## Engineering Design Optimization using Calculus Level Methods: A Casebook Approach (Manual)

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**Engineering Design Optimization
using Calculus Level Methods: A Casebook Approach**

By Phil B Brubaker

# Welcome to Calculus-level Problem Solving!

Engineers in industry wanted to ‘tweak’ their parameters. So this textbook was written to show the simplicity of ‘tweaking’ parameters in algebraic through differential equation problems when using a Calculus-level language like PROSE or FortranCalculus. FortranCalculus (FC) is available on the web.

*Automatic Differentiation *(AD) and *Operator overloading *were key technologies that allowed numerical methods, now called solvers, to be stored in a FC library. A user will use a solver by stating a solver name in a ‘find’ statement using the ‘by’ clause. Want to switch solvers? Just change the solver name (e.g. from ‘Ajax’ to ‘Jupiter’) and you are ready to try a different numerical method! It is that easy to code. (See the FortranCalculus manual for suggestions on what solver to use for a given problem.)

Help spread the word about Calculus-level thinking and problem solving. Do you know any engineering or science professors that might have a problem that could be solved and shown to their future students?

This textbook tries to move today’s thinking from solving one problem at a time, to solving all of their project’s problems at once while tweaking parameters in order to achieve an optimum solution. This requires Calculus-level thinking. An analogy might be thinking in terms of Machine code, one bit at a time. Today, computer simulations have people thinking in terms of Algebraic code, one problem at a time. We are trying to move people to Calculus-level code, solving entire projects at a time. This will reduce development time and improve accuracy of their math models. (Future CEOs should study the Oil Refinery Production problem in order to see future possibilities with Calculus-level thinking.)

**Mission**** Statement**:

Get the FortranCalculus compiler operational and in use via the internet. It’s a free compiler that simplifies solving math problems by minimizing code necessary to state & solve a problem. Some new thinking is necessary for those wanting to get the most for their buck; convert from simulation to optimization thinking.

What’s the different between simulation and optimization? Picture a saw horse construction project. A Simulation would yield A saw horse where Optimization would yield an Optimal saw horse. If the objective (function) was good and proper then the Optimal saw horse would be the best solution, right? For example, the objective might be lightest & strongest saw horse. A wrong objective might be just the strongest saw horse. This might yield a strong horse but a very heavy one!

If you were a manager or CEO and had the choice of a simulation design versus an optimization design, which would you pick?

Modeling & Simulation’s next step is (Mathematical) Optimizations. Optimizations require an Objective (function). Today's Engineers & Scientists solve problems with a “Find X” mind-set. With some Operational Research training they could expand their thinking to a “Find X to Optimize Y” mind-set. Then they would be ready for Optimizations, Calculus-level programming and software. (This would drop today’s design times that require months even man years to one or two days! Manufacturing processes could be optimized to the days demand and thus maximize their profits.)

“Find X to Optimize Y” thinking among professors will cause most Engineering & Science textbooks to be rewritten with optimization examples and discussions. This will be great stuff for industries and government; applied engineering and/or science not just theories.

**Table of Content**

**Introduction**. 5 1 General Algebraic Equations. 9 Background of TFH Math Model for a Readback Pulse from Magnetic Recording. 9 A

*Typical*Readback Pulse from Magnetic Recording. 12 An

*Unusual*Readback Pulse from Magnetic Recording. 15 A

*Typical*Readback Pulse from Magnetic Recording with Improved Model 17 An

*Unusual*Readback Pulse from Magnetic Recording with Improved Model 19 Curve fitting: A Sinusoidal Signal 21 Curve fitting: A Damped Sinusoidal Signal 23 1.4 Conclusion on Curve Fitting. 25 Pharmacokinetics. 26 Slack Variable Techniques. 29 Paper Bicycle Design. 31 Chapter 1 Exercises. 33 2 La Place Transforms. 34 Optimum Matched Filter (Transfer Function). 34 Chapter 2 Exercises. 44 3 Ordinary Differential Equations. 46 Second Order Non-Linear ODE.. 47 A Third Order Non-Linear ODE.. 50 A Bang-Bang Control Problem.. 53 Non-Linear Equations of Motion. 62 4 System of Differential Equations. 65 The Lorentz Equations, a System of ODEs. 66 The Convection Reaction Equations, a System of PDEs. 69 Body Plasma Chemistry. 71 Modeling a Nanostructured Solar Cell 76 Chapter 4 Exercises. 82 5 Partial Differential Equations. 83 PDEs: Stock Market to Biology. 84 Burgers’ Equation. 86 Telegrapher’s Equation. 89 6 Inverse Problems. 92 Custom Thermistor Design. 93 Drug Development 95 Heat Transfer over 1D Slab. 96 Robot Arm Movement 99 Plane Crash Locator. 102 7 Implicit Equations. 104 System of Implicit Algebraic Equations. 105 2

^{nd}Order Implicit Differential Equation. 108 8 Nesting Solvers. 110 Nesting … Matched Filter. 111 Oil Refinery Production. 113 9 Miscellaneous. 118 Monte Carlo Simulation OR Total Derivative?. 118 Stiff Equations & Trouble Shooting. 119 10 Conclusions. 121 10.1 Future: Thinking outside the box. 121 11 Appendix. 125 Picking the right Solver. 125 ‘aplot’ source code. 125 Spectral Estimation (freeware) Software. 126 ‘readrit1.100’ File Listing. 126 ‘readrit2.200’ File Listing. 127 Arbitrary Equalization with Simple LC Structures. 130 Incomplete Problems: can you help complete one or more?. 133 Index. 134

Preface

How to teach new problem solving technology to engineers and scientists? Problem solving requires a broad based knowledge in math and science as well as discernment and flexibility to challenge the way it has always been done in the past. Generally, an objective driven design will yield the best design in the least amount of time. Companies need engineers trained in setting objectives before they begin the time- consuming process of formulating and testing new concepts and designs.

This textbook considers design from the pragmatic concerns of industry. It utilizes casebook studies of math problems with their solutions in real life situations. Because it encourages students to view themselves as part of the design team, this text is the next best thing to an on-the-job training. It shows how setting objectives to problem solving assignments can help students complete work quickly and efficiently, but it also stresses that while every situation is different, the approach remains the same: objective-driven engineers state a math model and an objective function for a given problem while leaving the solving to a calculus-level computer language/compiler.

The text attempts to fill a gap in educational material in the mathematical problem solving arena. Traditional texts leave students in a simulation thinking mode. Simulations require many computer runs causing delays in solution and little gain, if any, in problem understanding. Simulations require a numerical algorithm to be meshed with their math model. In such form, math models are hard to recognize and discuss. Besides slowing their understanding, users lose confidence in program solutions.

This textbook tries to move today’s thinking from solving one problem at a time, to solving all of their project’s problems at once while tweaking parameters in order to achieve an optimum solution. This requires Calculus-level thinking. An analogy might be thinking in terms of Machine code, one bit at a time. Today, computer simulations have people thinking in terms of Algebraic code, one problem at a time. We are trying to move people to Calculus-level code, solving entire projects at a time. This will reduce development time and improve accuracy of their math models.

NASA funded the development of the first Calculus-level language through TRW called Prose. Prose became available to the public in 1974 through a national computer time-sharing network. Prose ran on large Control Data Corporation (CDC) 6600 computers. *Automatic differentiation* and *operator overloading* were key technologies for this project. I taught the Prose language to Engineers & Scientists in the San Francisco Bay Area from 1975 through 1979. Most national time-sharing computer networks died in the 1980s and thus went Prose. FortranCalculus is the next Calculus language on the horizon. It is in testing mode now and will soon be released on the web.

Things to learn from this textbook include:

- How Calculus-level programming simplifies problem solving;
- Use of Lorentzian (function) series for curve fitting;
- How to find frequency parameters when curve fitting sine series to data;
- Manage by Objective; and,
- How to ‘tweak’ hundreds of parameters at once.

For more, download casebook for the above example problems solved.

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