def erro_maximo_uds(numberOfNodes,angularVelocity):
##############################
# Setting input paramethers
##############################
yNumberOfNodes = numberOfNodes
xNumberOfNodes = numberOfNodes
re = 0.1 #[m]
ri = 0.04 #[m]
L = mt.sin(0.25*mt.pi)*(re-ri)#[m]
k = 15.0#[W/mºC]
Ti = 250.0 #[ºC]
Te = 30.0 #[ºC]
omega = angularVelocity #[rad/s]
rho = 800.0 #[kg/m^3]
cp = 2000.0 #[J/kgK]
gamma = k/cp
#############################
# Mesh generation in y
#############################
yNodesPositions = []
deltaY = L/(yNumberOfNodes - 1);
ySum = mt.sin(0.25*mt.pi)*ri
for i in range(yNumberOfNodes):
yNodesPositions.append(ySum)
ySum += deltaY
ySurfacePositions = []
ySum = mt.cos(0.25*mt.pi)*ri + 0.5*deltaY
for i in range(yNumberOfNodes - 1):
ySurfacePositions.append(ySum)
ySum += deltaY
#############################
# Mesh generation in x
#############################
xNodesPositions = []
deltaX = L/(float(xNumberOfNodes) - 1);
xSum = mt.cos(0.25*mt.pi)*ri;
for i in range(xNumberOfNodes):
xNodesPositions.append(xSum)
xSum += deltaX
xSurfacePositions = []
xSum = mt.cos(0.25*mt.pi)*ri + 0.5*deltaX
for i in range(xNumberOfNodes - 1):
xSurfacePositions.append(xSum)
xSum += deltaX
##############################################################
# Generating the matrix of coefficients and independent vector
##############################################################
A = []
b = []
numberOfNodes = xNumberOfNodes * yNumberOfNodes
for i in range(numberOfNodes):
A.append([])
for j in range(numberOfNodes):
A[i].append(0.0)
b.append(0.0)
#bottom
for j in range(xNumberOfNodes):
ap = 1.0
A[j][j] = ap
b[j] = aux.analyticSolution(xNodesPositions[j],yNodesPositions[0])
#center
for i in range(1,yNumberOfNodes-1):
for j in range(xNumberOfNodes):
if j == 0 or j == (xNumberOfNodes - 1):
ap = 1.0
A[i*xNumberOfNodes+j][i*xNumberOfNodes+j] = ap
b[i*xNumberOfNodes+j] = aux.analyticSolution(xNodesPositions[j],yNodesPositions[i])
else:
uw = aux.uVelocity(omega,xSurfacePositions[j-1],yNodesPositions[i])
ue = aux.uVelocity(omega,xSurfacePositions[j],yNodesPositions[i])
vs = aux.vVelocity(omega,xNodesPositions[j],ySurfacePositions[i-1])
vn = aux.vVelocity(omega,xNodesPositions[j],ySurfacePositions[i])
Mw = rho*uw*deltaY
Me = rho*ue*deltaY
Ms = rho*vs*deltaX
Mn = rho*vn*deltaX
De = gamma*deltaY/deltaX
Dw = De
Dn = gamma*deltaY/deltaX
Ds = Dn
aEast = De - Me
aWest = Dw
aSouth = Ds + Ms
aNorth = Dn
ap = aEast + aWest + aSouth + aNorth
A[i*xNumberOfNodes+j][j+xNumberOfNodes*(i-1)] = -aSouth
A[i*xNumberOfNodes+j][i*xNumberOfNodes+j-1] = -aWest
A[i*xNumberOfNodes+j][i*xNumberOfNodes+j] = ap
A[i*xNumberOfNodes+j][i*xNumberOfNodes+j+1] = -aEast
A[i*xNumberOfNodes+j][j+xNumberOfNodes*(i+1)] = -aNorth
#top
for j in range(xNumberOfNodes):
ap = 1.0
A[(yNumberOfNodes-1)*xNumberOfNodes+j][(yNumberOfNodes-1)*xNumberOfNodes+j] = ap
b[(yNumberOfNodes-1)*xNumberOfNodes+j] = aux.analyticSolution(xNodesPositions[j],yNodesPositions[yNumberOfNodes - 1])
#############################
# Solving the linear system
############################
solution = np.linalg.solve(np.array(A),np.array(b))
temperatureField = []
for i in range(yNumberOfNodes):
temperatureField.append([])
for j in range(xNumberOfNodes):
temperatureField[i].append(solution[i*xNumberOfNodes+j])
#########################################
# Error with respect to the 1D solution
#########################################
errors = []
for i in range(yNumberOfNodes):
errors.append([])
for j in range(xNumberOfNodes):
exactTemperature = aux.analyticSolution(xNodesPositions[i],yNodesPositions[j])
aproxTemperature = temperatureField[i][j]
diff = (exactTemperature - aproxTemperature)/(Ti - Te)
errors[i].append(abs(diff))
errors = np.array(errors)
maximumError = errors.max()
return maximumError
#bottom
for j in range(xNumberOfNodes):
ap = 1.0
A[j][j] = ap
b[j] = aux.analyticSolution(xNodesPositions[j], yNodesPositions[0])
#center
for i in range(1, yNumberOfNodes - 1):
for j in range(xNumberOfNodes):
if j == 0 or j == (xNumberOfNodes - 1):
ap = 1.0
A[i * xNumberOfNodes + j][i * xNumberOfNodes + j] = ap
b[i * xNumberOfNodes + j] = aux.analyticSolution(
xNodesPositions[j], yNodesPositions[i])
else:
uw = aux.uVelocity(omega, xSurfacePositions[j - 1],
yNodesPositions[i])
ue = aux.uVelocity(omega, xSurfacePositions[j], yNodesPositions[i])
vs = aux.vVelocity(omega, xNodesPositions[j],
ySurfacePositions[i - 1])
vn = aux.vVelocity(omega, xNodesPositions[j], ySurfacePositions[i])
Mw = rho * uw * deltaY
Me = rho * ue * deltaY
Ms = rho * vs * deltaX
Mn = rho * vn * deltaX
De = gamma * deltaY / deltaX
Dw = De
Dn = gamma * deltaY / deltaX
Ds = Dn
aEast = De - 0.5 * Me
aWest = Dw + 0.5 * Mw
aSouth = Ds + 0.5 * Ms