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# Computational Methods CE 30125

ND

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This 0 page Class Notes was uploaded by Ms. Shaniya Gutmann on Sunday November 1, 2015. The Class Notes belongs to CE 30125 at University of Notre Dame taught by Joannes Westerink in Fall. Since its upload, it has received 12 views. For similar materials see /class/232701/ce-30125-university-of-notre-dame in Civil Engineering at University of Notre Dame.

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Date Created: 11/01/15

CE 30125 Lecture 14 LECTURE 14 PRACTICAL ISSUES IN APPLYING FINITE DIFFERENCE APPROXIMATIONS Derivatives of Variable Coefficients Consider the term am where g is a coef cient which is a function of x 0 Proceed in steps First let df u 356 Now evaluate Q 2 1 dx p 141 CE 30125 Lecture 14 0 Hence 1 gi12 gi12 55712 55712 u f f f71 t 1 havghi1 l havghi1 havghi l havghi hh where hanE T H 0 When I h H hi then the approximation reduces to 1 1 up fi1g 1 fig 1g1fi71g 1 h2 l 1 17 17 2 2 2 2 0 Example Applications with variable coef cients Fluids spatially varying viscositiesdiffusivities Solids spatially varying elasticities p 143 CE 30125 Lecture 14 Two Dimensional Finite Difference Approximations De ne nodes in a twodimensional plane with 0 i spatial index in the xdirection 7 spatial index in the ydirection fi j2 fi2 j2 o o i j2 o o i2 j2 p 144 CE 30125 Lecture 14 0 When taking partial derivatives wrt x we hold y constant and Similarly g Zfiljfiilj 3x 1 2Ax 6 2f Zfilj2fijfiilj 6x2 1 sz 6 2f 2 fijl2fijfijil 2 2 3y w Ay p 145 CE 30125 Lecture 14 Examgle 1 Consider Laplace s equation and Ax Ay h 52f 52f 2 V i j 3x2 332 2 1 Vflij Pm1jfi71jfij1 j714 j This may be Viewed as follows in terms of a molecule fi j1 p 146 Examgle 2 Let s consider CE 30125 Lecture 14 4 V4f V2V2 a2 6x Draw a molecule diagram for the 2nd order central difference formulation 4 Obtain 5 1 by nding 6x a Zl l 6x2 6x2 i 2 11 6x4 6x2 6x2 gt 4 2 a a 4 7 Z 11 2 1 711 6x 6x p147 CE 30125 Lecture 14 4 2 2 a 1 a a a f 1 2 f f1 gt1 6x4 Ax2 391 2 1 quot391 6x2 1 6x2 I j gt 4 a 1 1 g E Zfi2 2fz1 721 2 2fi1j2fijfiilj Ax 1 2fij2fiiljfz2j Ax gt a4 1 4 4fz3972j4 71j6 j4 1j 2j 3x Ax 09690 p 148 CE 30125 Lecture 14 0 Similarly 4fz39j724fij71 6fij4fij1fij2 QJH 9099 p 149 CE 30125 Lecture 14 0 Evaluation of the mixed derivative axzay2 6y 5x2 gt 4 2 a a 1 2f2 2 2f171j fijfi1f ax ay 6y Ax gt 64f 1 1 a 25 2 i 112 1i 71W x y x V 2 2fz39j71 2fz39jfij1 Ay 1 31j712 1j 1j1 Ay2 p 1410 CE 30125 Lecture 14 4 a l 2f2 WWI711712J71j137111 ax ay Ax Ay 2fijil 4fij2fijl fz391jgt1 2 1j 1j o This then produces the following module p 1411

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