MATLAB: How to set time-varying boundary conditions using the Partial Differential Equation Toolbox

Partial Differential Equation Toolbox

I want to solve a partial differential equation (PDE) that has a time-varying boundary condition using the Partial Differential Equation Toolbox.

Best Answer

You can use the Partial Differential Equation Toolbox to solve PDEs with time-varying boundary conditions. The following example describes how to define these boundary conditions :

1. Open the PDETOOL GUI by enetering

pdetool

at the MATLAB command prompt.

2. From the PDE menu, select PDE Specifications. In the dialog box, click Hyperbolic. Then click OK to close the dialog box.

3. From the Boundary menu, select Boundary Mode.

4. Double-click one of the boundaries in the figure.

5. In the Boundary Condition dialog box, enter values for the boundary condition parameters. The default values are constants. You can enter any function of time for these parameters, e.g. "sin(t)". You can also enter user-defined functions, e.g. "myBoundary(t)". Note that the function argument must be named "t". Any function you use must

a. Return a scalar for each value of "t" passed to it.
b. Return NaN for any "t" value of NaN.

An example of such a function is shown below:

function f = myFunc(t)
if isnan(t)
      f=NaN;
else
      f = t;
end

6. From the Solve menu, select Solve PDE.

7. You can visualize the time-varying boundary condition as follows:

a. From the Plot menu, select Parameters.
b. Select the Animation checkbox, and click Plot.

You can make the boundary condition a function of space as well as time by specifying a boundary condition such as "myFunc2(t,x,y)" in the Boundary Condition dialog box. The function "myFunc2" must

a. Return NaN for a "t" value of NaN.
b. Return a vector the same size as "x" or "y" for every numerical value of "t".

An example of such a function is shown below:

function f = myFunc2(t, x, y)
if isnan(t)
      f=NaN;
else
      f = (x+y) * t;
end

Related Solutions

MATLAB: Does the Partial Differential Equation Toolbox provide the functionality of “assembleFEMatrices” for the new “modal-solid” workflow introduced in R2018a

For the vertical workflow using "createpde('structural','modal-solid')", the Partial Differential Equation (PDE) Toolbox does not provide the functionality of "assembleFEMatrices". As an alternative, you could solve the 'modal-solid' problem by setting up a PDE model with "createpde()", where you can get access to the matrices with all the checks and internal validatation in place. You can think of PDE model workflow as a super workflow which can also solve ‘modal-solid’ systems. However, the setup is different because of the generality of the approach.

This example shows you how to compute the modes of a cantilever beam using the PDEModel workflow. In the end, you get access to the matrices and can you use them for further calculations, e.g. finding the Frequency Response Function.

% Create PDEModel to solve for 3 PDEs of equilibrium for a solid body
PDEmodel = createpde(3);
% Create a cantilever beam geometry
PDEmodel.Geometry = multicuboid(0.45, 0.02, 0.003);
% Plot geometry
figure(1)
    pdegplot(PDEmodel, 'FaceLabels', 'on'); 
    title('Geometry');
    % NOTE: the beam will be fixed on the left face ([5])
% Generate mesh
PDEmodel.generateMesh;
% Plot mesh
figure(2)
    pdemesh(PDEmodel);
    title('Mesh');
% Set material properties (steel)
E = 210E9;
nu = 0.3;
rho = 7800;
% Compute c-coefficients in the required form using material properties
c = elasticityC3D(E,nu);
PDEmodel.specifyCoefficients('m', rho, 'd', 0, 'c', c, 'a', 0, 'f', [0;0;0]);
% Apply boundary conditions:
% % B.C. 1: fix beam on left face
PDEmodel.applyBoundaryCondition('dirichlet', 'Face', 5, 'u', [0;0;0]);
% % B.C. 2: one face is applied with a traction
%applyBoundaryCondition(model,'neumann','Face',3,'g',[0;0;1000]); 
% NOTE: line is commented out, as this BC is ineffective as loads are not 
%       accounted for in modal analysis.
% Solve for modes and frequency
modalSol = PDEmodel.solvepdeeig([0,3E5]);
% Plot mode shapes
figure(3)
subplot(2,1,1)
    pdeplot3D(PDEmodel, 'ColorMapData', modalSol.Eigenvectors(:,3,1))
    title(['Z-Disp, Mode = 1, frequency = ' num2str(sqrt(modalSol.Eigenvalues(1))/2/pi) ' Hz'])
subplot(2,1,2)
    pdeplot3D(PDEmodel, 'ColorMapData', modalSol.Eigenvectors(:,3,2))
    title(['Z-Disp, Mode = 2, frequency = ' num2str(sqrt(modalSol.Eigenvalues(2))/2/pi) ' Hz'])
% Get access to the matrices used in modal solution
FEmodel = assembleFEMatrices(PDEmodel)

MATLAB: How to apply Dirichlet boundary condition to a part of a face

Please find the example script, "partialDirichletBCExample.m" attached with this email that demonstrates how to apply Dirichlet boundary condition only on a portion of a face.. In the attached example, we are solving two problems with two PDE models as follows:

1. Simulate the requirement of applying Dirichlet boundary condition to a part of a face. In the PDE model, "model1", we apply y-displacement on half of the face numbered 6, using functional form of the boundary condition specification.

2. Verify the simulated model. In the PDE model, "model2", we create the geometry such that there are two faces in place of face 6 in "model1". Here, we apply mixed boundary conditions on face 11, which is equivalent to applying Dirichlet boundary condition on half of the face numbered 6 on "model1".

As you can observe from the results plots, both these models give the same results except near the half way partition. Here "model1" has an imperfect face, unlike "model2", where the geometry has been created such that there are two faces in place of face 6 of "model1". Similarly, you can use this approach to apply Dirichlet boundary condition only on a portion of a face to your PDE model.

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Related Solutions

MATLAB: Does the Partial Differential Equation Toolbox provide the functionality of “assembleFEMatrices” for the new “modal-solid” workflow introduced in R2018a

MATLAB: How to apply Dirichlet boundary condition to a part of a face

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