How to set the boundary conditions of 3D Poisson Equation
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Anthony Owusu-Mensah
il 2 Mar 2020
Commentato: David Martín Oliva Zavala
il 12 Nov 2020
I am trying to compute the electric potential at point (x,y,z) by solving the 3D Poisson equation below using finite difference method.
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/274754/image.png)
I have the charge densities at various positions of (x,y,z).
Below are the boundary condtions;
- Dirichlet boundary condition are applied at the top and bottom of the planes of the rectangular grid.
- Electric potential is to be incorporated by setting
and
, where h is the height of the simulation box.
- Neumann boundary conditions are also enforced at the remaining box interfaces by setting
at faces with constant x ,
at faces with constant y, and
at faces with constant z.
I did the implementation of the boundary conditions with this code and I would want to ascertain if
the implementation of the boundary condtions is right.
x1 = linspace(0,10,20);
y1 = linspace(0,10,20);
z1 = linspace(0,10,20);
V = zeros(length(x1),length(y1),length(z1));
%Dirichlet Boundary Conditions
%Top plane
V(:,end,end) = 0;
V(end,:,end) = 0;
V(end,end,:) = 0;
% Bottom plane
V(:,1,1) = 0;
V(1,:,1) = 0;
V(1,1,:) = 0;
%Incoporated electric potential
V(:,:,1) = 0;
V(:,:,end) = -40*z1(end);
%Neumann Boundary Condition
i = 2:length(x1)-1;
j = 2:length(y1)-1;
k = 2:length(z1)-1;
V(i+1,j,k) = V(i-1,j,k);
V(i,j+1,k) = V(i,j-1,k);
V(i,j,k+1) = V(i,j,k-1);
1 Commento
David Martín Oliva Zavala
il 12 Nov 2020
Hey Anthony, have you finished solving poisson eqn in 3D? I´m just starting this project and I may need some help in the future jeje:)
Risposta accettata
Dinesh Yadav
il 5 Mar 2020
Modificato: Dinesh Yadav
il 5 Mar 2020
Hi Anthony I think you have wrongly defined the top and bottom planes.A plane is a 2-D sheet structure however the line of code
V(:,end,end);
represent a single row vector, similar to a line and not a plane. As V is a 20x20x20 3-D matrix, lets assume the first plane facing us is top plane i.e (assuming x direction increases no of columns and y direction increases no of rows, i.e origin is at (20,1,1))
V(:,:,1); % top plane face with constant z
V(:,:,end); % bottom plane face with constant z
V(:,1,:); % face with constant x
V(:,end,:) % face with constant x
V(1,:,:); % face with constant y
V(end,:,:) % face with constant y
dx = 2:length(x1)-1;
dy = 2:length(y1)-1;
dz = 2:length(z1)-1;
V(dy,dx+1,dz) = V(dy,dx-1,dz); % dv/dx = 0
V(dy+1,dx,dz) = V(dy-1,dx,dz); % dv/dy = 0
V(dy,dx,dz+1) = V(dy,dx,dz-1); % dv/dz = 0
3 Commenti
Dinesh Yadav
il 5 Mar 2020
Yes, replace i with dy j with dx and k with dz. Just to make it more readable. I have corrected the above code.
Anthony Owusu-Mensah
il 5 Mar 2020
Modificato: Anthony Owusu-Mensah
il 5 Mar 2020
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