Solve PDEs with Nonconstant Boundary Conditions

This example shows how to write functions for a nonconstant boundary condition specification.

Geometry

All the specifications use the same 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

Scalar Problem

• Edge 3 has Dirichlet conditions with value 32.

• Edge 1 has Dirichlet conditions with value 72.

• Edges 2 and 4 have Dirichlet conditions that linearly interpolate between edges 1 and 3.

• The circular edges (5 through 8) have Neumann conditions with q = 0, g = -1.

applyBoundaryCondition(model,'dirichlet','Edge',3,'u',32);
applyBoundaryCondition(model,'dirichlet','Edge',1,'u',72);
applyBoundaryCondition(model,'neumann','Edge',5:8,'g',-1); % q = 0 by default

Edges 2 and 4 need functions that perform the linear interpolation. Each edge can use the same function that returns the value $$u(x,y)=52+20x$$.

You can implement this simple interpolation in an anonymous function.

myufunction = @(location,state)52 + 20*location.x;

Include the function for edges 2 and 4. To help speed the solver, allow a vectorized evaluation.

applyBoundaryCondition(model,'dirichlet','Edge',[2,4],...
                                         'u',myufunction,...
                                         'Vectorized','on');

Solve an elliptic PDE with these boundary conditions, using the parameters 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)

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].

• Edges 2 and 4 have Dirichlet conditions that interpolate between the conditions on edges 1 and 3, and include a sinusoidal variation.

• Circular edges (edges 5 through 8) have q = 0 and g = -10.

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

applyBoundaryCondition(model,'dirichlet','Edge',3,'u',[32,72]);
applyBoundaryCondition(model,'dirichlet','Edge',1,'u',[72,32]);
applyBoundaryCondition(model,'neumann','Edge',5:8,'g',[-10,-10]);

The first component of edges 2 and 4 satisfies the equation $$u_1\left(x\right)=52+20x+10\sin \left(\pi x^3\right)$$.

The second component satisfies $$u_2\left(x\right)=52-20x-10\sin \left(\pi x^3\right)$$.

Write a function file myufun.m that incorporates these equations in the syntax described in Nonconstant Boundary Conditions.

function bcMatrix = myufun(location,state)
bcMatrix = [52 + 20*location.x + 10*sin(pi*(location.x.^3));
    52 - 20*location.x - 10*sin(pi*(location.x.^3))]; % OK to vectorize
end

Include this function in the edge 2 and edge 4 boundary condition.

applyBoundaryCondition(model,'dirichlet','Edge',[2,4],...
                                         'u',@myufun,...
                                         'Vectorized','on');

Solve an elliptic PDE with these boundary conditions, with the parameters c = 1, a = 0, and f = (10,-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;-10]);
generateMesh(model,'Hmax',0.1);
results = solvepde(model);
u = results.NodalSolution;

subplot(1,2,1)
pdeplot(model,'XYData',u(:,1),'ZData',u(:,1),'ColorBar','off')
view(-9,24)
subplot(1,2,2)
pdeplot(model,'XYData',u(:,2),'ZData',u(:,2),'ColorBar','off')
view(-9,24)

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