"""
Implementation of the equations used for surface tension modelling,
for example in KHI simulations. The references are as under:

 - M. Shadloo, M. Yildiz, "Numerical modelling of Kelvin-Helmholtz
   isntability using smoothed particle hydrodynamics", IJNME, 2011,
   87, pp 988--1006 [SY11]

 - Joseph P. Morris "Simulating surface tension with smoothed particle
   hydrodynamics", JCP, 2000, 33, pp 333--353 [JM00]

 - Adami et al. "A new surface-tension formulation for multi-phase SPH
   using a reproducing divergence approximation", JCP 2010, 229, pp
   5011--5021 [A10]

"""
from pysph.sph.equation import Equation

from math import sqrt

class SmoothedColor(Equation):
    r"""Smoothed color function. Eq. (17) in [JM00]

    .. math::
    
        c_a = \sum_b \frac{m_b}{\rho_b} c_b^i \nabla_a W_{ab}\,,

    where, :math:`c_b^i` is the color index associated with a
    particle.

    """
    def __init__(self, dest, sources=None, smooth=False):
        self.smooth = smooth
        super(SmoothedColor, self).__init__(dest, sources)

    def initialize(self, d_idx, d_scolor):
        d_scolor[d_idx]  = 0.0

    def loop(self, d_idx, s_idx, d_rho, s_rho, s_m,
             s_color, d_scolor, WIJ):

        # Smoothed color Eq. (17) in [JM00]
        d_scolor[d_idx] += s_m[s_idx]/s_rho[s_idx] * s_color[s_idx] * WIJ

    def post_loop(self, d_idx, d_color, d_scolor):
        # overwrite the smoothed color if not needed
        if not self.smooth:
            d_scolor[d_idx] = d_color[d_idx]

class ColorGradientUsingNumberDensity(Equation):
    r"""Gradient of the color function using Eq. (13) of [SY11]:

    .. math::

        \nabla C_a = \sum_b \frac{2 C_b - C_a}{\psi_a + \psi_a}
        \nabla_{a} W_{ab}


    Using the gradient of the color function, the normal and
    discretized dirac delta is calculated in the post
    loop. 
    
    Singularities are avoided as per the recommendation by [JM00] (see eqs
    20 & 21) using the parameter :math:`\epsilon`

    """
    def __init__(self, dest, sources=None, epsilon=1e-6):
        self.epsilon2 = epsilon*epsilon
        super(ColorGradientUsingNumberDensity, self).__init__(dest, sources)

    def initialize(self, d_idx, d_cx, d_cy, d_cz, d_nx, d_ny, d_nz,
                   d_ddelta, d_N):

        # color gradient
        d_cx[d_idx] = 0.0
        d_cy[d_idx] = 0.0
        d_cz[d_idx] = 0.0

        # interface normals
        d_nx[d_idx] = 0.0
        d_ny[d_idx] = 0.0
        d_nz[d_idx] = 0.0

        # discretized dirac delta
        d_ddelta[d_idx] = 0.0

        # reliability indicator for normals
        d_N[d_idx] = 0.0

    def loop(self, d_idx, s_idx, d_scolor, s_scolor, d_cx, d_cy, d_cz,
             d_V, s_V, DWIJ):
        
        # average particle volume
        psiab1 = 2.0/( d_V[d_idx] + s_V[s_idx] )

        # difference in color divided by psiab. Eq. (13) in [SY11]
        Cba = (s_scolor[s_idx] - d_scolor[d_idx]) * psiab1
        
        # color gradient
        d_cx[d_idx] += Cba * DWIJ[0]
        d_cy[d_idx] += Cba * DWIJ[1]
        d_cz[d_idx] += Cba * DWIJ[2]

    def post_loop(self, d_idx, d_cx, d_cy, d_cz,
                  d_nx, d_ny, d_nz, d_N, d_ddelta):
        # absolute value of the color gradient
        mod_gradc2 = d_cx[d_idx]*d_cx[d_idx] + \
            d_cy[d_idx]*d_cy[d_idx] + \
            d_cz[d_idx]*d_cz[d_idx]

        # avoid sqrt computations on non-interface particles
        # (particles for which the color gradient is zero) Eq. (19,
        # 20) in [JM00]
        if mod_gradc2 > self.epsilon2:
            # this normal is reliable in the sense of [JM00]
            d_N[d_idx] = 1.0

            # compute the normals
            mod_gradc = 1./sqrt( mod_gradc2 )

            d_nx[d_idx] = d_cx[d_idx] * mod_gradc
            d_ny[d_idx] = d_cy[d_idx] * mod_gradc
            d_nz[d_idx] = d_cz[d_idx] * mod_gradc

            # discretized Dirac Delta function
            d_ddelta[d_idx] = 1./mod_gradc

class MorrisColorGradient(Equation):
    r"""Gradient of the color function using Eq. (17) of [JM00]:

    .. math::

        \nabla c_a = \sum_b \frac{m_b}{\rho_b}(c_b - c_a) \nabla_{a}
        W_{ab}\,,

    where a smoothed representation of the color is used in the
    equation. Using the gradient of the color function, the normal and
    discretized dirac delta is calculated in the post loop.
    
    Singularities are avoided as per the recommendation by [JM00] (see eqs
    20 & 21) using the parameter :math:`\epsilon`

    """
    def __init__(self, dest, sources=None, epsilon=1e-6):
        self.epsilon2 = epsilon*epsilon
        super(MorrisColorGradient, self).__init__(dest, sources)

    def initialize(self, d_idx, d_cx, d_cy, d_cz, d_nx, d_ny, d_nz,
                   d_ddelta, d_N):

        # color gradient
        d_cx[d_idx] = 0.0
        d_cy[d_idx] = 0.0
        d_cz[d_idx] = 0.0

        # interface normals
        d_nx[d_idx] = 0.0
        d_ny[d_idx] = 0.0
        d_nz[d_idx] = 0.0

        # reliability indicator for normals and dirac delta
        d_N[d_idx] = 0.0
        d_ddelta[d_idx] = 0.0        

    def loop(self, d_idx, s_idx, d_scolor, s_scolor, d_cx, d_cy, d_cz,
             s_m, s_rho, DWIJ):
        
        # Eq. (17) in [JM00]
        Cba = (s_scolor[s_idx] - d_scolor[d_idx]) * s_m[s_idx]/s_rho[s_idx]
        
        # color gradient
        d_cx[d_idx] += Cba * DWIJ[0]
        d_cy[d_idx] += Cba * DWIJ[1]
        d_cz[d_idx] += Cba * DWIJ[2]

    def post_loop(self, d_idx, d_cx, d_cy, d_cz,
                  d_nx, d_ny, d_nz, d_N, d_ddelta):
        # absolute value of the color gradient
        mod_gradc2 = d_cx[d_idx]*d_cx[d_idx] + \
            d_cy[d_idx]*d_cy[d_idx] + \
            d_cz[d_idx]*d_cz[d_idx]

        # avoid sqrt computations on non-interface particles
        # (particles for which the color gradient is zero) Eq. (19,
        # 20) in [JM00]
        if mod_gradc2 > self.epsilon2:
            # this normal is reliable in the sense of [JM00]
            d_N[d_idx] = 1.0

            # compute the normals
            mod_gradc = 1./sqrt( mod_gradc2 )

            d_nx[d_idx] = d_cx[d_idx] * mod_gradc
            d_ny[d_idx] = d_cy[d_idx] * mod_gradc
            d_nz[d_idx] = d_cz[d_idx] * mod_gradc

            # discretized Dirac Delta function
            d_ddelta[d_idx] = 1./mod_gradc

class SY11ColorGradient(Equation):
    r"""Gradient of the color function using Eq. (13) of [SY11]:

    .. math::

        \nabla C_a = \sum_b \frac{2 C_b - C_a}{\psi_a + \psi_a}
        \nabla_{a} W_{ab}


    Using the gradient of the color function, the normal and
    discretized dirac delta is calculated in the post
    loop. 

    """
    def __init__(self, dest, sources=None, epsilon=1e-6):
        self.epsilon2 = epsilon*epsilon
        super(SY11ColorGradient, self).__init__(dest, sources)

    def initialize(self, d_idx, d_cx, d_cy, d_cz, d_nx, d_ny, d_nz,
                   d_ddelta, d_N):

        # color gradient
        d_cx[d_idx] = 0.0
        d_cy[d_idx] = 0.0
        d_cz[d_idx] = 0.0

        # interface normals
        d_nx[d_idx] = 0.0
        d_ny[d_idx] = 0.0
        d_nz[d_idx] = 0.0

        # discretized dirac delta
        d_ddelta[d_idx] = 0.0

        # reliability indicator for normals
        d_N[d_idx] = 0.0

    def loop(self, d_idx, s_idx, d_color, s_color, d_cx, d_cy, d_cz,
             d_V, s_V, DWIJ):
        
        # average particle volume
        psiab1 = 2.0/( d_V[d_idx] + s_V[s_idx] )

        # difference in color divided by psiab. Eq. (13) in [SY11]
        Cba = (s_color[s_idx] - d_color[d_idx]) * psiab1
        
        # color gradient
        d_cx[d_idx] += Cba * DWIJ[0]
        d_cy[d_idx] += Cba * DWIJ[1]
        d_cz[d_idx] += Cba * DWIJ[2]

    def post_loop(self, d_idx, d_cx, d_cy, d_cz,
                  d_nx, d_ny, d_nz, d_N, d_ddelta):
        # absolute value of the color gradient
        mod_gradc2 = d_cx[d_idx]*d_cx[d_idx] + \
            d_cy[d_idx]*d_cy[d_idx] + \
            d_cz[d_idx]*d_cz[d_idx]

        # avoid sqrt computations on non-interface particles
        if mod_gradc2 > self.epsilon2:
            # this normal is reliable in the sense of [JM00]
            d_N[d_idx] = 1.0

            # compute the normals
            mod_gradc = 1./sqrt( mod_gradc2 )

            d_nx[d_idx] = d_cx[d_idx] * mod_gradc
            d_ny[d_idx] = d_cy[d_idx] * mod_gradc
            d_nz[d_idx] = d_cz[d_idx] * mod_gradc

            # discretized Dirac Delta function
            d_ddelta[d_idx] = 1./mod_gradc

class SY11DiracDelta(Equation):
    r"""Discretized dirac-delta for the SY11 formulation Eq. (14) in [SY11]

    This is essentially the same as computing the color gradient, the
    only difference being that this might be called with a reduced
    smoothing length.

    Note that the normals should be computed using the
    SY11ColorGradient equation. This function will effectively
    overwrite the color gradient.

    """
    def __init__(self, dest, sources=None, epsilon=1e-6):
        self.epsilon2 = epsilon*epsilon
        super(SY11DiracDelta, self).__init__(dest, sources)

    def initialize(self, d_idx, d_cx, d_cy, d_cz, d_ddelta):
        # color gradient
        d_cx[d_idx] = 0.0
        d_cy[d_idx] = 0.0
        d_cz[d_idx] = 0.0

        # discretized dirac delta
        d_ddelta[d_idx] = 0.0

    def loop(self, d_idx, s_idx, d_color, s_color, d_cx, d_cy, d_cz,
             d_V, s_V, DWIJ):
        
        # average particle volume
        psiab1 = 2.0/( d_V[d_idx] + s_V[s_idx] )

        # difference in color divided by psiab. Eq. (13) in [SY11]
        Cba = (s_color[s_idx] - d_color[d_idx]) * psiab1
        
        # color gradient
        d_cx[d_idx] += Cba * DWIJ[0]
        d_cy[d_idx] += Cba * DWIJ[1]
        d_cz[d_idx] += Cba * DWIJ[2]

    def post_loop(self, d_idx, d_cx, d_cy, d_cz,
                  d_nx, d_ny, d_nz, d_N, d_ddelta):
        # absolute value of the color gradient
        mod_gradc2 = d_cx[d_idx]*d_cx[d_idx] + \
            d_cy[d_idx]*d_cy[d_idx] + \
            d_cz[d_idx]*d_cz[d_idx]

        # avoid sqrt computations on non-interface particles
        if mod_gradc2 > self.epsilon2:
            mod_gradc = sqrt( mod_gradc2 )

            # discretized Dirac Delta function
            d_ddelta[d_idx] = mod_gradc

class InterfaceCurvatureFromNumberDensity(Equation):
    r"""Interface curvature using number density. Eq. (15) in [SY11]:

    .. math::

        \kappa_a = \sum_b \frac{2.0}{\psi_a + \psi_b}
        \left(\boldsymbol{n_a} - \boldsymbol{n_b}\right) \cdot
        \nabla_a W_{ab}

    """
    def __init__(self, dest, sources=None, with_morris_correction=True):
        self.with_morris_correction = with_morris_correction

        super(InterfaceCurvatureFromNumberDensity,self).__init__(dest, sources)

    def initialize(self, d_idx, d_kappa, d_wij_sum):
        d_kappa[d_idx] = 0.0
        d_wij_sum[d_idx] = 0.0

    def loop(self, d_idx, s_idx, d_kappa, d_nx, d_ny, d_nz, s_nx, s_ny, s_nz, 
             d_V, s_V, d_N, s_N, d_wij_sum, s_rho, s_m, WIJ, DWIJ):
        
        nijdotdwij = (d_nx[d_idx] - s_nx[s_idx]) * DWIJ[0] + \
            (d_ny[d_idx] - s_ny[s_idx]) * DWIJ[1] + \
            (d_nz[d_idx] - s_nz[s_idx]) * DWIJ[2]
        
        # averaged particle number density
        psiij1 = 2.0/(d_V[d_idx] + s_V[s_idx])

        # local number density with reliable normals Eq. (24) in [JM00]
        tmp = 1.0
        if self.with_morris_correction:
            tmp = min(d_N[d_idx], s_N[s_idx])

        d_wij_sum[d_idx] += tmp * s_m[s_idx]/s_rho[s_idx] * WIJ

        # Eq. (15) in [SY11] with correction Eq. (22) in [JM00]
        d_kappa[d_idx] += tmp * psiij1 * nijdotdwij

    def post_loop(self, d_idx, d_wij_sum, d_nx, d_kappa):
        # correct the curvature estimate. Eq. (23) in [JM00]
        if self.with_morris_correction:
            if d_wij_sum[d_idx] > 1e-12:
                d_kappa[d_idx] /= d_wij_sum[d_idx]

class ShadlooYildizSurfaceTensionForce(Equation):
    r"""Acceleration due to surface tension force Eq. (7,9) in [SY11]:

    .. math:

        \frac{d\boldsymbol{v}_a} = \frac{1}{m_a} \sigma \kappa_a
        \boldsymbol{n}_a \delta_a^s\,,

    where, :math:`\delta^s` is the discretized dirac delta function,
    :math:`\boldsymbol{n}` is the interface normal, :math:`\kappa` is
    the discretized interface curvature and :math:`\sigma` is the
    surface tension force constant.
    
    """
    def __init__(self, dest, sources=None, sigma=0.1):
        self.sigma = sigma
        
        # base class initialization
        super(ShadlooYildizSurfaceTensionForce, self).__init__(dest, sources)

    def initialize(self, d_idx, d_au, d_av, d_aw):
        d_au[d_idx] = 0.0
        d_av[d_idx] = 0.0
        d_aw[d_idx] = 0.0

    def loop(self, d_idx, d_au, d_av, d_aw, d_kappa,
             d_nx, d_ny, d_nz, d_m, d_rho, d_ddelta):

        mi = 1./d_m[d_idx]
        rhoi = 1./d_rho[d_idx]

        # acceleration per uint mass term Eq. (7) in [SY11]
        tmp = self.sigma * d_kappa[d_idx] * d_ddelta[d_idx] * rhoi

        d_au[d_idx] += tmp * d_nx[d_idx]
        d_av[d_idx] += tmp * d_ny[d_idx]
        d_aw[d_idx] += tmp * d_nz[d_idx]

class CSFSurfaceTensionForce(Equation):
    r"""Acceleration due to surface tension force Eq. (25) in [JM00]:

    .. math:

        \frac{d\boldsymbol{v}_a}{dt} = \frac{1}{rho_a} \sigma_b \kappa_a
        \boldsymbol{n}_a

    Note that as per Eq. (17) in [JM00], the un-normalized normal is
    basically the gradient of the color function. The acceleration
    term therefore depends on the gradient of the color field.
    
    """
    def __init__(self, dest, sources=None, sigma=0.1):
        self.sigma = sigma
        
        # base class initialization
        super(CSFSurfaceTensionForce, self).__init__(dest, sources)

    def initialize(self, d_idx, d_au, d_av, d_aw):
        d_au[d_idx] = 0.0
        d_av[d_idx] = 0.0
        d_aw[d_idx] = 0.0

    def loop(self, d_idx, d_au, d_av, d_aw, d_kappa,
             d_cx, d_cy, d_cz, d_rho):

        rhoi = 1./d_rho[d_idx]

        # acceleration per uint mass term Eq. (25) in [JM00]
        tmp = self.sigma * d_kappa[d_idx] * rhoi

        d_au[d_idx] += tmp * d_cx[d_idx]
        d_av[d_idx] += tmp * d_cy[d_idx]
        d_aw[d_idx] += tmp * d_cz[d_idx]

class AdamiColorGradient(Equation):
    r"""Gradient of color Eq. (14) in [A10]

    .. math::

        \nabla c_a = \frac{1}{V_a}\sum_b \left[V_a^2 + V_b^2
        \right]\tilde{c}_{ab}\nabla_a W_{ab}\,,

    where, the average :math:`\tilde{c}_{ab}` is defined as

    .. math::

        \tilde{c}_{ab} = \frac{\rho_b}{\rho_a + \rho_b}c_a +
        \frac{\rho_a}{\rho_a + \rho_b}c_b

    """
    def initialize(self, d_idx, d_cx, d_cy, d_cz, d_nx, d_ny, d_nz, d_ddelta, d_N):
        d_cx[d_idx] = 0.0
        d_cy[d_idx] = 0.0
        d_cz[d_idx] = 0.0

        d_nx[d_idx] = 0.0
        d_ny[d_idx] = 0.0
        d_nz[d_idx] = 0.0

        # reliability indicator for normals
        d_N[d_idx] = 0.0

        # Discretized dirac-delta
        d_ddelta[d_idx] = 0.0

    def loop(self, d_idx, s_idx, d_V, s_V, d_rho, s_rho, 
             d_cx, d_cy, d_cz, d_color, s_color, DWIJ):

        # particle volumes
        Vi = 1./d_V[d_idx]; Vj = 1./s_V[s_idx]
        Vi2 = Vi * Vi; Vj2 = Vj * Vj

        # averaged particle color
        rhoi = d_rho[d_idx]
        rhoj = s_rho[s_idx]
        rhoij1 = 1./(rhoi + rhoj)
        
        # Eq. (15) in [A10]
        cij = rhoj*rhoij1*d_color[d_idx] + rhoi*rhoij1*s_color[s_idx]
        
        # comute the gradient
        tmp = cij * (Vi2 + Vj2)/Vi

        d_cx[d_idx] += tmp * DWIJ[0]
        d_cy[d_idx] += tmp * DWIJ[1]
        d_cz[d_idx] += tmp * DWIJ[2]

    def post_loop(self, d_idx, d_cx, d_cy, d_cz,
                  d_nx, d_ny, d_nz, d_ddelta, d_N):
        # absolute value of the color gradient
        mod_gradc2 = d_cx[d_idx]*d_cx[d_idx] + \
            d_cy[d_idx]*d_cy[d_idx] + \
            d_cz[d_idx]*d_cz[d_idx]

        # avoid sqrt computations on non-interface particles
        if mod_gradc2 > 1e-6:
            # this normal is reliable in the sense of [JM00]
            d_N[d_idx] = 1.0

            # compute the normals
            mod_gradc = 1./sqrt( mod_gradc2 )

            d_nx[d_idx] = d_cx[d_idx] * mod_gradc
            d_ny[d_idx] = d_cy[d_idx] * mod_gradc
            d_nz[d_idx] = d_cz[d_idx] * mod_gradc

            # discretized dirac delta
            d_ddelta[d_idx] = 1./mod_gradc

# FIXME: The implementation based on the formulation presented in
# [A10] seems to be incorrect.
class AdamiReproducingDivergence(Equation):
    r"""Reproducing divergence approximation Eq. (20) in [A10] to
    compute the curvature

    .. math::

        \nabla \cdot \boldsymbol{\phi}_a = d\frac{\sum_b
        \boldsymbol{\phi}_{ab}\cdot \nabla_a
        W_{ab}V_b}{\sum_b\boldsymbol{x}_{ab}\cdot \nabla_a W_{ab} V_b}

    """
    def __init__(self, dest, sources=None, dim=2):
        self.dim = dim
        super(AdamiReproducingDivergence,self).__init__(dest, sources)

    def initialize(self, d_idx, d_kappa, d_wij_sum):
        d_kappa[d_idx] = 0.0
        d_wij_sum[d_idx] = 0.0

    def loop(self, d_idx, s_idx, d_kappa, d_wij_sum, 
             d_nx, d_ny, d_nz, s_nx, s_ny, s_nz, d_V, s_V,
             DWIJ, XIJ, RIJ, EPS):
        # particle volumes
        Vi = 1./d_V[d_idx]; Vj = 1./s_V[s_idx]
        
        # dot product in the numerator of Eq. (20)
        nijdotdwij = (d_nx[d_idx] - s_nx[s_idx]) * DWIJ[0] + \
            (d_ny[d_idx] - s_ny[s_idx]) * DWIJ[1] + \
            (d_nz[d_idx] - s_nz[s_idx]) * DWIJ[2]
        
        # dot product in the denominator of Eq. (20)
        xijdotdwij = XIJ[0]*DWIJ[0] + XIJ[1]*DWIJ[1] + XIJ[2]*DWIJ[2]
        xijdotdwij /= (RIJ + EPS)
        
        # accumulate the contributions
        d_kappa[d_idx] += nijdotdwij * Vj
        d_wij_sum[d_idx] += RIJ * xijdotdwij * Vj

    def post_loop(self, d_idx, d_kappa, d_wij_sum):
        # normalize the curvature estimate
        d_kappa[d_idx] /= d_wij_sum[d_idx]
        d_kappa[d_idx] *= self.dim
