Dateien nach „Project NGSolve“ hochladen

Project NGSolve/q2_time_temp.txt

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+Q2 with tau=2.5; theta = 0.5
+average temperature of the liquid (=coffee) for specific times
+
+time: 100.0; temperature: 81.388
+time: 200.0; temperature: 79.947
+time: 300.0; temperature: 78.664
+time: 400.0; temperature: 77.451
+time: 500.0; temperature: 76.282
+time: 600.0; temperature: 75.148
+time: 700.0; temperature: 74.042
+time: 800.0; temperature: 72.962
+time: 900.0; temperature: 71.905
+time: 1000.0; temperature: 70.870
+time: 1100.0; temperature: 69.855
+time: 1200.0; temperature: 68.861
+time: 1300.0; temperature: 67.886
+time: 1400.0; temperature: 66.930
+time: 1500.0; temperature: 65.993
+time: 1600.0; temperature: 65.074
+time: 1700.0; temperature: 64.172
+time: 1800.0; temperature: 63.288
+time: 1900.0; temperature: 62.421
+time: 2000.0; temperature: 61.570
+time: 2100.0; temperature: 60.736
+time: 2200.0; temperature: 59.918
+time: 2300.0; temperature: 59.115
+time: 2400.0; temperature: 58.328
+time: 2500.0; temperature: 57.555
+time: 2600.0; temperature: 56.798
+time: 2700.0; temperature: 56.055
+time: 2800.0; temperature: 55.326
+time: 2900.0; temperature: 54.611
+time: 3000.0; temperature: 53.910
+time: 3100.0; temperature: 53.223
+time: 3200.0; temperature: 52.548
+time: 3300.0; temperature: 51.887
+time: 3400.0; temperature: 51.238
+time: 3500.0; temperature: 50.601
+end-time: 3597.5; temperature: 49.993

Project NGSolve/scicomp_project.py

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+# Scientific Computing and FEM
+# Project C: mug simulation with ngsolve
+
+
+from ngsolve import *
+# from netgen.csg import *
+from netgen.geom2d import SplineGeometry
+import matplotlib.pyplot as plt
+import time
+
+""" It's not possible to generate a 3D model of the mug. I tried several ways.
+One can build the geometry and have a look on it. However, the computer is not able to generate a
+mesh from it. It got stuck. Using the jupyterhub from university does not change the situation.
+So we will continue solving this example in 2D instead of 3D. """
+
+# # generating the mesh 3D
+# geo = CSGeometry()
+
+# # parameter
+# h = 105     # height
+# r_bottom = 55   # radius bottom
+# r_top = 83  # radius top
+# thickness = 5   # thickness of cup
+# fill_height = 66
+
+# # outer cone (mug outside)
+# outer = Cone(Pnt(0,0,0), Pnt(0,0,h), r_bottom, r_top)
+
+# # inner cone
+# inner = Cone(Pnt(0,0,thickness), Pnt(0,0,h), r_bottom - thickness, r_top - thickness)
+
+# # ceramic (=difference)
+# #cup = outer - inner
+
+# # cutting liquid / air
+# cyl_liquid = Cylinder(Pnt(0,0,0), Pnt(0,0,fill_height), r_top+10)
+# cyl_air = Cylinder(Pnt(0,0,fill_height), Pnt(0,0,h), r_top+10)
+
+# inside = inner
+# # liquid (lower part)
+# liquid = inside * cyl_liquid
+
+# # air (upper part)
+# air = inside * cyl_air
+
+# cup = outer - (liquid + air)
+
+# # whole geometry + material
+# geo.Add(cup.mat("ceramic"))
+# geo.Add(liquid.mat("liquid"))
+# geo.Add(air.mat("air"))
+
+# mesh = Mesh(geo.GenerateMesh(maxh=40.5))
+# print("Elements:", mesh.ne)
+
+# Draw(mesh, mesh.Materials())
+# geo.Draw()
+
+# -----------------------------------
+
+# generating the 2D model and the corresponding mesh
+geo = SplineGeometry()
+
+midpoint = 22+66/99*(83/2-25)
+points_mm = [(25,0), (83/2,105), (-83/2,105), (-25,0),     # outer points of ceramic
+            (22,6), (midpoint,72), (-midpoint,72), (-22,6), # points of liquid
+            (83/2-3,105), (-83/2+3,105)] # upper points of air
+factor = 10**-3  # transformation from mm to m
+points = [[x * factor, y * factor] for x, y in points_mm]
+
+p0,p1,p2,p3,p4,p5,p6,p7,p8,p9 = [geo.AppendPoint(*pnt) for pnt in points]
+
+curves = [[["line",p0,p1],"right",1,0],
+            [["line",p1,p8],"top_mug",1,0],
+            [["line",p8,p5],"rest",1,3],
+            [["line",p5,p4],"rest",1,2],
+            [["line",p4,p7],"rest",1,2],
+            [["line",p7,p6],"rest",1,2],
+            [["line",p6,p9],"rest",1,3],
+            [["line",p9,p2],"top_mug",1,0],
+            [["line",p2,p3],"left",1,0],
+            [["line",p3,p0],"bottom",1,0],
+            [["line",p5,p6],"rest",2,3],
+            [["line",p8,p9],"top_air",3,0]]
+
+for geom_part, name, left, right in curves:
+    if name != "rest":  	# only on the boundary we need a name for bc
+        geo.Append(geom_part, leftdomain=left, rightdomain=right, bc=name)
+    else:
+        geo.Append(geom_part, leftdomain=left, rightdomain=right)
+
+geo.SetMaterial (1, "ceramic")
+geo.SetMaterial (2, "liquid")
+geo.SetMaterial (3, "air")
+
+mesh = Mesh(geo.GenerateMesh(maxh=0.0007))
+#print (f"segments of the boundary: {mesh.GetBoundaries()}")
+#print (f"name of the materials: {mesh.GetMaterials()}")
+
+#Draw(mesh) # plotting the mesh
+
+# printing the domains of the mug
+print_domains = False
+if print_domains == True:
+    cf = CoefficientFunction([
+        1 if mat=="ceramic" else
+        2 if mat=="liquid" else
+        3 if mat=="air" else 0
+        for mat in mesh.GetMaterials()
+    ])
+
+    Draw(cf, mesh, "domains")
+
+# --------------------------------------
+
+# defining the parameters
+# heat storage capacity (J/(m^3*K))
+c_domain_values = {'ceramic': 1240*2.3*10**3, 'liquid': 4.138*10**6, 'air': 1.21*10**3}
+c = mesh.MaterialCF(c_domain_values)
+# thermal conductivity (W/(m*K))
+lam_domain_values = {'ceramic': 5, 'liquid': 0.6, 'air': 0.026}
+lam = mesh.MaterialCF(lam_domain_values)
+# heat transfer coefficient (W/(m^2*K))
+alpha = 8
+# outside temperature
+u_out = 18
+
+# -----------------------------------
+
+# setting up the FEM
+# H1-conforming finite element space
+fes = H1(mesh, order=1, autoupdate=True)
+
+# define trial- and test-functions
+u = fes.TrialFunction()
+v = fes.TestFunction()
+
+# define the bilinear form (stiffness + Robin part)
+a = BilinearForm(fes, symmetric=True)
+a += (lam*grad(u)*grad(v))*dx
+a += (alpha*u*v)*ds
+
+# define the bilinear form (mass)
+m = BilinearForm(fes, symmetric=True)
+m += c*u*v*dx
+
+# define the right-hand side (Robin part)
+f = LinearForm(fes)
+f += (alpha*u_out*v)*ds
+    
+# assemble the system of linear equation
+a.Assemble()
+m.Assemble()
+f.Assemble()
+minv = m.mat.Inverse(fes.FreeDofs(), inverse="pardiso")
+
+# solution field
+gfu = GridFunction(fes)
+
+# ------------------------------------
+
+# stationary problem (no time aspect)
+# gfu.vec.data = a.mat.Inverse(fes.FreeDofs(), inverse="pardiso")*f.vec.data
+# Draw(gfu)
+
+# -----------------------------------
+
+# functions that will be helpful
+
+# calculatin the average temperatur in a given subdomain
+def mean_temp(gfu,mesh,domain):
+    temp_int = Integrate(gfu, mesh, definedon=mesh.Materials(domain))
+    vol = Integrate(1, mesh, definedon=mesh.Materials(domain))
+
+    return temp_int/vol
+
+# -----------------------------------
+
+# Q1
+run_Q1 = False
+
+if run_Q1 == True:
+
+    # initial data for Q1
+    init_u_liquid = 80
+    init_u_domain_values = {'ceramic': u_out, 'liquid': init_u_liquid, 'air': u_out}
+    init_u = mesh.MaterialCF(init_u_domain_values)
+    gfu.Set(init_u)
+    Draw(gfu)
+
+    time.sleep(7)
+    #Redraw(blocking=True)
+
+    # setting up the FEM environment
+    time_max = 10*60    # breaking condition for simulation
+    tau = 0.5  # duration of 1 timestep
+    temp_opt = 67
+
+    theta = 0.5 # paramater for time discretization
+    # 0 (explicit Euler), 0.5 (Crank Nicolson scheme), 1 (implicit Euler)
+
+    mstar = m.mat.CreateMatrix()
+    mstar.AsVector().data = m.mat.AsVector() + theta*tau*a.mat.AsVector()
+    mstarinv = mstar.Inverse(fes.FreeDofs(), inverse="pardiso")
+    r = gfu.vec.CreateVector()
+
+    time_act = 0    # starting time =0 for loop condition
+    temp = 0    # here just for starting condition in while loop
+    data_q1 = [[time_act,u_out]]
+
+    # calculating T_warm, temperature when ceramic has optimal temperature temp_opt 
+    while temp<temp_opt and time_act<time_max:
+        time_act = time_act + tau   # actual time point
+        r.data = (m.mat - (1 - theta)*tau*a.mat)*gfu.vec + tau*f.vec
+        gfu.vec.data = mstarinv * r
+        temp = mean_temp(gfu,mesh,'ceramic')
+        data_q1.append([time_act,temp])
+        if time_act % 5 == 0:
+            Draw(gfu, min=u_out,max=init_u_liquid,autoscale=False)
+        if time_act == 451: # time of maximal temperature (ceramic); for u_0 = 80 !!!
+            gfu.Save("solution_Q1.vol") 
+    # finding the maximal temperature of the ceramic cup
+    time_temp_max, temp_ceramic_max = max(data_q1, key=lambda x: x[1])
+    print(f"u_0^liquid = {init_u_liquid} °C.")
+    print(f"After {time_temp_max} s the ceramic part reaches its maximal temperature of {temp_ceramic_max:.3f} °C.")
+
+
+# -----------------------
+
+# Q2
+run_Q2 = True
+
+if run_Q2 == True:
+
+    # initial data for Q2
+    init_u_liquid = 85  # now we have coffee (nearly the same behavior as water)
+    gfu_old = GridFunction(fes)
+    gfu_old.Load("solution_Q1.vol")
+    chi_liquid = mesh.MaterialCF({'liquid': 1}, default=0)
+    cf_init = gfu_old * (1 - chi_liquid) + init_u_liquid * chi_liquid
+    gfu.Set(cf_init)
+    Draw(gfu, min=u_out,max=init_u_liquid,autoscale=False)
+    time.sleep(7)
+
+    # setting up the FEM environment
+    time_max = 10*60*7    # breaking condition for simulation
+    tau = 2.5  # duration of 1 timestep
+    temp_opt = 50   # coffee should cool down
+
+    theta = 0.5 # paramater for time discretization
+    # 0 (explicit Euler), 0.5 (Crank Nicolson scheme), 1 (implicit Euler)
+
+    mstar = m.mat.CreateMatrix()
+    mstar.AsVector().data = m.mat.AsVector() + theta*tau*a.mat.AsVector()
+    mstarinv = mstar.Inverse(fes.FreeDofs(), inverse="pardiso")
+    r = gfu.vec.CreateVector()
+
+    time_act = 0    # starting time =0 for loop condition
+    temp = init_u_liquid    # here just for starting condition in while loop
+    data_q2 = [[time_act,init_u_liquid]]
+
+    # calculating T_cof, when coffee drops to temperature temp_opt 
+    while temp>temp_opt and time_act<time_max:
+        time_act = time_act + tau   # actual time point
+        r.data = (m.mat - (1 - theta)*tau*a.mat)*gfu.vec + tau*f.vec
+        gfu.vec.data = mstarinv * r
+        temp = mean_temp(gfu,mesh,'liquid')
+        data_q2.append([time_act,temp])
+        if time_act % 25 == 0:
+            Draw(gfu, min=u_out,max=init_u_liquid,autoscale=False)
+        if time_act % 100 == 0:
+            print(f'time: {time_act}; temperature: {temp:.3f}') 
+        
+    print(f'end-time: {time_act}; temperature: {temp:.3f}')
+
+# -----------------------
+
+# -----------------------
+
+# plotting of the temperature curves
+
+# Q1: heating the ceramic
+plot_1 = False
+if plot_1 == True:
+    x,y = zip(*data_q1)
+    plt.figure()
+    plt.plot(x, y)
+    plt.grid(True)
+    plt.axhline(y=temp_ceramic_max, color='red', linestyle='--', label=f'T_max={temp_ceramic_max:.3f}') # horizontal line
+    plt.axvline(x=time_temp_max, color='black', linestyle='--', label=f't={time_temp_max}') # vertical line
+    plt.title('temperature curve of ceramic')
+    plt.xlabel('time (in s)')
+    plt.ylabel('temperature (°C)')
+    plt.legend()
+    plt.savefig('temperature_ceramic_q1.png', dpi=600)
+    plt.show()
+    plt.close()
+
+# Q2: coffee cool down
+plot_2 = False
+if plot_2 == True:
+    x,y = zip(*data_q2)
+    plt.figure()
+    plt.plot(x, y)
+    plt.grid(True)
+    plt.title('temperature curve of coffee')
+    plt.xlabel('time (in s)')
+    plt.ylabel('temperature (°C)')
+    plt.savefig('temperature_coffee_q2.png', dpi=600)
+    plt.show()
+    plt.close()