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All rights reserved. 32 AN INTRODUCTION TO THE FINITE ELEMENT METHOD and the solution uh so that the resulting expression would be symmetric in wi and uh . The integration-by-parts gives rise to two pairs of primary and secondary variables. We have 0= Z xb xa = " µ d duh wi (x) − a dx dx µ ¶ Z xb " dwi duh a dx xa ∙ dx µ duh + −wi · a dx = xa ( dx a " dx d2 + 2 dx à xa " = xa ( dx a " dx d2 uh b 2 dx à ! #)xb d2 uh b 2 dx d duh d2 uh + b 2 + wi · −a dx dx dx ! #)xb xa " dwi d2 uh ·b 2 + dx dx #xb (2b) xa From the boundary expressions of the last equation, we identify the primary and secondary variables.

D2 wi d2 uh +b 2 a + cwi uh − wi f dx dx dx dx dx2 dwi dwi (xa ) − Qb (xb ) − Pa wi (xa ) − Pb wi (xb ) − Qa dx dx (3) The primary variables include the dependent variable u and its derivative duh /dx. As a rule, the primary variables must be continuous across elements. Therefore, the finite element interpolation be such that both of the variables are treated as nodal variables so that the continuity conditions can be used during the assembly elements. Thus an element with two nodes (which is the minimum) will have four unknowns (u and du/dx at each of the two ends of the element), requiring a four-term polynomial - a cubic uh (x) = c1 + c2 x + c3 x2 + c4 x3 (4) The constants c1 through c4 can be expressed in terms of the nodal degrees of freedom µ uh (xa ) ≡ ∆1 , duh dx ¶ xa ≡ ∆2 , uh (xb ) ≡ ∆3 , µ duh dx ¶ xb ≡ ∆4 (5) Thus we will have uh (x) = c1 + c2 x + c3 x2 + c4 x3 = ∆1 φ1 (x) + ∆2 φ2 (x) + ∆3 φ3 (x) + ∆4 φ4 (x) = 4 X ∆j φj (x) (6) j=1 Note that ∆1 and ∆3 denote the values of the function u at the two nodes while ∆2 and ∆4 denote the values of derivative of u at the two nodes.

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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2nd Tac. Air Force [V. 2 - Breakout to Bodenplatte, Jul 1944 - Jan. 1945] by C. Shores, et. al.,

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