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Lecture 28j, PDE toroid geometry 

Extending PDE's to toroid geometry



solve PDE in toroid boundary




First refinement


Another view of refinement

Toroid center at  0,0,0  can be translated and rotated
  r1, r2, theta, phi  mapped to x,y,z               
  x = (r1+r2*sin(phi))*cos(theta)                   
  y = (r1+r2*sin(phi))*sin(theta)                   
  z = r2 * cos(phi)                                 
  0 ≤ theta ≤ 2Pi    0 ≤ phi ≤ 2Pi   0 < r2 < r1 

  Area = (2*Pi*r1)*(2*Pi*r2)     infinitesimal area = dtheta*r1*dphi*r2
  Volume = (2*Pi*r1)*(Pi*r2*r2)  infinitesimal volume = dtheta*r1*dphi*r2*dr2
  Equation   (r1-sqrt(x^2+y^2))^2 + z^2 = r2^2

  r2, x, y, z mapped to r1, theta, phi (r2 taken as constant)
  theta = arctan(y/x)
  phi = arccos(z/r2)
  r1 = x/cos(theta) - r2*sin(phi)   or
  r1 = y/sin(theta) - r2*sin(phi)   no divide by zero

  r1, x, y, z mapped to r2, theta, phi (r1 taken as constant)
  theta = arctan(y/x)    fix angle by quadrant
  x1 = r1*cos(theta)
  y1 = r1*sin(theta)
  phi = arctan(sqrt((x-x1)^2+(y-y1)^2)/z)  fix by quadrant
  r2 = sqrt((x-x1)^2+(y-y1)^2+z^2)

PDE for testing: r1 constant, r2 is r, theta is t, phi is p

 dU^2(r1,r2,t,p)/dr2^2 + dU^2(r1,r2,t,p)/dt^2 +
 dU^2(r1,r2,t,p)/dp^2  = f(r1,r2,t,p)

 U(r,t,p):=r*r*(1+sin(t))*(1+cos(p));
                                 2
                  U(r, t, p) := r  (1 + sin(t)) (1 + cos(p))

 Urr(r,t,p):=diff(diff(U(r,t.p),r),r);
                  Urr(r, t, p) := D[1, 1](U)(r, t . p)

 Urr(r,t,p):=diff(diff(U(r,t,p),r),r);
                  Urr(r, t, p) := 2 (1 + sin(t)) (1 + cos(p))

 Utt(r,t,p):=diff(diff(U(r,t,p),t),t);
                                      2
                  Utt(r, t, p) := -r  sin(t) (1 + cos(p))

 Upp(r,t,p):=diff(diff(U(r,t,p),p),p);
                                      2
                  Upp(r, t, p) := -r  (1 + sin(t)) cos(p)

 f(r,t,p):=Urr(r,t,p)+Utt(r,t,p)+Upp(r,t,p);
                                             2
f(r, t, p) := 2 (1 + sin(t)) (1 + cos(p)) - r  sin(t) (1 + cos(p))

        2
     - r  (1 + sin(t)) cos(p)

 simplify(f(r,t,p));
                                             2             2
2 + 2 cos(p) + 2 sin(t) + 2 sin(t) cos(p) - r  sin(t) - 2 r  sin(t) cos(p)

        2
     - r  cos(p)

f(r2,t,p) =
2.0*(1.0 + cos(p) + sin(t) + sin(t)*cos(p)) - 
r2*r2*(cos(p) + sin(t) + 2.0*sin(t)*cos(p))

Ub(r2,t,p):=r2*r2*(1.0+sin(t))*(1.0+cos(p));  Dirichlet boundary



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