sfepy.terms.terms_elastic module¶
- class sfepy.terms.terms_elastic.CauchyStrainTerm(name, arg_str, integral, region, **kwargs)[source]¶
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 [11, 22, 33, 12, 13, 23], in 2D it has 3 components with the indices ordered as [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:
\int_{\cal{D}} \ull{e}(\ul{w})
- Call signature:
ev_cauchy_strain
(parameter)
- Arguments:
parameter : \ul{w}
- arg_shapes = {'parameter': 'D'}¶
- arg_types = ('parameter',)¶
- integration = ('cell', 'facet_extra')¶
- name = 'ev_cauchy_strain'¶
- class sfepy.terms.terms_elastic.CauchyStressETHTerm(name, arg_str, integral, region, **kwargs)[source]¶
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 [11, 22, 33, 12, 13, 23], in 2D it has 3 components with the indices ordered as [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:
\int_{\Omega} \int_0^t \Hcal_{ijkl}(t-\tau)\,e_{kl}(\ul{w}(\tau)) \difd{\tau}
- Call signature:
ev_cauchy_stress_eth
(ts, material_0, material_1, parameter)
- Arguments:
ts :
TimeStepper
instancematerial_0 : \Hcal_{ijkl}(0)
material_1 : \exp(-\lambda \Delta t) (decay at t_1)
parameter : \ul{w}
- arg_shapes = {'material_0': 'S, S', 'material_1': '1, 1', 'parameter': 'D'}¶
- arg_types = ('ts', 'material_0', 'material_1', 'parameter')¶
- get_eval_shape(ts, mat0, mat1, parameter, mode=None, term_mode=None, diff_var=None, **kwargs)[source]¶
- name = 'ev_cauchy_stress_eth'¶
- class sfepy.terms.terms_elastic.CauchyStressTHTerm(name, arg_str, integral, region, **kwargs)[source]¶
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 [11, 22, 33, 12, 13, 23], in 2D it has 3 components with the indices ordered as [11, 22, 12].
Supports ‘eval’, ‘el_avg’ and ‘qp’ evaluation modes.
- Definition:
\int_{\Omega} \int_0^t \Hcal_{ijkl}(t-\tau)\,e_{kl}(\ul{w}(\tau)) \difd{\tau}
- Call signature:
ev_cauchy_stress_th
(ts, material, parameter)
- Arguments:
ts :
TimeStepper
instancematerial : \Hcal_{ijkl}(\tau)
parameter : \ul{w}
- arg_shapes = {'material': '.: N, S, S', 'parameter': 'D'}¶
- arg_types = ('ts', 'material', 'parameter')¶
- name = 'ev_cauchy_stress_th'¶
- class sfepy.terms.terms_elastic.CauchyStressTerm(name, arg_str, integral, region, **kwargs)[source]¶
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 [11, 22, 33, 12, 13, 23], in 2D it has 3 components with the indices ordered as [11, 22, 12].
Supports ‘eval’, ‘el_avg’ and ‘qp’ evaluation modes.
- Definition:
\int_{\cal{D}} D_{ijkl} e_{kl}(\ul{w})
- Call signature:
ev_cauchy_stress
(material, parameter)
- Arguments:
material : D_{ijkl}
parameter : \ul{w}
- arg_shapes = {'material': 'S, S', 'parameter': 'D'}¶
- arg_types = ('material', 'parameter')¶
- integration = ('cell', 'facet_extra')¶
- name = 'ev_cauchy_stress'¶
- class sfepy.terms.terms_elastic.ElasticWaveCauchyTerm(name, arg_str, integral, region, **kwargs)[source]¶
Elastic dispersion term involving the wave strain g_{ij}, g_{ij}(\ul{u}) = \frac{1}{2}(u_i \kappa_j + \kappa_i u_j), with the wave vector \ul{\kappa} and the elastic strain e_{ij}. D_{ijkl} is given in the usual matrix form exploiting symmetry: in 3D it is 6\times6 with the indices ordered as [11, 22, 33, 12, 13, 23], in 2D it is 3\times3 with the indices ordered as [11, 22, 12].
- Definition:
\int_{\Omega} D_{ijkl}\ g_{ij}(\ul{v}) e_{kl}(\ul{u})\\ \int_{\Omega} D_{ijkl}\ g_{ij}(\ul{u}) e_{kl}(\ul{v})
- Call signature:
dw_elastic_wave_cauchy
(material_1, material_2, virtual, state)
(material_1, material_2, state, virtual)
- Arguments 1:
material_1 : D_{ijkl}
material_2 : \ul{\kappa}
virtual : \ul{v}
state : \ul{u}
- Arguments 2:
material_1 : D_{ijkl}
material_2 : \ul{\kappa}
state : \ul{u}
virtual : \ul{v}
- arg_shapes = {'material_1': 'S, S', 'material_2': '.: D', 'state': 'D', 'virtual': ('D', 'state')}¶
- arg_types = (('material_1', 'material_2', 'virtual', 'state'), ('material_1', 'material_2', 'state', 'virtual'))¶
- geometries = ['2_3', '2_4', '3_4', '3_8']¶
- modes = ('ge', 'eg')¶
- name = 'dw_elastic_wave_cauchy'¶
- class sfepy.terms.terms_elastic.ElasticWaveTerm(name, arg_str, integral, region, **kwargs)[source]¶
Elastic dispersion term involving the wave strain g_{ij}, g_{ij}(\ul{u}) = \frac{1}{2}(u_i \kappa_j + \kappa_i u_j), with the wave vector \ul{\kappa}. D_{ijkl} is given in the usual matrix form exploiting symmetry: in 3D it is 6\times6 with the indices ordered as [11, 22, 33, 12, 13, 23], in 2D it is 3\times3 with the indices ordered as [11, 22, 12].
- Definition:
\int_{\Omega} D_{ijkl}\ g_{ij}(\ul{v}) g_{kl}(\ul{u})
- Call signature:
dw_elastic_wave
(material_1, material_2, virtual, state)
- Arguments:
material_1 : D_{ijkl}
material_2 : \ul{\kappa}
virtual : \ul{v}
state : \ul{u}
- arg_shapes = {'material_1': 'S, S', 'material_2': '.: D', 'state': 'D', 'virtual': ('D', 'state')}¶
- arg_types = ('material_1', 'material_2', 'virtual', 'state')¶
- geometries = ['2_3', '2_4', '3_4', '3_8']¶
- name = 'dw_elastic_wave'¶
- class sfepy.terms.terms_elastic.LinearDRotSpringTerm(name, arg_str, integral, region, **kwargs)[source]¶
Linear spring element with the stiffness transformed into the element direction.
- Definition:
f^{(i)}_k = -f^{(j)}_k = K_{kl} (u^{(j)}_l - u^{(i)}_l)\\ \quad \forall \mbox{ elements } T_K^{i,j}\\ \mbox{ in a region connecting nodes } i, j
- Call signature:
dw_lin_dspring_rot
(opt_material, material, virtual, state)
- Arguments:
opt_material : \ul{d}
material : \ul{k}
virtual: \ul{v}
state: \ul{u}
Stiffness matrix \ul{K} = \ul{T(\ul{d})}^T \ul{K(\ul{k})} \ul{T(\ul{d})} is defined by 6 components \ul{k} = [k_{u1}, k_{u2}, k_{u3}, k_{r1}, k_{r2}, k_{r3}] in 3D and by 3 components \ul{k} = [k_{u1}, k_{u2}, k_{r1}], where k_{ui} is the stiffness for the displacement DOF and r_{ui} is for the rotational DOF. Note that the components of \ul{k} are in the local coordinates system specified by a given direction \ul{d} or by the vector \ul{d} = \ul{x}^{(j)} - \ul{x}^{(i)} for non-coincidental end nodes. The stiffness parameter \ul{K} can also be defined as a 6x6 matrix in 3D or a 3x3 matrix in 2D.
- arg_shapes = [{'material': 'S, 1', 'opt_material': 'D, 1', 'state': 'S', 'virtual': ('S', 'state')}, {'material': 'S, S'}, {'opt_material': None}]¶
- arg_types = ('opt_material', 'material', 'virtual', 'state')¶
- name = 'dw_lin_dspring_rot'¶
- class sfepy.terms.terms_elastic.LinearDSpringTerm(name, arg_str, integral, region, **kwargs)[source]¶
Linear spring element with the stiffness transformed into the element direction.
- Definition:
f^{(i)}_k = -f^{(j)}_k = K_{kl} (u^{(j)}_l - u^{(i)}_l)\\ \quad \forall \mbox{ elements } T_K^{i,j}\\ \mbox{ in a region connecting nodes } i, j
- Call signature:
dw_lin_dspring
(opt_material, material, virtual, state)
- Arguments:
opt_material : \ul{d}
material : \ul{k}
virtual: \ul{v}
state: \ul{u}
Stiffness matrix \ul{K} = \ul{T(\ul{d})}^T \ul{K(\ul{k})} \ul{T(\ul{d})} is defined by 6 components \ul{k} = [k_{u1}, k_{u2}, k_{u3}, k_{r1}, k_{r2}, k_{r3}] in 3D and by 3 components \ul{k} = [k_{u1}, k_{u2}, k_{r1}], where k_{ui} is the stiffness for the displacement DOF and r_{ui} is for the rotational DOF. Note that the components of \ul{k} are in the local coordinates system specified by a given direction \ul{d} or by the vector \ul{d} = \ul{x}^{(j)} - \ul{x}^{(i)} for non-coincidental end nodes. The stiffness parameter \ul{K} can also be defined as a 6x6 matrix in 3D or a 3x3 matrix in 2D.
- arg_shapes = [{'material': 'D, 1', 'opt_material': 'D, 1', 'state': 'D', 'virtual': ('D', 'state')}, {'material': 'D, D'}, {'opt_material': None}]¶
- arg_types = ('opt_material', 'material', 'virtual', 'state')¶
- geometries = ['1_2', '2_1_2', '3_1_2']¶
- integration_order = 0¶
- name = 'dw_lin_dspring'¶
- class sfepy.terms.terms_elastic.LinearElasticETHTerm(name, arg_str, integral, region, **kwargs)[source]¶
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:
\int_{\Omega} \left [\int_0^t \Hcal_{ijkl}(t-\tau)\,e_{kl}(\ul{u}(\tau)) \difd{\tau} \right]\,e_{ij}(\ul{v})
- Call signature:
dw_lin_elastic_eth
(ts, material_0, material_1, virtual, state)
- Arguments:
ts :
TimeStepper
instancematerial_0 : \Hcal_{ijkl}(0)
material_1 : \exp(-\lambda \Delta t) (decay at t_1)
virtual : \ul{v}
state : \ul{u}
- arg_shapes = {'material_0': 'S, S', 'material_1': '1, 1', 'state': 'D', 'virtual': ('D', 'state')}¶
- arg_types = ('ts', 'material_0', 'material_1', 'virtual', 'state')¶
- static function(out, coef, strain, mtx_d, cmap, is_diff)¶
- get_fargs(ts, mat0, mat1, virtual, state, mode=None, term_mode=None, diff_var=None, **kwargs)[source]¶
- name = 'dw_lin_elastic_eth'¶
- class sfepy.terms.terms_elastic.LinearElasticIsotropicTerm(name, arg_str, integral, region, **kwargs)[source]¶
Isotropic linear elasticity term.
- Definition:
\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}
- Call signature:
dw_lin_elastic_iso
(material_1, material_2, virtual, state)
(material_1, material_2, parameter_1, parameter_2)
- Arguments:
material_1: \lambda
material_2: \mu
virtual/parameter_1: \ul{v}
state/parameter_2: \ul{u}
- arg_shapes = {'material_1': '1, 1', 'material_2': '1, 1', 'parameter_1': 'D', 'parameter_2': 'D', 'state': 'D', 'virtual': ('D', 'state')}¶
- arg_types = (('material_1', 'material_2', 'virtual', 'state'), ('material_1', 'material_2', 'parameter_1', 'parameter_2'))¶
- geometries = ['2_3', '2_4', '3_4', '3_8']¶
- get_eval_shape(mat1, mat2, virtual, state, mode=None, term_mode=None, diff_var=None, **kwargs)[source]¶
- name = 'dw_lin_elastic_iso'¶
- class sfepy.terms.terms_elastic.LinearElasticTHTerm(name, arg_str, integral, region, **kwargs)[source]¶
Fading memory linear elastic (viscous) term. Can use derivatives.
- Definition:
\int_{\Omega} \left [\int_0^t \Hcal_{ijkl}(t-\tau)\,e_{kl}(\ul{u}(\tau)) \difd{\tau} \right]\,e_{ij}(\ul{v})
- Call signature:
dw_lin_elastic_th
(ts, material, virtual, state)
- Arguments:
ts :
TimeStepper
instancematerial : \Hcal_{ijkl}(\tau)
virtual : \ul{v}
state : \ul{u}
- arg_shapes = {'material': '.: N, S, S', 'state': 'D', 'virtual': ('D', 'state')}¶
- arg_types = ('ts', 'material', 'virtual', 'state')¶
- static function(out, coef, strain, mtx_d, cmap, is_diff)¶
- name = 'dw_lin_elastic_th'¶
- class sfepy.terms.terms_elastic.LinearElasticTerm(name, arg_str, integral, region, **kwargs)[source]¶
General linear elasticity term, with D_{ijkl} given in the usual matrix form exploiting symmetry: in 3D it is 6\times6 with the indices ordered as [11, 22, 33, 12, 13, 23], in 2D it is 3\times3 with the indices ordered as [11, 22, 12]. Can be evaluated. Can use derivatives.
- Definition:
\int_{\Omega} D_{ijkl}\ e_{ij}(\ul{v}) e_{kl}(\ul{u})
- Call signature:
dw_lin_elastic
(material, virtual, state)
(material, parameter_1, parameter_2)
- Arguments 1:
material : D_{ijkl}
virtual : \ul{v}
state : \ul{u}
- Arguments 2:
material : D_{ijkl}
parameter_1 : \ul{w}
parameter_2 : \ul{u}
- arg_shapes = {'material': 'S, S', 'parameter_1': 'D', 'parameter_2': 'D', 'state': 'D', 'virtual': ('D', 'state')}¶
- arg_types = (('material', 'virtual', 'state'), ('material', 'parameter_1', 'parameter_2'))¶
- modes = ('weak', 'eval')¶
- name = 'dw_lin_elastic'¶
- class sfepy.terms.terms_elastic.LinearPrestressTerm(name, arg_str, integral, region, **kwargs)[source]¶
Linear prestress term, with the prestress \sigma_{ij} given either in the usual vector form exploiting symmetry: in 3D it has 6 components with the indices ordered as [11, 22, 33, 12, 13, 23], in 2D it has 3 components with the indices ordered as [11, 22, 12], or in the matrix (possibly non-symmetric) form. Can be evaluated.
- Definition:
\int_{\Omega} \sigma_{ij} e_{ij}(\ul{v})
- Call signature:
dw_lin_prestress
(material, virtual)
(material, parameter)
- Arguments 1:
material : \sigma_{ij}
virtual : \ul{v}
- Arguments 2:
material : \sigma_{ij}
parameter : \ul{u}
- arg_shapes = [{'material': 'S, 1', 'parameter': 'D', 'virtual': ('D', None)}, {'material': 'D, D'}]¶
- arg_types = (('material', 'virtual'), ('material', 'parameter'))¶
- modes = ('weak', 'eval')¶
- name = 'dw_lin_prestress'¶
- class sfepy.terms.terms_elastic.LinearSpringTerm(name, arg_str, integral, region, **kwargs)[source]¶
Linear spring element.
- Definition:
\ul{f}^{(i)} = - \ul{f}^{(j)} = k (\ul{u}^{(j)} - \ul{u}^{(i)})\\ \quad \forall \mbox{ elements } T_K^{i,j}\\ \mbox{ in a region connecting nodes } i, j
- Call signature:
dw_lin_spring
(material, virtual, state)
- Arguments 1:
material : k
virtual : \ul{v}
state : \ul{u}
- arg_shapes = {'material': '1, 1', 'state': 'D', 'virtual': ('D', 'state')}¶
- arg_types = ('material', 'virtual', 'state')¶
- geometries = ['1_2', '2_1_2', '3_1_2']¶
- integration_order = 0¶
- name = 'dw_lin_spring'¶
- class sfepy.terms.terms_elastic.LinearStrainFiberTerm(name, arg_str, integral, region, **kwargs)[source]¶
Linear (pre)strain fiber term with the unit direction vector \ul{d}.
- Definition:
\int_{\Omega} D_{ijkl} e_{ij}(\ul{v}) \left(d_k d_l\right)
- Call signature:
dw_lin_strain_fib
(material_1, material_2, virtual)
- Arguments:
material_1 : D_{ijkl}
material_2 : \ul{d}
virtual : \ul{v}
- arg_shapes = {'material_1': 'S, S', 'material_2': 'D, 1', 'virtual': ('D', None)}¶
- arg_types = ('material_1', 'material_2', 'virtual')¶
- static function(out, mtx_d, mat, cmap)¶
- name = 'dw_lin_strain_fib'¶
- class sfepy.terms.terms_elastic.LinearTrussInternalForceTerm(name, arg_str, integral, region, **kwargs)[source]¶
Evaluate internal force in the element direction. To be used with ‘el_avg’ or ‘qp’ evaluation modes which give the same results. The material parameter EA is equal to Young modulus times element coss-section. The internal force is given by F^{(i)} = -F^{(j)} = EA / l (U^{(j)} - U^{(i)}), where l is the element length and U, F are the nodal displacements and the nodal forces in the element direction.
- Definition:
F = EA / l (U^{(j)} - U^{(i)})\\ \quad \forall \mbox{ elements } T_K^{i,j}\\ \mbox{ in a region connecting nodes } i, j
- Call signature:
ev_lin_truss_force
(material, parameter)
- Arguments:
material : EA
parameter : \ul{w}
- arg_shapes = {'material': '1, 1', 'parameter': 'D'}¶
- arg_types = ('material', 'parameter')¶
- geometries = ['1_2', '2_1_2', '3_1_2']¶
- integration_order = 0¶
- name = 'ev_lin_truss_force'¶
- class sfepy.terms.terms_elastic.LinearTrussTerm(name, arg_str, integral, region, **kwargs)[source]¶
Evaluate internal force in the element direction. To be used with ‘el_avg’ or ‘qp’ evaluation modes which give the same results. The material parameter EA is equal to Young modulus times element coss-section. The internal force is given by F^{(i)} = -F^{(j)} = EA / l (U^{(j)} - U^{(i)}), where l is the element length and U, F are the nodal displacements and the nodal forces in the element direction.
- Definition:
F^{(i)} = -F^{(j)} = EA / l (U^{(j)} - U^{(i)})\\ \quad \forall \mbox{ elements } T_K^{i,j}\\ \mbox{ in a region connecting nodes } i, j
- Call signature:
dw_lin_truss
(material, virtual, state)
- Arguments:
material : EA
parameter : \ul{w}
- arg_shapes = {'material': '1, 1', 'state': 'D', 'virtual': ('D', 'state')}¶
- arg_types = ('material', 'virtual', 'state')¶
- geometries = ['1_2', '2_1_2', '3_1_2']¶
- integration_order = 0¶
- name = 'dw_lin_truss'¶
- class sfepy.terms.terms_elastic.NonsymElasticTerm(name, arg_str, integral, region, **kwargs)[source]¶
Elasticity term with non-symmetric gradient. The indices of matrix D_{ijkl} are ordered as [11, 12, 13, 21, 22, 23, 31, 32, 33] in 3D and as [11, 12, 21, 22] in 2D.
- Definition:
\int_{\Omega} \ull{D} \nabla\ul{u} : \nabla\ul{v}
- Call signature:
dw_nonsym_elastic
(material, virtual, state)
(material, parameter_1, parameter_2)
- Arguments 1:
material : \ull{D}
virtual : \ul{v}
state : \ul{u}
- Arguments 2:
material : \ull{D}
parameter_1 : \ul{w}
parameter_2 : \ul{u}
- arg_shapes = {'material': 'D2, D2', 'parameter_1': 'D', 'parameter_2': 'D', 'state': 'D', 'virtual': ('D', 'state')}¶
- arg_types = (('material', 'virtual', 'state'), ('material', 'parameter_1', 'parameter_2'))¶
- geometries = ['2_3', '2_4', '3_4', '3_8']¶
- modes = ('weak', 'eval')¶
- name = 'dw_nonsym_elastic'¶
- class sfepy.terms.terms_elastic.SDLinearElasticTerm(name, arg_str, integral, region, **kwargs)[source]¶
Sensitivity analysis of the linear elastic term.
- Definition:
\int_{\Omega} \hat{D}_{ijkl}\ e_{ij}(\ul{v}) e_{kl}(\ul{u})
\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}
- Call signature:
ev_sd_lin_elastic
(material, parameter_w, parameter_u, parameter_mv)
- Arguments:
material : D_{ijkl}
parameter_w : \ul{w}
parameter_u : \ul{u}
parameter_mv : \ul{\Vcal}
- arg_shapes = {'material': 'S, S', 'parameter_mv': 'D', 'parameter_u': 'D', 'parameter_w': 'D'}¶
- arg_types = ('material', 'parameter_w', 'parameter_u', 'parameter_mv')¶
- static function(out, coef, grad_v, grad_u, grad_w, mtx_d, cmap)¶
- geometries = ['2_3', '2_4', '3_4', '3_8']¶
- get_eval_shape(mat, par_w, par_u, par_mv, mode=None, term_mode=None, diff_var=None, **kwargs)[source]¶
- name = 'ev_sd_lin_elastic'¶