1. Finite Element Method Monday, 11/11/2002 Equations assembling Displacement boundary conditions Gaussian elimination

2. Corrections to Homework (node coordinates) We have 6 nodes, their coordinates are coor=(0.1,0.1; 0,0.6; 0.6,0; 0.5,0.4; 1.2,0.2; 1.2,0.5);

3. Corrections to Homework (element area) In MATLAB with the above definite of elmt and coor, the area can be calculated with A=0.5*det([1, coor(elmt(e,1),1), coor(elmt(e,1),2); … 1, coor(elmt(e,2),1), coor(elmt(e,2),2); … 1, coor(elmt(e,3),1), coor(elmt(e,3),2)])

4. System Equations

5. System Equations (matrix form)

6. Assembled System Equations N: number of nodes

7. Symmetric Stiffness Matrix

8. Proof of Symmetric Stiffness Matrix

9. MATLAB form F = [T_x+F_x; T_y+F_y] K = [Kxx,Kxy; Kyx,Kyy] If T_x, F_x, T_y, and F_y are column vector

10. MATLAB Matrix Equation

11. Displacement Boundary Conditions To solve: F = K * u u = gaussElim(K,F) u(5) u(6) u(11) u(12)

12. Introducing Boundary Conditions

13. Keep Stiffness Matrix Symmetric % apply displacement boundary condition for node 5 % in y direction ibound = 5 + nNode; for i = 1:2*nNode if(i ~= ibound) K(i,ibound)=0; K(ibound,i)=0; end F(ibound)=0;

14. Gaussian Elimination MATLAB Tutorial, Tutorial on subroutines. gaussElim.m:

15. Tractions on Displacement Boundaries