1. 2D Finite Difference Method

A 2D square potential box with W = H = 2μm, have a grid with 2x2 internal nodes, with the following boundary conditions for potentials:

V(0, y) = V(W, y) = 10V

V(x, 0) = V(x, H) = 0V

If the potential inside the box follows a 2D Laplace equation, plot the 2D grid, and start from all zeros initial solution for internal nodes, then:

a) Find the potentials at the internal grid nodes, using Standard Liebmann's Method (i.e. λ = ??), after 3 iterations

b) Repeat part a, if the Over relaxation Factor λ is equal to 1.5

c) Solve directly using LU factorization

d) Compare the final results for the three previous methods for all internal nodes in a table.

2. 1D Finite Difference Method

Assume a 1D rod with length L=10μm follows the following Poisson's Equation:

d²V/dx² = -2v/μm

with the following boundary conditions:

V(0) = -1V, V(L) = 2V

a) If the rod is divided into two internal nodes, find the potentials at the internal grid nodes, using Standard Liebmann's Method (i.e. λ = 1), after 3 iterations

b) Repeat part a, if the Over-relaxation Factor λ is equal to 1.5

c) Solve directly using LU factorization

d) Compare the final results for the three previous methods for all internal nodes in a table.

3. 1D Finite Element Method

If the rod in Question 2 is divided into 3 finite elements of equal length the rod, assume that:

Electric Field = dV(x)/dx = continuous at Element Boundaries

a) Find the Linear Interpolation functions for each element describing the potential distribution within the element as a function of its local boundary values.

b) Use Galerkin's Method to express the residues in each of these elements.

c) Write down the Local Matrix equations for these elements.

d) Assemble the Global Matrix equations.

e) Find the potential at the center of each of these elements.

f) Compare the potentials at the elements boundaries with those obtained from Question 2 using 1D Finite Difference Method.

4. 2D Finite Element Method

4) For Problem 1):

a) Redraw the grid after labeling for 2D square Finite Elements Method

b) Write down the Linear Interpolation Functions for any of these elements describing the potential distribution within the element as a function of its local boundary values

c) If one of these elements has the following local boundary conditions, as function of its local x-y coordinates:

V(0, 0) = 0V, V(0, w) = 1V, V(0, l) = 0.5V, V(w, l) = 0.8V

Where l and w are the size of the element in the ?? and ?? local coordinates, respectively.

Find the voltage in the center of this element.