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All rights reserved. 38 AN INTRODUCTION TO THE FINITE ELEMENT METHOD 11 , K 12 , K 21 , K 22 , F 1 , and F 2 in terms of the (a) Define the coefficients Kij ij ij ij i i interpolation functions, known data, and secondary variables, and (b) comment on the choice of the interpolation functions (what type, Lagrange or Hermite, and why). Solution: The weighted-residual statements of Eqs. (1) are 0= Z xb xa v1 Ã d2 w0 M − 2 − dx EI ! dx, 0= Z xb xa v2 Ã ! d2 M − 2 − q dx dx (4) where (v1 , v2 ) are the weight functions.

5 cm 3 4 • L = 55 m D = 5 cm 4 L = 60 m D = 8 cm 5 • • L = 70 m D = 5 cm 5 6• 6 Fig. 6 Solution: The assembled equations are ⎡ 1 R1 ⎢− 1 ⎢ R1 ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ ⎣ 0 0 1 R1 − R11 + R12 + − R12 − R13 0 0 1 R3 0 − R12 1 1 R2 + R4 0 − R14 0 0 − R13 0 1 1 R3 + R5 − R15 0 = PROPRIETARY MATERIAL. 1 R4 0 0 − R14 − R15 + R15 + − R16 ⎧ ⎫ 5 × 10−4 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ 0 ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎩ ⎭ −4 5 × 10 c The McGraw-Hill Companies, Inc. ° 1 R6 0 0 0 0 − R16 1 R6 ⎤ ⎧ ⎫ P1 ⎪ ⎪ ⎥⎪ ⎪ ⎪ P2 ⎪ ⎥⎪ ⎪ ⎪ ⎥⎪ ⎪ ⎪ ⎥ ⎨ P3 ⎬ ⎥ ⎥ ⎪ P4 ⎪ ⎥⎪ ⎪ ⎪ ⎪ ⎥⎪ ⎪ ⎪ ⎪ ⎦⎪ ⎩ P5 ⎪ ⎭ P6 All rights reserved.

All rights reserved. 13: Solve the problem described by the following equations − d2 u = cos πx, 0 < x < 1; dx2 u(0) = 0, u(1) = 0 Use the uniform mesh of three linear elements to solve the problem and compare against the exact solution u(x) = 1 (cos πx + 2x − 1) π2 Solution: The main part of the problem is to compute the source vector for an element. We have fie = Z xb x cos πx ψie dx Z xa b µ ¶ xb − x = cos πx dx he xb ∙ µ ¶¸xb 1 xb 1 x sin πx − = cos πx + sin πx he π π2 π xa 1 1 = − sin πxa − (cos πxb − cos πxa ) π he π 2 µ ¶ Z xb x − xa cos πx dx f2e = he xa 1 1 (cos πxb − cos πxa ) + sin πxb = 2 he π π f1e The element equations are ∙ 3 −3 −3 3 ¸½ e ¾ u 1 ue2 = ½ e¾ f 1 f2e + ½ Qe1 Qe2 ¾ with the element source terms are given as follows.

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