source: inundation/ga/storm_surge/pyvolution/shallow_water.py @ 232

"""Class Domain -
2D triangular domains for finite-volume computations of
the shallow water wave equation.

This module contains a specialisation of class Domain from module domain.py
consisting of methods specific to the Shallow Water Wave Equation

FIXME: Write equations here!


Conserved quantities are w (water level or stage), uh (x momentum)
and vh (y momentum).


Ole Nielsen, Stephen Roberts, Duncan Gray, Christopher Zoppou
Geoscience Australia, 2004   
"""

from domain import *
Generic_domain = Domain #Rename

class Domain(Generic_domain):

    def __init__(self, coordinates, vertices, boundary = None):

        conserved_quantities = ['level', 'xmomentum', 'ymomentum']
        other_quantities = ['elevation', 'friction'] 
        
        Generic_domain.__init__(self, coordinates, vertices, boundary,
                                conserved_quantities, other_quantities)

        from config import minimum_allowed_height
        self.minimum_allowed_height = minimum_allowed_height

        self.forcing_terms.append(gravity)
        self.forcing_terms.append(manning_friction)


    def check_integrity(self):
        Generic_domain.check_integrity(self)

        #Check that we are solving the shallow water wave equation

        msg = 'First conserved quantity must be "level"'
        assert self.conserved_quantities[0] == 'level', msg
        msg = 'Second conserved quantity must be "xmomentum"'
        assert self.conserved_quantities[1] == 'xmomentum', msg
        msg = 'Third conserved quantity must be "ymomentum"'
        assert self.conserved_quantities[2] == 'ymomentum', msg
        

        #Check that levels are >= bed elevation
        from Numeric import alltrue, greater_equal
        
        level = self.quantities['level']
        bed = self.quantities['elevation']        

        msg = 'All water levels must be greater than the bed elevation'
        assert alltrue( greater_equal(
            level.vertex_values, bed.vertex_values )), msg

        assert alltrue( greater_equal(
            level.edge_values, bed.edge_values )), msg

        assert alltrue( greater_equal(
            level.centroid_values, bed.centroid_values )), msg        


    def compute_fluxes(self):
        #Call correct module function
        #(either from this module or C-extension)
        compute_fluxes(self)

    def distribute_to_vertices_and_edges(self):
        #Call correct module function
        #(either from this module or C-extension)        
        distribute_to_vertices_and_edges(self)
        
####################################
# Flux computation        
def flux_function(normal, ql, qr, zl, zr): 
    """Compute fluxes between volumes for the shallow water wave equation
    cast in terms of w = h+z using the 'central scheme' as described in

    Kurganov, Noelle, Petrova. 'Semidiscrete Central-Upwind Schemes For
    Hyperbolic Conservation Laws and Hamilton-Jacobi Equations'.
    Siam J. Sci. Comput. Vol. 23, No. 3, pp. 707-740.

    The implemented formula is given in equation (3.15) on page 714

    Conserved quantities w, uh, vh are stored as elements 0, 1 and 2
    in the numerical vectors ql an qr.

    Bed elevations zl and zr.
    #FIXME: Remove those and pass in height directly
    #(unless we'll include a bed elevation discontinuity later)
    
    """

    from config import g, epsilon
    from math import sqrt
    from Numeric import array
    from util import rotate

    #Align momentums with x-axis
    q_left  = rotate(ql, normal, direction = 1)
    q_right = rotate(qr, normal, direction = 1)

    z = (zl+zr)/2 #Take average of field values
    #FIXME: Maybe allow discontinuity later

    w_left  = q_left[0]   #w=h+z
    h_left  = w_left-z
    uh_left = q_left[1]

    if h_left < epsilon:
        u_left = 0.0  #Could have been negative
        h_left = 0.0
    else:    
        u_left  = uh_left/h_left


    w_right  = q_right[0]  #w=h+z
    h_right  = w_right-z
    uh_right = q_right[1]


    if h_right < epsilon:
        u_right = 0.0  #Could have been negative
        h_right = 0.0
    else:    
        u_right  = uh_right/h_right

    vh_left  = q_left[2]
    vh_right = q_right[2]        

    soundspeed_left  = sqrt(g*h_left) 
    soundspeed_right = sqrt(g*h_right)

    #Maximal wave speed
    s_max = max(u_left+soundspeed_left, u_right+soundspeed_right, 0)
    
    #Minimal wave speed        
    s_min = min(u_left-soundspeed_left, u_right-soundspeed_right, 0)

    #Flux computation
    flux_left  = array([u_left*h_left,
                        u_left*uh_left + 0.5*g*h_left**2,
                        u_left*vh_left])
    flux_right = array([u_right*h_right,
                        u_right*uh_right + 0.5*g*h_right**2,
                        u_right*vh_right])    

    denom = s_max-s_min
    if denom == 0.0:
        edgeflux = array([0.0, 0.0, 0.0])
        max_speed = 0.0
    else:    
        edgeflux = (s_max*flux_left - s_min*flux_right)/denom
        edgeflux += s_max*s_min*(q_right-q_left)/denom

        edgeflux = rotate(edgeflux, normal, direction=-1)
        max_speed = max(abs(s_max), abs(s_min))

    return edgeflux, max_speed        


def compute_fluxes(domain):
    """Compute all fluxes and the timestep suitable for all volumes
    in domain.

    Compute total flux for each conserved quantity using "flux_function"

    Fluxes across each edge are scaled by edgelengths and summed up
    Resulting flux is then scaled by area and stored in
    domain.explicit_update

    The maximal allowable speed computed by the flux_function for each volume
    is converted to a timestep that must not be exceeded. The minimum of
    those is computed as the next overall timestep.

    Post conditions:
      domain.explicit_update is reset to computed flux values
      domain.timestep is set to the largest step satisfying all volumes. 
    """

    import sys
    from Numeric import zeros, Float
    from config import max_timestep    

    N = domain.number_of_elements
    
    neighbours = domain.neighbours
    neighbour_edges = domain.neighbour_edges
    normals = domain.normals

    areas = domain.areas
    radii = domain.radii
    edgelengths = domain.edgelengths
        
    timestep = max_timestep #FIXME: Get rid of this

    #Shortcuts
    Level = domain.quantities['level']
    Xmom = domain.quantities['xmomentum']
    Ymom = domain.quantities['ymomentum']
    Bed = domain.quantities['elevation']        

    #Arrays
    level = Level.edge_values
    xmom =  Xmom.edge_values
    ymom =  Ymom.edge_values
    bed =   Bed.edge_values    

    level_bdry = Level.boundary_values
    xmom_bdry =  Xmom.boundary_values
    ymom_bdry =  Ymom.boundary_values
    
    flux = zeros(3, Float) #Work array for summing up fluxes

    #Loop
    for k in range(N):
        optimal_timestep = float(sys.maxint)

        flux[:] = 0.  #Reset work array
        for i in range(3):
            #Quantities inside volume facing neighbour i
            ql = [level[k, i], xmom[k, i], ymom[k, i]]
            zl = bed[k, i]

            #Quantities at neighbour on nearest face
            n = neighbours[k,i] 
            if n < 0:
                m = -n-1 #Convert neg flag to index
                qr = [level_bdry[m], xmom_bdry[m], ymom_bdry[m]]
                zr = zl #Extend bed elevation to boundary
            else:    
                m = neighbour_edges[k,i]
                qr = [level[n, m], xmom[n, m], ymom[n, m]]
                zr = bed[n, m]                

                
            #Outward pointing normal vector   
            normal = normals[k, 2*i:2*i+2]

            #Flux computation using provided function
            edgeflux, max_speed = flux_function(normal, ql, qr, zl, zr)
            flux -= edgeflux * edgelengths[k,i]
            
            #Update optimal_timestep
            try:
                optimal_timestep = min(optimal_timestep, radii[k]/max_speed)
            except ZeroDivisionError:
                pass

        #Normalise by area and store for when all conserved
        #quantities get updated
        flux /= areas[k]
        Level.explicit_update[k] = flux[0]
        Xmom.explicit_update[k] = flux[1]
        Ymom.explicit_update[k] = flux[2]

        timestep = min(timestep, optimal_timestep)

    domain.timestep = timestep    

        

####################################
# Module functions for gradient limiting (distribute_to_vertices_and_edges)

def distribute_to_vertices_and_edges(domain):

    ##print 'Calling distrib within SW'
    if domain.order == 1:
        #FIXME: This should be cleaned up, but we try to follow 
        #pyvolution 2 strictly for now
        protect_against_negative_heights_centroid(domain)        
        protect_against_infinitesimal_heights_centroid(domain)        
        extrapolate_first_order(domain)
    elif domain.order == 2:
        #protect_against_negative_heights_centroid(domain)        
        extrapolate_second_order(domain)        
    else:
        raise 'Unknown order'

    #Compute edge values
    for name in domain.conserved_quantities:
        Q = domain.quantities[name]
        Q.interpolate_from_vertices_to_edges()            


def protect_against_infinitesimal_heights_centroid(domain):
    """Adjust height and momentum at centroid if height is less than
    minimum allowed height
    """
    #FIXME: Used only in first order
    
    #Water levels at centroids
    wc = domain.quantities['level'].centroid_values

    #Bed elevations at centroids
    zc = domain.quantities['elevation'].centroid_values

    #Water depths at centroids    
    hc = wc - zc

    #Momentums at centroids
    xmomc = domain.quantities['xmomentum'].centroid_values
    ymomc = domain.quantities['ymomentum'].centroid_values        

    for k in range(domain.number_of_elements):
        #Protect against infinitesimal heights and high velocities
        if hc[k] < domain.minimum_allowed_height:
            #Control level and height
            if hc[k] < 0.0:
                wc[k] = zc[k]; hc[k] = 0.0

            #Control momentum
            xmomc[k] = ymomc[k] = 0.0
                

def protect_against_negative_heights_centroid(domain):
    """Adjust height and momentum at centroid if height is less than zero
    """

    
    #Water levels at centroids
    wc = domain.quantities['level'].centroid_values

    #Bed elevations at centroids
    zc = domain.quantities['elevation'].centroid_values

    #Water depths at centroids    
    hc = wc - zc

    #Momentums at centroids
    xmomc = domain.quantities['xmomentum'].centroid_values
    ymomc = domain.quantities['ymomentum'].centroid_values        

    for k in range(domain.number_of_elements):
        #Protect against infinitesimal heights and high velocities
        if hc[k] < 0.0:
            #Control level and height
            wc[k] = zc[k]; hc[k] = 0.0

            #Control momentum
            xmomc[k] = ymomc[k] = 0.0
                



def extrapolate_first_order(domain):
    """First order extrapolator function, specific
    to the shallow water wave equation.
    
    It will ensure that h (w-z) is always non-negative even in the
    presence of steep bed-slopes (see comment in code)
    In addition, momemtums get distributed as constant values.

    Precondition:
      All quantities defined at centroids and bed elevation defined at
      vertices.

    Postcondition
      Conserved quantities defined at vertices
    """

    
    #Update conserved quantities using straight first order
    for name in domain.conserved_quantities:
        Q = domain.quantities[name]
        Q.extrapolate_first_order()

     
  
    #Water levels at centroids
    wc = domain.quantities['level'].centroid_values

    #Bed elevations at centroids
    zc = domain.quantities['elevation'].centroid_values

    #Water depths at centroids    
    hc = wc - zc

    #Water levels at vertices
    wv = domain.quantities['level'].vertex_values

    #Bed elevations at vertices
    zv = domain.quantities['elevation'].vertex_values
                
    #Water depths at vertices
    hv = wv-zv
    
    
    #Computed weighted balance between constant levels and and
    #levels parallel to the bed elevation.     
    for k in range(domain.number_of_elements):                
                
        #Compute maximal variation in bed elevation
        z_range = max(abs(zv[k,0]-zc[k]),
                      abs(zv[k,1]-zc[k]),
                      abs(zv[k,2]-zc[k]))


        #Weighted balance between stage parallel to bed elevation 
        #(wvi = zvi + hc) and constant stage (wvi = wc = zc+hc)
        #where i=0,1,2 denotes the vertex ids
        #
        #It follows that 
        #  wvi = (1-alpha)*(zvi+hc) + alpha*(zc+hc) = 
        #  (1-alpha)*zvi + alpha*zc + hc = 
        #  zvi + hc + alpha*(zc-zvi)
        #
        #where alpha in [0,1] and defined as the ratio between hc and 
        #the maximal difference from zc to zv0, zv1 and zv2
        #
        #Mathematically the following can be continued on using hc as
        #  wvi = 
        #  zvi + hc + alpha*(zc+hc-zvi-hc) =
        #  zvi + hc + alpha*(hvi-hc)
        #since hvi = zc+hc-zvi in the constant case

        
        if z_range > 0.0:
            alpha = min(hc[k]/z_range, 1.0)
        else: 
            alpha = 1.0
            
        #Update water levels
        for i in range(3):
            #FIXME: Use the original first-order one first, then switch
            wv[k,i] = zv[k,i] + hc[k] + alpha*(zc[k]-zv[k,i])
            #wv[k,i] = zv[k,i] + hc[k] + alpha*(hv[k,i]-hc[k])

        #FIXME: What about alpha weighting of momentum??
            
        
def extrapolate_second_order(domain):
    """Second order limiter function, specific to the shallow water wave
    equation.
    
    It will ensure that h (w-z) is always non-negative even in the
    presence of steep bed-slopes (see comment in code)

    A weighted average between shallow 
    and deep cases is as in the first order case.
    
    In addition, all conserved quantities get distributed as per a
    piecewise linear function.
    FIXME: more explanation about removal of artificial variability etc

    Precondition:
      All quantities defined at centroids and bed elevation defined at
      vertices.

    Postcondition
      Conserved quantities defined at vertices
    
    """
    
    #FIXME: first and second order might merge

    #Update conserved quantities using straight second order
    for name in domain.conserved_quantities:
        Q = domain.quantities[name]
        Q.extrapolate_second_order()
        
    #print 'y1', Q.vertex_values[1,:]   #OK
    
    #FIXME - like pyvolution 2 .......................
    protect_against_negative_heights_centroid(domain)   

    #print 'y1', Q.vertex_values[1,:]   #OK    
    
    for name in domain.conserved_quantities:
        Q = domain.quantities[name]     
        Q.limit()

  
    #print 'y1', Q.vertex_values[1,:]   #OK    
      
    #Water levels at centroids
    wc = domain.quantities['level'].centroid_values

    #Bed elevations at centroids
    zc = domain.quantities['elevation'].centroid_values

    #Water depths at centroids    
    hc = wc - zc

    #Water levels at vertices
    wv = domain.quantities['level'].vertex_values

    #Bed elevations at vertices
    zv = domain.quantities['elevation'].vertex_values
                
    #Water depths at vertices
    hv = wv-zv


    
    #Computed linear combination between constant levels and and
    #levels parallel to the bed elevation.     
    for k in range(domain.number_of_elements):                
                
        #Compute maximal variation in bed elevation
        #  This quantitiy is
        #    dz = max_i abs(z_i - z_c)
        #  and it is independent of dimension
        #  In the 1d case zc = (z0+z1)/2
        #  In the 2d case zc = (z0+z1+z2)/3
        #  
        
        z_range = max(abs(zv[k,0]-zc[k]),
                      abs(zv[k,1]-zc[k]),
                      abs(zv[k,2]-zc[k]))


        hmin = min( hv[k, :] )

        #Create alpha in [0,1], where alpha==0 means using shallow 
        #first order scheme and alpha==1 means using the limited
        #second order scheme for the stage w
        #If hmin < 0 then alpha=0 reverting to first order.
      
        if z_range > 0.0:
            alpha = max( min( 2*hmin/z_range, 1.0), 0.0)
        else: 
            alpha = 1.0
   
        ##if k==1: print 'alpha', alpha #OK
        
        #Update water levels
        #  (1-alpha)*(zvi+hc) + alpha*(zvi+hvi) = 
        #   zvi + hc + alpha*(hvi - hc)
        if alpha < 1:         
            for i in range(3):
                wv[k,i] = zv[k,i] + hc[k] + alpha*(hv[k,i]-hc[k])
            

            #Momentums at centroids
            xmomc = domain.quantities['xmomentum'].centroid_values
            ymomc = domain.quantities['ymomentum'].centroid_values        

            #Momentums at vertices
            xmomv = domain.quantities['xmomentum'].vertex_values
            ymomv = domain.quantities['ymomentum'].vertex_values         

            # Update momentum as a linear combination of 
            # xmomc and ymomc (shallow) and momentum
            # from extrapolator xmomv and ymomv (deep).
        
            xmomv[k,:] = (1-alpha)*xmomc[k] + alpha*xmomv[k,:];            
            ymomv[k,:] = (1-alpha)*ymomc[k] + alpha*ymomv[k,:];                     
            
    #print 'y1', Q.vertex_values[1,:]   #OK    
            
    #Finally, protect against infinitesimal heights and high speeds
    #Water depths at vertices
    hv = wv-zv
    hc = wc-zc    
    for k in range(domain.number_of_elements):
        hmax = max(hv[k,:])
        
        if hmax < domain.minimum_allowed_height:        
            #Reset negative heights to bed elevation        
            if hc[k] < 0.0:
                wc[k] = zc[k]
            for i in range(3):    
                if hv[k,i] < 0.0:
                    wv[k,i] = zv[k,i]

    


###############################################
#Boundary - specific to the shallow water wave equation
class Reflective_boundary(Boundary):
    """Reflective boundary returns same conserved quantities as
    those present in its neighbour volume but reflected.

    This class is specific to the shallow water equation as it
    works with the momentum quantities assumed to be the second
    and third conserved quantities.
    """
    
    def __init__(self, domain = None):        
        Boundary.__init__(self)

        if domain is None:
            msg = 'Domain must be specified for reflective boundary'
            raise msg
        
        self.domain = domain
        
        
    def __repr__(self):
        return 'Reflective_boundary'


    def evaluate(self, vol_id, edge_id):
        """Reflective boundaries reverses the outward momentum
        of the volume they serve.
        """
        
        q = self.domain.get_conserved_quantities(vol_id, edge = edge_id)
        normal = self.domain.get_normal(vol_id, edge_id)
        
        r = rotate(q, normal, direction = 1)
        r[1] = -r[1]
        q = rotate(r, normal, direction = -1)

        return q


#########################
#Standard forcing terms:
#
def gravity(domain):
    """Implement forcing function for bed slope working with
    consecutive data structures of class Volume
    """

    from config import g
    from util import gradient
    from Numeric import zeros, Float, array, sum

    Level = domain.quantities['level']
    Xmom = domain.quantities['xmomentum']
    Ymom = domain.quantities['ymomentum']

    Elevation = domain.quantities['elevation']    
    h = Level.edge_values - Elevation.edge_values
    V = domain.get_vertex_coordinates()
    
    for k in range(domain.number_of_elements):    
        avg_h = sum( h[k,:] )/3
        
        #Compute bed slope
        x0, y0, x1, y1, x2, y2 = V[k,:]    
        z0, z1, z2 = Elevation.vertex_values[k,:]

        zx, zy = gradient(x0, y0, x1, y1, x2, y2, z0, z1, z2)

        #Update momentum
        Xmom.explicit_update[k] += -g*zx*avg_h
        Ymom.explicit_update[k] += -g*zy*avg_h        


def manning_friction(domain):
    """Apply (Manning) friction to water momentum
    """

    import Numeric
    from math import sqrt
    from config import g, minimum_allowed_height

    Level = domain.quantities['level']
    Xmom = domain.quantities['xmomentum']
    Ymom = domain.quantities['ymomentum']

    Friction = domain.quantities['friction']    
    
    for k in range(domain.number_of_elements):    
        w = Level.centroid_values[k]
        uh = Xmom.centroid_values[k]
        vh = Ymom.centroid_values[k]
        manning = Friction.centroid_values[k]
        
        if w >= minimum_allowed_height:
            S = -g * manning**2 * sqrt((uh**2 + vh**2))
            S /= w**(7.0/3)

            #Update momentum
            Xmom.explicit_update[k] += S*uh
            Ymom.explicit_update[k] += S*vh
            

###########################
###########################
#Geometries


#FIXME: Rethink this way of creating values.


class Weir:
    """Set a bathymetry for weir with a hole and a downstream gutter
    x,y are assumed to be in the unit square
    """
    
    def __init__(self, stage):
        self.inflow_stage = stage

    def __call__(self, x, y):    
        from Numeric import zeros, Float
        from math import sqrt
    
        N = len(x)
        assert N == len(y)    

        z = zeros(N, Float)
        for i in range(N):
            z[i] = -x[i]/2  #General slope

            #Flattish bit to the left
            if x[i] < 0.3:
                z[i] = -x[i]/10 
            
            #Weir
            if x[i] >= 0.3 and x[i] < 0.4:
                z[i] = -x[i]+0.9

            #Dip
            x0 = 0.6
            #depth = -1.3
            depth = -1.0
            #plateaux = -0.9
            plateaux = -0.6            
            if y[i] < 0.7:
                if x[i] > x0 and x[i] < 0.9:
                    z[i] = depth

                #RHS plateaux
                if x[i] >= 0.9:
                    z[i] = plateaux
                    
                    
            elif y[i] >= 0.7 and y[i] < 1.5:
                #Restrict and deepen
                if x[i] >= x0 and x[i] < 0.8:
                    z[i] = depth-(y[i]/3-0.3)
                    #z[i] = depth-y[i]/5
                    #z[i] = depth
                elif x[i] >= 0.8:    
                    #RHS plateaux
                    z[i] = plateaux
                    
            elif y[i] >= 1.5:
                if x[i] >= x0 and x[i] < 0.8 + (y[i]-1.5)/1.2:
                    #Widen up and stay at constant depth
                    z[i] = depth-1.5/5
                elif x[i] >= 0.8 + (y[i]-1.5)/1.2:                        
                    #RHS plateaux
                    z[i] = plateaux                                    
                    

            #Hole in weir (slightly higher than inflow condition)
            if x[i] >= 0.3 and x[i] < 0.4 and y[i] > 0.2 and y[i] < 0.4:
                z[i] = -x[i]+self.inflow_stage + 0.02

            #Channel behind weir
            x0 = 0.5
            if x[i] >= 0.4 and x[i] < x0 and y[i] > 0.2 and y[i] < 0.4:
                z[i] = -x[i]+self.inflow_stage + 0.02
                
            if x[i] >= x0 and x[i] < 0.6 and y[i] > 0.2 and y[i] < 0.4:
                #Flatten it out towards the end
                z[i] = -x0+self.inflow_stage + 0.02 + (x0-x[i])/5

            #Hole to the east
            x0 = 1.1; y0 = 0.35
            #if x[i] < -0.2 and y < 0.5:
            if sqrt((2*(x[i]-x0))**2 + (2*(y[i]-y0))**2) < 0.2:
                z[i] = sqrt(((x[i]-x0))**2 + ((y[i]-y0))**2)-1.0

            #Tiny channel draining hole
            if x[i] >= 1.14 and x[i] < 1.2 and y[i] >= 0.4 and y[i] < 0.6:
                z[i] = -0.9 #North south
                
            if x[i] >= 0.9 and x[i] < 1.18 and y[i] >= 0.58 and y[i] < 0.65:
                z[i] = -1.0 + (x[i]-0.9)/3 #East west
            
            

            #Stuff not in use
            
            #Upward slope at inlet to the north west
            #if x[i] < 0.0: # and y[i] > 0.5:
            #    #z[i] = -y[i]+0.5  #-x[i]/2
            #    z[i] = x[i]/4 - y[i]**2 + 0.5

            #Hole to the west
            #x0 = -0.4; y0 = 0.35 # center
            #if sqrt((2*(x[i]-x0))**2 + (2*(y[i]-y0))**2) < 0.2:
            #    z[i] = sqrt(((x[i]-x0))**2 + ((y[i]-y0))**2)-0.2





        return z/2

class Weir_simple:
    """Set a bathymetry for weir with a hole and a downstream gutter
    x,y are assumed to be in the unit square
    """
    
    def __init__(self, stage):
        self.inflow_stage = stage

    def __call__(self, x, y):    
        from Numeric import zeros, Float 
    
        N = len(x)
        assert N == len(y)    

        z = zeros(N, Float)
        for i in range(N):
            z[i] = -x[i]  #General slope

            #Flat bit to the left
            if x[i] < 0.3:
                z[i] = -x[i]/10  #General slope
            
            #Weir
            if x[i] > 0.3 and x[i] < 0.4:
                z[i] = -x[i]+0.9

            #Dip
            if x[i] > 0.6 and x[i] < 0.9:
                z[i] = -x[i]-0.5  #-y[i]/5

            #Hole in weir (slightly higher than inflow condition)
            if x[i] > 0.3 and x[i] < 0.4 and y[i] > 0.2 and y[i] < 0.4:
                z[i] = -x[i]+self.inflow_stage + 0.05       
            

        return z/2
        

            
class Constant_height:
    """Set an initial condition with constant water height, e.g
    stage s = z+h
    """
    def __init__(self, W, h):
        self.W = W
        self.h = h

    def __call__(self, x, y):
        if self.W is None:
            from Numeric import ones, Float
            return self.h*ones(len(x), Float)
        else:
            return self.W(x,y) + self.h



##############################################
#Initialise module