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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 18.085 Computational Science and Engineering I
Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.085 Quiz 2 November 3, 2006 Professor Strang
Grading 1 2 3 4 Your PRINTED name is: SOLUTIONS 1) (25 pts.) This network (square grid) has 12 edges and 9 nodes. 1 4 7 2 5 3 6 9 3/4 1/2 1/4 u2 = 1 1/2 3/4 1/2 1/4 8 u8 = 0 (a) Do not write the incidence matrix ! Do not give me a MATLAB code ! Just tell me: (1) How many indep endent columns in A ? 8 12 − 8 = 4 (2) How many indep endent solutions to AT y = 0 ? (b) Suppose the node 2 has voltage u2 = 1, and node 8 has voltage u8 = 0 (ground). All edges have the same conductance c. On the second picture write all of the other voltages u1 to u9 . Check equation 5 of AT Au = 0 (at the middle node). �� 1 1 1 − − +4 −1−0=0 2 2 2 (3) What is row 5 (coming from node 5) of AT A ? � ⎦ I do want the whole of row 5. 0 −1 0 −1 4 −1 0 −1 0 1 2) (25 pts.) Suppose that square grid becomes a plane truss (usual pin joints at the 9 H V H V nodes). Nodes 2 and 8 now have supports so u = u = u = u = 0. 2 2 8 8 edge 3 1 4 5 2 Any numbering of edges (a) Think about the straindisplacement matrix A. Are there any mecha nisms that solve Au = 0 ? If there are, tell me carefully how many and draw a complete set. I believe 3. (b) Suppose now that all 8 of the outside nodes are ﬁxed. Only node 5 is free to move. There are forces f
H and f
V on that node. The bars 5 5 connected to it (North East South West) have constants cN cE cS cW . What is the (reduced) matrix A for this truss on the right ? What is
V the reduced matrix AT CA? What are the displacements uH and u ? 5 5 � ⎢ � ⎢ �0 1 ⎢ (any row ⎢ A=� � ⎢ � −1 0 ⎢ order is OK) � ⎣ 0 −1 � 1 0 ⎡ AT CA = � � cE + c W 0 0 cN + cS ⎡ ⎣ H u =
f
H /(cE + cW ) 5 5 uV =
f
V /(cN + cS ) 5 5 For 1 p oint, is that truss (ﬁxed at 8 nodes) statically determinate or indeterminate ? indeterminate 2 3) (25 pts.) This question is about the velocity ﬁeld v (x, y ) = (0, x) = w(x, y ). (a) Check that div w = 0 and ﬁnd a stream function s(x, y ). Draw the streamlines in the xy plane and show some velocity vectors. (b) Is this shear ﬂow a gradient ﬁeld (v = grad u) or is there rotation ? If you b elieve u exists, ﬁnd it. If you believe there is rotation, explain how this is p ossible with the streamlines you drew in part (a). Solution. (a) div(0, x) = 0 + 0 = 0 0= �s �y x=− �s �x 1 s = − x2 (+C ) 2 x = constant are vertical streamlines. � � � � � � � � � � � � (b) Not a gradient ﬁeld b ecause � v1 � v2 =0 = 1. �y �x � v2 � v1 Possible explanation for vorticity (rotation) − = 1. �x �y Right side going faster than left side produces rotation. 3 4) (25 pts.) Suppose I use linear ﬁnite elements (hat functions �(x) = trial functions V (x)). The equation has c(x) = 1 + x and a point load: Fixedfree � � � � d du 1 − (1 + x) =� x− dx dx 2 with u(0) = 0 and u � (1) = 0 . Take h = 1/3 with two hats and a halfhat as in the notes. (a) On the middle interval from 1/3 to 2/3, U (x) goes linearly from U1 to U2 . Compute � 2/3 ⎥ ⎤2 c(x) U � (x) dx and
1/3 � 2/3 1/3 � 1 � x− U (x) dx . 2 � Write those answers as � ⎡� ⎡ � � 4.5 −4.5 U ⎣� 1⎣ U1 U2 � −4.5 4.5 U2 “element load vector.” and � U1 U2 � �⎡
1 1 2 � 2 ⎣. You have found the 2 by 2 “element stiﬀness matrix” and the 2 by 1 (b) On the ﬁrst and third intervals, similar integrations give � �� �� � � �� � and U1 0 ; U1 3.5 U1 � ⎡� ⎡ �⎡ � � � �0 5.5 −5.5 U2 ⎣ � ⎣ and U2 U3 � U2 U3 � ⎣ . 0 −5.5 5.5 U3 Assuming your calculations and mine are correct, what would be the overall ﬁnite element equation K U = F ? (Not to solve) 4 Solution. (a) �
2/3 1/3 U2 − U 1 (1 + x) 1/3 � 2 �2/3 (1 + x)2 � � 9(U2 − U1 )2 dx = � 2 1/3 = 4 ( 5 )2 − ( 3 )2 3 9(U2 − U1 )2 2 � ⎡� ⎡ � � 4.5 −4.5 U ⎣� 1⎣ = U1 U2 � −4.5 4.5 U2 � 2/3 1/3 1 � x− 2 � � �� � 1 u(x) dx = U (NOT U (x) dx !!) 2 1 = (U1 + U2 ) (halfway up) 2 �⎡ � �1 = U1 U2 � 2 ⎣
1 2 (b) � 1/3 0 U1 (1 + x) 1/3 � � �2 �1/3 � (1 + x)2 2� 2 = 9 U1 � = 3.5 U1 2 0 0 ⎡ �2⎢ �⎢ F =�1⎢ �2⎣ 0 �⎡
1 8 −4.5 � ⎢ � ⎢ K = � −4.5 10 −5.5 ⎢ � ⎣ 0 −5.5 5.5 5 ...
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This note was uploaded on 04/12/2011 for the course MATH 18.085 taught by Professor Staff during the Fall '10 term at MIT.
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