lecture 14

lecture 14 - CE 30125 Lecture 14 LECTURE 14 PRACTICAL...

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Unformatted text preview: CE 30125 - Lecture 14 LECTURE 14 PRACTICAL ISSUES IN APPLYING FINITE DIFFERENCE APPROXIMATIONS Derivatives of Variable Coefficients • Consider the term d df x ----- g x ------------ dx dx where g is a coefficient which is a function of x • Proceed in steps. First let: df u g x ----dx • Now evaluate: du 1 ----- = u i dx p. 14.1 CE 30125 - Lecture 14 hi i-1 hi+1 x i-1/2 u 1 ui • Now evaluate u 1 i + -2 and u x i+1/2 i+1 i 1 i + -2 –u 1 i – -2 = -----------------------------1 -- h i + h i + 1 2 1 i – -2 u u 1 i + -2 1 i – -2 =g =g 1f 1 1 i + -- i + -2 2 1 1 i – -- i – -2 2 1f =g =g p. 14.2 fi + 1 – fi 1 ------------------ i + -- h i + 1 2 fi – fi – 1 1 ------------------ hi i – -- 2 CE 30125 - Lecture 14 • Hence 1 ui where g i + 1/2 g i – 1/2 g i – 1/2 g i + 1/2 = f i + 1 ---------------------- – f i ---------------------- + -------------- + f i – 1 -------------h avg h i + 1 h avg h i h avg h i + 1 h avg h i hi + hi + 1 h avg --------------------2 • When h = h i + 1 = h i , then the approximation reduces to: 1 ui 1 = ----- f i + 1 g 1 – f i g 1 + g 1 + f i – 1 g 1 2 i + -i + -i – -- i – -h 2 2 2 2 • Example Applications with variable coefficients: • Fluids: spatially varying viscosities/diffusivities • Solids: spatially varying elasticities p. 14.3 CE 30125 - Lecture 14 Two Dimensional Finite Difference Approximations • Define nodes in a two-dimensional plane with • i = spatial index in the x -direction • j = spatial index in the y -direction f y fi , j+2 i, j+2 fi+2 , j+2 i+2, j+2 i, j+1 Dy i-2, j i-1, j fi , j i, j i, j-1 i, j-2 p. 14.4 Dx fi+2, j i+1, j i+2, j x CE 30125 - Lecture 14 • When taking partial derivatives w.r.t. x , we hold y constant f ----x i j f i + 1 j – f i – 1 j = -------------------------------2x and 2 f ------2 x i j f i + 1 j – 2 f i j + f i – 1 j = -----------------------------------------------2 x • Similarly 2 f ------2 y i j f i j + 1 – 2 f i j + f i j – 1 = -----------------------------------------------2 y p. 14.5 CE 30125 - Lecture 14 Example 1 • Consider Laplace’s equation and x = y = h : 2f 2f 2f = ------- + ------x2 y2 i j 2 f i j 1 = ----- f i + 1 j + f i – 1 j + f i j + 1 + f i j – 1 – 4 f i j 2 h • This may be viewed as follows in terms of a molecule: fi , j+1 1 1 h2 fi-1, j 1 fi, j -4 1 p. 14.6 fi+1, j 1 fi , j-1 CE 30125 - Lecture 14 Example 2 • Let’s consider: 4 4 4 f f f f = f = ------- + 2 ---------------- + ------4 22 4 x x y y 4 2 2 • Draw a molecule diagram for the 2nd order central difference formulation. 4 2 2 f f • Obtain ------- by finding ------- ------- 4 2 2 x x x 4 2 2 f f ------- = ------- ------- 4 2 2 x x x 4 2 f 1 ------- = ------- -------- f i + 1 j – 2 f i j + f i – 1 j 4 2 2 x x x p. 14.7 CE 30125 - Lecture 14 4 2 2 2 f 1 ------- = -------- ------- f i + 1 j – 2 ------- f i j + ------- f i – 1 j 4 2 2 2 2 x x x x x 4 f 1 1 ------- = -------- -------- f i + 2 j – 2 f i + 1 j + f i j 4 2 2 x x x 2 – -------- f i + 1 j – 2 f i j + f i – 1 j 2 x 1 + -------- f i j – 2 f i – 1 j + f i – 2 j 2 x 4 f 1 ------- = -------- f i – 2 j – 4 f i – 1 j + 6 f i j – 4 f i + 1 j + f i + 2 j 4 4 x x 1 Dx4 1 -4 6 p. 14.8 -4 1 CE 30125 - Lecture 14 • Similarly: 4 f 1 ------- = -------- f i j – 2 – 4 f i j – 1 + 6 f i j – 4 f i j + 1 + f i j + 2 4 4 dy y 1 -4 1 Dy4 6 -4 1 p. 14.9 CE 30125 - Lecture 14 • Evaluation of the mixed derivative: 4 2 2 f f ---------------- = ------- ------- 22 2 x 2 x y y 4 2 f 1 ---------------- = ------- -------- f i – 1 j – 2 f i j + f i + 1 j 22 2 2 x y y x 4 f 1 1 ---------------- = -------- -------- f i – 1 j – 1 – 2 f i – 1 j + f i – 1 j + 1 22 2 2 x y x y 2 – -------- f i j – 1 – 2 f i j + f i j + 1 2 y 1 + -------- f i + 1 j – 1 – 2 f i + 1 j + f i + 1 j + 1 2 y p. 14.10 CE 30125 - Lecture 14 4 f 1 ---------------- = ----------------- f i – 1 j – 1 – 2 f i – 1 j + f i – 1 j + 1 22 2 2 x y x y – 2 f i j – 1 + 4 f i j – 2 f i j + 1 + f i + 1 j – 1 – 2 f i + 1 j + f i + 1 j • This then produces the following module 1 1 Dx2 Dy2 -2 1 -2 4 -2 1 -2 1 p. 14.11 CE 30125 - Lecture 14 • We can also obtain this module as follows: ¶ 2f ¶x2 = ¶ 2f ¶y2 1 Dx2 = [1 -2 1] 1 1 Dy2 -2 1 • Therefore 1 -2 1 -2 4 -2 1 -2 1 1 ¶2 ¶ 2f ¶y2 ¶x2 = 1 Dx2 Dy2 -2 x 1 -2 1= 1 Dx2 Dy2 1 p. 14.12 CE 30125 - Lecture 14 4 • To obtain f , we must add up modules: 4 4 4 f f f f = ------- + 2 ---------------- + ------4 22 4 x x y y 4 • If x = y = h then: 1 2 1 h4 1 -8 2 -8 20 -8 2 -8 2 1 p. 14.13 1 ...
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This note was uploaded on 03/02/2012 for the course CE 30125 taught by Professor Westerink,j during the Fall '08 term at Notre Dame.

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