
[docs] def check_shapes(self, mat, virtual): n_el, n_qp, dim, n_en, n_c = self.get_data_shape(virtual) sym = (dim + 1) * dim / 2 assert_(mat.shape == (n_el, n_qp, sym, 1))
[docs] def get_fargs(self, mat, virtual, mode=None, term_mode=None, diff_var=None, **kwargs): vg, _ = self.get_mapping(virtual) if mode == 'weak': return mat, vg 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 check_shapes(self, mat1, mat2, virtual): n_el, n_qp, dim, n_en, n_c = self.get_data_shape(virtual) sym = (dim + 1) * dim / 2 assert_(mat1.shape == (n_el, n_qp, sym, sym)) assert_(mat2.shape == (n_el, n_qp, dim, 1))
[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'} @staticmethod
[docs] 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'} @staticmethod
[docs] 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 = {}
[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 = {}
[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