By Irikura K.K., Frurip D.J.

Computational Thermochemistry is the 1st publication to hide this subject, and it combines available introductory fabric with cutting-edge advances. the quantity comprises chapters on response charges for gas-phase reactions, solvation versions, and phase-change enthalpies. The options span empirical estimation throughout the highest-level ab initio tools, and the appendices supply important info on present databases and software program, besides a thesaurus and various labored examples.

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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.

E. condensed equations) for the unknown voltages and currents. 3 R = 30 Ω 2 R = 35 Ω 1 V1= 10 volts R = 7 R=5Ω Ω 4 R = 15 Ω R = 10 Ω 6 V6= 200 volts 5 R=5Ω Fig. 3 for the direct current electric network shown in Fig. 4. PROPRIETARY MATERIAL. c The McGraw-Hill Companies, Inc. ° All rights reserved. 52 AN INTRODUCTION TO THE FINITE ELEMENT METHOD R=5Ω 3 R = 20 Ω 6 R=0Ω 8 R = 10 Ω R=5Ω 2 R=5Ω 5 R = 15 Ω R = 10 Ω 1 R = 20 Ω 4 R = 50 Ω V1= 110 volts 7 V7 = 40 volts Fig. 4 Solution: The assembled coeﬃcient matrix is ⎡ 1 5 + 1 20 ⎢ −1 ⎢ 5 ⎢ ⎢ 0 ⎢ 1 [K] = ⎢ ⎢ − 20 ⎢ 0 ⎢ ⎢ ⎣ 0 1 5 0 − 15 + 15 + 1 − 20 0 − 15 0 0 1 − 20 0 0 1 1 + 20 10 + 1 − 10 0 1 − 50 ⎤ 0 ⎥ 0 ⎥ ⎥ 0 ⎥ 1 − 50 ⎥ ⎥ ⎥ ⎥ 0 ⎥ 1 − 15 ⎦ 1 1 15 + 50 0 1 − 20 1 1 20 + 5 0 0 − 15 0 0 0 − 15 0 1 − 10 1 1 + 10 + 15 1 − 15 1 20 1 5 1 50 1 5 0 − 15 0 1 − 10 1 + 10 + 1 − 10 0 1 10 The condensed equations are ⎡ 9 20 ⎢− 1 ⎢ 20 ⎢ ⎢ 0 ⎢ 1 ⎣− 5 0 1 − 20 0 0 0 0 − 15 17 100 1 − 10 1 4 I1 = 0 − 15 0 1 − 10 2 5 1 − 10 1 5 0 − 15 0 1 − 10 1 + 10 + V1 − V2 V1 − V4 + , 5 20 PROPRIETARY MATERIAL.

E. condensed equations) for the unknown voltages and currents. 3 R = 30 Ω 2 R = 35 Ω 1 V1= 10 volts R = 7 R=5Ω Ω 4 R = 15 Ω R = 10 Ω 6 V6= 200 volts 5 R=5Ω Fig. 3 for the direct current electric network shown in Fig. 4. PROPRIETARY MATERIAL. c The McGraw-Hill Companies, Inc. ° All rights reserved. 52 AN INTRODUCTION TO THE FINITE ELEMENT METHOD R=5Ω 3 R = 20 Ω 6 R=0Ω 8 R = 10 Ω R=5Ω 2 R=5Ω 5 R = 15 Ω R = 10 Ω 1 R = 20 Ω 4 R = 50 Ω V1= 110 volts 7 V7 = 40 volts Fig. 4 Solution: The assembled coeﬃcient matrix is ⎡ 1 5 + 1 20 ⎢ −1 ⎢ 5 ⎢ ⎢ 0 ⎢ 1 [K] = ⎢ ⎢ − 20 ⎢ 0 ⎢ ⎢ ⎣ 0 1 5 0 − 15 + 15 + 1 − 20 0 − 15 0 0 1 − 20 0 0 1 1 + 20 10 + 1 − 10 0 1 − 50 ⎤ 0 ⎥ 0 ⎥ ⎥ 0 ⎥ 1 − 50 ⎥ ⎥ ⎥ ⎥ 0 ⎥ 1 − 15 ⎦ 1 1 15 + 50 0 1 − 20 1 1 20 + 5 0 0 − 15 0 0 0 − 15 0 1 − 10 1 1 + 10 + 15 1 − 15 1 20 1 5 1 50 1 5 0 − 15 0 1 − 10 1 + 10 + 1 − 10 0 1 10 The condensed equations are ⎡ 9 20 ⎢− 1 ⎢ 20 ⎢ ⎢ 0 ⎢ 1 ⎣− 5 0 1 − 20 0 0 0 0 − 15 17 100 1 − 10 1 4 I1 = 0 − 15 0 1 − 10 2 5 1 − 10 1 5 0 − 15 0 1 − 10 1 + 10 + V1 − V2 V1 − V4 + , 5 20 PROPRIETARY MATERIAL.

### Computational Thermochemistry: Prediction and Estimatoin of Molecular Thermodynamics by Irikura K.K., Frurip D.J.

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