Please see the attached picture. I have the x-displacements and y-displacements for node 13, 14 and 15. (I do not have force boundary condition). I want to apply the given nodal displacements as boundary conditions for the finite element and use it to compute the x and y displacements on each of all the other nodes (that is, node 1,2,3,4,5,6,7,8,9,10,11,12,16).
In general, I have non-zero displacements for certain nodes in finite elements. I want to compute the displacements on all the other nodes in the finite elements by applying the given displacements as the boundary condition. I understand that I can model this kind of problem in Finite element software like Abaqus. What I need either the mathematical formulation or MATLAB code that can solve this kind of problem. Thank you.
This can be found in any standard FEA book. In short the classroom procedure is:
In a similar way, to solve purely displacement control run the procedure doesn't change. In absence of external load, all dofs with unknown displacement will have zero force (equilibrium) in the global load vector. So follow steps 1 to 4 above.
Thank you so much, Abhimanyu Kumar. Please, can we correspond on email? My email address: [email protected]
In case you want to apply nonhomogeneous Dirichlet boundary conditions the stiffness matrix and the force vector have to be changed in the following way:
Force vector:
1) Multiply the column of the stiffness matrix that corresponds to a certain prescribed degree of freedom by the prescribed value.
2) The "force" vector obtained in the first step has to substracted from the original force vector of the system.
3) Exchange the value in the force vector that corresponds to the prescribed degree of freedom by the value of the main diagonal component of the stiffness matrix multiplied by the prescribed value.#
In Matlab syntax that would be:
F = F - K(:,CDoF)*PrePrimary;
Kdiag = diag(K(CDoF,CDoF));
F(CDoF) = Kdiag.*PrePrimary;
where F is the force vector, K is the stiffness matrix, CDoF s a vector containing the global number of the constraint degrees of freedom and PrePrimary is the prescribed value of the primary variable (here the prescribed displacement for a certain degree of freedom)
Stiffness matrix:
1) Save the components of the stiffness matrix that are on the main diagonal
2) Clear all rows and columns that are corresponding to the constraint degrees of freedom
3) Insert the original main diagonal component (that you saved in the first step)
In Matlab syntax that would be:
% Adjust the stiffness matrix
Kdiag = diag(K(CDoF,CDoF));
K(CDoF,:) = 0;
K(:,CDoF) = 0;
K(CDoF,CDoF) = diag(Kdiag);
where K ist the stiffness matrix and CDoF is a vector containing the global number of the constraint degrees of freedom.
After you have adjusted both the stiffness matrix and the force vector you can easily solve the linear system of equations using Gaussian elimination, Cholesky decompositon etc.
Hope this helps
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