applyBoundaryCondition

R2026b

Add boundary condition to PDEModel container

Syntax

applyBoundaryCondition(model,"dirichlet",RegionType=RegionID,Name=Value)

applyBoundaryCondition(model,"neumann",RegionType=RegionID,Name=Value)

applyBoundaryCondition(model,"mixed",RegionType=RegionID,Name=Value)

bc = applyBoundaryCondition(___)

Description

applyBoundaryCondition(model,"dirichlet",RegionType=RegionID,Name=Value) adds a Dirichlet boundary condition to model. The boundary condition applies to boundary regions of type RegionType with ID numbers in RegionID, and with arguments r, h, u, EquationIndex specified in the Name=Value arguments. For Dirichlet boundary conditions, specify either both arguments r and h, or the argument u. When specifying u, you can also use EquationIndex.

applyBoundaryCondition(model,"neumann",RegionType=RegionID,Name=Value) adds a Neumann boundary condition to model. The boundary condition applies to boundary regions of type RegionType with ID numbers in RegionID, and with values g and q specified in the Name=Value arguments.

applyBoundaryCondition(model,"mixed",RegionType=RegionID,Name=Value) adds an individual boundary condition for each equation in a system of PDEs. The boundary condition applies to boundary regions of type RegionType with ID numbers in RegionID, and with values specified in the Name,Value pairs. For mixed boundary conditions, you can use Name=Value arguments from both Dirichlet and Neumann boundary conditions as needed.

bc = applyBoundaryCondition(___) returns the boundary condition object.

Examples

Dirichlet Boundary Conditions

Create a PDE model and geometry.

model = createpde;
R1 = [3,4,-1,1,1,-1,-.4,-.4,.4,.4]';
g = decsg(R1);
geometryFromEdges(model,g);

View the edge labels.

pdegplot(model,EdgeLabels="on")
xlim([-1.2,1.2])
axis equal

Apply zero Dirichlet condition on the edge 1.

applyBoundaryCondition(model,"dirichlet",Edge=1,u=0);

On other edges, apply Dirichlet condition h*u = r, where h = 1 and r = 1.

applyBoundaryCondition(model,"dirichlet",Edge=2:4,r=1,h=1);

Neumann Boundary Conditions

Create a PDE model and geometry.

model = createpde(2);
R1 = [3,4,-1,1,1,-1,-.4,-.4,.4,.4]';
g = decsg(R1);
geometryFromEdges(model,g);

View the edge labels.

pdegplot(model,EdgeLabels="on")
xlim([-1.2,1.2])
axis equal

Apply the following Neumann boundary conditions on the edge 4.

applyBoundaryCondition(model,"neumann",Edge=4,g=[0;.123],q=[0;0;0;0]);

Dirichlet and Neumann Boundary Conditions for Different Boundaries

Apply both types of boundary conditions to a scalar problem. First, create a PDE model and import a simple block geometry.

model = createpde;
importGeometry(model,"Block.stl");

View the face labels.

pdegplot(model,FaceLabels="on",FaceAlpha=0.5)

Set zero Dirichlet conditions on the narrow faces, which are labeled 1 through 4.

applyBoundaryCondition(model,"dirichlet",Face=1:4,u=0);

Set Neumann boundary conditions with opposite signs on faces 5 and 6.

applyBoundaryCondition(model,"neumann",Face=5,g=1);
applyBoundaryCondition(model,"neumann",Face=6,g=-1);

Solve an elliptic PDE with these boundary conditions, and plot the result.

specifyCoefficients(model,m=0,d=0,c=1,a=0,f=0);
generateMesh(model);
results = solvepde(model);
u = results.NodalSolution;
pdeplot3D(model,ColorMapData=u)

Individual Boundary Conditions for Equations in a System

Create a PDE model and import a simple block geometry.

model = createpde(3);
importGeometry(model,"Block.stl");

View the face labels.

pdegplot(model,FaceLabels="on",FaceAlpha=0.5)

Set zero Dirichlet conditions on faces 1 and 2.

applyBoundaryCondition(model,"dirichlet",Face=1:2,u=[0,0,0]);

Set Neumann boundary conditions with opposite signs on faces 4, 5, and 6.

applyBoundaryCondition(model,"neumann",Face=4:5,g=[1;1;1]);
applyBoundaryCondition(model,"neumann",Face=6,g=[-1;-1;-1]);

For face 3, apply generalized Neumann boundary condition for the first equation and Dirichlet boundary conditions for the second and third equations.

applyBoundaryCondition(model,"mixed",Face=3, ...
                       h=[0 0 0;0 1 0;0 0 1], ...
                       r=[0;3;3], ...
                       g=[10;0;0], ...
                       q=[1 0 0;0 0 0;0 0 0]);

Solve an elliptic PDE with these boundary conditions, and plot the result.

specifyCoefficients(model,m=0,d=0,c=1,a=0,f=[0;0;0]);
generateMesh(model);
results = solvepde(model);
u = results.NodalSolution;
pdeplot3D(model,ColorMapData=u(:,1))

Boundary Conditions on Internal Edges

There are two types of edges in 2-D geometries:

• External boundary edges. These edges separate the geometry from the rest of the 2-D space.

• Internal boundary edges. These edges separate faces of the geometry.

By default, boundary conditions, either Dirichlet or generalized Neumann, apply only to external boundary edges. For example, look at a rectangular region with a circular subdomain. The red numbers are the subdomain labels, the black numbers are the edge segment labels.

% 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 model and geometry
model = createpde;
gd = decsg(geom,sf,ns);
geometryFromEdges(model,gd);

% View geometry
pdegplot(model,EdgeLabels="on",FaceLabels="on")
xlim([-1.1 1.1])
axis equal

The applyBoundaryCondition function does not set boundary conditions on the edges forming the circle because these are internal edges. To apply boundary conditions on internal edges, use the InternalBC argument.

specifyCoefficients(model,m=0,d=0,c=1,a=0,f=0);
applyBoundaryCondition(model,"dirichlet", ...
                       Edge=1:4,u=0, ...
                       InternalBC=true);
applyBoundaryCondition(model,"dirichlet", ...
                       Edge=5:8,u=10, ...
                       InternalBC=true);
generateMesh(model);
R = solvepde(model);
pdeplot(model,XYData=R.NodalSolution)
axis equal

Input Arguments

model — PDE model

PDEModel object

PDE model, specified as a PDEModel object.

Example: model = createpde

RegionType — Geometric region type

Face for 3-D geometry | Edge for 2-D geometry

Geometric region type, specified as Face for 3-D geometry or Edge for 2-D geometry.

Example: applyBoundaryCondition(model,"dirichlet",Face=3,u=0)

RegionID — Geometric region ID

vector of positive integers

Geometric region ID, specified as a vector of positive integers. Find the region IDs using pdegplot with the FaceLabels (3-D) or EdgeLabels (2-D) value set to "on".

Example: applyBoundaryCondition(model,"dirichlet",Face=3:6,u=0)

Data Types: double

Name-Value Arguments

Example: applyBoundaryCondition(model,"dirichlet",Face=1:4,u=0)

r — Dirichlet condition h*u = r

zeros(N,1) (default) | vector with N elements | function handle

Dirichlet condition h*u = r, specified as a vector with N elements or a function handle. N is the number of PDEs in the system. For the syntax of the function handle form of r, see Nonconstant Boundary Conditions.

Example: r=[0;4;-1]

Data Types: double | function_handle
Complex Number Support: Yes

h — Dirichlet condition h*u = r

eye(N) (default) | N-by-N matrix | vector with N^2 elements | function handle

Dirichlet condition h*u = r, specified as an N-by-N matrix, a vector with N^2 elements, or a function handle. N is the number of PDEs in the system. For the syntax of the function handle form of h, see Nonconstant Boundary Conditions.

Example: h=[2,1;1,2]

Data Types: double | function_handle
Complex Number Support: Yes

g — Generalized Neumann condition n·(c×∇u) + qu = g

zeros(N,1) (default) | vector with N elements | function handle

Generalized Neumann condition n·(c×∇u) + qu = g, specified as a vector with N elements or a function handle. N is the number of PDEs in the system. For scalar PDEs, the generalized Neumann condition is n·(c∇u) + qu = g. For the syntax of the function handle form of g, see Nonconstant Boundary Conditions.

Example: g=[3;2;-1]

Data Types: double | function_handle
Complex Number Support: Yes

q — Generalized Neumann condition n·(c×∇u) + qu = g

zeros(N) (default) | N-by-N matrix | vector with N^2 elements | function handle

Generalized Neumann condition n·(c×∇u) + qu = g, specified as an N-by-N matrix, a vector with N^2 elements, or a function handle. N is the number of PDEs in the system. For the syntax of the function handle form of q, see Nonconstant Boundary Conditions.

Example: q=eye(3)

Data Types: double | function_handle
Complex Number Support: Yes

u — Dirichlet conditions

zeros(N,1) (default) | vector of up to N elements | function handle

Dirichlet conditions, specified as a vector of up to N elements or as a function handle. If u has less than N elements, then you must also use EquationIndex. The u and EquationIndex arguments must have the same length. If u has N elements, then specifying EquationIndex is optional.

For the syntax of the function handle form of u, see Nonconstant Boundary Conditions.

Example: applyBoundaryCondition(model,"dirichlet",Face=[2,4,11],u=0)

Data Types: double
Complex Number Support: Yes

EquationIndex — Index of the known u components

1:N (default) | vector of integers with entries from 1 to N

Index of the known u components, specified as a vector of integers with entries from 1 to N. EquationIndex and u must have the same length.

When using EquationIndex to specify Dirichlet boundary conditions for a subset of components, use the mixed argument instead of dirichlet. The remaining components satisfy the default Neumann boundary condition with the zero values for g and q.

Example: applyBoundaryCondition(model,"mixed",Face=[2,4,11],u=[3,-1],EquationIndex=[2,3])

Data Types: double

Vectorized — Vectorized function evaluation

"off" (default) | "on"

Vectorized function evaluation, specified as "on" or "off". This evaluation applies when you pass a function handle as an argument. To save time in function handle evaluation, specify "on", assuming that your function handle computes in a vectorized fashion. See Vectorization. For details of this evaluation, see Nonconstant Boundary Conditions.

Example: applyBoundaryCondition(model,"dirichlet",Face=[2,4,11],u=@ucalculator,Vectorized="on")

Data Types: char | string

InternalBC — Apply boundary conditions on internal edges in 2-D geometries

false (default) | true

Apply boundary conditions on internal edges in 2-D geometries for all models, specified as true or false.

There are two types of edges in 2-D geometries:

• External boundary edges. These edges separate the geometry from the rest of the 2-D space.

• Internal boundary edges. These edges separate faces of the geometry.

By default, boundary conditions, either Dirichlet or generalized Neumann, apply only to external boundary edges.

Data Types: logical

Output Arguments

bc — Boundary condition

BoundaryCondition object

Boundary condition, returned as a BoundaryCondition Properties object. The model object contains a vector of BoundaryCondition objects. bc is the last element of this vector.

Tips

• When there are multiple boundary condition assignments to the same geometric region, the toolbox uses the last applied setting.

• To avoid assigning boundary conditions to a wrong region, ensure that you are using the correct geometric region IDs by plotting and visually inspecting the geometry.

• If you do not specify a boundary condition for an edge or face, the default is the Neumann boundary condition with the zero values for g and q.

Version History

Introduced in R2015a

R2025a: Apply boundary conditions on internal edges for 2-D geometries by using InternalBC

In previous releases boundary conditions for PDEmodel applied only to external boundary edges. So the toolbox ignored boundary conditions on internal edges.

Apply boundary conditions on internal edges by calling applyBoundaryCondition with the InternalBC argument set to true.