1 / 38

Analytical Methods (2 sessions)

Analytical Methods (2 sessions) . Separation of Variables Method. Separation of variables sometimes called the method of Fourier for solving a PDE: Separation of variables in rectangular coordinates for Laplace’s Equations: Dirichlet boundary conditions:

abby
Télécharger la présentation

Analytical Methods (2 sessions)

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Analytical Methods(2 sessions)

  2. Separation of Variables Method • Separation of variablessometimes called the method of Fourier for solving a PDE: • Separation of variables in rectangular coordinates for Laplace’s Equations: • Dirichlet boundary conditions: • After applying boundary condition: • Using superposition:

  3. Separation of Variables Method • Separation of variables in rectangular coordinates for wave equation: • After applying boundary condition:

  4. Separation of Variables Method • Another Example: • Method of separation of variables for a problem with four inhomogeneousboundary conditions: • Using superposition: = + + +

  5. Separation of Variables Method • Separation of Variables in Cylindrical Coordinates: • Laplace’s Equation in Cylindrical Coordinates : • After ... We have: • by the superposition principle to form a complete series solution:

  6. Separation of Variables Method • Wave Equation in Cylindrical Coordinates: • Dividing both sides by ρ2: Using: Using: is Bessel’s equation Assuming x=λρand R=y

  7. Separation of Variables Method • Jn & Ynarefirst and second kinds of ordern • Ynis also called the Neumann function • Bessel’s equation has a general solution: • We may replace xby jx, modified Bessel’s equation is generated: • ModifiedBessel’s equation has a general solution: • In & Knarefirst and second kinds of ordern

  8. Separation of Variables Method Gamma function • Final solution is:

  9. Separation of Variables Method • Asymptotic expressions:

  10. Separation of Variables Method • Example: • A plane wave E=azEo e−jkxis incident on an infinitely long conducting cylinder: • Determine the scattered field. • Solution: • Integrating over 0≤φ≤2π gives: • Taking the m-thderivative of both sides: • Since Eszmust consist of outgoing waves that vanish at infinity, it contains: • Hence:

  11. Separation of Variables Method • Solution (cont.): • The total field in medium 2 is: • While the total field in medium 1 is zero, At ρ=a, the boundary condition requires that the tangential components of E1 and E2 be equal. Hence:

  12. Separation of Variables in Spherical Coordinates • Laplace’s equation for finding potential due to an uncharged conducting spherelocated in an external uniform electric field: • To solve (1): • To solve (2): (1) is Cauchy-Euler equation (2)

  13. Separation of Variables in Spherical Coordinates Legendre differential equation • Making these substitutions: • Its solution is obtained by the method of Frobenius as: Legendre functions of the first and second kind

  14. Separation of Variables in Spherical Coordinates • Qnare not useful since they are singular at θ=0,π • We use Qnin problems having conical boundaries that exclude the axisfrom the solution region. • For this problem, θ=0,πis included so that: • After determining Anand Bn, using boundary conditions:

  15. Separation of Variables in Spherical Coordinates • Wave Equation: • is one of: • is one of: spherical Bessel functions Using: ordinary Bessel functions

  16. Separation of Variables in Spherical Coordinates • By replacing H=y & cosθ=x, thesecond equation, which is Legendre’s associated differential equation, can be rewritten: • General solution is: • Using ordinary Legendre functions: Legendre functions of the first and second kind

  17. Separation of Variables in Spherical Coordinates • Example: A thin ring of radius a carries charge of density ρ. • Find V(P(r, θ,φ) at: (a)point P(0, 0, z) on the axis of the ring, (b) point P(r, θ,φ) in space. • Solution: From elementary electrostatics, at P(0, 0, z): • Using ∇2V=0 havingboundary-value solution in P(0, 0, z): • Since Qnis singular at θ=0,π,Bn=0, Thus: • For 0≤r≤a, Dn=0 since V must be finite at r=0: • Using boundary-value:

  18. Separation of Variables in Spherical Coordinates • Solution (cont.): • For r≥a, Cn=0since V must be finite as r→∞: • Therefore: ?

  19. Separation of Variables in Spherical Coordinates • Example: • A PEC spherical shell of radius is maintained at potential Vocos2φ • Determine Vat any point inside the sphere. • Solution: • Solution is similar to that of previous problem • Using the boundary condition:

  20. Orthogonal Functions • Orthogonal functions, including Bessel, Legendre, Hermite, Laguerre, and Chebyshev, are useful in solution of PDE. • They are very useful in series expansion of functions such as Fourier-Bessel series, Legendre series, etc. • Besseland Legendre functions are importance in EM problems. • A system of real functions φn(n=0,1,2,…) is said to be orthogonal with weight w(x) on (a, b) if: • For example, the system of functions cos(nx)is orthogonal:

  21. Orthogonal Functions • An arbitrary function f(x), defined over interval (a, b), can be expressed in terms of any complete, orthogonal set of functions: Where: Others are in Table 2.5

  22. Orthogonal Functions • Example: • Expand f(x) in a series of Chebyshevpolynomials. • Let: • Since f(x)is an even function, odd terms in expansion vanish: • By multiply both sides by: • and integrate over −1≤ x≤1:

  23. Series Expansion • As noticed, PDE can be solved with the aid of infinite series and generally, with series of orthogonal functions. • An example: • Let the solution be of the form: • by substituting this equation into PDE:

  24. Practical Applications Using: • Scattering by Dielectric Sphere: • It is illuminated by a plane wave propagating in z direction and Epolarized in xdirection. • As a similar way: … … = =

  25. Practical Applications • Scattering by Dielectric Sphere (cont.): • Form of the incident fields: • Form of the scattered fields:

  26. Practical Applications • Scattering by Dielectric Sphere (cont.): • Similarly, transmitted field inside the sphere: • Continuity of tangential components at surface: • Radar Cross Section (RCS): • By using asymptotic expression (R=∞) for spherical Bessel functions: • Where amplitude functions S1(θ)and S2(θ)are given as:

  27. Practical Applications • Scattering by Dielectric Sphere (cont.): • Using: • Similarly, forward-scattering cross section:

  28. Practical Applications • Attenuation Due to Raindrops: • At f>10GHz, attenuation caused by atmospheric particles (oxygen, ice crystals, rain, fog, and snow) can reduce the reliability and performance of radar and space communication links. • We will examine propagating through rain drops: • By Mie solution, the attenuation and phase shift of an EM wave by raindrops having spherical shape (for low rate intensity) is investigated. • For high rain intensity, an oblate spheroidal model would be more realistic as: • EM attenuation due to travel a wave through a homogeneous medium (with Nidentical spherical particles/m3) in a distance is given by e−γ. • γ is attenuation coefficient as [11]: • To relate attenuation and phase shift to a realistic rainfall, drop-size distribution (D) for a given rate intensity N(D) must be known. • Total attenuation and phase shift over the entire volume: [11] H.C. Van de Hulst, “Light Scattering of Small Particles”. New York: JohnWiley,1957, pp. 28–37, 114–136, 284.

  29. Practical Applications Laws and parsons drop-size distributions for various rain rates • Attenuation Due to Raindrops (cont.): Raindrop terminal velocity

  30. Numerical Integrations • Numerical integration is used whenever a function cannot easily be integrated in closed form. • The common ones include: • Euler’s rule (Riemann sum), • Trapezoidal rule, • Simpson’s rule, • Newton-Cotes rules, and • Gaussian (quadrature) rules. • Euler’s Rule (Rectangular Rule) (Riemann Sum): • We divide the curve into n equal intervals as shown: • This method gives an inaccurate result since each Ai is less or greater than the true area introducing negative or positive error, respectively.

  31. Numerical Integrations • Euler’s Rule (cont.): • Other forms:

  32. Numerical Integrations • Trapezoidal Rule:

  33. Numerical Integrations • Simpson’s Rule: • Simpson’s rule gives more accurate result than the trapezoidal rule. • While trapezoidal approximates curve by connecting successive points on curve by straight lines, Simpson’s rule connects successive groups of three points on curve by a second-degree polynomial (i.e., a parabola). • where n is even. • Computational molecules for integration:

  34. Numerical Integrations Where • Newton-Cotes Rules: • To apply, we divide interval a<x<b into m equal intervals so that: • Where m is a multiple of n, and n is number of intervals covered at a time or order of approximating polynomial. • Subarea in the interval xn(i−1)<x<xni: • Integration is: • Most widely known Newton-Cotes formulas are: • n=1 (2-point; trapezoidal rule): • n=2(3-point; Simpson’s 1/3 rule): • n=3(4-point; Newton’s rule): Newton-Cotes Numbers

  35. Numerical Integrations • Gaussian Rules: • Idea of integration rules using unequally spaced abscissa points stems from Gauss. • Gaussian rules are more complicated but more accurate than Newton-Cotes. • A Gaussian rule: • i is a sequence of orthogonal polynomials and xiare zeros of i. • Any of orthogonal polynomials discussed in CH2can be used to give a particular Gaussian rule. • Commonly used rules are: • Gauss-Legendre, • Gauss-Chebyshev, etc., • Points xi are the roots of the Legendre, Chebyshev, etc., of degree of n. • For Legendre(n=1 to 16) and Laguerre (n=1 to 16) polynomials, the zeros xiand weights wihave been tabulated in the paper:

  36. Numerical Integrations where • Using Gauss-Legendrerule: • Gauss-Chebyshev rule is similar to Gauss-Legendre rule. • We use above eq. except that sample points xi, roots of Chebyshev polynomial Tn(x), are: • The weights are all equal: • When limits of integration are ±∞: • A major drawback is that to improve accuracy, n must be increased which means that wi & ximust be included in program for each value of n. • Another disadvantage is that function f(x) must be explicit since sample points xiare unassigned. are transformation of roots xiof Legendre polynomials Abscissas (Roots of Legendre Polynomials) and Weights for Gauss-Legendre Integration

  37. Numerical Integrations • Multiple Integration: • This is an extension of one-dimensional (1D) integration. • Substitution: • Procedure can be extended to a 3D integral:

  38. Numerical Integrations • Integration Molecule: Triple integration molecule for Simpson’s 1/3 rule Double integration molecule for Simpson’s 1/3 rule

More Related