# Source code for pysph.sph.wc.transport_velocity

    """
    Transport Velocity Formulation
    ##############################

    References
    ----------
        .. [Adami2012] S. Adami et. al "A generalized wall boundary condition for
            smoothed particle hydrodynamics", Journal of Computational Physics
            (2012), pp. 7057--7075.

        .. [Adami2013] S. Adami et. al "A transport-velocity formulation for
            smoothed particle hydrodynamics", Journal of Computational Physics
            (2013), pp. 292--307.

    """

    from pysph.sph.equation import Equation
    from math import sin, pi

    # constants
    M_PI = pi

    class SummationDensity(Equation):
        r"""**Summation density with volume summation**

        In addition to the standard summation density, the number density
        for the particle is also computed. The number density is important
        for multi-phase flows to define a local particle volume
        independent of the material density.

        .. math::

            \rho_a = \sum_b m_b W_{ab}\\

            \mathcal{V}_a = \frac{1}{\sum_b W_{ab}}

        Notes
        -----

        Note that in the pysph implementation, V is the inverse volume of a
        particle, i.e. the equation computes V as follows:

        .. math::

           \mathcal{V}_a = \sum_b W_{ab}

        For this equation, the destination particle array must define the
        variable `V` for particle volume.
        """

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

        def loop(self, d_idx, d_V, d_rho, d_m, WIJ):
            d_V[d_idx] += WIJ
            d_rho[d_idx] += d_m[d_idx]*WIJ

    class VolumeSummation(Equation):
        r"""**Number density for volume computation**

        See `SummationDensity`

        Note that the quantity `V` is really :math:`\sigma` of the original paper,
        i.e. inverse of the particle volume.

        """

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

        def loop(self, d_idx, d_V, WIJ):
            d_V[d_idx] += WIJ

    class VolumeFromMassDensity(Equation):
        """**Set the inverse volume using mass density**"""

        def loop(self, d_idx, d_V, d_rho, d_m):
            d_V[d_idx] = d_rho[d_idx]/d_m[d_idx]

    class SetWallVelocity(Equation):
        r"""**Extrapolating the fluid velocity on to the wall**

        Eq. (22) in [Adami2012]:

        .. math::

            \tilde{\boldsymbol{v}}_a = \frac{\sum_b\boldsymbol{v}_b W_{ab}}
            {\sum_b W_{ab}}

        Notes
        -----
        The destination particle array for this equation should define the
        *filtered* velocity variables :math:`uf, vf, wf`.

        """

        def initialize(self, d_idx, d_uf, d_vf, d_wf, d_wij):
            d_uf[d_idx] = 0.0
            d_vf[d_idx] = 0.0
            d_wf[d_idx] = 0.0
            d_wij[d_idx] = 0.0

        def loop(self, d_idx, s_idx, d_uf, d_vf, d_wf,
                 s_u, s_v, s_w, d_wij, WIJ):

            # normalisation factor is different from 'V' as the particles
            # near the boundary do not have full kernel support
            d_wij[d_idx] += WIJ

            # sum in Eq. (22)
            # this will be normalized in post loop
            d_uf[d_idx] += s_u[s_idx] * WIJ
            d_vf[d_idx] += s_v[s_idx] * WIJ
            d_wf[d_idx] += s_w[s_idx] * WIJ

        def post_loop(self, d_uf, d_vf, d_wf, d_wij, d_idx,
                      d_ug, d_vg, d_wg, d_u, d_v, d_w):

            # calculation is done only for the relevant boundary particles.
            # d_wij (and d_uf) is 0 for particles sufficiently away from the
            # solid-fluid interface
            if d_wij[d_idx] > 1e-12:
                d_uf[d_idx] /= d_wij[d_idx]
                d_vf[d_idx] /= d_wij[d_idx]
                d_wf[d_idx] /= d_wij[d_idx]

            # Dummy velocities at the ghost points using Eq. (23),
            # d_u, d_v, d_w are the prescribed wall velocities.
            d_ug[d_idx] = 2*d_u[d_idx] - d_uf[d_idx]
            d_vg[d_idx] = 2*d_v[d_idx] - d_vf[d_idx]
            d_wg[d_idx] = 2*d_w[d_idx] - d_wf[d_idx]

    class ContinuityEquation(Equation):
        r"""**Conservation of mass equation**

        Eq (6) in [Adami2012]:

        .. math::

            \frac{d\rho_a}{dt} = \rho_a \sum_b \frac{m_b}{\rho_b}
            \boldsymbol{v}_{ab} \cdot \nabla_a W_{ab}

        """

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

        def loop(self, d_idx, s_idx, d_arho, s_m, s_rho, d_rho, VIJ, DWIJ):
            vijdotdwij = VIJ[0] * DWIJ[0] + VIJ[1] * DWIJ[1] + VIJ[2] * DWIJ[2]
            d_arho[d_idx] += d_rho[d_idx] * vijdotdwij * s_m[s_idx] / s_rho[s_idx]

    class ContinuitySolid(Equation):
        """Continuity equation for the solid's ghost particles.

        The key difference is that we use the ghost velocity ug, and not the
        particle velocity u.

        """

        def loop(self, d_idx, s_idx, d_rho, d_u, d_v, d_w, d_arho,
                 s_m, s_rho, s_ug, s_vg, s_wg, DWIJ):
            Vj = s_m[s_idx] / s_rho[s_idx]
            rhoi = d_rho[d_idx]
            uij = d_u[d_idx] - s_ug[s_idx]
            vij = d_v[d_idx] - s_vg[s_idx]
            wij = d_w[d_idx] - s_wg[s_idx]
            vij_dot_dwij = uij*DWIJ[0] + vij*DWIJ[1] + wij*DWIJ[2]

            d_arho[d_idx] += rhoi*Vj*vij_dot_dwij

    class StateEquation(Equation):
        r"""**Generalized Weakly Compressible Equation of State**

        .. math::

            p_a = p_0\left[ \left(\frac{\rho}{\rho_0}\right)^\gamma - b
            \right] + \mathcal{X}

        Notes
        -----
        This is the generalized Tait's equation of state and the suggested values
        in [Adami2013] are :math:`\mathcal{X} = 0`, :math:`\gamma=1` and
        :math:`b = 1`.

        The reference pressure :math:`p_0` is calculated from the artificial
        sound speed and reference density:

        .. math::

            p_0 = \frac{c^2\rho_0}{\gamma}
        """

        def __init__(self, dest, sources, p0, rho0, b=1.0):
            r"""
            Parameters
            ----------
            p0 : float
                reference pressure
            rho0 : float
                reference density
            b : float
                constant (default 1.0).
            """

            self.b = b
            self.p0 = p0
            self.rho0 = rho0
            super(StateEquation, self).__init__(dest, sources)

        def loop(self, d_idx, d_p, d_rho):
            d_p[d_idx] = self.p0 * (d_rho[d_idx]/self.rho0 - self.b)

    class MomentumEquationPressureGradient(Equation):
        r"""**Momentum equation for the Transport Velocity Formulation: Pressure**

        Eq. (8) in [Adami2013]:

        .. math::

            \frac{d \boldsymbol{v}_a}{dt} = \frac{1}{m_a}\sum_b (V_a^2 +
            V_b^2)\left[-\bar{p}_{ab}\nabla_a W_{ab} \right]

        where

        .. math::

            \bar{p}_{ab} = \frac{\rho_b p_a + \rho_a p_b}{\rho_a + \rho_b}
        """

        def __init__(self, dest, sources, pb, gx=0., gy=0., gz=0.,
                     tdamp=0.0):

            r"""
            Parameters
            ----------
            pb : float
                background pressure
            gx : float
                Body force per unit mass along the x-axis
            gy : float
                Body force per unit mass along the y-axis
            gz : float
                Body force per unit mass along the z-axis
            tdamp : float
                damping time

            Notes
            -----
            This equation should have the destination as fluid and sources as
            fluid and boundary particles.

            This function also computes the contribution to the background
            pressure and accelerations due to a body force or gravity.

            The body forces are damped according to Eq. (13) in [Adami2012] to
            avoid instantaneous accelerations. By default, damping is neglected.
            """

            self.pb = pb
            self.gx = gx
            self.gy = gy
            self.gz = gz
            self.tdamp = tdamp
            super(MomentumEquationPressureGradient, self).__init__(dest, sources)

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

            d_auhat[d_idx] = 0.0
            d_avhat[d_idx] = 0.0
            d_awhat[d_idx] = 0.0

        def loop(self, d_idx, s_idx, d_m, d_rho, s_rho,
                 d_au, d_av, d_aw, d_p, s_p,
                 d_auhat, d_avhat, d_awhat, d_V, s_V, DWIJ):

            # averaged pressure Eq. (7)
            rhoi = d_rho[d_idx]
            rhoj = s_rho[s_idx]
            p_i = d_p[d_idx]
            p_j = s_p[s_idx]

            pij = rhoj * p_i + rhoi * p_j
            pij /= (rhoj + rhoi)

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

            # inverse mass of destination particle
            mi1 = 1.0/d_m[d_idx]

            # accelerations 1st term in Eq. (8)
            tmp = -pij * mi1 * (Vi2 + Vj2)

            d_au[d_idx] += tmp * DWIJ[0]
            d_av[d_idx] += tmp * DWIJ[1]
            d_aw[d_idx] += tmp * DWIJ[2]

            # contribution due to the background pressure Eq. (13)
            tmp = -self.pb * mi1 * (Vi2 + Vj2)

            d_auhat[d_idx] += tmp * DWIJ[0]
            d_avhat[d_idx] += tmp * DWIJ[1]
            d_awhat[d_idx] += tmp * DWIJ[2]

        def post_loop(self, d_idx, d_au, d_av, d_aw, t):
            # damped accelerations due to body or external force
            damping_factor = 1.0
            if t < self.tdamp:
                damping_factor = 0.5 * (sin((-0.5 + t/self.tdamp)*M_PI) + 1.0)

            d_au[d_idx] += self.gx * damping_factor
            d_av[d_idx] += self.gy * damping_factor
            d_aw[d_idx] += self.gz * damping_factor

    class MomentumEquationViscosity(Equation):
        r"""**Momentum equation for the Transport Velocity Formulation: Viscosity**

        Eq. (8) in [Adami2013]:

        .. math::

               \frac{d \boldsymbol{v}_a}{dt} = \frac{1}{m_a}\sum_b (V_a^2 +
               V_b^2)\left[ \bar{\eta}_{ab}\hat{r}_{ab}\cdot \nabla_a W_{ab}
               \frac{\boldsymbol{v}_{ab}}{|\boldsymbol{r}_{ab}|}\right]

        where

        .. math::

            \bar{\eta}_{ab} = \frac{2\eta_a \eta_b}{\eta_a + \eta_b}
        """

        def __init__(self, dest, sources, nu):
            r"""
            Parameters
            ----------
            nu : float
                kinematic viscosity
            """

            self.nu = nu
            super(MomentumEquationViscosity, 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, s_idx, d_rho, s_rho, d_m, d_V, s_V,
                 d_au, d_av, d_aw,
                 R2IJ, EPS, DWIJ, VIJ, XIJ):

            # averaged shear viscosity Eq. (6)
            etai = self.nu * d_rho[d_idx]
            etaj = self.nu * s_rho[s_idx]

            etaij = 2 * (etai * etaj)/(etai + etaj)

            # scalar part of the kernel gradient
            Fij = DWIJ[0]*XIJ[0] + DWIJ[1]*XIJ[1] + DWIJ[2]*XIJ[2]

            # particle volumes, d_V is inverse volume.
            Vi = 1./d_V[d_idx]
            Vj = 1./s_V[s_idx]
            Vi2 = Vi * Vi
            Vj2 = Vj * Vj

            # accelerations 3rd term in Eq. (8)
            tmp = 1./d_m[d_idx] * (Vi2 + Vj2) * etaij * Fij/(R2IJ + EPS)

            d_au[d_idx] += tmp * VIJ[0]
            d_av[d_idx] += tmp * VIJ[1]
            d_aw[d_idx] += tmp * VIJ[2]

    class MomentumEquationArtificialViscosity(Equation):
        r"""**Artificial viscosity for the momentum equation**

        Eq. (11) in [Adami2012]:

        .. math::

            \frac{d \boldsymbol{v}_a}{dt} = -\sum_b m_b \alpha h_{ab}
            c_{ab} \frac{\boldsymbol{v}_{ab}\cdot
            \boldsymbol{r}_{ab}}{\rho_{ab}\left(|r_{ab}|^2 + \epsilon
            \right)}\nabla_a W_{ab}

        where

        .. math::

            \rho_{ab} = \frac{\rho_a + \rho_b}{2}\\

            c_{ab} = \frac{c_a + c_b}{2}\\

            h_{ab} = \frac{h_a + h_b}{2}
        """
        def __init__(self, dest, sources, c0, alpha=0.1):
            r"""
            Parameters
            ----------
            alpha : float
                constant
            c0 : float
                speed of sound
            """

            self.alpha = alpha
            self.c0 = c0
            super(MomentumEquationArtificialViscosity, 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, s_idx, s_m, d_au, d_av, d_aw,
                 RHOIJ1, R2IJ, EPS, DWIJ, VIJ, XIJ, HIJ):

            # v_{ab} \cdot r_{ab}
            vijdotrij = VIJ[0]*XIJ[0] + VIJ[1]*XIJ[1] + VIJ[2]*XIJ[2]

            # scalar part of the accelerations Eq. (11)
            piij = 0.0
            if vijdotrij < 0:
                muij = (HIJ * vijdotrij)/(R2IJ + EPS)

                piij = -self.alpha*self.c0*muij
                piij = s_m[s_idx] * piij*RHOIJ1

            d_au[d_idx] += -piij * DWIJ[0]
            d_av[d_idx] += -piij * DWIJ[1]
            d_aw[d_idx] += -piij * DWIJ[2]

    class MomentumEquationArtificialStress(Equation):
        r"""**Artificial stress contribution to the Momentum Equation**

        .. math::

              \frac{d\boldsymbol{v}_a}{dt} = \frac{1}{m_a}\sum_b (V_a^2 +
              V_b^2)\left[ \frac{1}{2}(\boldsymbol{A}_a +
              \boldsymbol{A}_b) : \nabla_a W_{ab}\right]

        where the artificial stress terms are given by:

        .. math::

               \boldsymbol{A} = \rho \boldsymbol{v} (\tilde{\boldsymbol{v}}
             - \boldsymbol{v})

        """

        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, s_idx, d_rho, d_u, d_v, d_w, d_V,
                 d_uhat, d_vhat, d_what, d_au, d_av, d_aw, d_m,
                 s_rho, s_u, s_v, s_w, s_V, s_uhat, s_vhat, s_what, DWIJ):
            rhoi = d_rho[d_idx]
            rhoj = s_rho[s_idx]

            # physical and advection velocities
            ui = d_u[d_idx]
            uhati = d_uhat[d_idx]
            vi = d_v[d_idx]
            vhati = d_vhat[d_idx]
            wi = d_w[d_idx]
            whati = d_what[d_idx]

            uj = s_u[s_idx]
            uhatj = s_uhat[s_idx]
            vj = s_v[s_idx]
            vhatj = s_vhat[s_idx]
            wj = s_w[s_idx]
            whatj = s_what[s_idx]

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

            # artificial stress tensor
            Axxi = rhoi*ui*(uhati - ui)
            Axyi = rhoi*ui*(vhati - vi)
            Axzi = rhoi*ui*(whati - wi)
            Ayxi = rhoi*vi*(uhati - ui)
            Ayyi = rhoi*vi*(vhati - vi)
            Ayzi = rhoi*vi*(whati - wi)
            Azxi = rhoi*wi*(uhati - ui)
            Azyi = rhoi*wi*(vhati - vi)
            Azzi = rhoi*wi*(whati - wi)

            Axxj = rhoj*uj*(uhatj - uj)
            Axyj = rhoj*uj*(vhatj - vj)
            Axzj = rhoj*uj*(whatj - wj)
            Ayxj = rhoj*vj*(uhatj - uj)
            Ayyj = rhoj*vj*(vhatj - vj)
            Ayzj = rhoj*vj*(whatj - wj)
            Azxj = rhoj*wj*(uhatj - uj)
            Azyj = rhoj*wj*(vhatj - vj)
            Azzj = rhoj*wj*(whatj - wj)

            # contraction of stress tensor with kernel gradient
            Ax = 0.5*(
                (Axxi + Axxj)*DWIJ[0] +
                (Axyi + Axyj)*DWIJ[1] +
                (Axzi + Axzj)*DWIJ[2]
            )

            Ay = 0.5*(
                (Ayxi + Ayxj)*DWIJ[0] +
                (Ayyi + Ayyj)*DWIJ[1] +
                (Ayzi + Ayzj)*DWIJ[2]
            )

            Az = 0.5*(
                (Azxi + Azxj)*DWIJ[0] +
                (Azyi + Azyj)*DWIJ[1] +
                (Azzi + Azzj)*DWIJ[2]
            )

            # accelerations 2nd part of Eq. (8)
            tmp = 1./d_m[d_idx] * (Vi2 + Vj2)

            d_au[d_idx] += tmp * Ax
            d_av[d_idx] += tmp * Ay
            d_aw[d_idx] += tmp * Az

    class SolidWallNoSlipBC(Equation):
        r"""**Solid wall boundary condition** [Adami2012]_

        This boundary condition is to be used with fixed ghost particles
        in SPH simulations and is formulated for the general case of
        moving boundaries.

        The velocity and pressure of the fluid particles is extrapolated
        to the ghost particles and these values are used in the equations
        of motion.

        No-penetration:

        Ghost particles participate in the continuity and state equations
        with fluid particles. This means as fluid particles approach the
        wall, the pressure of the ghost particles increases to generate a
        repulsion force that prevents particle penetration.

        No-slip:

        Extrapolation is used to set the `dummy` velocity of the ghost
        particles for viscous interaction. First, the smoothed velocity
        field of the fluid phase is extrapolated to the wall particles:

        .. math::

            \tilde{v}_a = \frac{\sum_b v_b W_{ab}}{\sum_b W_{ab}}

        In the second step, for the viscous interaction in Eqs. (10) in [Adami2012]
        and Eq. (8) in [Adami2013], the velocity of the ghost particles is
        assigned as:

        .. math::

           v_b = 2v_w -\tilde{v}_a,

        where :math:`v_w` is the prescribed wall velocity and :math:`v_b`
        is the ghost particle in the interaction.
        """

        def __init__(self, dest, sources, nu):
            r"""
            Parameters
            ----------
            nu : float
                kinematic viscosity

            Notes
            -----
            For this equation the destination particle array should be the
            fluid and the source should be ghost or boundary particles. The
            boundary particles must define a prescribed velocity :math:`u_0,
            v_0, w_0`
            """

            self.nu = nu
            super(SolidWallNoSlipBC, 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, s_idx, d_m, d_rho, s_rho, d_V, s_V,
                 d_u, d_v, d_w,
                 d_au, d_av, d_aw,
                 s_ug, s_vg, s_wg,
                 DWIJ, R2IJ, EPS, XIJ):

            # averaged shear viscosity Eq. (6).
            etai = self.nu * d_rho[d_idx]
            etaj = self.nu * s_rho[s_idx]

            etaij = 2 * (etai * etaj)/(etai + etaj)

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

            # scalar part of the kernel gradient
            Fij = XIJ[0]*DWIJ[0] + XIJ[1]*DWIJ[1] + XIJ[2]*DWIJ[2]

            # viscous contribution (third term) from Eq. (8), with VIJ
            # defined appropriately using the ghost values
            tmp = 1./d_m[d_idx] * (Vi2 + Vj2) * (etaij * Fij/(R2IJ + EPS))

            d_au[d_idx] += tmp * (d_u[d_idx] - s_ug[s_idx])
            d_av[d_idx] += tmp * (d_v[d_idx] - s_vg[s_idx])
            d_aw[d_idx] += tmp * (d_w[d_idx] - s_wg[s_idx])

    class SolidWallPressureBC(Equation):
        r"""**Solid wall pressure boundary condition** [Adami2012]_

        This boundary condition is to be used with fixed ghost particles
        in SPH simulations and is formulated for the general case of
        moving boundaries.

        The velocity and pressure of the fluid particles is extrapolated
        to the ghost particles and these values are used in the equations
        of motion.

        Pressure boundary condition:

        The pressure of the ghost particle is also calculated from the
        fluid particle by interpolation using:

        .. math::

            p_g = \frac{\sum_f p_f W_{gf} + \boldsymbol{g - a_g} \cdot
            \sum_f \rho_f \boldsymbol{r}_{gf}W_{gf}}{\sum_f W_{gf}},

        where the subscripts `g` and `f` relate to the ghost and fluid
        particles respectively.

        Density of the wall particle is then set using this pressure

        .. math::

            \rho_w=\rho_0\left(\frac{p_w - \mathcal{X}}{p_0} +
            1\right)^{\frac{1}{\gamma}}
        """

        def __init__(self, dest, sources, rho0, p0, b=1.0, gx=0.0, gy=0.0, gz=0.0):
            r"""
            Parameters
            ----------
            rho0 : float
                reference density
            p0 : float
                reference pressure
            b : float
                constant (default 1.0)
            gx : float
                Body force per unit mass along the x-axis
            gy : float
                Body force per unit mass along the y-axis
            gz : float
                Body force per unit mass along the z-axis

            Notes
            -----
            For a two fluid system (boundary, fluid), this equation must be
            instantiated with boundary as the destination and fluid as the
            source.

            The boundary particle array must additionally define a property
            :math:`wij` for the denominator in Eq. (27) from [Adami2012]. This
            array sums the kernel terms from the ghost particle to the fluid
            particle.
            """

            self.rho0 = rho0
            self.p0 = p0
            self.b = b
            self.gx = gx
            self.gy = gy
            self.gz = gz

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

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

        def loop(self, d_idx, s_idx, d_p, s_p, d_wij, s_rho,
                 d_au, d_av, d_aw, WIJ, XIJ):

            # numerator of Eq. (27) ax, ay and az are the prescribed wall
            # accelerations which must be defined for the wall boundary
            # particle
            gdotxij = (self.gx - d_au[d_idx])*XIJ[0] + \
                (self.gy - d_av[d_idx])*XIJ[1] + \
                (self.gz - d_aw[d_idx])*XIJ[2]

            d_p[d_idx] += s_p[s_idx]*WIJ + s_rho[s_idx]*gdotxij*WIJ

            # denominator of Eq. (27)
            d_wij[d_idx] += WIJ

        def post_loop(self, d_idx, d_wij, d_p, d_rho):
            # extrapolated pressure at the ghost particle
            if d_wij[d_idx] > 1e-14:
                d_p[d_idx] /= d_wij[d_idx]

            # update the density from the pressure Eq. (28)
            d_rho[d_idx] = self.rho0 * (d_p[d_idx]/self.p0 + self.b)