Skip to main content
Erschienen in: Structural and Multidisciplinary Optimization 5/2015

09.07.2015 | RESEARCH PAPER

Pareto front spacing with differential geometry in multidisciplinary systems

verfasst von: Craig Bakker, Geoffrey T. Parks

Erschienen in: Structural and Multidisciplinary Optimization | Ausgabe 5/2015

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

Multidisciplinary Design Optimization, also known as MDO, deals with the optimization of complex but typically single-objective design problems. As such, it would be valuable to combine it with Multi-Objective Optimization (MOO). Within MOO, Weighted Sums (WS) is a standard solution method, but it may not produce a good spacing of solution points along the Pareto Front (PF). Using our differential geometry MDO framework, we present a technique for intelligently spacing out Pareto solutions produced with WS in a multidisciplinary MOO formulation. Following an initial demonstration, we present modifications to our method to handle ill-conditioning better, to produce fewer dominated points in PF refinement, and to use MOO solution methods other than WS.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Literatur
Zurück zum Zitat Athan TW, Papalambros PY (1996) A note on weighted criteria methods for compromise solutions in multi-objective optimization. Eng Optim 27:155–176CrossRef Athan TW, Papalambros PY (1996) A note on weighted criteria methods for compromise solutions in multi-objective optimization. Eng Optim 27:155–176CrossRef
Zurück zum Zitat Bakker C, Parks GT (2014a) Differential geometry tools for multidisciplinary design optimization, part I: Theory Structural and Multidisciplinary Optimization Publication Pending Bakker C, Parks GT (2014a) Differential geometry tools for multidisciplinary design optimization, part I: Theory Structural and Multidisciplinary Optimization Publication Pending
Zurück zum Zitat Bakker C, Parks GT (2014b) Differential geometry tools for multidisciplinary design optimization, part II: Application to QSD. Struct Multidiscip Optim Publication Pending Bakker C, Parks GT (2014b) Differential geometry tools for multidisciplinary design optimization, part II: Application to QSD. Struct Multidiscip Optim Publication Pending
Zurück zum Zitat Bakker C, Parks GT, Jarrett JP (2012) Geometric perspectives on MDO and MDO architectures. In: 12 t h Aviation Technology, Integration and Operations (ATIO) Conference and 14 t h AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, AIAA, Indianapolis, Indiana Bakker C, Parks GT, Jarrett JP (2012) Geometric perspectives on MDO and MDO architectures. In: 12 t h Aviation Technology, Integration and Operations (ATIO) Conference and 14 t h AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, AIAA, Indianapolis, Indiana
Zurück zum Zitat Bakker C, Parks GT, Jarrett JP (2013a) Differential geometry and design coupling in MDO. In: 54 t h AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, AIAA, Boston,Massachusetts Bakker C, Parks GT, Jarrett JP (2013a) Differential geometry and design coupling in MDO. In: 54 t h AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, AIAA, Boston,Massachusetts
Zurück zum Zitat Bakker C, Parks GT, Jarrett JP (2013b) Optimization algorithms and ODE’s in MDO. In: ASME 2013 Design Engineering Technical Conferences and Computers and Information in Engineering Conference, ASME, Portland, Oregon Bakker C, Parks GT, Jarrett JP (2013b) Optimization algorithms and ODE’s in MDO. In: ASME 2013 Design Engineering Technical Conferences and Computers and Information in Engineering Conference, ASME, Portland, Oregon
Zurück zum Zitat Borouchaki H, George PL, Hecht F, Laug P, Saltel E (1997) Delaunay mesh generation governed by metric specifications Part I. Algorithms. Finite Elem Anal Des 25:61–83CrossRefMathSciNetMATH Borouchaki H, George PL, Hecht F, Laug P, Saltel E (1997) Delaunay mesh generation governed by metric specifications Part I. Algorithms. Finite Elem Anal Des 25:61–83CrossRefMathSciNetMATH
Zurück zum Zitat Cao W (2007) An interpolation error estimate on anisotropic meshes in R n and optimal metrics for mesh refinement. SIAM J Numer Anal 45:2368–2391CrossRefMathSciNetMATH Cao W (2007) An interpolation error estimate on anisotropic meshes in R n and optimal metrics for mesh refinement. SIAM J Numer Anal 45:2368–2391CrossRefMathSciNetMATH
Zurück zum Zitat Cramer E, Dennis JE Jr, Frank PD, Lewis RM, Shubin GR (1994) Problem formulation for multidisciplinary optimization problems. SIAM J Optim 4:754–776CrossRefMathSciNetMATH Cramer E, Dennis JE Jr, Frank PD, Lewis RM, Shubin GR (1994) Problem formulation for multidisciplinary optimization problems. SIAM J Optim 4:754–776CrossRefMathSciNetMATH
Zurück zum Zitat Das I (1999) On characterizing the “knee” of the Pareto curve based on normal-boundary intersection. Struct Optim 18:107–115CrossRef Das I (1999) On characterizing the “knee” of the Pareto curve based on normal-boundary intersection. Struct Optim 18:107–115CrossRef
Zurück zum Zitat Das I, Dennis JE (1997) A closer look at drawbacks of minimizing weighted sums of objectives for Pareto set generation in multicriteria optimization problems. Struct Optim 14:63–69CrossRef Das I, Dennis JE (1997) A closer look at drawbacks of minimizing weighted sums of objectives for Pareto set generation in multicriteria optimization problems. Struct Optim 14:63–69CrossRef
Zurück zum Zitat Delaunay B (1934) Sur la sphere vide. a la memoire de georges voronoi. Bulletin de l’Academie des Sciences de l’URSS. Classe des Science Mathematique et Naturelles 6:793–800 Delaunay B (1934) Sur la sphere vide. a la memoire de georges voronoi. Bulletin de l’Academie des Sciences de l’URSS. Classe des Science Mathematique et Naturelles 6:793–800
Zurück zum Zitat Dervieux A, Leservoisier D, George PL, Coudière Y (2003) About theoretical and practical impact of mesh adaptation on approximation of functions and PDE solutions. Int J Numer Methods Fluids 43:507–516MATH Dervieux A, Leservoisier D, George PL, Coudière Y (2003) About theoretical and practical impact of mesh adaptation on approximation of functions and PDE solutions. Int J Numer Methods Fluids 43:507–516MATH
Zurück zum Zitat Guggenheimer HW (1977) Differential Geometry. Dover Publications Inc., New YorkMATH Guggenheimer HW (1977) Differential Geometry. Dover Publications Inc., New YorkMATH
Zurück zum Zitat Haimes YY, Lasdon LS, Wismer DA (1971) On a bicriterion formulation of the problem of integrated system identification and system optimization. IEEE Trans Syst Man Cybern 1:296–297CrossRefMathSciNetMATH Haimes YY, Lasdon LS, Wismer DA (1971) On a bicriterion formulation of the problem of integrated system identification and system optimization. IEEE Trans Syst Man Cybern 1:296–297CrossRefMathSciNetMATH
Zurück zum Zitat Hancock BJ, Mattson CA (2013) The smart normal constraint method for directly generating a smart Pareto set. Struct Multidiscip Optim 48:763–775CrossRefMathSciNet Hancock BJ, Mattson CA (2013) The smart normal constraint method for directly generating a smart Pareto set. Struct Multidiscip Optim 48:763–775CrossRefMathSciNet
Zurück zum Zitat Izui K, Yamada T, Nishiwaki S (2013) A gradient-based multiobjective optimization technique using an adaptive weighting method. In: 10 t h World Congress on Structural and Multidisciplinary Optimization, ISSMO, Orlando, Florida Izui K, Yamada T, Nishiwaki S (2013) A gradient-based multiobjective optimization technique using an adaptive weighting method. In: 10 t h World Congress on Structural and Multidisciplinary Optimization, ISSMO, Orlando, Florida
Zurück zum Zitat Kang N, Kokkolaras M, Papalambros PY (2014) Solving multiobjective optimization problems using quasi-separable MDO formulations and analytical target cascading. Struct Multidiscip Optim 50:849–859CrossRefMathSciNet Kang N, Kokkolaras M, Papalambros PY (2014) Solving multiobjective optimization problems using quasi-separable MDO formulations and analytical target cascading. Struct Multidiscip Optim 50:849–859CrossRefMathSciNet
Zurück zum Zitat Kim IY (2006) Adaptive weighted sum method for multiobjective optimization A new method for Pareto front generation. Struct Multidiscip Optim 31:105–116CrossRefMathSciNetMATH Kim IY (2006) Adaptive weighted sum method for multiobjective optimization A new method for Pareto front generation. Struct Multidiscip Optim 31:105–116CrossRefMathSciNetMATH
Zurück zum Zitat Loseille A, Alauzet F (2011) Continuous mesh framework part I Well-posed continuous interpolation error. SIAM J Numer Anal 49:38–60CrossRefMathSciNetMATH Loseille A, Alauzet F (2011) Continuous mesh framework part I Well-posed continuous interpolation error. SIAM J Numer Anal 49:38–60CrossRefMathSciNetMATH
Zurück zum Zitat Madetoja E, Miettinen K, Tarvainen P (2006) Issues related to the computer realization of a multidisciplinary and multiobjective optimization system, vol 22, pp 33–46 Madetoja E, Miettinen K, Tarvainen P (2006) Issues related to the computer realization of a multidisciplinary and multiobjective optimization system, vol 22, pp 33–46
Zurück zum Zitat Marler RT, Arora JS (2004) Survey of multi-objective optimization methods for engineering. Struct Multidiscip Optim 26 :369–395CrossRefMathSciNetMATH Marler RT, Arora JS (2004) Survey of multi-objective optimization methods for engineering. Struct Multidiscip Optim 26 :369–395CrossRefMathSciNetMATH
Zurück zum Zitat Mattson CA, Mullur A A, Messac A (2004) Smart Pareto filter Obtaining a minimal representation of multiobjective design space, vol 36, pp 721–740 Mattson CA, Mullur A A, Messac A (2004) Smart Pareto filter Obtaining a minimal representation of multiobjective design space, vol 36, pp 721–740
Zurück zum Zitat Mesmer BL, Bloebaum CL, Kannan H (2013) Incorporation of value-driven design in multidisciplinary design optimization. In: 10 t h World Congress on Structural and Multidisciplinary Optimization, ISSMO, Orlando, Florida Mesmer BL, Bloebaum CL, Kannan H (2013) Incorporation of value-driven design in multidisciplinary design optimization. In: 10 t h World Congress on Structural and Multidisciplinary Optimization, ISSMO, Orlando, Florida
Zurück zum Zitat Messac A, Mattson CA (2002) Generating well-distributed sets of Pareto points for engineering design using physical programming. Optim Eng 3:431–450CrossRefMATH Messac A, Mattson CA (2002) Generating well-distributed sets of Pareto points for engineering design using physical programming. Optim Eng 3:431–450CrossRefMATH
Zurück zum Zitat Pagnutti D, Ollivier-Gooch O (2009) A generalized framework for high order anisotropic mesh adaptation. Comput Struct 87:670–679CrossRef Pagnutti D, Ollivier-Gooch O (2009) A generalized framework for high order anisotropic mesh adaptation. Comput Struct 87:670–679CrossRef
Zurück zum Zitat Pennec X, Fillard P, Ayache N (2006) A Riemannian framework for tensor computing. Int J Comput Vis 66:41–66CrossRefMATH Pennec X, Fillard P, Ayache N (2006) A Riemannian framework for tensor computing. Int J Comput Vis 66:41–66CrossRefMATH
Zurück zum Zitat Potschka A, Logist F, Van Impe JF, Bock HG (2011) Tracing the Pareto frontier in bi-objective optimization problems by ODE techniques. Numer Algorithm 57:217–233CrossRefMATH Potschka A, Logist F, Van Impe JF, Bock HG (2011) Tracing the Pareto frontier in bi-objective optimization problems by ODE techniques. Numer Algorithm 57:217–233CrossRefMATH
Zurück zum Zitat Rakowska J, Haftka RT, Watson LT (1991) Tracing the efficient curve for multi-objective control-structure optimization. Comput Syst Eng 2:461–471CrossRef Rakowska J, Haftka RT, Watson LT (1991) Tracing the efficient curve for multi-objective control-structure optimization. Comput Syst Eng 2:461–471CrossRef
Zurück zum Zitat Smale S (2000) Global analysis and economics V: Pareto theory with constraints. In: Cucker F, Wong R (eds) The Collected Papers of Stephen Smale, vol 1, World Scientific Publishing Co. Pte. Ltd., Singapore, pp 305–313 Smale S (2000) Global analysis and economics V: Pareto theory with constraints. In: Cucker F, Wong R (eds) The Collected Papers of Stephen Smale, vol 1, World Scientific Publishing Co. Pte. Ltd., Singapore, pp 305–313
Zurück zum Zitat Sobieszczanski-Sobieski J, Venter G (2005) Imparting desired attributes in structural design by means of multi-objective optimization. Struct Multidiscip Optim 29 :432–444CrossRef Sobieszczanski-Sobieski J, Venter G (2005) Imparting desired attributes in structural design by means of multi-objective optimization. Struct Multidiscip Optim 29 :432–444CrossRef
Zurück zum Zitat Tappeta R V, Renaud JE, Rodríguez JF (2000) An interactive multiobjective optimization design strategy for multidisciplinary systems. In: 41 s t AIAA/ASME/ASCE/ASC Structures, Structural Dynamics, and Materials Conference, AIAA, Atlanta, Georgia Tappeta R V, Renaud JE, Rodríguez JF (2000) An interactive multiobjective optimization design strategy for multidisciplinary systems. In: 41 s t AIAA/ASME/ASCE/ASC Structures, Structural Dynamics, and Materials Conference, AIAA, Atlanta, Georgia
Zurück zum Zitat Yano M (2012) An optimization framework for adaptive higher-order discretizations of partial differential equations on anisotropic simplex meshes. PhD thesis, Massachusetts Institute of Technology, Cambridge, Massachusetts Yano M (2012) An optimization framework for adaptive higher-order discretizations of partial differential equations on anisotropic simplex meshes. PhD thesis, Massachusetts Institute of Technology, Cambridge, Massachusetts
Zurück zum Zitat Yano M, Darmofal DL (2012) An optimization-based framework for anisotropic simplex mesh adaptation. J Comput Phys 231:7626–7649CrossRefMathSciNetMATH Yano M, Darmofal DL (2012) An optimization-based framework for anisotropic simplex mesh adaptation. J Comput Phys 231:7626–7649CrossRefMathSciNetMATH
Metadaten
Titel
Pareto front spacing with differential geometry in multidisciplinary systems
verfasst von
Craig Bakker
Geoffrey T. Parks
Publikationsdatum
09.07.2015
Verlag
Springer Berlin Heidelberg
Erschienen in
Structural and Multidisciplinary Optimization / Ausgabe 5/2015
Print ISSN: 1615-147X
Elektronische ISSN: 1615-1488
DOI
https://doi.org/10.1007/s00158-015-1289-x

Weitere Artikel der Ausgabe 5/2015

Structural and Multidisciplinary Optimization 5/2015 Zur Ausgabe

    Marktübersichten

    Die im Laufe eines Jahres in der „adhäsion“ veröffentlichten Marktübersichten helfen Anwendern verschiedenster Branchen, sich einen gezielten Überblick über Lieferantenangebote zu verschaffen.