Computational Approach to Materials Science and Engineering

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1 Computational Approach to Materials Science and Engineering Prita Pant and M. P. Gururajan October, 2012 Copyright c 2012, Prita Pant and M P Gururajan. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back- Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License. 1

2 Module: Interpolation 1 Pre-requisites The modules in Part II of this course material. 2 Learning goals Given a set of data, to interpolate between the given data points 3 Interpolation Interpolation is a procedure that is used to obtain the values of y for any x in the range x 1 to x 2 given the data points (x 1,y 1 ) and (x 2,y 2 ). There are many different interpolations that are possible. In this module, we shall discuss some examples. 4 Air bubbles in polar ice It is possible to track the atmospheric concentrations of CO 2 on a millennial timescale by analysing the air bubbles trapped inthepolar ice. Forexample, consider the data of the age of air bubbles found in two different ice cores, namely GRID in Greenland and Byrd in Antarctica, shown in Table. 1. The age is given in units of years Before Present (BP), that is, years calculated with 1 January 1950 as the origin (after 1950, the years can not be reliably calculated using radioactive isotope abundance data due to nuclear weapons testing). Given this data, let us consider the following questions: 1. To what depth in GRID ice core should one go to find air bubbles of age years BP? 2. At a depth of 1600 m in the Byrd ice core, what would be the age of air bubbles? 2

3 3. Adepth of 2000min GRID ice core corresponds to what depth in Byrd ice core? All these questions can be answered using interpolation. 4.1 Linear interpolation The simplest interpolation to consider is the linear interpolation. In this, one assumes that the data at the two points (x 1,y 1 ) and (x 2,y 2 ) can be connected through a straight line so that for any intermediate value of x, the corresponding y value can be obtained and vice versa. The GNU Octave command interp1 can be used to carry out such a linear interpolation. In the script below, for example, we show how interp1 can be used to answer all the three questions above: and the answers for the three questions are, m, years BP and m, repsectively. DepthGasAge.oct Copyright (C) 2011 Prita Pant and M P Gururajan This program is free software; you can redistribute it and/or modify it under the terms of GNU General Public License as published by the Free Software Foundation; either version 3 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program; if not, write to the Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA , USA % Load the given data from the data file X = load("depthgasage.dat"); 3

4 Age (in years BP) Depth (in m) Depth (in m) GRID: Greenland Byrd: Antarctica Table 1: The age of air bubbles at various depths in two ice cores, namely GRID in Greenland and Byrd in Antarctica. The data is taken from Atmospheric CO 2 concentration and millennial-scale 4 climate change during the last glacial period, B. Stauffer et al, Nature, 392, pp , 1954.

5 % Read the first column of the data as age A = X(:,1); % Read the second column of the data as depth for GRID dgrid = X(:,2); % Read the third column of the data as depth for Byrd dbyrd = X(:,3); % Interpolate the GRID data for the age of air bubble to be interp1(a,dgrid,20500) % Interpolate the Byrd data for the depth of 1600 interp1(dbyrd,a,1600) % Interpolate the Byrd data to a depth of 2000 m of GRID interp1(dgrid,dbyrd,2000) From the above script, it is clear that the x and y data are given as the first two parameters of interp1 and the third parameter is the x for which we want to calculate the y value. For example, a command such as interp1(dbyrd, dgrid, 1880) will give the depth of GRID that correpsonds to 1880 m of Byrd while interp1(dgrid,dbyrd,1880) gave the depth of Byrd that corresponds to 1880 m of GRID. Here are two important points to remember while using interp1. The sample points x, that is, the first set of parameters passed on to interp1 should be strictly monotonic that is, either continuously increasing or decreasing. The point at which we want to calculate the y value should lie between the given data; calculating the y values for x values that lie outside the given range of sample data is known as extrapolation. It is possible to exptrapolate using interp1. For that, we need to explicitly mention extrap as the fourth input parameter in the command: for example, interp1(dbyrd,dgrid,1900, extrap ) will give answers while without the extrap string, the program will return NA. 5 Error function evaluation Consider a diffusion couple, consisting of two very long rods, welded face to face as shown. The composition in such a diffusion couple, as a fun- 5

6 Diffusion couple c c 1 2 c t=0 Figure 1: The schematic of a diffusion couple. Two long rods of different compositions, c 1 and c 2 ae welded face to face. The initial composition profile looks as shown below the couple. If this assembly is kept at high enough temperatures, then the diffusion profile evovles and the profile is described by error function. x tion of time, is known to be described by the expression c(x,t) = c 1+c 2 2 c 1 c 2 erf ( ) x 2 2 Dt where c1 and c 2 are the initial compositions of the two rods, and D is the diffusivity. Theerrorfunction, erf(z), foranyz hastoreadofffromtables. Typically, the tabulation is done for a certain number of z values and for any intermediate values, the error function has to be obtained using interpolation. Let us suppose that we know the error function values for nine different points as shown in Table. 2. For any intermediate values of z, one can then calculate the error function values by interpolation. 6

7 z erf(z) Table 2: Tabulation of error function values in the range x = 2 to x = 2. See Materials Science and Engineering: a first course, V Raghavan, Third edition, Prentice-Hall of India Pvt. Ltd, 1995, for example. With the given data, if we try a linear interpolation (See Problem 1), the interpolation looks as shown in Figure. 2. As is clear from the figure, the interpolation is not satisfactory since it gives rise to a jagged profile. It indeed is possible to get a much better plot of error function using a table with much finer sampling: see Table. 3 and Fig Spline interpolation It is also possible to get a smooth curve for the error function by carrying out spline interpolation instead of linear interpolation. In the script below we show how to fit even a small number of data (as given in Table. 2) using spline interpolation to achieve a smoother profile, which is comparable to what is achieved using a much larger data set. The plot obtained using the spline interpolation is shown in Fig. 4 7

8 1 0.5 erf(z) z Figure 2: The linear interpolation for the error function. 8

9 z erf(z) z erf(z) Table3: Tabulationoferrorfunctionvaluesintherangex = 2.8tox = 2.8. See Materials Science and Engineering: 9 a first course, V Raghavan, Third edition, Prentice-Hall of India Pvt. Ltd, 1995, for example.

10 1 0.5 erf(z) z Figure 3: The plot of error function with a much finer tabulation. 10

11 ErfSplineInterpolation.oct Copyright (C) 2011 Prita Pant and M P Gururajan This program is free software; you can redistribute it and/or modify it under the terms of GNU General Public License as published by the Free Software Foundation; either version 3 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program; if not, write to the Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA , USA X = load( errorfunction.dat ); a = X(:,1); b = X(:,2); plot(a,b, o ); hold on x = [-2.:0.1:2.]; xx = interp1(a,b,x, spline ); plot(x,xx, r ) axis( square ) xlabel("z") ylabel("erf(z)") print -depsc../figures/erfsplineinterpolation.eps 11

12 1 0.5 erf(z) z Figure 4: The plot of error function with very few points of tabulated data and spline interpolation. The spline interpolation results in a smoother profile. 12

13 6 The specific heat data: polynomial interpolation In the previous section, we have used either polyfit or the design matrix to fit the given data to polynomials. It is also possible to use the fitted polynomials for interpolation. Thus the command polyval can be used for interpolation if polyfit was used for fitting data; in the case a design matrix wasused, thecoefficients canbeusedforinterpolation. Seethemoduleonfitting for further examples of polynomial interpolation, with specific reference to specific heats. 7 Self-assessment questions 1. What is the command in GNU Octave for interpolation? 2. interp1(x,y,a) means the x value corresponding to y = a is obtained. True or false? 3. polyval can be used for interpolation. True or false? 4. Can interp1 be used for exptrapolation? 5. In interp1(x,y,a), should the y also be monotonic? 8 Answers to self-assessment questions 1. interp1 2. False. It means the y value corresponding to x = a is obtained. 3. True. If polyfit is used on the given data to fit a polynomial, then, polyval can be used for interpolation. 4. Yes; by passing an optional parameter extrap, extrapolation can be carried out. 5. No. See the Problem 2, for example. 13

14 x y Table 4: Tabulation of x and y values in which only x values are monotonic. The y values are oscillatory. 9 Exercises Problem 1. Use the data given in Table. 2 to do linear interpolation for z values in the range z = 2 to z = 2 (in increments of 0.1) and calculate erf(z). Problem 2. Do a linear and spline interpolation for the data given in Table. 4. Problem 3. The stress-strain data for a brass sample is as shown in Table. 5. Calculate the strain at a stress of 50 MPa and the stress for a strain of Solution to exercises Solution to Problem 1. 14

15 Stress (in MPa) Strain Table 5: Tabulation of stress strain data for a brass sample. X = load( errorfunction.dat ); a = X(:,1); b = X(:,2); plot(a,b, o ); hold on x = [-2.:0.1:2.]; xx = interp1(a,b,x); plot(x,xx, r ) axis( square ) xlabel("z") ylabel("erf(z)") print -depsc../figures/erflinearinterpolation.eps The plot generated by this script is shown in Fig. 2. Solution to Problem 2 The script for carrying out linear interpolation on the given data is given below and the figure generated is shown in Fig

16 y x Figure 5: Linear interpolation for the data given in Problem 2. Note that even if the y values are non-monotonic, the interpolation can be carried out as long as x is monotonic. From the figure, it is clear that the given data is for a sinusoidal curve. X = load( SineData.dat ); a = X(:,1); b = X(:,2); plot(a,b, o ); hold on x = [0.0:0.01:6.3]; xx = interp1(a,b,x); plot(x,xx, r ) axis( square ) xlabel("x") ylabel("y") print -depsc../figures/sinelinear.eps 16

17 The script for carrying out linear interpolation on the given data is given below and the figure generated is shown in Fig. 6. X = load( SineData.dat ); a = X(:,1); b = X(:,2); plot(a,b, o ); hold on x = [0.0:0.01:6.3]; xx = interp1(a,b,x, spline ); plot(x,xx, r ) axis( square ) xlabel("x") ylabel("y") print -depsc../figures/sinespline.eps Note that even though the spline fitting gives the sinusoidal curve much better than the linear fit, it is not exact. This can be clearly seen if we compare the calculated value from spline and compare it with the actual value of sine at the given point. Solution to Problem 3 By plotting the data, one can see that the stress strain plot is linear; so, we are within the elastic limit. Hence, for finding the strain, one can use linear interpolation. Using linear interpolation, we get a strain of for a stress of 50 MPa and a stress of MPa for a strain of The code for calculation is as shown below. X = load( StressStrain.dat ); a = X(:,1); b = X(:,2); interp1(a,b,50) interp1(b,a, ) 17

18 y x Figure 6: Spline interpolation for the data given in Problem 2. Note that even if the y values are non-monotonic, the interpolation can be carried out as long as x is monotonic. From the figure, it is clear that the given data is for a sinusoidal curve. 18

19 11 References and further reading 1. Advanced Engineering Mathematics, E. Kreyszig, 8th edition, John Wiley and Sons, Elementary Numerical Analysis, K. E. Atkinson, 3rd edition, Wiley India,

Exercise(s) Solution(s) to the exercise(s)

Exercise(s) Solution(s) to the exercise(s) Exercise(s) Problem 1. Counting configurations Consider two different types of atoms, say A and B (represented by red and blue, respectively in the figures). Let A atoms and B atoms be distributed on N