 Research article
 Open Access
Dimension reduction in heterogeneous parametric spaces with application to naval engineering shape design problems
 Marco Tezzele^{1},
 Filippo Salmoiraghi^{1},
 Andrea Mola^{1} and
 Gianluigi Rozza^{1}Email authorView ORCID ID profile
https://doi.org/10.1186/s4032301801183
© The Author(s) 2018
 Received: 4 May 2018
 Accepted: 1 September 2018
 Published: 10 September 2018
Abstract
We present the results of the first application in the naval architecture field of a methodology based on active subspaces properties for parameter space reduction. The physical problem considered is the one of the simulation of the hydrodynamic flow past the hull of a ship advancing in calm water. Such problem is extremely relevant at the preliminary stages of the ship design, when several flow simulations are typically carried out by the engineers to assess the dependence of the hull total resistance on the geometrical parameters of the hull, and others related with flows and hull properties. Given the high number of geometric and physical parameters which might affect the total ship drag, the main idea of this work is to employ the active subspaces properties to identify possible lower dimensional structures in the parameter space. Thus, a fully automated procedure has been implemented to produce several small shape perturbations of an original hull CAD geometry, in order to exploit the resulting shapes and to run high fidelity flow simulations with different structural and physical parameters as well, and then collect data for the active subspaces analysis. The free form deformation procedure used to morph the hull shapes, the high fidelity solver based on potential flow theory with fully nonlinear free surface treatment, and the active subspaces analysis tool employed in this work have all been developed and integrated within SISSA mathLab as open source tools. The contribution will also discuss several details of the implementation of such tools, as well as the results of their application to the selected target engineering problem.
Keywords
 Parametric studies
 Reduction in parameter space
 Free form deformation
 Active subspaces
 BEM
 Response surface method
Introduction
The content of this contribution is organized as follows. “Fully nonlinear potential model” section introduces the ship resistance prediction problem, its dependence on hull shape deformations, and equations of the fluid structure interaction model used for the simulations. In “Shape morphing based on free form deformation” section we recall the free form deformation technique and we show the main features of the developed tool to manage parametric shapes. “Implementation of high fidelity potential solver based on the boundary element method” section has the purpose of introducing some detail about the high fidelity solver implementation. In “Parameter space reduction by active subspaces” section we present the active subspaces properties and its features, with a numerical recipe to identify them. Then “Numerical results” section shows the numerical results obtained by coupling the three methods in sequence. Finally conclusions and perspectives are drawn in “Conclusions and perspectives” section.
A model naval problem: wave resistance estimation of a hull advancing in calm water
Shape morphing based on free form deformation
As already mentioned, we are interested in problems characterized by both physical and geometrical parameters. In such framework, the free form deformation (FFD) approach is adopted to implement the hull deformations corresponding to each geometrical parameter set considered.
A very detailed description of FFD is beyond the scope of the present work. In the following we will give only a brief overview. For a further insight see [40] for the original formulation and [18, 26, 36, 38, 39] for more recent works.
We decided to adopt free form deformation among other possibilities (including, for instance, radial basis functions or inverse distance weighting) because it allows to have global deformations with a few parameters. For the complexity of the problem at hand, by trying to reduce the number of parameters starting from hundreds of them can be infeasible for the number of Monte Carlo simulations required. One of the possible drawbacks of FFD is generally that the parameters do not have a specific geometric meaning, like, for instance, a prescribed length or distance. In the case of application to active subspaces (AS) this is not a problem since AS itself identifies new parameters, obtained by combination of the original ones, meaningless from the geometric and physical point of view.

Mapping the physical domain \(\varOmega \) to the reference one \({\widehat{\varOmega }}\) with the map \(\varvec{\psi }\).

Moving some control points \(\varvec{P}\) to deform the lattice with \({\widehat{T}}\). The movement of the control points is given by the weights of FFD, and represent our geometrical parameters \(\varvec{\mu }^{\text {GEOM}}\).

Mapping back to the physical domain \(\varOmega (\varvec{\mu })\) with the map \(\varvec{\psi }^{1}\).
In order to exemplify the equations above to our case, let us consider Fig. 6, where the control points we are going to move are marked with numbers. As geometrical parameters we select six components of these four control points of the FFD lattice over one side wall of the hull. Then we apply the same deformation to the other side. This because one of our hypothesis is the symmetry of the deformed hull. In particular Table 1 summarizes the set of design variables, the associated FFDnode coordinate modified (y is the span of the hull, x its length and z its depth) and the lower and upper bound of the modification. There are also two more parameters that do not affect the geometry, and are related to the physics of the problem, that is the displacement and the velocity of the hull. From now on we denote with \(\varvec{\mu } := \{ \mu _i \}_{i \in [1, \dots , 8]}\) the column vector of the parameters, where \(\mu _i\) are defined in Table 1. To denote only the parameters affecting the geometrical deformation we use \(\varvec{\mu }^{\text {GEOM}} := \{ \mu _i \}_{i \in [1, \dots , 6]}\). For sake of clarity we underline that the undeformed original domain is obtained setting all the geometrical parameters to 0. All the upper and lower bounds are chosen in order to satisfy physical constraints.
Design space (FFD lattice \(2 \times 2 \times 2\)) with eight design parameters
Parameter  Nature  Lower bound  Upper bound 

\(\mu _1\)  FFD point 1 y  \(\) 0.2  0.3 
\(\mu _2\)  FFD point 2 y  \(\) 0.2  0.3 
\(\mu _3\)  FFD point 3 y  \(\) 0.2  0.3 
\(\mu _4\)  FFD point 4 y  \(\) 0.2  0.3 
\(\mu _5\)  FFD point 3 z  \(\) 0.2  0.5 
\(\mu _6\)  FFD point 4 z  \(\) 0.2  0.5 
\(\mu _7\)  Weight (kg)  500  800 
\(\mu _8\)  Velocity (m/s)  1.87  2.70 
Implementation of high fidelity potential solver based on the boundary element method
Parameter space reduction by active subspaces
The active subspaces (AS) approach represents one of the emerging ideas for dimension reduction in the parameter studies and it is based on the homonymous properties. The concept was introduced by Constantine in [10], for example, and employed in different real world problems. We mention, among others, aerodynamic shape optimization [27], the parameter reduction for the HyShot II scramjet model [12], active subspaces for integrated hydrologic model [23], and in combination with PODGalerkin method in cardiovascular problems [44].
We underline that the size of the eigenvalue problem is the limiting factor. We need to compute eigenvalue decompositions with \(m \times m\) matrices, where m is the dimension of the simulation, that is the number of inputs.
Active subspaces can be seen in the more general context of ridge approximation (see [25, 34]). In particular it can be proved that, under certain conditions, the active subspace is nearly stationary and it is a good starting point in optimal ridge approximation as shown in [11, 22].
Numerical results
In this section we present the results of the complete pipeline, presented in the previous sections and in Fig. 1, applied to the DTMB 5415 hull.
The mesh is discretized with quadrilateral cells. The BEM uses bilinear quadrilateral elements. This results in roughly 4000 degrees of freedom for each simulation realized. The high fidelity solver described in “Implementation of high fidelity potential solver based on the boundary element method” section is implemented in WaveBEM [29] using the deal.II library [2].
Let us recall that the parameter space is a \(m = 8\) dimensional space. The parameters are showed in Table 1. We remark that the first six are geometrical parameters that produce the deformation of the original domain, while the last two are structural and physical parameters—the displacement and the velocity of the hull —. The PyGeM open source package is used to perform the free form deformation [35].
We create a dataset with 130 different couples of input/output data. We split the dataset in two, creating a train dataset with 80% of the data, and a test dataset with the remaining 20%. That means that \(N_{\text {train}}^{\text {AS}} = 104\) in Eq. (19). Even though it may be challenging to explore a 8 dimensional space, as reported in [10], heuristics suggest that this choice of \(N_{\text {train}}^{\text {AS}}\) is enough for the purposes of the active subspaces identification described in “Parameter space reduction by active subspaces” section.
In order to construct the uncentered covariance matrix \(\varvec{\varSigma }\), defined in Eq. (18), we use a Monte Carlo method as shown in Eq. (19), employing the software in [9]. Since we have only pairs of input/output data we need to approximate the gradients of the total wave resistance with respect to the parameters, that is \(\nabla _{\varvec{\mu }} f\). We use a local linear model that approximates the gradients with the best linear approximation using 14 nearest neighbors. After constructing the matrix we calculate its real eigenvalue decomposition. Recalling “Parameter space reduction by active subspaces” section, since \(m = 8\), we have that \(\varvec{\varSigma } \in {\mathbb {M}} (8, {\mathbb {R}})\).
We can compare the decay of the eigenvalues with the decay of surrogate model error on the test dataset shown in Fig. 12a. We project the data onto active subspaces of varying dimension, and construct a surrogate model with a leastsquaresfit, global, multivariate polynomial approximation of different orders. Then we calculate the rootmeansquare error of the test data against the surrogate. This error is scaled with respect to the range of the function evaluations, making it a relative error. We repeat this procedure 20 times constructing every time the uncentered covariance matrix of Eq. (19), since a Monte Carlo approximation is involved. Finally we take the average of the errors computed. Because we have a large amount of training data, we can expect the surrogate model constructed in a low dimension to be accurate if the data collapses into a manifold. Thus the test error is an indication of how well the active subspace has collapsed the data. In Fig. 12a are depicted the errors with respect to the active subspace dimension and the order of the response surface. The subspace dimension varies from 1 to 3, while the order of the response surface from 1 to 4. The minimum error is achieved using a two dimensional active variable and a response surface of degree 4 and it is \(\approx 2.5\%\). Further investigations show that increasing the dimension of the active variable does not decrease significantly the error committed while the time to construct the corresponding response surface increases. This is confirmed by the marginal gains in the decay of the eigenvalues for active variables of dimension greater then three as shown in Fig. 10a. We can affirm that the active subspace of dimension one is sufficient to model the wave resistance of the DTMB 5415 if we can afford an error of approximately 4.5%. In particular in Fig. 12b we can see the predictions made with the surrogate model of dimension one and the actual observations. Otherwise, we can achieve a \(\approx 2.5\%\) error if we take advantage of two active dimensions and a response surface of order four, preserving a fast evaluation of the surrogate model.
We want to stress the fact that the result is remarkable if we consider the heterogeneous nature of all the parameters involved. In the case of only geometrical parameters one can easily expect such a behaviour but considering also physical and structural ones make the result not straightforward at all. Moreover the evaluation of the response surface takes less than one s compared to the 12 h of a full simulation per single set of parameters on the same computing machine. This opens new potential approaches to optimization problems.
Conclusions and perspectives
In this work we presented a numerical framework for the reduction of the parameter space and the construction of an optimized response surface to calculate the total wave resistance of the DTMB 5415 advancing in calm water. We integrate heterogeneous parameters in order to have insights on the more important parameters. The reduction both in terms of cost and time, remaining below the 4.2% of error on new unprocessed data, is very remarkable and promising as a new design interpreted tool. The methodological and computational pipeline is also easily extensible to different hulls and/or different parameters. This allows a fast preprocessing and a very good starting point to minimize the quantities of interest in the field of optimal shape design.
This work is directed in the development of reduced order methods (ROMs) and efficient parametric studies. Among others we would like to cite [7, 20, 37, 39] for a comprehensive overview on ROM and geometrical deformation. Future developments involve the application of the POD after the reduction of the parameter space through the active subspaces approach.
Declarations
Author's contributions
The coauthors worked to the manuscript and the research topics and projects giving each of them an equivalent amount of contribution. MT worked on active subspace developments, AM on CFD simulations, FS on geometrical parametrisation, GR supervised and integrated the activities, carried out the biblio research and brought he computational reduction expertise. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Availability of data and materials
Open source software (pyGeM, WaveBem) available on the website mathlab.sissa.it/csesoftware.
Ethics approval and consent to participate
Not applicable.
Funding
This work was partially supported by the project OpenViewSHIP, Sviluppo di un ecosistema computazionale per la progettazione idrodinamica del sistema elicacarena and Underwater Blue Efficiency, supported by Regione FVG—PAR FSC 20072013, Fondo per lo Sviluppo e la Coesione, by the project TRIM Tecnologia e Ricerca Industriale per la Mobilit‘a Marina, CTN0100176163601, supported by MIUR, the italian Ministry of Education, University and Research, by the INDAMGNCS 2017 project Advanced numerical methods combined with computational reduction techniques for parameterised PDEs and applications and by European Union Funding for Research and Innovation Horizon 2020 Program in the framework of European Research Council Executive Agency: H2020 ERC CoG 2015 AROMACFD project 681447 Advanced Reduced Order Methods with Applications in Computational Fluid Dynamics P.I. Gianluigi Rozza.
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
 Azcueta R. Computation of turbulent freesurface flows around ships and floating bodies., Schriftenreihe Schiffbau, Bericht Nr. 612; 2001. ISBN 3892206120.Google Scholar
 Bangerth W, Davydov D, Heister T, Heltai L, Kanschat G, Kronbichler M, Maier M, Turcksin B, Wells D. The deal.II library, version 8.4. J Numer Math. 2016;24(3):135–41.MathSciNetView ArticleMATHGoogle Scholar
 Bangerth W, Hartmann R, Kanschat G. Deal.II—a general purpose object oriented finite element library. ACM Trans Math Softw. 2007;33(4):24/1–27.MathSciNetView ArticleMATHGoogle Scholar
 Beck RF. Timedomain computations for floating bodies. Appl Ocean Res. 1994;16:267–82.View ArticleGoogle Scholar
 Box GE, Draper NR. Empirical modelbuilding and response surfaces, vol. 424. New York: Wiley; 1987.MATHGoogle Scholar
 Brebbia CA. The boundary element method for engineers. London: Pentech Press; 1978.Google Scholar
 Chinesta F, Huerta A, Rozza G, Willcox K. Model order reduction. Encyclopedia of computational mechanics. 2nd ed. New York: John Wiley & Sons Ltd; 2017.Google Scholar
 Constantine P, Gleich D. Computing active subspaces with Monte Carlo. arXiv preprint; 2015. arXiv:1408.0545.
 Constantine P, Howard R, Glaws A, Grey Z, Diaz P, Fletcher L. Python activesubspaces utility library. J Open Source Softw. 2016;1(5):79. https://doi.org/10.21105/joss.00079.
 Constantine PG. Active subspaces: emerging ideas for dimension reduction in parameter studies, vol. 2. Philadelphia: SIAM; 2015.View ArticleMATHGoogle Scholar
 Constantine PG, Eftekhari A, Ward R. A nearstationary subspace for ridge approximation. arXiv preprint; 2016. arXiv:1606.01929.
 Constantine PG, Emory M, Larsson J, Iaccarino G. Exploiting active subspaces to quantify uncertainty in the numerical simulation of the HyShot II scramjet. J Comput Phys. 2015;302:1–20.MathSciNetView ArticleMATHGoogle Scholar
 Cook RD. Regression graphics: ideas for studying regressions through graphics, vol. 482. New York: John Wiley & Sons; 2009.MATHGoogle Scholar
 Dambrine J, Pierre M, Rousseaux G. A theoretical and numerical determination of optimal ship forms based on michells wave resistance. ESAIM Control Optim Calc Var. 2016;22(1):88–111.MathSciNetView ArticleMATHGoogle Scholar
 Devore JL. Probability and statistics for engineering and the sciences. Boston: Cengage Learning; 2015.Google Scholar
 Diez M, Campana EF, Stern F. Designspace dimensionality reduction in shape optimization by KarhunenLoève expansion. Comput Methods Appl Mech Eng. 2015;283:1525–44.View ArticleMATHGoogle Scholar
 Formaggia L, Mola A, Parolini N, Pischiutta M. A threedimensional model for the dynamics and hydrodynamics of rowing boats. Proc Inst Mech Eng Part P J Sports Eng Technol. 2010;224(1):51–61.Google Scholar
 Forti D, Rozza G. Efficient geometrical parametrisation techniques of interfaces for reducedorder modelling: application to fluidstructure interaction coupling problems. Int J Comput Fluid Dyn. 2014;28(3–4):158–69.MathSciNetView ArticleGoogle Scholar
 Giuliani N, Mola A, Heltai L, Formaggia L. FEM SUPG stabilisation of mixed isoparametric BEMs: application to linearised free surface flows. Eng Anal Bound Elem. 2015;8–22:59.MathSciNetMATHGoogle Scholar
 Hesthaven JS, Rozza G, Stamm B. Certified reduced basis methods for parametrized partial differential equations. SpringerBriefs in mathematics. Berlin: Springer; 2016.View ArticleMATHGoogle Scholar
 Hindmarsh AC, Brown PN, Grant KE, Lee SL, Serban R, Shumaker DE, Woodward CS. Sundials: Suite of nonlinear and differential/algebraic equation solvers. ACM Trans Math Softw. 2005;31(3):363–96.View ArticleMATHGoogle Scholar
 Hokanson JM, Constantine PG. Datadriven polynomial ridge approximation using variable projection. arXiv preprint; 2017. arXiv:1702.05859.
 Jefferson JL, Gilbert JM, Constantine PG, Maxwell RM. Reprint of: active subspaces for sensitivity analysis and dimension reduction of an integrated hydrologic model. Comput Geosci. 2016;90:78–89.View ArticleGoogle Scholar
 Kaipio J, Somersalo E. Statistical and computational inverse problems, vol. 160. Berlin: Springer Science & Business Media; 2006.MATHGoogle Scholar
 Keiper S. Analysis of generalized ridge functions in high dimensions. In: 2015 international conference on sampling theory and applications (SampTA). New York: IEEE; 2015. p. 259–63.Google Scholar
 Lombardi M, Parolini N, Quarteroni A, Rozza G. Numerical simulation of sailing boats: dynamics, FSI, and shape optimization. In: Buttazzo G, Frediani A, editors. Variational analysis and aerospace engineering: mathematical challenges for aerospace design Boston: Springer; 2012. p. 339–77.Google Scholar
 Lukaczyk TW, Constantine P, Palacios F, Alonso JJ. Active subspaces for shape optimization. In: 10th AIAA multidisciplinary design optimization conference; 2014. p. 1171.Google Scholar
 Metropolis N, Ulam S. The monte carlo method. J Am Stat Assoc. 1949;44(247):335–41.View ArticleMATHGoogle Scholar
 Mola A, Heltai L, De Simone A. Wet and dry transom stern treatment for unsteady and nonlinear potential flow model for naval hydrodynamics simulations. J Ship Res. 2017;61(1):1–14.View ArticleGoogle Scholar
 Mola A, Heltai L, DeSimone A. A stable and adaptive semilagrangian potential model for unsteady and nonlinear shipwave interactions. Eng Anal Bound Elem. 2013;128–143:37.MathSciNetMATHGoogle Scholar
 Mola A, Heltai L, DeSimone A, et al. Ship sinkage and trim predictions based on a CAD interfaced fully nonlinear potential model. In: The 26th international ocean and polar engineering conference, vol. 3, Mountain View: International Society of Offshore and Polar Engineers; 2016. p. 511–8.Google Scholar
 Morrall A. 1957 ITTC modelship correlation line values of frictional resistance coefficient. Ship report. National Physical Laboratory, Ship Division; 1970.Google Scholar
 Olivieri A, Pistani F, Avanzini A, Stern F, Penna R. Towing tank experiments of resistance, sinkage and trim, boundary layer, wake, and free surface flow around a naval combatant insean 2340 model. Technical report, DTIC Document; 2001.Google Scholar
 Pinkus A. Ridge functions, vol. 205. Cambridge: Cambridge University Press; 2015.View ArticleMATHGoogle Scholar
 PyGeM: Python geometrical morphing. https://github.com/mathLab/PyGeM.
 Rozza G, Koshakji A, Quarteroni A. Free form deformation techniques applied to 3D shape optimization problems. Commun Appl Ind Math. 2013;4:1–26.MathSciNetMATHGoogle Scholar
 Salmoiraghi F, Ballarin F, Corsi G, Mola A, Tezzele M, Rozza G. Advances in geometrical parametrization and reduced order models and methods for computational fluid dynamics problems in applied sciences and engineering: overview and perspectives. In: Papadrakakis M, Papadopoulos V, Stefanou G, Plevris V, editors. Proceedings of the VII European congress on computational methods in applied sciences and engineering, Crete, Greece, vol. 1; 510 June 2016. p. 1013–31.Google Scholar
 Salmoiraghi F, Ballarin F, Heltai L, Rozza G. Isogeometric analysisbased reduced order modelling for incompressible linear viscous flows in parametrized shapes. Adv Model Simul Eng Sci. 2016;3(1):21.View ArticleGoogle Scholar
 Salmoiraghi F, Scardigli A, Telib H, Rozza G. Freeform deformation, mesh morphing and reducedorder methods: enablers for efficient aerodynamic shape optimisation. Int J Comput Fluid Dyn. 2018. https://doi.org/10.1080/10618562.2018.1514115.
 Sederberg TW, Parry SR. Freeform deformation of solid geometric models. In: ACM SIGGRAPH computer graphics, vol. 20. New York: ACM; 1986. p. 151–60.Google Scholar
 Shoemake K. Animating rotation with quaternion curves. In: ACM computer graphics (Proc. SIGGRAPH); 1985. p. 245–54.Google Scholar
 Stern F, Longo J, Penna R, Olivieri A, Ratcliffe T, Coleman H. International collaboration on benchmark CFD validation data for surface combatant DTMB model 5415. In: Twentythird symposium on naval hydrodynamics office of naval research Bassin d’Essais des Carenes National Research Council; 2001.Google Scholar
 Tahara Y, Kobayashi H, Kandasamy M, He W, Peri D, Diez M, Campana E, Stern F. CFDbased multiobjective stochastic optimization of a waterjet propelled high speed ship. In: 29th symposium on naval hydrodynamics, Gothenburg, Sweden; 2012.Google Scholar
 Tezzele M, Ballarin F, Rozza G. Combined parameter and model reduction of cardiovascular problems by means of active subspaces and PODGalerkin methods. In: Mathematical and Numerical Modeling of the Cardiovascular System and Applications. SEMA SIMAI Springer Series 16; 2018.Google Scholar
 Tezzele M, Demo N, Gadalla M, Mola A, Rozza G. Model order reduction by means of active subspaces and dynamic mode decomposition for parametric hull shape design hydrodynamics. In: Technology and science for the ships of the future: proceedings of NAV 2018: 19th international conference on ship & maritime research. Amsterdam: IOS Press; 2018. p. 569–76. https://doi.org/10.3233/9781614998709569
 Volpi S, Diez M, Gaul NJ, Song H, Iemma U, Choi K, Campana EF, Stern F. Development and validation of a dynamic metamodel based on stochastic radial basis functions and uncertainty quantification. Struct Multidiscip Optim. 2014;51(2):347–68.View ArticleGoogle Scholar