Study questions for the ECEN5154 in-class quiz, April 1st, 2009

  1. How would you classify the boundary value problems that arise in EM. Provide a brief description for all categories. What is the difference between open and closed domain problems. Which boundary conditions do you know?
  2. Write a general operator equation and describe what is what. Give an example for differential and integral equations.
  3. Write Maxwell’s equations in differential form. Provide constitutive relations and continuity equation as well. Describe in details the physics of the Faraday’s law.
  4. Write Maxwell’s equations in integral form. Give continuity equation as well. Give a couple of consequences of the continuity equation. Describe in details the physics of Maxwell-Ampere’s law.
  5. What is a wave equation? Why is it important? How do we derive it (give an example for a source free VWE for the electric field in linear, isotropic, homogeneous medium)? Denote the operator.
  6. Why do we need boundary conditions (BCs)? How do we classify BCs based on the physics? State natural BCs in words and mathematically (you do not need to derive those).
  7. Why do we need approximate BCs? When would you use standard impedance BCs? What about sheet transition BCs? What is the radiation BC and why is it important?
  8. Why do we care about the numerical integration (NI)? What is the basic premises of the NI?
  9. Describe the Euler’s rule. Provide all the steps needed for the numerical implementation. What do you observe with respect to this method?
  10. Describe the trapezoidal rule. Provide the steps needed for the numerical implementation. What do you observe with respect to this method?
  11. Describe the Simpson’s rule. Provide the steps needed for the numerical implementation. What do you observe with respect to this method? Do we know what is the error in this method?
  12. Describe the Gauss integration. Provide the steps needed for the numerical implementation. Derive the weights and sampling points for the one point rule.
  13. Derive the weights and sampling points for the two point Gauss integration rule.
  14. Which factors determine the accuracy of a numerical method? What should we do to improve the accuracy? Which types of errors are used in this process? What would you do if the exact solution is not known apriori?
  15. Provide the steps needed to determine the order of convergence. Are there any other way for determining the convergence order?
  16. Why do we care about matrices? Which kinds arise in PDE and IE BVPs? What is the positive definiteness? What are the norms?
  17. Which errors occur in solving linear systems of equations (LSE)? How do we understand/study these errors? How would you estimate a condition number?
  18. What could happen if the condition number is high? Why is the variation of solution vector high when you have large condition numbers (derive mathematically)?
  19. How do we invert system matrix A (in Ax=b)? Which solvers should we use? What do you know about the Gauss elimination (do not need to go step by step)? What are the major difficulties with GE?
  20. What is the LU decomposition (conceptually)? Why is it good? What should we do if the elements on the main diagonal are zero (aii=0)?
  21. What is a Cholesky decomposition? Can you always apply it? Is it more efficient (computationally) than the LU decomposition?
  22. Why do we care about iterative methods (IM)? How would you classify IMs?
  23. Describe the Jacobi’s algorithm? What are the differences between the Jacobi and Gauss-Saidel’s algorithms?
  24. Why do we care about the non-stationary methods? Which methods belong to this class? Sketch the convergence history for CG, BiCG and GMRES.
  25. What is the preconditioning and why do we care about it? For a matrix A=[…] determine the diagonal preconditioner. Will diagonal preconditioner always help?
  26. What is a finite difference method (FDM)? What are the steps in FDM?
  27. Describe in as many details as you can the discretization step?
  28. Describe the basic premises of the steps: ‘setting-up FD equations’, ‘Solve’, ‘Extraction of parameters’.
  29. For a general 1D function derive the forward, backward and central finite difference (the lowest order). Provide a sketch for the function. What is the order of accuracy for these approximations, and how would you establish that order (just say in the words you don’t need to derive)?
  30. Derive the lowest order forward difference for the 2nd order derivative on a 1D function f(x).
  31. How do we define/derive the finite difference for 2D functions?
  32. For a 2D Poisson equation provide the basic finite difference equation (sketch the arbitrary 2D domain as well).
  33. FD example (similar or the same like we had in the class – with 3 nodes in the enclosure….).
  34. Derive the FD equation for a node point on the boundary between two electrically dissimilar media.
  35. What is accuracy? What is stability? What are the error sources in FD, and how does the round off and discretization errors depend on the sampling size?
  36. What is the outcome/result of a FD method? How do you compute the capacitance? How do you compute the stored energy? What about the characteristic impedance? What about the phase velocity? Effective dielectric constant?
  37. What is the general procedure for computing the attenuation constant?
  38. What is the extraction to the zero grid size? Can you use FDM for the eigenvalue problems and if you can how?
  39. What are the basis functions (BFs)? What are some general properties of BFs? What are the practical factors in selecting BFs?
  40. Describe fully Dirac delta BFs.
  41. Describe fully piecewise constant BFs.
  42. Describe fully piecewise linear BFs.
  43. Describe fully piecewise sinusoidal BFs.
  44. Describe how would you generate the higher-order spline BFs. Which splines ensure continuity of a function and its first derivative over the domain of the BF? Are spline BFs orthogonal?
  45. Derive symbolically the weighting residual method (WRM). Which WRMs are in the general use?
  46. Describe fully point matching WRM.
  47. Describe fully subdomain WRM.
  48. Describe fully Galerkin WRM.
  49. Describe fully Least square WRM.
  50. Provide detail description, including the breakdown in the steps, for the theoretical approach in the development of a WRM.
  51. How can we use MoM in electrostatics? Describe the procedure for obtaining the potential distribution everywhere in the space.
  52. What are the singularities and how do we deal with those in, for example, BVPs that require finding the charge distribution on finite diameter wires.
  53. Given is a straight wire tessellated in 5 uniform segments and set on a uniform potential. Fill the coefficient matrix and excitation vector with symbolic letters that would represent the real numbers you would compute using the MoM (I want to see if you locate some symmetries, if you make some elements in the matrix equal, etc..), and order these in descending order. Hint: you will need five numbers A, B, C, D, E for the coefficient matrix.
  54. Formulate the example from the class (Point matching for a strip TRL). You do not need to memorize explicit formulas for matrix entries.
  55. Give the schematics of the steps for the scattering from the wires and denote pre-processing, analysis and post-processing parts.
  56. What is the difference between TMz and TEz excitations, and what should we do if the incident plane wave is of arbitrary polarization? Which integral equation do we use for TEz and TMz cases (just name those)?
  57. Provide description of the steps used in solving the 2D TMz scattering from the wires.
  58. For which practical problems can we use the formulation for the 2D TMz scattering from the wires? Which issues should your coding address?
  59. How should we model scatterer that has curved segments?
  60. What are the error sources in the analysis? Describe where from do they arise?
  61. Which assumptions do we have to include to derive the Pocklington integral equation? How do we include boundary conditions in the MoM formulation for the radiation/scattering problems that involve Pocklington’s integral equation?
  62. What are the source models that arise in wire problems? What is the fundamental difference in solving antenna boundary value problem as radiator vs scatterer?
  63. Describe the Delta gap source.
  64. Describe the magnetic-frill source.
  65. What are the memory requirements for a MoM solution of a problem that has N expansion coefficients? If N=106 how much memory do we need to store the coefficient matrix which arise from the positive-definite and self-adjoint operator. We need complex double precision for the elements.
  66. Provide approximate breakdown of the timing requirements for the MoM formulation.