Radial basis functions (RBFs) are approximate mathematical models that can mimic the behaviour of rapidly changing and computationally expensive simulations, such as finite element simulations for predicting automobile crash responses. The most popular way of selecting optimal RBF shape parameters is based on minimizing the global cross-validation error (CVE). Solving this optimization problem may lead to the construction of globally accurate RBF models, but the shape parameters are assumed to be constant over the entire design space. On the other hand, having flexible shape parameters that can change over the design space may allow the local behaviour to be captured better, thereby improving the accuracy. Thus, optimizing the RBF shape parameters based on minimization of the pointwise CVE rather than the global CVE is proposed in this paper. Three benchmark mathematical functions followed by an automobile crash problem are used to evaluate the effectiveness of the proposed method. It is found that the RBF models based on the minimum pointwise CVE outperform the RBF models based on the minimum global CVE.
MullurA. A.MessacA.Extended radial basis functions: More flexible and effective metamodeling. AIAA J.200543, (6), 1306–1315
2.
HardyR. L.Multiquadric equations of topography and other irregular surfaces. J. Geophys. Res.197176, 1905–1915
3.
HardyR. L.Theory and applications of the multiquadric–biharmonic method. Comput. Math. Applic.199019, (8–9), 163–208
4.
HardyR. L.A contribution of the multiquadric method: Interpolation of potential inside the Earth: Advances in the theory and applications of radial basis functions. Comput. Math. Applic.199224, (12), 81–97
5.
AradN.DynN.ReisfeldD.YeshurunY.Image warping by radial basis functions: Applications to facial expressions. Comput. Vision, Graphics, Image Processing199456, (2), 161–172
6.
TuC. H.BartonR. R.Production yield estimation by the metamodel method with a boundary-focused experiment design. In Proceedings of the ASME Design Theory and Methodology Conference, Sacramento, California, USA 1997, paper DETC97/DTM3870 ASME International, New York
7.
ZalaC. A.BarrodaleI.Warping aerial photographs to orthomaps using thin plate splines. Adv. Comput. Math.199911, 211–227
8.
KremperA.SchanzeT.EckhornR.Classification of neural signals by a generalized correlation classifier based on radial basis functions. J. Neurosci. Meth.2002116, 179–187
9.
PapilaN.ShyyW.GriffinL.DorneyD. J.Shape optimization of supersonic turbines using global approximation methods. J. Propulsion Power200218, (3), 509–518
10.
ReddyR. R. K.GanguliR.Structural damage detection in a helicopter rotor blade using radial basis function neural networks. Smart Mater. Structs200312, 232–241
11.
LaiW.TseP. W.ZhangG.ShiT.Classification of gear faults using cumulants and the radial basis function network. Mech. Systems Signal Processing200418, (2), 381–389
12.
SonarD. K.DixitU. S.OjhaD. K.The application of a radial basis function neural network for predicting the surface roughness in a turning process. Int. J. Advd Mfg Technol.200627, 661–666
13.
ZhangT.ChoiK. K.RahmanS.ChoK.BakerP.ShakilM.HeitkampD.A hybrid surrogate and pattern search optimization method and application to microelectronics. Struct. Multidisciplinary Optimization200632, (4), 327–345
14.
YoungA.CaoC.PatelV.HovakimyanN.LavertskyE.Adaptive control design methodology for nonlinear-in-control systems in aircraft application. J. Guidance, Control Dynamics200730, (6), 1770–1782
15.
SjögrenD.Statistical methods for improving surrogate models in antenna optimization, Master's Thesis, Chalmers University of Technology, Göteborg, Sweden 2009
16.
JinR.ChenW.SimpsonT. W.Comparative studies of metamodelling techniques under multiple modelling criteria. Struct. Multidisciplinary Optimization200123, (1), 1–13
17.
ChenW.GarimellaR.MichelenaN.Robust design for improved vehicle handling under a range of maneuver conditions. Engng Optimization200133, (3), 303–326
18.
LanziL.CastellettiL. M. L.AnghileriM.Multi-objective optimisation of composite absorber shape under crashworthiness requirements. Composite Structs200465, 433–441
19.
HamzaK.SaitouK.Crashworthiness design using meta-models for approximating the response of structural members. In Proceedings of the Eighth Cairo University Conference on Mechanical design and production (MDP-8), Cairo, Egypt, 4–6 January 2004, vol. 1 591–602Cairo University, Cairo
20.
FangA.Rais-RohaniM.LiuZ.HorstemeyerM. F.A comparative study of metamodeling methods for multiobjective crashworthiness optimization. Comput. Structs200583, 2121–2136
21.
YangR. J.WangN.ThoC. H.BobineauJ. P.WangB. P.Metamodeling development for vehicle frontal impact simulation. J. Mech. Des.2005127, (11), 1014–1020
22.
Rais-RohaniM.SolankiK.EamonC.Reliability-based optimization of lightweight automotive structures for crashworthiness. In Proceedings of the 11th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Portsmouth, Virginia, USA, 6–8 September 2006, AIAA paper 2006-7004 AIAA, Reston, Virginia
23.
FrankeR.Scattered data interpolation: Tests of some methods. Math. Comput.198238, (157), 181–200
24.
KansaE. J.Multiquadrics – a scattered data approximation scheme with applications to computational fluid-dynamics – I surface approximations and partial derivative estimates. Comput. Math. Applic.199019, (8–9), 127–145
25.
KansaE. J.Multiquadrics – a scattered data approximation scheme with applications to computational fluid-dynamics – II solutions to parabolic, hyperbolic and elliptic partial differential equations. Comput. Math. Applic.199019, (8–9), 147–161
26.
FasshauerG. E.Newton iteration with multiquadrics for the solution of nonlinear PDEs. Comput. Math. Applic.200243, (3–5), 423–438
RoqueC. M. C.FerreiraA. J. M.Numerical experiments on optimal shape parameters for radial basis functions. Numer. Meth. Partial Differential Equations201026, (3), 675–689
32.
BuhmannM. D.Radial basis functions: Theory and implementations2003Cambridge University Press, Cambridge
33.
WangL.BeesonD.WiggsG.RayasamM.A comparison of meta-modeling methods using practical industry requirements. In Proceedings of the 47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Newport, Rhode Island, USA, 1–4 May 2006, AIAA paper 2006-1811 AIAA, Reston, Virginia
34.
BogomolnyA.Fundamental solutions method for elliptic boundary value problems. SIAM J. Numer. Analysis198522, 644–669
35.
ChengA. H. D.GolbergM. A.KansaE. J.ZammitoG.Exponential convergence and h–c multiquadric collocation method for partial differential equations. Numer. Meth. Partial Differential Equations200319, (5), 2003, 571–594
36.
HuangC. S.LeeC. F.ChengA. H. D.Error estimate, optimal shape factor, and high precision computation of multiquadric collocation method. Engng Analysis Boundary Elements200731, 614–623
37.
GoelT.HaftkaR. T.ShyyW.QueipoN. V.Ensemble of surrogates. Struct. Multidisciplinary Optimization200733, (3), 199–216
38.
AcarE.Rais-RohaniM.Ensemble of Metamodels with optimized weight factors. Struct. Multidisciplinary Optimization200937, (3), 279–294
39.
VianaF. A. C.HaftkaR. T.SteffenV.Multiple surrogates: How cross-validation errors can help us to obtain the best predictor. Struct. Multidisciplinary Optimization200939, (4), 439–457
40.
DixonL. C. W.SzegöG. P.Towards global optimization 21978North-Holland, Amsterdam
41.
Rais-RohaniM.SolankiK.AcarE.EamonC. D.Shape and sizing optimization of automotive structures with deterministic and probabilistic design constraints. Int. J. Veh. Des.2010, (in press)
