Design optimization: Difference between revisions

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Revision as of 05:50, 27 July 2018

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'''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:<ref name="pyp2017">{{Cite book|url=http://principlesofoptimaldesign.org/|title=Principles of Optimal Design: Modeling and Computation|last=Papalambros|first=Panos Y.|last2=Wilde|first2=Douglass J.|date=2017-01-31|publisher=Cambridge University Press|year=|isbn=9781316867457|___location=|pages=|language=en}}</ref>:

# [[Variable (mathematics)|Variables]]: Describe the design alternatives

# Objective: Elected functional combination of variables (to be maximized or minimized)

# Constraints: Combination of Variables expressed as equalities or inequalities that must be satisfied for any acceptable design alternative

# Feasibility: Values for set of variables that satisfies all constraints and minimizes/maximizes Objective.

== Design optimization problem ==

{{Main|Optimization problem}}

The formal mathematical ([[Canonical form|standard form]]) statement of the design optimization problem is <ref>{{Cite book|url=https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf|title=Convex Optimization|last=Boyd|first=Stephen|last2=Boyd|first2=Stephen P.|last3=California)|first3=Stephen (Stanford University Boyd|last4=Vandenberghe|first4=Lieven|last5=Angeles)|first5=Lieven (University of California Vandenberghe, Los|date=2004-03-08|publisher=Cambridge University Press|year=|isbn=9780521833783|___location=|pages=|language=en}}</ref> 

<math>\begin{align}

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* <math>h_i(x)</math> are <math>m_1</math>'''equality constraints'''

* <math>g_j(x)</math> are <math>m_2</math> '''inequality constraints'''

* <math>X</math> is a set constraint that includes additional restrictions on <math>x</math> 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

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\end{align}

</math> ''  ''

to rewrite the above statement in the compact expression  

<math>\begin{align}

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\end{align}</math>

We call <math>h, g</math> the ''set'' or ''system of'' (''functional'') ''constraints'' and <math>X</math> 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-valued function|vector]] rather than a [[Scalar field|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''' <ref name="pyp2017" />

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A detailed and rigorous description of the stages and practical applications with examples can be found in the book [http://www.cambridge.org/gh/academic/subjects/engineering/control-systems-and-optimization/principles-optimal-design-modeling-and-computation-3rd-edition?format=HB&isbn=9781107132672#C3r6r6aLRe2bUoef.97 Principles of Optimal Design].

Practical design optimization problems are typically solved numerically and many [[:Category:Mathematical optimization software|optimization software]] exist in academic and commercial forms.<ref>{{Cite book|url=https://books.google.com/books?hl=en&lr=&id=wy5hBwAAQBAJ&oi=fnd&pg=PR17&dq=design+optimization+with+matlab+achille+messac&ots=n9WXaRte_T&sig=DVBwayB-6W1b2YyruSz6O2fMKAM#v=onepage&q=design%20optimization%20with%20matlab%20achille%20messac&f=false|title=Optimization in Practice with MATLAB®: For Engineering Students and Professionals|last=Messac|first=Achille|date=2015-03-19|publisher=Cambridge University Press|isbn=9781316381373|language=en}}</ref>. 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 (EDA)|power optimization]]. Several books, articles and journal publications are listed below for reference.

== Journals ==

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* [[Journal of Product Innovation Management]]

* [[International Journal of Research in Marketing]]

== See also ==

* [https://wiki.ece.cmu.edu/ddl/index.php/Main_Page 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 ==

<references />

== See also ==

* [https://wiki.ece.cmu.edu/ddl/index.php/Main_Page 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

== Further reading ==

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* P., Bendsøe, Martin (2003). ''Topology optimization : theory, methods, and applications''. Sigmund, O. (Ole), 1966-. Berlin: Springer. {{ISBN|3540429921}}. [[OCLC]]&nbsp;50448149.

* ''Topology optimization in structural and continuum mechanics''. Rozvany, G. I. N.,, Lewiński, T.,. Wien. {{ISBN|9783709116432}}. [[OCLC]]&nbsp;859524179.

*