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Finding the Best: Mathematical Optimization Based on Product and Process Requirements

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Abstract

The challenge of finding the best solution for a given problem plays a central role in many fields and disciplines. In mathematics, best solutions can be found by formulating and solving optimization problems. An optimization problem consists of an objective function, optimization variables, and optimization constraints, all of which define the solution space. Finding the optimal solution within this space means minimizing or maximizing the objective function by finding the optimal variables of the solution. Problems, such as geometry optimization of profiles (Hess and Ulbrich 2012), process control for stringer sheet forming (Bäcker et al. 2015) and optimization of the production sequence for branched sheet metal products (Günther and Martin 2006) are solved using mathematical optimization methods (Sects. 5.2 and 5.3). A variety of mathematical optimization methods is comprised within the field of engineering design optimization (EDO) (Roy et al. 2008).

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References

  • Andreassen E, Clausen A, Schevenels M, Lazarov BS, Sigmund O (2011) Efficient topology optimization in MATLAB using 88 lines of code. Struct Multidiscip Optim 43:1–16

    Article  Google Scholar 

  • Arian E, Fahl M, Sachs EW (2000) Trust-region proper orthogonal decomposition for flow control. Technical Report, Institute for Computer Applications in Science and Engineering

    Google Scholar 

  • Bäcker F, Bratzke D, Groche P, Ulbrich S (2015) Time-varying process control for the stringer sheet forming by a deterministic derivative-free optimization approach. Int J Adv Manuf Technol 80(5):817–828

    Article  Google Scholar 

  • Bartels S, Carstensen C (2002) Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. Part II: higher order FEM. Math Comput 71(239):971–994

    Article  Google Scholar 

  • Birkhofer H (1980) Analyse und Synthese der Funktionen technischer Produkte. Dissertation, TU Braunschweig

    Google Scholar 

  • Birkhofer H (2005) In fünf Minuten von der Aufgabe zur optimalen Lösung: Ein Beitrag zur Algorithmisierung der Frühen Phasen. In: Meerkamm H (ed) Proceedings of the 16th Symposium on Design for X, Neukirchen, Erlangen, 13–14 October 2005, p 47–58

    Google Scholar 

  • Birkhofer H, Wäldele M (2005) Applied engineering design science: the missing link between design science and design in industry. In: Hosnedl S (ed) AEDS 2005 Workshop, Pilsen, 3–4 November 2005

    Google Scholar 

  • Birkhofer H, Wäldele M (2008) Properties and characteristics and attributes and …—an approach on structuring the description of technical systems. In: Vanek V, Hosnedl S, Bartak J (eds) Proceedings of AEDS 2008 Workshop, Pilsen, p 19–34

    Google Scholar 

  • Bornemann F, Erdmann B, Kornhuber R (1993) Adaptive multilevel methods in three space dimensions. Int J Numer Methods Eng 36(18):3187–3203

    Article  Google Scholar 

  • Braess D (1986) On the combination of multigrid method and conjugate gradients. In: Hackbusch W, Trottenberg U (eds) Lecture notes in mathematics, vol 1228. Springer, Heidelberg, pp 52–64

    Google Scholar 

  • Braess D (2007) Finite elemente. Springer, Heidelberg

    Google Scholar 

  • Bratzke D (2015) Optimal control of deep drawing processes based on reduced order models. Dissertation, TU Darmstadt

    Google Scholar 

  • Brenner SC, Scott LR (2007) The mathematical theory of finite element methods. Springer, Heidelberg

    Google Scholar 

  • Cartis C, Gould NIM, Toint PL (2011a) Adaptive cubic regularisation methods for unconstrained optimization. Part I: motivation, convergence and numerical results. Math Program 2(127):245–295

    Article  Google Scholar 

  • Cartis C, Gould NIM, Toint PL (2011b) Adaptive cubic regularisation methods for unconstrained optimization. Part II: worst-case function- and derivative-evaluation complexity. Math Program 2(130):295–319

    Article  Google Scholar 

  • Curtis FE, Johnson TC, Robinson DP, Wächter A (2013) An inexact sequential quadratic optimization algorithm for large-scale nonlinear optimization. Technical report, Department of ISE, Lehigh University

    Google Scholar 

  • Deutsches Institut für Normung (2012) Sachmerkmal-Listen—Teil 1: Begriffe und Grundsätze, DIN 4000-1:2012-09. Beuth, Berlin

    Google Scholar 

  • Ehrlenspiel K (2009) Integrierte Produktenwicklung: Denkabläufe, Methodeneinsatz, Zusammenarbeit, 4. akt. Auflage. Hanser, München

    Google Scholar 

  • Franke, HJ (1976) Untersuchungen zur Algorithmisierbarkeit des Konstruktionsprozesses. Dissertation, TU Braunschweig

    Google Scholar 

  • Fügenschuh A, Hess W, Schewe L, Martin A, Ulbrich S (2008) Verfeinerte Modelle zur Topologie- und Geometrie-Optimierung von Blechprofilen mit Kammern. In: Groche P (ed) 2. Zwischenkolloquium des Sonderforschungsbereichs 666: Integrale Blechbauweise höherer Verzweigungsordnung—Entwicklung, Fertigung, Bewertung, Darmstadt, 12–13 November 2008

    Google Scholar 

  • Göllner T (2014) Geometry optimization of branched sheet metal structures with a globalization strategy by adaptive cubic regularization. Dissertation, TU Darmstadt

    Google Scholar 

  • Göllner T; Günther U, Hess W, Martin A, Ulbrich S (2010) Form- und Topologieoptimierung verzweigter Blechbauteile. In: Groche, P (ed) 3. Zwischenkolloquium des Sonderforschungsbereichs 666, Darmstadt, 29–30 September 2010

    Google Scholar 

  • Göllner T, Günther U, Hess W, Pfetsch M, Ulbrich S (2012) Optimierung der Geometrie und Topologie flächiger verzweigter Blechbauteile und von Mehrkammerprofilen. In: Groche, P (ed) 4. Zwischenkolloquium des Sonderforschungsbereichs 666, Darmstadt, 14–15 November 2012

    Google Scholar 

  • Gramlich S (2013) Vom fertigungsgerechten Konstruieren zum produktionsintegrierenden Entwickeln: Durchgängige Modelle und Methoden im Produktlebenszyklus. Dissertation, Technische Universität Darmstadt

    Google Scholar 

  • Gramlich S, Birkhofer H, Bohn A (2011) Design process automation: a structured product description by properties and development of optimization algorithms. In: Culley SJ, Hicks BJ, McAloone TC, Howard TJ, Clarkson PJ (eds) Proceedings of the 18th International Conference on Engineering Design (ICED 11), Lyngby, Copenhagen, 15–19 August 2011, p 299–309

    Google Scholar 

  • Groche P, Schmitt W, Bohn A, Gramlich S, Ulbrich S, Günther U (2012) Integration of manufacturing-induced properties in product design. CIRP Ann 61(1):163–166

    Article  Google Scholar 

  • Günther U (2010) Integral sheet metal design by discrete optimization. Dissertation, TU Darmstadt

    Google Scholar 

  • Günther U, Martin A (2006) Mixed integer models for branched sheet metal products. Proc Appl Math Mech 6(1):697–698

    Article  Google Scholar 

  • Haasdonk B, Salomon J, Wohlmuth B (2011) A reduced basis method for the simulation of american options. In: Cangiani A, Davidchack R, Georgoulis E, Gorban A, Levesley J, Tretyakov M (eds) Numerical mathematics and advanced applications 2011. Springer, Berlin

    Google Scholar 

  • Hager C, Wohlmuth BI (2009) Nonlinear complementarity functions for plasticity problems with frictional contact. Comput Methods Appl Mech Eng 198:3411–3427

    Article  Google Scholar 

  • Haslinger J, Mäkinen RAE (2003) Introduction to shape optimization. SIAM, Philadelphia

    Book  Google Scholar 

  • Heidemann B (2001) Trennende Verknüpfung: Ein Prozessmodell als Quelle für Produktideen. Dissertation, Technische Universität Darmstadt

    Google Scholar 

  • Heinkenschloss M, Vicente LN (2001) Analysis of inexact trust-region SQP algorithms. SIAM J Optim 12(2):283–302

    Article  Google Scholar 

  • Hess W (2010) Geometry optimization with PDE constraints and applications to the design of branched sheet metal products. Dissertation, TU Darmstadt

    Google Scholar 

  • Hess W, Ulbrich S (2012) An inexact ℓ1 penalty SQP algorithm for PDE-constrained optimization with an application to shape optimization in linear elasticity. Optim Mehtod Softw 28(5):943–968

    Article  Google Scholar 

  • Hinze M, Pinnau R, Ulbrich M, Ulbrich S (2009) Optimization with PDE constraints. Springer, Heidelberg

    Google Scholar 

  • Hubka V (1973) Theorie der Maschinensysteme: Grundlagen einer wissenschaftlichen Konstruktionslehre. Springer, Berlin

    Book  Google Scholar 

  • Johnson D (1985) The NP-completeness column: an ongoing guide. J Algorithms 6(1):145–159

    Article  Google Scholar 

  • Johnson SG (2016) The NLOpt nonlinear optimization package. http://ab-initio.mit.edu/nlopt. Accessed 25 Jun 2015

  • Kikuchi N, Oden JT (1988) Contact problems in elasticity: a study of variational inequalities and finite element methods. SIAM, Philadelphia

    Book  Google Scholar 

  • Laursen TA (2002) Computational contact and impact mechanics. Springer, Heidelberg

    Google Scholar 

  • Lindemann U (2009) Methodische Entwicklung technischer Produkte: Methoden flexibel und situationsgerecht anwenden, 3. korr. Auflage. Springer, Berlin

    Google Scholar 

  • Lüthen H, Pfetsch ME (2016) On the connected subgraph polytope. Manuscript, TU Darmstadt

    Google Scholar 

  • Mattmann I, Roos M, Gramlich S (2014) Transformation und Integration von Marktanforderungen und fertigungstechnologischen Erkenntnissen in die Produktentwicklung. In: Groche P (ed) Tagungsband 5. Zwischenkolloquium SFB 666, Mörfelden-Walldorf, 19–20 November 2014, pp 5–14

    Google Scholar 

  • Mattmann I, Gramlich S, Kloberdanz H (2015a) The malicious labyrinth of requirements: three types of requirements for a systematic determination of product properties. In: Weber C, Husung S, Cascini G, Cantamessa M, Marjanovic D, Rotini F (eds) Proceedings of the 20th International Conference on Engineering Design (ICED 15), Milan, 27–30 July 2015, p 31–40

    Google Scholar 

  • Mattmann I, Gramlich S, Kloberdanz H (2015b) The inscrutable jungle of quality criteria: how to formulate requirements for a successful product development. In: Shpitalni M, Fischer A, Molcho G (eds) Procedia CIRP, vol 36, CIRP 25th Design Conference Innovative Product Creation, p 153–158

    Google Scholar 

  • Mattmann I, Gramlich S, Kloberdanz H (2016a) Mapping requirements to product properties: the mapping model. In: Marjanovic D, Storga M, Pavkovic N, Bojcetic N, Skec S (eds) Proceedings of the DESIGN 2016 14th International Design Conference, Dubrovnik, p 33–44

    Google Scholar 

  • Mattmann I, Gramlich S, Kloberdanz H (2016b) Getting requirements fit for purpose—improvement of requirement quality for requirement standardization. In: Wang L, Kjellberg R (eds) Procedia CIRP, vol 50, 26th CIRP Design Conference, p 466–471

    Google Scholar 

  • Nemhauser GL, Wolsey LA (1988) Integer and combinatorial optimization. Wiley-Interscience, New York

    Book  Google Scholar 

  • Nocedal J, Wright SJ (2006) Numerical optimization. Springer, Heidelberg

    Google Scholar 

  • Outrata J, Kocvara M, Zowe J (1998) Nonsmooth approach to optimization problems with equilibrium constraints, vol. 28 of Nonconvex Optimization and Its Applications. Kluwer Academic Publishers, Heidelberg

    Google Scholar 

  • Pahl G, Beitz W, Feldhusen J, Grote KH (2007) Engineering design: a systematic approach, 3rd edn. Springer, London

    Book  Google Scholar 

  • Piegl L, Tiller W (1997) The NURBS book. Springer, Heidelberg

    Book  Google Scholar 

  • Popp A, Gee MW, Wall WA (2009) A finite deformation mortar contact formulation using a primal-dual active set strategy. Int J Numer Methods Eng 79(11):1354–1391

    Article  Google Scholar 

  • Röder J (2014) Entwicklung einer clusterbasierten Methodik zur Anforderungserfassung auf Basis eines Modellraums zur Kategorisierung von Anforderungen. Dissertation, TU Darmstadt

    Google Scholar 

  • Röder B, Birkhofer H, Bohn A (2011) Clustering customer dreams: an approach for more efficient requirement acquisition. In: Culley SJ, Hicks BJ, McAloone TC, Howard TJ, Dong A (eds) Proceedings of the 18th International Conference on Engineering Design (ICED 11), Lyngby, Copenhagen, 15–19 August 2011, p 11–20

    Google Scholar 

  • Röder B, Gamlich S, Birkhofer H (2012) Von der abstrakten Anforderung zur formalisierten Entwicklungsaufgabe: Algorithmenbasierte Entwicklung am Beispiel komplexer, spaltprofilierter Blechbaugruppen. In: Groche P (ed) Tagungsband 4. Zwischenkolloquium SFB 666, Darmstadt, 14–15 November 2012, p 5–14

    Google Scholar 

  • Roos M, Horn B, Gramlich S, Ulbrich S, Kloberdanz H (2016) Manufacturing integrated algorithm-based product design: case study of a snap-fit fastening. In: Wang L, Kjellberg R (eds) Procedia CIRP, vol 50, 26th CIRP Design Conference, p 123–128

    Google Scholar 

  • Roy R, Hinduja S, Teti R (2008) Recent advances in engineering design optimisation: challenges and future trends. CIRP Ann 57(2):697–715

    Article  Google Scholar 

  • Schramm H, Zowe J (1992) A version of the bundle idea for minimizing a nonsmooth function: conceptual idea, convergence analysis, numerical results. SIAM J Optim 2(1):121–152

    Article  Google Scholar 

  • Seitz A, Popp A, Wall WA (2014) A Semismooth Newton method for orthotropic plasticity and frictional contact at finite strains. Comput Methods Appl Mech Eng 285:228–254

    Article  Google Scholar 

  • Suh NP (1998) Axiomatic design theory for systems. Res Eng Des 10(4):189–209

    Article  Google Scholar 

  • Suh NP (2001) Axiomatic design: advances and applications. MIT-Pappalardo series in mechanical engineering, Oxford University Press

    Google Scholar 

  • Tekkaya AE, Allwood JM, Bariani PF, Bruschi S, Cao J, Gramlich S, Groche P, Hirt G, Ishikawa T, Löbbe C, Lueg-Althoff J, Merklein M, Misiolek WZ, Pietrzyk M, Shivpuri R, Yanagimoto J (2015) Metal forming beyond shaping: predicting and setting product properties. CIRP Ann 64(2):629–653

    Article  Google Scholar 

  • Ulbrich M (2002) Semismooth Newton methods for operator equations in function spaces. SIAM J Optim 13(3):805–841

    Article  Google Scholar 

  • Ulbrich M, Ulbrich S, Bratzke D (2016) A multigrid semismooth Newton method for semilinear contact problems. Technical report, TU Darmstadt, TU München

    Google Scholar 

  • Verein Deutscher Ingenieure (1993) Methodik zum Entwickeln und Konstruieren technischer Systeme und Produkte, Richtlinie VDI 2221. VDI, Düsseldorf

    Google Scholar 

  • Volkwein S (2013) Proper orthogonal decomposition: theory and reduced-order modelling. Lecture Notes

    Google Scholar 

  • Wagner C, Roos M, Gramlich S, Kloberdanz H (2016) Process integrated design guidelines: systematically linking manufacturing processes to product design. In: Marjanovic D, Storga M, Pavkovic N, Bojcetic N, Skec S (eds) Proceedings of the DESIGN 2016 14th International Design Conference, Dubrovnik, p 739–748

    Google Scholar 

  • Wäldele M (2012) Erarbeitung einer Theorie der Eigenschaften technischer Produkte: Ein Beitrag für die konventionelle und algorithmenbasierte Produktentwicklung. Dissertation, TU Darmstadt

    Google Scholar 

  • Wang Y, Buchanan A, Butenko S (2015) On imposing connectivity constraints in integer programs. Optimization Online

    Google Scholar 

  • Wohlmuth BI (2000) A mortar finite element method using dual spaces for the lagrange multiplier. SIAM J Num Analysis 38(3):989–1012

    Article  Google Scholar 

  • Wriggers P, Laursen TA (2006) Computational contact mechanics, 2nd edn. Springer, Heidelberg

    Book  Google Scholar 

  • Yserentant H (1993) Old and new convergence proofs for multigrid methods. Acta Numerica 2:285–326

    Article  Google Scholar 

  • Ziems JC, Ulbrich S (2011) Adaptive multilevel inexact SQP methods for PDE-constrained optimization. SIAM J Optim 21(1):1–40

    Article  Google Scholar 

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Authors and Affiliations

  1. Research Group Discrete Optimization (DOpt), Technische Universität Darmstadt, Darmstadt, Germany

    H. Lüthen & M. Pfetsch

  2. Institute for Product Development and Machine Elements (pmd), Technische Universität Darmstadt, Darmstadt, Germany

    S. Gramlich, I. Mattmann, M. Roos & C. Wagner

  3. Research Group Nonlinear Optimization (NOpt), Technische Universität Darmstadt, Darmstadt, Germany

    B. Horn, S. Ulbrich & A. Walter

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  1. H. Lüthen
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  2. S. Gramlich
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  4. I. Mattmann
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  7. S. Ulbrich
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Correspondence to H. Lüthen .

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Editors and Affiliations

  1. Institute for Production Engineering and Forming Machines (PtU), Technische Universität Darmstadt, Darmstadt, Germany

    Peter Groche

  2. Materials Science Department, Physical Metallurgy Division , Technische Universität Darmstadt, Darmstadt, Germany

    Enrico Bruder

  3. Institute for Product Development and Machine Elements (pmd), Technische Universität Darmstadt, Darmstadt, Germany

    Sebastian Gramlich

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Lüthen, H. et al. (2017). Finding the Best: Mathematical Optimization Based on Product and Process Requirements. In: Groche, P., Bruder, E., Gramlich, S. (eds) Manufacturing Integrated Design. Springer, Cham. https://doi.org/10.1007/978-3-319-52377-4_5

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