speccon1d

Class summary

Speccon1d([reader, pkg_for_version])

Solve 1D parabolic partial differential equation using spectral method.

Function summary

dim1sin_E_Igamv_the_BC_D_aDf_linear(drn, m, ...)	Calculate E and theta parts and assemble E_Igamv_the matrix that arises from homogenising D[a(Z)*D[u(Z, t),Z],Z] for non_zero top and bottom boundary conditions.
dim1sin_E_Igamv_the_BC_aDfDt_linear(drn, m, ...)	Calculate E and theta parts and assemble E_Igamv_the matrix that arises from homogenising a(Z)*D[u(Z, t), t] for non_zero top and bottom boundary conditions.
dim1sin_E_Igamv_the_BC_abDfDt_linear(drn, m, ...)	Calculate E and theta parts and assemble E_Igamv_the matrix that arises from homogenising a(Z)*b(Z)*D[u(Z, t), t] for non_zero top and bottom boundary conditions.
dim1sin_E_Igamv_the_BC_abf_linear(drn, m, ...)	Calculate E and theta parts and assemble E_Igamv_the matrix that arises from homogenising a(Z)*b(Z)*u(Z,t) for non_zero top and bottom boundary conditions.
dim1sin_E_Igamv_the_BC_deltaf_linear(drn, m, ...)	Calculate E and theta parts and assemble E_Igamv_the matrix that arises from homogenising delta(Z-zd)*u(Z,t) for non_zero top and bottom boundary conditions.
dim1sin_E_Igamv_the_aDmagDt_bilinear(m, ...)	Calculate E and theta parts and assemble E_Igamv_the matrix for loading terms of the form a(Z) * D[mag(t, Z),t] where mag is piecewise linear in depth and time and multiplied by cos(omega * t + phase).
dim1sin_E_Igamv_the_abmag_bilinear(m, eigs, ...)	Calculate E and theta parts and assemble E_Igamv_the matrix for loading terms of the form a(Z)*b(Z)*mag(t, Z) where mag is piecewise linear in depth and time and multiplied by cos(omega * t + phase).
dim1sin_E_Igamv_the_deltamag_linear(m, eigs, ...)	Calculate E and theta parts and assemble E_Igamv_the matrix for loading terms of the form a(Z) * delta(Z-Zd)*mag(t) where mag is piecewise linear in time multiplied by cos(omega * t + phase).
dim1sin_avgf(m, z, tvals, v_E_Igamv_the, drn)	Average u(Z,t) between Z1 and Z2 where u(Z,t) = phi * v_E_Igam_v_the + utop(t) * (1-Z) + ubot(t)*Z.
dim1sin_f(m, outz, tvals, v_E_Igamv_the, drn)	Assemble output u(Z,t) = phi * v_E_Igam_v_the + utop(t) * (1-Z) + ubot(t)*Z.
dim1sin_foft_Ipsiw_the_BC_D_aDf_linear(drn, ...)	Calculate the f(t) and theta parts and assemble the foft_Ipsiw_the matrix that arises from homgenising D[a(Z)*D[u(Z, t),Z],Z] terms with non_zero top and bottom boundary conditions when modelling drains/wells/columns with finite permeability.
dim1sin_integrate_af(m, z, tvals, ...[, ...])	Integrate u(Z,t) between Z1 and Z2 where u(Z,t) = phi * v_E_Igam_v_the + utop(t) * (1-Z) + ubot(t)*Z.

Module listing

This module has functions classes and common functionality for one dimensinal Spectral Galerkin methods.

class geotecha.speccon.speccon1d.Speccon1d(reader=None, pkg_for_version='geotecha')

Bases: geotecha.inputoutput.inputoutput.InputFileLoaderCheckerSaver

Solve 1D parabolic partial differential equation using spectral method.

Basically a base class to provide a broad template for one dimensional spectral method consolidation problems.

See also

geotecha.inputoutput.InputFileLoaderCheckerSaver

Details on how to initialize the object and attribute checks.

Methods

check_input_attributes()	Perform checks on attributes
make_all()	Run checks, make all arrays, make output
make_output()	Make all output i.e.
make_time_dependent_arrays()	Make all time-independent arrays; To be overridden in subclasses.
make_time_independent_arrays()	Make all time-independent arrays; To be overridden in subclasses.

make_all()

Run checks, make all arrays, make output

Generally run this after attributes have been entered

See also

check_input_attributes, make_time_independent_arrays, make_time_dependent_arrays, make_output

make_output()

Make all output i.e. data tables; To be overridden in subclasses.

make_time_dependent_arrays()

Make all time-independent arrays; To be overridden in subclasses.

make_time_independent_arrays()

Make all time-independent arrays; To be overridden in subclasses.

geotecha.speccon.speccon1d.dim1sin_E_Igamv_the_BC_D_aDf_linear(drn, m, eigs, tvals, Igamv, a, top_vs_time, bot_vs_time, top_omega_phase=None, bot_omega_phase=None, dT=1.0, theta_zero_indexes=None, implementation='vectorized')

Calculate E and theta parts and assemble E_Igamv_the matrix that arises from homogenising D[a(Z)*D[u(Z, t),Z],Z] for non_zero top and bottom boundary conditions.

When accounting for non-zero boundary conditions we homogenise the governing equation by letting u(Z,t) = v(Z,t) + utop(t)*(1-Z) + ubot(t)*Z and solving for v(Z, t). This function calculates the time dependent E part, the depth dependent theta part, and then assembles E*inverse(gam*v)*theta which forms part of solution v(Z,t)=phi*v*E*inverse(gam*v)*theta. The E and theta parts arise by subbing the boundary conditions into into governing equation terms of the form D[a(Z)*D[u(Z, t),Z],Z].

The contribution of each mag_vs_time-omega_phase pairing are superposed. The result is an array of size (neig, len(tvals)). So each column is the are the column vector E*inverse(gam*v)*theta calculated at each output time. This will allow us later to do v(Z,t) = phi*v*E_Igamv_the.

Uses sin(m*z) in the calculation of theta.

Parameters:

drn : [0,1]

Drainage condition, drn=0 for Pervious top pervious bottom (PTPB). drn=1 for Pervious top impoervious bottom (PTIB).

m : list of float

Eigenvalues of BVP, the m in sin(m*Z). Generate with geotecha.speccon.m_from_sin_mx.

eigs : 1d numpy.ndarray

List of eigenvalues of the spectral matrix i.e. Eigenvalues of the square Igam_psi matrix.

tvals : 1d numpy.ndarray`

List of time values to evaluate E matrix at.

Igamv : ndarray

Speccon matrix. Igamv = inverse of [gam * v])

a : PolyLine

Piewcewise linear function. e.g. for 1d consolidation surcharge radial draiange term is D[kv(z)*D[u(Z,t), Z],Z] so a would be kv. be et.

top_vs_time, bot_vs_time : list of PolyLine

Piecewise linear magnitude vs time for the top and bottom boundary. Use None if there is no variation.

top_omega_phase, bot_omega_phase : list of 2 element tuples, optional

(omega, phase) for use in cos(omega * t + phase) * mag_vs_time if omega_phase is None then mag_vs_time will not be multiplied by a cosine. If any element of omega_phase is None then in that particular loading combo, mag_vs_time will not be multiplied by a cosine.

dT : float, optional

Time factor multiple for numerical convieniece. Default dT=1.0.

theta_zero_indexes : slice/list etc., optional

A slice object, list, etc that can be used for numpy fancy indexing. Any specified index of the theta vector will be set to zero. This is useful when using the spectral method with block matrices and the loading term only refers to a subset of the equations. When using block matrices m should be the same size as the block matrix. Default theta_zero_indexes=None i.e. no elements of theta will be set to zero.

Returns:

E_Igamv_the : ndarray

Loading matrix of size (neig, len(tvals)).

Notes

Assuming the loads are formulated as the product of separate time and depth dependant functions as well as a cyclic component:

\[\sigma\left({Z,t}\right)= \sigma\left({Z}\right) \sigma\left({t}\right) \cos\left(\omega t + \phi\right)\]

the solution to the consolidation equation using the spectral method has the form:

\[u\left(Z,t\right)= \mathbf{\phi} \mathbf{v} \mathbf{E} \left(\mathbf{\Gamma v}\right)^{-1} \mathbf{\theta}\]

When we consider non-zero boundary conditions, additional loading terms are created when we sub in the following into the original governing equation.

\[u\left({Z,t}\right)= v\left({Z,t}\right) + u_{top}\left({t}\right)\left({1-Z}\right) + u_{bot}\left({b}\right)Z\]

Two additional loading terms are created with each substitution, one for the top boundary condition and one for the bottom boundary condition.

This function calculates \(\mathbf{E}\left(\mathbf{\Gamma v}\right)^{-1}\mathbf{\theta}\) when substitutions are made in terms of the following form:

\[\frac{\partial}{\partial Z} \left( {a\left({Z}\right) \frac{\partial u\left({Z,t}\right)}{\partial Z}} \right)\]

It is assumed that \(u_{top}\left({t}\right)\) and \(u_{bot}\left({t}\right)\) are piecewise linear in time with a cyclic component, and that multiple functions are superposed. Also \(a\left(Z\right)\) is a piecewise linear function with respect to \(Z\)

For this particular function the \(\mathbf{\theta}\) vector for each load is given by:

\[\mathbf{\theta}_{i}= \int_{0}^1{ \frac{\partial}{\partial Z} \left( {a\left({Z}\right) \frac{\partial \sigma\left({Z}\right)}{\partial Z}} \right) f\left({Z}\right) \phi_i\,dZ}\]

Where \(f\left({Z}\right)\) is the appropriate z-dependent term corresponding to either \(u_{top}\) or \(u_{bot}\) homogenisations.

The \(\mathbf{E}\) matrix for each load is given by:

\[\mathbf{E}_{i,j}= \int_{0}^{t_j}{ {\cos\left(\omega\tau+\textrm{phase}\right)} {\sigma\left(\tau\right)} {\exp\left({(dT\left(t-\tau\right)\lambda_i}\right)} \,d\tau}\]

where

• \(\lambda_i\) is the ith eigenvalue of the problem,
• \(dT\) is a time factor for numerical convienience,
• \(\sigma\left(\tau\right)\) is the piecewise linear time dependant load.

Note that the listed equations above are in terms of normalised depth Z, with depth integrations between [0, 1]. However, IF YOU KNOW WHAT YOU ARE DOING the integrations can be done using non-normalised depths. The first z value in the piecewise definition a(z) must still be 0 however the end point for integration will be the final z value in the definition of a(z). If you are doing this then your m values will include the normalising Factor. e.g. m = [pi/2/H, 3*pi/2/H] and a(z) is defined in two layers [0, z1], [z1, zend] as opposed to m = [pi/2, 3*pi/2] and a(Z) is two layers [0, z1/H], [z1/H, zend/H].

geotecha.speccon.speccon1d.dim1sin_E_Igamv_the_BC_aDfDt_linear(drn, m, eigs, tvals, Igamv, a, top_vs_time, bot_vs_time, top_omega_phase=None, bot_omega_phase=None, dT=1.0, theta_zero_indexes=None, implementation='vectorized')

Calculate E and theta parts and assemble E_Igamv_the matrix that arises from homogenising a(Z)*D[u(Z, t), t] for non_zero top and bottom boundary conditions.

When accounting for non-zero boundary conditions we homogenise the governing equation by letting u(Z,t) = v(Z,t) + utop(t)*(1-Z) + ubot(t)*Z and solving for v(Z, t). This function calculates the time dependent E part, the depth dependent theta part, and then assembles E*inverse(gam*v)*theta which forms part of solution v(Z,t)=phi*v*E*inverse(gam*v)*theta. The E and theta parts arise by subbing the boundary conditions into into governing equation terms of the form a(z)*D[u(Z,t), t].

The contribution of each mag_vs_time-omega_phase pairing are superposed. The result is an array of size (neig, len(tvals)). So each column is the are the column vector E*inverse(gam*v)*theta calculated at each output time. This will allow us later to do v(Z,t) = phi*v*E_Igamv_the.

Uses sin(m*z) in the calculation of theta.

Parameters:

drn : [0,1]

Drainage condition, drn=0 for Pervious top pervious bottom (PTPB). drn=1 for Pervious top impoervious bottom (PTIB).

m : list of float

Eigenvalues of BVP, the m in sin(m*Z). Generate with geotecha.speccon.m_from_sin_mx.

eigs : 1d numpy.ndarray

List of eigenvalues of the spectral matrix i.e. Eigenvalues of the square Igam_psi matrix.

tvals : 1d numpy.ndarray`

List of time values to evaluate E matrix at.

Igamv : ndarray

Speccon matrix. Igamv = inverse of [gam * v])

a : PolyLine

Piewcewise linear function. e.g. for 1d consolidation surcharge loading term is mv*D[sigma(z, t), t] so a would be mv.

top_vs_time, bot_vs_time : list of PolyLine

Piecewise linear magnitude vs time for the top and bottom boundary. Use None if there is no variation.

top_omega_phase, bot_omega_phase : list of 2 element tuples, optional

(omega, phase) for use in cos(omega * t + phase) * mag_vs_time if omega_phase is None then mag_vs_time will not be multiplied by a cosine. If any element of omega_phase is None then in that particular loading combo, mag_vs_time will not be multiplied by a cosine.

dT : float, optional

Time factor multiple for numerical convieniece. Default dT=1.0.

theta_zero_indexes : slice/list etc., optional

A slice object, list, etc that can be used for numpy fancy indexing. Any specified index of the theta vector will be set to zero. This is useful when using the spectral method with block matrices and the loading term only refers to a subset of the equations. When using block matrices m should be the same size as the block matrix. Default theta_zero_indexes=None i.e. no elements of theta will be set to zero.

Returns:

E_Igamv_the : ndarray

Loading matrix of size (neig, len(tvals)).

Notes

Assuming the loads are formulated as the product of separate time and depth dependant functions as well as a cyclic component:

\[\sigma\left({Z,t}\right)= \sigma\left({Z}\right) \sigma\left({t}\right) \cos\left(\omega t + \phi\right)\]

the solution to the consolidation equation using the spectral method has the form:

\[u\left(Z,t\right)= \mathbf{\phi v E} \left(\mathbf{\Gamma v}\right)^{-1} \mathbf{\theta}\]

When we consider non-zero boundary conditions, additional loading terms are created when we sub in the following into the original governing equation.

\[u\left({Z,t}\right)= v\left({Z,t}\right) + u_{top}\left({t}\right)\left({1-Z}\right) + u_{bot}\left({b}\right)Z\]

Two additional loading terms are created with each substitution, one for the top boundary condition and one for the bottom boundary condition.

This function calculates \(\mathbf{E}\left(\mathbf{\Gamma v}\right)^{-1}\mathbf{\theta}\) when substitutions are made in terms of the following form:

\[a\left({Z}\right)\frac{\partial u}{\partial t}\]

It is assumed that \(u_{top}\left({t}\right)\) and \(u_{bot}\left({t}\right)\) are piecewise linear in time with a cyclic component, and that multiple functions are superposed. Also \(a\left(Z\right)\) is a piecewise linear function w.r.t. \(Z\)

For this particular function the \(\mathbf{\theta}\) vector for each load is given by:

\[\mathbf{\theta}_{i}= \int_{0}^1{ {a\left(Z\right)} {\sigma\left(Z\right)} f\left({Z}\right) \phi_i\,dZ}\]

Where \(f\left({Z}\right)\) is the appropriate z-dependent term corresponding to either \(u_{top}\) or \(u_{bot}\) homogenisations.

The \(\mathbf{E}\) matrix for each load is given by:

\[\mathbf{E}_{i,j}= \int_{0}^{t_j}{ \frac{d{ {\cos\left(\omega\tau+\textrm{phase}\right)} \sigma\left(\tau\right)}} {d\tau} {\exp\left({(dT\left(t-\tau\right)\lambda_i}\right)} \,d\tau}\]

where

• \(\lambda_i\) is the ith eigenvalue of the problem,
• \(dT\) is a time factor for numerical convienience,
• \(\sigma\left(\tau\right)\) is the piecewise linear time dependant load.

Note that the listed equations above are in terms of normalised depth Z, with depth integrations between [0, 1]. However, IF YOU KNOW WHAT YOU ARE DOING the integrations can be done using non-normalised depths. The first z value in the piecewise definition a(z) must still be 0 however the end point for integration will be the final z value in the definition of a(z). If you are doing this then your m values will include the normalising Factor. e.g. m = [pi/2/H, 3*pi/2/H] and a(z) is defined in two layers [0, z1], [z1, zend] as opposed to m = [pi/2, 3*pi/2] and a(Z) is two layers [0, z1/H], [z1/H, zend/H].

geotecha.speccon.speccon1d.dim1sin_E_Igamv_the_BC_abDfDt_linear(drn, m, eigs, tvals, Igamv, a, b, top_vs_time=None, bot_vs_time=None, top_omega_phase=None, bot_omega_phase=None, dT=1.0, theta_zero_indexes=None, implementation='vectorized')

Calculate E and theta parts and assemble E_Igamv_the matrix that arises from homogenising a(Z)*b(Z)*D[u(Z, t), t] for non_zero top and bottom boundary conditions.

When accounting for non-zero boundary conditions we homogenise the governing equation by letting u(Z,t) = v(Z,t) + utop(t)*(1-Z) + ubot(t)*Z and solving for v(Z, t). This function calculates the time dependent E part, the depth dependent theta part, and then assembles E*inverse(gam*v)*theta which forms part of solution v(Z,t)=phi*v*E*inverse(gam*v)*theta. The E and theta parts arise by subbing the boundary conditions into into governing equation terms of the form a(Z)*b(Z)*D[u(Z,t), t].

The contribution of each mag_vs_time-omega_phase pairing are superposed. The result is an array of size (neig, len(tvals)). So each column is the are the column vector E*inverse(gam*v)*theta calculated at each output time. This will allow us later to do v(Z,t) = phi*v*E_Igamv_the.

Uses sin(m*Z) in the calculation of theta.

Parameters:

drn : [0,1]

Drainage condition, drn=0 for Pervious top pervious bottom (PTPB). drn=1 for Pervious top impoervious bottom (PTIB).

m : list of float

Eigenvalues of BVP, the m in sin(m*Z). Generate with geotecha.speccon.m_from_sin_mx.

eigs : 1d numpy.ndarray

List of eigenvalues of the spectral matrix i.e. Eigenvalues of the square Igam_psi matrix.

tvals : 1d numpy.ndarray`

List of time values to evaluate E matrix at.

Igamv : ndarray

Speccon matrix. Igamv = inverse of [gam * v])

a, b : PolyLine

Piewcewise linear function. e.g. for 1d consolidation surcharge radial draiange term is dTh*kh*et*u(Z,t) a would be kh, b would be et.

top_vs_time, bot_vs_time : list of PolyLine

Piecewise linear magnitude vs time for the top and bottom boundary. Use None if there is no variation.

top_omega_phase, bot_omega_phase : list of 2 element tuples, optional

(omega, phase) for use in cos(omega * t + phase) * mag_vs_time if omega_phase is None then mag_vs_time will not be multiplied by a cosine. If any element of omega_phase is None then in that particular loading combo, mag_vs_time will not be multiplied by a cosine.

dT : float, optional

Time factor multiple for numerical convieniece. Default dT=1.0.

theta_zero_indexes : slice/list etc., optional

A slice object, list, etc that can be used for numpy fancy indexing. Any specified index of the theta vector will be set to zero. This is useful when using the spectral method with block matrices and the loading term only refers to a subset of the equations. When using block matrices m should be the same size as the block matrix. Default theta_zero_indexes=None i.e. no elements of theta will be set to zero.

Returns:

E_Igamv_the : ndarray

Loading matrix of size (neig, len(tvals)).

Notes

Assuming the loads are formulated as the product of separate time and depth dependant functions as well as a cyclic component:

\[\sigma\left({Z,t}\right)= \sigma\left({Z}\right) \sigma\left({t}\right) \cos\left(\omega t + \phi\right)\]

the solution to the consolidation equation using the spectral method has the form:

\[u\left(Z,t\right)= \mathbf{\phi v E} \left(\mathbf{\Gamma v}\right)^{-1} \mathbf{\theta}\]

When we consider non-zero boundary conditions, additional loading terms are created when we sub in the following into the original governing equation.

\[u\left({Z,t}\right)= v\left({Z,t}\right) + u_{top}\left({t}\right)\left({1-Z}\right) + u_{bot}\left({b}\right)Z\]

Two additional loading terms are created with each substitution, one for the top boundary condition and one for the bottom boundary condition.

This function calculates \(\mathbf{E}\left(\mathbf{\Gamma v}\right)^{-1}\mathbf{\theta}\) when substitutions are made in terms of the following form:

\[a\left({Z}\right) b\left({Z}\right) \frac{\partial u}{\partial t}\]

It is assumed that \(u_{top}\left({t}\right)\) and \(u_{bot}\left({t}\right)\) are piecewise linear in time with a cyclic component, and that multiple functions are superposed. Also \(a\left(Z\right)\) and \(b\left(Z\right)\) are piecewise linear functions w.r.t. \(Z\)

For this particular function the \(\mathbf{\theta}\) vector for each load is given by:

\[\mathbf{\theta}_{i}= \int_{0}^1{ {a\left(Z\right)} {b\left(Z\right)} {\sigma\left(Z\right)} f\left({Z}\right) \phi_i\,dZ}\]

Where \(f\left({Z}\right)\) is the appropriate z-dependent term corresponding to either \(u_{top}\) or \(u_{bot}\) homogenisations.

The \(\mathbf{E}\) matrix for each load is given by:

\[\mathbf{E}_{i,j}= \int_{0}^{t_j}{ \frac{d{ {\cos\left(\omega\tau+\textrm{phase}\right)} \sigma\left(\tau\right)}} {d\tau} {\exp\left({(dT\left(t-\tau\right)\lambda_i}\right)} \,d\tau}\]

where

• \(\lambda_i\) is the ith eigenvalue of the problem,
• \(dT\) is a time factor for numerical convienience,
• \(\sigma\left(\tau\right)\) is the piecewise linear time dependant load.

Note that the listed equations above are in terms of normalised depth Z, with depth integrations between [0, 1]. However, IF YOU KNOW WHAT YOU ARE DOING the integrations can be done using non-normalised depths. The first z value in the piecewise definition a(z) must still be 0 however the end point for integration will be the final z value in the definition of a(z). If you are doing this then your m values will include the normalising Factor. e.g. m = [pi/2/H, 3*pi/2/H] and a(z) is defined in two layers [0, z1], [z1, zend] as opposed to m = [pi/2, 3*pi/2] and a(Z) is two layers [0, z1/H], [z1/H, zend/H].

geotecha.speccon.speccon1d.dim1sin_E_Igamv_the_BC_abf_linear(drn, m, eigs, tvals, Igamv, a, b, top_vs_time=None, bot_vs_time=None, top_omega_phase=None, bot_omega_phase=None, dT=1.0, theta_zero_indexes=None, implementation='vectorized')

Calculate E and theta parts and assemble E_Igamv_the matrix that arises from homogenising a(Z)*b(Z)*u(Z,t) for non_zero top and bottom boundary conditions.

When accounting for non-zero boundary conditions we homogenise the governing equation by letting u(Z,t) = v(Z,t) + utop(t)*(1-Z) + ubot(t)*Z and solving for v(Z, t). This function calculates the time dependent E part, the depth dependent theta part, and then assembles E*inverse(gam*v)*theta which forms part of solution v(Z,t)=phi*v*E*inverse(gam*v)*theta. The E and theta parts arise by subbing the boundary conditions into into governing equation terms of the form a(z)*b(z)*u(Z,t)

The contribution of each mag_vs_time-omega_phase pairing are superposed. The result is an array of size (neig, len(tvals)). So each column is the are the column vector E*inverse(gam*v)*theta calculated at each output time. This will allow us later to do v(Z,t) = phi*v*E_Igamv_the.

Uses sin(m*z) in the calculation of theta.

Parameters:

drn : [0,1]

Drainage condition, drn=0 for Pervious top pervious bottom (PTPB). drn=1 for Pervious top impoervious bottom (PTIB).

m : list of float

Eigenvalues of BVP, the m in sin(m*Z). Generate with geotecha.speccon.m_from_sin_mx.

eigs : 1d numpy.ndarray

List of eigenvalues of the spectral matrix i.e. Eigenvalues of the square Igam_psi matrix.

tvals : 1d numpy.ndarray`

List of time values to evaluate E matrix at.

Igamv : ndarray

Speccon matrix. Igamv = inverse of [gam * v])

a, b : PolyLine

Piewcewise linear function. e.g. for 1d consolidation surcharge radial draiange term is dTh*kh*et*u(Z,t) a would be kh, b would be et.

top_vs_time, bot_vs_time : list of PolyLine

Piecewise linear magnitude vs time for the top and bottom boundary. Use None if there is no variation.

top_omega_phase, bot_omega_phase : list of 2 element tuples, optional

(omega, phase) for use in cos(omega * t + phase) * mag_vs_time if omega_phase is None then mag_vs_time will not be multiplied by a cosine. If any element of omega_phase is None then in that particular loading combo, mag_vs_time will not be multiplied by a cosine.

dT : float, optional

Time factor multiple for numerical convieniece. Default dT=1.0.

theta_zero_indexes : slice/list etc., optional

A slice object, list, etc that can be used for numpy fancy indexing. Any specified index of the theta vector will be set to zero. This is useful when using the spectral method with block matrices and the loading term only refers to a subset of the equations. When using block matrices m should be the same size as the block matrix. Default theta_zero_indexes=None i.e. no elements of theta will be set to zero.

Returns:

E_Igamv_the : ndarray

Loading matrix of size (neig, len(tvals)).

Notes

Assuming the loads are formulated as the product of separate time and depth dependant functions as well as a cyclic component:

\[\sigma\left({Z,t}\right)= \sigma\left({Z}\right) \sigma\left({t}\right) \cos\left(\omega t + \phi\right)\]

the solution to the consolidation equation using the spectral method has the form:

\[u\left(Z,t\right)= \mathbf{\phi v E} \left(\mathbf{\Gamma v}\right)^{-1} \mathbf{\theta}\]

When we consider non-zero boundary conditions, additional loading terms are created when we sub in the following into the original governing equation.

\[u\left({Z,t}\right)= v\left({Z,t}\right) + u_{top}\left({t}\right)\left({1-Z}\right) + u_{bot}\left({b}\right)Z\]

Two additional loading terms are created with each substitution, one for the top boundary condition and one for the bottom boundary condition.

This function calculates \(\mathbf{E}\left(\mathbf{\Gamma v}\right)^{-1}\mathbf{\theta}\) when substitutions are made in terms of the following form:

\[a\left({Z}\right) b\left({Z}\right) u\left({Z,t}\right)\]

It is assumed that \(u_{top}\left({t}\right)\) and \(u_{bot}\left({t}\right)\) are piecewise linear in time with a cyclic component, and that multiple functions are superposed. Also \(a\left(Z\right)\) and \(b\left(Z\right)\) are piecewise linear functions with respect to \(Z\)

For this particular function the \(\mathbf{\theta}\) vector for each load is given by:

\[\mathbf{\theta}_{i}= \int_{0}^1{ {a\left(Z\right)} {b\left(Z\right)} {\sigma\left(Z\right)} f\left({Z}\right) \phi_i\,dZ}\]

Where \(f\left({Z}\right)\) is the appropriate z-dependent term corresponding to either \(u_{top}\) or \(u_{bot}\) homogenisations.

The \(\mathbf{E}\) matrix for each load is given by:

\[\mathbf{E}_{i,j}= \int_{0}^{t_j}{ {\cos\left(\omega\tau+\textrm{phase}\right)} {\sigma\left(\tau\right)} {\exp\left({(dT\left(t-\tau\right)\lambda_i}\right)} \,d\tau}\]

where

• \(\lambda_i\) is the ith eigenvalue of the problem,
• \(dT\) is a time factor for numerical convienience,
• \(\sigma\left(\tau\right)\) is the piecewise linear time dependant load.

Note that the listed equations above are in terms of normalised depth Z, with depth integrations between [0, 1]. However, IF YOU KNOW WHAT YOU ARE DOING the integrations can be done using non-normalised depths. The first z value in the piecewise definition a(z) must still be 0 however the end point for integration will be the final z value in the definition of a(z). If you are doing this then your m values will include the normalising Factor. e.g. m = [pi/2/H, 3*pi/2/H] and a(z) is defined in two layers [0, z1], [z1, zend] as opposed to m = [pi/2, 3*pi/2] and a(Z) is two layers [0, z1/H], [z1/H, zend/H].

geotecha.speccon.speccon1d.dim1sin_E_Igamv_the_BC_deltaf_linear(drn, m, eigs, tvals, Igamv, zvals, pseudo_k, top_vs_time, bot_vs_time, top_omega_phase=None, bot_omega_phase=None, dT=1.0, theta_zero_indexes=None, implementation='vectorized')

Calculate E and theta parts and assemble E_Igamv_the matrix that arises from homogenising delta(Z-zd)*u(Z,t) for non_zero top and bottom boundary conditions.

When accounting for non-zero boundary conditions we homogenise the governing equation by letting u(Z,t) = v(Z,t) + utop(t)*(1-Z) + ubot(t)*Z and solving for v(Z, t). This function calculates the time dependent E part, the depth dependent theta part, and then assembles E*inverse(gam*v)*theta which forms part of solution v(Z,t)=phi*v*E*inverse(gam*v)*theta. The E and theta parts arise by subbing the boundary conditions into into governing equation terms of the form delta(Z-zd)*u(Z,t).

The contribution of each mag_vs_time-omega_phase pairing are superposed. The result is an array of size (neig, len(tvals)). So each column is the are the column vector E*inverse(gam*v)*theta calculated at each output time. This will allow us later to do v(Z,t) = phi*v*E_Igamv_the.

Uses sin(m*z) in the calculation of theta.

Parameters:

drn : [0,1]

Drainage condition, drn=0 for Pervious top pervious bottom (PTPB). drn=1 for Pervious top impoervious bottom (PTIB).

m : list of float

Eigenvalues of BVP, the m in sin(m*Z). Generate with geotecha.speccon.m_from_sin_mx.

eigs : 1d numpy.ndarray

List of eigenvalues of the spectral matrix i.e. Eigenvalues of the square Igam_psi matrix.

tvals : 1d numpy.ndarray`

List of time values to evaluate E matrix at.

Igamv : ndarray

Speccon matrix. Igamv = inverse of [gam * v])

zvals : list of float

z values defining each delta function, zd.

pseudo_k : list of float

Coefficients to multiply each delta function by.

top_vs_time, bot_vs_time : list of PolyLine

Piecewise linear magnitude vs time for the top and bottom boundary. Use None if there is no variation.

top_omega_phase, bot_omega_phase : list of 2 element tuples, optional

(omega, phase) for use in cos(omega * t + phase) * mag_vs_time if omega_phase is None then mag_vs_time will not be multiplied by a cosine. If any element of omega_phase is None then in that particular loading combo, mag_vs_time will not be multiplied by a cosine.

dT : float, optional

Time factor multiple for numerical convieniece. Default dT=1.0.

theta_zero_indexes : slice/list etc., optional

A slice object, list, etc that can be used for numpy fancy indexing. Any specified index of the theta vector will be set to zero. This is useful when using the spectral method with block matrices and the loading term only refers to a subset of the equations. When using block matrices m should be the same size as the block matrix. Default theta_zero_indexes=None i.e. no elements of theta will be set to zero.

Returns:

E_Igamv_the : ndarray

Loading matrix of size (neig, len(tvals)).

Notes

Assuming the loads are formulated as the product of separate time and depth dependant functions as well as a cyclic component:

\[\sigma\left({Z,t}\right)= \sigma\left({Z}\right) \sigma\left({t}\right) \cos\left(\omega t + \phi\right)\]

the solution to the consolidation equation using the spectral method has the form:

\[u\left(Z,t\right)= \mathbf{\phi v E} \left(\mathbf{\Gamma v}\right)^{-1} \mathbf{\theta}\]

When we consider non-zero boundary conditions, additional loading terms are created when we sub in the following into the original governing equation.

\[u\left({Z,t}\right)= v\left({Z,t}\right) + u_{top}\left({t}\right)\left({1-Z}\right) + u_{bot}\left({b}\right)Z\]

Two additional loading terms are created with each substitution, one for the top boundary condition and one for the bottom boundary condition.

This function calculates \(\mathbf{E}\left(\mathbf{\Gamma v}\right)^{-1}\mathbf{\theta}\) when substitutions are made in terms of the following form:

\[k_{\textrm{pseudo}} \delta\left({Z-Z_d}\right) u\left({Z,t}\right)\]

It is assumed that \(u_{top}\left({t}\right)\) and \(u_{bot}\left({t}\right)\) are piecewise linear in time with a cyclic component, and that multiple functions are superposed.

For this particular function the \(\mathbf{\theta}\) vector for each load is given by:

\[\mathbf{\theta}_{i}= \int_{0}^1{ k_{\textrm{pseudo}} \delta\left({Z-Z_d}\right) {\sigma\left(Z\right)} f\left({Z}\right) \phi_i\,dZ}\]

Where \(f\left({Z}\right)\) is the appropriate z-dependent term corresponding to either \(u_{top}\) or \(u_{bot}\) homogenisations.

The \(\mathbf{E}\) matrix for each load is given by:

\[\mathbf{E}_{i,j}= \int_{0}^{t_j}{ {\cos\left(\omega\tau+\textrm{phase}\right)} {\sigma\left(\tau\right)} {\exp\left({(dT\left(t-\tau\right)\lambda_i}\right)} \,d\tau}\]

where

• \(\lambda_i\) is the ith eigenvalue of the problem,
• \(dT\) is a time factor for numerical convienience,
• \(\sigma\left(\tau\right)\) is the piecewise linear time dependant load.

Note that this function, unlike many similar functions in this module, has only been formulated for Normalised depths between [0,1]

geotecha.speccon.speccon1d.dim1sin_E_Igamv_the_aDmagDt_bilinear(m, eigs, tvals, Igamv, a, mag_vs_depth, mag_vs_time, omega_phase=None, dT=1.0, theta_zero_indexes=None, implementation='vectorized')

Calculate E and theta parts and assemble E_Igamv_the matrix for loading terms of the form a(Z) * D[mag(t, Z),t] where mag is piecewise linear in depth and time and multiplied by cos(omega * t + phase).

Make the E*inverse(gam*v)*theta part of solution u(Z,t)=phi*v*E*inverse(gam*v)*theta for terms of the form a(Z) * D[mag(t, Z),t]. The contribution of each mag_vs_time-omega_phase pairing and are superposed. The result is an array of size (neig, len(tvals)). So each column is the column vector E*inverse(gam*v)*theta calculated at each output time. This will allow us later to do u(Z,t) = phi*v*E_Igamv_the.

Uses sin(m*z) in the calculation of theta.

Parameters:

drn : [0,1]

Drainage condition, drn=0 for Pervious top pervious bottom (PTPB). drn=1 for Pervious top impoervious bottom (PTIB).

m : list of float

Eigenvalues of BVP, the m in sin(m*Z). Generate with geotecha.speccon.m_from_sin_mx.

eigs : 1d numpy.ndarray

List of eigenvalues of the spectral matrix i.e. Eigenvalues of the square Igam_psi matrix.

tvals : 1d numpy.ndarray`

List of time values to evaluate E matrix at.

Igamv : ndarray

Speccon matrix. Igamv = inverse of [gam * v])

a : PolyLine

Piewcewise linear function. e.g. for 1d consolidation surcharge loading term is mv*D[sigma(z, t), t] so a would be mv.

dT : float, optional

Time factor multiple for numerical convieniece. Default dT=1.0.

theta_zero_indexes : slice/list etc., optional

A slice object, list, etc that can be used for numpy fancy indexing. Any specified index of the theta vector will be set to zero. This is useful when using the spectral method with block matrices and the loading term only refers to a subset of the equations. When using block matrices m should be the same size as the block matrix. Default theta_zero_indexes=None i.e. no elements of theta will be set to zero.

Returns:

E_Igamv_the : ndarray

Loading matrix of size (neig, len(tvals)).

Notes

Assuming the loads are formulated as the product of separate time and depth dependant functions as well as a cyclic component:

\[\sigma\left({Z,t}\right)= \sigma\left({Z}\right) \sigma\left({t}\right) \cos\left(\omega t + \phi\right)\]

the solution to the consolidation equation using the spectral method has the form:

\[u\left(Z,t\right)= \mathbf{\phi v E} \left(\mathbf{\Gamma v}\right)^{-1} \mathbf{\theta}\]

This function calculates \(\mathbf{E}\left(\mathbf{\Gamma v}\right)^{-1}\mathbf{\theta}\) for terms of the following form:

\[a\left({Z}\right)\frac{\partial \sigma}{\partial t}\]

It is assumed that \(\sigma\left({t}\right)\) is piecewise linear in time with a cyclic component, and that multiple functions are superposed. Also \(a\left(Z\right)\) is a piecewise linear function with respect to \(Z\).

For this particular function the \(\mathbf{\theta}\) vector for each load is given by:

\[\mathbf{\theta}_{i}= \int_{0}^1{ {a\left(Z\right)} {\sigma\left(Z\right)} \phi_i\,dZ}\]

The \(\mathbf{E}\) matrix for each load is given by:

\[\mathbf{E}_{i,j}= \int_{0}^{t_j}{ \frac{d{ {\cos\left(\omega\tau+\textrm{phase}\right)} \sigma\left(\tau\right)}} {d\tau} {\exp\left({(dT\left(t-\tau\right)\lambda_i}\right)} \,d\tau}\]

where

• \(\lambda_i\) is the ith eigenvalue of the problem,
• \(dT\) is a time factor for numerical convienience,
• \(\sigma\left(\tau\right)\) is the piecewise linear time dependant load.

Note that the listed equations above are in terms of normalised depth Z, with depth integrations between [0, 1]. However, IF YOU KNOW WHAT YOU ARE DOING the integrations can be done using non-normalised depths. The first z value in the piecewise definition a(z) must still be 0 however the end point for integration will be the final z value in the definition of a(z). If you are doing this then your m values will include the normalising Factor. e.g. m = [pi/2/H, 3*pi/2/H] and a(z) is defined in two layers [0, z1], [z1, zend] as opposed to m = [pi/2, 3*pi/2] and a(Z) is two layers [0, z1/H], [z1/H, zend/H].

geotecha.speccon.speccon1d.dim1sin_E_Igamv_the_abmag_bilinear(m, eigs, tvals, Igamv, a, b, mag_vs_depth, mag_vs_time, omega_phase=None, dT=1.0, theta_zero_indexes=None, implementation='vectorized')

Calculate E and theta parts and assemble E_Igamv_the matrix for loading terms of the form a(Z)*b(Z)*mag(t, Z) where mag is piecewise linear in depth and time and multiplied by cos(omega * t + phase).

Make the E*inverse(gam*v)*theta part of solution u(Z,t)=phi*v*E*inverse(gam*v)*theta for terms of the form a(Z)*b(Z)*mag(t, Z). The contribution of each mag_vs_time-omega_phase pairing and are superposed. The result is an array of size (neig, len(tvals)). So each column is the column vector E*inverse(gam*v)*theta calculated at each output time. This will allow us later to do u(Z,t) = phi*v*E_Igamv_the.

Uses sin(m*Z) in the calculation of theta.

Parameters:

m : list of float

eigenvlaues of BVP. generate with geoteca.speccon.m_from_sin_mx

eigs : 1d numpy.ndarray

list of eigenvalues

tvals : 1d numpy.ndarray`

list of time values to calculate integral at

Igamv : ndarray

speccon matrix

a : PolyLine

Piewcewise linear function. e.g. for 1d consolidation surcharge loading term is mv*D[sigma(z, t), t] so a would be mv.

b : PolyLine

Piewcewise linear function. e.g. for 1d consolidation vacuum term is

kh*et*w(z,t) so a would be kh, b would be et

mag_vs_depth : list of PolyLine

Piecewise linear magnitude vs depth.

mag_vs_time : list of PolyLine

Piecewise linear magnitude vs time

omega_phase : list of 2 element tuples, optional

(omega, phase) for use in cos(omega * t + phase) * mag_vs_time if omega_phase is None then mag_vs_time will not be multiplied by a cosine. If any element of omega_phase is None then in that particular loading combo, mag_vs_time will not be multiplied by a cosine.

dT : float, optional

time factor multiple (default = 1.0)

theta_zero_indexes : slice/list etc., optional=None

a slice object, list, etc that can be used for numpy fancy indexing. Any specified index of the theta vector will be set to zero. This is useful when using the spectral method with block matrices and the loading term only refers to a subset of the equations. When using block matrices m should be the same size as the block matrix. default=None i.e. no elements of theta will be set to zero.

Returns:

E_Igamv_the: ndarray

loading matrix

Notes

Assuming the loads are formulated as the product of separate time and depth dependant functions as well as a cyclic component:

\[\sigma\left({Z,t}\right)=\sigma\left({Z}\right)\sigma\left({t}\right)\cos\left(\omega t + \phi\right)\]

the solution to the consolidation equation using the spectral method has the form:

\[u\left(Z,t\right)=\mathbf{\Phi v E}\left(\mathbf{\Gamma v}\right)^{-1}\mathbf{\theta}\]

In this instance \(\sigma\left({Z}\right)\) and \(\sigma\left({t}\right)\) are piecewise linear in depth and time (hence the ‘bilinear’ in the function name) there is also a cyclic component.

dim1sin_E_Igamv_the_abmag_bilinear will calculate \(\mathbf{E}\left(\mathbf{\Gamma v}\right)^{-1}\mathbf{\theta}\) for terms with the form:

\[a\left({z}\right)b\left({z}\right)\frac{\partial\sigma\left({Z,t}\right)}{\partial t}\]

where \(a\left(z\right)\), \(b\left(z\right)\) are piecewise linear functions w.r.t. \(z\).

geotecha.speccon.speccon1d.dim1sin_E_Igamv_the_deltamag_linear(m, eigs, tvals, Igamv, zvals, pseudo_k, mag_vs_time, omega_phase=None, dT=1.0, theta_zero_indexes=None, implementation='vectorized')

Calculate E and theta parts and assemble E_Igamv_the matrix for loading terms of the form a(Z) * delta(Z-Zd)*mag(t) where mag is piecewise linear in time multiplied by cos(omega * t + phase).

Make the E*inverse(gam*v)*theta part of solution u(Z,t)=phi*v*E*inverse(gam*v)*theta for terms of the form a(Z) * delta(Z-Zd)*mag(t). The contribution of each mag_vs_time-omega_phase pairing and each zval are superposed. The result is an array of size (neig, len(tvals)). So each column is the are the column vector E*inverse(gam*v)*theta calculated at each output time. This will allow us later to do u(Z,t) = phi*v*E_Igamv_the.

Uses sin(m*Z) in the calculation of theta.

Parameters:

drn : [0,1]

Drainage condition, drn=0 for Pervious top pervious bottom (PTPB). drn=1 for Pervious top impoervious bottom (PTIB).

m : list of float

Eigenvalues of BVP, the m in sin(m*Z). Generate with geotecha.speccon.m_from_sin_mx.

eigs : 1d numpy.ndarray

List of eigenvalues of the spectral matrix i.e. Eigenvalues of the square Igam_psi matrix.

tvals : 1d numpy.ndarray`

List of time values to evaluate E matrix at.

Igamv : ndarray

Speccon matrix. Igamv = inverse of [gam * v])

zvals : list of float

z values defining each delta function, zd.

pseudo_k : list of float

Coefficients to multiply each delta function by.

dT : float, optional

Time factor multiple for numerical convieniece. Default dT=1.0.

theta_zero_indexes : slice/list etc., optional

A slice object, list, etc that can be used for numpy fancy indexing. Any specified index of the theta vector will be set to zero. This is useful when using the spectral method with block matrices and the loading term only refers to a subset of the equations. When using block matrices m should be the same size as the block matrix. Default theta_zero_indexes=None i.e. no elements of theta will be set to zero.

Returns:

E_Igamv_the : ndarray

Loading matrix of size (neig, len(tvals)).

Notes

Assuming the loads are formulated as the product of separate time and depth dependant functions as well as a cyclic component:

\[\sigma\left({Z,t}\right)= \sigma\left({Z}\right) \sigma\left({t}\right) \cos\left(\omega t + \phi\right)\]

the solution to the consolidation equation using the spectral method has the form:

\[u\left(Z,t\right)= \mathbf{\phi v E} \left(\mathbf{\Gamma v}\right)^{-1} \mathbf{\theta}\]

This function calculates \(\mathbf{E}\left(\mathbf{\Gamma v}\right)^{-1}\mathbf{\theta}\) for terms of the following form:

\[k_{\textrm{pseudo}} \delta\left({Z-Z_d}\right) \sigma\left({t}\right)\]

It is assumed that \(\sigma\left({t}\right)\) is piecewise linear in time with a cyclic component, and that multiple functions are superposed.

For this particular function the \(\mathbf{\theta}\) vector for each load is given by:

\[\mathbf{\theta}_{i}= \int_{0}^1{ k_{\textrm{pseudo}} \delta\left({Z-Z_d}\right) \phi_i\,dZ}\]

The \(\mathbf{E}\) matrix for each load is given by:

\[\mathbf{E}_{i,j}= \int_{0}^{t_j}{ {\cos\left(\omega\tau+\textrm{phase}\right)} {\sigma\left(\tau\right)} {\exp\left({(dT\left(t-\tau\right)\lambda_i}\right)} \,d\tau}\]

where

• \(\lambda_i\) is the ith eigenvalue of the problem,
• \(dT\) is a time factor for numerical convienience,
• \(\sigma\left(\tau\right)\) is the piecewise linear time dependant load.

geotecha.speccon.speccon1d.dim1sin_avgf(m, z, tvals, v_E_Igamv_the, drn, top_vs_time=None, bot_vs_time=None, top_omega_phase=None, bot_omega_phase=None)

Average u(Z,t) between Z1 and Z2 where u(Z,t) = phi * v_E_Igam_v_the + utop(t) * (1-Z) + ubot(t)*Z.

Basically calculates the average phi part for each z pair value, then dot product with v_E_Igamv_the (which has elsewhere been calculated at each tvals value). Then account for non-zero boundary conditions by adding average of utop(t)*(1-Z) and ubot(t)*Z parts for each z, tvals pair.

Use sin(m*Z) for the phi part.

Parameters:

m : list of float

Eigenvalues of BVP, the m in sin(m*Z). Generate with geotecha.speccon.m_from_sin_mx.

z : size (n, 2) 2d numpy.ndarray

Depths to evaluate average between.

tvals : 1d numpy.ndarray

Time values to evaluate at.

eigs : 1d numpy.ndarray

List of eigenvalues of the spectral matrix i.e. Eigenvalues of the square Igam_psi matrix.

tvals : 1d numpy.ndarray`

List of time values to evaluate E matrix at.

v_E_Igamv_the : ndarray of size (neig, len(tvals))

Speccon matrix.

drn : [0,1]

Drainage condition, drn=0 for Pervious top pervious bottom (PTPB). drn=1 for Pervious top impoervious bottom (PTIB).

top_vs_time, bot_vs_time : list of PolyLine

Piecewise linear magnitude vs time for the top and bottom boundary. Use None if there is no variation.

top_omega_phase, bot_omega_phase : list of 2 element tuples, optional

(omega, phase) for use in cos(omega * t + phase) * mag_vs_time if omega_phase is None then mag_vs_time will not be multiplied by a cosine. If any element of omega_phase is None then in that particular loading combo, mag_vs_time will not be multiplied by a cosine.

Returns:

uavg : np.ndarray

Average pore pressure between depths at each time. Array of size (len(z), len(tvals)).

Notes

The average of the \(\phi\) term is:

\[\mathbf{\phi}_{i\textrm{average}}= \frac{1}{Z_2-Z_1} \int_{z_1}^{z_2}{\sin\left({m_i Z}\right)\,dZ}\]

geotecha.speccon.speccon1d.dim1sin_f(m, outz, tvals, v_E_Igamv_the, drn, top_vs_time=None, bot_vs_time=None, top_omega_phase=None, bot_omega_phase=None)

Assemble output u(Z,t) = phi * v_E_Igam_v_the + utop(t) * (1-Z) + ubot(t)*Z.

Basically calculates the phi part for each outz value, then dot product with v_E_Igamv_the (which has elsewhere been calculated at each tvals value). Then account for non-zero boundary conditions by adding utop(t)*(1-Z) and ubot(t)*Z parts for each outz, tvals pair.

Use sin(m*Z) for the phi part.

Parameters:

m : list of float

Eigenvalues of BVP, the m in sin(m*Z). Generate with geotecha.speccon.m_from_sin_mx.

outz : 1d numpy.ndarray

Depths to evaluate at.

tvals : 1d numpy.ndarray

Time values to evaluate at

eigs : 1d numpy.ndarray

List of eigenvalues of the spectral matrix i.e. Eigenvalues of the square Igam_psi matrix.

tvals : 1d numpy.ndarray`

List of time values to evaluate E matrix at.

v_E_Igamv_the : ndarray of size (neig, len(tvals))

Speccon matrix.

drn : [0,1]

Drainage condition, drn=0 for Pervious top pervious bottom (PTPB). drn=1 for Pervious top impoervious bottom (PTIB).

top_vs_time, bot_vs_time : list of PolyLine

Piecewise linear magnitude vs time for the top and bottom boundary. Use None if there is no variation.

top_omega_phase, bot_omega_phase : list of 2 element tuples, optional

(omega, phase) for use in cos(omega * t + phase) * mag_vs_time if omega_phase is None then mag_vs_time will not be multiplied by a cosine. If any element of omega_phase is None then in that particular loading combo, mag_vs_time will not be multiplied by a cosine.

Returns:

u : np.ndarray

Pore pressure at depth and time. Array of size (len(outz), len(tvals)).

Notes

The \(\phi\) term is simply: \(\sin\left({m Z}\right)\) evaluated at each required depth.

geotecha.speccon.speccon1d.dim1sin_foft_Ipsiw_the_BC_D_aDf_linear(drn, m, eigs, tvals, Ipsiw, a, top_vs_time, bot_vs_time, top_omega_phase=None, bot_omega_phase=None, theta_zero_indexes=None)

Calculate the f(t) and theta parts and assemble the foft_Ipsiw_the matrix that arises from homgenising D[a(Z)*D[u(Z, t),Z],Z] terms with non_zero top and bottom boundary conditions when modelling drains/wells/columns with finite permeability.

When accounting for non-zero boundary conditions we homogenise the governing equation by letting u(Z,t) = v(Z,t) + utop(t)*(1-Z) + ubot(t)*Z and solving for v(Z, t). For some problems the homogenisation process produces an additional term that has t be added to the usual solution of v(Z,t)=phi*v*E*inverse(gam*v)*theta i.e. v(Z,t)=phi*v*E*inverse(gam*v)*theta + f(t)*Ipsiw*theta This function calculates the f(t) and theta parts of the second term and then assembles the foft_Ipsiw_the matrix. These parts arise by subbing the boundary conditions into into governing equation terms of the form D[a(Z)*D[u(Z, t),Z],Z].

Uses sin(m*Z) in the calculation of theta.

Parameters:

drn : [0,1]

Drainage condition, drn=0 for Pervious top pervious bottom (PTPB). drn=1 for Pervious top impoervious bottom (PTIB).

m : list of float

Eigenvalues of BVP, the m in sin(m*Z). Generate with geotecha.speccon.m_from_sin_mx.

eigs : 1d numpy.ndarray

List of eigenvalues of the spectral matrix i.e. Eigenvalues of the square Igam_psi matrix.

tvals : 1d numpy.ndarray`

List of time values to evaluate E matrix at.

Ipsiw : ndarray, square matrix

Speccon matrix. Ipsiw was originally formulated/denoted for vertical peremability term in the well resistance flow equation for vertical drain consolidatoin with well resistance. It is different, but still a square matrix, for the stone column consolidation problem. As long as you know what you are doing Ipsiw can be any appropriate square matrix.

a : PolyLine

Piewcewise linear function. e.g. for 1d consolidation surcharge radial draiange term is D[kv(z)*D[u(Z,t), Z],Z] so a would be kv. be et.

top_vs_time, bot_vs_time : list of PolyLine

Piecewise linear magnitude vs time for the top and bottom boundary. Use None if there is no variation.

top_omega_phase, bot_omega_phase : list of 2 element tuples, optional

(omega, phase) for use in cos(omega * t + phase) * mag_vs_time if omega_phase is None then mag_vs_time will not be multiplied by a cosine. If any element of omega_phase is None then in that particular loading combo, mag_vs_time will not be multiplied by a cosine.

dT : float, optional

Time factor multiple for numerical convieniece. Default dT=1.0.

theta_zero_indexes : slice/list etc., optional

A slice object, list, etc that can be used for numpy fancy indexing. Any specified index of the theta vector will be set to zero. This is useful when using the spectral method with block matrices and the loading term only refers to a subset of the equations. When using block matrices m should be the same size as the block matrix. Default theta_zero_indexes=None i.e. no elements of theta will be set to zero.

Returns:

foft_Ipsiw_the: ndarray

Additional homgenising term of size (neig, len(t)).

Notes

Assuming the loads are formulated as the product of separate time and depth dependant functions as well as a cyclic component:

\[\sigma\left({Z,t}\right)= \sigma\left({Z}\right) \sigma\left({t}\right) \cos\left(\omega t + \phi\right)\]

the solution to consolidation equation using the spectral method has a similar form to the following:

\[u\left(Z,t\right)= \mathbf{\phi} \mathbf{v} \mathbf{E} \left(\mathbf{\Gamma v}\right)^{-1} \mathbf{\theta}\]

When we consider non-zero boundary conditions, additional loading terms are created when we sub in the following into the original governing equation.

\[u\left({Z,t}\right)= v\left({Z,t}\right) + u_{top}\left({t}\right)\left({1-Z}\right) + u_{bot}\left({b}\right)Z\]

As well as loading terms the process may produce an extra term to be added to the final solution of the following form:

\[\mathbf{\phi} \sigma\left({t}\right) \mathbf{\psi}_{w}^{1} \mathbf{\theta}_w\]

There are actually two of these terms one for the top boundary condition and one for the bottom boundary condition.

This function calculates \(\sigma\left({t}\right)\mathbf{\psi}_{w}^{-1}\mathbf{\theta}_w\)

when substitutions are made in terms of the following form:

\[\frac{\partial}{\partial Z} \left( {a\left({Z}\right) \frac{\partial u\left({Z,t}\right)}{\partial Z}} \right)\]

It is assumed that \(u_{top}\left({t}\right)\) and \(u_{bot}\left({t}\right)\) are piecewise linear in time with a cyclic component, and that multiple functions are superposed. Also \(a\left(Z\right)\) is a piecewise linear function with respect to \(Z\)

For this particular function the \(\mathbf{\theta}\) vector for each load is given by:

\[\mathbf{\theta}_{i}= \int_{0}^1{ \frac{\partial}{\partial Z} \left( {a\left({Z}\right) \frac{\partial \sigma\left({Z}\right)}{\partial Z}} \right) f\left({Z}\right) \phi_i\,dZ}\]

Where \(f\left({Z}\right)\) is the appropriate z-dependent term corresponding to either \(u_{top}\) or \(u_{bot}\) homogenisations.

The time dependent function evaluated at each time produces a matrix that we will call \(\mathbf{E}\) (not to be confused with other E matrices) which is given by:

\[\mathbf{E}_{j}= {\sigma\left(t_j\right)} {\cos\left(\omega t_j+\textrm{phase}\right)}\]

Note that the listed equations above are in terms of normalised depth Z, with depth integrations between [0, 1]. However, IF YOU KNOW WHAT YOU ARE DOING the integrations can be done using non-normalised depths. The first z value in the piecewise definition a(z) must still be 0 however the end point for integration will be the final z value in the definition of a(z). If you are doing this then your m values will include the normalising Factor. e.g. m = [pi/2/H, 3*pi/2/H] and a(z) is defined in two layers [0, z1], [z1, zend] as opposed to m = [pi/2, 3*pi/2] and a(Z) is two layers [0, z1/H], [z1/H, zend/H].

geotecha.speccon.speccon1d.dim1sin_integrate_af(m, z, tvals, v_E_Igamv_the, drn, a, top_vs_time=None, bot_vs_time=None, top_omega_phase=None, bot_omega_phase=None)

Integrate u(Z,t) between Z1 and Z2 where u(Z,t) = phi * v_E_Igam_v_the + utop(t) * (1-Z) + ubot(t)*Z.

Basically calculates the integral phi part for each z pair value, then dot product with v_E_Igamv_the (which has elsewhere been calculated at each tvals value). Then account for non-zero boundary conditions by adding average of utop(t)*(1-Z) and ubot(t)*Z parts for each z, tvals pair.

Use sin(m*Z) for the phi part.

Parameters:

m : list of float

Eigenvalues of BVP, the m in sin(m*Z). Generate with geotecha.speccon.m_from_sin_mx.

z : size (n, 2) 2d numpy.ndarray

Depths to evaluate integral between.

tvals : 1d numpy.ndarray

Time values to evaluate at.

eigs : 1d numpy.ndarray

List of eigenvalues of the spectral matrix i.e. Eigenvalues of the square Igam_psi matrix.

tvals : 1d numpy.ndarray`

List of time values to evaluate E matrix at.

v_E_Igamv_the : ndarray of size (neig, len(tvals))

Speccon matrix.

drn : [0,1]

Drainage condition, drn=0 for Pervious top pervious bottom (PTPB). drn=1 for Pervious top impoervious bottom (PTIB).

top_vs_time, bot_vs_time : list of PolyLine

Piecewise linear magnitude vs time for the top and bottom boundary. Use None if there is no variation.

top_omega_phase, bot_omega_phase : list of 2 element tuples, optional

(omega, phase) for use in cos(omega * t + phase) * mag_vs_time if omega_phase is None then mag_vs_time will not be multiplied by a cosine. If any element of omega_phase is None then in that particular loading combo, mag_vs_time will not be multiplied by a cosine.

Returns:

uintegral : np.ndarray

Integral of pore pressure between depths at each time. Array of size (len(z), len(tvals)).

Notes

The integral of the \(\phi\) term is:

\[\mathbf{\phi}_{i\textrm{integral}}= \int_{z_1}^{z_2}{\sin\left({m_i Z}\right)\,dZ}\]