Multipoint Boundary Value Problem with 16 ODEs and 3 regions

Hey guys,

I hope you can help me 'cause I got stuck with this problem for days.

I want to do a calculation of two adhesive-bonded plates. The overlap bond is just in region 2. So in region 1 it is just plate 1 and plate 2 is in region 3. So ODE 1-8 are refer to plate 1 and ODE 9-16 refer to plate 2.

I want to use BVP5C to solve this MBVP. My Problem is that i have 8 ODEs in region 1 und 3 and 16 in region 2. How can I mange this type of multipoint bvp. If i want to run bvp5c the solver wants to have 48 boundary conditions because it is calculating 16 equations in all 3 regions. I think a possible solution could be to have 3 different mesh for the initial guess or i have to add boundary conditions for the non existing plates in regions 1 and 3.

Thanks for your help, all the best, bernardo

ODEs:

 dydx = zeros(16,1);
   switch region
      case 1                                % x in [-L1 0]
         dydx(1)=    k_11*y(5)+k_21*y(6)+k_31*y(7);
         dydx(2)=   -y(3);
         dydx(3)=   -k_41*y(5)-k_51*y(6)-k_61*y(7);
         dydx(4)=   k_71*y(5)+k_81*y(6)+k_91*y(7);
         dydx(5)=   0;
         dydx(6)=   0;
         dydx(7)=   y(8);
         dydx(8)=   0;

      case 2                                % x in [0 L]
        dydx(1)=    k_11*y(5)+k_21*y(6)+k_31*y(7);  
        dydx(2)=    -y(3);
        dydx(3)=    -k_41*y(5)-k_51*y(6)-k_61*y(7); 
        dydx(4)=    k_71*y(5)+k_81*y(6)+k_91*y(7);
        dydx(5)=    (G_a/t_a)*(y(1)+t_1/2*y(3)-y(9)+t_2/2*y(11));
        dydx(6)=    G_a/t_a*(y(4)-y(12));
        dydx(7)=    y(8)+G_a*(t_1+t_a)/(2*t_a)*(y(1)+ t_1/2*y(3)-y(9)+t_2/2*y(11));
        dydx(8)=    E_a/t_a*(y(2)-y(10));
        dydx(9)=    k_12*y(13)+k_22*y(14)+k_32*y(15);
        dydx(10)=   -y(11);
        dydx(11)=   -k_42*y(13)-k_52*y(14)-k_62*y(15);
        dydx(12)=    k_72*y(13)+k_82*y(14)+k_92*y(15);
        dydx(13)=    G_a/t_a*(-y(1)-t_1/2*y(3)+y(9)-t_2/2*y(11));
        dydx(14)=    G_a/t_a*(-y(4)+y(12));
        dydx(15)=    y(16)+G_a*(t_2+t_a)/(2*t_a)*(y(1)+t_1/2*y(3)-y(9)+t_2/2*y(11));
        dydx(16)=    E_a/t_a*(y(10)-y(2));

      case 3                %x in [L L+L1]   

         dydx(9)=    k_12*y(13)+k_22*y(14)+k_32*y(15);
         dydx(10)=   -y(11);
         dydx(11)=   -k_42*y(13)-k_52*y(14)-k_62*y(15);
         dydx(12)=   k_72*y(13)+k_82*y(14)+k_92*y(15);
         dydx(13)=   0;
         dydx(14)=   0;
         dydx(15)=   y(16);
         dydx(16)=   0; 
  end

boundary conditions:

    res_bc=  [      YL(5,1)-Nxx       %Force x
                YL(6,1)           %Force Y=0
                YL(7,1)       %moment of force=0
                YL(8,1)-Qxx       %Force z
                YR(1,1) - YL(1,2) % Continuity bei X=0
                YR(2,1) - YL(2,2) % Continuity bei X=0
                YR(3,1) - YL(3,2) % Continuity bei X=0
                YR(4,1) - YL(4,2) % Continuity bei X=0
                YR(5,1) - YL(5,2) % Continuity bei X=0
                YR(6,1) - YL(6,2) % Continuity bei X=0
                YR(7,1) - YL(7,2) % Continuity bei X=0
                YR(8,1) - YL(8,2) % Continuity bei X=0  
                YL(13,2)        %free edge plate 2
                YL(14,2)        %free edge plate 2
                YL(15,2)        %free edge plate 2
                YL(16,2)        %free edge plate 2
                YR(5,2)         %free edge plate 1
                YR(6,2)         %free edge plate 1
                YR(7,2)         %free edge plate 1
                YR(8,2)         %free edge plate 1
                YR(9,2) -  YL(9,3)  % Continuity X=L
                YR(10,2) - YL(10,3) % Continuity X=L
                YR(11,2) - YL(11,3) % Continuity X=L
                YR(12,2) - YL(12,3) % Continuity X=L
                YR(13,2) - YL(13,3) % Continuity X=L
                YR(14,2) - YL(14,3) % Continuity X=L
                YR(15,2) - YL(15,3) % Continuity X=L
                YR(16,2) - YL(16,3) % Continuity X=L
                YR(13,3)-Nxx   
                YR(14,3)       
                YR(15,3)-Mxx       
                YR(16,3)-Qxx ];    

end

[Torsten]

Your problem is non-standard.

So I'd discretize the equations on my own, incorporate the transmission conditions and solve the resulting system of algebraic equations using "fsolve".

This way, you have optimum control and flexibility.

Best wishes

Torsten.

[Bernardo]

thanks for the quick answer, so yyou mean i can not solve this with bvp4c or bvp5c? I think my maths an matlab skills are not good enough to solve such a problem without a toolbox

[Torsten]

I don't know if one can find a way to make bvp4c/bvp5c solve your problem, but each artificial construction to make a solver work on a problem it is not designed for has the risk that garbage is produced.

Best wishes

Torsten.

[Torsten]

As far as I can see, there are no compliciated transmission conditions at the contact points. So why don't you call bvp4c three times in series with different ODEs and boundary conditions ?

Best wishes

Torsten.

[LIU YAO]

Hi Torsten, I have a same problem like Bernardo. I saw you suggested that we can discrete the equations and use fsolve to solve it. Could you show me an example to explain how to use fsolve to deal with BVPS?

[Torsten]

Google

boundary value problem & finite difference method

Best wishes

Torsten.