Introduction

Aerospace engineering students soon begin to ask when they can start to solve actual problems relevant to aircraft, rockets, or spacecraft. Of course, it is natural to ask such questions, even early on in an engineering education. However, practical problem-solving in engineering is a serious business that requires engineers-in-training to become well-versed in the fundamental subjects appropriate to their field. Besides the usual engineering disciplines, this process of learning and becoming proficient will inevitably include having a solid background in physics, chemistry, mathematics, numerical methods, and computer programming.

In general, engineering problem-solving is an established and well-proven process in which the bigger problem is first dissected into smaller, more manageable, and perhaps better digestible parts, as illustrated in the figure below for a flight vehicle. Then, each vehicle component can be analyzed separately, at least initially, perhaps with experiments, mathematical models, numerical models, or all these working approaches working hand-in-hand. Finally, the understanding and functionality of the parts can be reassembled using a synthesis approach to understand the flight vehicle as a whole, with interactions considered, too. Of course, such an approach is always flawed, but it forms at least one rational basis for understanding and designing complex engineering systems, including aircraft and spacecraft.

Today, aerospace engineers must also have increasingly multidisciplinary technical skill sets, which means they must follow a broader-based educational path and become more knowledgeable and versatile in using a more comprehensive range of subject matter. It is no longer sufficient to be an aerodynamics specialist or “aerodynamicist,” or a “structural dynamicist” or an “acoustician.” Successful aerospace engineers must be technically broad and increasingly versatile to work with others in interdisciplinary contexts.

Furthermore, using artificial intelligence (or AI), machine learning, and big data analytics is increasingly prevalent in the aerospace industry, for which engineers must know the benefits and limitations. As the industry evolves and solves more challenging problems, it is also crucial that aerospace engineers continue to develop their problem-solving skills throughout their careers to stay current with the latest technological advancements.

Hypothetico-Deductive Method

Many have argued that the engineering design process must follow the hypothetico-deductive method (or H-D method), a primary method for testing hypotheses or conjectures. The “hypothetico” (or hypothetical) part is where a hypothesis or theory is proposed, which needs to be tested, and the “deductive” part is where the consequences are drawn from the hypothesis or hypotheses.^[1] The H-D method is sometimes called THE scientific method, but it is not the only method used in scientific and engineering work. The H-D method can be divided into the four stages outlined in the figure below.

1. Identify the theory, the hypothesis, or the conjecture to be tested. This approach does not necessarily need to rely on facts and allows for “imaginative preconceptions, intuition, and even luck.” However, the hypothesis usually relies on prior understanding or awareness of pervasive laws.

2. Generate predictions from the theory. The theories are used to make predictions about what we see, i.e., we proceed to imitate what we perceive as the “real world.” These predictions would also encompass the range of conditions perceived as the theory’s domain of applicability, which cannot necessarily have limitless bounds and will trade cost with theoretical complexity and value with predictive fidelity.

3. Use various types of experiments and measurements to test whether predictions are, in fact, correct. If done correctly, the measurements represent the truth. The data acquired, assuming the quantities needed can be measured to the necessary levels of fidelity, provides the evidence to test the proposed theories. Replication of an experiment by others and repeatability of the data over time are critical to doing good science. The data may often uncover unexpected outcomes and spawn new directions if sufficiently comprehensive and high-quality.

4. Expose the theory to criticism, then reject or modify the theory or declare that the theory has been validated or otherwise proved. This process may take significant time and often depends on specific experimental measurements and other data availability. However, it is the most crucial part of doing scientific research. Alternative hypotheses may be pursued after criticism to see which is more likely to explain the predictions. In this regard, the principle of parsimony, or Ockham’s Razor, is essential in mathematical modeling, i.e., in generating equations and mathematical models to represent any given physical behavior.

The goal is to keep modeling complexity and predictive fidelity in equilibrium. This can be achieved through careful, systematic validation studies and requires a degree of engineering common sense. After all, as the figure below suggests, this is the ultimate foundation on which the scientific method is based.

Other methods are used in scientific and engineering work. For example, descriptive methods observe and describe phenomena, while experimental methods manipulate variables to establish causal relationships. Correlational studies examine the relationships between variables without manipulation. Qualitative research gathers in-depth insights through non-numerical data, while mixed-methods research combines quantitative and qualitative approaches. Longitudinal studies track changes over time, meta-analysis synthesizes findings from multiple studies, and action research collaboratively addresses practical issues. Each method offers unique strengths suited to different research questions and objectives, guiding scientific exploration across diverse disciplines and contexts. Nevertheless, the hypothetico-deductive method remains the primary method for testing scientific and engineering hypotheses.

Starting Out

As students think more about engineering concepts and various types of problem-solving in aerodynamics, structures, flight vehicle performance, and other areas, some words of caution are appropriate. First, it must be appreciated that there are few “handy equations” for solving engineering problems, especially in aerodynamics. Instead, the relevant equations for problem-solving must be selected carefully in terms of the specific equations that most appropriately govern the problem, called the governing equations. Choosing the governing equations is one issue, but solving them using the correct boundary conditions, and perhaps with any appropriate simplifications, involves considerable skill that comes only from much practice, i.e., consistent, purposeful, and focused practice over time. The solutions to some of these problems become homework exemplars of the field. “Those who study a scientific discipline are expected to know its exemplars.” There is no fixed set of exemplars in aerospace engineering; however, this ebook is filled with a few hundred for starters.

Aerodynamics

Remember that aerodynamics is the underpinning of flight, so some form of aerodynamic analysis, such as shown in the figure below, comes into almost all types of problem-solving with aircraft and even with spacecraft, i.e., launch and re-entry vehicles. Therefore, it is essential that the selected aerodynamic models be sufficiently comprehensive and detailed to predict what is needed and for the right reason. Ultimately, they will provide a more substantial basis for decision-making, resource allocation, risk management, and adaptability. By considering a wide range of factors and understanding the underlying reasons, it becomes possible to navigate options and reduce uncertainties to make informed choices.

Value-Based Analysis

How comprehensive and detailed any mathematical model needs to be will depend on the specific type and complexity of the physical problem, which also affects the time and effort (i.e., cost) to solve the problem. If the model is intended for precise predictions, design optimization, or critical engineering applications, a higher level of detail and accuracy is typically required, and the penalty is time for a solution. In the workplace, time equates to money, i.e., a value-based decision process is needed, so it is essential to balance the technical level of detail in the model and the resulting computational cost.

Mathematical models must consider additional factors such as non-linearities, coupling effects, boundary interactions, or time-dependent behavior. These complexities demand more comprehensive models that incorporate more detail to represent the physical problem and its behavior accurately at the price of longer execution time and higher costs. For example, aerodynamic equations may need to be solved consistently and simultaneously with other sets of equations describing the behavior of different aspects of the aircraft, such as its dynamic flight motion, aeroelasticity, acoustics, etc.

It is soon concluded that problem-solving in engineering, especially concerning flight vehicles, can be very challenging, time-consuming, and quite costly because the forms of the governing equations for different disciplines will inevitably be different. This issue means different solution methods, including the various specific techniques, numerical methods, etc., may be required in each case, e.g., numerically solving together or coupling the governing equations that are applicable and traditionally used in the different engineering fields.

Governing Equations

As students learn more about aerospace engineering disciplines, they will be exposed to more general forms of potentially applicable governing equations in each field. In many cases, the relevant governing equations that apply to a given problem may be subsets of more general governing equations intended to apply to a broader range of problems and conditions. In other cases, the governing equations may need to be developed from first principles, e.g., directly invoking the conservation principles of mechanics and thermodynamics. However, the approach must be systematic when choosing or creating the equations that apply to or govern a specific problem. The approach of picking a few equations and hoping they will apply is called an ad hoc approach. However, such an approach will inevitably prove disastrous; engineering case histories and experience speak for themselves.

Problem-solving also requires that the meaning of the governing equations must be understood, as well as each of the terms that comprise these equations. There may be terms or groups of terms in these equations that have different levels of complexity and then may also have interdependencies, so dropping one term may have unintended consequences on the evaluation of other terms. In many cases, simplified (reduced) or otherwise particular governing equations may be appropriate because they simplify the solution process. Indeed, an approximate (and fast) solution might be adequate in the initial design phases. However, a more accurate (and likely more time-consuming) solution might be needed later. Part of the skill that must be developed in engineering problem-solving is to decide what terms in the equations must be retained and what terms can be eliminated without substantially affecting the outcomes of the final solution.

Setting Up Aerodynamic Models

Solving problems in aerodynamics requires that appropriate mathematical models of the flow are set up correctly. The derivation of the mathematical equations that describe fluid dynamic and aerodynamic flows is a systematic process that has become well-established in engineering practice. However, because air (like all fluids) will continuously deform as it flows, its behavior must naturally be expected to be more difficult to describe than a solid material.

The starting point of any aerodynamic analysis is the statement of the physical problem, a definition of the appropriate boundary conditions, and making justifiable assumptions and approximations about how the flow develops. An example of a boundary condition is one that defines the values of the free-stream flow conditions or how the flow behaves on a boundary or surface, e.g., it flows parallel to the surface. The three fundamental conservation principles of mechanics must then be applied to the aerodynamic problem, namely:

1. Conservation of mass, i.e., mass is neither created nor destroyed.
2. Conservation of momentum, i.e., a force applied to a mass equals its time rate of change of momentum.
3. Conservation of energy, i.e., energy is neither created nor destroyed and can only be converted from one form into another.

The resulting mathematical equations should then describe the aerodynamic behavior of the flow of interest, at least within the bounds of the stated assumptions and approximations. The solution to these equations can proceed analytically, numerically, or both, hopefully giving an engineer the desired results.

All practical problems will inevitably require some assumptions and approximations to obtain solutions. A common assumption is that air behaves as an ideal gas. Other assumptions might include two-dimensional flow, steady flow, incompressible flow, inviscid flow, etc. However, all such assumptions must be justified, and such justifications often take skill and experience. Skill and experience are obtained through solving engineering problems, which first develops from diligently doing homework problems.

For example, the figure below shows a hierarchy of aerodynamic methods that could be used to model the real flow, starting from lifting line theory, passing through lifting surface theory to a panel method that also models the effects of wing thickness, and finally to a full computational fluid dynamics (CFD) model. Each method is accompanied by a commensurate increase in fidelity and an increase in execution time and computational cost. Whether the problem is aerodynamics or otherwise, the idea is often to start with a more straightforward method to get some initial understanding of the problem at a relatively low cost and then progress to a more complex model with higher fidelity for the final calculations. For example, a CFD calculation may take up to five orders of magnitude more time and cost than the lifting line theory.

Verification and Validation

Inevitably, the outcomes from any model (i.e., the computed results) would be checked to make sure that they make sense, such as by comparing them with measurements, i.e., to establish how well the model works, a process called verification and validation or “V & V.” If the outcomes are positive, then the process inevitably also involves checking it for different input scenarios to ensure that the model predicts what it needs to predict and for the right reason. However, the process may conclude that the results do not agree. The V & V of all mathematical models is critically important, especially if such models are to be used in the design process. Confidence in the design tools being used is critical if the design is to prove viable or successful.

An example is shown in the figure below, where predictions from four mathematical models (i.e., different competing methods or “theories”) are compared against measurements. The question is: “Which mathematical model works the best compared to measurements?” All methods work reasonably well over some ranges, but one or more may be better than the others. The exact theory fails for higher values of the independent variable. Subjectively, Method 3 seems to compare best to the measurement points. However, when measurement uncertainty is accounted for, Methods 1 and 2 may be just as good regarding their predicted values. Unfortunately, in this case, the final answer as to the “best” model may be in the eye of the beholder! The other issue to examine is to ensure that the best method predicts the behavior for the right reason, which may take additional measurements and/or analysis to answer fully. Predicting the correct outcome for the wrong reason is not a verification or validation of the method. Indeed, some may argue that such models cannot be validated, only disproved, i.e., the essence of Popper’s falsification principle.

Finally, these equations (i.e., the “model,” in general) can be used to analyze similar problems of interest, such as in a parametric study where one quantity is varied using the model to establish the corresponding effects on the other quantities. Such parametric studies with mathematical and/or numerical models are critical in engineering design. For example, a particular combination of parameters may be sought to optimize the system performance or for another reason. However, it should always be remembered that no mathematical description of a physical problem can be perfect for its behavior. It can only be an approximation whose accuracy depends on how diligently the model is set up, including the nature of the assumptions and approximations.

Multi-Disciplinary Problems

As previously discussed, aircraft and spacecraft structures are lightweight, thin-walled structures of various beams, columns, shafts, plates, shells, etc. All must be modeled in aggregate with aerodynamics using a multi-disciplinary approach, the idea being shown in the figure below. Coupling aerodynamics with the structures and structural dynamics is called aeroelasticity because the action of the aerodynamic loads will elastically deform the structure and so will feed back to the aerodynamic loads. Combining models of aerodynamics and structural loads and deformations usually requires an iterative approach, adding significantly to the computational time and overall effort.

Like aerodynamics, the governing equations of structural elasticity comprise sets of partial differential equations. Computational Fluid Dynamics (CFD) can help predict the detailed aerodynamic loads if sufficient computing power is available, i.e., the need for memory and execution speed. For the structure, the finite element method (FEM) is usually necessary to predict the behavior of complex structural geometries. The FEM is highly developed and sophisticated enough to handle just about any aerospace structure, but like CFD also requires significant computing power. Sometimes, subsets of the governing equations may be adequate to speed up the computational process, depending on the assumptions and the approximations that can be justified, e.g., small displacements or isotropic structural properties.

Flutter is a form of aeroelasticity and is a potentially catastrophic dynamic phenomenon that can happen with the inherently flexible structures of aircraft and spacecraft. Flutter usually occurs when the forces created on an object cause it to displace or deform, elastically return to where it was, but also overshoot and then repeat the process and begin to oscillate, i.e., it is a dynamic process requiring an inherent coupled aerodynamic and structural dynamic solution process. If the forces subsequently increase in magnitude, the oscillations also increase until the object eventually fails structurally, e.g., the tail or a wing may break off during flight. Therefore, flutter must be avoided, and great efforts must be undertaken to ensure a flight vehicle is flutter-free. Unfortunately, flutter can still occur on flight vehicles, and it can also happen with buildings, bridges, and other flexible objects exposed to the wind.

Problem-Solving Process

The figure below indicates that a general procedure for solving a physical problem requires many steps. The steps are not unique and will vary from problem to problem, but the iterative process is typical to all aspects of design, particularly for flight vehicles. The process requires many specialized activities and inevitably takes considerable time.

1. Specify or define and then describe the nature of the physical problem, which can often be done relatively quickly using an appropriately annotated sketch. Typically, for an aerodynamic problem, the annotations could include the size and shape of a control volume, the general flow directions, and the specifications of relevant boundary conditions.

2. Mathematically specify any relevant known boundary conditions, such as upstream free-stream conditions. On the surface of a solid body placed in the flow, the flow would not pass through that body, so another boundary condition is that the flow is parallel to the body surface.

3. Decide on the primary form of the needed model, i.e., whether an integral or differential approach is required. For example, if detailed properties are needed at all points, an integral approach is unlikely to be appropriate, and the problem should be approached using a differential model.

4. For aerodynamic problems, decide whether an Eulerian or Lagrangian flow model is required, i.e., whether the aerodynamic behavior at a fixed point or over a volume in space is needed or whether the identical fluid particles need to be tracked as they move through the flow.

5. Make any justifiable assumptions about the problem. The idea here is to take the actual physical problem and derive a simplified but still relevant mathematical version of the physical problem. By drawing on experience or from experiments on similar problems, it may be possible to make reasonable assumptions. For example, for aerodynamic problems, it may be possible to assume that the flow is steady and/or in predominantly two dimensions, which usually results in considerable simplifications of the mathematics. These assumptions are then used to help develop the appropriate governing equations for the model.

6. Use the conservation principles to establish the model’s mathematical form that describes the physical problem. Because there are three physical principles to invoke (i.e., mass, momentum, and energy), most problems will involve three governing equations. However, auxiliary equations (e.g., an equation of state) may also be needed. These equations then need to be solved consistently and concurrently.

7. Conduct the solution process where the relevant equations are solved for the desired physical quantities, e.g., for flow problems, velocities, pressures, etc., may be needed. In some cases, the equations may be solved analytically in closed form, meaning that the resulting solutions are pure mathematics and the creation of final sets of descriptive equations. The equations will likely need to be solved numerically, i.e., a computer program must be written with numbers as the outputs.

8. Verify and validate the model to determine the model’s accuracy and correctness, a process often called “V & V,” i.e., use predicted outcomes to determine how good the model is in representing the physical behavior that was the desired outcome. The model’s validity can be established by comparing the results against measurements if such measurements are already available or can be conducted, as shown in the figure below. This step is essential in aerodynamic modeling and is one reason wind tunnels are critical in understanding all types of aerodynamic flows. If experimental data are unavailable, sometimes other theories can be used for validation, but validation is rarely conducted without reference to appropriate measurements. In all cases, experience and good judgment must be used to establish that the predictive credibility of the model has been obtained. This is rarely a test that one person can objectively conduct because confirmational bias, also known as confirmation bias, can play a role.^[2]

9. Improve upon the capabilities of the model to broaden its capabilities. As experience is gained in the validation process regarding what the model predicts adequately and what it does not, the limitations and assumptions within the model can be progressively removed, or other enhancements can be made. For example, extending the range of validity of an aerodynamic model may be possible by including unsteady effects or with a representation of turbulence. In a structural model, it may be necessary to include non-linear effects, e.g., from large deflections.

10. Finally, balance the model’s complexities and capabilities against the time and cost of obtaining solutions from the model. In this case, questions will have to be asked about how the model will be used and whether the full fidelity of the model is needed. For example, the need to model compressibility effects in the flow might not be required if the problem is restricted to low Mach numbers. Furthermore, if the model is to be used exclusively for research, then computational time and cost will be less crucial than for use in the industry, where a short turnaround time is always needed.

Generalization of Data & Data Fitting

The ability to generalize data from one or more sources, whether experiments or computations, is an essential skill for effective engineering modeling. Many engineering problems are far too complicated to have theoretical models in the form of parsimonious equations. Therefore, mathematical models used for design at the system level are often empirical or semi-empirical, meaning that they are either based entirely on observations from experiments, i.e., based on “curve-fits,” or based primarily on theory but augmented by key data taken from experiments, which is called a semi-empirical model. Semi-empirical models are common in engineering analyses, an example being shown in the figure below. Dimensional analysis is helpful here, too, because the entire domain can be more readily generalized if the measured data from different experiments are expressed and compared in terms of dimensionless parameters or groupings, e.g., force coefficients, Reynolds numbers, Mach numbers, etc.

For example, a pure theoretical model may be of the form

(1)  

which may show good qualitative agreement with measurements but not quantitive agreement. However, a semi-empirical model of the form

(2)  

may show much better agreement with experiments where the numerical values of and are semi-empirical coefficients derived by matching the available data in some manner, such as on the basis of a least-squares fit. Experienced modelers may even establish the coefficient based on subjective assessments, but this is not recommended. Nevertheless, the measurements used to establish the semi-empirical model must be comprehensive enough to ensure that reasonable predictive confidence has been achieved. Finally, reasonable enforcement of the scientific method will limit the predictions to within the bounds of the measurements; significant extrapolation (up or down) using a semi-empirical model is always a dangerous practice.

It is rarely possible to cover the entire domain of the parameter space in one single experiment. In this regard, not all test facilities are created equally. One experiment may cover one limited range of the domain, and another experiment may cover another range, perhaps with some overlap, as suggested in the figure below. Each experiment may be inadequate in establishing any useful, robust, causal correlation. However, if the data is taken collectively and used intelligently in the process of generalization, then a semi-empirical model can show a more robust and positive correlation.

Repetition of experiments is always a good practice, which can also benefit from the advancements in measurement technologies. Indeed, the scientific method requires that any one experiment can only be fully trusted once it has been independently repeated and the results confirmed.  Unfortunately, repeating experiments is often difficult to justify based on cost, time, or both. Inherent differences in the testing facilities can also be a consideration, i.e., wind tunnel interference effects, measurement limitations, etc. However, the logical rationalization here from the perspective of the scientific method is that the ability to make better measurements will lead to lower errors and inaccuracies in the acquired data. So, a better and stronger causal correlation is likely to be shown. Therefore, more confidence in prediction can then be gained. Nevertheless, for any given dataset, there is always considerable uncertainty in extrapolation beyond the range of the measured data.

Inevitably, new experiments on a given problem are performed in time, and some experiments may be ambitious enough to extend the range of the domain. Such data may confirm existing trends or be disruptive, suggesting a different correlation, as shown in the figure above. On the one hand, such disruptive data leads to an improved correlation and more confident mathematical models that show enhanced predictive confidence at the system level. On the other hand, predictions at the system level may be worse, in which case one can suspect that one modeling error has previously acted to cancel another, i.e., duo mala faciunt unum bonum, and so point to further modeling deficiencies. The stakes in such outcomes are high in that the validity of the entire system model can be thrown into question until the weaknesses in the other parts of the model are identified and corrected. In this regard, complex models, especially those with significant empiricism, can remain perpetually tentative until more and/or confirmatory measurements are obtained.

Ockham’s Razor – The KIS² Principle

One issue with complex engineering models for multidisciplinary aerospace applications is that significant empiricism may be needed. Some physical problems are difficult to model without resorting to substantial empiricism, an unavoidable artifact of representing complex physical processes with parsimonious models with practical levels of computational efficiency. In this regard, there is always a need to balance the complexity of the mathematical model against the model’s predictive accuracy while aiming to minimize the variability and maximize the intelligibility of the resulting simulations. For complex mathematical models, history has proved that predictive accuracy increases with increasing modeling complexity only up to a point where the cumulative uncertainties in the components of the model (particularly those with significant empiricism) begin to increase the “noise” in the predictions. Then, beyond a certain level of complexity, the predictive accuracy decreases again, the system exhibiting a classic “Ockham’s Hill,” as shown in the figure below.

In this regard, it is essential to remember the principle of Ockham’s Razor, i.e., given two sets of solutions from methods of equivalent accuracy, one should side with the simpler or parsimonious method, i.e., Frustra fit per plura quod potest fieri per pauciora. This approach is sometimes called the KISS or KIS2 principle, which means “Keep it Short and simple.” Ernst Mach was also an advocate of a similar principle he called the “Principle of Economy,” stating: “Scientists must use the simplest means of arriving at their results and exclude everything not perceived by the senses.” The message is evident in that the goal for engineers is to keep modeling complexity and predictive fidelity in balance, something that cannot just be achieved through careful, systematic validation studies but also requires a degree of common sense.

Approaching Homework Problems

The skills and abilities to solve real engineering problems develop from the exemplar problems encountered in the classroom. To this end, budding engineers must first be good at doing homework problems. Results generated in homework problems may be single numbers, tables, or graphs in any or all combinations that must be adequately presented. Below are some general guidelines to help new students tackle homework problems and build up the skills they need to learn as engineers.

Homework is NOT a quiz!

The idea of homework for engineering students is to begin to learn the process of solving real problems, which becomes a lifelong endeavor for an engineer. Homework problems expand upon what can be done in the classroom, which are usually simpler and straightforward. In doing the homework, students can use all the resources available, including exemplar problems and solutions given in the textbook or ebook, for which many are former exam or quiz questions. Students should use these exemplars to help them understand the process of solving similar problems that may come up on homework and future exams.

If you are a student and need help understanding how to go about a specific homework problem, then ask your instructor before or after class, during office hours, or by email. Refrain from guessing! Also, do not copy down former “solutions” to similar problems and present them as your own – use the exemplar solutions to understand the process of solving the current homework questions, then write out your own solutions. Finally, do not assume that the current question(s) is (are) exactly the same as a previous question you may have seen; submitting the correct answer to the wrong homework problem will not have a good outcome.

A student’s homework score may be a significant fraction of their total grade in an engineering course. This fraction reflects the importance instructors place on learning the methods and techniques of solving engineering problems and developing and maintaining other skills, including presentation formats, time management, MATLAB use, drawing graphs, etc. Remember that students may see exam questions very similar to homework questions, which are often easier! So, if students approach homework problems seriously and put in the required time to understand the processes of solving them, they can expect to do well on the exams. History speaks for itself in this regard.

Pointers for Doing Good Homework

Preparations

1. 1. Start to review the homework problem set as soon as the professor or course instructor sets it. Remember: Start working on the problem set early! Time management is essential.
   2. Don’t guess! Homework is not a quiz. It is a learning exercise, so don’t consider homework as a quiz or exam.
   3. If you need help with the homework questions, attend the professor’s, instructor’s, or TA’s office hours. Come prepared and ask specific questions; other students may be waiting to ask their questions.
   4. Don’t post your homework problems online, hoping that someone in cyberspace knows better than the person who wrote the question. Ask the professor or whoever wrote the problem(s) for advice.
   5. Check whether a similar problem is solved in the textbook, the ebook, or the notes. Review old problem sets and their solutions – professors usually make them freely available to students.
   6. Meet with other students, your study group, the TA, the grader, or anyone else and try to understand the problem and potential solution method. An effective learning method is for one student in a study group, for example, to try to teach the solution method to the others.

Starting on the Problems

1. 1. State the problem statement briefly and concisely based on the information given. It is often helpful to restate the information in the question, clearly and straightforwardly laying out what is known and what is not.
   2. If appropriate, draw a sketch or schematic of the problem/approach. In most cases, an annotated sketch of the physical problem will help you decide the nature of the mathematical model that needs to be adopted, e.g., control volume approach or otherwise.
   3. Write down the appropriate mathematical equations necessary to solve the problem. These equations should not be expected to be given in the homework problem, at least not in all cases. Choosing (or deriving) a set of simpler equations from a broader, more general set of governing equations may be necessary.
   4. List and develop any simplifying assumptions appropriate to the problem. Sometimes, the assumptions will be specified; other times, they may be left as part of the problem. For some more challenging problems, it may not be apparent initially what assumptions are needed. Several attempts may be necessary until the correct assumptions can be confirmed and verified.

Working the Problems

1. 1. Be sure to work carefully and systematically through each of the problems. It is better to work slowly but get the correct answer than to work faster and make silly mistakes.
   2. Complete the analysis in algebraic form (i.e., equations with symbols and variables) before substituting the specific numerical values. Sometimes, the answer will be presented as an equation rather than a numerical value.
   3. Substitute known numerical values (using a consistent set of engineering units) to obtain a numerical answer or answers. If the problem is given in SI units, it is best to work it entirely in SI units; conversely, if it is given in USC units, then work it entirely in USC units. For example, switching back and forth between USC and SI is inadvisable because this approach is often a source of mistakes and numerical errors.
   4. All numerical answers must have appropriate units unless they are in non-dimensional form. Always double-check the engineering units of the solution (s). Units should be used consistently throughout, ideally in base units.
   5. Check that the number of significant digits in the answer(s) is/are consistent with the given data. For example, if you are given information to 3 significant digits, it would not be appropriate to calculate your final results to 5 significant digits. However, it is good practice to round off numerical values at the end of the problem.
   6. Review the answer(s) for correctness. In some cases, it will be evident if the result is wrong; it may be challenging in other cases. In many cases, the question is: Does that result seem physically correct? Check with someone else if in doubt, such as a course instructor.
   7. Draw a box around the final answer to clarify that this is your definitive answer. The answer will often be an equation (formula) or number, but it could be a table or a graph. You do not need to draw a box around tables or graphs.
   8. Import the results into the appropriate software if a graph needs to be drawn. Never draw graphs freehand!! Use MATLAB, Excel, or your favorite graphing program. Kaleidagraph was used for many of the plots in this ebook. As appropriate, all graphs need to have proper legends, labels, and other annotations.

Submission

1. 1. Your submitted homework must be neat and easy to follow. You will also need to follow the submission rules. For example, you may have to use squared engineering paper, put your name and student identifying number on each page, staple together your pages, etc. If not, you will likely get less credit regardless of the correct solution. In industry, great emphasis is placed on the clarity and presentation of reports and papers; for the same reasons, clarity in homework is a place to start.
   2. Write down clearly and unambiguously the names of the student(s) you worked with on the homework, if any. Working with others is acceptable, but you should submit your OWN answers to the questions. For example, you may state, “I cross-checked my final answers with John and Kevin, and my answers agreed with theirs.”
   3. Be sure to submit your homework on time and follow your professor’s requirements, such as submitting it on paper or as a file upload.
      Need some help drawing a decent graph for your homework submission? Here’s a short video lesson on drawing good graphs from Dr. Leishman’s “Math & Physics Hints and Tips” series.
      

All those Equations!

In engineering (or any science field), students do not have to memorize hundreds of equations or “formulas.” The questions and concerns from new students of the field inevitably soon start to flow, such as: ” What equation do I use?” or “Where do I find the equation?” or “What is this symbol in this equation?” or “Where do I get the solution to this integral?” Such questions are natural, and guidance from experienced engineers and professors is essential. As taught in many educational contexts, the “plug and chug” paradigm of plugging a numerical value into some equation and chugging out an answer is a surefire recipe for disaster in actual engineering problem-solving. Bona fide engineering students need to learn to do much better.

The number of equations in science and engineering could be as many as the number of stars in the galaxy! No exaggeration. As the figure below suggests, most successful engineers and professors remember the details of the biggest stars and the most general form of the equations or the “governing” equations. They also understand the concepts and have done enough problems to know how to reduce and simplify the equations under certain assumptions and conditions to produce subsets of equations with specific applicability to the problem at hand. The equations they can’t remember can usually be derived! The governing equations can then be adapted and applied successfully to various situations by grasping the underlying concepts without memorizing dozens of problem-specific equations. Science and engineering is not a memory game.

The risk in memorizing without understanding is that the wrong equation (or equations) is (are) applied, so the answer obtained is inevitably wrong too, which is always a disastrous outcome in engineering problem-solving. Therefore, rather than rote memorization or hunting down a specific formula that may or may not be applicable, understanding the fundamental principles allows engineers to derive and apply the relevant equations to particular problems. While some foundational equations may be essential at one’s fingertips, the focus should still be on understanding the underlying physical principles and fundamental concepts expressed by the equations, appreciating the meaning of all the terms in the equations, and rationalizing the best solution strategies for these equations.

Moreover, with access to the Internet, lots of information and resources are readily available for students to find the more general form of the equation(s) that might be needed. Unfortunately, there is a lot of opinionated absurdity on the Internet, especially regarding engineering topics, so using authoritative resources is critical. Utilizing selective online resources, as well as peer-reviewed publications, textbooks, and sanctioned computational tools, can help new students and practicing engineers retrieve or verify specific equations and/or solution methods when needed. Labora sapienter, non strenue.

What is Brainstorming?

Brainstorming is an informal but highly effective approach to engineering problem-solving that works for homework problems. Usually conducted with a small group of engineers or students and a single moderator, the process encourages all participants to think laterally and develop ideas that might initially seem unusual or even sound slightly crazy!

A group of five to seven people is usually the most influential, with a mix of experienced and less experienced engineers. The moderator should be someone other than the chief or lead engineer or an engineer in management, and the group itself should select the moderator. The people in the group should come from several technical disciplines to foster and develop the most effective and productive brainstorming environment. Excellent ideas may ultimately flow from engineers who see an avenue of opportunity in a discipline different from their own, i.e., when they start looking at the problem with a fresh mind. Some of these ideas may be developed into a rational basis for engineering problem-solving, often following a new or innovative path that nobody else had considered previously. The basic idea is to get all participants to think “out of the box,” be creative, and divert their attention away from using their “conventional wisdom” to solve problems.

Brainstorming is usually very effective for solving complex engineering problems requiring multi-disciplinary engineering. The best ideas in a brainstorming session will often flow from the less experienced engineers, who are not so encumbered by conventional wisdom. Brainstorming is best conducted in an informal, relaxed environment away from the typical day-to-day work environment, often at a retreat location. Brainstorming can also be fun, an excellent environment for team building, and a way of getting to know other engineers outside one’s primary technical discipline or organization. It is not unusual for companies to work together on solving complex engineering problems in the aerospace field, and brainstorming sessions can foster more substantial inter-company dialog where everyone works more effectively together.

For brainstorming to be effective, all group members must be active participants, and personal criticism is inappropriate. Often, some quirky idea from one group member makes no sense at first, but after discussion, there can be an “aha!” or “I never thought of that!” moment, and the idea can be built upon by the group after that. Alternatively, the idea may lead to some other revelation and a different path to solving the problem that nobody thought about before.

In preparing for the brainstorming session, a location with no distractions must be found, all phones and computers should be turned off, the doors locked, and a whiteboard should be available to write down the ideas. Everyone should have an opportunity to speak when they want to. The moderator must refrain from allowing any one member of the group to dominate the brainstorming session. At the end of the session, the group decides on the best ideas and then moves forward, as needed, to follow up and pursue them. The history of engineering suggests that many of the best and most innovative ideas can come from brainstorming sessions.

Summary & Closure

Developing mathematical models that can be used to study and solve various problems is an integral part of engineering. However, the selection process must be conducted carefully and systematically to choose or develop the relevant governing equations that apply to the specific problem (or problems) of interest. In some cases, it may be possible to down-select the appropriate governing equations for a particular problem from more general forms of governing equations that are intended to apply to a broader range of conditions. In other cases, the governing equations may need to be developed from first principles.

Once a mathematical model or set of models has/have been developed, it is necessary to solve the equations to make predictions about the system being modeled. The solution process can involve analytical methods, such as closed-form solutions, or numerical methods, such as finite element, finite difference, or boundary element methods. The choice of solution method will depend on the nature of the problem, the available resources, and the desired accuracy of the solution. In all cases, the justification of assumptions and/or approximations is needed. In this regard, any reason may require reliance on outcomes from experiments, i.e., for verification and validation of the modeling. A combination of solution methods may often arrive at an acceptable solution. Also, post-processing and visualization techniques may help interpret the results and behavior of the modeled system.