16 Problem Solving & Modeling
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 problemsolving in engineering is a serious business that requires that engineersintraining become wellversed 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 problemsolving is an established and wellproven process in which the bigger problem is first dissected into smaller, more manageable, and 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 the aid of experiments, mathematical models, numerical models, or usually all of these. 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 never perfect, 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 broaderbased 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,” and successful aerospace engineers must be technically broad and increasingly versatile to operate in interdisciplinary contexts.
Furthermore, using artificial intelligence (or AI), machine learning, and big data analytics is increasingly prevalent in the aerospace industry, for which new 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 problemsolving skills throughout their careers to stay current with the latest technological advancements.
Learning Objectives
 Begin to understand the fundamental processes involved in engineering problemsolving.
 Appreciate the significance of governing equations and how such equations can be reduced so that they apply to specific problems.
 Understand the ideas of modeling complexity, and the trades between fidelity and cost.
 Better understand the importance and expectations of doing homework problems in engineering classes.
HypotheticoDeductive Method
Many have argued that the engineering design process must follow the hypotheticodeductive method (or HD), 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. The HD method is sometimes called THE scientific method, but it is not the only method used in scientific work. The HD 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 we perceive as the domain of applicability for the theory, 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 highquality.
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 the physics.
The goal is to keep in equipoise modeling complexity and predictive fidelity, something that cannot just be achieved through careful, systematic validation studies but also requires a degree of engineering common sense. After all, this is the ultimate foundation on which the scientific method is based.
Starting Out
As students think more about engineering concepts and various types of problemsolving 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 problemsolving must be selected carefully in terms of the specific equations that most appropriately govern the problem, which are 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. Skill in problemsolving is the culmination of repeated effort, deliberate refinement, and a commitment to continuous learning and improvement.
“When you can measure what you are speaking about and express it in numbers, you know something about it; but when you cannot measure it when you cannot express it in numbers, your knowledge is of a meager and unsatisfactory kind: it may be the beginning of knowledge, but you have scarcely, in your thoughts, advanced to the stage of science, whatever the matter may be.” Source: Sir William Thompson (Lord Kevin), Popular Lectures and Addresses, Vol. 1 (1889) “Electrical Units of Measurement,” from a presentation delivered on May 3, 1883.
Remember that aerodynamics is the underpinning of flight, so some form of aerodynamic analysis, as shown in the figure below, comes into almost all types of problemsolving with aircraft and even with spacecraft, i.e., launch and reentry 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 decisionmaking, 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.
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 valuebased decision process is needed, so it is essential to strike a balance between the technical level of detail in the model and the resulting computational cost.
Furthermore, mathematical models must consider additional factors such as nonlinearities, coupling effects, boundary interactions, or timedependent behavior. These complexities demand more comprehensive models incorporating more detail to accurately represent the physical problem and its behavior 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 problemsolving in engineering, especially concerning flight vehicles, can be very challenging, timeconsuming, 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.
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.
Problemsolving 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 timeconsuming) solution might be needed later. Part of the skill that must be developed in engineering problemsolving 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 wellestablished 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 freestream 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:
 Conservation of mass, i.e., mass is neither created nor destroyed.
 Conservation of momentum, i.e., a force applied to a mass equals its time rate of change of momentum.
 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 allow solutions to be obtained, a common assumption being that air behaves as an ideal gas. Other assumptions might include twodimensional 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. 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. 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 but for the wrong reason is not verification of validation of the method.
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 diligent the model is set up to begin with, including the nature of the assumptions and approximations.
MultiDisciplinary Problems
As previously discussed, aircraft and spacecraft structures are lightweight, thinwalled structures made of various beams, columns, shafts, plates, shells, etc. All must be modeled in aggregate with the aerodynamics in a multidisciplinary 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. Coupling together models of the 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 happens 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 flutterfree. Unfortunately, flutter can still occur on flight vehicles, and it can also happen with buildings, bridges, and other flexible objects exposed to the wind.
ProblemSolving 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 known boundary conditions relevant, such as upstream freestream 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 set down 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 be needed too. 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, good judgment must be used to establish that the predictive credibility of the model has been obtained.
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 nonlinear 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, 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 are less crucial than for use in the industry, where a short turnaround time is always needed.
Ockham’s Razor – The KIS^{2} 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 having to represent 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. 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 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 in equipoise modeling complexity and predictive fidelity, something that cannot just be achieved through careful, systematic validation studies but also requires a degree of common sense.
Attention Students! 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 to them, 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.
20Pointers for Doing Good Homework

 Start to review the homework problem set as soon as the professor or course instructor sets it. Remember: Start to try to solve the problem set early! Again, start the problem set early!! Time management is essential.
 Don’t guess! Homework is not a quiz. It is a learning exercise, so don’t consider homework as a quiz or exam.
 Attend the professor’s, instructor’s, or TA’s office hours if you need help with the homework questions. Come prepared and ask specific questions – other students may be waiting to ask their questions.
 Don’t post your homework problems online, hoping that someone out there knows better than the person who wrote the question. Ask the professor or whoever wrote the problem(s) for advice.
 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.
 State briefly and concisely the problem statement from the information given. It is often helpful to restate the information in the question, laying out what is known and what is not straightforwardly.
 Draw a sketch or schematic of the problem/approach if appropriate. 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.
 Write down the appropriate mathematical equations necessary to solve the problem. It should not be expected that these equations will 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.
 List and develop any simplifying assumptions appropriate to the problem. Sometimes, the assumptions will be specified, and other times, that may be left as part of the problem. For some more challenging problems, it may not be so apparent initially as to what assumptions are needed. Several attempts may be necessary until the correct assumptions can be confirmed and verified.
 Please be sure to work carefully and systematically through the problems. It is better to work slowly but get the correct answer than to work faster and make silly mistakes.
 Meet with other students, your study group, the TA, the grader, or whomever, and try to understand the solution method. An effective way of learning is that one student in a study group, for example, tries to teach the solution method to the others.
 Carry the analysis to completion in an 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.
 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.
 All numerical answers must have appropriate units unless they are in nondimensional form. Always doublecheck the engineering units of the solution (s). Units should be used consistently throughout, ideally in base units.
 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.
 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.
 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.
 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.
 Your submitted homework must be neat and easy to follow. You must also follow the submission rules, e.g., 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.
 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 crosschecked my final answers with John and Kevin, and my answers agreed with theirs.”
all those Equations!
In engineering (or any science field), it is completely necessary for students to memorize hundreds or thousands of equations or “formulas.” The questions and concerns from students inevitably 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?” 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 problemsolving. Bona fide 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 remember the biggest stars and the most general form of the equations or the “governing” equations. They also understand the concepts and have done enough homework and related problems to know how to reduce and simplify the equations under certain assumptions and conditions to produce subsets of equations with specific applicability. 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 problemspecific equations. 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. Therefore, rather than rote memorization or hunting down a specific formula, understanding the fundamental principles allows engineers to derive and apply the relevant equations to specific problems. While some foundational equations may be essential to have 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 also 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. While there is a lot of opinionated absurdity on the internet, especially regarding engineering topics, using authoritative resources is critical. Utilizing selective online resources, as well as peerreviewed publications, textbooks, and sanctioned computational tools, can help students and 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 problemsolving, which also 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. The excellent ideas may ultimately flow from engineers who see an avenue of opportunity in a different discipline to 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 problemsolving, 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 multidisciplinary 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 daytoday 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 intercompany dialog where everyone works more effectively together.
All group members need to be active participants for brainstorming to be effective, and personal criticism is not appropriate. 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 after that by the group. 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 downselect 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 closedform 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, postprocessing and visualization techniques may help interpret the results and behavior of the modeled system.
5Question SelfAssessment Quickquiz
For Further Thought or Discussion
 The “KISS” or “KIS^{2} principle refers to the acronym for “Keep It Short & Simple.” Discuss the meaning of this principle as it might apply to engineering modeling.
 What is meant by a parametric study? Why, in engineering design, might we want to conduct parametric studies?
 How might assumptions impact the accuracy of models? Can overly simplifying a problem lead to inaccurate results
 What might be the tradeoffs between making assumptions to simplify a model and retaining complexity for better accuracy?
 Why is it essential to validate and verify models? What are some standard methods for validation, and how do they contribute to model credibility?
 How might advances in computational power and simulation techniques influence how we set up and solve flow models in the future?
Additional Online Resources
 An excellent video on the use of mathematical and computer models in engineering.
 Video on the use of models and simulation in engineering.
 View a video on Eulerian and Lagrangian flow models.
 Navigate here to watch a video on types of flow models from the National Science Foundation.
 Review the KIS^{2} Principle here and a video here.