# Design Optimization

Design optimization is an engineering design methodology using a mathematical formulation of a design problem to support selection of the optimal design among many alternatives. Design optimization involves the following stages:[1]

1. Variables: Describe the design alternatives
2. Objective: Elected functional combination of variables (to be maximized or minimized)
3. Constraints: Combination of Variables expressed as equalities or inequalities that must be satisfied for any acceptable design alternative
4. Feasibility: Values for set of variables that satisfies all constraints and minimizes/maximizes Objective.

## Design optimization problem

The formal mathematical (standard form) statement of the design optimization problem is [2]

{\displaystyle {\begin{aligned}&{\operatorname {minimize} }&&f(x)\\&\operatorname {subject\;to} &&h_{i}(x)=0,\quad i=1,\dots ,m_{1}\\&&&g_{j}(x)\leq 0,\quad j=1,\dots ,m_{2}\\&\operatorname {and} &&x\in X\subseteq R^{n}\end{aligned}}}

where

• ${\displaystyle x}$ is a vector of n real-valued design variables ${\displaystyle x_{1},x_{2},...,x_{n}}$
• ${\displaystyle f(x)}$ is the objective function
• ${\displaystyle h_{i}(x)}$ are ${\displaystyle m_{1}}$equality constraints
• ${\displaystyle g_{j}(x)}$ are ${\displaystyle m_{2}}$ inequality constraints
• ${\displaystyle X}$ is a set constraint that includes additional restrictions on ${\displaystyle x}$ besides those implied by the equality and inequality constraints.

The problem formulation stated above is a convention called the negative null form, since all constraint function are expressed as equalities and negative inequalities with zero on the right-hand side. This convention is used so that numerical algorithms developed to solve design optimization problems can assume a standard expression of the mathematical problem.

We can introduce the vector-valued functions

{\displaystyle {\begin{aligned}&&&{h=(h_{1},h_{2},\dots ,h_{m1})}\\\operatorname {and} \\&&&{g=(g_{1},g_{2},\dots ,g_{m2})}\end{aligned}}}

to rewrite the above statement in the compact expression

{\displaystyle {\begin{aligned}&{\operatorname {minimize} }&&f(x)\\&\operatorname {subject\;to} &&h(x)=0,\quad g(x)\leq 0,\quad x\in X\subseteq R^{n}\\\end{aligned}}}

We call ${\displaystyle h,g}$ the set or system of (functional) constraints and ${\displaystyle X}$ the set constraint.

## Application

Design optimization applies the methods of mathematical optimization to design problem formulations and it is sometimes used interchangeably with the term engineering optimization. When the objective function f is a vector rather than a scalar, the problem becomes a multi-objective optimization one. If the design optimization problem has more than one mathematical solutions the methods of global optimization are used to identified the global optimum.

Optimization Checklist [1]

• Problem Identification
• Initial Problem Statement
• Analysis Models
• Optimal Design Model
• Model Transformation
• Local Iterative Techniques
• Global Verification
• Final Review

A detailed and rigorous description of the stages and practical applications with examples can be found in the book Principles of Optimal Design.

Practical design optimization problems are typically solved numerically and many optimization software exist in academic and commercial forms.[3] There are several domain-specific applications of design optimization posing their own specific challenges in formulating and solving the resulting problems; these include, shape optimization, wing-shape optimization, topology optimization, architectural design optimization, power optimization. Several books, articles and journal publications are listed below for reference.

## Journals

• Design Decisions Wiki (DDWiki) : Established by the Design Decisions Laboratory at Carnegie Mellon University in 2006 as a central resource for sharing information and tools to analyze and support decision-making

## References

1. ^ a b Papalambros, Panos Y.; Wilde, Douglass J. (2017-01-31). Principles of Optimal Design: Modeling and Computation. Cambridge University Press. ISBN 9781316867457.
2. ^ Boyd, Stephen; Boyd, Stephen P.; California), Stephen (Stanford University Boyd; Vandenberghe, Lieven; Angeles), Lieven (University of California Vandenberghe, Los (2004-03-08). Convex Optimization (PDF). Cambridge University Press. ISBN 9780521833783.
3. ^ Messac, Achille (2015-03-19). Optimization in Practice with MATLAB®: For Engineering Students and Professionals. Cambridge University Press. ISBN 9781316381373.

• Rutherford., Aris, ([2016], ©1961). The optimal design of chemical reactors : a study in dynamic programming. Saint Louis: Academic Press/Elsevier Science. ISBN 9781483221434. OCLC 952932441
• Jerome., Bracken, ([1968]). Selected applications of nonlinear programming. McCormick, Garth P.,. New York,: Wiley. ISBN 0471094404. OCLC 174465
• L., Fox, Richard ([1971]). Optimization methods for engineering design. Reading, Mass.,: Addison-Wesley Pub. Co. ISBN 0201020785. OCLC 150744
• Johnson, Ray C. Mechanical Design Synthesis With Optimization Applications. New York: Van Nostrand Reinhold Co, 1971.
• 1905-, Zener, Clarence, ([1971]). Engineering design by geometric programming. New York,: Wiley-Interscience. ISBN 0471982008. OCLC 197022
• H., Mickle, Marlin ([1972]). Optimization in systems engineering. Sze, T. W., 1921-2017,. Scranton,: Intext Educational Publishers. ISBN 0700224076. OCLC 340906.
• Optimization and design; [papers]. Avriel, M.,, Rijckaert, M. J.,, Wilde, Douglass J.,, NATO Science Committee., Katholieke Universiteit te Leuven (1970- ). Englewood Cliffs, N.J.,: Prentice-Hall. [1973]. ISBN 0136380158. OCLC 618414.
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• Structural optimization : recent developments and applications. Lev, Ovadia E., American Society of Civil Engineers. Structural Division., American Society of Civil Engineers. Structural Division. Committee on Electronic Computation. Committee on Optimization. New York, N.Y.: ASCE. 1981. ISBN 0872622819. OCLC 8182361.
• Foundations of structural optimization : a unified approach. Morris, A. J. Chichester [West Sussex]: Wiley. 1982. ISBN 0471102008. OCLC 8031383.
• N., Siddall, James (1982). Optimal engineering design : principles and applications. New York: M. Dekker. ISBN 0824716337. OCLC 8389250.
• 1944-, Ravindran, A., (2006). Engineering optimization : methods and applications. Reklaitis, G. V., 1942-, Ragsdell, K. M. (2nd ed ed.). Hoboken, N.J.: John Wiley & Sons. ISBN 0471558141. OCLC 61463772.
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• T., Haftka, Raphael (1990). Elements of Structural Optimization. Gürdal, Zafer., Kamat, Manohar P. (Second rev. edition ed.). Dordrecht: Springer Netherlands. ISBN 9789401578622. OCLC 851381183.
• S., Arora, Jasbir (2011). Introduction to optimum design (3rd ed ed.). Boston, MA: Academic Press. ISBN 9780123813756. OCLC 760173076.
• S.,, Janna, William. Design of fluid thermal systems (SI edition ; fourth edition ed.). Stamford, Connecticut. ISBN 9781285859651. OCLC 881509017.
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• Mathematical programming for industrial engineers. Avriel, M., Golany, B. New York: Marcel Dekker. 1996. ISBN 0824796209. OCLC 34474279.
• Hans., Eschenauer, (1997). Applied structural mechanics : fundamentals of elasticity, load-bearing structures, structural optimization : including exercises. Olhoff, Niels., Schnell, W. Berlin: Springer. ISBN 3540612327. OCLC 35184040.
• 1956-, Belegundu, Ashok D., (2011). Optimization concepts and applications in engineering. Chandrupatla, Tirupathi R., 1944- (2nd ed ed.). New York: Cambridge University Press. ISBN 9781139037808. OCLC 746750296.
• Okechi., Onwubiko, Chinyere (2000). Introduction to engineering design optimization. Upper Saddle River, NJ: Prentice-Hall. ISBN 0201476738. OCLC 41368373.
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### Structural Topology Optimization

• "Generating optimal topologies in structural design using a homogenization method". Computer Methods in Applied Mechanics and Engineering. 71 (2): 197–224. 1988-11-01. doi:10.1016/0045-7825(88)90086-2. ISSN 0045-7825.
• Bendsøe, Martin P (1995). Optimization of structural topology, shape, and material. Berlin; New York: Springer. ISBN 3540590579.
• Behrooz., Hassani, (1999). Homogenization and Structural Topology Optimization : Theory, Practice and Software. Hinton, E. (Ernest). London: Springer London. ISBN 9781447108917. OCLC 853262659.
• P., Bendsøe, Martin (2003). Topology optimization : theory, methods, and applications. Sigmund, O. (Ole), 1966-. Berlin: Springer. ISBN 3540429921. OCLC 50448149.
• Topology optimization in structural and continuum mechanics. Rozvany, G. I. N.,, Lewiński, T.,. Wien. ISBN 9783709116432. OCLC 859524179.