# Author: <NAME>

'''
Hierarchy of subroutines::

 hh_param_from_profile
   |--- _calc_rho()
           |--- helper_site_response.thk2dep()
   |--- _calc_Gmax()
   |--- _calc_vertical_stress()
   |--- _calc_OCR()
   |--- _calc_K0()
   |--- _calc_PI()
   |--- _calc_shear_strength()
           |--- _calc_K0()
   |--- produce_HH_G_param()
           |---  _calc_mean_confining_stress()
           |---  produce_Darendeli_curves()
                      |--- helper_generic.assert_1D_numpy_array()
                      |--- helper_generic.check_length_or_extend_to_array()
                      |--- _calc_K0()
                      |--- _calc_mean_confining_stress()
           |---  helper_mkz_model.fit_MKZ()
           |---  _optimization_kernel()
                      |
                      |--- hlper_mkz_model.tau_MKZ()
                      |--- helper_generic.find_closest_index()
                      |--- __calc_area()
                                 |--- helper_hh_model.tau_FKZ()
                                 |--- helper_generic.find_closest_index()
                      |--- __find_x_t_and_d()
                                 |--- helper_hh_model.tau_FKZ()
                                 |--- helper_generic.find_closest_index()
'''

import os
import numpy as np
import matplotlib.pyplot as plt

from . import helper_generic as hlp
from . import helper_site_response as sr
from . import helper_mkz_model as mkz
from . import helper_hh_model as hh

#%%----------------------------------------------------------------------------
def hh_param_from_profile(vs_profile, *, Tmax=None, show_fig=False,
                          save_fig=False, fig_output_dir=None,
                          save_HH_G_file=False, HH_G_file_dir=None,
                          profile_name=None, verbose=True):
    '''
    Get HH parameters of each soil layer from the Vs values of every layer.

    Parameters
    ----------
    vs_profile : numpy.ndarray
        Shear-wave velocity profile, as a 2D numpy array. It should have the
        following columns:

            +---------------+----------+---------+---------+-----------------+
            | Thickness (m) | Vs (m/s) | Damping | Density | Material Number |
            +===============+==========+=========+=========+=================+
            |      ...      |   ...    |   ...   |   ...   |      ...        |
            +---------------+----------+---------+---------+-----------------+

    Tmax : numpy.ndarray or ``None``
        Shear strength of each layer of soil. If ``None``, it will be
        calculated using a combination of Ladd (1991) and Mohr-Coulomb criteria.
    show_fig : bool
        Whether or not to show figures G/Gmax and stress-strain curves of MKZ,
        FKZ, and HH for each layer.
    save_fig : bool
        Whether or not to save the figures to the hard drive. Only effective
        if ``show_fig`` is set to ``True``.
    fig_output_dir : str
        The output directory for the figures. Only effective if ``show_fig``
        and ``save_fig`` are both ``True``.
    save_HH_G_file : bool
        Whether or not to save the HH parameters to the hard drive (as a
        "HH_G" file).
    HH_G_file_dir : str
        The output directory for the "HH_G" file. Only effective if
        ``save_HH_G_file`` is ``True``.
    profile_name : str or ``None``
        The name of the Vs profile, such as "CE.12345". If ``None``, a string
        of current date and time will be used as the profile name.
    verbose : bool
        Whether or not to print progresses on the console.

    Returns
    -------
    HH_G_param : numpy.ndarray
        The HH parameters of each layer. It's a 2D array of shape
        ``(9, n_layer)``. For each layer (i.e., column), the values are in
        this order:
            gamma_t, a, gamma_ref, beta, s, Gmax, mu, Tmax, d
    '''
    phi = 30.  # friction angle (choose 30 degrees, because there is no better info)

    hlp.check_Vs_profile_format(vs_profile)
    h = vs_profile[:-1, 0]
    Vs = vs_profile[:-1, 1]  # exclude the last layer (i.e., half space)

    n_layer = len(Vs)
    if Tmax is not None:
        hlp.assert_array_length(Tmax, n_layer)

    rho      = _calc_rho(h, Vs)
    Gmax     = _calc_Gmax(Vs, rho)
    sigma_v0 = _calc_vertical_stress(h, rho)
    OCR      = _calc_OCR(Vs, rho, sigma_v0)
    K0       = _calc_K0(OCR, phi=phi)
    PI       = _calc_PI(Vs)

    if Tmax is None:
        Tmax = _calc_shear_strength(Vs, OCR, sigma_v0, K0=K0, phi=phi)

    HH_G_param = produce_HH_G_param(Vs, Gmax, Tmax, OCR, sigma_v0, K0,
                                    curves=None, PI=PI, phi=phi,
                                    show_fig=show_fig, save_fig=save_fig,
                                    fig_output_dir=fig_output_dir,
                                    verbose=verbose)

    if save_HH_G_file:
        if HH_G_file_dir is None:
            raise ValueError('Please specify `HH_G_file_dir`.')
        if profile_name is None:
            profile_name = hlp.get_current_time(for_filename=True)
        np.savetxt(os.path.join(HH_G_file_dir, 'HH_G_%s.txt' % profile_name),
                   HH_G_param, delimiter='\t', fmt='%.6g')

    return HH_G_param

#%%----------------------------------------------------------------------------
def hh_param_from_curves(vs_profile, curves, *, Tmax=None, show_fig=False,
                         save_fig=False, fig_output_dir=None, save_HH_G_file=False,
                         HH_G_file_dir=None, profile_name=None, verbose=True):
    '''
    Get HH parameters of each soil layer from the Vs profile and G/Gmax curves.

    Parameters
    ----------
    vs_profile : numpy.ndarray
        Shear-wave velocity profile, as a 2D numpy array. It should have the
        following columns:
            +---------------+----------+---------+---------+-----------------+
            | Thickness (m) | Vs (m/s) | Damping | Density | Material Number |
            +===============+==========+=========+=========+=================+
            |      ...      |   ...    |   ...   |   ...   |      ...        |
            +---------------+----------+---------+---------+-----------------+

    curves : numpy.ndarray
        A 2D numpy array that represents G/Gmax and damping curves of each
        layer, in the following format:
         +------------+--------+------------+-------------+-------------+--------+-----+
         | strain [%] | G/Gmax | strain [%] | damping [%] |  strain [%] | G/Gmax | ... |
         +============+========+============+=============+=============+========+=====+
         |    ...     |  ...   |    ...     |    ...      |    ...      |  ...   | ... |
         +------------+--------+------------+-------------+-------------+--------+-----+

        The damping information is neglected in this function, so users can
        supply some dummy values.
    Tmax : numpy.ndarray or ``None``
        Shear strength of each layer of soil. If ``None``, it will be
        calculated using a combination of Ladd (1991) and Mohr-Coulomb criteria.
    show_fig : bool
        Whether or not to show figures G/Gmax and stress-strain curves of MKZ,
        FKZ, and HH for each layer.
    save_fig : bool
        Whether or not to save the figures to the hard drive. Only effective
        if ``show_fig`` is set to ``True``.
    fig_output_dir : str
        The output directory for the figures. Only effective if ``show_fig``
        and ``save_fig`` are both ``True``.
    save_HH_G_file : bool
        Whether or not to save the HH parameters to the hard drive (as a
        "HH_G" file).
    HH_G_file_dir : str
        The output directory for the "HH_G" file. Only effective if
        ``save_HH_G_file`` is ``True``.
    profile_name : str or ``None``
        The name of the Vs profile, such as "CE.12345". If ``None``, a string
        of current date and time will be used as the profile name.
    verbose : bool
        Whether or not to print progresses on the console.

    Returns
    -------
    HH_G_param : numpy.ndarray
        The HH parameters of each layer. It's a 2D array of shape
        ``(9, n_layer)``. For each layer (i.e., column), the values are in
        this order:
            gamma_t, a, gamma_ref, beta, s, Gmax, mu, Tmax, d
    '''
    phi = 30.0

    hlp.check_Vs_profile_format(vs_profile)
    h = vs_profile[:-1, 0]
    Vs = vs_profile[:-1, 1]  # exclude the last layer (i.e., half space)
    n_layer = len(Vs)

    if vs_profile.shape[1] == 5:  # there can only be 5 or 2 columns
        mat = vs_profile[:-1, -1]
        rho = vs_profile[:-1, 3]
    else:  # only 2 columns
        mat = np.arange(1, n_layer+1)
        rho = _calc_rho(h, Vs)

    if Tmax is not None:
        hlp.assert_array_length(Tmax, n_layer)

    Gmax = _calc_Gmax(Vs, rho)
    sigma_v0 = _calc_vertical_stress(h, rho)
    OCR = _calc_OCR(Vs, rho, sigma_v0)
    K0 = _calc_K0(OCR, phi=phi)

    if Tmax is None:
        Tmax = _calc_shear_strength(Vs, OCR, sigma_v0, K0=K0, phi=phi)

    curves_old = curves.copy()
    curves_expanded = None
    for j in range(n_layer):
        tmp = curves_old[:, int(mat[j]) * 4 - 4: int(mat[j]) * 4]
        if curves_expanded is None:
            curves_expanded = tmp
        else:
            curves_expanded = np.column_stack((curves_expanded, tmp))
    curves = curves_expanded

    HH_G_param = produce_HH_G_param(Vs, Gmax, Tmax, OCR, sigma_v0, K0,
                                    curves=curves, PI=None, phi=None,
                                    show_fig=show_fig, save_fig=save_fig,
                                    fig_output_dir=fig_output_dir,
                                    verbose=verbose)

    if save_HH_G_file:
        if HH_G_file_dir is None:
            raise ValueError('Please specify `HH_G_file_dir`.')
        if profile_name is None:
            profile_name = hlp.get_current_time(for_filename=True)
        np.savetxt(os.path.join(HH_G_file_dir, 'HH_G_%s.txt' % profile_name),
                   HH_G_param, delimiter='\t', fmt='%.6g')

    return HH_G_param

#%%----------------------------------------------------------------------------
def produce_HH_G_param(Vs, Gmax, Tmax, OCR, sigma_v0, K0, curves=None,
                       PI=None, phi=None, show_fig=False, save_fig=False,
                       fig_output_dir=None, verbose=True):
    '''
    Produce HH_G parameters from profiles of Vs, Tmax, OCR, etc.

    Parameters
    ----------
    Vs : numpy.ndarray
        Vs values of each layer. Shape: ``(n_layer, )``, where ``n_layer`` is
        the length of ``Vs``. Unit: m/s.
    Gmax : numpy.ndarray
        Initial stiffness of each layer. Shape: ``(n_layer, )``. Unit: Pa.
    Tmax : numpy.ndarray
        The shear strength of each layer. Shape: ``(n_layer, )``. Unit: Pa.
    OCR : numpy.ndarray
        Over-consolidation ratio of each layer. Shape: ``(n_layer, )``.
    sigma_v0 : numpy.ndarray
        Vertical effective confining stress of each layer. Shape:
        ``(n_layer, )``. Unit: Pa.
    K0 : numpy.ndarray or float
        Lateral soil pressure coefficient. If an array, it must have shape
        ``(n_layer, )``. If a single value, it means that all layers share
        this same value.
    curves : numpy.ndarray or ``None``
        A 2D numpy array that represents G/Gmax and damping curves of each
        layer, in the following format:
         +------------+--------+------------+-------------+-------------+--------+-----+
         | strain [%] | G/Gmax | strain [%] | damping [%] |  strain [%] | G/Gmax | ... |
         +============+========+============+=============+=============+========+=====+
         |    ...     |  ...   |    ...     |    ...      |    ...      |  ...   | ... |
         +------------+--------+------------+-------------+-------------+--------+-----+

        The damping information is neglected in this function, so users can
        supply some dummy values. If ``None``, it means that the users do not
        have G/Gmax curve information, so this function will calculate the
        MKZ G/Gmax curves from the empirical formulas in Darendeli (2001).
    PI : float or numpy.ndarray or ``None``
        Plasticity index of the soils. It is not necessary (can be ``None``) if
        ``curves`` is provided (i.e., not ``None``). If an array, it must have
        shape ``(n_layer, )``. If a single value, it means that all layers
        share this same value.
    phi : float or numpy.ndarray or ``None``
        Effective internal frictional angle (in degrees). It is not necessary
        (can be ``None``) if ``curve`` is provided (i.e., not ``None``). If
        an array, it must have shape ``(n_layer, )``. If a single value, it
        means that all layers share this same value.
    show_fig : bool
        Whether or not to show figures G/Gmax and stress-strain curves of MKZ,
        FKZ, and HH for each layer.
    save_fig : bool
        Whether or not to save the figures to the hard drive. Only effective
        if ``show_fig`` is set to ``True``.
    fig_output_dir : str
        The output directory for the figures. Only effective if ``show_fig``
        and ``save_fig`` are both ``True``.
    verbose : bool
        Whether or not to print progresses on the console.

    Returns
    -------
    parameters : numpy.ndarray
        The HH parameters of each layer. It's a 2D array of shape
        ``(9, n_layer)``. For each layer (i.e., column), the values are in
        this order:
            gamma_t, a, gamma_ref, beta, s, Gmax, mu, Tmax, d

    Notes
    -----
    This function is based on ``hybridParaKernel_FKZ.m``.
    '''
    hlp.assert_1D_numpy_array(Vs, '`Vs`')
    n_layer = len(Vs)

    hlp.assert_array_length(Gmax, n_layer, name='`Gmax`')
    hlp.assert_array_length(Tmax, n_layer, name='`Tmax`')
    hlp.assert_array_length(OCR, n_layer, name='`OCR`')
    hlp.assert_array_length(sigma_v0, n_layer, name='`sigma_v0`')

    K0 = hlp.check_length_or_extend_to_array(K0, n_layer, name='`K0`')
    p0 = _calc_mean_confining_stress(sigma_v0, K0)

    if verbose:
        print('========== Start optimizing for HH_G parameters ===========')

    # ============= MKZ fit ===================================================
    if curves is None:  # user does not provide curves
        if verbose:
            print('------ G/Gmax not provided; will generate MKZ curves using '
                  'Darendeli (2001): ------')

        strain_ = np.geomspace(1e-4, 10, 400)  # unit: percent
        GGmax, _, gamma_ref = produce_Darendeli_curves(sigma_v0, PI, OCR=OCR,
                                                       K0=K0, phi=phi,
                                                       strain_in_pct=strain_)
        strain = np.tile(strain_, (n_layer, 1)).T  # strain matrix for all layers
        beta = np.ones(n_layer)
        s = 0.9190 * np.ones(n_layer)
    else:  # user provides own curves
        if verbose:
            print('------ G/Gmax provided; fitting MKZ curves to data: ------')
        hlp.assert_2D_numpy_array(curves)
        assert(curves.shape[1] == n_layer * 4)

        # ----------- Extract G/Gmax information ------------------------------
        strain = curves[:, 0::4]  # unit: percent
        GGmax = curves[:, 1::4]

        # ----------- Fit MKZ parameters --------------------------------------
        param_MKZ, _ = mkz.fit_MKZ(curves, show_fig=show_fig)
        gamma_ref = param_MKZ[:, 0]
        s = param_MKZ[:, 2]
        beta = param_MKZ[:, 3]

        # ----------- Show results on console ---------------------------------
        if verbose:
            print('****** MKZ parameters: ******')
        for j in range(n_layer):
            if verbose:
                print('Layer %d: gamma_ref = %.3g, s = %.3g, beta = %.3g' \
                      % (j, gamma_ref[j], s[j], beta[j]))

    # ========== Stress-strain curve implied by G/Gmax ========================
    sigma = np.zeros_like(GGmax)
    for j in range(n_layer):
        sigma[0, j] = 0
        for k in range(1, GGmax.shape[0]):
            sigma[k, j] = GGmax[k, j] * Gmax[j] * strain[k, j] / 100.0
        # END FOR
    # END FOR

    # ========== Estimate mu using empirical correlations =====================
    p0 = p0 / 1000.0  # unit: Pa --> kPa

    mu = np.zeros_like(OCR)
    for j in range(n_layer):
        if Vs[j] <= 760:  # softer soil: use Vardanega & Bolton (2011) CGJ formula
            mu[j] = 1.0 / (0.000872 * Gmax[j]/Tmax[j] * OCR[j]**0.47 * p0[j]**0.28)
            if mu[j] <= 0.02:  # mu too small --> too low tau_FKZ --> sharply decreasing tau_HH
                mu[j] = mu[j] * 10.0 ** (0.236 * 3)  # 0.236 is the standard error suggested in Vardanega & Bolton (2011)
            elif mu[j] <= 0.03:
                mu[j] = mu[j] * 10.0 ** (0.236 * 2)
            elif mu[j] <= 0.04:
                mu[j] = mu[j] * 10.0 ** (0.236 * 1)
            # END IF
        else:  # stiffer soils: set mu to 1 for lack of better information
            mu[j] = 1.0
        # END IF
    # END FOR

    # ========== Start FKZ optimization =======================================
    if verbose:
        print('-----------  FKZ optimization  -----------------------------')
    parameters = np.zeros((9, n_layer))

    lw = 1.0
    muted_blue = np.array([107, 174, 214]) / 255.
    muted_green = np.array([120, 198, 121]) / 255.
    muted_red = np.array([222, 45, 38]) / 255.

    for j in range(n_layer):
        strain_j = strain[:, j]
        a, gamma_t, d = _optimization_kernel(strain_j / 100.0, gamma_ref[j],
                                             beta[j], s[j], Gmax[j], Tmax[j],
                                             mu[j])
        if verbose:
            print('%d/%d: mu = %.3f, a = %.1f, gamma_t = %.3f%%, d = %.3f' \
                  % (j + 1, n_layer, mu[j], a, gamma_t * 100, d))
        T_FKZ = hh.tau_FKZ(strain_j / 100.0, Gmax=Gmax[j], mu=mu[j], d=d,
                           Tmax=Tmax[j])
        T_HH = hh.tau_HH(strain_j / 100.0, gamma_t=gamma_t, a=a,
                         gamma_ref=gamma_ref[j], beta=beta[j], s=s[j],
                         Gmax=Gmax[j], mu=mu[j], Tmax=Tmax[j], d=d)

        parameters[0, j] = gamma_t
        parameters[1, j] = a
        parameters[2, j] = gamma_ref[j]
        parameters[3, j] = beta[j]
        parameters[4, j] = s[j]
        parameters[5, j] = Gmax[j]
        parameters[6, j] = mu[j]
        parameters[7, j] = Tmax[j]
        parameters[8, j] = d

        GGmax_HH = T_HH / (Gmax[j] * (strain_j / 100.0))

        if show_fig:
            fig = plt.figure(figsize=(4.2, 6.0))

            plt.subplot(211)
            if curves is None:
                plt.semilogx(strain_j, sigma[:, j] / 1000., c=muted_blue,
                             lw=lw*2.5, label='MKZ')  # Darendeli's curve
                plt.semilogx(strain_j, T_FKZ / 1000., c=muted_green, lw=lw*1.75,
                             label='FKZ')
            else:
                plt.semilogx(strain_j, sigma[:, j] / 1000., c=muted_blue,
                             marker='o', ls='-', lw=lw*2.5,
                             label='Given $G/G_{\max}$')
                plt.semilogx(strain_j, T_FKZ / 1000., c=muted_green, lw=lw*1.75,
                             label='FKZ')
            plt.grid(ls=':', lw=0.5)
            plt.plot([np.min(strain_j), np.max(strain_j)],
                     np.array([Tmax[j], Tmax[j]]) / 1000., lw=lw, c='gray',
                     ls='--')
            plt.plot(strain_j, T_HH / 1000., c=muted_red, lw=lw, label='HH')
            plt.plot([gamma_t * 100] * 2, plt.ylim(), ls='--', c='gray')
            plt.ylabel('Stress [kPa]')
            plt.xlim(np.min(strain_j), np.max(strain_j))
            plt.legend(loc='upper left')
            plt.title('$V_S$ = %.1f m/s, $G_{\max}$ = %.3f MPa,'
                      '\n$\\tau_{\mathrm{ff}}$ = %.3f kPa, '
                      '$\gamma_{\mathrm{ref}}$ = %.3f%%' \
                      % (Vs[j], Gmax[j]/1e6, Tmax[j]/1e3, gamma_ref[j]*100))

            plt.subplot(212)
            if curves is None:
                plt.semilogx(strain_j, GGmax[:, j], c=muted_blue, lw=lw*2.5)
            else:
                plt.semilogx(strain_j, GGmax[:, j], c=muted_blue, ls='-',
                             marker='o', lw=lw*2.5)
            plt.grid(ls=':', lw=0.5)
            plt.plot(strain_j,
                     mu[j] / (1 + Gmax[j]/Tmax[j]*mu[j]*np.abs(strain_j/100.) ),
                     c=muted_green, lw=lw*1.75)
            plt.plot(strain_j, GGmax_HH, c=muted_red, lw=lw)
            plt.plot([gamma_t * 100] * 2, plt.ylim(), ls='--', c='gray')
            plt.ylabel('$G/G_{\max}$')
            plt.xlabel('Strain [%]')
            plt.xlim(np.min(strain_j), np.max(strain_j))
            plt.title("$\mu$ = %.3f, a = %.1f, $\gamma_{\mathrm{t}}$ = %.4f%%\n"
                      "d = %.4f, $p'_{\mathrm{m0}}$ = %.2f kPa" \
                      % (mu[j], a, gamma_t * 100, d, p0[j]))

            fig.tight_layout(pad=0.5, h_pad=1.2, w_pad=0.3)

            if save_fig:
                if fig_output_dir is None:
                    raise ValueError('Please specify `fig_output_dir`.')
                fig.savefig(os.path.join(fig_output_dir,
                                         'Stress_GGmax_of_Layer_#%d.png' % (j+1)))

    return parameters

#%%----------------------------------------------------------------------------
def _calc_shear_strength(Vs, OCR, sigma_v0, K0=None, phi=30.0):
    '''
    Calculate shear strength of soils.

    Parameters
    ----------
    Vs : numpy.ndarray
        A 1D array of Vs values of each layer. Unit: m/s.
    OCR : numpy.ndarray
        A 1D array of OCR (over-consolidation ratio) of each layer. Unit: 1.
    sigma_v0 : numpy.ndarray
        A 1D array of vertical overburden pressure. Unit: Pa.
    K0 : float, int, numpy.ndarray, or ``None``
        Lateral soil pressure coefficient. If a single value is given, it is
        assumed to be the value for all layers. If ``None``, it will be
        determined from OCR via an empirical formula by Mayne & Kulhawy (1982).
    phi : float, int, or numpy.ndarray
        Effective internal friction angle of soils (in degrees).

    Returns
    -------
    Tmax : numpy.ndarray
        Shear strength of soils of each layer. Unit: Pa.
    '''
    dyna_coeff = 1.2  # assume a strain rate of 0.01 sec^(-1), from Vardanega & Bolton (2013)

    phi = hlp.check_length_or_extend_to_array(phi, len(Vs), name='`phi`')
    if K0 is None:
        K0 = _calc_K0(OCR, phi=phi)
    else:
        K0 = hlp.check_length_or_extend_to_array(K0, len(Vs), name='`K0`')

    Tmax = np.zeros(len(Vs))
    for j in range(len(Vs)):
        if Vs[j] <= 760:  # for softer soils, calculate undrained shear strength
            Tmax[j] = dyna_coeff * 0.28 * OCR[j]**0.8 * sigma_v0[j]  # Ladd (1991)
        else:  # stiffer soils: Mohr-Coulomb criterion
            sigma_h0 = K0[j] * sigma_v0[j]  # horizontal stress
            sigma_1 = np.max([sigma_v0[j], sigma_h0])  # largest principal stress
            sigma_3 = np.min([sigma_v0[j], sigma_h0])  # smallest principal stress
            # normal effective stress on the slip plane
            sigma_n = (sigma_1 + sigma_3)/2.0 \
                      - (sigma_1 - sigma_3)/2.0 * np.sin(np.deg2rad(phi[j]))
            Tmax[j] = dyna_coeff * sigma_n * np.tan(np.deg2rad(phi[j]))

    return Tmax

#%%----------------------------------------------------------------------------
def _calc_Gmax(Vs, rho):
    '''
    Calculate initial stiffness of each soil layer.

    Parameters
    ----------
    Vs : numpy.ndarray
        1D array of Vs of layers. Unit: m/s.
    rho : numpy.ndarray
        1D array of mass density of layers. Unit: kg/m^3.

    Returns
    -------
    Gmax : numpy.ndarray
        1D array of initial stiffness. Unit: Pa
    '''
    Gmax = rho * Vs**2
    return Gmax

#%%----------------------------------------------------------------------------
def _calc_OCR(Vs, rho, sigma_v0, OCR_upper_limit=None):
    '''
    Calculate OCR (over-consolidation ratio) of each layer from the Vs profile.

    Parameters
    ----------
    Vs : numpy.ndarray
        1D array of Vs of layers. Unit: m/s.
    rho : numpy.ndarray
        1D array of mass density of layers. Unit: kg/m^3.
    sigma_v0 : numpy.ndarray
        Vertical overburden stress at the mid-point of each layer. Unit: Pa.
    OCR_upper_limit : float or ``None``
        The maximum acceptable OCR value. If ``None``, there is no limit.

    Returns
    -------
    OCR : numpy.ndarray
        1D array of OCR value, for each soil layer. (Unitless.)
    '''
    sigma_p0 = 0.106 * Vs**1.47  # Mayne, <NAME> (1998) "Clay stress history evaluated fromseismic piezocone tests"
    sigma_p0 = sigma_p0 * 1000  # kPa --> Pa
    OCR = sigma_p0 / sigma_v0
    OCR = np.minimum(OCR, np.inf if OCR_upper_limit is None else OCR_upper_limit)
    return OCR

#%%----------------------------------------------------------------------------
def _calc_vertical_stress(h, rho):
    '''
    Calculate vertical overburden stress at the mid-point of each layer.

    Parameters
    ----------
    h : numpy.ndarray
        1D array of layer thickness. Unit: m.
    rho : numpy.ndarray
        1D array of mass density of each layer. Unit: kg/m^3.

    Returns
    -------
    stress : numpy.ndarray
        Vertical overburden stress at the mid-point of each layer. Unit: Pa.
    '''
    g = 9.81  # unit: m/s/s
    n = len(h)
    stress = np.zeros_like(h)

    if np.mean(rho) < 1000:
        print('Warning in __calc_vertical_stress(): It looks like the unit '
              'of mass density is g/cm^3. The correct unit should be kg/m^3.')

    if h[-1] == 0:  # zero thickness, i.e., half space
        h[-1] = 1

    stress[0] = rho[0] * g * h[0]/2  # divided by 2: middle of layer
    for i in range(1, n):
        stress[i] = stress[i-1] + rho[i-1] * g * h[i-1]/2 + rho[i] * g * h[i]/2

    return stress

#%%----------------------------------------------------------------------------
def _calc_rho(h, Vs):
    '''
    Calculate mass density of soils from Vs values, using the empirical formula
    by Mayne, Schneider & Martin (1999) and Burns & Mayne (1996).

    Parameters
    ----------
    h : numpy.ndarray
        The thickness of each soil layer. Unit: m.
    Vs : numpy.ndarray
        The shear-wave velocity for each layer. It needs to be a 1D numpy array.
        Unit: m/s.

    Returns
    -------
    rho : numpy.ndarray
        Mass density of each soil layer. Unit: kg/m^3.

    References
    ----------
    1. Mayne, Schneider & Martin (1999) "Small- and large-strain soil
       properties from seismic flat dilatometer tests." Pre-failure
       deformation characteristics of geomaterials, 1999 Balkema, Rotterdam.
    2. Burns & Mayne (1996) "Small- and high-strain soil properties using the
       seismic piezocone." Transportation Research Record 1548, National
       Acad. Press, Washington DC, 81-88.
    '''
    z = sr.thk2dep(h, midpoint=False)
    z[z == 0] = 0.0001  # avoid error of dividing by zero
    lb = 1.65  # lower bound of density: 1.65 g/cm^3
    rho = np.maximum(lb, 1 + 1. / (0.614 + 58.7 * (np.log(z) + 1.095) / Vs))
    rho *= 1000  # unit: g/cm^3 --> kg/m^3
    return rho

#%%----------------------------------------------------------------------------
def _calc_PI(Vs):
    '''
    Calculate PI (plasticity index) from Vs values.

    Parameters
    ----------
    Vs : numpy.ndarray
        The shear-wave velocity for each layer. It needs to be a 1D numpy array.
        Unit: m/s.

    Returns
    -------
    PI : numpy.ndarray
        The plasticity index for each layer. Unit: %.
    '''
    PI = np.zeros_like(Vs)
    for j in range(len(Vs)):
        if Vs[j] <= 200:
            PI[j] = 10
        elif Vs[j] <= 360:
            PI[j] = 5
        else:
            PI[j] = 0

    return PI

#%%----------------------------------------------------------------------------
def _calc_K0(OCR, phi=30.):
    '''
    Calculate K0 (lateral earth pressure coefficient at rest) from OCR using
    the empirical formula by Mayne & Kulhawy (1982).

    Parameters
    ----------
    OCR : float, int, or numpy.ndarray
        Over-consolidation ratio of each layer of soils. If it is a float/int,
        it means only one layer, or all the layers have the same OCR.
    phi : float, int, or numpy.ndarray
        Internal effective friction angle of soils. If it is a float/int, it
        means only one soil layer, or all the layers have the same angle.
        Unit: deg.

    Returns
    -------
    K0 : float or numpy.ndarray
        K0 value(s). If either ``OCR`` or ``phi`` is an array, ``K0`` will be
        an array of the same length.
    '''
    K0 = (1 - np.sin(np.deg2rad(phi))) * OCR ** (np.sin(np.deg2rad(phi)))
    return K0

#%%----------------------------------------------------------------------------
def produce_Darendeli_curves(sigma_v0, PI=20., OCR=1., K0=0.5, phi=30.0,
                             strain_in_pct=None):
    '''
    Produce G/Gmax and damping curves using empirical correlations by
    Darendeli (2001).

    Parameters
    ----------
    sigma_v0 : numpy.ndarray
        Effective vertical confining stress of each layer. Unit: Pa.
    PI : int, float, or numpy.ndarray
        Plasticity index of each layer. Unit: %. If a single value is given,
        it is assumed to be the PI for all layers.
    OCR : int, float, or numpy.ndarray
        Over-consolidation ratio of each layer. If a single value is given,
        it is assumed to be the value for all layers.
    K0 : int, float, numpy.ndarray, or ``None``
        Lateral soil pressure coefficient. If a single value is given, it is
        assumed to be the value for all layers. If ``None``, it will be
        determined from OCR via an empirical formula by Mayne & Kulhawy (1982).
    phi : int, float, or numpy.ndarray
        Internal effective friction angle of soils. If it is a float/int, it
        means all the layers have the same angle. Unit: deg.
    strain_in_pct : numpy.ndarray or ``None``
        The strain values at which to calculate G/Gmax and damping values. If
        ``None``, numpy.geomspace(1e-4, 10, 400) will be used. Unit: percent.

    Returns
    -------
    GGmax : numpy.ndarray
        G/Gmax curves for each layer. It is a 2D numpy array. Each column of it
        represents the G/Gmax curve of a particular layer. Unit: 1
    xi : numpy.ndarray
        Damping curves for each layer. Same shape as ``GGmax``. Unit: 1.
    gamma_r : numpy.ndarray
        Reference strain for each layer. It is a 1D numpy array, corresponding
        to each soil layer. Unit: 1.
    '''
    hlp.assert_1D_numpy_array(sigma_v0)
    n_layer = len(sigma_v0)

    phi = hlp.check_length_or_extend_to_array(phi, n_layer, name='`phi`')
    PI = hlp.check_length_or_extend_to_array(PI, n_layer, name='`PI`')
    OCR = hlp.check_length_or_extend_to_array(OCR, n_layer, name='`OCR`')

    if K0 is None:
        K0 = _calc_K0(OCR, phi=phi)
    else:
        K0 = hlp.check_length_or_extend_to_array(K0, n_layer, name='`K0`')

    if strain_in_pct is None:
        gamma = np.geomspace(1e-4, 10, 400)
    else:
        gamma = strain_in_pct.copy()

    # Define all constants
    nr_cycle = 10
    frq = 1
    N = nr_cycle

    phi1 = 0.0352
    phi2 = 0.0010
    phi3 = 0.3246
    phi4 = 0.3483
    phi5 = 0.9190
    phi6 = 0.8005
    phi7 = 0.0129
    phi8 = -0.1069
    phi9 = -0.2889
    phi10 = 0.2919
    phi11 = 0.6329
    phi12 = -0.0057
    a = phi5

    c1 = -1.1143*a**2 + 1.8618*a + 0.2523  # from Darendeli (2001), page 226
    c2 = 0.0805*a**2 - 0.0710*a - 0.0095
    c3 = -0.0005*a**2 + 0.0002*a + 0.0003
    b = phi11 + phi12 * np.log(N)  # Darendeli (2001) Eq 9.1d

    # Confinine stress
    sigma_0 = _calc_mean_confining_stress(sigma_v0, K0)  # octahedral stress
    sigma_0 = sigma_0 / 101325.0  # unit: Pa --> atm
    n_strain_pts = len(strain_in_pct)

    # Reference strain for each layer (Eq 9.1a). Unit: percent
    gamma_r = (phi1 + phi2 * PI * OCR**phi3) * sigma_0**phi4

    GGmax = np.zeros((n_strain_pts, n_layer))
    xi = np.zeros_like(GGmax)
    for i in range(n_layer):
        GGmax[:, i] = 1. / (1 + (gamma / gamma_r[i])**a)  # G of i-th layer (Eq 9.2a)
        D_masing_1 = (100. / np.pi) \
            * (4 * (gamma - gamma_r[i] * np.log((gamma + gamma_r[i]) / gamma_r[i])) \
                / (gamma**2 / (gamma + gamma_r[i])) - 2)  # Unit: percent (page 226)
        D_masing = c1 * D_masing_1 + c2 * D_masing_1**2 + c3 * D_masing_1**3  # Unit: percent (page 226)
        D_min = (phi6 + phi7 * PI[i] * OCR[i]**phi8) * sigma_0[i]**phi9 * (1 + phi10 * np.log(frq))  # Eq 9.1c (page 221)
        xi[:, i] = b * GGmax[:, i]**0.1 * D_masing + D_min  # Eq 9.2b (page 224). Unit: percent

    xi /= 100.0
    gamma_r /= 100.0
    return GGmax, xi, gamma_r

#%%----------------------------------------------------------------------------
def _calc_mean_confining_stress(sigma_v0, K0):
    '''
    Calculate mean (of three directions) confining stress.

    Parameters
    ----------
    sigma_v0 : numpy.ndarray
        (Effective) vertical stress of each layer. Unit: Pa.
    K0 : numpy.ndarray
        Lateral stress coefficient of each layer. Unit: 1.

    Returns
    -------
    sigma_m0 : numpy.ndarray
        Mean effective confining stress (of three directions). Unit: Pa.
    '''
    sigma_m0 = (2 * K0 + 1)/3.0 * sigma_v0
    return sigma_m0

#%%----------------------------------------------------------------------------
def _optimization_kernel(x, x_ref, beta, s, Gmax, tau_f, mu):
    '''
    Optimization process to find FKZ parameters.

    Parameters
    ----------
    x : numpy.ndarray
        An 1D array of shear strain. Unit: 1.
    x_ref : float
        The "reference strain" parameter (in MKZ) of the soil. Unit: 1.
    beta : float
        A shape parameter of the FKZ model.
    s : float
        A shape parameter of the FKZ model.
    Gmax : float
        Initial shear modulus. Unit: Pa.
    tau_f : float
        The shear strength of the current soil layer. Unit: Pa.
    mu : float
        The "shape parameter" of the FKZ model.

    Returns
    -------
    a : float
        A parameter of the HH model that defines the "speed" of transition from
        MKZ to FKZ
    gamma_t : float
        The shear strain at which the transition from MKZ to FKZ happens.
        Unit: 1
    d : float
        The "shape power" parameter in the FKZ model.

    Notes
    -----
    Based on optHybFKZ.m
    '''
    T_MKZ = mkz.tau_MKZ(x, gamma_ref=x_ref, beta=beta, s=s, Gmax=Gmax)
    if mu <= 0.03:  # when mu is too small, there may be some numerical issues
        gamma_t_LB = 0.001  # therefore gamma_t lower bound is relaxed
    else:
        gamma_t_LB = 0.01

    gamma_t_UB = 3.0  # unit: percent

    index_gamma_t_LB, _ = hlp.find_closest_index(x, gamma_t_LB / 100.0)
    if T_MKZ[index_gamma_t_LB] >= 0.85 * tau_f:
        gamma_t_LB = 0.005  # for very deep layers, tau_MKZ may be larger than tau_f at gamma_t_LB

    index_gamma_t_LB, _ = hlp.find_closest_index(x, gamma_t_LB / 100.0)  # do it again
    if T_MKZ[index_gamma_t_LB] >= 0.85 * tau_f:
        gamma_t_LB = 0.001

    range_d = np.linspace(0.67, 1.39, 200)
    area = __calc_area(range_d, x, Gmax, mu, tau_f, gamma_t_LB, gamma_t_UB, T_MKZ)
    if np.min(area) < np.inf:  # it means that a proper d value is found
        gamma_t, d = __find_x_t_and_d(area, range_d, x, Gmax, mu, tau_f, T_MKZ)
    else:  # cannot find a proper d value
        range_d = np.linspace(0.67, 1.39, 400)  # increase grid density to 400
        area = __calc_area(range_d, x, Gmax, mu, tau_f, gamma_t_LB,
                           gamma_t_UB, T_MKZ)
        if np.min(area) < np.inf:
            gamma_t, d = __find_x_t_and_d(area, range_d, x, Gmax, mu, tau_f, T_MKZ)
        else:
            range_d = np.linspace(0.67, 1.39, 1000)  # increase grid density
            new_gamma_t_LB = 0.005  # further relax
            area = __calc_area(range_d, x, Gmax, mu, tau_f, new_gamma_t_LB,
                               gamma_t_UB, T_MKZ)
            if np.min(area) < np.inf:
                gamma_t, d = __find_x_t_and_d(area, range_d, x, Gmax, mu, tau_f, T_MKZ)
            else:
                d = 1.03
                gamma_t = 1e-3 / 100.0  # further ralax to 0.001%
            # END IF
        # END IF
    #END IF

    a = 100.0  # always use a fast transition
    return a, gamma_t, d

#%%----------------------------------------------------------------------------
def __find_x_t_and_d(area, range_d, x, Gmax, mu, tau_f, T_MKZ):
    '''
    Find the ``x_t`` (transition strain) that minimizes the "area" between
    the MKZ stress curve and the FKZ stress curve.

    Parameters
    ----------
    area : numpy.ndarray
        The "area" between the MKZ stress curve and the FKZ stress curve. It
        has the same shape as ``range_d``, because each element of ``area`` is
        the area corresponding to a ``d`` value within ``range_d``.
    range_d : numpy.ndarray
        The range of ``d`` to search from. Must be a 1D numpy array.
    x : numpy.ndarray
        An 1D array of shear strain. Unit: 1.
    Gmax : float
        Initial shear modulus. Unit: Pa.
    mu : float
        The "shape parameter" of the FKZ model.
    tau_f : float
        The shear strength of the current soil layer. Unit: Pa.
    T_MKZ : numpy.ndarray
        The MKZ stress curve, which has the same shape as ``x``. Unit: Pa.

    Returns
    -------
    x_t : float
        The ``x_t`` value that minimizes the "area" between the MKZ stress
        curve and the FKZ stress curve. Unit: 1.
    d : float
        The ``d`` value that minimizes the "area" between the MKZ stress curve
        and the FKZ stress curve. (No unit.)
    '''
    j_ = np.argmin(area)
    d = range_d[j_]
    T_FKZ = hh.tau_FKZ(x, Gmax=Gmax, mu=mu, d=d, Tmax=tau_f)
    copt, _ = hlp.find_closest_index(np.abs(T_MKZ - T_FKZ), 0)
    x_t = x[copt]

    return x_t, d

#%%----------------------------------------------------------------------------
def __calc_area(range_d, x, Gmax, mu, tau_f, gamma_t_LB, gamma_t_UB, T_MKZ):
    '''
    Calculate the "area" between the MKZ stress curve and the FKZ stress curve.
    The MKZ stress curve is supplied as a parameter, and the FKZ stress curve
    is determined by ``x``, ``Gmax``, ``mu``, ``d``, ``tau_f``, and ``gamma_t``.

    Parameters
    ----------
    range_d : numpy.ndarray
        The range of ``d`` to search from. Must be a 1D numpy array.
    x : numpy.ndarray
        An 1D array of shear strain. Unit: 1.
    Gmax : float
        Initial shear modulus. Unit: Pa.
    mu : float
        The "shape parameter" of the FKZ model.
    tau_f : float
        The shear strength of the current soil layer. Unit: Pa.
    gamma_t_LB : float
        The lower bound of ``gamma_t`` (:math:`\gamma_t`), i.e., the transition
        strain. Unit: %.
    gamma_t_UB : float
        The upper bound of ``gamma_t``. Unit: %.
    T_MKZ : numpy.ndarray
        The MKZ stress curve, which has the same shape as ``x``. Unit: Pa.

    Returns
    -------
    area : numpy.ndarray
        The "area" between the MKZ stress curve and the FKZ stress curve. It
        has the same shape as ``range_d``, because each element of ``area`` is
        the area corresponding to a ``d`` value within ``range_d``.
    '''
    area = np.zeros_like(range_d)
    for j in range(len(range_d)):
        d = range_d[j]
        T_FKZ = hh.tau_FKZ(x, Gmax=Gmax, mu=mu, d=d, Tmax=tau_f)
        range_gamma_t = np.geomspace(gamma_t_LB, gamma_t_UB, 200) / 100.0  # unit: 1

        copt, _ = hlp.find_closest_index(np.abs(T_MKZ - T_FKZ), 0)  # "copt" = cross-over point
        gamma_t = x[copt]
        if (gamma_t >= range_gamma_t[0]) and (gamma_t <= range_gamma_t[-1]):
            diff_T = np.abs(T_MKZ[:copt+1] - T_FKZ[:copt+1])
            area[j] = np.linalg.norm(diff_T) / (copt + 1.0)
        else:
            area[j] = np.inf
        # END IF
    # END FOR
    return area

# Author: <NAME>

'''
Hierarchy of subroutines::

 hh_param_from_profile
   |--- _calc_rho()
           |--- helper_site_response.thk2dep()
   |--- _calc_Gmax()
   |--- _calc_vertical_stress()
   |--- _calc_OCR()
   |--- _calc_K0()
   |--- _calc_PI()
   |--- _calc_shear_strength()
           |--- _calc_K0()
   |--- produce_HH_G_param()
           |---  _calc_mean_confining_stress()
           |---  produce_Darendeli_curves()
                      |--- helper_generic.assert_1D_numpy_array()
                      |--- helper_generic.check_length_or_extend_to_array()
                      |--- _calc_K0()
                      |--- _calc_mean_confining_stress()
           |---  helper_mkz_model.fit_MKZ()
           |---  _optimization_kernel()
                      |
                      |--- hlper_mkz_model.tau_MKZ()
                      |--- helper_generic.find_closest_index()
                      |--- __calc_area()
                                 |--- helper_hh_model.tau_FKZ()
                                 |--- helper_generic.find_closest_index()
                      |--- __find_x_t_and_d()
                                 |--- helper_hh_model.tau_FKZ()
                                 |--- helper_generic.find_closest_index()
'''

import os
import numpy as np
import matplotlib.pyplot as plt

from . import helper_generic as hlp
from . import helper_site_response as sr
from . import helper_mkz_model as mkz
from . import helper_hh_model as hh

#%%----------------------------------------------------------------------------
def hh_param_from_profile(vs_profile, *, Tmax=None, show_fig=False,
                          save_fig=False, fig_output_dir=None,
                          save_HH_G_file=False, HH_G_file_dir=None,
                          profile_name=None, verbose=True):
    '''
    Get HH parameters of each soil layer from the Vs values of every layer.

    Parameters
    ----------
    vs_profile : numpy.ndarray
        Shear-wave velocity profile, as a 2D numpy array. It should have the
        following columns:

            +---------------+----------+---------+---------+-----------------+
            | Thickness (m) | Vs (m/s) | Damping | Density | Material Number |
            +===============+==========+=========+=========+=================+
            |      ...      |   ...    |   ...   |   ...   |      ...        |
            +---------------+----------+---------+---------+-----------------+

    Tmax : numpy.ndarray or ``None``
        Shear strength of each layer of soil. If ``None``, it will be
        calculated using a combination of Ladd (1991) and Mohr-Coulomb criteria.
    show_fig : bool
        Whether or not to show figures G/Gmax and stress-strain curves of MKZ,
        FKZ, and HH for each layer.
    save_fig : bool
        Whether or not to save the figures to the hard drive. Only effective
        if ``show_fig`` is set to ``True``.
    fig_output_dir : str
        The output directory for the figures. Only effective if ``show_fig``
        and ``save_fig`` are both ``True``.
    save_HH_G_file : bool
        Whether or not to save the HH parameters to the hard drive (as a
        "HH_G" file).
    HH_G_file_dir : str
        The output directory for the "HH_G" file. Only effective if
        ``save_HH_G_file`` is ``True``.
    profile_name : str or ``None``
        The name of the Vs profile, such as "CE.12345". If ``None``, a string
        of current date and time will be used as the profile name.
    verbose : bool
        Whether or not to print progresses on the console.

    Returns
    -------
    HH_G_param : numpy.ndarray
        The HH parameters of each layer. It's a 2D array of shape
        ``(9, n_layer)``. For each layer (i.e., column), the values are in
        this order:
            gamma_t, a, gamma_ref, beta, s, Gmax, mu, Tmax, d
    '''
    phi = 30.  # friction angle (choose 30 degrees, because there is no better info)

    hlp.check_Vs_profile_format(vs_profile)
    h = vs_profile[:-1, 0]
    Vs = vs_profile[:-1, 1]  # exclude the last layer (i.e., half space)

    n_layer = len(Vs)
    if Tmax is not None:
        hlp.assert_array_length(Tmax, n_layer)

    rho      = _calc_rho(h, Vs)
    Gmax     = _calc_Gmax(Vs, rho)
    sigma_v0 = _calc_vertical_stress(h, rho)
    OCR      = _calc_OCR(Vs, rho, sigma_v0)
    K0       = _calc_K0(OCR, phi=phi)
    PI       = _calc_PI(Vs)

    if Tmax is None:
        Tmax = _calc_shear_strength(Vs, OCR, sigma_v0, K0=K0, phi=phi)

    HH_G_param = produce_HH_G_param(Vs, Gmax, Tmax, OCR, sigma_v0, K0,
                                    curves=None, PI=PI, phi=phi,
                                    show_fig=show_fig, save_fig=save_fig,
                                    fig_output_dir=fig_output_dir,
                                    verbose=verbose)

    if save_HH_G_file:
        if HH_G_file_dir is None:
            raise ValueError('Please specify `HH_G_file_dir`.')
        if profile_name is None:
            profile_name = hlp.get_current_time(for_filename=True)
        np.savetxt(os.path.join(HH_G_file_dir, 'HH_G_%s.txt' % profile_name),
                   HH_G_param, delimiter='\t', fmt='%.6g')

    return HH_G_param

#%%----------------------------------------------------------------------------
def hh_param_from_curves(vs_profile, curves, *, Tmax=None, show_fig=False,
                         save_fig=False, fig_output_dir=None, save_HH_G_file=False,
                         HH_G_file_dir=None, profile_name=None, verbose=True):
    '''
    Get HH parameters of each soil layer from the Vs profile and G/Gmax curves.

    Parameters
    ----------
    vs_profile : numpy.ndarray
        Shear-wave velocity profile, as a 2D numpy array. It should have the
        following columns:
            +---------------+----------+---------+---------+-----------------+
            | Thickness (m) | Vs (m/s) | Damping | Density | Material Number |
            +===============+==========+=========+=========+=================+
            |      ...      |   ...    |   ...   |   ...   |      ...        |
            +---------------+----------+---------+---------+-----------------+

    curves : numpy.ndarray
        A 2D numpy array that represents G/Gmax and damping curves of each
        layer, in the following format:
         +------------+--------+------------+-------------+-------------+--------+-----+
         | strain [%] | G/Gmax | strain [%] | damping [%] |  strain [%] | G/Gmax | ... |
         +============+========+============+=============+=============+========+=====+
         |    ...     |  ...   |    ...     |    ...      |    ...      |  ...   | ... |
         +------------+--------+------------+-------------+-------------+--------+-----+

        The damping information is neglected in this function, so users can
        supply some dummy values.
    Tmax : numpy.ndarray or ``None``
        Shear strength of each layer of soil. If ``None``, it will be
        calculated using a combination of Ladd (1991) and Mohr-Coulomb criteria.
    show_fig : bool
        Whether or not to show figures G/Gmax and stress-strain curves of MKZ,
        FKZ, and HH for each layer.
    save_fig : bool
        Whether or not to save the figures to the hard drive. Only effective
        if ``show_fig`` is set to ``True``.
    fig_output_dir : str
        The output directory for the figures. Only effective if ``show_fig``
        and ``save_fig`` are both ``True``.
    save_HH_G_file : bool
        Whether or not to save the HH parameters to the hard drive (as a
        "HH_G" file).
    HH_G_file_dir : str
        The output directory for the "HH_G" file. Only effective if
        ``save_HH_G_file`` is ``True``.
    profile_name : str or ``None``
        The name of the Vs profile, such as "CE.12345". If ``None``, a string
        of current date and time will be used as the profile name.
    verbose : bool
        Whether or not to print progresses on the console.

    Returns
    -------
    HH_G_param : numpy.ndarray
        The HH parameters of each layer. It's a 2D array of shape
        ``(9, n_layer)``. For each layer (i.e., column), the values are in
        this order:
            gamma_t, a, gamma_ref, beta, s, Gmax, mu, Tmax, d
    '''
    phi = 30.0

    hlp.check_Vs_profile_format(vs_profile)
    h = vs_profile[:-1, 0]
    Vs = vs_profile[:-1, 1]  # exclude the last layer (i.e., half space)
    n_layer = len(Vs)

    if vs_profile.shape[1] == 5:  # there can only be 5 or 2 columns
        mat = vs_profile[:-1, -1]
        rho = vs_profile[:-1, 3]
    else:  # only 2 columns
        mat = np.arange(1, n_layer+1)
        rho = _calc_rho(h, Vs)

    if Tmax is not None:
        hlp.assert_array_length(Tmax, n_layer)

    Gmax = _calc_Gmax(Vs, rho)
    sigma_v0 = _calc_vertical_stress(h, rho)
    OCR = _calc_OCR(Vs, rho, sigma_v0)
    K0 = _calc_K0(OCR, phi=phi)

    if Tmax is None:
        Tmax = _calc_shear_strength(Vs, OCR, sigma_v0, K0=K0, phi=phi)

    curves_old = curves.copy()
    curves_expanded = None
    for j in range(n_layer):
        tmp = curves_old[:, int(mat[j]) * 4 - 4: int(mat[j]) * 4]
        if curves_expanded is None:
            curves_expanded = tmp
        else:
            curves_expanded = np.column_stack((curves_expanded, tmp))
    curves = curves_expanded

    HH_G_param = produce_HH_G_param(Vs, Gmax, Tmax, OCR, sigma_v0, K0,
                                    curves=curves, PI=None, phi=None,
                                    show_fig=show_fig, save_fig=save_fig,
                                    fig_output_dir=fig_output_dir,
                                    verbose=verbose)

    if save_HH_G_file:
        if HH_G_file_dir is None:
            raise ValueError('Please specify `HH_G_file_dir`.')
        if profile_name is None:
            profile_name = hlp.get_current_time(for_filename=True)
        np.savetxt(os.path.join(HH_G_file_dir, 'HH_G_%s.txt' % profile_name),
                   HH_G_param, delimiter='\t', fmt='%.6g')

    return HH_G_param

#%%----------------------------------------------------------------------------
def produce_HH_G_param(Vs, Gmax, Tmax, OCR, sigma_v0, K0, curves=None,
                       PI=None, phi=None, show_fig=False, save_fig=False,
                       fig_output_dir=None, verbose=True):
    '''
    Produce HH_G parameters from profiles of Vs, Tmax, OCR, etc.

    Parameters
    ----------
    Vs : numpy.ndarray
        Vs values of each layer. Shape: ``(n_layer, )``, where ``n_layer`` is
        the length of ``Vs``. Unit: m/s.
    Gmax : numpy.ndarray
        Initial stiffness of each layer. Shape: ``(n_layer, )``. Unit: Pa.
    Tmax : numpy.ndarray
        The shear strength of each layer. Shape: ``(n_layer, )``. Unit: Pa.
    OCR : numpy.ndarray
        Over-consolidation ratio of each layer. Shape: ``(n_layer, )``.
    sigma_v0 : numpy.ndarray
        Vertical effective confining stress of each layer. Shape:
        ``(n_layer, )``. Unit: Pa.
    K0 : numpy.ndarray or float
        Lateral soil pressure coefficient. If an array, it must have shape
        ``(n_layer, )``. If a single value, it means that all layers share
        this same value.
    curves : numpy.ndarray or ``None``
        A 2D numpy array that represents G/Gmax and damping curves of each
        layer, in the following format:
         +------------+--------+------------+-------------+-------------+--------+-----+
         | strain [%] | G/Gmax | strain [%] | damping [%] |  strain [%] | G/Gmax | ... |
         +============+========+============+=============+=============+========+=====+
         |    ...     |  ...   |    ...     |    ...      |    ...      |  ...   | ... |
         +------------+--------+------------+-------------+-------------+--------+-----+

        The damping information is neglected in this function, so users can
        supply some dummy values. If ``None``, it means that the users do not
        have G/Gmax curve information, so this function will calculate the
        MKZ G/Gmax curves from the empirical formulas in Darendeli (2001).
    PI : float or numpy.ndarray or ``None``
        Plasticity index of the soils. It is not necessary (can be ``None``) if
        ``curves`` is provided (i.e., not ``None``). If an array, it must have
        shape ``(n_layer, )``. If a single value, it means that all layers
        share this same value.
    phi : float or numpy.ndarray or ``None``
        Effective internal frictional angle (in degrees). It is not necessary
        (can be ``None``) if ``curve`` is provided (i.e., not ``None``). If
        an array, it must have shape ``(n_layer, )``. If a single value, it
        means that all layers share this same value.
    show_fig : bool
        Whether or not to show figures G/Gmax and stress-strain curves of MKZ,
        FKZ, and HH for each layer.
    save_fig : bool
        Whether or not to save the figures to the hard drive. Only effective
        if ``show_fig`` is set to ``True``.
    fig_output_dir : str
        The output directory for the figures. Only effective if ``show_fig``
        and ``save_fig`` are both ``True``.
    verbose : bool
        Whether or not to print progresses on the console.

    Returns
    -------
    parameters : numpy.ndarray
        The HH parameters of each layer. It's a 2D array of shape
        ``(9, n_layer)``. For each layer (i.e., column), the values are in
        this order:
            gamma_t, a, gamma_ref, beta, s, Gmax, mu, Tmax, d

    Notes
    -----
    This function is based on ``hybridParaKernel_FKZ.m``.
    '''
    hlp.assert_1D_numpy_array(Vs, '`Vs`')
    n_layer = len(Vs)

    hlp.assert_array_length(Gmax, n_layer, name='`Gmax`')
    hlp.assert_array_length(Tmax, n_layer, name='`Tmax`')
    hlp.assert_array_length(OCR, n_layer, name='`OCR`')
    hlp.assert_array_length(sigma_v0, n_layer, name='`sigma_v0`')

    K0 = hlp.check_length_or_extend_to_array(K0, n_layer, name='`K0`')
    p0 = _calc_mean_confining_stress(sigma_v0, K0)

    if verbose:
        print('========== Start optimizing for HH_G parameters ===========')

    # ============= MKZ fit ===================================================
    if curves is None:  # user does not provide curves
        if verbose:
            print('------ G/Gmax not provided; will generate MKZ curves using '
                  'Darendeli (2001): ------')

        strain_ = np.geomspace(1e-4, 10, 400)  # unit: percent
        GGmax, _, gamma_ref = produce_Darendeli_curves(sigma_v0, PI, OCR=OCR,
                                                       K0=K0, phi=phi,
                                                       strain_in_pct=strain_)
        strain = np.tile(strain_, (n_layer, 1)).T  # strain matrix for all layers
        beta = np.ones(n_layer)
        s = 0.9190 * np.ones(n_layer)
    else:  # user provides own curves
        if verbose:
            print('------ G/Gmax provided; fitting MKZ curves to data: ------')
        hlp.assert_2D_numpy_array(curves)
        assert(curves.shape[1] == n_layer * 4)

        # ----------- Extract G/Gmax information ------------------------------
        strain = curves[:, 0::4]  # unit: percent
        GGmax = curves[:, 1::4]

        # ----------- Fit MKZ parameters --------------------------------------
        param_MKZ, _ = mkz.fit_MKZ(curves, show_fig=show_fig)
        gamma_ref = param_MKZ[:, 0]
        s = param_MKZ[:, 2]
        beta = param_MKZ[:, 3]

        # ----------- Show results on console ---------------------------------
        if verbose:
            print('****** MKZ parameters: ******')
        for j in range(n_layer):
            if verbose:
                print('Layer %d: gamma_ref = %.3g, s = %.3g, beta = %.3g' \
                      % (j, gamma_ref[j], s[j], beta[j]))

    # ========== Stress-strain curve implied by G/Gmax ========================
    sigma = np.zeros_like(GGmax)
    for j in range(n_layer):
        sigma[0, j] = 0
        for k in range(1, GGmax.shape[0]):
            sigma[k, j] = GGmax[k, j] * Gmax[j] * strain[k, j] / 100.0
        # END FOR
    # END FOR

    # ========== Estimate mu using empirical correlations =====================
    p0 = p0 / 1000.0  # unit: Pa --> kPa

    mu = np.zeros_like(OCR)
    for j in range(n_layer):
        if Vs[j] <= 760:  # softer soil: use Vardanega & Bolton (2011) CGJ formula
            mu[j] = 1.0 / (0.000872 * Gmax[j]/Tmax[j] * OCR[j]**0.47 * p0[j]**0.28)
            if mu[j] <= 0.02:  # mu too small --> too low tau_FKZ --> sharply decreasing tau_HH
                mu[j] = mu[j] * 10.0 ** (0.236 * 3)  # 0.236 is the standard error suggested in Vardanega & Bolton (2011)
            elif mu[j] <= 0.03:
                mu[j] = mu[j] * 10.0 ** (0.236 * 2)
            elif mu[j] <= 0.04:
                mu[j] = mu[j] * 10.0 ** (0.236 * 1)
            # END IF
        else:  # stiffer soils: set mu to 1 for lack of better information
            mu[j] = 1.0
        # END IF
    # END FOR

    # ========== Start FKZ optimization =======================================
    if verbose:
        print('-----------  FKZ optimization  -----------------------------')
    parameters = np.zeros((9, n_layer))

    lw = 1.0
    muted_blue = np.array([107, 174, 214]) / 255.
    muted_green = np.array([120, 198, 121]) / 255.
    muted_red = np.array([222, 45, 38]) / 255.

    for j in range(n_layer):
        strain_j = strain[:, j]
        a, gamma_t, d = _optimization_kernel(strain_j / 100.0, gamma_ref[j],
                                             beta[j], s[j], Gmax[j], Tmax[j],
                                             mu[j])
        if verbose:
            print('%d/%d: mu = %.3f, a = %.1f, gamma_t = %.3f%%, d = %.3f' \
                  % (j + 1, n_layer, mu[j], a, gamma_t * 100, d))
        T_FKZ = hh.tau_FKZ(strain_j / 100.0, Gmax=Gmax[j], mu=mu[j], d=d,
                           Tmax=Tmax[j])
        T_HH = hh.tau_HH(strain_j / 100.0, gamma_t=gamma_t, a=a,
                         gamma_ref=gamma_ref[j], beta=beta[j], s=s[j],
                         Gmax=Gmax[j], mu=mu[j], Tmax=Tmax[j], d=d)

        parameters[0, j] = gamma_t
        parameters[1, j] = a
        parameters[2, j] = gamma_ref[j]
        parameters[3, j] = beta[j]
        parameters[4, j] = s[j]
        parameters[5, j] = Gmax[j]
        parameters[6, j] = mu[j]
        parameters[7, j] = Tmax[j]
        parameters[8, j] = d

        GGmax_HH = T_HH / (Gmax[j] * (strain_j / 100.0))

        if show_fig:
            fig = plt.figure(figsize=(4.2, 6.0))

            plt.subplot(211)
            if curves is None:
                plt.semilogx(strain_j, sigma[:, j] / 1000., c=muted_blue,
                             lw=lw*2.5, label='MKZ')  # Darendeli's curve
                plt.semilogx(strain_j, T_FKZ / 1000., c=muted_green, lw=lw*1.75,
                             label='FKZ')
            else:
                plt.semilogx(strain_j, sigma[:, j] / 1000., c=muted_blue,
                             marker='o', ls='-', lw=lw*2.5,
                             label='Given $G/G_{\max}$')
                plt.semilogx(strain_j, T_FKZ / 1000., c=muted_green, lw=lw*1.75,
                             label='FKZ')
            plt.grid(ls=':', lw=0.5)
            plt.plot([np.min(strain_j), np.max(strain_j)],
                     np.array([Tmax[j], Tmax[j]]) / 1000., lw=lw, c='gray',
                     ls='--')
            plt.plot(strain_j, T_HH / 1000., c=muted_red, lw=lw, label='HH')
            plt.plot([gamma_t * 100] * 2, plt.ylim(), ls='--', c='gray')
            plt.ylabel('Stress [kPa]')
            plt.xlim(np.min(strain_j), np.max(strain_j))
            plt.legend(loc='upper left')
            plt.title('$V_S$ = %.1f m/s, $G_{\max}$ = %.3f MPa,'
                      '\n$\\tau_{\mathrm{ff}}$ = %.3f kPa, '
                      '$\gamma_{\mathrm{ref}}$ = %.3f%%' \
                      % (Vs[j], Gmax[j]/1e6, Tmax[j]/1e3, gamma_ref[j]*100))

            plt.subplot(212)
            if curves is None:
                plt.semilogx(strain_j, GGmax[:, j], c=muted_blue, lw=lw*2.5)
            else:
                plt.semilogx(strain_j, GGmax[:, j], c=muted_blue, ls='-',
                             marker='o', lw=lw*2.5)
            plt.grid(ls=':', lw=0.5)
            plt.plot(strain_j,
                     mu[j] / (1 + Gmax[j]/Tmax[j]*mu[j]*np.abs(strain_j/100.) ),
                     c=muted_green, lw=lw*1.75)
            plt.plot(strain_j, GGmax_HH, c=muted_red, lw=lw)
            plt.plot([gamma_t * 100] * 2, plt.ylim(), ls='--', c='gray')
            plt.ylabel('$G/G_{\max}$')
            plt.xlabel('Strain [%]')
            plt.xlim(np.min(strain_j), np.max(strain_j))
            plt.title("$\mu$ = %.3f, a = %.1f, $\gamma_{\mathrm{t}}$ = %.4f%%\n"
                      "d = %.4f, $p'_{\mathrm{m0}}$ = %.2f kPa" \
                      % (mu[j], a, gamma_t * 100, d, p0[j]))

            fig.tight_layout(pad=0.5, h_pad=1.2, w_pad=0.3)

            if save_fig:
                if fig_output_dir is None:
                    raise ValueError('Please specify `fig_output_dir`.')
                fig.savefig(os.path.join(fig_output_dir,
                                         'Stress_GGmax_of_Layer_#%d.png' % (j+1)))

    return parameters

#%%----------------------------------------------------------------------------
def _calc_shear_strength(Vs, OCR, sigma_v0, K0=None, phi=30.0):
    '''
    Calculate shear strength of soils.

    Parameters
    ----------
    Vs : numpy.ndarray
        A 1D array of Vs values of each layer. Unit: m/s.
    OCR : numpy.ndarray
        A 1D array of OCR (over-consolidation ratio) of each layer. Unit: 1.
    sigma_v0 : numpy.ndarray
        A 1D array of vertical overburden pressure. Unit: Pa.
    K0 : float, int, numpy.ndarray, or ``None``
        Lateral soil pressure coefficient. If a single value is given, it is
        assumed to be the value for all layers. If ``None``, it will be
        determined from OCR via an empirical formula by Mayne & Kulhawy (1982).
    phi : float, int, or numpy.ndarray
        Effective internal friction angle of soils (in degrees).

    Returns
    -------
    Tmax : numpy.ndarray
        Shear strength of soils of each layer. Unit: Pa.
    '''
    dyna_coeff = 1.2  # assume a strain rate of 0.01 sec^(-1), from Vardanega & Bolton (2013)

    phi = hlp.check_length_or_extend_to_array(phi, len(Vs), name='`phi`')
    if K0 is None:
        K0 = _calc_K0(OCR, phi=phi)
    else:
        K0 = hlp.check_length_or_extend_to_array(K0, len(Vs), name='`K0`')

    Tmax = np.zeros(len(Vs))
    for j in range(len(Vs)):
        if Vs[j] <= 760:  # for softer soils, calculate undrained shear strength
            Tmax[j] = dyna_coeff * 0.28 * OCR[j]**0.8 * sigma_v0[j]  # Ladd (1991)
        else:  # stiffer soils: Mohr-Coulomb criterion
            sigma_h0 = K0[j] * sigma_v0[j]  # horizontal stress
            sigma_1 = np.max([sigma_v0[j], sigma_h0])  # largest principal stress
            sigma_3 = np.min([sigma_v0[j], sigma_h0])  # smallest principal stress
            # normal effective stress on the slip plane
            sigma_n = (sigma_1 + sigma_3)/2.0 \
                      - (sigma_1 - sigma_3)/2.0 * np.sin(np.deg2rad(phi[j]))
            Tmax[j] = dyna_coeff * sigma_n * np.tan(np.deg2rad(phi[j]))

    return Tmax

#%%----------------------------------------------------------------------------
def _calc_Gmax(Vs, rho):
    '''
    Calculate initial stiffness of each soil layer.

    Parameters
    ----------
    Vs : numpy.ndarray
        1D array of Vs of layers. Unit: m/s.
    rho : numpy.ndarray
        1D array of mass density of layers. Unit: kg/m^3.

    Returns
    -------
    Gmax : numpy.ndarray
        1D array of initial stiffness. Unit: Pa
    '''
    Gmax = rho * Vs**2
    return Gmax

#%%----------------------------------------------------------------------------
def _calc_OCR(Vs, rho, sigma_v0, OCR_upper_limit=None):
    '''
    Calculate OCR (over-consolidation ratio) of each layer from the Vs profile.

    Parameters
    ----------
    Vs : numpy.ndarray
        1D array of Vs of layers. Unit: m/s.
    rho : numpy.ndarray
        1D array of mass density of layers. Unit: kg/m^3.
    sigma_v0 : numpy.ndarray
        Vertical overburden stress at the mid-point of each layer. Unit: Pa.
    OCR_upper_limit : float or ``None``
        The maximum acceptable OCR value. If ``None``, there is no limit.

    Returns
    -------
    OCR : numpy.ndarray
        1D array of OCR value, for each soil layer. (Unitless.)
    '''
    sigma_p0 = 0.106 * Vs**1.47  # Mayne, <NAME> (1998) "Clay stress history evaluated fromseismic piezocone tests"
    sigma_p0 = sigma_p0 * 1000  # kPa --> Pa
    OCR = sigma_p0 / sigma_v0
    OCR = np.minimum(OCR, np.inf if OCR_upper_limit is None else OCR_upper_limit)
    return OCR

#%%----------------------------------------------------------------------------
def _calc_vertical_stress(h, rho):
    '''
    Calculate vertical overburden stress at the mid-point of each layer.

    Parameters
    ----------
    h : numpy.ndarray
        1D array of layer thickness. Unit: m.
    rho : numpy.ndarray
        1D array of mass density of each layer. Unit: kg/m^3.

    Returns
    -------
    stress : numpy.ndarray
        Vertical overburden stress at the mid-point of each layer. Unit: Pa.
    '''
    g = 9.81  # unit: m/s/s
    n = len(h)
    stress = np.zeros_like(h)

    if np.mean(rho) < 1000:
        print('Warning in __calc_vertical_stress(): It looks like the unit '
              'of mass density is g/cm^3. The correct unit should be kg/m^3.')

    if h[-1] == 0:  # zero thickness, i.e., half space
        h[-1] = 1

    stress[0] = rho[0] * g * h[0]/2  # divided by 2: middle of layer
    for i in range(1, n):
        stress[i] = stress[i-1] + rho[i-1] * g * h[i-1]/2 + rho[i] * g * h[i]/2

    return stress

#%%----------------------------------------------------------------------------
def _calc_rho(h, Vs):
    '''
    Calculate mass density of soils from Vs values, using the empirical formula
    by Mayne, Schneider & Martin (1999) and Burns & Mayne (1996).

    Parameters
    ----------
    h : numpy.ndarray
        The thickness of each soil layer. Unit: m.
    Vs : numpy.ndarray
        The shear-wave velocity for each layer. It needs to be a 1D numpy array.
        Unit: m/s.

    Returns
    -------
    rho : numpy.ndarray
        Mass density of each soil layer. Unit: kg/m^3.

    References
    ----------
    1. Mayne, Schneider & Martin (1999) "Small- and large-strain soil
       properties from seismic flat dilatometer tests." Pre-failure
       deformation characteristics of geomaterials, 1999 Balkema, Rotterdam.
    2. Burns & Mayne (1996) "Small- and high-strain soil properties using the
       seismic piezocone." Transportation Research Record 1548, National
       Acad. Press, Washington DC, 81-88.
    '''
    z = sr.thk2dep(h, midpoint=False)
    z[z == 0] = 0.0001  # avoid error of dividing by zero
    lb = 1.65  # lower bound of density: 1.65 g/cm^3
    rho = np.maximum(lb, 1 + 1. / (0.614 + 58.7 * (np.log(z) + 1.095) / Vs))
    rho *= 1000  # unit: g/cm^3 --> kg/m^3
    return rho

#%%----------------------------------------------------------------------------
def _calc_PI(Vs):
    '''
    Calculate PI (plasticity index) from Vs values.

    Parameters
    ----------
    Vs : numpy.ndarray
        The shear-wave velocity for each layer. It needs to be a 1D numpy array.
        Unit: m/s.

    Returns
    -------
    PI : numpy.ndarray
        The plasticity index for each layer. Unit: %.
    '''
    PI = np.zeros_like(Vs)
    for j in range(len(Vs)):
        if Vs[j] <= 200:
            PI[j] = 10
        elif Vs[j] <= 360:
            PI[j] = 5
        else:
            PI[j] = 0

    return PI

#%%----------------------------------------------------------------------------
def _calc_K0(OCR, phi=30.):
    '''
    Calculate K0 (lateral earth pressure coefficient at rest) from OCR using
    the empirical formula by Mayne & Kulhawy (1982).

    Parameters
    ----------
    OCR : float, int, or numpy.ndarray
        Over-consolidation ratio of each layer of soils. If it is a float/int,
        it means only one layer, or all the layers have the same OCR.
    phi : float, int, or numpy.ndarray
        Internal effective friction angle of soils. If it is a float/int, it
        means only one soil layer, or all the layers have the same angle.
        Unit: deg.

    Returns
    -------
    K0 : float or numpy.ndarray
        K0 value(s). If either ``OCR`` or ``phi`` is an array, ``K0`` will be
        an array of the same length.
    '''
    K0 = (1 - np.sin(np.deg2rad(phi))) * OCR ** (np.sin(np.deg2rad(phi)))
    return K0

#%%----------------------------------------------------------------------------
def produce_Darendeli_curves(sigma_v0, PI=20., OCR=1., K0=0.5, phi=30.0,
                             strain_in_pct=None):
    '''
    Produce G/Gmax and damping curves using empirical correlations by
    Darendeli (2001).

    Parameters
    ----------
    sigma_v0 : numpy.ndarray
        Effective vertical confining stress of each layer. Unit: Pa.
    PI : int, float, or numpy.ndarray
        Plasticity index of each layer. Unit: %. If a single value is given,
        it is assumed to be the PI for all layers.
    OCR : int, float, or numpy.ndarray
        Over-consolidation ratio of each layer. If a single value is given,
        it is assumed to be the value for all layers.
    K0 : int, float, numpy.ndarray, or ``None``
        Lateral soil pressure coefficient. If a single value is given, it is
        assumed to be the value for all layers. If ``None``, it will be
        determined from OCR via an empirical formula by Mayne & Kulhawy (1982).
    phi : int, float, or numpy.ndarray
        Internal effective friction angle of soils. If it is a float/int, it
        means all the layers have the same angle. Unit: deg.
    strain_in_pct : numpy.ndarray or ``None``
        The strain values at which to calculate G/Gmax and damping values. If
        ``None``, numpy.geomspace(1e-4, 10, 400) will be used. Unit: percent.

    Returns
    -------
    GGmax : numpy.ndarray
        G/Gmax curves for each layer. It is a 2D numpy array. Each column of it
        represents the G/Gmax curve of a particular layer. Unit: 1
    xi : numpy.ndarray
        Damping curves for each layer. Same shape as ``GGmax``. Unit: 1.
    gamma_r : numpy.ndarray
        Reference strain for each layer. It is a 1D numpy array, corresponding
        to each soil layer. Unit: 1.
    '''
    hlp.assert_1D_numpy_array(sigma_v0)
    n_layer = len(sigma_v0)

    phi = hlp.check_length_or_extend_to_array(phi, n_layer, name='`phi`')
    PI = hlp.check_length_or_extend_to_array(PI, n_layer, name='`PI`')
    OCR = hlp.check_length_or_extend_to_array(OCR, n_layer, name='`OCR`')

    if K0 is None:
        K0 = _calc_K0(OCR, phi=phi)
    else:
        K0 = hlp.check_length_or_extend_to_array(K0, n_layer, name='`K0`')

    if strain_in_pct is None:
        gamma = np.geomspace(1e-4, 10, 400)
    else:
        gamma = strain_in_pct.copy()

    # Define all constants
    nr_cycle = 10
    frq = 1
    N = nr_cycle

    phi1 = 0.0352
    phi2 = 0.0010
    phi3 = 0.3246
    phi4 = 0.3483
    phi5 = 0.9190
    phi6 = 0.8005
    phi7 = 0.0129
    phi8 = -0.1069
    phi9 = -0.2889
    phi10 = 0.2919
    phi11 = 0.6329
    phi12 = -0.0057
    a = phi5

    c1 = -1.1143*a**2 + 1.8618*a + 0.2523  # from Darendeli (2001), page 226
    c2 = 0.0805*a**2 - 0.0710*a - 0.0095
    c3 = -0.0005*a**2 + 0.0002*a + 0.0003
    b = phi11 + phi12 * np.log(N)  # Darendeli (2001) Eq 9.1d

    # Confinine stress
    sigma_0 = _calc_mean_confining_stress(sigma_v0, K0)  # octahedral stress
    sigma_0 = sigma_0 / 101325.0  # unit: Pa --> atm
    n_strain_pts = len(strain_in_pct)

    # Reference strain for each layer (Eq 9.1a). Unit: percent
    gamma_r = (phi1 + phi2 * PI * OCR**phi3) * sigma_0**phi4

    GGmax = np.zeros((n_strain_pts, n_layer))
    xi = np.zeros_like(GGmax)
    for i in range(n_layer):
        GGmax[:, i] = 1. / (1 + (gamma / gamma_r[i])**a)  # G of i-th layer (Eq 9.2a)
        D_masing_1 = (100. / np.pi) \
            * (4 * (gamma - gamma_r[i] * np.log((gamma + gamma_r[i]) / gamma_r[i])) \
                / (gamma**2 / (gamma + gamma_r[i])) - 2)  # Unit: percent (page 226)
        D_masing = c1 * D_masing_1 + c2 * D_masing_1**2 + c3 * D_masing_1**3  # Unit: percent (page 226)
        D_min = (phi6 + phi7 * PI[i] * OCR[i]**phi8) * sigma_0[i]**phi9 * (1 + phi10 * np.log(frq))  # Eq 9.1c (page 221)
        xi[:, i] = b * GGmax[:, i]**0.1 * D_masing + D_min  # Eq 9.2b (page 224). Unit: percent

    xi /= 100.0
    gamma_r /= 100.0
    return GGmax, xi, gamma_r

#%%----------------------------------------------------------------------------
def _calc_mean_confining_stress(sigma_v0, K0):
    '''
    Calculate mean (of three directions) confining stress.

    Parameters
    ----------
    sigma_v0 : numpy.ndarray
        (Effective) vertical stress of each layer. Unit: Pa.
    K0 : numpy.ndarray
        Lateral stress coefficient of each layer. Unit: 1.

    Returns
    -------
    sigma_m0 : numpy.ndarray
        Mean effective confining stress (of three directions). Unit: Pa.
    '''
    sigma_m0 = (2 * K0 + 1)/3.0 * sigma_v0
    return sigma_m0

#%%----------------------------------------------------------------------------
def _optimization_kernel(x, x_ref, beta, s, Gmax, tau_f, mu):
    '''
    Optimization process to find FKZ parameters.

    Parameters
    ----------
    x : numpy.ndarray
        An 1D array of shear strain. Unit: 1.
    x_ref : float
        The "reference strain" parameter (in MKZ) of the soil. Unit: 1.
    beta : float
        A shape parameter of the FKZ model.
    s : float
        A shape parameter of the FKZ model.
    Gmax : float
        Initial shear modulus. Unit: Pa.
    tau_f : float
        The shear strength of the current soil layer. Unit: Pa.
    mu : float
        The "shape parameter" of the FKZ model.

    Returns
    -------
    a : float
        A parameter of the HH model that defines the "speed" of transition from
        MKZ to FKZ
    gamma_t : float
        The shear strain at which the transition from MKZ to FKZ happens.
        Unit: 1
    d : float
        The "shape power" parameter in the FKZ model.

    Notes
    -----
    Based on optHybFKZ.m
    '''
    T_MKZ = mkz.tau_MKZ(x, gamma_ref=x_ref, beta=beta, s=s, Gmax=Gmax)
    if mu <= 0.03:  # when mu is too small, there may be some numerical issues
        gamma_t_LB = 0.001  # therefore gamma_t lower bound is relaxed
    else:
        gamma_t_LB = 0.01

    gamma_t_UB = 3.0  # unit: percent

    index_gamma_t_LB, _ = hlp.find_closest_index(x, gamma_t_LB / 100.0)
    if T_MKZ[index_gamma_t_LB] >= 0.85 * tau_f:
        gamma_t_LB = 0.005  # for very deep layers, tau_MKZ may be larger than tau_f at gamma_t_LB

    index_gamma_t_LB, _ = hlp.find_closest_index(x, gamma_t_LB / 100.0)  # do it again
    if T_MKZ[index_gamma_t_LB] >= 0.85 * tau_f:
        gamma_t_LB = 0.001

    range_d = np.linspace(0.67, 1.39, 200)
    area = __calc_area(range_d, x, Gmax, mu, tau_f, gamma_t_LB, gamma_t_UB, T_MKZ)
    if np.min(area) < np.inf:  # it means that a proper d value is found
        gamma_t, d = __find_x_t_and_d(area, range_d, x, Gmax, mu, tau_f, T_MKZ)
    else:  # cannot find a proper d value
        range_d = np.linspace(0.67, 1.39, 400)  # increase grid density to 400
        area = __calc_area(range_d, x, Gmax, mu, tau_f, gamma_t_LB,
                           gamma_t_UB, T_MKZ)
        if np.min(area) < np.inf:
            gamma_t, d = __find_x_t_and_d(area, range_d, x, Gmax, mu, tau_f, T_MKZ)
        else:
            range_d = np.linspace(0.67, 1.39, 1000)  # increase grid density
            new_gamma_t_LB = 0.005  # further relax
            area = __calc_area(range_d, x, Gmax, mu, tau_f, new_gamma_t_LB,
                               gamma_t_UB, T_MKZ)
            if np.min(area) < np.inf:
                gamma_t, d = __find_x_t_and_d(area, range_d, x, Gmax, mu, tau_f, T_MKZ)
            else:
                d = 1.03
                gamma_t = 1e-3 / 100.0  # further ralax to 0.001%
            # END IF
        # END IF
    #END IF

    a = 100.0  # always use a fast transition
    return a, gamma_t, d

#%%----------------------------------------------------------------------------
def __find_x_t_and_d(area, range_d, x, Gmax, mu, tau_f, T_MKZ):
    '''
    Find the ``x_t`` (transition strain) that minimizes the "area" between
    the MKZ stress curve and the FKZ stress curve.

    Parameters
    ----------
    area : numpy.ndarray
        The "area" between the MKZ stress curve and the FKZ stress curve. It
        has the same shape as ``range_d``, because each element of ``area`` is
        the area corresponding to a ``d`` value within ``range_d``.
    range_d : numpy.ndarray
        The range of ``d`` to search from. Must be a 1D numpy array.
    x : numpy.ndarray
        An 1D array of shear strain. Unit: 1.
    Gmax : float
        Initial shear modulus. Unit: Pa.
    mu : float
        The "shape parameter" of the FKZ model.
    tau_f : float
        The shear strength of the current soil layer. Unit: Pa.
    T_MKZ : numpy.ndarray
        The MKZ stress curve, which has the same shape as ``x``. Unit: Pa.

    Returns
    -------
    x_t : float
        The ``x_t`` value that minimizes the "area" between the MKZ stress
        curve and the FKZ stress curve. Unit: 1.
    d : float
        The ``d`` value that minimizes the "area" between the MKZ stress curve
        and the FKZ stress curve. (No unit.)
    '''
    j_ = np.argmin(area)
    d = range_d[j_]
    T_FKZ = hh.tau_FKZ(x, Gmax=Gmax, mu=mu, d=d, Tmax=tau_f)
    copt, _ = hlp.find_closest_index(np.abs(T_MKZ - T_FKZ), 0)
    x_t = x[copt]

    return x_t, d

#%%----------------------------------------------------------------------------
def __calc_area(range_d, x, Gmax, mu, tau_f, gamma_t_LB, gamma_t_UB, T_MKZ):
    '''
    Calculate the "area" between the MKZ stress curve and the FKZ stress curve.
    The MKZ stress curve is supplied as a parameter, and the FKZ stress curve
    is determined by ``x``, ``Gmax``, ``mu``, ``d``, ``tau_f``, and ``gamma_t``.

    Parameters
    ----------
    range_d : numpy.ndarray
        The range of ``d`` to search from. Must be a 1D numpy array.
    x : numpy.ndarray
        An 1D array of shear strain. Unit: 1.
    Gmax : float
        Initial shear modulus. Unit: Pa.
    mu : float
        The "shape parameter" of the FKZ model.
    tau_f : float
        The shear strength of the current soil layer. Unit: Pa.
    gamma_t_LB : float
        The lower bound of ``gamma_t`` (:math:`\gamma_t`), i.e., the transition
        strain. Unit: %.
    gamma_t_UB : float
        The upper bound of ``gamma_t``. Unit: %.
    T_MKZ : numpy.ndarray
        The MKZ stress curve, which has the same shape as ``x``. Unit: Pa.

    Returns
    -------
    area : numpy.ndarray
        The "area" between the MKZ stress curve and the FKZ stress curve. It
        has the same shape as ``range_d``, because each element of ``area`` is
        the area corresponding to a ``d`` value within ``range_d``.
    '''
    area = np.zeros_like(range_d)
    for j in range(len(range_d)):
        d = range_d[j]
        T_FKZ = hh.tau_FKZ(x, Gmax=Gmax, mu=mu, d=d, Tmax=tau_f)
        range_gamma_t = np.geomspace(gamma_t_LB, gamma_t_UB, 200) / 100.0  # unit: 1

        copt, _ = hlp.find_closest_index(np.abs(T_MKZ - T_FKZ), 0)  # "copt" = cross-over point
        gamma_t = x[copt]
        if (gamma_t >= range_gamma_t[0]) and (gamma_t <= range_gamma_t[-1]):
            diff_T = np.abs(T_MKZ[:copt+1] - T_FKZ[:copt+1])
            area[j] = np.linalg.norm(diff_T) / (copt + 1.0)
        else:
            area[j] = np.inf
        # END IF
    # END FOR
    return area

# Author: <NAME> Hierarchy of subroutines::   hh_param_from_profile    |--- _calc_rho()            |--- helper_site_response.thk2dep()    |--- _calc_Gmax()    |--- _calc_vertical_stress()    |--- _calc_OCR()    |--- _calc_K0()    |--- _calc_PI()    |--- _calc_shear_strength()            |--- _calc_K0()    |--- produce_HH_G_param()            |---  _calc_mean_confining_stress()            |---  produce_Darendeli_curves()                       |--- helper_generic.assert_1D_numpy_array()                       |--- helper_generic.check_length_or_extend_to_array()                       |--- _calc_K0()                       |--- _calc_mean_confining_stress()            |---  helper_mkz_model.fit_MKZ()            |---  _optimization_kernel()                       |                       |--- hlper_mkz_model.tau_MKZ()                       |--- helper_generic.find_closest_index()                       |--- __calc_area()                                  |--- helper_hh_model.tau_FKZ()                                  |--- helper_generic.find_closest_index()                       |--- __find_x_t_and_d()                                  |--- helper_hh_model.tau_FKZ()                                  |--- helper_generic.find_closest_index() #%%---------------------------------------------------------------------------- Get HH parameters of each soil layer from the Vs values of every layer.      Parameters     ----------     vs_profile : numpy.ndarray         Shear-wave velocity profile, as a 2D numpy array. It should have the         following columns:              +---------------+----------+---------+---------+-----------------+             | Thickness (m) | Vs (m/s) | Damping | Density | Material Number |             +===============+==========+=========+=========+=================+             |      ...      |   ...    |   ...   |   ...   |      ...        |             +---------------+----------+---------+---------+-----------------+      Tmax : numpy.ndarray or ``None``         Shear strength of each layer of soil. If ``None``, it will be         calculated using a combination of Ladd (1991) and Mohr-Coulomb criteria.     show_fig : bool         Whether or not to show figures G/Gmax and stress-strain curves of MKZ,         FKZ, and HH for each layer.     save_fig : bool         Whether or not to save the figures to the hard drive. Only effective         if ``show_fig`` is set to ``True``.     fig_output_dir : str         The output directory for the figures. Only effective if ``show_fig``         and ``save_fig`` are both ``True``.     save_HH_G_file : bool         Whether or not to save the HH parameters to the hard drive (as a         "HH_G" file).     HH_G_file_dir : str         The output directory for the "HH_G" file. Only effective if         ``save_HH_G_file`` is ``True``.     profile_name : str or ``None``         The name of the Vs profile, such as "CE.12345". If ``None``, a string         of current date and time will be used as the profile name.     verbose : bool         Whether or not to print progresses on the console.      Returns     -------     HH_G_param : numpy.ndarray         The HH parameters of each layer. It's a 2D array of shape         ``(9, n_layer)``. For each layer (i.e., column), the values are in         this order:             gamma_t, a, gamma_ref, beta, s, Gmax, mu, Tmax, d # friction angle (choose 30 degrees, because there is no better info) # exclude the last layer (i.e., half space) #%%---------------------------------------------------------------------------- Get HH parameters of each soil layer from the Vs profile and G/Gmax curves.      Parameters     ----------     vs_profile : numpy.ndarray         Shear-wave velocity profile, as a 2D numpy array. It should have the         following columns:             +---------------+----------+---------+---------+-----------------+             | Thickness (m) | Vs (m/s) | Damping | Density | Material Number |             +===============+==========+=========+=========+=================+             |      ...      |   ...    |   ...   |   ...   |      ...        |             +---------------+----------+---------+---------+-----------------+      curves : numpy.ndarray         A 2D numpy array that represents G/Gmax and damping curves of each         layer, in the following format:          +------------+--------+------------+-------------+-------------+--------+-----+          | strain [%] | G/Gmax | strain [%] | damping [%] |  strain [%] | G/Gmax | ... |          +============+========+============+=============+=============+========+=====+          |    ...     |  ...   |    ...     |    ...      |    ...      |  ...   | ... |          +------------+--------+------------+-------------+-------------+--------+-----+          The damping information is neglected in this function, so users can         supply some dummy values.     Tmax : numpy.ndarray or ``None``         Shear strength of each layer of soil. If ``None``, it will be         calculated using a combination of Ladd (1991) and Mohr-Coulomb criteria.     show_fig : bool         Whether or not to show figures G/Gmax and stress-strain curves of MKZ,         FKZ, and HH for each layer.     save_fig : bool         Whether or not to save the figures to the hard drive. Only effective         if ``show_fig`` is set to ``True``.     fig_output_dir : str         The output directory for the figures. Only effective if ``show_fig``         and ``save_fig`` are both ``True``.     save_HH_G_file : bool         Whether or not to save the HH parameters to the hard drive (as a         "HH_G" file).     HH_G_file_dir : str         The output directory for the "HH_G" file. Only effective if         ``save_HH_G_file`` is ``True``.     profile_name : str or ``None``         The name of the Vs profile, such as "CE.12345". If ``None``, a string         of current date and time will be used as the profile name.     verbose : bool         Whether or not to print progresses on the console.      Returns     -------     HH_G_param : numpy.ndarray         The HH parameters of each layer. It's a 2D array of shape         ``(9, n_layer)``. For each layer (i.e., column), the values are in         this order:             gamma_t, a, gamma_ref, beta, s, Gmax, mu, Tmax, d # exclude the last layer (i.e., half space) # there can only be 5 or 2 columns # only 2 columns #%%---------------------------------------------------------------------------- Produce HH_G parameters from profiles of Vs, Tmax, OCR, etc.      Parameters     ----------     Vs : numpy.ndarray         Vs values of each layer. Shape: ``(n_layer, )``, where ``n_layer`` is         the length of ``Vs``. Unit: m/s.     Gmax : numpy.ndarray         Initial stiffness of each layer. Shape: ``(n_layer, )``. Unit: Pa.     Tmax : numpy.ndarray         The shear strength of each layer. Shape: ``(n_layer, )``. Unit: Pa.     OCR : numpy.ndarray         Over-consolidation ratio of each layer. Shape: ``(n_layer, )``.     sigma_v0 : numpy.ndarray         Vertical effective confining stress of each layer. Shape:         ``(n_layer, )``. Unit: Pa.     K0 : numpy.ndarray or float         Lateral soil pressure coefficient. If an array, it must have shape         ``(n_layer, )``. If a single value, it means that all layers share         this same value.     curves : numpy.ndarray or ``None``         A 2D numpy array that represents G/Gmax and damping curves of each         layer, in the following format:          +------------+--------+------------+-------------+-------------+--------+-----+          | strain [%] | G/Gmax | strain [%] | damping [%] |  strain [%] | G/Gmax | ... |          +============+========+============+=============+=============+========+=====+          |    ...     |  ...   |    ...     |    ...      |    ...      |  ...   | ... |          +------------+--------+------------+-------------+-------------+--------+-----+          The damping information is neglected in this function, so users can         supply some dummy values. If ``None``, it means that the users do not         have G/Gmax curve information, so this function will calculate the         MKZ G/Gmax curves from the empirical formulas in Darendeli (2001).     PI : float or numpy.ndarray or ``None``         Plasticity index of the soils. It is not necessary (can be ``None``) if         ``curves`` is provided (i.e., not ``None``). If an array, it must have         shape ``(n_layer, )``. If a single value, it means that all layers         share this same value.     phi : float or numpy.ndarray or ``None``         Effective internal frictional angle (in degrees). It is not necessary         (can be ``None``) if ``curve`` is provided (i.e., not ``None``). If         an array, it must have shape ``(n_layer, )``. If a single value, it         means that all layers share this same value.     show_fig : bool         Whether or not to show figures G/Gmax and stress-strain curves of MKZ,         FKZ, and HH for each layer.     save_fig : bool         Whether or not to save the figures to the hard drive. Only effective         if ``show_fig`` is set to ``True``.     fig_output_dir : str         The output directory for the figures. Only effective if ``show_fig``         and ``save_fig`` are both ``True``.     verbose : bool         Whether or not to print progresses on the console.      Returns     -------     parameters : numpy.ndarray         The HH parameters of each layer. It's a 2D array of shape         ``(9, n_layer)``. For each layer (i.e., column), the values are in         this order:             gamma_t, a, gamma_ref, beta, s, Gmax, mu, Tmax, d      Notes     -----     This function is based on ``hybridParaKernel_FKZ.m``. # ============= MKZ fit =================================================== # user does not provide curves # unit: percent # strain matrix for all layers # user provides own curves # ----------- Extract G/Gmax information ------------------------------ # unit: percent # ----------- Fit MKZ parameters -------------------------------------- # ----------- Show results on console --------------------------------- # ========== Stress-strain curve implied by G/Gmax ======================== # END FOR # END FOR # ========== Estimate mu using empirical correlations ===================== # unit: Pa --> kPa # softer soil: use Vardanega & Bolton (2011) CGJ formula # mu too small --> too low tau_FKZ --> sharply decreasing tau_HH # 0.236 is the standard error suggested in Vardanega & Bolton (2011) # END IF # stiffer soils: set mu to 1 for lack of better information # END IF # END FOR # ========== Start FKZ optimization ======================================= # Darendeli's curve #%d.png' % (j+1))) #%%---------------------------------------------------------------------------- Calculate shear strength of soils.      Parameters     ----------     Vs : numpy.ndarray         A 1D array of Vs values of each layer. Unit: m/s.     OCR : numpy.ndarray         A 1D array of OCR (over-consolidation ratio) of each layer. Unit: 1.     sigma_v0 : numpy.ndarray         A 1D array of vertical overburden pressure. Unit: Pa.     K0 : float, int, numpy.ndarray, or ``None``         Lateral soil pressure coefficient. If a single value is given, it is         assumed to be the value for all layers. If ``None``, it will be         determined from OCR via an empirical formula by Mayne & Kulhawy (1982).     phi : float, int, or numpy.ndarray         Effective internal friction angle of soils (in degrees).      Returns     -------     Tmax : numpy.ndarray         Shear strength of soils of each layer. Unit: Pa. # assume a strain rate of 0.01 sec^(-1), from Vardanega & Bolton (2013) # for softer soils, calculate undrained shear strength # Ladd (1991) # stiffer soils: Mohr-Coulomb criterion # horizontal stress # largest principal stress # smallest principal stress # normal effective stress on the slip plane #%%---------------------------------------------------------------------------- Calculate initial stiffness of each soil layer.      Parameters     ----------     Vs : numpy.ndarray         1D array of Vs of layers. Unit: m/s.     rho : numpy.ndarray         1D array of mass density of layers. Unit: kg/m^3.      Returns     -------     Gmax : numpy.ndarray         1D array of initial stiffness. Unit: Pa #%%---------------------------------------------------------------------------- Calculate OCR (over-consolidation ratio) of each layer from the Vs profile.      Parameters     ----------     Vs : numpy.ndarray         1D array of Vs of layers. Unit: m/s.     rho : numpy.ndarray         1D array of mass density of layers. Unit: kg/m^3.     sigma_v0 : numpy.ndarray         Vertical overburden stress at the mid-point of each layer. Unit: Pa.     OCR_upper_limit : float or ``None``         The maximum acceptable OCR value. If ``None``, there is no limit.      Returns     -------     OCR : numpy.ndarray         1D array of OCR value, for each soil layer. (Unitless.) # Mayne, <NAME> (1998) "Clay stress history evaluated fromseismic piezocone tests" # kPa --> Pa #%%---------------------------------------------------------------------------- Calculate vertical overburden stress at the mid-point of each layer.      Parameters     ----------     h : numpy.ndarray         1D array of layer thickness. Unit: m.     rho : numpy.ndarray         1D array of mass density of each layer. Unit: kg/m^3.      Returns     -------     stress : numpy.ndarray         Vertical overburden stress at the mid-point of each layer. Unit: Pa. # unit: m/s/s # zero thickness, i.e., half space # divided by 2: middle of layer #%%---------------------------------------------------------------------------- Calculate mass density of soils from Vs values, using the empirical formula     by Mayne, Schneider & Martin (1999) and Burns & Mayne (1996).      Parameters     ----------     h : numpy.ndarray         The thickness of each soil layer. Unit: m.     Vs : numpy.ndarray         The shear-wave velocity for each layer. It needs to be a 1D numpy array.         Unit: m/s.      Returns     -------     rho : numpy.ndarray         Mass density of each soil layer. Unit: kg/m^3.      References     ----------     1. Mayne, Schneider & Martin (1999) "Small- and large-strain soil        properties from seismic flat dilatometer tests." Pre-failure        deformation characteristics of geomaterials, 1999 Balkema, Rotterdam.     2. Burns & Mayne (1996) "Small- and high-strain soil properties using the        seismic piezocone." Transportation Research Record 1548, National        Acad. Press, Washington DC, 81-88. # avoid error of dividing by zero # lower bound of density: 1.65 g/cm^3 # unit: g/cm^3 --> kg/m^3 #%%---------------------------------------------------------------------------- Calculate PI (plasticity index) from Vs values.      Parameters     ----------     Vs : numpy.ndarray         The shear-wave velocity for each layer. It needs to be a 1D numpy array.         Unit: m/s.      Returns     -------     PI : numpy.ndarray         The plasticity index for each layer. Unit: %. #%%---------------------------------------------------------------------------- Calculate K0 (lateral earth pressure coefficient at rest) from OCR using     the empirical formula by Mayne & Kulhawy (1982).      Parameters     ----------     OCR : float, int, or numpy.ndarray         Over-consolidation ratio of each layer of soils. If it is a float/int,         it means only one layer, or all the layers have the same OCR.     phi : float, int, or numpy.ndarray         Internal effective friction angle of soils. If it is a float/int, it         means only one soil layer, or all the layers have the same angle.         Unit: deg.      Returns     -------     K0 : float or numpy.ndarray         K0 value(s). If either ``OCR`` or ``phi`` is an array, ``K0`` will be         an array of the same length. #%%---------------------------------------------------------------------------- Produce G/Gmax and damping curves using empirical correlations by     Darendeli (2001).      Parameters     ----------     sigma_v0 : numpy.ndarray         Effective vertical confining stress of each layer. Unit: Pa.     PI : int, float, or numpy.ndarray         Plasticity index of each layer. Unit: %. If a single value is given,         it is assumed to be the PI for all layers.     OCR : int, float, or numpy.ndarray         Over-consolidation ratio of each layer. If a single value is given,         it is assumed to be the value for all layers.     K0 : int, float, numpy.ndarray, or ``None``         Lateral soil pressure coefficient. If a single value is given, it is         assumed to be the value for all layers. If ``None``, it will be         determined from OCR via an empirical formula by Mayne & Kulhawy (1982).     phi : int, float, or numpy.ndarray         Internal effective friction angle of soils. If it is a float/int, it         means all the layers have the same angle. Unit: deg.     strain_in_pct : numpy.ndarray or ``None``         The strain values at which to calculate G/Gmax and damping values. If         ``None``, numpy.geomspace(1e-4, 10, 400) will be used. Unit: percent.      Returns     -------     GGmax : numpy.ndarray         G/Gmax curves for each layer. It is a 2D numpy array. Each column of it         represents the G/Gmax curve of a particular layer. Unit: 1     xi : numpy.ndarray         Damping curves for each layer. Same shape as ``GGmax``. Unit: 1.     gamma_r : numpy.ndarray         Reference strain for each layer. It is a 1D numpy array, corresponding         to each soil layer. Unit: 1. # Define all constants # from Darendeli (2001), page 226 # Darendeli (2001) Eq 9.1d # Confinine stress # octahedral stress # unit: Pa --> atm # Reference strain for each layer (Eq 9.1a). Unit: percent # G of i-th layer (Eq 9.2a) # Unit: percent (page 226) # Unit: percent (page 226) # Eq 9.1c (page 221) # Eq 9.2b (page 224). Unit: percent #%%---------------------------------------------------------------------------- Calculate mean (of three directions) confining stress.      Parameters     ----------     sigma_v0 : numpy.ndarray         (Effective) vertical stress of each layer. Unit: Pa.     K0 : numpy.ndarray         Lateral stress coefficient of each layer. Unit: 1.      Returns     -------     sigma_m0 : numpy.ndarray         Mean effective confining stress (of three directions). Unit: Pa. #%%---------------------------------------------------------------------------- Optimization process to find FKZ parameters.      Parameters     ----------     x : numpy.ndarray         An 1D array of shear strain. Unit: 1.     x_ref : float         The "reference strain" parameter (in MKZ) of the soil. Unit: 1.     beta : float         A shape parameter of the FKZ model.     s : float         A shape parameter of the FKZ model.     Gmax : float         Initial shear modulus. Unit: Pa.     tau_f : float         The shear strength of the current soil layer. Unit: Pa.     mu : float         The "shape parameter" of the FKZ model.      Returns     -------     a : float         A parameter of the HH model that defines the "speed" of transition from         MKZ to FKZ     gamma_t : float         The shear strain at which the transition from MKZ to FKZ happens.         Unit: 1     d : float         The "shape power" parameter in the FKZ model.      Notes     -----     Based on optHybFKZ.m # when mu is too small, there may be some numerical issues # therefore gamma_t lower bound is relaxed # unit: percent # for very deep layers, tau_MKZ may be larger than tau_f at gamma_t_LB # do it again # it means that a proper d value is found # cannot find a proper d value # increase grid density to 400 # increase grid density # further relax # further ralax to 0.001% # END IF # END IF #END IF # always use a fast transition #%%---------------------------------------------------------------------------- Find the ``x_t`` (transition strain) that minimizes the "area" between     the MKZ stress curve and the FKZ stress curve.      Parameters     ----------     area : numpy.ndarray         The "area" between the MKZ stress curve and the FKZ stress curve. It         has the same shape as ``range_d``, because each element of ``area`` is         the area corresponding to a ``d`` value within ``range_d``.     range_d : numpy.ndarray         The range of ``d`` to search from. Must be a 1D numpy array.     x : numpy.ndarray         An 1D array of shear strain. Unit: 1.     Gmax : float         Initial shear modulus. Unit: Pa.     mu : float         The "shape parameter" of the FKZ model.     tau_f : float         The shear strength of the current soil layer. Unit: Pa.     T_MKZ : numpy.ndarray         The MKZ stress curve, which has the same shape as ``x``. Unit: Pa.      Returns     -------     x_t : float         The ``x_t`` value that minimizes the "area" between the MKZ stress         curve and the FKZ stress curve. Unit: 1.     d : float         The ``d`` value that minimizes the "area" between the MKZ stress curve         and the FKZ stress curve. (No unit.) #%%---------------------------------------------------------------------------- Calculate the "area" between the MKZ stress curve and the FKZ stress curve.     The MKZ stress curve is supplied as a parameter, and the FKZ stress curve     is determined by ``x``, ``Gmax``, ``mu``, ``d``, ``tau_f``, and ``gamma_t``.      Parameters     ----------     range_d : numpy.ndarray         The range of ``d`` to search from. Must be a 1D numpy array.     x : numpy.ndarray         An 1D array of shear strain. Unit: 1.     Gmax : float         Initial shear modulus. Unit: Pa.     mu : float         The "shape parameter" of the FKZ model.     tau_f : float         The shear strength of the current soil layer. Unit: Pa.     gamma_t_LB : float         The lower bound of ``gamma_t`` (:math:`\gamma_t`), i.e., the transition         strain. Unit: %.     gamma_t_UB : float         The upper bound of ``gamma_t``. Unit: %.     T_MKZ : numpy.ndarray         The MKZ stress curve, which has the same shape as ``x``. Unit: Pa.      Returns     -------     area : numpy.ndarray         The "area" between the MKZ stress curve and the FKZ stress curve. It         has the same shape as ``range_d``, because each element of ``area`` is         the area corresponding to a ``d`` value within ``range_d``. # unit: 1 # "copt" = cross-over point # END IF # END FOR