"""Class Quantity - Implements values at each 1d element To create: Quantity(domain, vertex_values) domain: Associated domain structure. Required. vertex_values: N x 2 array of values at each vertex for each element. Default None If vertex_values are None Create array of zeros compatible with domain. Otherwise check that it is compatible with dimenions of domain. Otherwise raise an exception """ class Quantity: def __init__(self, domain, vertex_values=None): #Initialise Quantity using optional vertex values. from domain import Domain from Numeric import array, zeros, Float msg = 'First argument in Quantity.__init__ ' msg += 'must be of class Domain (or a subclass thereof)' assert isinstance(domain, Domain), msg if vertex_values is None: N = domain.number_of_elements self.vertex_values = zeros((N, 2), Float) else: self.vertex_values = array(vertex_values, Float) N, V = self.vertex_values.shape assert V == 2,\ 'Two vertex values per element must be specified' msg = 'Number of vertex values (%d) must be consistent with'\ %N msg += 'number of elements in specified domain (%d).'\ %domain.number_of_elements assert N == domain.number_of_elements, msg self.domain = domain #Allocate space for other quantities self.centroid_values = zeros(N, Float) self.centroid_backup_values = zeros(N, Float) #self.edge_values = zeros((N, 2), Float) #edge values are values of the ends of each interval #Intialise centroid values self.interpolate() from Numeric import zeros, Float #Allocate space for boundary values #L = len(domain.boundary) self.boundary_values = zeros(2, Float) #assumes no parrellism #Allocate space for updates of conserved quantities by #flux calculations and forcing functions N = domain.number_of_elements self.explicit_update = zeros(N, Float ) self.semi_implicit_update = zeros(N, Float ) self.gradients = zeros(N, Float) self.qmax = zeros(self.centroid_values.shape, Float) self.qmin = zeros(self.centroid_values.shape, Float) self.beta = domain.beta def __len__(self): """ Returns number of intervals. """ return self.centroid_values.shape[0] def interpolate(self): """ Compute interpolated values at centroid Pre-condition: vertex_values have been set """ N = self.vertex_values.shape[0] for i in range(N): v0 = self.vertex_values[i, 0] v1 = self.vertex_values[i, 1] self.centroid_values[i] = (v0 + v1)/2.0 def set_values(self, X, location='vertices'): """Set values for quantity X: Compatible list, Numeric array (see below), constant or function location: Where values are to be stored. Permissible options are: vertices, centroid Default is "vertices" In case of location == 'centroid' the dimension values must be a list of a Numerical array of length N, N being the number of elements in the mesh. Otherwise it must be of dimension Nx2 The values will be stored in elements following their internal ordering. If values are described a function, it will be evaluated at specified points If selected location is vertices, values for centroid and edges will be assigned interpolated values. In any other case, only values for the specified locations will be assigned and the others will be left undefined. """ if location not in ['vertices', 'centroids']: msg = 'Invalid location: %s, (possible choices vertices, centroids)' %location raise msg if X is None: msg = 'Given values are None' raise msg import types if callable(X): #Use function specific method self.set_function_values(X, location) elif type(X) in [types.FloatType, types.IntType, types.LongType]: if location == 'centroids': self.centroid_values[:] = X else: self.vertex_values[:] = X else: #Use array specific method self.set_array_values(X, location) if location == 'vertices': #Intialise centroid and edge values self.interpolate() def set_function_values(self, f, location='vertices'): """Set values for quantity using specified function f: x -> z Function where x and z are arrays location: Where values are to be stored. Permissible options are: vertices, centroid Default is "vertices" """ if location == 'centroids': P = self.domain.centroids self.set_values(f(P), location) else: #Vertices P = self.domain.get_vertices() for i in range(2): self.vertex_values[:,i] = f(P[:,i]) def set_array_values(self, values, location='vertices'): """Set values for quantity values: Numeric array location: Where values are to be stored. Permissible options are: vertices, centroid, edges Default is "vertices" In case of location == 'centroid' the dimension values must be a list of a Numerical array of length N, N being the number of elements in the mesh. Otherwise it must be of dimension Nx2 The values will be stored in elements following their internal ordering. If selected location is vertices, values for centroid will be assigned interpolated values. In any other case, only values for the specified locations will be assigned and the others will be left undefined. """ from Numeric import array, Float values = array(values).astype(Float) N = self.centroid_values.shape[0] msg = 'Number of values must match number of elements' assert values.shape[0] == N, msg if location == 'centroids': assert len(values.shape) == 1, 'Values array must be 1d' self.centroid_values = values #elif location == 'edges': # assert len(values.shape) == 2, 'Values array must be 2d' # msg = 'Array must be N x 2' # self.edge_values = values else: assert len(values.shape) == 2, 'Values array must be 2d' msg = 'Array must be N x 2' assert values.shape[1] == 2, msg self.vertex_values = values def get_values(self, location='vertices', indices = None): """get values for quantity return X, Compatible list, Numeric array (see below) location: Where values are to be stored. Permissible options are: vertices, centroid and unique vertices. Default is 'vertices' In case of location == 'centroids' the dimension values must be a list of a Numerical array of length N, N being the number of elements. Otherwise it must be of dimension Nx3 The returned values with be a list the length of indices (N if indices = None). Each value will be a list of the three vertex values for this quantity. Indices is the set of element ids that the operation applies to. """ from Numeric import take if location not in ['vertices', 'centroids', 'unique vertices']: msg = 'Invalid location: %s' %location raise msg import types, Numeric assert type(indices) in [types.ListType, types.NoneType, Numeric.ArrayType],\ 'Indices must be a list or None' if location == 'centroids': if (indices == None): indices = range(len(self)) return take(self.centroid_values,indices) elif location == 'unique vertices': if (indices == None): indices=range(self.domain.coordinates.shape[0]) vert_values = [] #Go through list of unique vertices for unique_vert_id in indices: cells = self.domain.vertexlist[unique_vert_id] #In case there are unused points if cells is None: msg = 'Unique vertex not associated with cells' raise msg # Go through all cells, vertex pairs # Average the values sum = 0 for cell_id, vertex_id in cells: sum += self.vertex_values[cell_id, vertex_id] vert_values.append(sum/len(cells)) return Numeric.array(vert_values) else: if (indices == None): indices = range(len(self)) return take(self.vertex_values,indices) def get_vertex_values(self, x=True, smooth = None, precision = None, reduction = None): """Return vertex values like an OBJ format The vertex values are returned as one sequence in the 1D float array A. If requested the coordinates will be returned in 1D arrays X. The connectivity is represented as an integer array, V, of dimension M x 2, where M is the number of volumes. Each row has two indices into the X, A arrays defining the element. if smooth is True, vertex values corresponding to one common coordinate set will be smoothed according to the given reduction operator. In this case vertex coordinates will be de-duplicated. If no smoothings is required, vertex coordinates and values will be aggregated as a concatenation of values at vertices 0, vertices 1 Calling convention if x is True: X,A,V = get_vertex_values else: A,V = get_vertex_values """ from Numeric import concatenate, zeros, Float, Int, array, reshape if smooth is None: smooth = self.domain.smooth if precision is None: precision = Float if reduction is None: reduction = self.domain.reduction #Create connectivity if smooth == True: V = self.domain.get_vertices() N = len(self.domain.vertexlist) #N = len(self.domain.vertices) A = zeros(N, precision) #Smoothing loop for k in range(N): L = self.domain.vertexlist[k] #L = self.domain.vertices[k] #Go through all triangle, vertex pairs #contributing to vertex k and register vertex value if L is None: continue #In case there are unused points contributions = [] for volume_id, vertex_id in L: v = self.vertex_values[volume_id, vertex_id] contributions.append(v) A[k] = reduction(contributions) if x is True: #X = self.domain.coordinates[:,0].astype(precision) X = self.domain.coordinates[:].astype(precision) #Y = self.domain.coordinates[:,1].astype(precision) #return X, Y, A, V return X, A, V #else: return A, V else: #Don't smooth #obj machinery moved to general_mesh # Create a V like [[0 1 2], [3 4 5]....[3*m-2 3*m-1 3*m]] # These vert_id's will relate to the verts created below #m = len(self.domain) #Number of volumes #M = 3*m #Total number of unique vertices #V = reshape(array(range(M)).astype(Int), (m,3)) #V = self.domain.get_triangles(obj=True) V = self.domain.get_vertices #FIXME use get_vertices, when ready A = self.vertex_values.flat #Do vertex coordinates if x is True: X = self.domain.get_vertex_coordinates() #X = C[:,0:6:2].copy() #Y = C[:,1:6:2].copy() return X.flat, A, V else: return A, V def get_integral(self): """Compute the integral of quantity across entire domain """ integral = 0 for k in range(self.domain.number_of_elements): area = self.domain.areas[k] qc = self.centroid_values[k] integral += qc*area return integral def update(self, timestep): """Update centroid values based on values stored in explicit_update and semi_implicit_update as well as given timestep """ from Numeric import sum, equal, ones, Float N = self.centroid_values.shape[0] #Explicit updates self.centroid_values += timestep*self.explicit_update #Semi implicit updates denominator = ones(N, Float)-timestep*self.semi_implicit_update if sum(equal(denominator, 0.0)) > 0.0: msg = 'Zero division in semi implicit update. Call Stephen :-)' raise msg else: #Update conserved_quantities from semi implicit updates self.centroid_values /= denominator def compute_gradients(self): """Compute gradients of piecewise linear function defined by centroids of neighbouring volumes. """ #print 'compute_gradient' from Numeric import array, zeros, Float N = self.centroid_values.shape[0] G = self.gradients Q = self.centroid_values X = self.domain.centroids for k in range(N): # first and last elements have boundaries if k == 0: #Get data k0 = k k1 = k+1 k2 = k+2 q0 = Q[k0] q1 = Q[k1] q2 = Q[k2] x0 = X[k0] #V0 centroid x1 = X[k1] #V1 centroid x2 = X[k2] #Gradient #G[k] = (q1 - q0)/(x1 - x0) G[k] = (q1 - q0)*(x2 - x0)*(x2 - x0) - (q2 - q0)*(x1 - x0)*(x1 - x0) G[k] /= (x1 - x0)*(x2 - x0)*(x2 - x1) elif k == N-1: #Get data k0 = k k1 = k-1 k2 = k-2 q0 = Q[k0] q1 = Q[k1] q2 = Q[k2] x0 = X[k0] #V0 centroid x1 = X[k1] #V1 centroid x2 = X[k2] #Gradient #G[k] = (q1 - q0)/(x1 - x0) G[k] = (q1 - q0)*(x2 - x0)*(x2 - x0) - (q2 - q0)*(x1 - x0)*(x1 - x0) G[k] /= (x1 - x0)*(x2 - x0)*(x2 - x1) ## q0 = Q[k0] ## q1 = Q[k1] ## ## x0 = X[k0] #V0 centroid ## x1 = X[k1] #V1 centroid ## ## #Gradient ## G[k] = (q1 - q0)/(x1 - x0) else: #Interior Volume (2 neighbours) #Get data k0 = k-1 k2 = k+1 q0 = Q[k0] q1 = Q[k] q2 = Q[k2] x0 = X[k0] #V0 centroid x1 = X[k] #V1 centroid (Self) x2 = X[k2] #V2 centroid #Gradient #G[k] = (q2-q0)/(x2-x0) G[k] = ((q0-q1)/(x0-x1)*(x2-x1) - (q2-q1)/(x2-x1)*(x0-x1))/(x2-x0) def compute_minmod_gradients(self): """Compute gradients of piecewise linear function defined by centroids of neighbouring volumes. """ #print 'compute_minmod_gradients' from Numeric import array, zeros, Float,sign def xmin(a,b): return 0.5*(sign(a)+sign(b))*min(abs(a),abs(b)) def xmic(t,a,b): return xmin(t*xmin(a,b), 0.50*(a+b) ) N = self.centroid_values.shape[0] G = self.gradients Q = self.centroid_values X = self.domain.centroids for k in range(N): # first and last elements have boundaries if k == 0: #Get data k0 = k k1 = k+1 k2 = k+2 q0 = Q[k0] q1 = Q[k1] q2 = Q[k2] x0 = X[k0] #V0 centroid x1 = X[k1] #V1 centroid x2 = X[k2] #Gradient #G[k] = (q1 - q0)/(x1 - x0) G[k] = (q1 - q0)*(x2 - x0)*(x2 - x0) - (q2 - q0)*(x1 - x0)*(x1 - x0) G[k] /= (x1 - x0)*(x2 - x0)*(x2 - x1) elif k == N-1: #Get data k0 = k k1 = k-1 k2 = k-2 q0 = Q[k0] q1 = Q[k1] q2 = Q[k2] x0 = X[k0] #V0 centroid x1 = X[k1] #V1 centroid x2 = X[k2] #Gradient #G[k] = (q1 - q0)/(x1 - x0) G[k] = (q1 - q0)*(x2 - x0)*(x2 - x0) - (q2 - q0)*(x1 - x0)*(x1 - x0) G[k] /= (x1 - x0)*(x2 - x0)*(x2 - x1) ## #Get data ## k0 = k ## k1 = k-1 ## ## q0 = Q[k0] ## q1 = Q[k1] ## ## x0 = X[k0] #V0 centroid ## x1 = X[k1] #V1 centroid ## ## #Gradient ## G[k] = (q1 - q0)/(x1 - x0) elif (self.domain.wet_nodes[k,0] == 2) & (self.domain.wet_nodes[k,1] == 2): G[k] = 0.0 else: #Interior Volume (2 neighbours) #Get data k0 = k-1 k2 = k+1 q0 = Q[k0] q1 = Q[k] q2 = Q[k2] x0 = X[k0] #V0 centroid x1 = X[k] #V1 centroid (Self) x2 = X[k2] #V2 centroid # assuming uniform grid d1 = (q1 - q0)/(x1-x0) d2 = (q2 - q1)/(x2-x1) #Gradient #G[k] = (d1+d2)*0.5 #G[k] = (d1*(x2-x1) - d2*(x0-x1))/(x2-x0) G[k] = xmic( self.domain.beta, d1, d2 ) def extrapolate_first_order(self): """Extrapolate conserved quantities from centroid to vertices for each volume using first order scheme. """ qc = self.centroid_values qv = self.vertex_values for i in range(2): qv[:,i] = qc def extrapolate_second_order(self): """Extrapolate conserved quantities from centroid to vertices for each volume using second order scheme. """ if self.domain.limiter == "pyvolution": #Z = self.gradients #print "gradients 1",Z self.compute_gradients() #print "gradients 2",Z #Z = self.gradients #print "gradients 1",Z #self.compute_minmod_gradients() #print "gradients 2", Z G = self.gradients V = self.domain.vertices qc = self.centroid_values qv = self.vertex_values #Check each triangle for k in range(self.domain.number_of_elements): #Centroid coordinates x = self.domain.centroids[k] #vertex coordinates x0, x1 = V[k,:] #Extrapolate qv[k,0] = qc[k] + G[k]*(x0-x) qv[k,1] = qc[k] + G[k]*(x1-x) self.limit_pyvolution() elif self.domain.limiter == "minmod_steve": self.limit_minmod() else: self.limit_range() def limit_minmod(self): #Z = self.gradients #print "gradients 1",Z self.compute_minmod_gradients() #print "gradients 2", Z G = self.gradients V = self.domain.vertices qc = self.centroid_values qv = self.vertex_values #Check each triangle for k in range(self.domain.number_of_elements): #Centroid coordinates x = self.domain.centroids[k] #vertex coordinates x0, x1 = V[k,:] #Extrapolate qv[k,0] = qc[k] + G[k]*(x0-x) qv[k,1] = qc[k] + G[k]*(x1-x) def limit_pyvolution(self): """ Limit slopes for each volume to eliminate artificial variance introduced by e.g. second order extrapolator This is an unsophisticated limiter as it does not take into account dependencies among quantities. precondition: vertex values are estimated from gradient postcondition: vertex values are updated """ from Numeric import zeros, Float N = self.domain.number_of_elements beta = self.domain.beta #beta = 0.8 qc = self.centroid_values qv = self.vertex_values #Find min and max of this and neighbour's centroid values qmax = self.qmax qmin = self.qmin for k in range(N): qmax[k] = qmin[k] = qc[k] for i in range(2): n = self.domain.neighbours[k,i] if n >= 0: qn = qc[n] #Neighbour's centroid value qmin[k] = min(qmin[k], qn) qmax[k] = max(qmax[k], qn) #Diffences between centroids and maxima/minima dqmax = qmax - qc dqmin = qmin - qc #Deltas between vertex and centroid values dq = zeros(qv.shape, Float) for i in range(2): dq[:,i] = qv[:,i] - qc #Phi limiter for k in range(N): #Find the gradient limiter (phi) across vertices phi = 1.0 for i in range(2): r = 1.0 if (dq[k,i] > 0): r = dqmax[k]/dq[k,i] if (dq[k,i] < 0): r = dqmin[k]/dq[k,i] phi = min( min(r*beta, 1), phi ) #Then update using phi limiter for i in range(2): qv[k,i] = qc[k] + phi*dq[k,i] def limit_range(self): import sys from Numeric import zeros, Float from util import minmod, minmod_kurganov, maxmod, vanleer, vanalbada limiter = self.domain.limiter #print limiter #print 'limit_range' N = self.domain.number_of_elements qc = self.centroid_values qv = self.vertex_values C = self.domain.centroids X = self.domain.vertices beta_p = zeros(N,Float) beta_m = zeros(N,Float) beta_x = zeros(N,Float) for k in range(N): n0 = self.domain.neighbours[k,0] n1 = self.domain.neighbours[k,1] if ( n0 >= 0) & (n1 >= 0): #SLOPE DERIVATIVE LIMIT beta_p[k] = (qc[k]-qc[k-1])/(C[k]-C[k-1]) beta_m[k] = (qc[k+1]-qc[k])/(C[k+1]-C[k]) beta_x[k] = (qc[k+1]-qc[k-1])/(C[k+1]-C[k-1]) dq = zeros(qv.shape, Float) for i in range(2): dq[:,i] =self.domain.vertices[:,i]-self.domain.centroids #Phi limiter for k in range(N): n0 = self.domain.neighbours[k,0] n1 = self.domain.neighbours[k,1] if n0 < 0: phi = (qc[k+1] - qc[k])/(C[k+1] - C[k]) elif n1 < 0: phi = (qc[k] - qc[k-1])/(C[k] - C[k-1]) #elif (self.domain.wet_nodes[k,0] == 2) & (self.domain.wet_nodes[k,1] == 2): # phi = 0.0 else: if limiter == "minmod": phi = minmod(beta_p[k],beta_m[k]) elif limiter == "minmod_kurganov":#Change this # Also known as monotonized central difference limiter # if theta = 2.0 theta = 2.0 phi = minmod_kurganov(theta*beta_p[k],theta*beta_m[k],beta_x[k]) elif limiter == "superbee": slope1 = minmod(beta_m[k],2.0*beta_p[k]) slope2 = minmod(2.0*beta_m[k],beta_p[k]) phi = maxmod(slope1,slope2) elif limiter == "vanleer": phi = vanleer(beta_p[k],beta_m[k]) elif limiter == "vanalbada": phi = vanalbada(beta_m[k],beta_p[k]) for i in range(2): qv[k,i] = qc[k] + phi*dq[k,i] def limit_steve_slope(self): import sys from Numeric import zeros, Float from util import minmod, minmod_kurganov, maxmod, vanleer N = self.domain.number_of_elements limiter = self.domain.limiter limiter_type = self.domain.limiter_type qc = self.centroid_values qv = self.vertex_values #Find min and max of this and neighbour's centroid values beta_p = zeros(N,Float) beta_m = zeros(N,Float) beta_x = zeros(N,Float) C = self.domain.centroids X = self.domain.vertices for k in range(N): n0 = self.domain.neighbours[k,0] n1 = self.domain.neighbours[k,1] if (n0 >= 0) & (n1 >= 0): # Check denominator not zero if (qc[k+1]-qc[k]) == 0.0: beta_p[k] = float(sys.maxint) beta_m[k] = float(sys.maxint) else: #STEVE LIMIT beta_p[k] = (qc[k]-qc[k-1])/(qc[k+1]-qc[k]) beta_m[k] = (qc[k+2]-qc[k+1])/(qc[k+1]-qc[k]) #Deltas between vertex and centroid values dq = zeros(qv.shape, Float) for i in range(2): dq[:,i] =self.domain.vertices[:,i]-self.domain.centroids #Phi limiter for k in range(N): phi = 0.0 if limiter == "flux_minmod": #FLUX MINMOD phi = minmod_kurganov(1.0,beta_m[k],beta_p[k]) elif limiter == "flux_superbee": #FLUX SUPERBEE phi = max(0.0,min(1.0,2.0*beta_m[k]),min(2.0,beta_m[k]))+max(0.0,min(1.0,2.0*beta_p[k]),min(2.0,beta_p[k]))-1.0 elif limiter == "flux_muscl": #FLUX MUSCL phi = max(0.0,min(2.0,2.0*beta_m[k],2.0*beta_p[k],0.5*(beta_m[k]+beta_p[k]))) elif limiter == "flux_vanleer": #FLUX VAN LEER phi = (beta_m[k]+abs(beta_m[k]))/(1.0+abs(beta_m[k]))+(beta_p[k]+abs(beta_p[k]))/(1.0+abs(beta_p[k]))-1.0 #Then update using phi limiter n = self.domain.neighbours[k,1] if n>=0: #qv[k,0] = qc[k] - 0.5*phi*(qc[k+1]-qc[k]) #qv[k,1] = qc[k] + 0.5*phi*(qc[k+1]-qc[k]) qv[k,0] = qc[k] + 0.5*phi*(qv[k,0]-qc[k]) qv[k,1] = qc[k] + 0.5*phi*(qv[k,1]-qc[k]) else: qv[k,i] = qc[k] def backup_centroid_values(self): # Call correct module function # (either from this module or C-extension) #backup_centroid_values(self) self.centroid_backup_values[:] = (self.centroid_values).astype('f') def saxpy_centroid_values(self,a,b): # Call correct module function # (either from this module or C-extension) self.centroid_values[:] = (a*self.centroid_values + b*self.centroid_backup_values).astype('f') class Conserved_quantity(Quantity): """Class conserved quantity adds to Quantity: storage and method for updating, and methods for extrapolation from centropid to vertices inluding gradients and limiters """ def __init__(self, domain, vertex_values=None): Quantity.__init__(self, domain, vertex_values) print "Use Quantity instead of Conserved_quantity" ## ##def newLinePlot(title='Simple Plot'): ## import Gnuplot ## g = Gnuplot.Gnuplot() ## g.title(title) ## g('set data style linespoints') ## g.xlabel('x') ## g.ylabel('y') ## return g ## ##def linePlot(g,x,y): ## import Gnuplot ## g.plot(Gnuplot.PlotItems.Data(x.flat,y.flat)) def newLinePlot(title='Simple Plot'): import pylab as g g.ion() g.hold(False) g.title(title) g.xlabel('x') g.ylabel('y') def linePlot(x,y): import pylab as g g.plot(x.flat,y.flat) def closePlots(): import pylab as g g.close('all') if __name__ == "__main__": #from domain import Domain from shallow_water_domain import Domain from Numeric import arange points1 = [0.0, 1.0, 2.0, 3.0] vertex_values = [[1.0,2.0],[4.0,5.0],[-1.0,2.0]] D1 = Domain(points1) Q1 = Quantity(D1, vertex_values) print Q1.vertex_values print Q1.centroid_values new_vertex_values = [[2.0,1.0],[3.0,4.0],[-2.0,4.0]] Q1.set_values(new_vertex_values) print Q1.vertex_values print Q1.centroid_values new_centroid_values = [20,30,40] Q1.set_values(new_centroid_values,'centroids') print Q1.vertex_values print Q1.centroid_values class FunClass: def __init__(self,value): self.value = value def __call__(self,x): return self.value*(x**2) fun = FunClass(1.0) Q1.set_values(fun,'vertices') print Q1.vertex_values print Q1.centroid_values Xc = Q1.domain.vertices Qc = Q1.vertex_values print Xc print Qc Qc[1,0] = 3 Q1.extrapolate_second_order() #Q1.limit_minmod() newLinePlot('plots') linePlot(Xc,Qc) raw_input('press return') points2 = arange(10) D2 = Domain(points2) Q2 = Quantity(D2) Q2.set_values(fun,'vertices') Xc = Q2.domain.vertices Qc = Q2.vertex_values linePlot(Xc,Qc) raw_input('press return') Q2.extrapolate_second_order() #Q2.limit_minmod() Xc = Q2.domain.vertices Qc = Q2.vertex_values print Q2.centroid_values print Qc linePlot(Xc,Qc) raw_input('press return') for i in range(10): import pylab as g g.hold(True) fun = FunClass(i/10.0) Q2.set_values(fun,'centroids') Q2.extrapolate_second_order() #Q2.limit_minmod() Qc = Q2.vertex_values linePlot(Xc,Qc) raw_input('press return') raw_input('press return to quit') closePlots()