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ENGR30003代写、代做Programming、C/C++编程语言调试、c+%2

转载 2019-10-20 09:53:45
ENGR30003: Numerical Programming for EngineersSemester 2, 2019 / Assignment 2Marks : This assignment is worth 35 of the overall assessment for this course.Due Date : Monday, 21st October 2019, 5:00pm, via submission. You will losepenalty marks at the rate of 10 per day or part day late, and the assignment will no be markedif it is more than 5 days late.1 Learning OutcomesIn this assignment you will demonstrate your understanding of solving engineering problemsusing numerical computations and assessing particular algorithms. The objectives of thisassignment are to program algorithms for root-finding, solving systems of linear algebraicequations, performing least-squares approximations and interpolations, regressing solutions andsolving a differential equation using different integration and differentiation schemes.2 Rootfinding [9/35 marks]Figure 1: Difference in shock wave type for different wedge angles.Imagine a wedge-shaped object flying through air at supersonic speeds (see Fig. 1). Incompressible flow theory (covered in MCEN90008 Fluid Dynamics), it is known that an obliqueshock wave forms at the front tip of this object under certain conditions.The equation relating the wedge half-angle θ to the shock oblique angle, β, and the Machnumber, M is given bywhere γ = 1.4 is the ratio of specific heats. For a given M, as you keep increasing θ, there is acritical angle, θmax where the shock becomes detached. We can also recast Eq. (1) into thefollowing formYour tasks are the following:© 2019 The University of Melbourne 1Figure 2: θ −β− M diagram for M = 5.0.For θ = 0, i.e. the object would be a flat plate, show that two possible solutions for β areβL = arcsinµ1M¶, βU = 90. (3)βU and βL are usually called the strong shock and weak shock solutions, respectively. Eventhough we can mathematically obtain two possible solutions, in reality, or physically, only theweak shock solution occurs. Note also that in order for the solution to be physically realisable,θ 2.2 Graphical solution [2 mark]Plot f (β) vs β fora. M = 1.5 and θ = 5, 10 and 15b. M = 5.0 and θ = 20, 30 and 45Indicate how βU and βL change with θ and M. Can you identify from your plots the approximatevalue of θmaxPLEASE REMEMBER to change the angle from degrees to radians when you use sinusoidalfunctions in your computer program.2.3 C program to solve shock-wave equation [6 marks]Your task is to write a C code that solves Eq. (2) using the Newton–Raphson method to find theroot of f (β), regarding θ and M as parameters and solving for β.(a) Write your C program such that it uses the Newton–Raphson method to solve f (β) = 0.What values of βL and βU do you think might be appropriate to use as an initial guess ForM = 5.0 and θ = 20, you should find thatβL = 29.80092..., and βU = 84.55625...(b) Extend your C program to find βU and βL values for M = 5.0 and 0 ≤ θ ≤ θmax. Rememberthat for θ = 0, βL = arcsin¡1M¢ and βU = 90. Plot values of θ on the horizontal axis andcorresponding values of β on the vertical axis. Your solution to this part of the assignmentshould look like Fig. 2. Note that you can plot your results obtained from your C code withyour program of choice, e.g. Matlab, Excel, etc.© 2019 The University of Melbourne 2(c) Use your computer program to solve f (β) = 0 for M = 2.0, 3.0, 4.0, 6.0, 7.0, 8.0. Plot β vs θfor all the M’s. This plot is called the θ −β− M diagram and you will find it very useful ifyou decide to do MCEN90008 Fluid Dynamics in the future.The implementation for this task is to be done in where input parameters are tobe read in from. The input file consists of three parts, as shown below:M, theta , be ta_ l , beta_u ,gammaThe first two lines corresponds to the part (a) where for M = 5.0 and θ = 200, you need to computethe values of βL and βU . You will notice the initial values set here are 00 apiece so that you canmodify them to find the right initial guess to compute the angles. The next line corresponds topart (b) where you will for M = 5 evaluate the values of βL and βU for different values of θ(increments of 10from 00 up to θmax). The next set of lines corresponds to part (c) where you willevaluate for different M, the values of βL and βU for different values of θ (increments of 10from 00 up to θmax). Outputting the results of this task are to be done only for part (c) into in the format as shown below. This example only shows part of the results forfor the set of Mach numbers chosen (you can use this to validate your code). The outputof M, βU ,βL is to be done up to 6 decimal places while θ as an integer:M, theta , beta_ lower , beta_upper3.000000 ,0 ,19.471221 ,90.0000003.000000 ,1 ,20.157561 ,89.6499124.000000 ,0 ,14.477513 ,90.0000004.000000 ,1 ,15.131265 ,89.719941You must dynamically allocate space for the Mach numbers so that your implementation maywork for a different set of M. For each Mach number, upon increasing θ by 10, you will reach themaximum θ beyond which the solutions of β are not physically relevant. You will write to fileonly up to this maximum θ per Mach number.© 2019 The University of Melbourne 33 Regression [4/35 marks]Here, an alternative way of solving a Least Squares Problem is considered. Recall from thelecture that(4)is the system of equations for linear regression yˆ = ax b.Show that:By considering Eq. 5, when will linear regression fail to find a solution4 Linear Algebraic Systems [5/35 marks]Consider the following tri-diagonal system:© 2019 The University of Melbourne 4Thus the solution to the original tri-diagonal matrix can be written asThe algorithm outlined above is called the Thomas algorithm. It is a very efficient method forsolving linear tri-diagonal systems.Write a C code using the Thomas algorithm to solve the tri-diagonal system shown in Eq. 6.Since the tri-diagonal system is a banded matrix, you need not store all the zeros! Instead, yourcode should take as an input the vectors ai, bi, ci and Qi. The output from your C code should bethe solution vector xi.Use your code to solve the following tri-diagonal system(8)You will implement the code for this task in the function, where you will read inas an input the vectors ai, bi, ci and Qi from the file. The output from yourimplementation should be the solution vector xi, written out to (up to 6decimal places). Your implementation should allocate dynamically the space for the values of ai,bi, ci and Qi such that your implementation would work for different problem sizes as well.5 Interpolation [4/35 marks]For this task, you will perform a cubic spline interpolation on the data provided in the file. First, plot the data (using MATLAB or Excel) in order to get an idea of thebehaviour f (x) should have. Next, write C code that uses cubic splines in order to estimate thefunction f (x) that cuts through the data. You will then use your code to calculate the value(s) off (x) at xo = 4, where xo is to be input at runtime from stdout. You will implement your method in, reading the values of x and f (x) from the file and outputting thevalues of the interpolated value to in the following format (up to 6 decimalplaces).xo , f ( xo )4.000000 ,0.2346554.000000 ,1.107865The output provided above is an example and does not constitute the solution. There may bemore than 1 solution possible here and you must be able to identify the correct interval(s) tocompute the interpolated value. For the C implementation, you must dynamically allocate spacefor x, f (x) and write your code such that it can work for a different set of input data and differentxo. Finally, you must plot the interpolated function using either MATLAB or Excel. How does theinterpolated function look compared with the actual datapoints© 2019 The University of Melbourne 56 Differentiation, differential equations [13/35 marks]Write a C program that solves the modified wave equationon the interval x = [0;1], and for times 0 ≤ t ≤ 0.2, using the phase speed c = 1.Most transport equations can also be written as∂f∂t= RHS(f ) (10)where RHS denotes the ‘right-hand-side’ of the equation to be solved, containing all spatialderivatives, in this case −c∂f∂x.The wave equation is to be solved on the spatial interval 0 ≤ x ≤ 1, that is discretized usingNx 1 points xi, with i = 0,1,2,..., Nx. The equidistant grid spacing therefore is x = 1/Nx.In order to discretize the time derivative ∂f∂t, use the second-order accurate Runge–Kutta, (11)where n is the time index so that fniis the function value at time level n at point xi and fn1the function value at time level n1 at point xi, superscript n0.5 denotes an intermediate(time) level, and t is the numerical timestep for the time integration.Two different finite-difference approximations for the spatial derivative are to be coded up:• The first-order accurate upwind scheme:for i = 1,..., NxFor i = 0, use the boundary stencil• The second-order accurate central scheme:for i = 1,..., Nx −1For i = 0, use the boundary stenciland for i = Nx use the boundary stencilThe initial condition for f is0 ≤ x 0.125 ≤ x ≤ 0.375 → f (x,t = 0) = 0.5[1−cos{8π(x−0.125)}]0.375 © 2019 The University of Melbourne 66.1 Suggested Code StructureIn the following possible steps for coding up the methods are suggested.a. Set up main program that does the following• Allocate arrays for the functions f, etc, where i = 0,1,2,..., Nx (keepNx a parameter so that you can change it).• Write the initial condition into the array fni, and write to a file for later visualization.• Set up loop over the number of timesteps you want to run (each with t) - within thisloop you want to call your RK2 routine.• Write intermediate solutions to file for later visualization at several time instances.b. Code up your RK2 routine:spatial derivatives are computed based on the function values at time level spatial derivatives are computed based on the function values at intermediate levelc. Code up your ‘RHS’ routine. In this routine, implement the two options for taking thespatial derivative, the upwinding scheme and the central scheme.For your code to run stably, your time steps need to satisfy the so-called CFL condition,specifying ENGR30003代写、代做Programming、C/C%that ctx≤ 1.You will implement your routines in the function , reading in the values of c, Nxand CFL from . This will allow you to compile your code just once and run it fordifferent values of c, Nx and CFL by just changing the infile. For code assessment, you will writeout the solutions from your two finite-difference approximations versions to and , respectively, the kth time the time loop executes(this depends on what Nx,CFL are) in the following format (up to 6 decimal places).x , f ( x )0.000000 ,0.0000000.010000 ,0.0003940.020000 ,0.000907The kth time will be read in from the infile from the parameter .6.2 TasksRun your code until a time of t = 0.2 for both finite-difference approximations (1st-order upwindand 2nd order central) for the following cases:© 2019 The University of Melbourne 7• Chose two different resolutions x, by setting Nx to 80 or 200.• For each resolution, chose the timestep t from the CFL number, using CFL=1, 0.75 and0.25.Output your results for the time-levels t = 0.05,0.1,0.15,0.2 and plot the exact solution as well,which is the same as the initial condition, shifted by c ·t in increasing x. How well does thesolution at t = 0.2 agree with the exact solution Discuss:• How does the agreement of the numerical prediction with the exact solution depend on thegrid resolution x• How does the agreement of the numerical prediction with the exact solution depend on theCFL number• What happens if you chose CFL>1 for either method, for example CFL=1.0027 SubmissionThis assignment, unlike assignment 1, consists of two partsa. A project report, detailing any derivations and solutions and displaying the required graphsb. C programs developed to solve some of the problemsYou need to submit your programs and report for assessment; Submission of the report andthe code will be done via . You may discuss your assignment work during yourworkshop, and with others in the class, but what gets typed into your programs and the reportmust be individual work, not copied from anyone else. So, do not give a hard or soft copy ofyour work to anyone; do not “lend” your “Uni backup” memory stick to others for any reason atall; and do not ask others to give you their programs “just so that I can take a look and get someideas, I won’t copy, honest”. The best way to help your friends in this regard is to say a very firm“no” when they ask for a copy of, or to see, your programs or report, pointing out that your “no”,and their acceptance of that decision, is the only thing that will preserve your friendship. Asophisticated program that undertakes deep structural analysis of C code identifying regions ofsimilarity will be run over all submissions in “compare every pair” mode. Students whoseprograms or reports are so identified will be referred to the Student Center. See for more information.7.1 Project ReportYour project report need not be a full technical report but should state all approximations madeand use figures of results to back up any conclusions. Be sure to include enough detail (usingappendices as necessary) so that your results could be reproduced by another researcher (oryourself at a future date!) wishing to check or extend your findings. Your report will be primarilyassessed on the completeness of the results, and the visual/logical effectiveness of the manner inwhich they are presented. Please type your report - scanned handwritten notes are sometimestoo difficult to read and mark and therefore will not be accepted.7.2 C programsThe C codes are to be submitted on where they would be tested on different inputs fromthe ones you were provided to test your implementations.© 2019 The University of Melbourne 87.2.1 Provided Code, where the parsing of data from command line is to be done and timing for eachtask implemented.where you will implement four functions (Question 2),which acts as a header file to link the C file to the main file. You may edit this toadd any or the input arguments to the functions.which you must use as input duringexecution of your program. During your submission, your code will be tested on differentinfiles from the ones you were provided. Please make sure any data structures used to readin these infiles are dynamically allocated to avoid any errors during runtime.Remember to fill in your details in each file you write, such as the student name and studentnumber. Key points about your code for this assignment you need to understand are as follows:• For the purposes of the report, you can output as many files or terminal outputs as youneed. These outputs can be used to generate graphs/plots and values for the different tasks.• Once you have all information you need for the report, your code must be made submissionworthy i.e. only output the outfiles described above (5 outfiles are expected). This meansyour code must not expect user input once you execute it, all inputs would come from theinfiles or the terminal before execution.• You have to parse the command line arguments (all infiles and any command line values)i.e. no hardcoding the names of the infiles or the value for interpolation task. This isbecause we will be using our own infiles, with different filenames and different locations.• Plan before you write your code. Cover all possibilities regarding different inputs.Dynamically allocating structures for the infile contents is a must so that for infiles withmore or less entries don’t end up giving you errors during submission.7.3 Running on DimefoxYou must first transfer your files from your home computer to the Dimefox server using the protocol. Then, log into dimefox and transfer the files to a relevant folder: perform thefollowing set of commands on the terminal from your home location on dimefox (making the rightfolders and transfering the files in the right location):© 2019 The University of Melbourne 9Remember to check this folder contains only the .c or .h files (if you use multiple c files and hfiles) you need for the assignment and the PDF report you wish to submit. Then try compilingyour code using on the terminal from the folder, to see if it works. The followingcompilation procedure must return no errors or warnings:You can test your compiled code then using:There are 5 command line arguments here, one for each of the 4 coding functions. The argumentnumber 4 () is part of the interpolation task: the value of xo at which the interpolated value is tobe outputted. Once your code works for the provided infiles, it would help you if you changed theinfiles and the 4th argument to different values and see if the code still works and outputsacceptable results. If this works, your code is now submission worthy.7.4 Submitting on DimefoxAll submissions must be made through the program. Assignments submitted throughany other method will not be marked. Transfer your files to your home drive on dimefox. Checkthe transfer was successful by logging into dimefox and doing the s command on the terminal.Once you’ve tested the code by running on Dimefox using the approach described in Running onDimefox section, you can then submit your files using the command as follows:Do NOT submit your infiles as this would likely corrupt your submission and would take time tofix. Wait for a few minutes and then carry out the verify and check steps:Look through this text file to check (a) your program compiled (b) it executed without error.Please read for confirmation about how to use, especially how to your submission. No special consideration will be given to any student who has not used properly.You must also check that you have no memory leaks in your code as loss of memory from yourimplementation will result in deductions. You can use for this as follows:It must be pointed out that will take longer time to run compared with the normalexecution, so, plan your submission accordingly. Incase your submission fails to pass the memorycheck, you will lose marks. There are two potential areas where you can lose marks: runtimeerror messages and heap/leak summary. Examples of runtime error messages include:Happens when you use a variable that has not© 2019 The University of Melbourne 10been defined or does not exist anymore or initialised. Happens whenusing an uninitialised variable to perform operations: Happens when trying to scan or write to file a variableto memory which you do not have access to.All submissions should be in C99, and use no functions outside of the C standard library andmaths library. Some key points to consider about your submission and verification are outlinedhere:• Submissions that can not be compiled or run by system will receive zero marksfor the programming part.• Submissions are also limited to a maximum runtime of 200 seconds and maximum file sizeper task of 2 MB. It would help if you don’t write any additional files through your code. Ifit is absolutely necessary, then make sure the files do not exceed 2 MB individually. Thiswould give errors during your submission.• Since each task can be assessed individually, you can work submit your code with just oneor two tasks implemented. The feedback will skip over tasks not implemented and onlylook at the outfile of the tasks implemented.• Only your last submission is stored in the system i.e. everytime you use, the newsubmission overwrites the previous version. Keep a backup of your previous versionssomewhere safe in case your latest submission works worse than the previous.8 Getting HelpThere are several ways for you to seek help with this assignment. First, check the Assignment 2Frequently Asked Questions wiki in the LMS (subsection Assignments). It is likely that yourquestion has been answered here already. You may also discuss the assignment on the“Assignment 2” discussion board. However, please do not post any source code on the discussionboard. You may also ask questions during the workshops or send me (Professor Sandberg, an email directly.Note: Students seeking extensions for medical or other “outside my control” reasons shouldemail Professor Sandberg, as soon as possible after thosecircumstances arise.© 2019 The University of Melbourne 11
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