Boundary conditions with non-normal derivatives - Thermoelastic deformation 2D-Axisymmetric

Can anyone help me to write properly the boundary conditions, specially the coupled boundary conditions, for the following 2D-Axisymmetric thermoelastic equations (a solid cylinder of radius rf and thickness L subjected to a point source pressure pz at r=0 and z=0):

eq1 = Derivative[2, 0, 0][ur][r, z, t] + (1/r)* Derivative[1, 0, 0][ur][r, z, t] +  Derivative[0, 2, 0][ur][r, z, t] +  c1*D[Derivative[1, 0, 0][ur][r, z, t] + (1/r)*ur[r, z, t] +  Derivative[0, 1, 0][uz][r, z, t], r] ==  c2*Derivative[0, 0, 2][ur][r, z, t]  eq2 = Derivative[2, 0, 0][uz][r, z, t] + (1/r)* Derivative[1, 0, 0][uz][r, z, t] +  Derivative[0, 2, 0][uz][r, z, t] +  c1*D[Derivative[1, 0, 0][ur][r, z, t] + (1/r)*ur[r, z, t] +  Derivative[0, 1, 0][uz][r, z, t], z] ==  c2*Derivative[0, 0, 2][uz][r, z, t]  ic = {uz[r, z, 0] == 0, ur[r, z, 0] == 0,  Derivative[0, 0, 1][uz][r, z, 0] == 0,  Derivative[0, 0, 1][ur][r, z, 0] == 0}  bc = {uz[rf, z, t] == 0, ur[rf, z, t] == 0, ur[0, z, t] == 0}  coupledbc1= {pz == c4*(Derivative[0, 1, 0][uz][r, 0, t] +  c3*(Derivative[1,0, 0][ur][r, 0, t] + (1/r)*ur[r, 0, t] +  Derivative[0, 1, 0][uz][r, 0, t]))}  coupledbc2= {0 == Derivative[0, 1, 0][uz][r, L,t] + c3* (Derivative[1, 0, 0][ur][r, L,t] + (1/r)*ur[r, L, t] +  Derivative[0, 1, 0][uz][r, L, t])}  coupledbc3= {0 == (Derivative[1, 0, 0][uz][r, 0, t] +  Derivative[0, 1, 0][ur][r, 0, t])}  coupledbc4= {0 == (Derivative[1, 0, 0][uz][r, L, t] +  Derivative[0, 1, 0][ur][r, L, t])}  coupledbc5= {0 == (Derivative[1, 0, 0][uz][rf, z, t] +  Derivative[0, 1, 0][ur][rf, z, t])}  coupledbc6= {0 == Derivative[1, 0, 0][ur][rf, z, t] +  c3*(Derivative[1, 0, 0][ur][rf, z, t] + (1/r)*ur[rf, z, t] +  Derivative[0, 1, 0][uz][rf, z, t])} 

c_i and pz are known parameters. Mathematica does not recognize non-normal derivatives as boundary conditions. I have tried the following (rf=0.005 e L=0.001):

NDSolve[{eq1, eq2, ic, bc1, coupledbc's}, {uz, ur}, {r, 0, rf}, {z,0,L},  {t, 0, tf}, Method -> {"PDEDiscretization" -> {"FiniteElement"}}] 

Thanks...

Nelson

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Category: differential equations Time: 2016-07-29 Views: 0

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