In the PDE toolbox, can I set my Dirichlet boundary condition to a single point rather than the edge?

In the PDEtoolbox, can I set my Dirichlet boundary condition to a single point rather than the edge?

We use the following code (a really small circle) to approximate the Dirichlet boundary condition for a single point (u=220 when x=0 and y=0). I would like to know if there is a method to directly define boundary conditions on a Point instead of an Edge.

Circle = [1,0,0,0.001]';

applyBoundaryCondition(GWModel,'dirichlet','Edge',[5,6,7,8],'u',220);

%Create FE analysis object
GWModel = createpde();

%Create 2-D geometry
%R = [3,4,200,200,-200,-200,100,-100,-100,100]';
R = [2,4,200,200,-200,-200,100,-100,-100,100]';
Circle = [1,0,0,0.001]';
Circle = [Circle;zeros(length(R) - length(Circle),1)];
gd = [R,Circle];
ns = char('RectangularDomain','Well'); ns = ns';
sf = 'RectangularDomain-Well';
[g,bt] = decsg(gd,sf,ns);
geometryFromEdges(GWModel,g); 

figure(1);pdegplot(GWModel,'EdgeLabels','on');xlim([-250,250]), ylim([-150,150]);axis equal

specifyCoefficients(GWModel, 'm',0,'d',0,'c',1,'a',0,'f',0);
%Assume Boundary conditions
applyBoundaryCondition(GWModel,'dirichlet','Edge',[5,6,7,8],'u',220);
applyBoundaryCondition(GWModel,'dirichlet','Edge',[1,2,3,4], 'u',200);
%applyBoundaryCondition(GWModel,'neumann', 'Edge',[1,4], 'q',0,'g',0);
%Create a mesh, max size
hmax = 1.25;
generateMesh(GWModel,'Hmax',hmax);
GWModel.Mesh

ans = 
  FEMesh with properties:

             Nodes: [2×233668 double]
          Elements: [6×116352 double]
    MaxElementSize: 1.2500
    MinElementSize: 0.6250
     MeshGradation: 1.5000
    GeometricOrder: 'quadratic'

figure(2);pdemesh(GWModel);xlim([-250,250]), ylim([-150,150]);axis equal;

%Solve PDE
results = solvepde(GWModel);
u = results.NodalSolution;

%Compute the flux of the solution and plot the results.
[cgradx,cgrady] = evaluateCGradient(results);

fluxx = -cgradx;
fluxy = -cgrady;

%Plot solution
figure(3)
pdeplot(GWModel,'XYData',u,'Contour','on','FlowData',[fluxx,fluxy])
axis equal
title('Numerical Solution');xlabel('x');ylabel('y')
xlim([-200,200]), ylim([-100,100])

Answer 1

[Avni Agrawal]

Hi,

I understand that you are trying to apply Dirichlet boundary conditions to single point instead of edge. But, in PDE Toolbox of MATLAB, you typically apply boundary conditions on the boundaries or edges of the geometry, not at individual points. This is because the solutions to PDEs are generally defined over a continuous space, and the value at a single point doesn't usually affect the solution across the rest of the domain unless that point is at the boundary.

If you need to enforce a condition at a single point within the domain, you might need to reconsider your model or explore using a different numerical method that allows for point-specific constraints.

The PDE Toolbox in MATLAB doesn't support setting Dirichlet boundary conditions at a single point within the domain. It supports Edges and Face only.

Take a look at this documentation for better understanding:

https://www.mathworks.com/help/pde/ug/pde.pdemodel.applyboundarycondition.html?s_tid=doc_ta#bupbdpw-2