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 3, 2004 Professor Strang Your name is: Grading 1. 2. 3. 1) (36 pts.) The 5 nodes in the network are at the corners of a square and the center. Node 5 is grounded so x5 = 0. All 8 edges have conductances c =1 so C = I. current source ⎡⎤ ⎢⎥ 2 R - 6 1 4 1 3 7 ? ? 2 5- R 8 3 4 f1 =3 z ⎢⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥A8×4 = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ 5 (a) Fill in the 8 by 4 incidence matrix A (node 5 grounded). What is ATA? Is ATA invertible (YES,NO)? (b) How many independent solutions to ATy = 0? Write down one nonzero solution y. (c) The current source f1 = 3 enters node 1 and exits at grounded node 5. In 2 by 2 block form (using A), what are the 12 equations for the 8 currents y and t he 4 potentials x? (d) Write out in full with numbers the 4 equations for the 4 potentials, after the currents y are eliminated. Using symmetry (or guessing or solving) what is the solution x1,x2,x3,x4?x 22) (24 pts.) The same 8 edges and 5 nodes form a square pin-jointed truss. The pin at = xV5 = 0. All 8 elastic constants are c =1node 5 isheldinpositionso xso C = I. (a) How many unknown displacements? What is the shape of the matrix A in e = Ax? Find the first column of A, corresponding to the stretching e in the 8 H5 edges from a small displacement xat node 1. H1 (b) Are there any nonzero solutions to Ax = 0? (YES,NO) How many independent solutions do you physically expect? Draw a picture of each independent solution (if any) to show the move-ment of the 4 nodes. (c) How many independent solutions to ATy = 0? Can you find them? 3xx 4 3) (40 pts.) (a) F ind a 4th degree polynomial s(x, y) with only 2 terms that solves Laplace’s equation. Please draw a box around your answer s(x, y). (b) In the xy plane draw all the solutions to s(x, y)=0. Then in the same picture roughly draw the curve s(x, y)= c that goes through the particular point (x, y)= (2, 1). (c) If the curves s(x, y)= c are the streamlines of a potential flow (in the usual framework), what is the corresponding velocity v(x, y)= w(x, y)? (d) (this Green’s formula question is not related to parts a, b, c) Suppose w(x, y)= (w1(x, y), 0) is a flow field. With w2 =0 write down the remaining (not zero) terms in Green’s formula for the integral (grad u)·wdx dy in the unit square 0 ≤ x ≤ 1, 0 ≤ y ≤ 1. Substitute for n and ds when you know what they are for this square. (e) A one-dimensional formula on any horizontal line y = y0 is integration by parts:  1 du  1 dw1 w1(x) dx = − u(x) dx + uw1(x =1)− uw1(x =0). dxx=0 dx x=0 Here u and w1 are u(x, y0)and w1(x, y0)since y = y0 is fixed. Question 1 How do you derive your Green’s formula in part (d) from this one-dimensional formula? ANSWER IN ONE SENTENCE, NO MATH SYMBOLS !! Question 2 (not related) Find all vector fields of this form (w1(x, y), 0) that can be velocity fields v = w =(w1(x, y), 0) in potential flow [so v =grad u and div w = 0 as usual]. 5xxx 6xxxx