Advanced Undergraduate Engineering Mathematics

Paper ID #11518 Advanced Undergraduate Engineering Mathematics Dr. Michael P. Hennessey, University of St. Thomas Michael P. Hennessey (Mike) joined the full-time faculty as an assistant professor in the fall of 2000. Mike gained 10 years of industrial and academic laboratory experience at 3M, FMC, and the University of Minnesota prior to embarking on an academic career at Rochester Institute of Technology (3 years) and Minnesota State University, Mankato (2 years). He has taught over 20 courses in mechanical engineering at the undergraduate and graduate level, advised 11 MSME graduates, and has written (or co-written) 45 technical papers (published or accepted), in either journals (11), conference proceedings (33), or in magazines (1). He also actively consults with industry and is a member of ASME, SIAM and ASEE. Page 26.161.1 c American Society for Engineering Education, 2015 Advanced Undergraduate Engineering Mathematics Abstract This paper presents the details of a course on advanced engineering mathematics taught several times to undergraduate engineering students at the University of St. Thomas. Additionally, it provides motivation for the selection of different topics and showcases related numerical and graphical work done mostly in MATLAB. Primary course topics covered in this survey course include: (1) vector integral Calculus, (2) an introduction to Fourier series, (3) an introduction to partial differential equations (PDEs), (4) an introduction to complex analysis, and (5) conformal mapping and applications. Also, examples of student project work are shown. Lastly, useful student feedback and lessons learned is shared that others involved in engineering mathematics instruction may find useful or be able to relate to. Keywords: Vector integral Calculus, Fourier series, partial differential equations, complex analysis, conformal mapping, engineering mathematics education 1. Introduction Due to increasing undergraduate enrollments in both electrical and mechanical engineering within the School of Engineering and interest in helping our graduates become better prepared to handle the applied mathematical rigors of engineering graduate school, especially at top institutions, a technical elective course entitled Advanced Engineering Mathematics was developed and has now been taught a total of 3 times. The prerequisites were both Multivariable Calculus (MATH 200) and Introduction to Differential Equations and Linear Algebra (MATH 210)1. A survey approach was adopted and topics were selected to appeal to both the needs of electrical and mechanical engineering students, and for which there are mainstream textbooks available. Page 26.161.2 As part of the content selection exercise, an effort was made to solicit input (via email) from directors of graduate studies at several well-known R1 engineering graduate schools, especially ones offering Ph.D.’s in both electrical and mechanical engineering, since those are the undergraduate programs that St. Thomas offers (3 requests in total). Unfortunately, none of them responded! That said, based on the author’s academic experience over many years along with discussions with other faculty members (including several from the Mathematics Department), the following core topics were selected: (1) vector integral Calculus, (2) an introduction to Fourier series, (3) an introduction to partial differential equations, (4) an introduction to complex analysis, and (5) conformal mapping and applications. Note that a high percentage of the material builds upon itself. More specifically, the solution of a PDE as a Fourier series via separation of variables follows the topic of Fourier series and complex analysis notions like conformal mappings and analytic functions follow the PDE topic in which harmonic functions are introduced. Much of the material is classic engineering mathematics that has admittedly been around for decades and hasn’t really changed. However, modern software such as MATLAB2,3 or Mathematica4, brings the material more to life and offers tremendous advantages for related numeric, graphic, and associated student project work. Originally, vector integral Calculus (specifically the 3 integral theorems due to Green, Gauss, and Stokes) was not going to be covered since theoretically these topics would be covered in Multivariable Calculus (MATH 210). However, based on discussions with the Mathematics Department, it was felt that it would be a good idea as most engineering students find it to be difficult material and it can’t be guaranteed that all 3 theorems would be covered in MATH 210. This decision had the effect of bumping another popular topic, namely, the Calculus of Variations, but the resultant slate of topics fits well with available texts on engineering mathematics currently in print (unlike the text by Wylie and Barrett5, which the author has taught out of), such as that written by Kreyszig in his 10th edition6. As an interesting side story, the author (who is clearly dating himself) was first exposed to Kreyszig’s 3rd edition decades ago7. The existing engineering education academic literature on the topic of applied mathematical preparation of students for engineering graduate school is very limited, as determined from Compendex and Google Scholar. This is perhaps due to the greater need associated with issues dealing with required undergraduate engineering mathematics courses (i.e. Calculus, introduction to differential equations and linear algebra) and the fact that the smaller population of graduate school-bound students are better at mathematics anyway. On the required undergraduate mathematics topic, the literature is quite extensive, dealing with issues like the mathematical preparation of freshman students, improving the performance and retention of students, especially demographically under-represented groups, projects, and use of technology, such as elearning, classroom technology, etc. One school, Wright State University, with NSF funding, even revamped their entire required engineering mathematics curriculum to improve program attributes such as student retention (Klingbeil and Bourne8). That said, Sun et. al9. advocate for the use of applied mathematical project work as means of better preparing students for engineering graduate school and Siegenthaler pushes for plowing through a rigorous text by Arfken and Weber on mathematical physics10. Specific engineering disciplines may have more focused or nuanced needs. For example, in chemical engineering, Kauffman11 makes the case that applied mathematics is one of the top 3 topical areas within a US Ph.D. program. Lastly, in mechanical engineering, Yerion12 uses finite differences and other numerical analysis techniques in a course that is a prerequisite for heat transfer. 2. Commentary on Topical Areas Vector Integral Calculus: It is the opinion of the author that, generally speaking, vector differential Calculus is an easier topic than vector integral Calculus, and that students have some proficiency with this topic from their Multivariable Calculus course, which of course is a prerequisite. The main focus then is on the 3 famous integral theorems from Green, Gauss, and Stokes, namely: Green: Gauss (Divergence): + � ∇∙� = �= � � � − � � �∙� � Page 26.161.3 Stokes: ∇ × � ∙ � = �� � ∙ � � Beyond coverage of an overview of the derivations and typical verification-style example problems (e.g. checking both sides) some other interesting discussions can take place such as the relationship between these theorems (e.g. Green’s is an obvious special case of Stokes), and for a given problem, which side would be easier to evaluate. With the internet, students were looking up this topic on Kahn Academy13 and learning a bit of history associated with these mathematicians, including their desire to achieve integral order reduction, which is really the mathematical spirit of these theorems, especially when further generalized (e.g. to higher dimensional manifolds), beyond the scope of this course however. Applications of these theorems were of primary interest, including calculation of area using a line integral (using Green’s Theorem), to fluid mechanics & electromagnetic field theory that involve notions such as flux, circulation, projections, and sources & sinks. It was noted that your standard GPS unit, such as an eTrex Legend14 has the capability of calculating area based on a closed track (or path) file. This topic also provided an excuse to purchase a planimeter for a few hundred dollars15 (see Fig. 1), a neat mechanical instrument used to calculate the area of closed curves (e.g. for use in quantifying experimental aspects of thermodynamic cycles) that, while not in common use today due to technical obsolescence, is of historical significance, much like the now obsolete slide rule. Exploring the details of its usage and operational mysteries was the basis for a student project. Fig. 1 Area measuring planimeter instrument (Model L-10 from LASICO Inc.). Page 26.161.4 Introduction to Fourier Series: Basic coverage of interest to both electrical and mechanical engineering students was achieved with a focus on applications that typically take the form of cosine and sine series representations of different periodic scalar-valued real functions, e.g. see Fig. 2. Other standard tricks of the trade, like odd/even extensions, non-periodic functions, functions of arbitrary periods, and Parseval’s Theorem for error estimates were covered as well. This topic lends itself to the use of MATLAB for practical numerical and graphical studies, such as convergence, both from a quantitative and qualitative point of view as well as studying the frequency content and how the oddness or evenness of different functions affects the coefficients, such as if they are zero or nonzero. Students were impressed with what can be done with only a few lines of code in a MATLAB script style M-file and it gave them some more practice with MATLAB, especially working with nested loops and arrays, which are used in other engineering courses such as Dynamics (ENGR 322) and Control Systems and Automation (ENGR 410). Lastly, the utility of the Fourier series representation was brought to use in solving linear ordinary differential equations (ODEs) with constant coefficients but a “non-friendly,” but periodic right hand side (RHS) that serves as a forcing function. This last topic was a good opportunity to refresh the student’s knowledge regarding different partitions of an ODE solution (i.e. homogeneous/particular and transient/steady-state) which have both mathematical and engineering significance. Introduction to Partial Differential Equations: The classic separation of variables method in which the solution is represented as a Fourier series was covered in detail and illustrated through a number of common engineering examples dealing with the PDEs associated with strings/membranes, vibrating beams, steady and unsteady heat conduction, and electro-statics and fluid mechanics. We’re talking about equations such as: Laplace (2D): ∇2 � = � � + � Heat Conduction (1D, 2D): � Wave (1D, 2D): � = � � � 2� � � , � � � = = = 2� � 2 � � , � � + = � � 2 � � + � � = 2 Vibrating Beam: � � And, of course appropriate boundary and/or initial conditions must be provided. Beyond asking students to plow through various solution steps, including all of the symbolic manipulations, dealing with boundary conditions, and well-posedness, which is important for establishing a good foundation, it’s a good excuse to do some more numerical work in MATLAB, as shown in Fig. 3. For problems involving time, MATLAB movies (“animation” vs. “movie” is perhaps a more accurate, and less over-stated term) can be made and with the creation of files with the proper format, played on a typical smart phone, such as an iPhone5-6 which are in common use by students. They seemed to really like this idea and could show video snippets lasting only a few seconds (and then looped) to others such as classmates, friends, and relatives. Again, they were impressed with the power of just a few lines of MATLAB code and as a fun class topic, they were shown how to solve the 1D wave equation and make an animation with only 10 essential lines of MATLAB code (see Appendix). Fig. 3 Steady-state solution of 2D thermal conduction problem. Page 26.161.5 Fig. 2 Fourier series approximation of a periodic sawtooth function with N = 15 terms using MATLAB. Introduction to Complex Analysis: This topic was covered primarily to lay the theoretical background for the rich set of applications in engineering using conformal mapping. A review of complex numbers and complex algebra served as a warm-up for introducing some old and new ideas in the complex realm such as limits, continuity, differentiation, analyticity (including the Cauchy-Riemann equations), and working with common elementary functions, all of which have familiar real versions. The tie-in to the notion of a harmonic function was a natural, having just covered a PDE introduction. Details and nuances that get into issues like multiplicity, branches of functions, clarification of domains and ranges, and elementary point-set topology concepts were covered. Coverage of the point-set topology ideas were needed to better understand certain theorems, such as their assumptions and limitations. In an effort to become somewhat proficient with basic knowledge, students were challenged with evaluating odd expressions such as � � , using the delta method to establish derivatives, differentiation of unusual functions, etc. Conformal Mapping and Applications: One of the big ideas on this topic, expressed as a consequence of the Harmonic Functions under Conformal Mapping Theorem, is that one can transform a problem on a difficult domain to a more favorable domain for which an analytical solution can be obtained through standard means, and then transform back to the original problem domain, thereby obtaining the solution to the original problem posed. Of course, prior to using the above mentioned theorem to solve some neat engineering analysis problems, one needs to understand what a conformal mapping is and to exercise one’s thinking about this idea, such as when it breaks down, etc., as it is only a local concept. One of the activities promoted was a conformal mapping artwork contest, in which students could create their own exotic artwork that graphically illustrated conformality, at least approximately through the mapping of orthogonal grids of non-infinitesimal resolution, as shown in Fig. 4. Fig. 4 Example conformal mapping “artwork” generated from applied to an origin centered square grid and from the origin to infinity. = + + = , including = 2 applied to a wedge emanating radially Page 26.161.6 As a prelude for solving some interesting engineering problems, the idea of a linear fractional transformation (LFT), or Mobius transformation was introduced, namely: + , − ≠ = = + with , , , being complex constants. LFTs are conformal, with the additional property of transforming circles or lines to circles or lines and often provide a convenient mechanism for transforming difficult problem domains into ones for which a solution is more easily obtained (and then back again to the original problem domain). As an example of this approach, and to showcase one of the author’s favorite problems from the course, consider a MATLAB version of Kreyszig’s Example 1 of Section 2 in Chapter 18 that aims to find the electro-static potential between 2 non-concentric circles. Its solution, complete with both orthogonal constant potential and constant stream function lines, is shown in Fig. 5. It entails applying an LFT to create a problem involving concentric circles, for which a solution is known, and then transforming back to the original geometry. This very problem, with the same conceptual solution approach using conformal mapping, is also discussed in Jeng-Tsong et. al16. Another interesting problem is a 2D heat conduction problem with mixed boundary conditions (see Fig. 6). Fig. 5 Electro-static equi-potential and equi-stream lines for asymmetric axial cables. Fig. 6 Mixed boundary value problem involving heat flow with both boundary insulation [ − , ] and temperature sources [both −∞, − & , ∞ ]. The last topic of the course introduces the idea of a complex potential function � and Figs. 78, with the aid of MATLAB, illustrate the famous examples involving 2D flow past a circular cylinder (� = + / ) and a family of Joukowski airfoils ( = + / applied to a special set of circles). Page 26.161.7 Project: Project work gives the students an opportunity to delve more deeply into a specific problem, focus on an application of interest to them, be creative, perform some numerical and graphical computing using MATLAB, possibly some other software like Mathematica and Simulink2,17 and lastly, present their work to the class in our own “conference.” The objective of the project is to gain expertise in using applied mathematics to solve an engineering problem. In the process the student teams are engaged in the following activities (as given to students):  Understand and/or clarify step-by-step (e.g. fill-in missing steps) how applied mathematics is used to model an engineering system of interest  Create appropriate and mathematically correct simulation models using MATLAB incorporating relevant parameters     With specific scenarios in mind, perform MATLAB simulation runs, plot results, and create animations Prepare a technical report (with CD of all relevant computer files) and make a short presentation to the class (you can use a few PPT slides, or not!) Demonstrate how the engineering device works (if applicable) Receive feedback on the technical report with an opportunity to correct/address any issues raised with the goal of improving the overall quality Fig. 7 Streamlines associated with flow over a 2D circular object with � = + / . Fig. 8 A family of 3 Joukowski airfoils plus a center curve. On the last day of class an “Advanced Engineering Mathematics Conference” was held. Grading of the project incorporated the following elements:  Title page (1%)  Abstract (4%)  Development of System Equations and Solution (30%)  MATLAB Numerical Work (30%)  Scenarios (5%)  Presentation (10%)  CD contents (20%, MATLAB graphics including any animations + storage of all files) Table 1 identifies specific projects undertaken by different student teams for all 3 semesters with Figs. 9-11 illustrating sample project work. Table 1 Student Project Listing Page 26.161.8 Spring 2011 Piston Motion in a Hydraulic Accumulator (Sean Engen, Frances Van Sloun) Solving Rectangular Vibrating Membranes using Double Fourier Series (Michael Cowdrey, Ryan Huynh) 1-D Transient Heat Conduction Application (Colin Grist, Chris Cogan) Spring 2012 Partial Differential Equations, Fourier Series, and Vibrating Membrane Equations (Perry Jagger, Luke LoPresto) Vibration of a Rectangular Membrane (Adam Gibson, Andy Edmunds) Electrostatics of Nonsymmetrical Semicircular Plates: Mapping constant potential lines using Linear Fractional Transformations (Artem Mosesov, Julie Olson) The Mechanical Planimeter and Green’s Theorem (David Bailly, Matthew Moore) Spring 2014 2D Steady Heat Flow through a Rectangular Plate (Noel Naughton, Alex Krause) Vibration of a Rectangular Membrane (Lucas Unger, Cole Hazelbaker) Longitudinal Vibrations in an Elastic Bar (Brendan O’Connell, Drew Stangler) Vibration of a Rectangular Membrane (Sam Miller) Fig. 9 Time-varying rectangular membrane deflection frame sequence (Lucas Unger, Cole Hazelbaker). Fig. 10 Constant potential lines between 2 nonsymmetrical semicircular plates at different potentials (red & dark blue; Artem Mosesov, Julie Olson). 3. Lessons Learned, Other Issues, and Student Feedback Having taught this course a total of 3 times, there have been some reoccurring issues and lessons learned that are worth mentioning, dealing with: (1) vector integral Calculus, (2) programming loops, (3) ODE solution structures, (4) the impact of friction on various mathematical theories and their practical usage, and (5) student feedback. Some of these (and related) issues have been previously shared with the Mathematics Department18. Page 26.161.9 Vector Integral Calculus: Theoretically, in their Multivariable Calculus course (MATH 210) the students have seen the 3 famous integral theorems due to Green, Gauss, and Stokes, and should be comfortable with this material. As conjectured by instructors of those courses in the Mathematics Department, another pass through the material, perhaps from a slightly different perspective, would be beneficial. I found this to be the case as well. They may have seen the material before but in talking to them and working through the material, both theory and application problems, it is clear that another pass is warranted. From a learning and pedagogy point of view, this practical observation is consistent with the idea that if you want to learn mathematics at level N very well, you need to see it at level N + 1! This situation also presents an opportunity to delve into some important engineering applications and interpretations of related theories. Fig. 11 Simulink model of piston dynamics (Sean Engen, Frances Van Sloun). Programming Loops: Loops and the generalization to nested loops (e.g. of order 2, 3, or even 4) are not that well understood by most of the students, in spite of having taken a 4 CR Introduction to Programming course (CISC 1301) based on MATLAB and C. Of course, when solving PDEs numerically using series expansions this becomes an issue, especially as the spatial dimension order is increased beyond 1, as the most straight forward approach is to use nested loops. Page 26.161.10 ODE Solution Structures: Maybe it’s just the author, but the “old school” method in which he learned elementary differential equations is unfamiliar to the students and consequently they have trouble following the construction of certain solution structures. More specifically, this refers to the solution of linear, non-homogeneous, ordinary differential equations with constant coefficients with a “friendly” RHS, as in Rainville and Bedient19, the author’s undergraduate ODE book. “Friendly” in the sense that conversion from a non-homogeneous ODE to a homogeneous ODE is made possible through the creation and application to both sides of a minimal-order polynomial-style differential operator with constant coefficients based on an analysis of the roots associated with the RHS. And of course, more generally, the “roots perspective” lays the groundwork for the structure of both the homogeneous and particular solutions that very clearly and readily deals with issues such as multiplicity (within the LHS, RHS, or hybrid LHS/RHS) and complex vs. real roots. Anyway, enough about refreshing the reader on the details of this method, the point being that the students lack the “roots perspective” and therefore are at a loss for where certain solution structures come from, relying only on a recipe for a few special cases, typically being lower-order (e.g. second order). Within the course this issue shows up when solving linear ordinary ODEs with constant coefficients on the LHS, but a “non-friendly” RHS. The solution strategy is to represent the RHS as a Fourier series and on a term by term basis solve an infinite series of implied non-homogeneous ODEs (indexed with the ℎ series term) and then add all of these pieces of the solution together to construct the entire solution. Impact of Friction on Various Mathematical Theories their Practical Usage: Critics of the topic of “potential flow” and its analogies in other energy domains (like electro-statics) accurately point out that this elegant mathematical theory which connects key ideas in both complex analysis and PDEs via analytic functions whose real and imaginary components are harmonic functions, doesn’t readily handle internal friction and therefore it is not that useful. Rather, numerical solutions to problems involving friction that form the basis of finite element analysis (FEA) is preferred, especially when powerful commercial software is available. The author’s view on this issue is that, while true, in many cases ignoring internal friction can be justified, it is important to follow the historical development of the applied mathematical theories and their application to engineering analysis problems, and that the topics (especially dealing with harmonic functions) are good for the mathematical maturity of the student who will undoubtedly encounter more advanced topics that use PDEs and/or complex analysis, should they go on to engineering graduate school in a rigorous program. Student Feedback: Feedback comes in 2 forms: (1) IDEA20 numbers and written comments, and (2) comments heard directly from students. As for IDEA feedback, the average course rating was “very good” and written comments have been generally positive. One student wrote “I feel much better at understanding the math behind engineering,” with the most negative comment being “More problems that have real-world applications would be beneficial.” As for anecdotal feedback, one student who was back home visiting from his new graduate school felt that he had an advantage over other engineering graduate students when studying the underlying principles of how Magnetic Resonance Imaging (MRI) machines work since he knew about conformal mapping. 4. Conclusions and Future Work The offering of the above described technical elective course on engineering mathematics makes good sense for both of our BSEE and BSME programs, especially for graduate-school bound students. Informal feedback from previous students who have moved on to graduate school has been that it has been useful in terms of better preparing students for the applied mathematical rigors encountered. Over time the hope is that the enrollment will grow and an effort is underway to make it an official course listed in the catalog. Acknowledgements Page 26.161.11 The author acknowledges Dr. Don Weinkauf, dean of the School of Engineering, who, based on his experience in working with senior-level chemical engineering students, advocated for the development and delivery of this course (by the author) and Dr. Cheri Shakiban of the Mathematics Department, who provided critical review and input regarding the development of this course. Additionally, acknowledgement of the efforts of the students who took this course and worked on interesting projects is expressed. Bibliography [1] University of St. Thomas, Undergraduate Catalog: 2012-2014, St. Paul, MN, 2012. [2] http://www.mathworks.com/. [3] Hanselman, D. and Littlefield, B., Mastering MATLAB 7TM, Pearson Prentice Hall, Upper Saddle River, NJ, 2004. [4] www.wolfram.com. [5] Wylie, C. R. and Barrett, L. C., Advanced Engineering Mathematics, 6th Edition, McGraw-Hill, New York, NY, 1995. [6] Kreyszig, Advanced Engineering Mathematics, 10th Edition, Wiley, Hoboken, NJ, 2011. [7] Kreyszig, Advanced Engineering Mathematics, 3th Edition, Wiley, Hoboken, NJ, 1972. [8] Klingbeil, N. W., and Bourne, A., “A national model for engineering mathematics education: Longitudinal impact at Wright State University,” 120th ASEE Annual Conference and Exposition, June 23-26, 2013. [9] Sun, C., Dusseay, R., Cleary, D., Sukumaran, B., and Gabauer, D., “Open-ended projects for graduate schoolbound undergraduate students in civil engineering,” ASEE Annual Conference and Exposition, p 7647-7656, June 24-27, 2001. [10] Siegenthaler, K., “Advanced mathematics preparation for graduate school of undergraduate science and engineering students,” ASEE Annual Conference and Exposition, p 297-308, June 20-23, 2004. [11] Kauffman, D., “The core graduate chemical engineering program: Does it exist?,” ASEE Annual Conference and Exposition, p 7435-7440, Montreal, Quebec, June 16-19, 2002. [12] Yerion, K. A., Computer experiments with diffusion: finite difference, round-off error and animal stripes?, International Journal of Mechanical Engineering Education, v 41, n 3, p 227-45, Manchester University Press, July 2013. [13] http://www.khanacademy.org/. [14] Garmin Inc., eTrex HC series: personal navigator Owner’s Manual, Olathe, KS, 2007 (www.garmin.com/products/etrexLegend). [15] LASICO (Los Angeles Scientific Instrument Company) Inc., Instruction Manual for Mechanical Polar Planimeters, Los Angeles, CA (www.lasico.com). [16] Jeng-Tzong, C., Ming-Hong, T., and Chein-Shan, L., Conformal mapping and bipolar coordinate for eccentric Laplace problems, Computer Applications in Engineering Education, v 17, n 3, p 314-22, September 2009. [17] Dabney, J. B. and Harman, T. L., Mastering SIMULINKTM, Pearson Prentice Hall, Upper Saddle River, NJ, 2003. [18] Hennessey, M. P., “Applied Mathematics in an Undergraduate Engineering Program,” presented to the statistics contingent of University of St. Thomas Mathematics Department, May 13, 2011. [19] Rainville, E. D. and Bedient, P. E., A Short Course in Differential Equations, 5th Edition, Macmillan, New York, 1974. [20] http://ideaedu.org. Appendix: MATLAB Script File Listing: StringMovieCompact.m Page 26.161.12 c=1.0;k=0.5;L=1.0;N=3;M=500;TT=5.0;x=[0.000:0.01:1.00]';hold off for T = 1:1:M t=(T-1)*(TT/(M-1));u = zeros(size(x)); for n = 1:1:N ln=c*n*pi/L;bn=(8*k/(n^2*pi^2))*sin(n*pi/2); u=u+bn*cos(ln*t)*sin(n*pi*x/L); end plot(x,u,'LineWidth',6,'Color','green');xlim([0.0 1.0]);ylim([-0.5 0.5]) String(:,T) = getframe; end