 Research Article
 Open Access
A hybrid RNNGPOD surrogate model for realtime settlement predictions in mechanised tunnelling
 BaTrung Cao^{1}Email authorView ORCID ID profile,
 Steffen Freitag^{1, 2} and
 Günther Meschke^{1}
https://doi.org/10.1186/s4032301600579
© Cao et al. 2016
 Received: 14 October 2015
 Accepted: 28 January 2016
 Published: 5 March 2016
Abstract
Realistic 3D simulations of the tunnelling process are increasingly required to investigate the interactions between machinedriven tunnel construction and the surrounding soil in order to provide reliable estimates of the expected settlements and associated risks of damage for existing structures, in particular in urban tunnelling projects. To accomplish the step from largescale computational analysis to realtime predictions of expected settlements during tunnel construction, the focus of this paper is laid on the generation of a numerically efficient hybrid surrogate modelling strategy, combining Gappy proper orthogonal decomposition (GPOD) and recurrent neural networks (RNN). In this hybrid RNNGPOD surrogate model, the RNN is employed to extrapolate the time variant settlements at several monitoring points within an investigated surface area and GPOD is utilised to predict the whole field of surface settlements based on the RNN predictions and a POD radial basis functions approximation. Both parts of the surrogate model are created based on results of finite element simulations from geotechnical and process parameters varied within the range of intervals given in the design stage of a tunnel project. In the construction stage, the hybrid surrogate model is applied for realtime reliability analyses of the mechanised tunnelling process to support the machine operator in steering the tunnel boring machine.
Keywords
 Proper orthogonal decomposition
 Recurrent neural network
 Mechanised tunnelling
 Surrogate model
 Realtime application
 Computational steering
Background
Mechanised tunnelling is a widely used construction method for underground infrastructure in particular in urban areas due to its effectiveness in controlling the advancement process and to limit the construction induced ground deformations. This construction method is characterised by a staged procedure of soil excavation at the tunnel face and lining erection, providing at the same time a continuous support of the soil by means of supporting fluids at the tunnel face and pressurised grouting of the tail gap. The interactions between the tunnel boring machine (TBM), the support measures and the soil, including the groundwater, are the determining factors for the efficiency, the safety of the tunnel advancement and the risk of damage on the existing infrastructure. Evidently, these interactions are complex and require computational simulation methods based upon sufficiently 3D realistic numerical models for the individual components and their interactions. However, for a sufficient degree of spatial resolution of this highly nonlinear, time dependent problem, adequate simulation models are characterized by a large number of unknowns in the range of \(10^6\)–\(10^8\) unknowns.
Up to date, numerical simulations in tunnelling are restricted to the design stage of a project. The support of the steering of TBMs during construction mainly relies upon the interpretation of monitoring data during the construction by experienced experts. If results from large scale numerical models are to be used during construction to provide additional information on the potential consequences of decisions taken for the steering of the TBM (e.g., the surface settlement field ahead of the tunnel face), a realtime system response is required. However, performing large scale, computationally expensive models on site is unrealistic for realtime applications, which demand obtaining the system response in the range of seconds to minutes. To accomplish the step from largescale computational analysis to realtime predictions of expected settlements during tunnel construction, model reduction strategies are required to substitute the original numerical model by surrogate models. Recently, the authors have proposed a hybrid surrogate model [1], which employs a combination of two different techniques: recurrent neural network (RNN) and proper orthogonal decomposition (POD), aiming to exploit the advantages of both methods. While RNNs are well suited for predictions using extrapolation of data (see [2]), the POD is able to deal with high dimensional outputs, see, e.g., [3, 4].
For the specific problem at hand, two different variants of the POD method have been employed in [1]. The first one is the POD with interpolation to approximate surface settlements from previous stages of the tunnel excavation (excavation steps 1 to n). Frequently, radial basis functions (RBF) are considered as interpolation functions, resulting in the PODRBF approach, see e.g., [3–5]. The second approach is the Gappy POD (GPOD) approach, first proposed in [6] and employed to tackle the missing data problem in image processing, or to reconstruct human faces from partial data. In [1], this approach has been utilised to approximate the field of settlements from data available from a limited number of monitoring points in step \(n+1\).
To allow for a monitoring and model based support of TBM steering, supplying information in realtime to the TBM driver on the expected (i.e., future) settlements, a model reduction technique which allows extrapolation in time with multiple outputs is proposed in this paper. To this end, the hybrid surrogate modelling strategy originally proposed in [1] is extended. More precisely, strategies to improve the predictions of hybrid surrogate models by means of problem specific enhancements for their components, PODRBF and GPOD, are proposed.
For the POD using interpolation functions, the concept of RBF is replaced by an extension called Extended RBF (ERBF) [7]. This concept is capable of improving the prediction capability of the surrogate model by combining the effectiveness of the RBF and the flexibility of nonRBF approaches. This leads to a PODERBF network which has been shown to produce better prediction results as compared to the PODRBF, see [8]. In association with the PODERBF algorithm, the reconstruction accuracy of the GPOD approach can be enhanced by adopting iterative schemes (IGPOD) to derive the POD basis, e.g., [9–12]. In the GPOD approach, an incomplete data vector is reconstructed based on the assumption that the vector inherits the characteristics of the known data set. The reconstruction procedure is performed in one step based on the POD modes of the data set, e.g., [6, 9]. A more precise approach is to include this partial missing vector into the known data set and to iterate until convergence. Everson and Sirovich [6] have performed an iterative approach based on an initial guess, which is denoted as the E–S method. The E–S method depends on the initial guess and therefore may lead to less accurate solutions because it only employs a certain number of POD modes in the reconstruction procedure. Based on the E–S method, Venturi and Karniadakis [11] proposed another extension, referred to as the V–K method, which is not dependent on the initial guess and improves the accuracy significantly by gradually increasing the number of POD modes. However, using the V–K method leads to a significant increase in computation time. Therefore, in order to replace classical GPOD, the IGPOD, based upon both the E–S and the V–K scheme is investigated in this paper in regards to prediction performance and computation time.
The remainder of this paper is organised as follows. “Numerical model for mechanised tunnelling” gives a brief description of the finite element model for processoriented numerical simulations of TBM driven tunnelling. “Hybrid surrogate model” first summarises the algorithmic scheme suggested for the hybrid surrogate model and subsequently provides a synopsis of methods employed in the hybrid surrogate model: RNN, POD, PODRBF, PODERBF, GPOD and IGPOD (E–S and V–K methods). “Application to mechanised tunnelling” is devoted to applications of the proposed surrogate model in mechanised tunnelling. Based upon the developed surrogate model, a software for realtime prognosis in TBM tunnelling is finally proposed in “Realtime simulation software for TBM steering support” to enable supporting decisions during mechanised tunnel construction.
Numerical model for mechanised tunnelling
The soil is modelled as a three (or two) phase material for partially (or fully) saturated soils. Two elastoplastic models are available for the consideration of the inelastic response of soft soils: the clay and sand model and the DruckerPrager model (Fig. 1(1)). The TBM is represented as a deformable body (Fig. 1(2)) moving through the soil with frictional surfacetosurface contact along the shield skin, allowing that the deformation of the soil naturally follows the real, tapered geometry of the TBM and that the effect of overcutting is captured. In order to realistically model the movement of the TBM and its interaction with the soil, an algorithm to control the individual jack thrusts is used to keep the TBM on the designed alignment path. After each TBM advance, the excavation at the cutting face, the tail void grouting and the erection of a new lining ring during standstill are taken into account by rezoning the finite element mesh and adjusting all boundary conditions to the new situation. The tail gap grouting (see Fig. 1(4)) is modelled as a fully saturated twophase material with a hydrating matrix phase, considering the temporal evolution of stiffness and permeability of the cementitious grout. In the simulation model ekate, buildings are represented by substitute models with appropriate stiffness and thickness to consider the soilstructure interaction (see Fig. 1(5)). In the presented FE formulation, isotropic shell, or, alternatively, volume elements with respective properties are adopted interacting with the soil through a mesh independent surfacetosurface contact algorithm, which prevents the penetration of the foundation of the building into the soil. For more details on the numerical model ekate and its ability to predict tunnelling induced settlements in the context of different applications we refer to [16–18].
For realtime applications during tunnel construction, the FE model must be substituted by a fast surrogate model which is able to approximate the physical behaviour of all relevant components involved in the simulation model for mechanised tunnelling. The surrogate model is developed to predict time variant settlement fields, which is realised by a mapping of time constant and time variant low dimensional inputs onto time variant high dimensional outputs. For this mapping, a hybrid surrogate modelling strategy is proposed in the next section.
Hybrid surrogate model
RNNGPOD approach to predict the complete field of surface displacements
Offline stage  
1. Generate a numerical model representing the tunnelling process within a selected section of the project from time step 1 to n  
2. Define investigated input parameters and the corresponding range of values  
3. Run numerical simulations with different input parameters  
4. Store the numerical results of displacements of the complete surface area of the analysis section  
5. Provide data of selected monitoring points for the training of a RNN  
6. Provide data of the complete displacements field for training of the PODRBF  
Online stage  
1. Input: an arbitrary set of input parameters  
2. Approximate the complete displacement field from time step 1 to n (by PODRBF)  
3. Predict the displacements of selected monitoring points for the next time step n+1 (by RNN)  
4. Predict the displacements field for the next time step n+1 (by GPOD)  
5. Update the complete displacements field from time step 1 to n+1  
6. Repeat steps 3–5 
It should be noted, that also monitoring data can be used to update the surrogate model, see [19]. However, the focus of this paper is on improving the prediction quality of the PODRBF and the GPOD components of the hybrid surrogate model. The individual components of the hybrid RNNGPOD surrogate model are explained below, highlighting the proposed extensions and improvements.
Recurrent neural network
Proper orthogonal decomposition
POD is a mathematical procedure that allows to perform a decomposition of a large set of data to describe the original system with a much smaller number of unknowns by projecting them onto subspaces. The method was proposed under different names for applications in various fields including principal component analysis (PCA) [22] in statistics, singular value decomposition (SVD) [23] in linear algebra and Karhunen–Loeve decomposition (KLD) [24, 25] in signal processing. Nowadays the method is applied extensively in various branches of computational mechanics, such as fluid dynamics [6, 26], aerodynamics [9] and nonlinear solid mechanics [27, 28], see also [29] for a comprehensive review of POD applications.
POD with interpolation
Nonradial basis functions [7]
Region  Range of \({\varvec{\xi }}^{{\varvec{i}}}_{{\varvec{j}}}\)  \({\varvec{\phi }}^{{\varvec{L}}}\)  \({\varvec{\phi }}^{{\varvec{R}}}\)  \({\varvec{\phi }}^{{\varvec{\beta }}}\) 

I  \(\xi ^i_j \le \gamma \)  \((n\gamma ^{n1})\xi ^i_j\) + \(\gamma ^n(1n)\)  0  \(\xi ^i_j\) 
II  \(\gamma \le \xi ^i_j\le 0\)  \((\xi ^i_j)^n\)  0  \(\xi ^i_j\) 
III  \(0\le \xi ^i_j\le \gamma \)  0  \((\xi ^i_j)^n\)  \(\xi ^i_j\) 
IV  \(\xi ^i_j \ge \gamma \)  0  \((n\gamma ^{n1})\xi ^i_j\) + \(\gamma ^n(1n)\)  \(\xi ^i_j\) 
POD with missing data
Iterative GPOD In the method described above, the reconstruction procedure is accomplished in one step with the assumption that the “repaired” vector can be characterised with the already known snapshots set. This method can be extended to the case, where the missing vector itself is included into the snapshots sets. This requires an iterative procedure to construct the POD basis. Two algorithms, the E–S and the V–K method proposed in the literature, are briefly summarised below.
 (a)
Use timeaverage values as initial guess at missing locations to obtain \(\widetilde{\mathbf{U }}^{*}\).
 (b)
Include \(\widetilde{\mathbf{U }}^{*}\) into snapshot U to form \(\widetilde{\mathbf{U }}\).
 (c)
Perform POD on \(\widetilde{\mathbf{U }}\) to obtain \(\varvec{\Phi }\).
 (d)
Choose the number of modes K to be employed in the reconstruction.
 (e)
Compute \(\overline{\mathbf{A }}^{*}\) from Eq. (32) with \(\mathbf M = ({\overline{\varvec{\Phi }}}^\mathrm{T} , \overline{\varvec{\Phi }})\) and \(\mathbf R = ({\overline{\varvec{\Phi }}}^\mathrm{T} , \mathbf U ^{*})\).
 (f)
Compute \(\widetilde{\mathbf{U }}\) from Eq. (31) and overwrite the previous guess at missing locations.
 (e)
Proceed until convergence, the eigenvalues no longer change. If no convergence is obtained, go to (b).
 (a)
Perform the standard E–S procedure, but employ only \(K = 2\) modes in the reconstruction.
 (b)
Take the converged result from the previous step as a new initial guess.
 (c)
Perform the E–S procedure, but employ now \(K = 3\) modes in the reconstruction.
 (d)
Proceed until convergence (the eigenvalues no longer change).
Application to mechanised tunnelling
In the present demonstration example, it is assumed that the current state of the TBM advance corresponds to the 22nd step of the excavation process and that the history of the settlements of the surface monitoring points are known. The proposed hybrid surrogate modelling approach is employed to predict the complete surface displacement field in the subsequent excavation step (step 23) from input parameter selected within a specific range. In this application example, 11 monitoring points are selected for predicting the future tunnelling induced settlements by the RNN approach. The number of monitoring points is selected to ensure that the RNN is capable to provide good predictions and to have an appropriate accuracy of the GPOD. In particular, the reconstruction quality of the GPOD would be better, if there are more available data points. However, the training and prediction of the RNN might be more complicated. In addition, the position of 11 points are chosen based on the usual position of measurement sensors on the surface in a real tunnel project. Subsequently, the complete surface displacement field will be approximated with the GPOD method.

In order to set up the surrogate model, 60 numerical simulations are performed to obtain data in the offline stage. Each simulation corresponds to a combination of two varying parameters: the elastic modulus \(E_{1}\) of soil layer 1 and the grouting pressure \(^{[n]}GP\).

The range of variation of these parameters are 20–110 MPa for \(E_{1}\) and 130–230 kPa for \(^{[n]}GP\), respectively. \(E_{1}\) can take 1 out of 10 particular values in the range from 20 to 110 MPa, whereas for \(^{[n]}GP\) one of the six scenarios of time varying pressures is taken, see Fig. 8.

The numerical results of 60 simulations are split randomly into two separate data sets. The first set containing results from 54 simulations is used to establish the surrogate model. In contrast, the results from six cases are stored for validation.

In addition, only the displacements of the complete surface field from time step 1–22 are kept in the first set for training and generating the surrogate model. Meanwhile, the validation set contains surface settlements from time step 1–23. The results from time step 23 are used to validate the prediction capability of the proposed surrogate modelling strategy. The performances of the PODRBF and the PODERBF networks are compared with numerical results from time step 1–22.

An arbitrary set of input parameters within the investigated ranges is selected: a value of \(E_{1}\) in the range from 20 to 110 MPa and values of \(^{[n]}GP\) from time step 1–22 in the range of 130–230 kPa.

Approximate the complete displacement field from time step 1–22 by PODRBF based on the data of 54 numerical simulations.

With an arbitrary value of \(^{[23]}GP\) in the range of 130–230 kPa, the RNN predicts the settlements at 11 selected monitoring points of time step 23.

The complete displacement field of time step 23 is predicted using GPOD.
Prediction performances of PODRBF and PODERBF for all validation cases
Error (%)  PODRBF  PODERBF 

Validation case 1  1.50  0.87 
Validation case 2  2.51  2.46 
Validation case 3  1.80  1.38 
Validation case 4  2.04  1.52 
Validation case 5  2.06  1.57 
Validation case 6  1.57  0.97 
Average  1.91  1.46 
Prediction performances of GPOD and IGPOD for all validation cases
Error (%)  PODRBF  PODERBF  

GPOD  E–S  V–K  GPOD  E–S  V–K  
Validation case 1  5.18  5.03  1.36  5.20  5.06  1.18 
Validation case 2  7.82  7.80  1.18  7.65  7.63  1.01 
Validation case 3  4.81  4.70  1.38  4.55  4.42  0.95 
Validation case 4  4.33  4.22  2.13  4.49  4.33  1.15 
Validation case 5  7.45  7.24  2.15  6.90  6.75  1.09 
Validation case 6  5.40  5.36  1.81  5.60  5.55  2.31 
Average  5.83  5.72  1.67  5.73  5.62  1.28 
Computation time of different surrogate models (in average)
PODRBF  PODERBF  

GPOD  E–S  V–K  GPOD  E–S  V–K  
Computation time [s]  0.01  0.03  0.06  0.61  0.73  0.92 
Realtime simulation software for TBM steering support
Based on the enhanced IGPODERBF algorithm described in ‘Hybrid surrogate model’, a realtime simulation software is developed with the aim to support the steering of the tunnel boring machine during mechanised tunnelling. The goal of this software is to predict the system response, i.e., the surface settlements (and/or tunnel lining forces etc.), resulting from the TBMsoil interaction in realtime as the response to changes of operational parameters, such as the face pressure or the grouting pressure.
The example described in this section is similar to the application example in the previous section. However, two varying operational parameters in mechanised tunnelling are considered here. They are the support pressure \(^{[n]}SP\) and the grouting pressure \(^{[n]}GP\) which constitute the main operational parameters to control the settlements caused by the tunnel drive. The software consists of three main modules: “Overview” module, “Monitoring” module and “Prediction” module. For the prediction of time variant surface settlement fields for the next excavation step, the user has to follow the sequence of all three modules.
The “Monitoring” module is used to store and visualise the locations of all monitoring points and the evolution of the settlements in the already constructed tunnel section, see Fig. 12. The settlement history of these monitoring points will be extrapolated with predicted values from RNN for the next time step corresponding to the chosen support pressure and grouting pressure. In practice, the predicted values from RNN are updated with real measurements at monitoring points. Nevertheless, in this example this step is skipped since there are only synthetic data from numerical simulations.
The tests presented in this paper are executed on a standard computer with IntelChip 2\(\,\times \,\)1.70 GHz, 2\(\,\times \,\)4 GB RAM. The software can run in almost every standard laptop or even tablet since it does not require any expensive hardware.
Conclusion
In this paper, a hybrid surrogate modelling strategy based upon the combination of a RNN and the GPOD has been proposed to enable realtime prognoses during mechanised tunnelling. The proposed approach is an extension of preliminary work by the authors, characterised by supplementing the previous surrogate model with Extended RBFs and an iterative Gappy POD. It was proven, that the suggested enhancements improve the prediction capability of the surrogate model. Although more complex formulations and an iterative concept are involved, the new hybrid surrogate model still provides predictions within a reasonable time in the context of applications in mechanised tunnelling, targeted to predict tunnelling induced settlements in realtime. In particular, the surrogate model combining the PODERBF and the V–K method yields the best prediction results in the presented application. The new approach has been integrated into a software which provides, in realtime, the expected tunnelling induced settlements for varying operational parameters chosen by the user and thus enables to support the steering of tunnel boring machines. The developed software provides a new option for tunnel engineers on the construction site to select the appropriate parameters such, that tolerated settlements will not be exceeded during the upcoming excavation steps.
The proposed hybrid surrogate model will be further extended to consider uncertain geotechnical parameters. First steps include interval and fuzzy data within the presented RNNGPOD approach to allow predicting time variant interval settlement fields by means of interval arithmetic operations computed in realtime. In contrast to more time consuming optimisation approaches for interval analysis with the hybrid RNNGPOD surrogate model, see [1], the problem of overestimation has to be considered within interval arithmetic. In addition, a strategy is currently developed to continuously reduce the predicted interval uncertainty by updating the settlement field with monitoring data.
Declarations
Authors’ contributions
All authors, BTC, SF and GM, participated equally in the manuscript writing. All authors read and approved the final manuscript.
Acknowlegements
Financial support was provided by the German Research Foundation (DFG) in the framework of project C1 of the Collaborative Research Center SFB 837 “Interaction Modelling in Mechanised Tunnelling”. This support is gratefully acknowledged.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Freitag S, Cao BT, Ninić J, Meschke G. Hybrid surrogate modelling for mechanised tunnelling simulations with uncertain data. Int J Reliab Saf. 2015;9(2/3):154–73.View ArticleGoogle Scholar
 Freitag S, Graf W, Kaliske M. Recurrent neural networks for fuzzy data. Integr Comput Aided Eng. 2011;18(3):265–80.Google Scholar
 Ostrowski Z, Bialecki RA, Kassab AJ. Solving inverse heat conduction problems using trained podrbf network inverse method. Inverse Probl Sci Eng. 2008;16(1):39–54.View ArticleMATHGoogle Scholar
 Buljak V, Maier G. Proper orthogonal decomposition and radial basis functions in material characterization based on instrumented indentation. Eng Struct. 2011;33:492–501.View ArticleGoogle Scholar
 Ostrowski Z, Bialecki RA, Kassab AJ. Estimation of constant thermal conductivity by use of proper orthogonal decomposition. Comput Mech. 2005;37:52–9.View ArticleMATHGoogle Scholar
 Everson R, Sivorich L. Karhunen–Loeve procedure for gappy data. J Opt Soc Am A Opt Image Sci Vis. 1995;12(8):1657–64.View ArticleGoogle Scholar
 Mullur AA, Messac A. Extended radial basis functions: more flexible and effective metamodeling. AIAA J. 2005;43(6):1306–15.View ArticleGoogle Scholar
 Khaledi K, Miro S, König M, Schanz T. Robust and reliable metamodels for mechanized tunnel simulations. Comput Geotech. 2014;61:1–12.View ArticleGoogle Scholar
 BuiThanh T, Damodaran M, Willcox K. Aerodynamic data reconstruction and inverse design using proper orthogonal decomposition. AIAA J. 2004;42:1505–16.View ArticleGoogle Scholar
 Willcox K. Unsteady flow sensing and estimation via the gappy proper orthogonal decomposition. Comput Fluids. 2006;35(2):208–26.View ArticleMATHGoogle Scholar
 Venturi D, Karniadakis GE. Gappy data and reconstruction procedures for flow past a cylinder. J Fluids Mech. 2004;519:315–36.MathSciNetView ArticleMATHGoogle Scholar
 Gunes H, Sirisup S, Karniadakis GE. Gappy data: to krig or not to krig? J Comput Phys. 2006;212:358–82.View ArticleMATHGoogle Scholar
 Nagel F, Stascheit J, Meschke G. Processoriented numerical simulation of shield tunneling in soft soils. Geomech Tunn. 2010;3(3):268–82.View ArticleGoogle Scholar
 Kasper T, Meschke G. A 3D finite element model for TBM tunneling in soft ground. Int J Numer Analyt Methods Geomech. 2004;28:1441–60.View ArticleMATHGoogle Scholar
 Stascheit J, Nagel F, Meschke G, Stavropoulou M, Exadaktylos G. An automatic modeller for finite element simulations of shield tunnelling. In: Eberhardsteiner J, Beer G, Hellmich C, Mang HA, Meschke G, Schubert W, editors. Computational modelling in tunnelling (EURO:TUN 2007). Vienna; 2007.Google Scholar
 Meschke G, Ninic J, Stascheit J, Alsahly A. Parallelized computational modeling of pilesoil interactions in mechanized tunneling. Eng Struct. 2013;47:35–44.View ArticleGoogle Scholar
 Cao BT, Chmelina K, Stascheit J, Meschke G. Enhanced monitoring and simulation assisted tunnelling (emsat). In: Meschke G, Eberhardsteiner J, Soga K, Schanz T, Thewes M, editors. Computational methods in tunnelling and subsurface engineering (EURO:TUN 2013). 2013. p. 41–50.Google Scholar
 Ninić J, Koch C, Freitag S, Meschke G, König M. Simulation and monitoringbased steering for mechanized tunneling using project data of WehrhahnLinie. In: Proceedings of the 11th World Congress on Computational Mechanics (WCCM 2014), Barcelona; 2014 (online available).Google Scholar
 Freitag S, Cao BT, Ninić J, Meschke G. Hybrid rnngpod surrogate model for simulation and monitoring supported tbm steering. In: Tsompanakis Y, Kruis J, Topping BHV, editors. Proceedings of the 4th International conference on soft computing technology in civil, structural and environmental engineering (CSC2015). Prague: CivilComp Press; 2015. paper 2. doi:10.4203/ccp.109.2.
 Adeli H. Neural networks in civil engineering: 1989–2000. Comput Aided Civ Infrastruct Eng. 2001;16:126–42.View ArticleGoogle Scholar
 Freitag S. Artificial neural networks in structural mechanics. In: Tsompanakis Y, Kruis J, Topping BHV, editors. Computational technology reviews, vol. 12. Stirlingshire: SaxeCoburg Publications; 2015. p. 1–26. doi:10.4203/ctr.12.1.
 Hotelling H. Analysis of a complex system of statistical variables into principal components. J Educ Psychol. 1933;24:417–441498520.View ArticleMATHGoogle Scholar
 Golub GH, Reinsch C. Singular value decomposition and least squares solutions. Numer Math. 1970;14(5):403–20.MathSciNetView ArticleMATHGoogle Scholar
 Karhunen K. Zur spektraltheorie stochastischer prozesse. Ann Acad Sci Fennicae 37. 1946.Google Scholar
 Loeve M. Probability theory II. 4th ed. Graduate texts in mathematics, vol. 46. New York: Springer; 1978.Google Scholar
 Holmes P, Lumley JL, Berkooz G. Turbulence, coherent structures. Dynamical systems and symmetry. New York: Cambridge University Press; 1996.Google Scholar
 Radermacher A, Reese S. Model reduction in elastoplasticity: proper orthogonal decomposition combined with adaptive substructuring. Comput Mech. 2014;54(3):677–87.MathSciNetView ArticleMATHGoogle Scholar
 Galland F, Gravouil A, Malvesin E, Rochette M. A global model reduction approach for 3d fatigue crack growth with confined plasticity. Comput Method Appl Mech Eng. 2011;200(5–8):699–716.MathSciNetView ArticleMATHGoogle Scholar
 Mifsud M. Reducedorder modelling for highspped aerial weapon aerodynamics. Ph.D. Thesis. Cranfield UniversityCollege of Aeronautics; 2008.Google Scholar
 Sirovich L. Turbulence and the dynamics of coherent structures: i, ii and iii. Q Appl Math. 1987;45:561–71.MathSciNetMATHGoogle Scholar
 Hardy RL. Multiquadric equations of topography and other irregular surfaces. J Geophys Res. 1971;76:1905–15.View ArticleGoogle Scholar
 Jin R, Chen W, Simpson T. Comparative studies of metamodelling techniques under multiple modelling criteria. Struct Multidiscip Optim. 2001;23(1):1–13.View ArticleGoogle Scholar
 Simpson TW, Peplinski JD, Koch PN, Allen JK. Metamodels for computerbased engineering design: survey and recommendations. Eng Comput. 2001;17:129–50.View ArticleMATHGoogle Scholar
 Hussain MF, Barton RR, Joshi SB. Metamodeling: radial basis functions, versus polynomials. Eur J Oper Res. 2002;138:142–54.View ArticleMATHGoogle Scholar
 Mullur AA, Messac A. Metamodeling using extended radial basis functions: a comparative approach. Eng Comput. 2006;21:203–17.View ArticleGoogle Scholar