# EMH 451 Numerical Methods For Engineers [Kaedah Berangka Untuk Jurutera]

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1 -1- [EMH 451/3] UNIVERSITI SAINS MALAYSIA First Semester Examination 2013/2014 Academic Session December 2013 / January 2014 EMH 451 Numerical Methods For Engineers [Kaedah Berangka Untuk Jurutera] Duration : 2 hours [Masa : 2 jam] INSTRUCTIONS TO CANDIDATE: ARAHAN KEPADA CALON: Please check that this paper contains FOUR (4) printed pages and FIVE (5) questions before you begin the examination. Sila pastikan bahawa kertas soalan ini mengandungi EMPAT (4) mukasurat bercetak dan LIMA (5) soalan sebelum anda memulakan peperiksaan. Answer ALL questions. Jawab SEMUA soalan. Appendix/Lampiran : 1. Useful formulas and pseudocodes [1 page/mukasurat] You may answer all questions in English OR Bahasa Malaysia OR a combination of both. Calon boleh menjawab semua soalan dalam Bahasa Malaysia ATAU Bahasa Inggeris ATAU kombinasi kedua-duanya. Answer to each question must begin from a new page. Jawapan untuk setiap soalan mestilah dimulakan pada mukasurat yang baru. In the event of any discrepancies, the English version shall be used. Sekiranya terdapat sebarang percanggahan pada soalan peperiksaan, versi Bahasa Inggeris hendaklah diguna pakai. 2/-

2 -2- Q1. [a] Consider the transient heat conduction problem in a rod of 1m length and unit thermal diffusivity. The rod experiences constant uniform heat source at 9W/m. The left and right ends of the rod are fixed at 0 C and 60 C, respectively. The initial temperature is uniform at 0 C. Use the FDM with implicit method to set up the linear algebra systems for the solution of temperature distribution after 0.2 seconds. You must use 4 grid points {x 0, x 1, x 2, x 3 } and 3 time levels {t 0, t 1, t 2 }. DO NOT solve for the unknowns. Pertimbangkan masalah aliran haba fana di dalam rod sepanjang 1m dan kemeresapan haba 1m. Rod itu dikenakan punca haba sekata pada 9W/m. Hujung-hujung kiri dan kanannya ditetapkan pada suhu masing-masing sebanyak 0 C dan 60 C. Suhu awal rod ialah seragam pada 0 C. Dengan menggunakan FDM beserta kaedah tersirat, tentukan sistem-sistem algebra linear bagi penyelesaian terhadap agihan suhu di dalam rod selepas 0.2 saat. Gunakan 4 titik grid {x0, x1, x2, x3} dan 3 tahap masa {t0, t1, t2}. JANGAN selesaikan semua pemboleh ubah tersebut. (70 marks/markah) [b] State ONE advantage of the implicit method over the explicit method for numerical solution of transient problems. Nyatakan SATU kelebihan kaedah tersirat ke atas kaedah tak tersirat bagi penyelesaian berangka untuk masalah fana. (30 marks/markah) Q2. [a] Discretization error in the solution of a numerical method for an engineering problem can be characterized with l 2 -norm of the error as: Ralat pendiskretan di dalam penyelesaian dengan kaedah berangka bagi sesuatu masalah kejuruteraan boleh ditetapkan dengan norma l 2 bagi ralat sebagai: where u is the numerical solution and u T is the true solution. State the problem with this approach and propose an alternative to the error characterization. di mana u ialah penyelesaian berangka dan u T ialah penyelesaian sebenar. Nyatakan masalah terhadap pendekatan ini dan cadangkan kaedah lain untuk pencirian ralat. (50 marks/markah) 3/-

4 -4- Q5. [a] A rod that is subjected to an arbitrary load along its axis can be described by the Poisson's equation with a constant coefficient. Write down the weak form of the problem ignoring the boundary conditions. Sebatang rod yang dikenakan beban sembarangan sepanjang paksinya boleh dinyatakan dengan persamaan Poisson berserta pekali malar. Nyatakan bentuk lemah masalah ini dengan mengabaikan keadaan-keadaan sempadan. (30 marks/markah) [b] Assume the following linear algebra system is obtained: Andaikan sistem algebra linear di bawah diperolehi: Write down a complete MATLAB code to solve the above using the Gauss- Seidel method. DO NOT evaluate the final answers. The pseudocode is provided in the appendix. Tuliskan kod MATLAB yang lengkap untuk menyelesaikan masalah tersebut dengan menggunakan kaedah Gauss-Seidel. JANGAN mengira jawapanjawapan akhir. Pseudokod yang berkaitan disediakan di dalam lampiran. (70 marks/markah) -oooooooo- 5/-

5 APPENDIX 1/LAMPIRAN 1 Useful formulas Centered Difference Quadratic interpolation at interface value of control volume ui+1/2 = (-ui-1 + 6ui + 3ui+1)/8 Pseudocode for Gauss-Seidel Iterative Method [1/1]

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