AC : TEACHING SPREADSHEET-BASED NUMERICAL ANAL- YSIS WITH VISUAL BASIC FOR APPLICATIONS AND VIRTUAL IN- STRUMENTS

Transcription

1 AC : TEACHING SPREADSHEET-BASED NUMERICAL ANAL- YSIS WITH VISUAL BASIC FOR APPLICATIONS AND VIRTUAL IN- STRUMENTS Nkunja Swan, South Carolna State Unversty Dr. Swan s currently a Professor at the South Carolna State Unversty. Dr. Swan has 25+ years of experence as an engneer and educator. He has more than 50 publcatons n journals and conference proceedngs, has procured research and development grants from the NSF, NASA, DOT, DOD, and DOE and revewed number of books on computer related areas. He s also a revewer for ACM Computng Revews, IJAMT, CIT, ASEE, and other conferences and journals. He s a regstered Professonal Engneer n South Carolna. c Amercan Socety for Engneerng Educaton, 2011 Page

2 Teachng Spreadsheet-Based Numercal Analyss wth Vsual Basc for Applcatons and Vrtual Instruments Abstract LabVIEW, EXCEL and VBA are currently used n a number of engneerng schools and ndustres for smulaton and analyss. By ntroducng vrtual nstrumentaton (LabVIEW) and EXCEL/VBA to the exstng laboratory facltes and course(s) the students can be well traned wth the latest desgn technques and computer aded nstrumentaton, desgn and process control used throughout ndustry. Ths wll also allow the students greater nteracton wth the subject matter and mprove hs/her sklls n the use of number of appled engneerng software packages. Ths paper wll dscuss desgn and development of nteractve nstructonal modules for Numercal Analyss Course usng LabVIEW, EXCEL and VBA. Introducton The students over relance upon formulas and routne use of technque n problem solvng too often lead to poor performance n advanced courses and a hgh attrton rate n the engneerng, technology, and scence programs. The students lack of comprehenson of mathematcal concepts results n tme wastage durng laboratory experments, msnterpretatons of lab data and underachevement n standardzed scence and engneerng tests that stress the fundamentals 1, 2, 3, 4. Ths problem can be effectvely addressed by mprovng the student s conceptual understandng and comprehenson of the topcs covered n ntroductory scence and technology courses. One way to acheve ths s through nteractve learnng and teachng and upgradng the exstng laboratores wth modern equpment. Ths wll requre ncreased fundng and resources. But n recent years there s a decrease n resource allocaton makng t ncreasngly dffcult to modernze the laboratores to provde adequate levels of laboratory and course work and unverstes are under pressure to look for alternatve cost effectve methods. One way to acheve ths s through nteractve learnng and teachng through the use of software packages lke LabVIEW (Vrtual Instruments), Excel and Vsual Basc for Applcatons (VBA). These programs along wth MatLAB, Mathematca and Maple are used to teach courses such as Numercal Analyss and Engneerng Problem Solvng. Ths paper dscusses some of the numercal analyss nstructonal modules usng LabVIEW, EXCEL and VBA. Numercal Analyss Instructonal Modules usng LabVIEW Approxmately 10 to 15 years ago, Natonal Instruments Corporaton ntroduced a new program called LabVIEW. The acronym stands for Laboratory Vrtual Instrumentaton Engneerng Work Bench. Orgnally desgned for test and measurement applcatons, the program has been modfed over the years to desgn and analyze varous complex systems. LabVIEW s a graphcal programmng envronment and s based on the concept of data flow programmng. The data flow programmng concept s dfferent from the sequental nature of tradtonal programmng languages, and t cuts down on the desgn and development tme of an applcaton. It s wdely accepted by ndustry, academa, and research laboratores around the world as a standard for data acquston and nstrument control software. Snce LabVIEW s based on graphcal programmng, users can buld nstrumentaton called vrtual nstruments (VIs) usng software objects. Wth proper hardware, these VIs can be used for remote data acquston, Page

3 analyss, desgn, and dstrbuted control. The bult-n lbrary of LabVIEW has a number of VIs that can be used to desgn and develop any system. LabVIEW can be used to address the needs of varous courses n a technology and scence currculum 6, 7, 8, 9. LabVIEW Applcaton Areas LabVIEW s extremely flexble and some of the applcaton areas of LabVIEW are Smulaton, Data Acquston, and Data Processng. The Data Processng lbrary ncludes sgnal generaton, dgtal sgnal processng (DSP), measurement, flters, wndows, curve fttng, probablty and statstcs, lnear algebra, numercal methods, nstrument control, program development, control systems, and fuzzy logc. These features of LabVIEW wll help provde an nterdscplnary, ntegrated teachng and learnng experence that ntegrates team-orented, hands-on learnng experences throughout the engneerng technology and scences currculum, engagng students n the desgn and analyss process begnnng wth ther frst year. The Mathematcs VIs of LabVIEW are located on the Functons»Analyze»Mathematcs palette. These VIs can be used to perform many dfferent knds of mathematcal calculatons. The followng s a lstng of VIs n LabVIEW: 1D and 2D Evaluaton VIs, Lnear Algebra VIs, Array Operatons VIs, Numerc Functons VIs Calculus VIs, Optmzaton VIs, Curve Fttng VIs Probablty and Statstcs VIs Formula VIs, and Zeroes VIs. The user has to provde approprate nputs and outputs n the LabVIEW Control Panel and make requred connectons n the LabVIEW dagram panel to smulate these VIs. Example VI to solve system of lnear equatons Ths VI solves the followng lnear equatons: 5x1 + x2 + 3x3 = 5 2x1 + 7x2 + 9x3 = 4 8x1 + 6x2+ 4x3 = 9 The lnear equatons are wrtten n Matrx Form (Ax = B) form and then A and B (known vector) are suppled as nputs to the VI. The VI solves for the roots and dsplays the results as shown n Fgure 1. The VI s flexble and can be easly modfed to accommodate more number of equatons by smply changng the dmenson of A, B, and soluton vector. Page

4 Example VI to perform LU Decomposton Fgure 1 VI to solve lnear equatons LU decomposton method has ts place n the soluton of smultaneous lnear equatons 5. Research shows that LU decomposton method s computatonally more effcent than Gaussan elmnaton. LU decomposton takes less computatonal tme to fnd the nverse of a matrx. Typcal values of the rato of the computatonal tme for dfferent values of n are gven n Table 1. Table 1 Comparng computatonal tmes of fndng nverse of a matrx usng LU decomposton and Gaussan elmnaton. n CT nversege / nverse LU CT Page

5 Fgure 2 below presents the LU decomposton VI. Ths VI s found n Lnear Algebra VI secton of Mathematcs palette of LabVIEW. The user has to provde the A matrx n control panel of LabVIEW and make approprate connectons n the dagram panel. Fgure 2 LU Decomposton VI Numercal Analyss Instructonal Module usng Vsual Basc for Applcatons (VBA) The Newton-Raphson method s based on the prncple that f the ntal guess of the root of f ( x) = 0 s at x, then f one draws the tangent to the curve at f ( x ), the pont x + 1 where the tangent crosses the x - axs s an mproved estmate of the root 5. Usng the defnton of the slope of a functon, at ( x ) = θ f tan ( x ) f 0 =, x x + 1 x = x whch gves ( x ) ( x ) f x +1 = x (1) f Page

6 Equaton (1) s called the Newton-Raphson formula for solvng nonlnear equatons of the form f ( x) = 0. So startng wth an ntal guess, x, one can fnd the next guess, x + 1, by usng Equaton (1). One can repeat ths process untl one fnds the root wthn a desrable tolerance. Algorthm The steps of the Newton-Raphson method to fnd the root of an equaton ( x) = 0 1. Evaluate f ( x) symbolcally 2. Use an ntal guess of the root, f ( x ) x +1 = x f ( x ) 3. Fnd the absolute relatve approxmate error a as a = x + 1 x + 1 x 100 f are x, to estmate the new value of the root, x + 1, as 4. Compare the absolute relatve approxmate error wth the pre-specfed relatve error tolerance, s. If a > s, then go to Step 2, else stop the algorthm. Also, check f the number of teratons has exceeded the maxmum number of teratons allowed. If so, one needs to termnate the algorthm and notfy the user. Fgure 3 represents the VBA screen shot for solvng f(x) = 3x3 + 3x 1 usng Newton s method, and Fgure 4 represents the smulaton results. The data for the smulaton s entered n EXCEL spread sheet. The VBA module gets the nput data from the EXCEL spreadsheet, performs the teratons, and dsplays the result n Excel spread sheet Page

7 Fgure 3 VBA code for Newton method Page

8 Fgure 4 Results of VBA smulaton Numercal Analyss Instructonal Module usng EXCEL Many a tmes, a functon f ( x) ( x 0, y0 ), ( x1, y1 ),...,( xn 1, yn 1),( xn, yn ) a contnuous functon ( x) y = s gven only at dscrete ponts such as. How does one fnd the value of y at any other value of x? Well, f may be used to represent the + 1 f x passng through the n data values wth ( ) n +1 ponts. Then one can fnd the value of y at any other value of x. Ths s called nterpolaton. Of course, f x falls outsde the range of x for whch the data s gven, t s no longer nterpolaton but nstead s called extrapolaton. Of course, f falls outsde the range of for whch the data s gven, t s no longer nterpolaton but nstead s called extrapolaton. So what knd of functon should one choose? A polynomal s a common choce for an nterpolatng functon because polynomals are easy to (A) evaluate, (B) dfferentate, and (C) ntegrate, relatve to other choces such as a trgonometrc and exponental seres. Polynomal nterpolaton nvolves fndng a polynomal of order n that passes through the n+1 data ponts. One of the methods used to fnd ths polynomal s called the Lagrangan method of nterpolaton. Other methods nclude Newton s dvded dfference polynomal method and the drect method. Page

9 Fgures 5 and 6 demonstrate the use of Drect Interploaton and Lagrangan method of nterpolaton n Excel. The results agree wth the theoretcal soluton provded n 5. The problem selected for ths example s form 5 mentoned above and s as follows: The upward velocty of a rocket s gven as a functon of tme n Table 1. Determne the value of the velocty at t = 16 seconds usng (a) Drect Method and (b) Lagrange polynomal method.. Table 1: Velocty as a functon of tme. t (s) v (t) (m/s) Page

10 Fgure 5 Excel spreadsheet results for Drect Method Page

11 Fgure 6 Excel Spreadsheet results for Lagrange polynomal. Summary and Conclusons We have developed number of modules usng EXCEL, VBA, and LabVIEW for numercal analyss and engneerng problem solvng courses. The sample modules presented above were user frendly and performed satsfactorly under varous nput condtons. These and other modules (Bsecton Method, Runga-Kutta method, Secant Method, Numercal Integraton, etc) helped the students to understand the concepts n more detal. These modules can be used n conjuncton wth other teachng ads to enhance student learnng n varous courses and wll provde a truly modern envronment n whch students and faculty members can study engneerng, technology, and scences at a level of detal. Acknowledgement Ths work was funded n part by a grant from the NSF-HBCU-UP/RISC grant. We are thankful to the NSF for provdng us wth ths help.. Page

12 References 1. Swan, N. K., Korrapat, R., Anderson, J. A. (1999) Revtalzng Undergraduate Engneerng, Technology, and Scence Educaton Through Vrtual Instrumentaton, NI Week Conference, Austn, TX.. 2. Elane L., Mack, Lynn G. (2001), Developng and Implementng an Integrated Problem-based Engneerng Technology Currculum n an Amercan Techncal College System Communty College Journal of Research and Practce, Vol. 25, No. 5-6, pp Bunyamn, N, Mohamad, Z., 2000 Engneerng Currculum Development: Balancng Employer Needs and Natonal Interest--A Case Study Retreved from ERIC database. 4. Kelle, Andrew C., And Others. (1984), Experence wth Computer-Asssted Instructon n Engneerng Technology, Engneerng Educaton, Vol. 74, No. 8, pp URL: 6. Anderson, J. A., Korrapat. R. B., & Swan. N. K., "Dgtal sgnal processng usng vrtual nstrumentaton". Proceedngs of SPIE Vol Korrapat, R. B. & Swan. N. K., "Study of Modulaton usng Vrtual Instruments". Proceedngs of Natonal Conference on Alled Academes, Sprng Swan, N. K., Anderson, J. A., & Korrapat. R. B. "Computer based vrtual engneerng Laboratory (CBVEL) and Engneerng Technology Educaton" Annual ASEE Conference Proceedngs. 9. Lsa Wells and Jeferey Travs, LabVIEW for Everyone, Graphcal Programmng Even Made Easer, Prentce Hall, NJ 07458, Page