Solve PDEs with Constant Boundary Conditions

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This example shows how to apply various constant boundary condition specifications for both scalar PDEs and systems of PDEs.

Geometry

All the specifications use the same 2-D geometry, which is a rectangle with a circular hole.

% Rectangle is code 3, 4 sides,
% followed by x-coordinates and then y-coordinates
R1 = [3,4,-1,1,1,-1,-.4,-.4,.4,.4]';
% Circle is code 1, center (.5,0), radius .2
C1 = [1,.5,0,.2]';
% Pad C1 with zeros to enable concatenation with R1
C1 = [C1;zeros(length(R1)-length(C1),1)];
geom = [R1,C1];

% Names for the two geometric objects
ns = (char('R1','C1'))';

% Set formula
sf = 'R1 - C1';

% Create geometry
g = decsg(geom,sf,ns);

% Create geometry model
model = createpde;

% Include the geometry in the model
% and view the geometry
geometryFromEdges(model,g);
pdegplot(model,EdgeLabels="on")
xlim([-1.1 1.1])
axis equal

Figure contains an axes object. The axes object contains 9 objects of type line, text.

Scalar Problem

Suppose that edge 3 has Dirichlet conditions with value 32, edge 1 has Dirichlet conditions with value 72, and all other edges have Neumann boundary conditions with q = 0, g = -1.

applyBoundaryCondition(model,"dirichlet", ...
                       Edge=3,u=32);
applyBoundaryCondition(model,"dirichlet", ...
                       Edge=1,u=72);
applyBoundaryCondition(model,"neumann", ...
                       Edge=[2,4:8],g=-1);

This completes the boundary condition specification.

Solve an elliptic PDE with these boundary conditions with c = 1, a = 0, and f = 10. Because the shorter rectangular side has length 0.8, to ensure that the mesh is not too coarse choose a maximum mesh size Hmax = 0.1.

specifyCoefficients(model,m=0,d=0,c=1,a=0,f=10);
generateMesh(model,Hmax=0.1);
results = solvepde(model);
u = results.NodalSolution;
pdeplot(model,XYData=u,ZData=u)
view(-23,8)

Figure contains an axes object. The axes object contains an object of type patch.

System of PDEs

Suppose that the system has N = 2.

Edge 3 has Dirichlet conditions with values [32,72].
Edge 1 has Dirichlet conditions with values [72,32].
Edge 4 has a Dirichlet condition for the first component with value 52, and has a Neumann condition for the second component with q = 0, g = -1.
Edge 2 has Neumann boundary conditions with q = [1,2;3,4] and g = [5,-6].
The circular edges (edges 5 through 8) have q = 0 and g = 0.

model = createpde(2);
geometryFromEdges(model,g);

applyBoundaryCondition(model,"dirichlet", ...
                       Edge=3,u=[32,72]);
applyBoundaryCondition(model,"dirichlet", ...
                       Edge=1,u=[72,32]);
applyBoundaryCondition(model,"mixed", ...
                       Edge=4,u=52, ...
                       EquationIndex=1,g=[0,-1]);
Q2 = [1,2;3,4];
G2 = [5,-6];
applyBoundaryCondition(model,"neumann", ...
                       Edge=2,q=Q2,g=G2);

% The next step is optional,
% because it sets "g" to its default value
applyBoundaryCondition(model,"neumann", ...
                       Edge=5:8,g=[0,0]);