nach oben

Structural and Multidisciplinary Optimization

Erschienen in:

13.03.2020 | Educational Paper

On three concepts in robust design optimization: absolute robustness, relative robustness, and less variance

verfasst von: Yoshihiro Kanno

Erschienen in: Structural and Multidisciplinary Optimization | Ausgabe 2/2020

Einloggen

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

search-config

KI-gestützte Suche

Aus

Abstract

This paper provides a clear perspective on existing several different approaches to robust design optimization of structures. We primarily consider three approaches: the worst-case optimization, the discrepancy (i.e., the maximum gap between the objective values in a nominal case and a possibly occurring case) minimization, and the variance minimization. Some other formulations can also be linked with one of these three approaches. To investigate how the solutions derived by these three approaches differ from each other, we present two numerical examples. This direct comparison clarifies different features of these approaches.

Vorheriger Artikel Crashworthiness optimisation for the rectangular tubes with axisymmetric and uniform thicknesses under offset loading

Nächster Artikel A filter-based level set topology optimization method using a 62-line MATLAB code

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!

Jetzt informieren

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!

Jetzt informieren

It should be clear again that, in general, $\hat {x}_{\text {b}}^{\prime }$ is different both from $x^{*}_{\text {b}}$ and $\hat {x}_{\text {b}}$.

Semidefinite programming (SDP) is the optimization of a linear objective function under a linear matrix inequality. Namely, it has the form

$$ \begin{array}{@{}rcl@{}} &&{\kern-.7pc} \text{{Minimize}} {\quad} \sum\limits_{i=1}^{m} b_{i} y_{i} \\ && {\kern-.7pc}\text{{subject~to}}\quad \sum\limits_{i=1}^{m} x_{i} A_{i} + C \succeq 0 , \end{array} $$

where $A_{1},\dots ,A_{m}$, $C \in \mathbb {R}^{n \times n}$ are constant symmetric matrices and $b_{1},\dots ,b_{m} \in \mathbb {R}$ are constants.

The stiffness matrix of a truss satisfies K(x + y) = K(x) + K(y) for any x, y ∈ X. Therefore, for any λ ∈ [0, 1], we have

$$ \begin{array}{@{}rcl@{}} &&{\kern-.6pc}{\pi}(\lambda {\boldsymbol{x}} + (1-\lambda){\boldsymbol{y}};{\boldsymbol{q}} ) \\ &&{\kern-.7pc}= \sup_{{\boldsymbol{u}}} \left\{ 2 {\boldsymbol{q}}^{\top} {\boldsymbol{u}} - \left[ \lambda {\boldsymbol{u}}^{\top} K({\boldsymbol{x}}) {\boldsymbol{u}} + (1-\lambda) {\boldsymbol{u}}^{\top} K({\boldsymbol{y}}) {\boldsymbol{u}} \right] \right\} \\ &&{\kern-.7pc}\le \lambda \sup_{{\boldsymbol{u}}} \{ 2 {\boldsymbol{q}}^{\top} {\boldsymbol{u}} - {\boldsymbol{u}}^{\top} K({\boldsymbol{x}}) {\boldsymbol{u}} \} + (1-\lambda) \sup_{{\boldsymbol{u}}} \{ 2 {\boldsymbol{q}}^{\top} {\boldsymbol{u}} - {\boldsymbol{u}}^{\top} K({\boldsymbol{y}}) {\boldsymbol{u}} \} \\ &&{\kern-.7pc}= \lambda {\pi}({\boldsymbol{x}} ;{\boldsymbol{q}} ) + (1-\lambda) {\pi}({\boldsymbol{y}};{\boldsymbol{q}} ) , \end{array} $$

which shows the convexity of π(⋅; q). See, e.g., Kanno (2011, Proposition 3.1.2) for more accounts.

Using a vector-valued function as the objective function in (5a) means that this problem is a multi-objective optimization problem (particularly, problem (5a) is a bi-objective optimization problem). That is, we attempt to minimize both ${\pi }(\textit {\textbf {x}}; \tilde {\textit {\textbf {q}}})$ and π^worst(x; α). We call x^∗∈ X a Pareto optimal solution of (5a) if there exists no ${\boldsymbol {x}}^{\prime } \in \mathbb {R}^{n}$$({\boldsymbol {x}}^{\prime } \not = {\boldsymbol {x}}^{*})$ satisfying ${\pi }({\boldsymbol {x}}^{\prime }; \tilde {{\boldsymbol {q}}}) \le {\pi }({\boldsymbol {x}}^{*}; \tilde {{\boldsymbol {q}}})$ and ${\pi }^{{\text {worst}}}({\boldsymbol {x}}^{\prime }; \alpha ) \le {\pi }^{{\text {worst}}}({\boldsymbol {x}}^{*}; \alpha )$. See, e.g., Boyd and Vandenberghe (2004, Section 4.7) and Chankong et al. (1985) for fundamentals of multi-objective optimization.

We adopt the constraint method (a.k.a. ε-constraint method) for scalarization of multi-objective optimization. Application of standard constraint method converts problem (5a) into problem (6a). See, e.g., Haimes et al. (1971), Haimes and Hall (1974), Cohon and Marks (1974), Hwang et al. (1980), Chankong et al. (1985) for details of the constraint method.

We make use of the equality

$$ \begin{array}{@{}rcl@{}} &&{\kern-.7pc}\sup\{ {\boldsymbol{x}}^{\top} {\boldsymbol{y}} \mid \| {\boldsymbol{x}} \|_{\infty} \le 1 \} = \sup\{ {\boldsymbol{x}}^{\top} {\boldsymbol{y}} \mid \| {\boldsymbol{x}} \|_{\infty} = 1 \} \\ &&~~~~~{\kern-.7pc}= \sum\limits_{i=1}^{n} \sup\{ x_{i} y_{i} \mid |x_{i}| = 1 \} = \| {\boldsymbol{y}} \|_{1} . \end{array} $$

Here, ∥⋅∥₁ is called the dual norm of $\| \cdot \|_{\infty }$.

Indeed, for any x, y ∈ X and for any λ ∈ [0, 1], we have

$$ \begin{array}{@{}rcl@{}} &&{\kern-.7pc}{\pi}^{{\text{worst}}}(\lambda {\boldsymbol{x}} + (1-\lambda) {\boldsymbol{y}} ;\alpha) \\ &&~~~~~~{\kern-.7pc}= \sup_{{\boldsymbol{q}} \in Q(\alpha)} \{ {\pi}(\lambda {\boldsymbol{x}} + (1-\lambda) {\boldsymbol{y}} ;{\boldsymbol{q}}) \} \\ &&~~~~~~{\kern-.7pc}\le \sup_{{\boldsymbol{q}} \in Q(\alpha)} \{ \lambda {\pi}({\boldsymbol{x}} ;{\boldsymbol{q}}) + (1-\lambda) {\pi}({\boldsymbol{y}} ;{\boldsymbol{q}}) \} \\ &&~~~~~~{\kern-.7pc}\le \lambda \sup_{{\boldsymbol{q}} \in Q(\alpha)} \{ {\pi}({\boldsymbol{x}} ;{\boldsymbol{q}}) \} + (1-\lambda) \sup_{{\boldsymbol{q}} \in Q(\alpha)} \{ {\pi}({\boldsymbol{y}} ;{\boldsymbol{q}}) \} \\ &&~~~~~~{\kern-.7pc}\le \lambda {\pi}^{{\text{worst}}}({\boldsymbol{x}} ;\alpha) + (1-\lambda) {\pi}^{{\text{worst}}}({\boldsymbol{y}} ;\alpha) , \end{array} $$

where the first inequality follows the convexity of π(⋅; q). See, e.g., Hiriart-Urruty and Lemaréchal (1993, Proposition IV.2.1.2) for more accounts.

See also Calafiore and Dabbene (2008, Proposition 2).

We have $\sup \{ {\boldsymbol {x}}^{\top } {\boldsymbol {y}} \mid \| {\boldsymbol {x}} \|_{1} \le 1 \} = \| {\boldsymbol {y}} \|_{\infty }$, because dual of the dual norm is the norm itself; see, e.g., Hiriart-Urruty and Lemaréchal (1993, Proposition V.3.2.1).

DC decomposition of a DC function is not unique. For example, for any convex function $f : \mathbb {R}^{n} \to \mathbb {R}$, we see that π^disc(⋅; α) can be decomposed as

$$ \begin{array}{@{}rcl@{}} {\pi}^{{\text{disc}}}({\boldsymbol{x}}; \alpha) = ({\pi}^{{\text{worst}}}({\boldsymbol{x}}; \alpha) + f({\boldsymbol{x}})) - ({\pi}({\boldsymbol{x}}; \tilde{{\boldsymbol{q}}}) + f({\boldsymbol{x}})) , \end{array} $$

which is also a DC decomposition of π^disc(⋅; α).

Indeed, with the first-order approximation of ${\pi }(\tilde {{\boldsymbol {q}}}+{\boldsymbol {\zeta }})$, (18) follows

$$ \begin{array}{@{}rcl@{}} {\mathrm{E}}[{\pi}(\tilde{{\boldsymbol{q}}}+{\boldsymbol{\zeta}})] \simeq {\mathrm{E}}[ {\pi}(\tilde{{\boldsymbol{q}}}) + \nabla{\pi}(\tilde{{\boldsymbol{q}}})^{\top} {\boldsymbol{\zeta}} ] = {\pi}(\tilde{{\boldsymbol{q}}}) + \nabla{\pi}(\tilde{{\boldsymbol{q}}})^{\top} {\mathrm{E}}[{\boldsymbol{\zeta}}] \end{array} $$

and (20) follows

$$ \begin{array}{@{}rcl@{}} &&{\kern-.7pc}{\text{Var}}[{\pi}(\tilde{{\boldsymbol{q}}}+{\boldsymbol{\zeta}})] \\ &&{\kern-.7pc}~~~~~~~= {\mathrm{E}} \left[ ({\pi}(\tilde{{\boldsymbol{q}}}+{\boldsymbol{\zeta}}) - {\mathrm{E}}[{\pi}(\tilde{{\boldsymbol{q}}}+{\boldsymbol{\zeta}})])^{2} \right] \\ &&{\kern-.7pc}~~~~~~~\simeq {\mathrm{E}} \left[ ({\pi}(\tilde{{\boldsymbol{q}}}) + \nabla{\pi}(\tilde{{\boldsymbol{q}}})^{\top} {\boldsymbol{\zeta}} - {\pi}(\tilde{{\boldsymbol{q}}}))^{2} \right] \\ &&{\kern-.7pc}~~~~~~~= {\mathrm{E}}\left[ \nabla{\pi}(\tilde{{\boldsymbol{q}}})^{\top} ({\boldsymbol{\zeta}} {\boldsymbol{\zeta}}^{\top}) \nabla{\pi}(\tilde{{\boldsymbol{q}}}) \right] \\ &&{\kern-.7pc}~~~~~~~= \nabla{\pi}(\tilde{{\boldsymbol{q}}})^{\top} {\mathrm{E}}\left[ {\boldsymbol{\zeta}} {\boldsymbol{\zeta}}^{\top} \right] \nabla{\pi}(\tilde{{\boldsymbol{q}}}) . \end{array} $$

Here, the explicit dependency of π on x has been suppressed for notational simplicity.

It is clear that $\max \limits \{ {\pi }(\textit {\textbf {x}};\textit {\textbf {q}}) \mid \textit {\textbf {q}} \in Q(\alpha ) \}$, as well as $\min \limits \{ {\pi }({\boldsymbol {x}};{\boldsymbol {q}}) \mid {\boldsymbol {q}} \in Q(\alpha ) \}$, is attained at a point on the boundary of Q(α).

For α = 10kN, there exists almost no difference between the solution with minimal nominal-case compliance and the solution with minimal worst-case compliance.

In the following we assume without loss of generality that the original design optimization problem is formulated as a minimization problem, as the case in the previous sections.

Alefeld G, Mayer G (2000) Interval analysis: theory and applications. J Comput Appl Math 121:421–464MathSciNetMATH

Asadpoure A, Tootkaboni M, Guest JK (2011) Robust topology optimization of structures with uncertainties in stiffness—application to truss structures. Comput Struct 89:1131–1141

Ben-Haim Y (1994) Fatigue lifetime with load uncertainty represented by convex model. J Eng Mech (ASCE) 120:445–462

Ben-Haim Y (1995) A non-probabilistic measure of reliability of linear systems based on expansion of convex models. Struct Saf 17:91–109

Ben-Haim Y, Elishakoff I (1990) Convex models of uncertainty in applied mechanics. Elsevier, New YorkMATH

Ben-Tal A, El Ghaoui L, Nemirovski A (2009) Robust optimization. Princeton University Press, PrincetonMATH

Beyer H-G, Sendhoff B (2007) Robust optimization—a comprehensive survey. Comput Methods Appl Mech Eng 196:3190–3218MathSciNetMATH

Boyd S, Vandenberghe L (2004) Convex optimization. Cambridge University Press, CambridgeMATH

Calafiore GC, Dabbene F (2008) Optimization under uncertainty with applications to design of truss structures. Struct Multidiscip Optim 35:189–200MathSciNetMATH

Chankong V, Haimes YY, Thadathil J, Zionts S (1985) Multiple criteria optimization; a state of the art review. In: Haimes YY, Chankong V (eds) Decision making with multiple objectives. Springer, Berlin, pp 36–90

Chen W, Allen JK, Tsui K-L, Mistree F (1996) A procedure for robust design: minimizing variations caused by noise factors and control factors. J Mech Des 118:478–485

Chen W, Fu W, Biggers SB, Latour RA (2000) An affordable approach for robust design of thick laminated composite structure. Optim Eng 1:305–322MATH

Chen S, Lian H, Yang X (2002) Interval static displacement analysis for structures with interval parameters. Int J Numer Methods Eng 53:393–407MATH

Cherkaev E, Cherkaev A (2003) Principal compliance and robust optimal design. J Elast 72:71–98MathSciNetMATH

Cherkaev E, Cherkaev A (2008) Minimax optimization problem of structural design. Comput Struct 86:1426–1435MATH

Choi JH, Lee WH, Park JJ, Youn BD (2008) A study on robust design optimization of layered plate bonding process considering uncertainties. Struct Multidiscip Optim 35:531–540

Cohon JL, Marks DH (1974) A review and evaluation of multiobjective programing techniques. Water Resour Res 11:208–220

Collobert R, Sinz F, Weston J, Bottou L (2006) Large scale transductive SVMs. J Mach Learn Res 7:1687–1712MathSciNetMATH

Cornuéjols G, Pena J, Tütüncü R (2018) Optimization methods in finance, 2nd edn. Cambridge University Press, CambridgeMATH

da Silva GA, Cardoso EL, Beck AT (2019) Non-probabilistic robust continuum topology optimization with stress constraints. Struct Multidiscip Optim 59:1181–1197MathSciNet

de Gournay F, Allaire G, Jouve F (2008) Shape and topology optimization of the robust compliance via the level set method. ESAIM: Control Optim Calc Var 14:43–70MathSciNetMATH

Ganzerli S, Pantelides CP (1999) Load and resistance convex models for optimum design. Struct Optim 17:259–268

Grant M, Boyd S (2008) Graph implementations for nonsmooth convex programs. In: Blondel V, Boyd S, Kimura H (eds) Recent advances in learning and control (a tribute to M. Vidyasagar). Springer, pp 95–110

Grant M, Boyd S (2019) CVX: Matlab software for disciplined convex programming, Ver. 2.1 http://cvxr.com/cvx/ (Accessed: July 2019)

Haimes YY, Hall WA (1974) Multiobjectives in water resource systems analysis: the surrogate worth trade off method. Water Resour Res 10:615–624

Haimes YY, Lasdon LS, Wismer DA (1971) On a bicriterion formulation of the problems of integrated system identification and system optimization. IEEE Trans Syst Man Cybern 1:296–297MathSciNetMATH

Han JS, Kwak BM (2004) Robust optimization using a gradient index: MEMS applications. Struct Multidiscip Optim 27:469–478

Hashimoto D, Kanno Y (2015) A semidefinite programming approach to robust truss topology optimization under uncertainty in locations of nodes. Struct Multidiscip Optim 51:439–461MathSciNet

Hiriart-Urruty J-B, Lemaréchal C (1993) Convex analysis and minimization algorithms, vol I. Springer, BerlinMATH

Holmberg E, Thore C-J, Klarbring A (2015) Worst-case topology optimization of self-weight loaded structures using semi-definite programming. Struct Multidiscip Optim 52:915–928MathSciNet

Huan Z, Zhenghong G, Fang X, Yidian Z (2019) Review of robust aerodynamic design optimization for air vehicles. Arch Comput Methods Eng 26:685–732MathSciNet

Hwang CL, Paidy SR, Yoon K, Masud ASM (1980) Mathematical programming with multiple objectives: a tutorial. Comput Oper Res 7:5–31

Ito M, Kogiso N, Hasegawa T (2018) A consideration on robust design optimization problem through formulation of multiobjective optimization. J Adv Mech Des Syst Manuf 12:18–00076

Kanno Y (2011) Nonsmooth mechanics and convex optimization. CRC Press, Boca RatonMATH

Kanno Y (2015) A note on formulations of robust compliance optimization under uncertain loads. J Struct Construct Eng (Trans AIJ) 80:601–607

Kanno Y (2018) Robust truss topology optimization via semidefinite programming with complementarity constraints: a difference-of-convex programming approach. Comput Optim Appl 71:403–433MathSciNetMATH

Kanno Y (2019) A data-driven approach to non-parametric reliability-based design optimization of structures with uncertain load. Struct Multidiscip Optim 60:83–97MathSciNet

Keshavarzzadeh V, Fernandez F, Tortorelli DA (2017) Topology optimization under uncertainty via non-intrusive polynomial chaos expansion. Comput Methods Appl Mech Eng 318:120–147MathSciNetMATH

Kim N-K, Kim D-H, Kim D-W, Kim H-G, Lowther DA, Sykulski JK (2010) Robust optimization utilizing the second-order design sensitivity information. IEEE Trans Magn 46:3117–3120

Kitayama S, Yamazaki K (2014) Sequential approximate robust design optimization using radial basis function network. Int J Mech Mater Des 10:313–328

Kogiso N, Ahn W-J, Nishiwaki S, Izui K, Yoshimura M (2008) Robust topology optimization for compliant mechanisms considering uncertainty of applied loads. J Adv Mech Des Syst Manuf 2:96–107

Kriegesmann B, Lüdeker JK (2019) Robust compliance topology optimization using the first-order second-moment method. Struct Multidiscip Optim 60:269–286MathSciNet

Lee K-H, Park G-J (2006) A global robust optimization using kriging based approximation model. JSME Int J Ser C Mech Syst Mach Element Manuf 49:779–788

Le Thi HA, Pham Dinh T (2005) The DC (difference of convex functions) programming and DCA revisited with DC models of real world nonconvex optimization problems. Ann Oper Res 133:23–46MathSciNetMATH

Le Thi HA, Pham Dinh T (2018) DC Programming and DCA: thirty years of developments. Math Program 169:5–68MathSciNetMATH

Lipp T, Boyd S (2016) Variations and extension of the convex–concave procedure. Optim Eng 17:263–287MathSciNetMATH

Luo Y, Kang Z, Luo Z, Li A (2009) Continuum topology optimization with non-probabilistic reliability constraints based on multi-ellipsoid convex model. Struct Multidiscip Optim 39:297–310MathSciNetMATH

Markowitz H (1952) Portfolio selection. J Financ 7:77–91

Nakazawa Y, Kogiso N, Yamada T, Nishiwaki S (2016) Robust topology optimization of thin plate structure under concentrated load with uncertain load position. J Adv Mech Des Syst Manuf 10:16–00232

Neumaier A, Pownuk A (2007) Linear systems with large uncertainties, with applications to truss structures. Reliab Comput 13:149–172MathSciNetMATH

Pantelides CP, Ganzerli S (1998) Design of trusses under uncertain loads using convex models. J Struct Eng (ASCE) 124:318–329

Park G-J, Lee T-H, Lee KH, Hwang K-H (2006) Robust design: an overview. AIAA J 44:181–191

Pólik I (2005) Addendum to the SeDuMi user guide: Version 1.1. Technical Report, Advanced Optimization Laboratory. McMaster University, Hamilton. http://sedumi.ie.lehigh.edu/sedumi/

Rao SS, Cao L (2002) Optimum design of mechanical systems involving interval parameters. J Mech Des (ASME) 124:465–472

Shimoyama K, Lim JN, Jeong S, Obayashi S, Koishi M (2009) Practical implementation of robust design assisted by response surface approximation and visual data-mining. J Mech Des 131:061007

Shin YS, Grandhi RV (2001) A global structural optimization technique using an interval method. Struct Multidiscip Optim 22:351–36

Sobieszczanski-Sobieski J, Morris A, van Tooren MJL (2015) Multidisciplinary design optimization supported by knowledge based engineering. Wiley, Chichester

Sriperumbudur BK, Lanckriet GRG (2009) On the convergence of the concave-convex procedure. Adv Neural Inf Process Syst 22:1759–1767

Sturm JF (1999) Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim Methods Softw 11–12:625–653MathSciNetMATH

Su J, Renaud JE (1997) Automatic differentiation in robust optimization. AIAA J 35:1072–1079MATH

Sun G, Song X, Baek S, Li Q (2014) Robust optimization of foam-filled thin-walled structure based on sequential Kriging metamodel. Struct Multidiscip Optim 49:897–913

Takezawa A, Nii S, Kitamura M, Kogiso N (2011) Topology optimization for worst load conditions based on the eigenvalue analysis of an aggregated linear system. Comput Methods Appl Mech Eng 200:2268–2281MathSciNetMATH

Tootkaboni M, Asadpoure A, Guest JK (2012) Topology optimization of continuum structures under uncertainty—a polynomial chaos approach. Comput Methods Appl Mech Eng 201–204:263– 275MathSciNetMATH

Toyoda M, Kogiso N (2015) Robust multiobjective optimization method using satisficing trade-off method. J Mech Sci Technol 29:1361–1367

Valdebenito MA, Schuëller GI (2010) A survey on approaches for reliability-based optimization. Struct Multidiscip Optim 42:645–663MathSciNetMATH

Yao W, Chen X, Luo W, van Tooren M, Guo J (2011) Review of uncertainty-based multidisciplinary design optimization methods for aerospace vehicles. Prog Aerosp Sci 47:450–479

Yonekura K, Kanno Y (2010) Global optimization of robust truss topology via mixed integer semidefinite programming. Optim Eng 11:355–379MathSciNetMATH

Yuille AL, Rangarajan A (2003) The concave-convex procedure. Neural Comput 15:915–936MATH

Zang C, Friswell MI, Mottershead JE (2005) A review of robust optimal design and its application in dynamics. Comput Struct 83:315–326

Zhang X, He J, Takezawa A, Kang Z (2018a) Robust topology optimization of phononic crystals with random field uncertainty. Int J Numer Methods Eng 115:1154–1173MathSciNet

Zhang W, Kang Z (2017) Robust shape and topology optimization considering geometric uncertainties with stochastic level set perturbation. Int J Numer Methods Eng 110:31–56MathSciNetMATH

Zhang Y, Li X, Guo S (2018b) Portfolio selection problems with Markowitz’s mean–variance framework: a review of literature. Fuzzy Optim Decis Making 17:125–158MathSciNetMATH

Zhao Z, Han X, Jiang C, Zhou X (2010) A nonlinear interval-based optimization method with local-densifying approximation technique. Struct Multidiscip Optim 42:559–573

Titel: On three concepts in robust design optimization: absolute robustness, relative robustness, and less variance
verfasst von: Yoshihiro Kanno
Publikationsdatum: 13.03.2020
Verlag: Springer Berlin Heidelberg
Erschienen in: Structural and Multidisciplinary Optimization / Ausgabe 2/2020
Print ISSN: 1615-147X
Elektronische ISSN: 1615-1488
DOI: https://doi.org/10.1007/s00158-020-02503-9

Premium Partner

Marktübersichten

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

Zur Marktübersicht

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

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

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Weitere Artikel der Ausgabe 2/2020

Comment on “Minimization of maximum failure criterion of laminated composite shell structure by optimizing distributed-material orientation” by Shimoda M., Muramatsu Y., Tsukihara R.

A projection approach for topology optimization of porous structures through implicit local volume control

Surrogate model-based reliability analysis for structural systems with correlated distribution parameters

Accelerating the convergence of steady adjoint equations by dynamic mode decomposition

Sequential approximate reliability-based design optimization for structures with multimodal random variables

An efficient method for estimating time-dependent global reliability sensitivity

Premium Partner

Marktübersichten