MATLAB: How to use the boundary conditions of the “assembleFEMatrices” function in the own numerical integration using the Crank-Nicolson method

Partial Differential Equation Toolbox

How do I use the boundary conditions of the "assembleFEMatrices" function in my own numerical integration using the Crank-Nicolson method? I cannot used the PDE Toolbox's "solve" method.

Best Answer

There is no built-in function for you to accomplish this, and our "solve" method uses several numerical integration processes to solve the FEM problem.

In general, you will have to adjust the equations you are using to take into account the boundary conditions, which you current integration does not account for. Take a look at this link for an example solution that has both stiffness and boundary value matrices:

https://en.wikipedia.org/wiki/Crank%E2%80%93Nicolson_method

Related Solutions

MATLAB: How to specify boundary conditions on subdomain borders using the Partial Differential Equation (PDE) Toolbox 1.0.13 (R2008b)

One should be able to specify boundary conditions on all external boundaries of any geometry.

 For example, look at the “A scattering problem” example in the documentation. It has a Neumann condition at the outer boundary and a Dirichlet condition at the inner boundary (that is formed by the square sub-region).

Boundary conditions on internal boundaries such as boundaries across which material properties or governing equations might change are not required by Finite Element method.

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)

Best Answer

Related Solutions

MATLAB: How to specify boundary conditions on subdomain borders using the Partial Differential Equation (PDE) Toolbox 1.0.13 (R2008b)

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

Related Question