# Introduction

Students of aerospace engineering soon ask when they will be able to solve actual problems relevant to aircraft, rockets, or spacecraft. Of course, it is natural to do so, even early in an engineering education. However, practical problem-solving in engineering is a serious business that requires engineers to become well-versed in the fundamental subjects appropriate to their field. Besides the usual engineering disciplines, this process will inevitably include having a solid background in physics, chemistry, mathematics, numerical methods, and computer programming.

In general, engineering problem-solving is a process in which the bigger problem is first dissected into smaller, more manageable, and digestible parts, as illustrated in the figure below for a flight vehicle. Then each part can be analyzed separately, at least initially, perhaps with the aid of experiments, mathematical models, or usually both. Finally, the parts can be reassembled using a synthesis approach to understand the problem as a whole. Of course, such an approach is never perfect, but it forms one rational basis for understanding and designing complex engineering systems, including aircraft and spacecraft. Problem solving in aerospace engineering most often involves the dissection of the bigger problem into smaller more analyzable parts, followed by synthesis to create an understanding of the entire problem. Needless to say, the smaller parts are not always mutually exclusive.

Today, aerospace engineers must also have increasingly inter-disciplinary technical skill sets, which means they must follow a broader-based educational path and become more versatile in using a more comprehensive range of subject matter. Furthermore, the use of artificial intelligence, machine learning, and big data analytics is becoming increasingly prevalent in the aerospace industry. As the industry continues to evolve, it is also crucial that aerospace engineers continue to develop their problem solving skills, both technical and non-technical, throughout their careers, to stay current with the latest advancements in the field.

Objectives of this Lesson

• Begin to understand the fundamental processes involved in engineering problem-solving.
• Appreciate the types of aerodynamic flow models that can be used to solve problems.
• Better understand the expectations when doing homework problems in engineering classes.

# Hypothetico-Deductive Method

Many have argued that the engineering design process follows the hypothetico-deductive method (or H-D), 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 H-D method is sometimes referred to as THE scientific method, but it is not the only method used in scientific work. The H-D method can be divided into the four stages outlined in the figure below. Engineering design is a complex syncretic affair, and various disciplines within one field can operate in different ways, but there are four main interconnected components that are vital to success.

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 that we perceive as the domain of applicability for the theory, which cannot necessarily have limitless bounds and will trade cost with complexity and value with fidelity.

3. Use various types of experiments to check whether predictions are, in fact, correct. If done correctly, experiments 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 are critical to good science. If sufficiently comprehensive and of high quality, the data itself may often uncover unexpected outcomes and spawn new directions.

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 a concept of great importance in modeling.

# Starting Out

As students of the field 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 problems in engineering, especially in the field of 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. The choice of the governing equations is one issue, but solving them using the correct boundary conditions, and perhaps with any appropriate simplifications, involves a considerable amount of skill that comes only from much practice.

Remember that aerodynamics is the underpinning of flight, so some form of aerodynamic analysis, as shown, for example, 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. The modeling of the flow over the wing of an aircraft, for example, may need to be comprehensive enough to predict all of the physical intricacies of the flow such as streamlines, pressures, etc.

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. In the workplace, time equates to money, i.e., a value-based decision process is needed. Furthermore, in many cases, the aerodynamic equations need to be solved consistently and simultaneously with other sets of equations describing the behavior of other 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 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 conservation principles. However, the approach must be systematic when choosing or developing 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 be understood and 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.

Also, 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 final solution.

For example, the figure below shows a hierarchy of 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, as well as an increase in execution time and computational cost. The ability to reduce the physical problem into mathematical models that have various levels of approximation is helpful for many approaches in engineering design. In increasing order of complexity and computational cost: Lifting line wing model; Lifting surface wing model; Panel method wing model; CFD method wing model.

Regardless of 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.

Inevitably the outcomes from any model (i.e., the “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 design tools 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 measurements. Although, when measurement uncertainty is accounted for, then perhaps Methods 1 and 2 are just as good in terms of 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. Verification and validation of the outcomes from mathematical models against measurements is critical if such models are to be successful as predictive design tools.

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.

# Setting Up Aerodynamic Models

Solving problems in aerodynamics requires that appropriate mathematical models of the flow field are set up correctly. The derivation of the mathematical equations that describe 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. All practical problems will inevitably require at least 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 two-dimensional flow, steady flow, incompressible flow, etc. However, all such assumptions need to be justified, and such justifications often take skill and experience. Skill and experience are obtained through solving engineering problems, and in the classroom, such practice develops from diligently doing homework problems.

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 is equal to the 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 then proceed analytically, numerically, or both, which will hopefully give an engineer the desired results.

There are two basic approaches used in aerodynamics:

1. An integral or finite control volume approach in which the equations are developed as they would apply to a finite control volume surrounding the problem.
2. The differential or infinitesimal fluid element approach in which the relevant equations apply at every flow point.

In both cases, the control volume or the fluid element may be fixed in space, and the flow moves through it, or it may move with the flow such that it contains the same group of fluid molecules. The former approach (i.e., fixed in space) is called an Eulerian model, and the latter approach (i.e., moves with the flow) is called a Lagrangian model. Each of these types of modeling approaches has certain advantages and disadvantages when applied to solving specific problems in aerodynamics, and in most cases there will be a preferred approach. Control volume or integral approach to solving aerodynamic problems.  A fixed control volume in space with the fluid elements flowing in and out of the control volume, which is called an Eulerian representation. A moving control volume carrying the same fluid elements is called a Lagrangian representation. A differential fluid element approach to solving aerodynamic problems. A fixed elemental volume in space with the fluid flowing in an out, which is called an Eulerian representation. A moving elemental volume carrying the same fluid elements is called a Lagrangian representation.

For example, on the one hand, an integral approach could be used to find total effects, such as the forces on a body in the flow, but without necessarily solving for all of the point properties in the flow. It may not be necessary to know what the air is doing at every point in the flow, so an integral approach may be more appropriate in this case. On the other hand, the differential approach would be needed if the local distributions of flow velocity and pressure at points in the flow and over the surface of a body were the desired outcome.

Likewise, a Lagrangian approach might be adopted over an Eulerian approach because it makes the problem description more manageable in modeling the physical problem and/or from a mathematical description and/or solution methodology. Part of the skills needed in aerodynamic problem solving (and engineering problem solving in general) is to decide which type of basic model to apply to specific problems. Sometimes, such decisions may not be obvious even for an experienced engineer, and different approaches may need to be tried.

For example, it may be desired to predict the velocity and pressure distribution over the surface of a wing. The question is then: What basic form of aerodynamic model should be used? The answer, in this case, is a differential model in that point properties could be determined. Integral forms of the equations would be appropriate only when the overall or integrated aerodynamic effects are needed. For example, the total lift on the wing is an integral quantity because it arises from the effects of the pressure distribution when it is resolved and integrated over the wing’s surface.

In practice, the integral approach is usually easier to work with, at least from a mathematics and/or numerical perspective. The differential form of the equations would be appropriate when the distributive quantities would be needed such as when the velocity and pressure distributions over the wing’s surface are needed. Again, the relative cost of obtaining a solution needs to be factored into the final choice of the model.

# Multi-Disciplinary Problems

As previously discussed, aircraft and spacecraft structures are lightweight, thin-walled structures made of various beams, columns, shafts, plates, shells, etc. All need to be modeled in aggregate with the aerodynamics in a multi-disciplinary approach. Like the field of aerodynamics, the governing equations of structural elasticity are comprised of sets of partial differential equations. Therefore, any exact analytical solutions may only be obtained for relatively simple shapes or geometries because of the mathematics involved (i.e., solving sets of partial differential equations).

Nevertheless, the basic approach to problem-solving in aircraft structures and structural design is to assess what is known and what needs to be known before embarking on a particular solution strategy. Sometimes subsets of the governing equations may be adequate, depending on the assumptions made and the approximations that can be justified, e.g., small displacements or isotropic structural properties. In many cases, however, approximations may not be possible.

Numerical solution methods such as the finite element method (FEM) are usually necessary for complex structural geometries. Where available, exact solutions to structural problems are always helpful and can also be used for V & V of FEM outcomes. The FEM requires an understanding of matrix methods and linear algebra. The FEM approximates an entire structure as an assembly of elements or components with various interconnections at a finite number of points or nodes. The behavior of the overall structure then becomes the aggregate of how all of the various elements behave, usually in the form of a matrix that represents the deformations of the entire structure (nodes) in response to a set of applied loads.

The FEM is highly developed and sophisticated enough to handle just about the type of aerospace structure if sufficient computing power is available, i.e., the need for memory and execution speed. However, V & V of FEM requires experimental measurements, and all airframe manufacturers invest heavily in this area to ensure that their methods can be used confidently for future design purposes. The most challenging cases are where large strains and displacements are involved.

Coupling aerodynamics with the field of 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, as shown in the figure below. 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. The avoidance of adverse aeroelastic effects such as flutter is a key goal in flight vehicle design, which requires closely integrated aerodynamic and structural and structural dynamic models.

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., its dynamic process. If the forces subsequently increase in magnitude, then the oscillations also increase until the object eventually fails structurally, e.g., the tail or a wing may break off during flight. Flutter is a phenomenon to be avoided. Unfortunately, flutter can occur on flight vehicles, and it can also happen with buildings, bridges, and other flexible objects exposed to the wind.

# Problem-Solving Process

A general procedure for the solution of a physical problem can be outlined in the following steps:

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 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 needed. 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 same 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. More likely, the equations will 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, i.e., use predicted outcomes from the model 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 outcomes against measurements if such measurements are already available or can be conducted. 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 not available, sometimes other types of 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 the credibility of the model 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 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, compressibility effects in the flow might not be needed 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 not as crucial as for use in the industry, where a short turnaround time is always needed.

# What is Brainstorming?

Brainstorming is an informal but highly effective approach to engineering problem-solving. The process, which is usually conducted with a small group of engineers and with a single moderator, encourages all participants to think laterally and come up with ideas that might initially seem unusual or may even sound a bit 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 really good 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 problem-solving, often following a new or innovative path that nobody else had even thought about before. The basic idea is to get all participants to think “out of the box,” be creative, and divert their attention away from using their own “conventional wisdom” to solve problems.

Brainstorming is usually very effective for solving complex engineering problems where multi-disciplinary engineering is needed. 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 stronger inter-company 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 is 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.

# Ockham’s Razor – The KIS2 Principle

One issue with complex engineering models used for multi-disciplinary aerospace applications is that significant empiricism may be needed. Some physical problems are difficult to model without resorting to significant empiricism, which is an unavoidable artifact of having to represent complex physical processes with parsimonious models that have 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 begins to decrease again, the system exhibiting a classic “Ockham’s Hill,” as shown in the figure below. Modeling accuracy is a function of the complexity of the model. Beyond a certain level of complexity, the predictive accuracy begins to decrease.

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 & 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 at first be good at doing homework problems. Results from homework problems may be single numbers, tables, or graphs in any or all combinations that must be adequately presented. Below are some general guidelines that will help students tackle homework problems and build up the skills they will need as engineers.

1. Submitted work must be neat and easy to follow. If it is not, you will likely get less credit regardless of whether your solution is correct. 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. State briefly and concisely the problem statement from the information given. It is often helpful to restate the information in the question, laying out simply what is known and what is not.

3. Draw a sketch or schematic of the problem/approach if needed and appropriate. In most cases, an annotated sketch of the physical problem helps decide the nature of the mathematical model that needs to be adopted, e.g., control volume approach or otherwise.

4. Write down the appropriate mathematical equations that are 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 and more general set of equations in other homework problems may be necessary.

5. List and/or develop any simplifying assumptions appropriate to the problem. Sometimes the assumptions to use will be specified, and other times that may be left as part of the problem. For some more challenging problems, then it may not be so apparent at the beginning as to what assumptions are needed. Several attempts may be needed until the correct assumptions can be confirmed and verified.

6. 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.

7. If asked for, 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 the problem entirely in SI units; conversely, if the problem is given in USC units, then work the problem entirely in USC units. For example, it is inadvisable to switch back and forth between USC and SI, as this approach is often a source of errors.

8. All numerical answers must have appropriate units unless they are in non-dimensional form. Always double-check the engineering units of the answer(s). Units should be used consistently throughout, ideally base units.

9. 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.

10. 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.

11. Draw a box around the final answer to make it clear that this is your final answer. The answer will often be a formula or number, but it could be a table or a graph. You do not need to draw a box around tables or graphs.

12. Import the results into the appropriate software if a graph needs to be drawn. Never ever draw graphs freehand – use MatLab or Excel or your favorite graphing program. As appropriate, all graphs need to have legends, labels, and other annotations.

Need some help drawing a decent graph for your homework submission? Here is a short video lesson on how to draw good graphs from Dr. Leishman’s “Math & Physics Hints and Tips” series.

# 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 relevant governing equations for a specific problem from more general forms of governing equations 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 set of mathematical models has been developed, it is necessary to solve them in order 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 methods, finite difference methods, 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 justification may require the reliance on outcomes from experiments, i.e., for verification and validation of the modeling.  In many cases, a combination of solution methods may be used to arrive at an acceptable solution. Additionally, post-processing and visualization techniques may be used to help interpret the results and understand the behavior of the system being modeled.

5-Question Self-Assessment Quickquiz

For Further Thought or Discussion
• The “KISS” or “KIS2 principle refers to the acronym for “Keep It Short & Simple.” Discuss the meaning of this principle as it might apply to engineering modeling.
• It is often argued that there must always be some trades in modeling fidelity versus cost. Discuss this perspective.
• What is meant by a parametric study? Why, in the process of engineering design, might we want to conduct parametric studies.

• A 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 KISS Principle here and a video here.
• Here is a short fun video on the history of bookbag calculators from Dr. Leishman’s “Math & Physics Hints and Tips” series. 