mws_civ_inp_txt_lagrange_examples

mws_civ_inp_txt_lagrange_examples - Chapter 05.04...

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05.04.1 Chapter 05.04 Lagrangian Interpolation – More Examples Civil Engineering Example 1 To maximize a catch of bass in a lake, it is suggested to throw the line to the depth of the thermocline. The characteristic feature of this area is the sudden change in temperature. We are given the temperature vs. depth data for a lake in Table 1. Table 1 Temperature vs. depth for a lake. Temperature,   C T Depth,   m z 19.1 0 19.1 –1 19 –2 18.8 –3 18.7 –4 18.3 –5 18.2 –6 17.6 –7 11.7 –8 9.9 –9 9.1 –10
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05.04.2 Chapter 05.0 Figure 1 Temperature vs. depth of a lake. Using the given data, we see the largest change in temperature is between m 8 z and m 7 z . Determine the value of the temperature at m 5 . 7 z using a first order Lagrange polynomial. Solution For first order Lagrange polynomial interpolation (also called linear interpolation), the temperature is given by 1 0 ) ( ) ( ) ( i i i z T z L z T ) ( ) ( ) ( ) ( 1 1 0 0 z T z L z T z L
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Lagrange Method of Interpolation-More Examples: Civil Engineering 05.04.3 Figure 2 Linear interpolation. Since we want to find the temperature at m 5 . 7 z , we need to choose the two data points that are closest to m 5 . 7 z that also bracket m 5 . 7 z to evaluate it. The two points are 8 0 z and 7 1 z . Then  7 . 11 , 8 0 0 z T z  6 . 17 , 7 1 1 z T z gives 1 0 0 0 0 ) ( j j j j z z z z z L 1 0 1 z z z z 1 1 0 1 1 ) ( j j j j z z z z z L 0 1 0 z z z z Hence ) ( ) ( ) ( 1 0 1 0 0 1 0 1 z T z z z z z T z z z z z T 7 8 ), 6 . 17 ( 8 7 8 ) 7 . 11 ( 7 8 7 z z z ) 6 . 17 ( 8 7 8 5 . 7 ) 7 . 11 ( 7 8 7 5 . 7 ) 5 . 7 ( T ( x 0 , y 0 ) ( x 1 , y 1 ) f 1 ( x ) x y
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05.04.4 Chapter 05.0 ) 6 . 17 ( 5 . 0 ) 7 . 11 ( 5 . 0 C 65 . 14 You can see that 5 . 0 ) ( 0 z L and 5 . 0 ) ( 1 z L are like weightages given to the temperatures at m 8 z and m 7 z to calculate the temperature at m 5 . 7 z . Example 2 To maximize a catch of bass in a lake, it is suggested to throw the line to the depth of the thermocline. The characteristic feature of this area is the sudden change in temperature. We are given the temperature vs. depth data for a lake in Table 2.
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mws_civ_inp_txt_lagrange_examples - Chapter 05.04...

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