Description

JINEETH N [EE20B051]
Abstract
• To solve for potential and currents in a system.
• To solve 2-D Laplace equations in an iterative manner.
• To plot graphs to understand the 2-D Laplace equation.
1 Introduction
A wire is soldered to the middle of a copper plate and its voltage is held at 1 Volt. One side of the plate is
Ohm’s law
J⃗ = σE⃗
Charge Continuity equation. (1)
(2)
From the above equations,
(3)
For DC currents, the right side is zero, and we obtain
∇2ϕ = 0 (4)
2 Tasks
2.1 Parameters
Niter = 1500
Ny = 25 Nx = 25 radius = 8
# The above parameters can also be given through the commmand line.
if len(sys.argv) > 1:
Nx = sys.argv[1]
Ny = sys.argv[2]
Niter = sys.argv[3] n = arange(Niter)
2.2 Variable initialization
Create a zero 2-D array of size Nx x Ny and assign 1 to cordinates within radius 1 from center phi = np.zeros((Ny, Nx))
rng = 0.5 y = np.linspace(-rng, rng, Ny) x = np.linspace(-rng, rng, Nx) X, Y = meshgrid(x, -y)
2.3 Allocating potential and plotting it
# Assign potential = 1 ii = where(((X * X) + (Y * Y)) <= (0.35 * 0.35)) phi[ii] = 1.0 plt.plot(x[ii[0]], y[ii[1]], “or”, label=”V = 1″)

Figure 1: semilogy scale
2.4 Updating Potential
After converting the equation(4) to discrete domain, we can update matrix through iterations
(5)
The bottom boundary is grounded. The other 3 boundaries have a normal potential of zero

Figure 2: Error (semilog) Figure 3: Error (loglog)
errors = np.zeros((Niter, 1))
for k in range(Niter):
oldphi = phi.copy() phi[1:-1, 1:-1] = (1/4) * ( phi[1:-1, 0:-2] + phi[1:-1, 2:] + phi[0:-2, 1:-1] + phi[2:, 1:-1]
)
# These lines will set the proper boundary conditions.
phi[1:-1, 0] = phi[1:-1, 1] phi[1:-1, -1] = phi[1:-1, -2] phi[0, 1:-1] = phi[1, 1:-1] phi[ii] = 1.0
errors[k] = (abs(phi – oldphi)).max()
• we can observe that error decreases linearly for higher no of iterations,so from this we conclude that for large iterations error decreases exponentially with No of iterations i.e it follows AeBx as it is a semilog plot
And if we observe loglog plot the error is almost linearly decreasing for smaller no of iterations so it follows ax form since it is loglog plot and follows some other pattern at larger iterations.
2.5 Plotting the errors
We will plot the errors on semi-log and log-log plots. We can observe the error decreases very slow
loglog(n, errors)
semilogy(n[500:], errors[500:])
2.5.1 Fitting exponential curve
We observed that the error is decaying exponentially for higher iterations. We will fit two curves.

Figure 4: Potting exponential curves
2.6 Plotting Potential
# Contour plot of potential plt.title(“Contour plot of V”, fontsize=16) plt.contourf(Y, X[::-1], phi) plt.plot(x[ii[0]], y[ii[1]], “or”, label=”V = 1″)
egin{verbatim}
# The exponent part of the error values can be got using the below piece of code.
y1 = log(errors) yfit = lstsq(c_[np.ones(Niter – 0), arange(Niter – 0)], y1, rcond=None)[0] #fitting a straight line in log scale and calculating actual values in normal scale a, b = exp(yfit[0]), yfit[1] y2 = log(errors[500:]) yfit = lstsq(c_[np.ones(Niter – 500), arange(500, Niter)], y2, rcond=None)[0] a_500, b_500 = exp(yfit[0]), yfit[1]
• Considering all iteration
• Considering after 500th iteration

Figure 5: Potting exponential curves

Figure 6: Contour plot of potential
2.7 Plotting Potential
figure(7, figsize=(6, 6)) contourf(X, Y, phi) plot(xp, yp, “ro”, label=”V = 1″) xlabel(r”X”) ylabel(r”Y”) title(“Contour plot of potential”) grid() legend()
fig1 = figure(8, figsize=(6, 6)) ax = p3.Axes3D(fig1) xlabel(r”X”) ylabel(r”Y”) title(“The 3-D surface plot of the potential”) surf = ax.plot_surface(X, Y, phi, rstride=1, cstride=1, cmap=cm.jet)

Figure 7: 3D Potential plot
2.8 Vector plot of currents
We use the below equations to calculate current,
) (6)
) (7)
Jx = np.zeros((Ny, Nx))
Jy = np.zeros((Ny, Nx))
Jx = 0.5 * (phi[1:-1, 0:-2] – phi[1:-1, 2:]) Jy = 0.5 * (phi[2:, 1:-1] – phi[0:-2, 1:-1])
figure(9, figsize=(6, 6)) quiver(X[1:-1, 1:-1], Y[1:-1, 1:-1], Jx, Jy) plot(xp, yp, “ro”) xlabel(r”X”) ylabel(r”Y”) title(“The vector plot of the current flow”)

Figure 8: Current vectors
3 Conclusion
• Most of the current is restricted to bottom part of the wire and it is normal to the surface of both wire and metal
• We can vectorize multiple ”for” loops to a single line in python
• Also we observed that the decrease in error is very slow after 500 iterations which makes this method inefficient