Assignment IV: Method of Moments

Exercise 1 - MoM for Poisson in 1D

In this exercise, we will solve the Poisson Equation numerically in the integral form to determine a 2D capacitor's capacitance.

As part of the exercise:

1.1 Describe the advantages of the MoM method with respect to the FD and FEM Methods.

1.2 What is a Green's function? How do we calculate the Green's function from a PDE, e.g., Poisson equation?

1.3 In the MoM methodology, we apply a FEM methodology (expansion with the basis functions) to solve an integral equation. What are the two basic approaches we use to minimize residual in MoM? Which approach is the generalization of the other approach? See section 7.1.3.2 of the textbook.

1.4 We solve the integral form of the Poisson equation numerically to determine the capacitance of a capacitor in free space using the two Matlab codes (capacitor2D.m, MoM2D.m), see the figure above.

As part of the exercise:

1.4.1 Which basis function does the code use?

1.4.2 Identify and report the Matlab code that calculates the integral for a single element? Is the integral calculated analytically or numerically?

1.4.3 Perform and plot a convergence test of the method (calculate the Capacitance), increasing the number of elements. Compare your results with Figure 7.5 in the textbook. What is the accuracy of the method?

1.4.4 Plot the charge density distribution for the top capacitor against the x-coordinate with 20 and 200 elements. What is the difference in the value of the first (or last) element in the two cases? Why is there such a difference (see section 2.3 of the textbook)?

Exercise 2 - Adaptive MoM

2.1 Implement an adaptive strategy with different element sizes. For instance, to have an equal charge in each element is a good strategy (see plot below but other adaptive grid strategies are also acceptable for this exercise.

A possible adaptive strategy is to go through either the left-sided or right-sided plate and alter the sizes of the elements. The new element sizes could be found as h_i = (hi₊₁ rho_i+1 )/ rho_i
This formula ensures that the charge at adjacent elements should be equal. 

Make sure to read section 7.2.4 of the textbook presenting this strategy.

2.2 Perform a convergence test using the new adaptive code. Compare the convergence results obtained in 1.4.3.  What is the advantage of using an adaptive strategy? What are the challenges for implementing it?

Bonus Exercise - Numerical Integration

Implement the numerical calculation of the integral per element

and substitute it in MoM2D.m. Implement the i) midpoint and ii) trapezoidal integration rules (see Section 7.2.5 of the textbook).

2.1 Compare the capacitance results obtained in exercise 1 with the results i) and ii) varying the number of elements. Discuss the differences between the results in exercises 1, i) and ii)