MIT OpenCourseWare http://ocw.mit.edu 18.085 Computational Science and Engineering IFall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.18.085 Quiz 1 October 5, 2007 Professor Strang Your PRINTED name is: Grading 1 2 3 1) (39 pts.) With h = 1 1 2there are 4 meshpoints 0, , , 1 and displacements u0, u1, u2, u3. 3 3 3 a) Write down the matrices A0, A1, A2 with three rows that produce the first differences ui − ui−1: A0 has 0 boundary conditions on u A1 has 1 boundary condition u0 = 0 (left end fixed) A2 has 2 boundary conditions u0 = u3 = 0. b) Write down all three matrices AT0 A0, AT1 A1, AT2 A2. CROSS OUT IF FALSE / GIVE REASON BASED ON COLUMNS OF A ! K0 = AT0 A0 is (singular) (invertible) (positive definite) Reason: K1 = AT1 A1 is (singular) (invertible) (positive definite) Reason: c) Find all solutions w = (w1, w2, w3) to each of these equations: A0T w = 0 A1T w = 0 A2T w = 0 12) (33 pts.) a) Find the eigenvalues �1, �2, �3 and unit eigenvectors y1, y2, y3 of B. Hint: one eigenvector is (1, 0, −1)/�2. � ⎡ � 1 −1 0 ⎢ B = � � � −1 2 −1 ⎢ ⎢ ⎣ . 0 −1 1 b) Factor B into Q�QT with Q−1 = QT . Draw a graph of the energy function f(u1, u2, u3) = 21 uTBu. This is a surface in 4-dimensional u1, u2, u3, f space so your graph may not be perfect—OK to describe it in 1 sentence. c) What differential equation with what boundary conditions on y(x) at x = 0 and 1 is the continuous analog of By = �y ? What are the eigenfunctions y(x) and eigenvalues � in this differential equation ? At which x’s would you sample the first three eigenfunctions to get the three eigenvectors in part (a) ? 23) (28 pts.) The fixed-fixed figure shows n = 2 masses and m = 4 springs. Displacements u1, u2. m1 m2 c1 c2 c4 c3 a) Write down the stretching-displacement matrix A in e = Au. b) What is the stiffness matrix K = ATCA for this system ? c) Theory question about any ATCA. C is symmetric positive definite. What condition on A assures that uTATCAu > 0 for every vector u = 0 ? Explain why this is greater than zero and where you → use your condition on A.
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