godunov.jl

#=
Copyright 2017 INSIGNEO Institute for in silico Medicine

Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at

   http://www.apache.org/licenses/LICENSE-2.0

Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
=#

# 1D blood flow within elastic vessels is ruled by a system of hyperbolic
# partial derivative equations
# $$
#   \begin{cases}
#     \dfrac{\partial A}{\partial t} + \dfrac{\partial Au}{\partial x} = 0, \\
#     \dfrac{\partial Au}{\partial t} + \dfrac{\partial Au^2}{\partial x}
#       \dfrac{A}{\rho}\dfrac{\partial P}{\partial x} = - Ru, \\
#     P(x, t) = P_e(x, t) + K_a \left[\left(\dfrac{A}{A_0} \right)^{1/2}
#     -1 \right],
#   \end{cases}
# $$
# By defining $\mathbf{U} = [A, u]^T$, the conservative form of the system
# can be written along with initial and boundary conditions as
# $$
#   \begin{cases}
#     \mathbf{U}_t + \mathbf{F}(\mathbf{U})_x = \mathbf{S}(\mathbf{U}),
#       \quad x \in \big[0, L \big], \\
#     \mathbf{U}(x, 0) = \mathbf{U}^{(0)}(x), \quad t>0, \\
#     \mathbf{U}(0, t) = \mathbf{U}_l(t), \mathbf{U}(L, t) = \mathbf{U}_r(t),
#   \end{cases}
# $$
# where $\mathbf{F}$ is the *flux* term, and $\mathbf{S}$ is the *source*
# term. This set of equations and initial and
# boundary conditions is also called Initial Boundary Value Problem (IBVP).
# The IBVP is solved by means of the Godunov's method [^1], a
# conservative first order finite volume method. The method is applied on
# a computational domain (red) subdivided in $M$ *cell*s $I_i$ (gray).

# <div style="text-align:center">
# <img src="images/comp_domain.pdf.png" width="400"></div>

# Domain length is
# equal to vessel length $L$, and cell length is given by $\Delta x$ size.
# The total simulation time $T$ defines domain height, hence cell height is
# given by $\Delta t$. In each cell $I_i$ three points are defined: center point $x_i$, left
# extreme $x_{i-1/2}$, and right extreme $x_{i+1/2}$.

# <div style="text-align:center">
# <img src="images/cell_i.pdf.png" width="350"></div>

# Godunov's method assumes a piece-wise constant distribution of $\mathbf{U}$
# at each $t = t^n$.

# <div style="text-align:center">
# <img src="images/piecewise.pdf.png" width="350"></div>

# For a *small enough* $\Delta t$, the method updates the solution as
# $$
#   \mathbf{U}_i^{n+1} = \mathbf{U}_i^{n} + \frac{\Delta t}{\Delta x}
#     \left(\mathbf{F}_{i-1/2} - \mathbf{F}_{i+1/2} \right),
# $$
# where $\mathbf{F}_{i\pm 1/2} = \mathbf{F}\left(\mathbf{U}_{i\pm 1/2}(0)
# \right)$ is the *numerical* flux. The argument of the numerical flux,
# $\mathbf{U}_{i\pm 1/2}(0)$, is computed as the solution of a homogeneous (
# the source term will be added later on) quasi-linear local Riemann
# problem (RP) at the interface of two cells. For $\mathbf{U}_{i + 1/2}(0)$
# we have
# $$
#   \begin{cases}
#     \mathbf{U}_t + \mathbf{F}_\mathbf{U} \mathbf{U}_x = 0, \\
#     \mathbf{U}(x_i, 0) = \mathbf{U}^{(0)}(x_i) = \begin{cases}
#       \mathbf{U}_L = \mathbf{U}(x_{i-1/2}), \quad x<0, \\
#       \mathbf{U}_R = \mathbf{U}(x_{i+1/2}), \quad x>0. \\
#     \end{cases}
#   \end{cases}
# $$
# The solution of the RP is depicted below. <a name="RP_solution"></a>

# <div style="text-align:center">
# <img src="images/rp_solution.pdf.png" width="400"></div>

# The solution structure includes two families of waves associated to the
# two system Riemann invariants $\lambda_1$ and $\lambda_2$, respectively.
# There are three constant states $\mathbf{U}_L$, $\mathbf{U}^*$, and
# $\mathbf{U}_R$, of which $\mathbf{U}^*$ is unknown and refers
# to the solution in the star region (yellow). We need *jump*
# conditions to cross the waves and connect these states. Depending on the
# nature of each wave we have *shock* or *rarefaction* conditions.

# [`Godunov`](godunov.html#Godunov) function glues together all the functions
# needed to [solve the RP](godunov.html#riemannSolver)
# at each interface and to compute the [numerical flux](godunov.html#flux).

# *function* __`Godunov`__
#
# ----------------------------------------------------------------------------
# Parameters:
# ---------------- -----------------------------------------------------------
# `i`              `::Int` cell index within the vessel numerical domain.
#
# `v`              `::Vessel` current vessel data structure.
#
# `dt`             `::Float` $\Delta t$, current time step.
#
# `b`              `::Blood` blood data structure.
# ----------------------------------------------------------------------------
#
# --------------------------------------------------------------------------
# Functioning:
# --------------------------------------------------------------------------
# In order to solve the local RP, local boundary conditions should be
# retrieved at both left ($i-1/2$) and right ($i+1/2$) interfaces.
# --------------------------------------------------------------------------
# <a name="Godunov"></a>
function Godunov(i :: Int64, v :: Vessel, dt :: Float64, b :: Blood)

  # <div style="text-align:center">
  # <img src="images/ghost_cells.pdf.png" width="400"></div>

  # First we compute the states $\mathbf{U}_L$ and $\mathbf{U}_R$ at the
  # left hand side interface ($i-1/2$). When we deal with the inlet node
  # ($i=1$), blue ghost cells are used to find left states. Right states are
  # always inside the numerical domain and they do not need ghost cells.
  #i-1/2 local left
  if i-1 == 0      #1st ghost cell
    UlA = v.U00A
    UlQ = v.U00Q

  elseif i-1 == -1 #2nd ghost cell
    UlA = v.U01A
    UlQ = v.U01Q

  else
    UlA = v.A[i-1]
    UlQ = v.Q[i-1]
  end

  #local right
  UrA = v.A[i]
  UrQ = v.Q[i]
  # Once the local states have been computed, the solution of the RP at the
  # left
  # interface is found by [`riemannSolver`](godunov.html#riemannSolver)
  # function.
  Al, Ql = riemannSolver(UlA, UlQ, UrA, UrQ, v)

  # The same process is applied to the local right interface.
  #i+1/2 local right
  if i+1 == v.M+1     #1st ghost cell
    UrA = v.UM1A
    UrQ = v.UM1Q

  elseif i+1 == v.M+2 #2nd ghost cell
    UrA = v.UM2A
    UrQ = v.UM2Q

  else
    UrA = v.A[i+1]
    UrQ = v.Q[i+1]
  end

  #local left
  UlA = v.A[i]
  UlQ = v.Q[i]

  Ar, Qr = riemannSolver(UlA, UlQ, UrA, UrQ, v)

  # When both interfaces are solved, left and right ($\mathbf{F}_{i\mp 1/2}$)
  # numerical fluxes are computed by [`flux`](godunov.html#flux) fucntion.
  Fl1, Fl2 = flux(Al, Ql, v.gamma)
  Fr1, Fr2 = flux(Ar, Qr, v.gamma)

  # Conservative variables `A` and `Q` are updated by Godunov's method
  # formula.
  v.A[i] += dt/v.dx * (Fl1-Fr1)
  v.Q[i] += dt/v.dx * (Fl2-Fr2)

  # The source term is solved by [`source`](godunov.html#source) function
  # and `Q` is directly updated. The source term is always zero for the
  # continuity equation, hence `A` is not updated.
  v.Q[i] += source(dt, v.A[i], v.u[i], b)

  # Eventually, primitive variables `P`, `u`, and `c` are updated accordingly
  # to their definitions by [`pressure`](converter.html#pressure) and
  # [`waveSpeed`](converter.html#waveSpeed) functions.
  v.P[i] = pressure(v.A[i], v.A0, v.beta, v.Pext)
  v.u[i] = v.Q[i]/v.A[i]
  v.c[i] = waveSpeed(v.A[i], v.gamma)

end

# The [RP solution](godunov.html#RP_solution) is herein coded. The Riemann
# solver computes the cross-sectional area in the star region
# $A^*$ starting from a guess solution (two rarefactions solution). The
# value of $A^*$ comes from the solution of the nonlinear equation
# $$
#   f(A^*) = f_L(A^*) + f_R(A^*) + \Delta u = 0,
# $$
# where $\Delta u = u_R - u_L$, and $f_L$ and $f_R$ are the local fluxes
# computed by [`fl`](godunov.html#fl) and [`fr`](godunov.html#fr),
# respectively.

# *function* __`riemannSolver`__ $\rightarrow$ `As::Float`, `Qs::Float`
#
# ----------------------------------------------------------------------------
# Parameters:
# ---------------- -----------------------------------------------------------
# `Al`             `::Float` `A` value at the left side of the local
#                  cell interface.
#
# `Ql`             `::Float` `Q` value at the left side of the local
#                  cell interface.
#
# `Ar`             `::Float` `A` value at the right side of the local
#                  cell interface.
#
# `Qr`             `::Float` `Q` value at the right side of the local
#                  cell interface.
#
# `v`              `::Vessel` current vessel data structure.
# ----------------------------------------------------------------------------
# <a name="riemannSolver"></a>
function riemannSolver(Al :: Float64, Ql :: Float64,
                       Ar :: Float64, Qr :: Float64,
                        v :: Vessel)
  # --------------------------------------------------------------------------
  # Functioning:
  # --------------------------------------------------------------------------
  # Primitives are initialised by their definitions, and the two rarefaction
  # solution is applied to compute initial values for $c^*$ and $A^*$.
  # --------------------------------------------------------------------------
  #initialise solution
  ur = Qr/Ar
  ul = Ql/Al
  cl = waveSpeed(Al, v.gamma)
  cr = waveSpeed(Ar, v.gamma)
  du = ur - ul

#     #vacuum
#   if ul + 4*cl <= ur - 4*cr
# #     println("v")
# #     return (Al+Ar)*0.5, (Ql+Qr)*0.5
#     return v.A0, (Ql+Qr)*0.5
#   end

  #two rarefaction solution
  cs = 0.5 * (cl + cr) - (ur - ul)*0.125
  As = 4/9 * (cs^4)/(v.gamma*v.gamma)
  # The nonlinear equation for the RP is defined and solved by Newton's
  # method.
  fun(As) = fL(As, Al, cs, cl, v) + fR(As, Ar, cs, cr, v) + du

  As = newton(fun, As)

  # Eventually, $u^*$ is computed as
  # $$
  #   \frac{1}{2}\left(u_L + u_R\right) + \frac{1}{2}\left[f_R\left(A^*\right)
  #     -f_L\left(A^* \right) \right],
  # $$
  us = 0.5*((ur+ul) + (fR(As, Ar, cs, cr, v) - fL(As, Al, cs, cl, v)))
  # $Q^*$ is calculated from its definition and returned.
  Qs = As * us
  # --------------------------------------------------------------------------
  # Returns:
  # ---------- ---------------------------------------------------------------
  # `As`       `::Float` cross-sectional area in the star region, solution
  #            of the RP at the interface.
  #
  # `Qs`       `::Float` flow rate in the star region, solution
  #            of the RP at the interface.
  # --------------------------------------------------------------------------
  return As, Qs
end

# Local fluxes (left and right) are computed depending on the nature of the
# wave system located at the interface. In particular
# $$
#   f_L = \begin{cases}
#       4\left(c^* - c_L\right), & A^*\leq A_L \quad \text{(rarefaction)}, \\
#       \sqrt{\dfrac{\gamma\left(A^*-A_L\right)\left(A^{* 3/2}-
#         A_L^{3/2}\right)}{A_L A^*}}, & A^*>A_L \quad \text{(shock)},
#     \end{cases}
# $$
# $$
#   f_R = \begin{cases}
#       4\left(c^* - c_R\right), & A^*\leq A_R \quad \text{(rarefaction)}, \\
#       \sqrt{\dfrac{\gamma\left(A^*-A_R\right)\left(A^{* 3/2}-
#         A_R^{3/2}\right)}{A_R A^*}}, & A^*>A_R \quad \text{(shock)},
#     \end{cases}
# $$

# *function* __`fL`__ $\rightarrow$ `left_flux::Float`
#
# ----------------------------------------------------------------------------
# Parameters:
# ------------- --------------------------------------------------------------
# `A`           `::Float` current value of $A^*$ at the left interface.
#
# `Al`          `::Float` cross sectional area of the left cell.
#
# `c`           `::Float` current value of $c^*$ at the left interface.
#
# `cl`          `::Float` wave speed in the left cell.
#
# `v`           `::Vessel` current vessel data structure.
# ----------------------------------------------------------------------------
#
# ----------------------------------------------------------------------------
# Returns:
# ------------- --------------------------------------------------------------
# `left_flux`   `::Float` local flux at the left interface.
# ----------------------------------------------------------------------------
# <a name="fL"></a>
function fL(A, Al :: Float64, c :: Float64, cl :: Float64, v :: Vessel)

  if A <= Al
    return 4*(c - cl)

  else
    return sqrt(
                (v.gamma*(A - Al)*(A^1.5 - Al^1.5)) /
                              (Al*A)
               )
  end

end

# *function* __`fR`__ $\rightarrow$ `right_flux::Float`
#
# ----------------------------------------------------------------------------
# Parameters:
# ------------- --------------------------------------------------------------
# `A`           `::Float` current value of $A^*$ at the right interface.
#
# `Ar`          `::Float` cross sectional area of the right cell.
#
# `c`           `::Float` current value of $c^*$ at the right interface.
#
# `cr`          `::Float` wave speed in the right cell.
#
# `v`           `::Vessel` current vessel data structure.
# ----------------------------------------------------------------------------
#
# ----------------------------------------------------------------------------
# Returns:
# ------------- --------------------------------------------------------------
# `right_flux`   `::Float` local flux at the right interface.
# ----------------------------------------------------------------------------
# <a name="fR"></a>
function fR(A, Ar :: Float64, c :: Float64, cr :: Float64, v :: Vessel)

  if A <= Ar
    return 4*(c - cr)

  else
    return sqrt(
                (v.gamma*(A - Ar)*(A^1.5 - Ar^1.5)) /
                              (Ar*A)
               )
  end

end

# The flux in conservative form reads
# $$
#   \mathbf{F}(\mathbf{U}) = \left\{ \begin{array}{c}
#       Au \\
#       Au^2 + \gamma A^{3/2}
#     \end{array} \right\},
# $$
# and its numerical form is computed by [`flux`](godunov.html#flux) function.

# *function* __`flux`__ $\rightarrow$ `f1::Float`, `f2::Float`
#
# ----------------------------------------------------------------------------
# Parameters:
# ------------- --------------------------------------------------------------
# `A`           `::Float` cross sectional area along the cell.
#
# `Q`           `::Float` flow rate within the cell.
#
# `gamma`       `::Vessel` elastic constant for `c` calculation.
# ----------------------------------------------------------------------------
#
# ----------------------------------------------------------------------------
# Returns:
# ---------- -----------------------------------------------------------------
# `f1`       `::Float` first component of the numerical flux.
#
# `f2`       `::Float` second component of the numerical flux.
# ----------------------------------------------------------------------------
# <a name="flux"></a>
function flux(A :: Float64, Q :: Float64, gamma :: Float64)

  f1 = Q
  f2 = Q*Q/A + gamma*A^1.5

  return f1, f2
end

# *function* __`source`__ $\rightarrow$ `S::Float`
#
# ----------------------------------------------------------------------------
# Parameters:
# ------------ ----------------------------------------------------------------
# `dt`         `::Float` $\Delta t$, current time step.
#
# `A`          `::Float` cross sectional area along the cell.
#
# `u`          `::Vessel` longitudinal velocity along the cell.
#
# `b`          `::Blood` blood data structure.
# ----------------------------------------------------------------------------
#
# ----------------------------------------------------------------------------
# Functioning
# ----------------------------------------------------------------------------
# The numerical source is updated by means of a first order implicit backward
# Euler method.
# ----------------------------------------------------------------------------
#
# ----------------------------------------------------------------------------
# Returns:
# ---------- -----------------------------------------------------------------
# `S`        `::Float` source term.
# ----------------------------------------------------------------------------
# <a name="source"></a>
function source(dt :: Float64, A :: Float64, u :: Float64, b :: Blood)

  return -b.Cf*u*dt

end

# *function* __`calculateDeltaT`__ $\rightarrow$ `DT::Float`
#
# ----------------------------------------------------------------------------
# Parameters:
# ---------------- -----------------------------------------------------------
# `vessels`        `::Array{Vessel, 1}`
# ----------------------------------------------------------------------------
#
# ----------------------------------------------------------------------------
# Functioning
# ----------------------------------------------------------------------------
# This function calculates the smallest $\Delta t$ allowed by all vessels
# geometries and properties.
# ----------------------------------------------------------------------------
# <a name="calculateDeltaT"></a>
function calculateDeltaT(vessels, dt :: Array{Float64, 1})

  # For each vessel the $\Delta t$ is computed as
  # $$
  #   \Delta t = \frac{\Delta x}{S_{max}} C_{cfl}
  # $$
  # where $S_{max}$ is the maximum between the forward and the backward
  # characteristics, and $C_{cfl}$ is the Courant-Friedrichs-Lewy condition
  # defined by the user.
  i = 1
  for v in vessels

    lambdap = zeros(Float64, v.M)
    for j in 1:v.M
      lambdap[j] = v.u[j] + v.c[j]
    end

    Smax = maximum(abs.(lambdap))

    dt[i] = v.dx*v.Ccfl/Smax

    i+=1
  end
  #
  # --------------------------------------------------------------------------
  # Returns:
  # ----------- --------------------------------------------------------------
  # `DT`        `::Float` $\Delta t$ computed as the smallest between
  #             the smallest $\Delta t$ associated with each vessel and
  #             the $\Delta t$ limiter $0.001$.
  # --------------------------------------------------------------------------
  return minimum(dt)

end

# *function* __`solveModel`__
#
# ----------------------------------------------------------------------------
# Parameters:
# --------------- ------------------------------------------------------------
# `grafo`         `::GenericGraph` graph data structure containing all the
#                 vessels.
#
# `edgess`        `::Array` collection of `edge` objects.
#
# `vessels`       `::Array{Vessel, 1}` collection of vessels in the system.
#
# `heart`         `::Heart` heart data structure.
#
# `blood`         `::Blood` blood data structure.
#
# `dt`            `::Float` $\Delta t$, current time step.
#
# `current_time`  `::Float` current simulation time.
# ----------------------------------------------------------------------------
#
# ----------------------------------------------------------------------------
# Functioning
# ----------------------------------------------------------------------------
# See the [detailed example](grafo.html).
#
# ----------------------------------------------------------------------------
# <a name="solveModel"></a>
function solveModel(vessels, heart :: Heart, edge_list,
                    blood :: Blood, dt :: Float64, current_time :: Float64)

  for j in 1:size(edge_list)[1]
    i = edge_list[j,1]
    s = edge_list[j,2]
    t = edge_list[j,3]
    v = vessels[i]
    lbl = v.label

    if size(find(edge_list[:,3] .== s))[1] == 0
      openBF.setInletBC(current_time, dt, v, heart)
    end

    openBF.MUSCL(v, dt, blood)

    if size(find(edge_list[:,2] .== t))[1] == 0
      openBF.setOutletBC(dt, v)
      # println("\t Outlet vessel - Compute outlet BC")

    elseif size(find(edge_list[:,2] .== t))[1] == 2
      d1_i = find(edge_list[:,2] .== t)[1]
      d2_i = find(edge_list[:,2] .== t)[2]

      openBF.joinVessels(blood, v, vessels[d1_i], vessels[d2_i])
      # println("\t\t Bifurcation at node $t between vessels $i, $d1_i, and $d2_i")

    elseif size(find(edge_list[:,3] .== t))[1] == 1
      d_i = find(edge_list[:,2] .== t)[1]

      openBF.joinVessels(blood, v, vessels[d_i])
      # println("\t\t Junction at node $t between vessels $i and $d_i")

    elseif size(find(edge_list[:,3] .== t))[1] == 2
      p1_i = find(edge_list[:,3] .== t)[1]
      p2_i = find(edge_list[:,3] .== t)[2]
      if maximum([p1_i, p2_i]) == i
        p2_i = minimum([p1_i, p2_i])
        d = find(edge_list[:,2] .== t)[1]
        openBF.solveAnastomosis(v, vessels[p2_i], vessels[d])
        # println("\t\t Anastomosis at node $s between vessels $i, $p2_i, and $d")
      end
    end
  end
end

# ### References
#
# [^1]: Toro, Eleuterio F. Riemann solvers and numerical methods for fluid
# dynamics: a practical introduction. Springer Science & Business Media, 2009.