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 2 November 2, 2007 Professor Strang Your PRINTED name is: Grading 1 2 3 1) (40 pts.) This problem is based on a 5-node graph. I have not included edge numbers and arrows. Add them if you want to: not needed. (a) Find ATA for this graph. A is the incidence matrix. (b) The sum of the eigenvalues of ATA is . The product of those eigenvalues is . (c) What is ATA for a graph with only one edge ? How can that small ATA be used in constructing ATA for a large graph ? (d) Suppose I want to solve Au = ones(8, 1) = b by least squares. What equation gives a best u�? For the incidence matrix A, is there exactly one best u�solving that equation ? (If your equation has more than one best u�, describ e the difference between any two solutions.) 1 2 3 4 5 12) (30 pts.) (a) Suppose A is an m by n matrix of rank r (so it has r independent columns). How many independent solutions to Au = 0 and ATw = 0 ? (b) Draw a full set of mechanisms (solutions to e = Au = 0 with no stretching) for this truss with unit length bars and 45◦ angles. 1 2 3 H V H V(c) Suppose a mechanism has u1 = .01. What are u1 and u3 and u3 ? What is the actual new length of the bar between joints 1 and 3 ? 23) (30 pts.) This problem is about the equation −u��(x) + u(x) = 1 with u(0) = 0 and u(1) = 0 . (a) Multiply by a test function v(x). Find the weak form of the equation, after an integration by parts. (b) With h = Δx = 13 draw the admissible piecewise linear trial functions φ1(x), . . . , φn(x). What is n ? With test functions = trial functions, give a formula for the entry K12 in the finite element equation KU = F . (c) Find all the numbers in K and F .
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