Question

(40 pts) 2a. Show that u(z) is the solution to the problem where k(x)-1 for x 1 /2. 2b. Set up the weak form for the differential equation above and the resulting element stiffness and element load vector and calculate the element stiffness matrix and load vector for 4 quadratic elements by using the Gaussian quadrature that is going to exactly calculate the integrals Then set up the global K and F vector and calculate the nodal values. Do they coincide with the analytical solution? Comment on the accuracy of the FEM solution with respect to the nodal values and the nature of the analytical solution. What is the mean square error for the specific quadratic basis? (40 pts) 2a. Show that 仔ー爷), 0S1-1/2. a(z) = is the solution to the problem u(0)-u(1) = 0. where k(x) = 1 for r 1 /2. 2b. Set up the weak form for the differential equation above and the resulting element stiffness and element load vector and calculate the element stiffness matrix and load vector for 4 quadratic elements by using the Gaussian quadrature that is going to exactly calculate the integrals. Then set up the global K and F vector and calculate the nodal values. Do they coincide with the analytical solution? Comment on the accuracy of the FEM solution with respect to the nodal values and the nature of the analytical solution. What is the mean square error for the specific quadratic basis ? is the solution to the problem where k(z) 1 for z1/2. ting element stiffness a element load vector and calculate the element stiffness matrix and load vector for 4 quadratic elements by using the Gaussian quadrature that is going to exactly calculate the integrals. Then set up the global K and F vector and calculate the nodal values. Do they coincide with the analytical solution? Comment on the accuracy of the FEM solution with respect to the nodal values and the nature of the analytical solution. What is the mean square error for the specific quadratic basis?

Answer 1

는) du 乙

1 I 2. 2 2. 2. t, e S 2 (2 2. 다 _