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Amouzgar, K. & Strömberg, N. (2017). Radial Basis Functions as Surrogate Models with A Priori Bias in Comparison with a Posteriori Bias. Structural and multidisciplinary optimization (Print), 55(4), 1453-1469
Open this publication in new window or tab >>Radial Basis Functions as Surrogate Models with A Priori Bias in Comparison with a Posteriori Bias
2017 (English)In: Structural and multidisciplinary optimization (Print), ISSN 1615-147X, E-ISSN 1615-1488, Vol. 55, no 4, p. 1453-1469Article in journal (Refereed) Published
Abstract [en]

In order to obtain a robust performance, the established approach when using radial basis function networks (RBF) as metamodels is to add a posteriori bias which is defined by extra orthogonality constraints. We mean that this is not needed, instead the bias can simply be set a priori by using the normal equation, i.e. the bias becomes the corresponding regression model. In this paper we demonstrate that the performance of our suggested approach with a priori bias is in general as good as, or even for many test examples better than, the performance of RBF with a posteriori bias. Using our approach, it is clear that the global response is modelled with the bias and that the details are captured with radial basis functions. The accuracy of the two approaches are investigated by using multiple test functions with different degrees of dimensionality. Furthermore, several modeling criteria, such as the type of radial basis functions used in the RBFs, dimension of the test functions, sampling techniques and size of samples, are considered to study their affect on the performance of the approaches. The power of RBF with a priori bias for surrogate based design optimization is also demonstrated by solving an established engineering benchmark of a welded beam and another benchmark for different sampling sets generated by successive screening, random, Latin hypercube and Hammersley sampling, respectively. The results obtained by evaluation of the performance metrics, the modeling criteria and the presented optimal solutions, demonstrate promising potentials of our RBF with a priori bias, in addition to the simplicity and straight-forward use of the approach.

Place, publisher, year, edition, pages
Springer Publishing Company, 2017
Keywords
metamodeling, radial basis function, design optimization, design of experiment
National Category
Applied Mechanics Computer Sciences
Research subject
Mechanical Engineering
Identifiers
urn:nbn:se:oru:diva-56960 (URN)10.1007/s00158-016-1569-0 (DOI)000398951100020 ()2-s2.0-84989170510 (Scopus ID)
Available from: 2017-04-05 Created: 2017-04-05 Last updated: 2018-01-13Bibliographically approved
Strömberg, N. (2017). Reliability-based design optimization using SORM and SQP. Structural and multidisciplinary optimization (Print), 56(3), 631-645
Open this publication in new window or tab >>Reliability-based design optimization using SORM and SQP
2017 (English)In: Structural and multidisciplinary optimization (Print), ISSN 1615-147X, E-ISSN 1615-1488, Vol. 56, no 3, p. 631-645Article in journal (Refereed) Published
Abstract [en]

In this work a second order approach for reliability-based design optimization (RBDO) with mixtures of uncorrelated non-Gaussian variables is derived by applying second order reliability methods (SORM) and sequential quadratic programming (SQP). The derivation is performed by introducing intermediate variables defined by the incremental iso-probabilistic transformation at the most probable point (MPP). By using these variables in the Taylor expansions of the constraints, a corresponding general first order reliability method (FORM) based quadratic programming (QP) problem is formulated and solved in the standard normal space. The MPP is found in the physical space in the metric of Hasofer-Lind by using a Newton algorithm, where the efficiency of the Newton method is obtained by introducing an inexact Jacobian and a line-search of Armijo type. The FORM-based SQP approach is then corrected by applying four SORM approaches: Breitung, Hohenbichler, Tvedt and a recent suggested formula. The proposed SORM-based SQP approach for RBDO is accurate, efficient and robust. This is demonstrated by solving several established benchmarks, with values on the target of reliability that are considerable higher than what is commonly used, for mixtures of five different distributions (normal, lognormal, Gumbel, gamma and Weibull). Established benchmarks are also generalized in order to study problems with large number of variables and several constraints. For instance, it is shown that the proposed approach efficiently solves a problem with 300 variables and 240 constraints within less than 20 CPU minutes on a laptop. Finally, a most well-know deterministic benchmark of a welded beam is treated as a RBDO problem using the proposed SORM-based SQP approach.

Place, publisher, year, edition, pages
Springer, 2017
Keywords
RBDO, FORM, SORM, SQP
National Category
Computer Sciences
Identifiers
urn:nbn:se:oru:diva-59289 (URN)10.1007/s00158-017-1679-3 (DOI)000406604700009 ()2-s2.0-85015679938 (Scopus ID)
Available from: 2017-08-29 Created: 2017-08-29 Last updated: 2018-01-13Bibliographically approved
Amouzgar, K. & Strömberg, N. (2014). An approach towards generating surrogate models by using RBFN with a priori bias. In: Proceedings of the ASME International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, 2014, Vol. 2B: . Paper presented at ASME, International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE, Buffalo, NY, USA, August 17-20, 2014. New York, USA: ASME Press, Article ID V02BT03A024.
Open this publication in new window or tab >>An approach towards generating surrogate models by using RBFN with a priori bias
2014 (English)In: Proceedings of the ASME International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, 2014, Vol. 2B, New York, USA: ASME Press, 2014, article id V02BT03A024Conference paper, Published paper (Refereed)
Abstract [en]

In this paper, an approach to generate surrogate modelsconstructed by radial basis function networks (RBFN) with a prioribias is presented. RBFN as a weighted combination of radialbasis functions only, might become singular and no interpolationis found. The standard approach to avoid this is to add a polynomialbias, where the bias is defined by imposing orthogonalityconditions between the weights of the radial basis functionsand the polynomial basis functions. Here, in the proposed a prioriapproach, the regression coefficients of the polynomial biasare simply calculated by using the normal equation without anyneed of the extra orthogonality prerequisite. In addition to thesimplicity of this approach, the method has also proven to predictthe actual functions more accurately compared to the RBFNwith a posteriori bias. Several test functions, including Rosenbrock,Branin-Hoo, Goldstein-Price functions and two mathematicalfunctions (one large scale), are used to evaluate the performanceof the proposed method by conducting a comparisonstudy and error analysis between the RBFN with a priori and aposteriori known biases. Furthermore, the aforementioned approachesare applied to an engineering design problem, that ismodeling of the material properties of a three phase sphericalgraphite iron (SGI) . The corresponding surrogate models arepresented and compared

Place, publisher, year, edition, pages
New York, USA: ASME Press, 2014
Keywords
Optimization, Response Surface, Surrogate Modelling, RBF, RBFN, Approximation Function
National Category
Applied Mechanics Mechanical Engineering
Research subject
Mechanical Engineering
Identifiers
urn:nbn:se:oru:diva-48247 (URN)000379987300024 ()2-s2.0-84961312861 (Scopus ID)978-0-7918-4632-2 (ISBN)
Conference
ASME, International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE, Buffalo, NY, USA, August 17-20, 2014
Available from: 2016-02-18 Created: 2016-02-15 Last updated: 2017-10-18Bibliographically approved
Amouzgar, K., Rashid, A. & Strömberg, N. (2013). Multi-objective optimization of a disc brake system by using SPEA2 and RBFN. In: ASME 2013 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference: Volume 3B: 39th Design Automation Conference. Paper presented at ASME 2013 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference (IDETC/CIE), Portland, Oregon, USA, August 4-7, 2013. New York: ASME Press
Open this publication in new window or tab >>Multi-objective optimization of a disc brake system by using SPEA2 and RBFN
2013 (English)In: ASME 2013 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference: Volume 3B: 39th Design Automation Conference, New York: ASME Press, 2013Conference paper, Published paper (Other academic)
Abstract [en]

Many engineering design optimization problems involve multiple conflicting objectives, which today often are obtained by computational expensive finite element simulations. Evolutionary multi-objective optimization (EMO) methods based on surrogate modeling is one approach of solving this class of problems. In this paper, multi-objective optimization of a disc brake system to a heavy truck by using EMO and radial basis function networks (RBFN) is presented. Three conflicting objectives are considered. These are: 1) minimizing the maximum temperature of the disc brake, 2) maximizing the brake energy of the system and 3) minimizing the mass of the back plate of the brake pad. An iterative Latin hypercube sampling method is used to construct the design of experiments (DoE) for the design variables. Next, thermo-mechanical finite element analysis of the disc brake, including frictional heating between the pad and the disc, is performed in order to determine the values of the first two objectives for the DoE. Surrogate models for the maximum temperature and the brake energy are created using RBFN with polynomial biases. Different radial basis functions are compared using statistical errors and cross validation errors (PRESS) to evaluate the accuracy of the surrogate models and to select the most accurate radial basis function. The multi-objective optimization problem is then solved by employing EMO using the strength Pareto evolutionary algorithm (SPEA2). Finally, the Pareto fronts generated by the proposed methodology are presented and discussed.

Place, publisher, year, edition, pages
New York: ASME Press, 2013
Keywords
Multi-objective Optimization, Disc Brake, RBF, RBFN, Surrogate Modelling, Response Surface, Pareto-front
National Category
Applied Mechanics Mechanical Engineering
Research subject
Mechanical Engineering
Identifiers
urn:nbn:se:oru:diva-48246 (URN)10.1115/DETC2013-12809 (DOI)000362380400029 ()2-s2.0-84896968665 (Scopus ID)978-0-7918-5589-8 (ISBN)
Conference
ASME 2013 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference (IDETC/CIE), Portland, Oregon, USA, August 4-7, 2013
Available from: 2013-06-25 Created: 2016-02-15 Last updated: 2017-10-18Bibliographically approved
Rashid, A. & Strömberg, N. (2013). Sequential simulation of thermal stresses in disc brakes for repeated braking. Proceedings of the Institution of mechanical engineers. Part J, journal of engineering tribology, 227(8), 919-929
Open this publication in new window or tab >>Sequential simulation of thermal stresses in disc brakes for repeated braking
2013 (English)In: Proceedings of the Institution of mechanical engineers. Part J, journal of engineering tribology, ISSN 1350-6501, E-ISSN 2041-305X, Vol. 227, no 8, p. 919-929Article in journal (Refereed) Published
Abstract [en]

In this paper, an efficient sequential approach for simulating thermal stresses in brake discs for repeated braking is presented. First, a frictional heat analysis is performed by using an Eulerian formulation of the disc. Then, by using the temperature history from the first step of the sequence, a plasticity analysis with temperature dependent material data is performed in order to determine the corresponding thermal stresses. Three-dimensional geometries of a disc and a pad to a heavy truck are considered in the numerical simulations. The contact forces are computed at each time step taking the thermal deformations of the disc and pad into account. In such manner, the frictional heat power distribution will also be updated in each time step, which in turn will influence the development of heat bands. The plasticity model is taken to be the von Mises yield criterion with linear kinematic hardening, where both the hardening and the yield limit are temperature dependent. The results show that during hard braking, high compressive stresses are generated on the disc surface in the circumferential direction which cause yielding. But when the disc cools down, these compressive stresses transform to tensile residual stresses. For repeated hard braking when this kind of stress history is repeated, we also show that stress cycles with high amplitudes are developed which might generate low cycle fatigue cracks after a few braking cycles.

Place, publisher, year, edition, pages
London, United Kingdom: Sage Publications, 2013
Keywords
Eulerian framework, frictional heat, thermal stresses, disc brake, repeated braking
National Category
Applied Mechanics Mechanical Engineering
Research subject
Mechanical Engineering
Identifiers
urn:nbn:se:oru:diva-48271 (URN)10.1177/1350650113481701 (DOI)000321702800014 ()2-s2.0-84884572691 (Scopus ID)
Available from: 2016-02-18 Created: 2016-02-15 Last updated: 2018-05-26Bibliographically approved
Strömberg, N. (2013). The Influence of Sliding Friction on Optimal Topologies. In: Recent Advances in Contact Mechanics: Papers Collected at the 5th Contact Mechanics International Symposium (CMIS2009). Paper presented at 5th Contact Mechanics International Symposium (CMIS2009), Chania, Greece, April 28-30, 2009 (pp. 327-336). Springer Berlin/Heidelberg, 56
Open this publication in new window or tab >>The Influence of Sliding Friction on Optimal Topologies
2013 (English)In: Recent Advances in Contact Mechanics: Papers Collected at the 5th Contact Mechanics International Symposium (CMIS2009), Springer Berlin/Heidelberg, 2013, Vol. 56, p. 327-336Conference paper, Published paper (Refereed)
Abstract [en]

In this paper the influence of sliding friction on optimal topologies is investigated and some preliminary results are presented. A design domain unilaterally constrained by a spinning support is considered. Most recently, Stromberg and Klarbring have developed methods for performing topology optimization of linear elastic structures with unilateral contact conditions. In this works sliding friction is also included in the contact model. In such manner it is possible to study how the spinning of the support will influence the optimal design. This was not possible before. The support is modeled by Signorini's contact conditions and Coulomb's law of friction. Signorini's contact conditions are regularized by a smooth approximation, which must not be confused with the well-known penalty approach. The state of the system, which is defined by the equilibrium equations and the smooth approximation, is solved by a Newton method. The design parametrization is obtained by using the SIMP-model. The minimization of compliance for a limited value of volume is considered. The optimization problem is solved by a nested approach where the equilibrium equations are linearized and sensitivities are calculated by the adjoint method. The problem is then solved by SLP, where the LP-problem is solved by an interior point method that is available in the package of Mat lab. In order to avoid mesh-dependency and patterns of checker-boards the sensitivities are filtered by Sigmund's filter. The method is implemented by using Mat lab and Visual Fortran, where the Fortran code is linked to Mat lab as mex-files. The implementation is done for a general design domain in 2D by using fully integrated isoparametric elements. The implementation seems to be very efficient and robust.

Place, publisher, year, edition, pages
Springer Berlin/Heidelberg, 2013
Series
Lecture Notes in Applied and Computational Mechanics, ISSN 1613-7736 ; 56
National Category
Mechanical Engineering Applied Mechanics
Research subject
Mechanical Engineering
Identifiers
urn:nbn:se:oru:diva-48284 (URN)10.1007/978-3-642-33968-4_20 (DOI)000315535300020 ()2-s2.0-84870737711 (Scopus ID)978-3-642-33967-7 (ISBN)
Conference
5th Contact Mechanics International Symposium (CMIS2009), Chania, Greece, April 28-30, 2009
Available from: 2007-12-17 Created: 2016-02-15 Last updated: 2017-10-17Bibliographically approved
Rashid, A. & Strömberg, N. (2013). Thermomechanical Simulation of Wear and Hot Bands in a Disc Brake by Adopting an Eulerian Approach. In: Eurobrake 2013: . Paper presented at Eurobrake, Dresden, Germany, 17-19 June, 2013.
Open this publication in new window or tab >>Thermomechanical Simulation of Wear and Hot Bands in a Disc Brake by Adopting an Eulerian Approach
2013 (English)In: Eurobrake 2013, 2013Conference paper, Published paper (Refereed)
Abstract [en]

In this paper frictional heating of a disc brake is simulated while taking wear into account. By performing thermomechanical finite element analysis, it is studied how the wear history will influence the development of hot bands. The frictional heat analysis is based on an Eulerian formulation of the disc, which requires significantly lower computational time as compared to a standard Lagrangian approach. A real disc-pad system to a heavy truck is considered, where complete three-dimensional geometries of the ventilated disc and pad are used in the simulations. A sequential approach is adopted, where the contact forces are computed at each time step taking the wear and thermal deformations of the mating parts into account. After each brake cycle, the wear profile of the pad is updated and used in subsequent analysis. The results show that when wear is considered, different distributions of the temperature on disc are obtained for each new brake cycle. After a few braking cycles two hot bands appear on the disc surface instead of only one. These results are in agreement with experimental observations.

Keywords
Eulerian framework, frictional heat, hot band, wear history, pad wear, repeated braking
National Category
Applied Mechanics Mechanical Engineering
Research subject
Mechanical Engineering
Identifiers
urn:nbn:se:oru:diva-48270 (URN)
Conference
Eurobrake, Dresden, Germany, 17-19 June, 2013
Available from: 2016-02-18 Created: 2016-02-15 Last updated: 2017-10-17Bibliographically approved
Klarbring, A. & Strömberg, N. (2013). Topology optimization of hyperelastic bodies including non-zero prescribed displacements. Structural and multidisciplinary optimization (Print), 47(1), 37-48
Open this publication in new window or tab >>Topology optimization of hyperelastic bodies including non-zero prescribed displacements
2013 (English)In: Structural and multidisciplinary optimization (Print), ISSN 1615-147X, E-ISSN 1615-1488, Vol. 47, no 1, p. 37-48Article in journal (Refereed) Published
Abstract [en]

Stiffness topology optimization is usually based on a state problem of linear elasticity, and there seems to be little discussion on what is the limit for such a small rotation-displacement assumption. We show that even for gross rotations that are in all practical aspects small (<3 deg), topology optimization based on a large deformation theory might generate different design concepts compared to what is obtained when small displacement linear elasticity is used. Furthermore, in large rotations, the choice of stiffness objective (potential energy or compliance), can be crucial for the optimal design concept. The paper considers topology optimization of hyperelastic bodies subjected simultaneously to external forces and prescribed non-zero displacements. In that respect it generalizes a recent contribution of ours to large deformations, but we note that the objectives of potential energy and compliance are no longer equivalent in the non-linear case. We use seven different hyperelastic strain energy functions and find that the numerical performance of the Kirchhoff–St.Venant model is in general significantly worse than the performance of the other six models, which are all modifications of this classical law that are equivalent in the limit of infinitesimal strains, but do not contain the well-known collapse in compression. Numerical results are presented for two different problem settings.

Place, publisher, year, edition, pages
Springer, 2013
Keywords
Hyperelasticity, Potential energy, Compliance, Optimality criteria
National Category
Applied Mechanics Mechanical Engineering
Research subject
Mechanical Engineering
Identifiers
urn:nbn:se:oru:diva-48269 (URN)10.1007/s00158-012-0819-z (DOI)000312878800004 ()2-s2.0-84871979511 (Scopus ID)
Note

Funding Agency:

Swedish Foundation for Strategic Research through the ProViking programme

Available from: 2012-01-26 Created: 2016-02-15 Last updated: 2017-11-30Bibliographically approved
Klarbring, A. & Strömberg, N. (2012). A note on the min-max formulation of stiffness optimization including non-zero prescribed displacements. Structural and multidisciplinary optimization (Print), 45(1), 147-149
Open this publication in new window or tab >>A note on the min-max formulation of stiffness optimization including non-zero prescribed displacements
2012 (English)In: Structural and multidisciplinary optimization (Print), ISSN 1615-147X, E-ISSN 1615-1488, Vol. 45, no 1, p. 147-149Article in journal (Refereed) Published
Abstract [en]

The present theoretical note shows how a naturalobjective function in stiffness optimization, including bothprescribed forces and non-zero prescribed displacements,is the equilibrium potential energy. It also shows how theresulting problem has a saddle point character that may beutilized when calculating sensitivities.

Place, publisher, year, edition, pages
Springer, 2012
Keywords
Stiffness optimization, Non-zero prescribed displacements, Saddle point problem, Sensitivity analysis
National Category
Applied Mechanics Mechanical Engineering
Research subject
Mechanical Engineering
Identifiers
urn:nbn:se:oru:diva-48268 (URN)10.1007/s00158-011-0674-3 (DOI)000298195000011 ()2-s2.0-84859606444 (Scopus ID)
Available from: 2012-01-26 Created: 2016-02-15 Last updated: 2017-11-30Bibliographically approved
Rashid, A. & Strömberg, N. (2012). An Efficient Sequential Approach for Simulation of Thermal Stresses in Disc Brakes. In: Nordtrib 2012, 15th Nordic Symposium on Tribology, 2012: . Paper presented at 15th Nordic Symposium on Tribology, NORDTRIB, Trondheim, Norway, 12-15 June, 2012.
Open this publication in new window or tab >>An Efficient Sequential Approach for Simulation of Thermal Stresses in Disc Brakes
2012 (English)In: Nordtrib 2012, 15th Nordic Symposium on Tribology, 2012, 2012Conference paper, Published paper (Other academic)
Abstract [en]

In this paper an efficient approach to simulate thermal stresses due to frictional heating of disc brakes is presented. Inthe approach thermal and stress analysis are performed sequentially. The frictional heat analysis is based on the Eulerianmethod, which requires significantly low computational time as compared to the Lagrangian approach. Completethree-dimensional geometries of a disc and a pad are considered for the numerical simulations. The contact forcesare computed at each time step taking the thermal deformations of the disc into account. The nodal temperaturehistory is recorded at each time step and is used in sequentially coupled stress analysis, where a temperature dependentelasto-plastic material model is used to compute the stresses in a disc brake. The results show that during hard braking,high compressive stresses are generated on the disc surface in circumferential direction which cause plastic yielding. Butwhen the disc cools down, the compressive stresses transform to tensile stresses. Such thermoplastic stress history maycause cracks on disc surface after a few braking cycles. These results are in agreement with experimental observationsavailable in the literature.

Keywords
Frictional heating, thermal stresses
National Category
Applied Mechanics Mechanical Engineering
Research subject
Mechanical Engineering
Identifiers
urn:nbn:se:oru:diva-48272 (URN)
Conference
15th Nordic Symposium on Tribology, NORDTRIB, Trondheim, Norway, 12-15 June, 2012
Available from: 2016-02-18 Created: 2016-02-15 Last updated: 2017-10-17Bibliographically approved
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Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0001-6821-5727

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