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] [2]}

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.

The formal mathematical (standard form) statement of the design optimization problem is ^[3]

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

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.

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 ^[2]

• 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.^[4] 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.

One modern application of design optimization is structural design optimization (SDO) is in building and construction sector. SDO emphasizes automating and optimizing structural designs and dimensions to satisfy a variety of performance objectives. These advancements aim to optimize the configuration and dimensions of structures to optimize augmenting strength, minimize material usage, reduce costs, enhance energy efficiency, improve sustainability, and optimize several other performance criteria. Concurrently, structural design automation endeavors to streamline the design process, mitigate human errors, and enhance productivity through computer-based tools and optimization algorithms. Prominent practices and technologies in this domain include the parametric design, generative design, building information modelling (BIM) technology, machine learning (ML), and artificial intelligence (AI), as well as integrating finite element analysis (FEA) with simulation tools.^[5]

• Journal of Engineering for Industry
• Journal of Mechanical Design
• Journal of Mechanisms, Transmissions, and Automation in Design
• Design Science
• Engineering Optimization
• Journal of Engineering Design
• Computer-Aided Design
• Journal of Optimization Theory and Applications
• Structural and Multidisciplinary Optimization
• Journal of Product Innovation Management
• International Journal of Research in Marketing

• 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

• 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.
• J., Wilde, Douglass (1978). Globally optimal design. New York: Wiley. ISBN 0471038989. OCLC 3707693.
• J., Haug, Edward (1979). Applied optimal design : mechanical and structural systems. Arora, Jasbir S.,. New York: Wiley. ISBN 047104170X. OCLC 4775674.
• Uri., Kirsch, (1981). Optimum structural design : concepts, methods, and applications. New York: McGraw-Hill. ISBN 0070348448. OCLC 6735289.
• Uri., Kirsch, (1993). Structural optimization : fundamentals and applications. Berlin: Springer-Verlag. ISBN 3540559191. OCLC 27676129.
• 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.). Hoboken, N.J.: John Wiley & Sons. ISBN 0471558141. OCLC 61463772.
• N.,, Vanderplaats, Garret (1984). Numerical optimization techniques for engineering design : with applications. New York: McGraw-Hill. ISBN 0070669643. OCLC 9785595.
• 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.). 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.
• Structural optimization : status and promise. Kamat, Manohar P. Washington, DC: American Institute of Aeronautics and Astronautics. 1993. ISBN 156347056X. OCLC 27918651.
• 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.). 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.
• Optimization in action : proceedings of the Conference on Optimization in Action held at the University of Bristol in January 1975. Dixon, L. C. W. (Laurence Charles Ward), 1935-, Institute of Mathematics and Its Applications. London: Academic Press. 1976. ISBN 0122185501. OCLC 2715969.
• P., Williams, H. (2013). Model building in mathematical programming (5th ed.). Chichester, West Sussex: Wiley. ISBN 9781118506189. OCLC 810039791.
• Integrated design of multiscale, multifunctional materials and products. McDowell, David L., 1956-. Oxford: Butterworth-Heinemann. 2010. ISBN 9781856176620. OCLC 610001448.
• M.,, Dede, Ercan. Multiphysics simulation : electromechanical system applications and optimization. Lee, Jaewook,, Nomura, Tsuyoshi,. London. ISBN 9781447156406. OCLC 881071474.
• 1962-, Liu, G. P. (Guo Ping), (2001). Multiobjective optimisation and control. Yang, Jian-Bo, 1961-, Whidborne, J. F. (James Ferris), 1960-. Baldock, Hertfordshire: Research Studies Press. ISBN 0585491941. OCLC 54380075.