Metamodel based high-fidelity stochastic analysis of composite laminates: A concise review with critical comparative assessment

Abstract This paper presents a concise state-of-the-art review along with an exhaustive comparative investigation on surrogate models for critical comparative assessment of uncertainty in natural frequencies of composite plates on the basis of computational efficiency and accuracy. Both individual and combined variations of input parameters have been considered to account for the effect of low and high dimensional input parameter spaces in the surrogate based uncertainty quantification algorithms including the rate of convergence. Probabilistic characterization of the first three stochastic natural frequencies is carried out by using a finite element model that includes the effects of transverse shear deformation based on Mindlin’s theory in conjunction with a layer-wise random variable approach. The results obtained by different metamodels have been compared with the results of traditional Monte Carlo simulation (MCS) method for high fidelity uncertainty quantification. The crucial issue regarding influence of sampling techniques on the performance of metamodel based uncertainty quantification has been addressed as an integral part of this article.

[1]  G. Wahba.  Smoothing noisy data with spline functions , 1975 .

[2]  Art B. Owen,et al.  9 Computer experiments , 1996, Design and analysis of experiments.

[3]  C. N. Madu,et al.  A fuzzy theoretic approach to simulation metamodeling , 1995 .

[4]  Xiaoping Du,et al.  The use of metamodeling techniques for optimization under uncertainty , 2001 .

[5]  Rekha S. Singhal,et al.  Comparison of artificial neural network (ANN) and response surface methodology (RSM) in fermentation media optimization: Case study of fermentative production of scleroglucan , 2008 .

[6]  Mohamad S. Qatu,et al.  Vibration studies for laminated composite twisted cantilever plates , 1991 .

[7]  T. Simpson,et al.  Analysis of support vector regression for approximation of complex engineering analyses , 2005, DAC 2003.

[8]  N. Cutland,et al.  On homogeneous chaos , 1991, Mathematical Proceedings of the Cambridge Philosophical Society.

[9]  Abbas S. Milani,et al.  Layout optimization of a multi-zoned, multi-layered composite wing under free vibration , 2009, Smart Structures and Materials + Nondestructive Evaluation and Health Monitoring.

[10]  Runze Li,et al.  Penalized Likelihood Kriging Model for Analysis of Computer Experiments , 2003, DAC 2003.

[11]  N. Cressie.  Spatial prediction and ordinary kriging , 1988 .

[12]  Kenny Q. Ye,et al.  Algorithmic construction of optimal symmetric Latin hypercube designs , 2000 .

[13]  Noel A Cressie,et al.  Statistics for Spatial Data, Revised Edition. , 1994 .

[14]  Richard O. Mason,et al.  Applying ethics to information technology issues , 1995, CACM.

[15]  E. Viola,et al.  Radial basis functions based on differential quadrature method for the free vibration analysis of laminated composite arbitrarily shaped plates , 2015 .

[16]  Sondipon Adhikari,et al.  A polynomial chaos expansion based molecular dynamics study for probabilistic strength analysis of nano-twinned copper , 2016 .

[17]  Jack P. C. Kleijnen.  Design and Analysis of Monte Carlo Experiments , 2004 .

[18]  Ya-Jung Lee,et al.  Stacking sequence optimization of laminated composite structures using genetic algorithm with local improvement , 2004 .

[19]  Tanmoy Mukhopadhyay,et al.  Optimisation of Fibre-Reinforced Polymer Web Core Bridge Deck—A Hybrid Approach , 2015 .

[20]  Jack P. C. Kleijnen,et al.  Design and Analysis of Monte Carlo Experiments , 2004 .

[21]  Damiano Pasini,et al.  Optimization of variable stiffness composites with embedded defects induced by Automated Fiber Placement , 2014 .

[23]  T. J. Mitchell,et al.  Bayesian Prediction of Deterministic Functions, with Applications to the Design and Analysis of Computer Experiments , 1991 .

[24]  Jaroslaw Sobieszczanski-Sobieski,et al.  Multidisciplinary aerospace design optimization - Survey of recent developments , 1996 .

[25]  Russell R. Barton,et al.  A review on design, modeling and applications of computer experiments , 2006 .

[26]  B. Manohar,et al.  An artificial neural network analysis of porcine pancreas lipase catalysed esterification of anthranilic acid with methanol , 2005 .

[27]  T. Simpson,et al.  Fuzzy clustering based hierarchical metamodeling for design space reduction and optimization , 2004 .

[28]  Sondipon Adhikari,et al.  A Response Surface Modelling Approach for Resonance Driven Reliability Based Optimization of Composite Shells , 2016 .

[29]  Chiara Bisagni,et al.  Metamodeling Methodology for Postbuckling Simulation of Damaged Composite Stiffened Structures with Physical Validation , 2010 .

[30]  T. Simpson,et al.  Comparative studies of metamodeling techniques under multiple modeling criteria , 2000 .

[31]  Helena Szczerbicka,et al.  Analysis of Simulation Models with Fuzzy Graph Based Metamodeling , 1996, Perform. Evaluation.

[32]  R. L. Hardy.  Multiquadric equations of topography and other irregular surfaces , 1971 .

[33]  Susmita Naskar,et al.  On quantifying the effect of noise in surrogate based stochastic free vibration analysis of laminated composite shallow shells , 2016 .

[34]  John Rasmussen,et al.  Nonlinear programming by cumulative approximation refinement , 1998 .

[35]  Sondipon Adhikari,et al.  Thermal uncertainty quantification in frequency responses of laminated composite plates , 2015 .

[36]  T. Simpson,et al.  Computationally Inexpensive Metamodel Assessment Strategies , 2002 .

[37]  M. E. Johnson,et al.  Minimax and maximin distance designs , 1990 .

[38]  H. Rabitz,et al.  High Dimensional Model Representations , 2001 .

[39]  S. W. Wang,et al.  Fully equivalent operational models for atmospheric chemical kinetics within global chemistry-transport models , 1999 .

[40]  Sondipon Adhikari,et al.  Rotational and ply-level uncertainty in response of composite shallow conical shells , 2015 .

[41]  A. Mellit,et al.  EPNN-based prediction of meteorological data for renewable energy systems , 2023, Journal of Renewable Energies.

[42]  L. Lessard,et al.  Surrogate-based multi-objective optimization of a composite laminate with curvilinear fibers , 2012 .

[43]  Karin Schwab,et al.  Best Approximation In Inner Product Spaces , 2016 .

[44]  Noel A Cressie,et al.  Statistics for Spatial Data. , 1992 .

[45]  Wei Chen,et al.  A robust concept exploration method for configuring complex systems , 1995 .

[46]  P. Villon,et al.  Moving least squares response surface approximation: Formulation and metal forming applications , 2005 .

[47]  Ramana V. Grandhi,et al.  MULTIVARIATE HERMITE APPROXIMATION FOR DESIGN OPTIMIZATION , 1996 .

[48]  Susmita Naskar,et al.  Stochastic natural frequency analysis of damaged thin-walled laminated composite beams with uncertainty in micromechanical properties , 2017 .

[49]  Raphael T. Haftka,et al.  Two-level composite wing structural optimization using response surfaces , 2000 .

[50]  Jennifer Werfel,et al.  Orthogonal Arrays Theory And Applications , 2016 .

[51]  Timothy W. Simpson,et al.  Analysis of support vector regression for approximation of complex engineering analyses , 2003, DAC 2003.

[52]  Sondipon Adhikari,et al.  Stochastic free vibration analysis of angle-ply composite plates – A RS-HDMR approach , 2015 .

[53]  Herschel Rabitz,et al.  General formulation of HDMR component functions with independent and correlated variables , 2011, Journal of Mathematical Chemistry.

[54]  Isaac M Daniel,et al.  Yield and failure criteria for composite materials under static and dynamic loading , 2016 .

[55]  Richard J. Beckman,et al.  A Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output From a Computer Code , 2000, Technometrics.

[56]  E. Viola,et al.  General higher-order shear deformation theories for the free vibration analysis of completely doubly-curved laminated shells and panels , 2013 .

[57]  Jiri Hlinka,et al.  Dynamic tests of composite panels of an aircraft wing , 2015 .

[58]  Nuri Ersoy,et al.  A Review on the Mechanical Modeling of Composite Manufacturing Processes , 2016, Archives of Computational Methods in Engineering.

[59]  Carolyn Conner Seepersad,et al.  Building Surrogate Models Based on Detailed and Approximate Simulations , 2004, DAC 2004.

[60]  Bernard Grossman,et al.  Noisy Aerodynamic Response and Smooth Approximations in HSCT Design , 1994 .

[61]  Timothy M. Mauery,et al.  COMPARISON OF RESPONSE SURFACE AND KRIGING MODELS FOR MULTIDISCIPLINARY DESIGN OPTIMIZATION , 1998 .

[62]  T. J. Mitchell,et al.  Bayesian design and analysis of computer experiments: Use of derivatives in surface prediction , 1993 .

[63]  Yao Lin,et al.  An Efficient Robust Concept Exploration Method and Sequential Exploratory Experimental Design , 2004 .

[64]  P. Lancaster,et al.  Surfaces generated by moving least squares methods , 1981 .

[65]  Urmila M. Diwekar,et al.  An efficient sampling technique for off-line quality control , 1997 .

[66]  R. Grandhi,et al.  A global structural optimization technique using an interval method , 2001 .

[67]  Wei Chen,et al.  An Efficient Algorithm for Constructing Optimal Design of Computer Experiments , 2005, DAC 2003.

[68]  Ronald L. Iman,et al.  Risk methodology for geologic disposal of radioactive waste: small sample sensitivity analysis techniques for computer models, with an application to risk assessment , 1980 .

[69]  Francesco Tornabene,et al.  Free vibrations of laminated composite doubly-curved shells and panels of revolution via the GDQ method , 2011 .

[70]  S. Rippa,et al.  Numerical Procedures for Surface Fitting of Scattered Data by Radial Functions , 1986 .

[71]  Susmita Naskar,et al.  Uncertain natural frequency analysis of composite plates including effect of noise – A polynomial neural network approach , 2016 .

[72]  Noel A Cressie,et al.  Spatial prediction and ordinary kriging , 1988 .

[73]  Anupam Chakrabarti,et al.  Structural damage identification: A random sampling-high dimensional model representation approach , 2016 .

[74]  H. Saunders,et al.  Finite element procedures in engineering analysis , 1982 .

[75]  Wei Shyy,et al.  Response surface techniques for diffuser shape optimization , 1997 .

[76]  Andrew D. Back,et al.  Radial Basis Functions , 2001 .

[77]  Herschel Rabitz,et al.  Global uncertainty assessments by high dimensional model representations (HDMR) , 2002 .

[78]  G. G. Wang,et al.  Adaptive Response Surface Method Using Inherited Latin Hypercube Design Points , 2003 .

[79]  A. J. Booker,et al.  A rigorous framework for optimization of expensive functions by surrogates , 1998 .

[80]  Sondipon Adhikari,et al.  Probabilistic characterisation for dynamics and stability of laminated soft core sandwich plates , 2017 .

[81]  Jack P. C. Kleijnen,et al.  Kriging for interpolation in random simulation , 2003, J. Oper. Res. Soc..

[82]  Anupam Chakrabarti,et al.  Structural Damage Identification Using Response Surface-Based Multi-objective Optimization: A Comparative Study , 2015, Arabian Journal for Science and Engineering.

[83]  Sondipon Adhikari,et al.  A Critical Assessment of Kriging Model Variants for High-Fidelity Uncertainty Quantification in Dynamics of composite Shells , 2016, Archives of Computational Methods in Engineering.

[84]  A. Ferreira,et al.  MLSDQ based on RBFs for the free vibrations of laminated composite doubly-curved shells , 2016 .

[85]  Luc Pronzato,et al.  Design of computer experiments: space filling and beyond , 2011, Statistics and Computing.

[86]  Manolis Papadrakakis,et al.  Structural optimization using evolution strategies and neural networks , 1998 .

[87]  Lih-Yuan Deng,et al.  Orthogonal Arrays: Theory and Applications , 1999, Technometrics.

[88]  Anupam Chakrabarti,et al.  Optimum design of FRP bridge deck: an efficient RS-HDMR based approach , 2015 .

[89]  Alfredo Liverani,et al.  FGM and laminated doubly curved shells and panels of revolution with a free-form meridian: A 2-D GDQ solution for free vibrations , 2011 .

[90]  Erik K. Antonsson,et al.  FITTING FUNCTIONS TO DATA IN HIGH DIMENSIONAL DESIGN SPACE , 1999 .

[91]  Boxin Tang.  Orthogonal Array-Based Latin Hypercubes , 1993 .

[92]  Mike Rees,et al.  5. Statistics for Spatial Data , 1993 .

[93]  Dimitri N. Mavris,et al.  AN INVESTIGATION OF METAMODELING TECHNIQUES FOR COMPLEX SYSTEMS DESIGN , 2002 .

[94]  Luca Lanzi,et al.  Post-buckling optimization of composite stiffened panels: Computations and experiments , 2006 .

[95]  Yunqian Ma,et al.  Practical selection of SVM parameters and noise estimation for SVM regression , 2004, Neural Networks.

[96]  C. D. Boor,et al.  On multivariate polynomial interpolation , 1990 .

[97]  Soo-Chang Kang,et al.  An efficient response surface method using moving least squares approximation for structural reliability analysis , 2010 .

[98]  J. H. Starnes,et al.  Construction of Response Surface Approximations for Design Optimization , 1998 .

[99]  Sudip Dey,et al.  Natural frequencies of delaminated composite rotating conical shells-A finite element approach , 2012 .

[100]  Nicholas Fantuzzi,et al.  Radial basis function method applied to doubly-curved laminated composite shells and panels with a General Higher-order Equivalent Single Layer formulation , 2013 .

[101]  S. Sriramula,et al.  Quantification of uncertainty modelling in stochastic analysis of FRP composites , 2009 .

[102]  Vladimir Vapnik,et al.  Statistical learning theory , 1998 .

[103]  Michael S. Eldred,et al.  OVERVIEW OF MODERN DESIGN OF EXPERIMENTS METHODS FOR COMPUTATIONAL SIMULATIONS , 2003 .

[104]  Yong Zhang,et al.  Uniform Design: Theory and Application , 2000, Technometrics.

[105]  Byeongdo Kim,et al.  Comparison study on the accuracy of metamodeling technique for non-convex functions , 2009 .

[106]  Sonja Kuhnt,et al.  Design and analysis of computer experiments , 2010 .

[107]  Ruichen Jin,et al.  On Sequential Sampling for Global Metamodeling in Engineering Design , 2002, DAC 2002.

[108]  Sondipon Adhikari,et al.  Probabilistic Analysis and Design of HCP Nanowires: An Efficient Surrogate Based Molecular Dynamics Simulation Approach , 2016 .

[109]  T. Simpson,et al.  Use of Kriging Models to Approximate Deterministic Computer Models , 2005 .

[110]  F.-X. Irisarri,et al.  Computational strategy for multiobjective optimization of composite stiffened panels , 2011 .

[111]  J. Freidman,et al.  Multivariate adaptive regression splines , 1991 .

[112]  Farrokh Mistree,et al.  Kriging Models for Global Approximation in Simulation-Based Multidisciplinary Design Optimization , 2001 .

[113]  Lee Margetts,et al.  Practical Application of the Stochastic Finite Element Method , 2016 .

[114]  Chiara Bisagni,et al.  Post-buckling optimisation of composite stiffened panels using neural networks , 2002 .

[115]  Murray Smith,et al.  Neural Networks for Statistical Modeling , 1993 .

[116]  T. Simpson,et al.  Fuzzy Clustering Based Hierarchical Metamodeling For Space Reduction and Design Optimization , 2004 .

[117]  Timothy W. Simpson,et al.  On the Use of Statistics in Design and the Implications for Deterministic Computer Experiments , 1997 .

[118]  Pierre Goovaerts,et al.  Adaptive Experimental Design Applied to Ergonomics Testing Procedure , 2002, DAC 2002.

[119]  Monika Richter,et al.  Finite Element Analysis Of Composite Laminates , 2016 .

[120]  Sung-Kwun Oh,et al.  Polynomial neural networks architecture: analysis and design , 2003, Comput. Electr. Eng..

[121]  A. Ghaffari,et al.  Performance comparison of neural network training algorithms in modeling of bimodal drug delivery. , 2006, International journal of pharmaceutics.

[122]  Robert Tibshirani,et al.  The Entire Regularization Path for the Support Vector Machine , 2004, J. Mach. Learn. Res..

[123]  Tanmoy Mukhopadhyay,et al.  A multivariate adaptive regression splines based damage identification methodology for web core composite bridges including the effect of noise , 2018 .

[124]  Leonard Meirovitch,et al.  Dynamics And Control Of Structures , 1990 .

[125]  Sondipon Adhikari,et al.  Stochastic natural frequency of composite conical shells , 2015 .

[126]  Peter Craven,et al.  Smoothing noisy data with spline functions , 1978 .

[127]  T. Simpson,et al.  Comparative studies of metamodelling techniques under multiple modelling criteria , 2001 .

[128]  Nicholas Fantuzzi,et al.  A new approach for treating concentrated loads in doubly-curved composite deep shells with variable radii of curvature , 2015 .

[129]  Thiagarajan Krishnamurthy,et al.  Response Surface Approximation with Augmented and Compactly Supported Radial Basis Functions , 2003 .

[130]  Douglas C. Montgomery,et al.  Response Surface Methodology: Process and Product Optimization Using Designed Experiments , 1995 .

[131]  Tarun Kant,et al.  Finite element analysis of laminated composite plates using a higher-order displacement model , 1988 .

[132]  Damiano Pasini,et al.  Defect layer method to capture effect of gaps and overlaps in variable stiffness laminates made by Automated Fiber Placement , 2013 .

[133]  F. Deutsch.  Best approximation in inner product spaces , 2001 .

[134]  Anupam Chakrabarti,et al.  Efficient lightweight design of FRP bridge deck , 2015 .

[135]  Sondipon Adhikari,et al.  Bottom up surrogate based approach for stochastic frequency response analysis of laminated composite plates , 2016 .

[136]  J. -F. M. Barthelemy,et al.  Approximation concepts for optimum structural design — a review , 1993 .

[137]  J. N. Reddy,et al.  Winkler–Pasternak foundation effect on the static and dynamic analyses of laminated doubly-curved and degenerate shells and panels , 2014 .

[138]  H. Fang,et al.  Global response approximation with radial basis functions , 2006 .

[139]  Ya-Jung Lee,et al.  Regression of the response surface of laminated composite structures , 2003 .

[140]  S. Adhikari,et al.  Stochastic free vibration analyses of composite shallow doubly curved shells – A Kriging model approach , 2015 .

[141]  John E. Renaud,et al.  Adaptive experimental design for construction of response surface approximations , 2001 .

[142]  Zafer Gürdal,et al.  Optimization of a composite cylinder under bending by tailoring stiffness properties in circumferential direction , 2010 .

[143]  Sondipon Adhikari,et al.  Uncertainty Quantification in Natural Frequency of Composite Plates - An Artificial Neural Network Based Approach: , 2016 .

[144]  Janis Auzins,et al.  Surrogate modeling in design optimization of stiffened composite shells , 2006 .

[145]  Jack P. C. Kleijnen,et al.  Kriging interpolation in simulation: a survey , 2004, Proceedings of the 2004 Winter Simulation Conference, 2004..

[146]  Min Xie,et al.  A systematic comparison of metamodeling techniques for simulation optimization in Decision Support Systems , 2010, Appl. Soft Comput..

[147]  Vishnu Pareek,et al.  Artificial neural network modeling of a multiphase photodegradation system , 2002 .

[148]  J. Kleijnen.  Statistical tools for simulation practitioners , 1986 .

[149]  Sudip Dey,et al.  Finite element analysis of bending-stiff composite conical shells with multiple delamination , 2012 .

[150]  Sondipon Adhikari,et al.  Effect of cutout on stochastic natural frequency of composite curved panels , 2016 .

[151]  H. Rabitz,et al.  Random sampling-high dimensional model representation (RS-HDMR) and orthogonality of its different order component functions. , 2006, The journal of physical chemistry. A.

[152]  Achille Messac,et al.  Extended Radial Basis Functions: More Flexible and Effective Metamodeling , 2004 .

[153]  T. W. Layne,et al.  A Comparison of Approximation Modeling Techniques: Polynomial Versus Interpolating Models , 1998 .

[154]  Herschel Rabitz,et al.  Observable-preserving control of quantum dynamics over a family of related systems , 2005 .

[155]  Sondipon Adhikari,et al.  Fuzzy uncertainty propagation in composites using Gram–Schmidt polynomial chaos expansion , 2016 .

[156]  Sudip Dey,et al.  Free vibration analyses of multiple delaminated angle-ply composite conical shells – A finite element approach , 2012 .

[157]  Jerome Sacks,et al.  Designs for Computer Experiments , 1989 .

[158]  M B Kasiri,et al.  Modeling and optimization of heterogeneous photo-Fenton process with response surface methodology and artificial neural networks. , 2008, Environmental science & technology.

[159]  Agus Sudjianto,et al.  Computer Aided Reliability and Robustness Assessment , 1998 .

[160]  Genichi Taguchi,et al.  Taguchi methods : design of experiments , 1993 .

[161]  Herbert A. Simon,et al.  Applications of machine learning and rule induction , 1995, CACM.

[162]  Kyung K. Choi,et al.  A new response surface methodology for reliability-based design optimization , 2004 .

[163]  Tarun Kant,et al.  Estimation of transverse/interlaminar stresses in laminated composites – a selective review and survey of current developments , 2000 .

[164]  Jeong‐Soo Park.  Optimal Latin-hypercube designs for computer experiments , 1994 .

[165]  Wei Chen,et al.  ROBUST CONCEPT EXPLORATION OF PROPULSION SYSTEMS WITH ENHANCED MODEL APPROXIMATION CAPABILITIES , 2000 .

[166]  Raphael T. Haftka,et al.  Making the Most Out of Surrogate Models: Tricks of the Trade , 2010, DAC 2010.

[167]  Hongzhe Dai,et al.  A multiwavelet support vector regression method for efficient reliability assessment , 2015, Reliab. Eng. Syst. Saf..

[168]  Richard H. Crawford,et al.  Selecting an Appropriate Metamodel: The Case for NURBs Metamodels , 2005, DAC 2005.

[169]  A. Lichtenstern,et al.  Kriging methods in spatial statistics , 2013 .