Treffer: Embedding numerical methods and MATLAB programming in a fluid mechanics course for undergraduates in engineering technology

Undergraduate students in engineering technology are typically not required to take any courses on numerical methods or computational techniques and thus have little or no knowledge of many basic numerical approaches commonly used in engineering disciplines, such as root finding, curve fitting, numerical integration, and numerical differentiation. In addition, they are only required to take one introductory level programming course and thus usually experience difficulty when working on course projects involving extensive programming. However, the industry is demanding different skillsets than the ones that were expected just a decade ago. Numerical and programming skills are becoming increasingly important. In this case study, the effectiveness of embedding numerical methods and MATLAB programming in MMET 303 Fluid Mechanics and Power, a four-credit junior-level required course offered every semester for undergraduates at the Department of Engineering Technology and Industrial Distribution at Texas A&M University, was assessed. A series of learning modules were purposefully designed and implemented as a trial test in the classes offered in the semester of Fall 2023. Instructor's observation, submitted assignments, and survey results were analyzed. The results suggested that embedding numerical methods and associated MATLAB programming into a required course enhanced students’ analytical skills of tackling practical problems, helping them become better prepared as they move on into the industrial companies or the graduate schools.

AN0185986047;8d801jul.25;2025Jun19.02:57;v2.2.500

Embedding numerical methods and MATLAB programming in a fluid mechanics course for undergraduates in engineering technology 

Undergraduate students in engineering technology are typically not required to take any courses on numerical methods or computational techniques and thus have little or no knowledge of many basic numerical approaches commonly used in engineering disciplines, such as root finding, curve fitting, numerical integration, and numerical differentiation. In addition, they are only required to take one introductory level programming course and thus usually experience difficulty when working on course projects involving extensive programming. However, the industry is demanding different skillsets than the ones that were expected just a decade ago. Numerical and programming skills are becoming increasingly important. In this case study, the effectiveness of embedding numerical methods and MATLAB programming in MMET 303 Fluid Mechanics and Power, a four-credit junior-level required course offered every semester for undergraduates at the Department of Engineering Technology and Industrial Distribution at Texas A&M University, was assessed. A series of learning modules were purposefully designed and implemented as a trial test in the classes offered in the semester of Fall 2023. Instructor's observation, submitted assignments, and survey results were analyzed. The results suggested that embedding numerical methods and associated MATLAB programming into a required course enhanced students' analytical skills of tackling practical problems, helping them become better prepared as they move on into the industrial companies or the graduate schools.

Keywords: Numerical methods; MATLAB programming; engineering technology; fluid mechanics; course development

Introduction

According to Engineering Technology Council of the American Society of Engineering Education (ASEE), engineering technology is a profession in which knowledge of the applied mathematical and natural sciences gained by higher education, practical experience, and competence developed in a specific field is devoted to application of engineering principles and the implementation of technological advances for the benefit of humanity through its focus on product improvement, manufacturing, and automation of technological processes and operational functions.[1] It is a unique blending of engineering and business. Compared with other engineering disciplines, engineering technology has a stronger focus on educating students with hands-on skills and entrepreneurship.

Undergraduate students in engineering technology programs are typically not required to take any courses on numerical methods or computational techniques and thus have little or no knowledge of many basic numerical approaches commonly used in engineering disciplines, such as root finding, curve fitting, numerical integration, and numerical differentiation. In addition, they are only required to take one introductory level programming course and thus usually experience difficulty when working on course projects involving extensive programming. Even worse, they are typically deterred by many negative perceptions about numerical methods and the associated programming activities, for example, it is difficult to learn and requires significant amounts of time, or it is irrelevant and only for researchers. However, this fast-changing world is demanding different skillsets than the ones that were expected just a decade ago. Many industrial companies, both large and small, explicitly require or prefer their job applicants to have numerical and programming skills. As technology advances in the industrial companies, the undergraduate curriculum must advance accordingly. Various innovative practices are ongoing, aiming at strengthening undergraduates' numerical and programming skills.

In this case study, the effectiveness of embedding numerical methods and MATLAB programming in MMET 303 Fluid Mechanics and Power, a four-credit junior-level required course offered every semester for undergraduates at the Department of Engineering Technology and Industrial Distribution at Texas A&M University, was assessed. A series of learning modules were purposefully designed and implemented as a trial test in the classes offered in the semester of Fall 2023. The course includes two 75-minute lectures and one 2-hour laboratory session each week. Laboratory activities are used to enhance lecture materials and provide students with hands-on experience in fluid mechanics testing. Typically, approximately 120 undergraduate students enroll in this course every semester, predominantly junior (approximately 80%) and senior (approximately 20%) engineering technology majors. In the semester of Fall 2023, 125 students were enrolled in the course, and the author served as the instructor. If proven useful, this practice could be employed in MMET 303 in future years; learning modules on more advanced numerical methods could be developed; and similar activities could be embedded into other required courses. Texas A&M University's Institutional Review Board reviewed and approved the research study under Exempt Category 1 at 45 CFR 46.104. The IRB number is 162456.

Background

Numerical methods, also referred to as computer mathematics, are techniques by which mathematical problems are formulated so that they can be solved with arithmetic and logical operations.[2] In the precomputer era, the time and drudgery of implementing such calculations seriously limited their practical use. As computers moved from laboratories to classrooms and homes, the role of numerical methods in engineering and scientific problem solving has exploded, becoming an essential part of every engineer's and scientist's basic education.

Numerical methods greatly expand the types of problems students can address. They are capable of handling large systems of equations, nonlinearities, and complicated geometries that are not uncommon in engineering and science and that are often impossible to solve analytically with standard calculus. Numerical methods also allow students to use commercially available software with insight, that is, the intelligent use of the software is greatly enhanced by a better understanding of the basic theory underlying the methods. On the other hand, when students are conversant with numerical methods and adept at programming, they can design their own programs to solve the problems that cannot be tackled using commercially available software.

Fluid mechanics courses often involve diverse numerical methods and plenty of valuable work has been done in the past. For example, Praks and Brkic[3] presented how the nonlinear Colebrook equation can be used for introducing in curricula various root-finding methods, including single fix-point iterative methods, two-point iterative methods, and multipoint iterative methods. Castilla and Pena[4] introduced various curve-fitting techniques when asking students to fit a set of experimental data on the rheological behavior of creams and lotions, that is, the viscosity versus shear stress data taken from the Center for Industrial Rheology,[5] to the classical Carreau model. Zamora et al.[6] introduced various numerical methods to solve partial differential equations, such as finite difference methods and finite volume methods, through a course project involving solving the Navier-Stokes equation governing a laminar, forced, two-dimensional flow between two flat plates. Campo et al.[7] introduced the numerical method of the transversal method of lines for the prediction of the friction factors and the axial descent of the convection coefficients in the upstream region of internally finned tubes for laminar regimes.

Purpose

The goal of this case study is to seamlessly integrate numerical methods and associated MATLAB programming into a fluid mechanics course in the form of learning modules and course projects. Four modules will be made available to the students, including root finding, curve fitting, numerical integration, and numerical solution of ordinary differential equations (ODEs). Each module is a concise yet self-sufficient teaching unit, and it takes the learn-by-example approach, emphasizing the practical application of the numerical methods rather than the complexity of the mathematics involved. Each learning module is associated with two course projects. Upon successful completion of the course projects, the students are expected to be able to develop scripts and graphics within MATLAB environment, use numerical methods to handle systems that are impossible to solve analytically with standard calculus, and design numerical procedures to tackle practical problems in various engineering disciplines that cannot be approached using commercially available software.

Without altering the current curricula, this practice has the effect of broadening the essential skillset of engineering technology graduates and stimulating students' interest in numerical methods and programming, timely addressing the demands from the industry. The practice also helps the Bachelor of Science programs offered by the Department of Engineering Technology and Industrial Distribution meet the Accreditation Board for Engineering and Technology (ABET) requirements, as the student outcomes in the recently updated ABET engineering accreditation criteria include "an ability to apply knowledge, techniques, skills and modern tools of mathematics, science, engineering, and technology to solve broadly defined engineering problems appropriate to the discipline."[8]

Method

The course aims to give students broad information on both the principles of fluid mechanics and the application of these principles to practical, applied problems. The book entitled "Applied Fluid Mechanics" by Mott and Untener has been used as the textbook.[9] The course content is tailored to fit a one-semester schedule. Four learning modules and eight course projects were developed to embed numerical methods and MATLAB programming into the course, as shown in Figure 1, each learning module being associated with two course projects.

Graph: Figure 1. Four learning modules and eight course projects were embedded into the course.

Curve-fitting module

During the 1st week, students obtain experimentally measured data of the dynamic viscosity of liquids <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#956;</mi></math> as a function of the absolute temperature T, which are supposed to be in excellent agreement with Andrade's equation,[2] as shown in equation (1). <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#956;</mi><mo>=</mo><mi>D</mi><msup><mi>e</mi><mrow><mi>B</mi><mo>/</mo><mi>T</mi></mrow></msup></math>

Graph

where both D and B are empirical parameters. This provides the instructor a valuable opportunity to introduce to curricula some of the curve-fitting techniques. The learning module, focusing on a curve-fitting technique called linearization of nonlinear relationships, is shown in Figure 2. It illustrates how to apply this technique to fit the experimental data to an exponential equation, a power equation, and a saturation-growth-rate equation, respectively. Students are asked to fit the experimental data to Andrade's equation, which is an exponential equation.

Graph: Figure 2. The learning module on linearization of nonlinear relationships.

During the 12th week, students obtain experimentally measured data of the velocity profile of a turbulent flow in an open channel. In a near wall flow, there exist three layers, that is, the viscous sublayer, the buffer layer, and the log-law layer. In the log-law layer, which occupies the majority of the water column, the time-averaged velocity at a certain point, U, is proportional to the logarithm of the distance from the wall, z, as shown in equation (2). <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi><mo>=</mo><mi>D</mi><mspace width="0.25em" /><mrow><mi mathvariant="normal">ln</mi></mrow><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mi>z</mi><mi>B</mi></mfrac></mrow></mstyle></math>

Graph

where both D and B are empirical parameters.[10] As a review, students are asked to fit the experimental data to the logarithm equation. They should first linearize the equation and then use linear regression to estimate D and B.

Numerical integration module

During the 3rd week, when students learn forces due to static fluids, they are asked to calculate the total force exerted by water to the upstream face of a dam, f, which can be determined by multiplying pressure times the area of the dam face.[9] Because both pressure and area vary with elevation z, the total force can be obtained by evaluating equation (3). <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>=</mo><munderover><mo>&#8747;</mo><mn>0</mn><mi>D</mi></munderover><mspace width="0.2em" /><mi>&#961;</mi><mi>g</mi><mi>w</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mrow><mi>D</mi><mo>&#8722;</mo><mi>z</mi></mrow><mo stretchy="false">)</mo><mi>d</mi><mi>z</mi></math>

Graph

where ρ is the density of water, g the acceleration due to gravity, D the total elevation of the water surface above reservoir bottom, and w(z) the width of the dam face at elevation z. w(z) may not be in the form of an analytical function and thus the instructor can take advantage of this opportunity to introduce in curricula various numerical integration techniques.

Numerical integration is used where an integral is impossible to solve analytically. The learning module on numerical integration is shown in Figure 3, including both composite trapezoidal rule and composite Simpson's 1/3 rule. The technique of composite trapezoidal rule is to subdivide the interval and approximate the integral of the function with a number of trapezoids, whereas the approach of composite Simpson's 1/3 rule is to subdivide the interval and approximate the integral of the function with a number of quadratics. During the 3rd week, students are asked to calculate the total force exerted by water to the upstream face of a dam using both methods. During the 8th week, students are asked to calculate the volumetric flow rate Q of a fluid flowing through a circular pipe based on the velocity profile,[9] as shown in equation (4). <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi><mo>=</mo><munderover><mo>&#8747;</mo><mn>0</mn><mi>R</mi></munderover><mspace width="0.2em" /><mi>v</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mrow><mn>2</mn><mi>&#960;</mi><mi>r</mi></mrow><mo stretchy="false">)</mo><mi>d</mi><mi>r</mi></math>

Graph

where R is the radius of the circular pipe, r the radial distance measured outward from the center of the pipe, and v(r) the velocity at radial distance r. v(r) may not be in the form of an analytical function. As a review, students should calculate the volumetric flow rate using the two numerical integration techniques and discuss the advantage and disadvantages of each method.

Graph: Figure 3. The learning module on numerical integration methods.

Numerical solution of ODEs module

The Roman natural philosopher, Pliny the Elder, purportedly had an intermittent fountain in his garden.[2] The water enters a cylindrical tank at a constant flow rate, Q, and fills until the water reaches a certain level, y<subs>high</subs>. At this point, water siphons out of the tank through a circular discharge pipe, producing a fountain at the pipe's exit. The fountain runs until the water level decreases to a certain level, y<subs>low</subs>, whereupon the siphon fills with air and the fountain stops. The cycle then repeats. Assuming an initial condition of an empty tank, that is, y(0) = 0, students are asked to compute and plot the level of the water in the tank as a function of time over 100 s, which can be derived based on equation (5). <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mi>Q</mi><mo>&#8722;</mo><mi>s</mi><mi>c</mi><msqrt><mn>2</mn><mi>g</mi><mi>y</mi></msqrt><mi>&#960;</mi><msup><mi>r</mi><mn>2</mn></msup></mrow><mrow><mi>&#960;</mi><msup><mi>R</mi><mn>2</mn></msup></mrow></mfrac></mrow></mstyle></mstyle></math>

Graph

where R is the radius of the cylindrical tank, r the radius of the pipe, g the acceleration due to gravity, c the discharge coefficient, and s a dimensionless variable that equals 0 when the fountain is off and equals 1 when it is flowing. This ODE is discontinuous at the point that the siphon switches on or off, and thus many MATLAB built-in ODE solvers, such as ode45, ode23tb, and ode23 s, fail to generate correct solutions. The instructor can take advantage of this opportunity to introduce in curricula various techniques to numerically solve ODEs.

The learning module on numerical techniques to solve ODEs is shown in Figure 4, including Euler's method, Heun's method, Ralston's method, and the classical fourth-order Runge-Kutta method. During the 5th week, students are asked to use these techniques to solve the intermittent fountain problem. As a review, during the 13th week, students are asked to compute the motion of a sphere moving relative to a fluid flow,[9] based on equations (6) and (7). <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mi>d</mi><mi>x</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><mo>=</mo><mi>u</mi></mstyle></math>

Graph

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mi>d</mi><mi>u</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><mspace width="0.25em" /><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mi>&#961;</mi><mi>c</mi><mi>&#960;</mi><msup><mi>R</mi><mn>2</mn></msup></mrow><mrow><mn>2</mn><mi>m</mi></mrow></mfrac></mrow><mo stretchy="false">(</mo><mrow><msup><mi>V</mi><mn>2</mn></msup><mo>&#8722;</mo><mn>2</mn><mi>u</mi><mi>V</mi><mo>+</mo><msup><mi>u</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mstyle></mstyle></math>

Graph

where x is the displacement of the sphere, u the velocity of the sphere, V the velocity of the fluid flow, ρ the density of the fluid, R the radius of the sphere, c the drag coefficient, and m the mass of the sphere. Under the guidance of the instructor, students should extend the techniques used for single equations to a system of ODEs. Students are also asked to discuss the advantage and disadvantages of each method.

Graph: Figure 4. The learning module on numerical methods to solve ordinary differential equations.

Root-finding module

The Colebrook equation is an empirical relationship among the friction factor f, the Reynolds number Re, and the relative roughness <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#949;</mi><mo>/</mo><mi>D</mi></math> , as shown in equation (8). It is often used to calculate the friction factor in gas/liquid pipelines in turbulent flow. It was developed by Colebrook in 1937,[11] when he was a Ph.D. student under the supervision of Professor White at Imperial College London. The experiment was conducted using a series of pipes whose inner surface was covered with sand of different grain sizes, creating various values of surface roughness. The Moody diagram represents a graphical interpretation of the relationship among the friction factor, the Reynolds number, and the relative roughness for all flow regimes, including laminar, critical, and turbulent. The turbulent portion should be solved based on the Colebrook equation.[12] <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mn>1</mn><mrow><msqrt><mi>f</mi></msqrt></mrow></mfrac></mrow><mo>=</mo><mo>&#8722;</mo><mn>2</mn><msub><mrow><mi mathvariant="normal">log</mi></mrow><mn>10</mn></msub><mrow><mo>(</mo><mrow><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mn>2.51</mn></mrow><mrow><mi>R</mi><mi>e</mi><msqrt><mi>f</mi></msqrt></mrow></mfrac></mrow><mo>+</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mi>&#1013;</mi><mo>/</mo><mi>D</mi></mrow><mrow><mn>3.71</mn></mrow></mfrac></mrow></mstyle></mstyle></mrow><mo>)</mo></mrow></mstyle></math>

Graph

One of the course projects is to ask students to numerically solve the Colebrook equation and plot the Moody diagram, which serves as a valuable opportunity to introduce in curricula various root-finding methods. The two parts of the learning module on root-finding methods are shown in Figures 5 and 6, respectively. The first part focuses on the bracketing methods, including the bisection method and the false position method. The second part focuses on the open methods, including the Newton-Raphson method and the secant method. In the 7th week, students are asked to solve the Colebrook equation using these methods. As a review, in the 12th week, students are asked to solve the nonlinear Manning equation for a rectangular open channel,[9] as shown in equation (9). <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><msqrt><mi>S</mi></msqrt><msup><mrow><mo stretchy="false">(</mo><mrow><mi>B</mi><mi>H</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>5</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow><mrow><mi>n</mi><msup><mrow><mo stretchy="false">(</mo><mrow><mi>B</mi><mo>+</mo><mn>2</mn><mi>H</mi></mrow><mo stretchy="false">)</mo></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></mfrac></mrow></mstyle></math>

Graph

where Q is volumetric flow rate, S the slope, H the depth, and n the Manning roughness coefficient. Students are asked to discuss the advantages and disadvantages of each method, including robustness, speed, and accuracy, etc.

Graph: Figure 5. Part one of the learning module on root-finding methods includes the bracketing methods, that is, the bisection method and the false position method.

Graph: Figure 6. Part two of the learning module on root-finding methods includes the open methods, that is, the Newton-Raphson method and the secant method.

It is extremely important to let the students know that different problems have distinct numerical characteristics and careful selection of the most appropriate numerical scheme is crucial. To plot the Moody diagram, either the bisection method or the false position method can be used. However, the bisection method is clearly superior to the false position method because the false position method has a major weakness, that is, its one-sidedness. As iterations are proceeding, one of the bracketing points will tend to stay fixed. This can lead to poor convergence, particularly for functions with significant curvature. For example, either the bisection method or the false position method can be used to locate the root of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mfrac><mn>1</mn><mrow><msqrt><mi>f</mi></msqrt></mrow></mfrac></mrow><mo>+</mo><mn>2</mn><msub><mrow><mi mathvariant="normal">log</mi></mrow><mn>10</mn></msub><mrow><mo>(</mo><mrow><mrow><mfrac><mrow><mn>2.51</mn></mrow><mrow><mi>R</mi><mi>e</mi><msqrt><mi>f</mi></msqrt></mrow></mfrac></mrow><mo>+</mo><mrow><mfrac><mrow><mi>&#1013;</mi><mo>/</mo><mi>D</mi></mrow><mrow><mn>3.71</mn></mrow></mfrac></mrow></mrow><mo>)</mo></mrow></math> between 0.0001 and 0.1 when Re = 10<sups>5</sups> and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#1013;</mi><mo>/</mo><mi>D</mi><mo>=</mo><mn>0.005</mn></math> . When eps_step = eps_abs = 10<sups>−7</sups>, the process converges after only 24 iterations using bisection method, whereas it converges after 579 iterations using the false position method. The slow convergence of the false position method is due to the shape of the function, as illustrated in Figure 7. Open methods such as the Newton-Raphson method and the secant method can also be used to plot the Moody diagram, and they usually converge much more quickly than the bracketing methods when they converge. However, their convergence is not guaranteed, and it highly depends on the accuracy of the initial guess.

Graph: Figure 7. Plot of g(f)=1f+2log10(2.51Ref+ϵ/D3.71), which illustrates the slow convergence of the false position method.

Course feedback

According to the instructor's classroom observation, students appeared to be engaged deeply with the course material, and classroom discussions on numerical methods and MATLAB programming were lively and frequent. In addition, some students expressed their interest in numerical methods via email or office hour visits during the semester. The instructor assisted numerous students in MATLAB scripting and debugging during office hours. Students' attendance rate and course performance were both satisfactory.

The submitted assignments were evaluated with rubrics against the ABET criterion "an ability to apply knowledge, techniques, skills and modern tools of mathematics, science, engineering, and technology to solve broadly defined engineering problems appropriate to the discipline."[8] There are 9%, 0%, 21%, 20%, 30%, and 20% of students meet the missing, emerging, developing, practicing, maturing, and mastering level, respectively. The overall outcome was considered as satisfactory.

To collect students' opinions, two different survey instruments have been used. First, there is the end-of-the-semester course survey, which is the anonymous survey uniformly conducted by Texas A&M University for each course at the end of each semester.[13] It is an in-depth survey designed by the university to collect the feelings and thoughts from the students on their overall learning experience, and thus it does not contain questions specifically related to numerical methods or MATLAB programming. Secondly, a Qualtrics survey designed by the instructor is used. The questions are directly related to numerical methods and MATLAB programming.

Students' feedback in the anonymous course survey was overall positive. Students' rating to the most related questions is presented in Table 1. In the semester of Fall 2023, 125 students were enrolled in the course, and 47 students responded to the end-of-the-semester course survey. This response rate is typical, since some of the strategies to encourage students to complete the survey, such as providing bonus point as incentives, are not allowed at Texas A&M University.[13]

Table 1. Students' rating in the end-of-the-semester course survey.

Graph

<table><colgroup><col align="left" /><col align="char" char="." /><col align="char" char="." /><col align="char" char="." /><col align="char" char="." /><col align="char" char="." /></colgroup><thead><tr><th align="left" /><th align="left"><bold>Average</bold></th><th align="left"><bold>Median</bold></th><th align="left"><bold>Mode</bold></th><th align="left"><bold>Standard deviation</bold></th><th align="left"><bold>Variance</bold></th></tr></thead><tbody><tr><td>What portion of the class preparation activities, e.g., readings, online modules, videos, and assignments did you complete? Scale (1&#8211;4) 1 = &#60;50% and 4 = &#62;90%</td><td>3.8298</td><td>4.0000</td><td>4.0000</td><td>0.4732</td><td>0.2234</td></tr><tr><td>I felt academically challenged by the rigor and workload of this course. Scale (1&#8211;5) 1 = Has serious deficiencies in this area which are detrimental to students and 5 =Deserves an award in this area; excellent</td><td>4.0851</td><td>4.0000</td><td>4.0000</td><td>0.4323</td><td>0.1987</td></tr><tr><td>Based on what the instructor communicated and the information provided in the syllabus, I understood what was expected of me. Scale (1&#8211;3) 1 = No, I did not understand what was expected of me and 3 = Yes, I understood what was expected of me</td><td>2.9149</td><td>3.0000</td><td>3.0000</td><td>0.3543</td><td>0.1232</td></tr><tr><td>I felt that this course prepared me well for follow-on courses as well as my eventual professional practice. Scale (1&#8211;5) 1 = Has serious deficiencies in this area which are detrimental to students and 5 = Deserves an award in this area; excellent</td><td>4.1489</td><td>4.0000</td><td>4.0000</td><td>0.6454</td><td>0.2432</td></tr><tr><td>This course helped me learn concepts or skills as stated in course objectives and outcomes. Scale (1&#8211;4) 1 = This course did not help me learn the concepts or skills and 4 = This course definitely helped me learn the concepts or skills</td><td>3.8936</td><td>4.0000</td><td>4.0000</td><td>0.4554</td><td>0.1932</td></tr><tr><td>In this course, I engaged in critical thinking and/or problem solving. Scale (1&#8211;4) 1 =Never and 4 = Frequently</td><td>3.8298</td><td>4.0000</td><td>4.0000</td><td>0.3774</td><td>0.2139</td></tr><tr><td>Please rate the organization of this course. Scale (1&#8211;4) 1 = Not at all organized and 4 = Very well organized</td><td>3.7234</td><td>4.0000</td><td>4.0000</td><td>0.3747</td><td>0.2892</td></tr><tr><td>The instructor fostered an effective learning environment. Scale (1&#8211;5) 1 = Strongly disagree and 5 = Strongly agree</td><td>4.5745</td><td>5.0000</td><td>5.0000</td><td>0.4967</td><td>0.2512</td></tr><tr><td>The instructor's teaching methods contributed to my learning. Scale (1&#8211;3) 1 = Did not contribute and 3 = Contributed a lot</td><td>2.8723</td><td>3.0000</td><td>3.0000</td><td>0.3543</td><td>0.1232</td></tr></tbody></table>

Students' rating in the Qualtrics survey designed by the instructor is presented in Table 2. A total of 40 students responded to the Qualtrics survey. The survey has only four questions. For each question, the students are asked to answer on a scale of five values: 1: Strongly disagree, 2: Somewhat disagree, 3: Neither disagree nor agree, 4: Somewhat agree, and 5: Strongly agree.

Table 2. Students' rating in the Qualtrics survey designed by the instructor.

Graph

<table><colgroup><col align="left" /><col align="char" char="." /><col align="char" char="." /><col align="char" char="." /><col align="char" char="." /><col align="char" char="." /></colgroup><thead><tr><th align="left" /><th align="left"><bold>Average</bold></th><th align="left"><bold>Median</bold></th><th align="left"><bold>Mode</bold></th><th align="left"><bold>Standard deviation</bold></th><th align="left"><bold>Variance</bold></th></tr></thead><tbody><tr><td>The classes where numerical methods and MATALB programming are introduced to the course is appropriate and well connected to the technical content of the course</td><td>3.5250</td><td>4.0000</td><td>5.0000</td><td>1.4320</td><td>2.0506</td></tr><tr><td>Overall, I enjoyed the numerical methods and MATALB programming provided by the course, as it enhanced my analytical skills and I feel more comfortable to develop scripts and graphics in MATLAB</td><td>3.0500</td><td>3.0000</td><td>5.0000</td><td>1.5351</td><td>2.3564</td></tr><tr><td>I will consider using numerical methods and MATALB programming as a problem-solving tool as I move on into the industry or the graduate school</td><td>3.3250</td><td>3.5000</td><td>5.0000</td><td>1.5087</td><td>2.2763</td></tr><tr><td>I recommend that the numerical methods and MATALB programming should be kept in MMET 303 in future years</td><td>3.1750</td><td>3.0000</td><td>5.0000</td><td>1.5671</td><td>2.4558</td></tr></tbody></table>

In addition, in the anonymous end-of-the-semester course survey, many students expressed positive attitudes toward the subject matter, the course materials, the teaching approach, and the instructor. All the text data were selected for analysis. When the method of qualitatively coding recommended by Creswell[14] was used to analyze the data, two strong themes emerged. Some representative comments are provided under each theme.

Theme #1: The course materials are interesting and enjoyable.

Theme #2: The teaching method is effective, and the course is well organized.

Conclusions

In this case study, the effectiveness of embedding numerical methods and MATLAB programming in MMET 303 Fluid Mechanics and Power, a four-credit junior-level required course offered every semester for undergraduates at the Department of Engineering Technology and Industrial Distribution at Texas A&M University, was assessed. A series of learning modules were purposefully designed and implemented as a trial test in the classes offered in the semester of Fall 2023. According to the instructor's classroom observation, students appeared to be engaged deeply with the course material, and classroom discussions on numerical methods and MATLAB programming were lively and frequent. In addition, some students expressed their interest in numerical methods and MATLAB programming via email or office hour visits during the semester. The submitted assignments were evaluated with rubrics against the ABET criterion and the results were overall satisfactory. To collect students' opinions, two different survey instruments were used, and the feedback was considered as positive. This study suggested that embedding numerical methods and associated MATLAB programming into a required course enhanced students' analytical skills of using numerical approaches to tackle practical problems, helping them become better prepared as they move on into the industrial companies or the graduate schools.

Acknowledgements

The editors and reviewers are thanked for their constructive comments that enhanced the quality of this manuscript.

By Congrui Jin

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