Parameter space and model order reduction for industrial optimization
Innovations in naval engineering
Marco Tezzele* and Gianluigi Rozza
*ECCOMAS Phd Award 2021, delivered in Oslo on June 6, 2022.
Surrogate based design optimization is a fundamental tool for modern industrial optimization, characterized by complex and sophisticated artifacts. In this regard, Reduced Order Methods (ROMs) [1, 2] play an important role in the creation of accurate and fast computational pipeline to evaluate surrogates. They allow to solve complex parametric problems, but high dimensional parameter spaces still represent a challenge.
Nowadays, the problem is even more emphasized due to the spread of high-performance computing facilities enabling the study of highly parametrized systems. On the other hand, industrial artifacts are also characterized by the presence of many input design parameters in order to represent a wide range of possible designs.
To handle a great amount of design parameters we developed several techniques for parameter space reduction based on active subspaces (AS) [3]. The scheme below depicts how non-intrusive model order reduction done with dynamic mode decomposition and proper orthogonal decomposition (POD) with interpolation [4] can be integrated with reduction in parameter space [5]. It also shows an enhancement of the genetic algorithm using AS, called ASGA [6, 7], which is able to accelerate the convergence for high-dimensional optimization tasks.
Parameter space reduction
Parameter space reduction [8] is a crucial aspect to fight the curse of dimensionality, which means that the complexity of the algorithms grows exponentially with the dimension of the input space.
Active subspaces method has emerged as a reliable and explainable technique for linear dimensionality reduction of input parameter space. Thanks to the decomposition of the uncentered covariance matrix of the gradients of the function of interest with respect to the input parameters, AS is able to identify a rotation of the domain and a projection of the data onto the so called active subspace. With this reduced variable we can build lower dimensional response surfaces while retaining most of the function’s variations. Despite its simplicity and interpretability, the linear restriction could be an issue for approximating complex nonlinear functions, possibly with radial symmetries. To overcome such a problem, we developed two techniques, which can be found in the open source Python package ATHENA [9].
Kernelization and localization
We present two main extensions based on kernel methods, and to localization methods, to go beyond linear reduction.
In the first approach the idea behind kernel-based AS or KAS [10], is to map the input parameters to an intermediate feature space, with higher dimension, over which we look for a linear active subspace. This idea is inspired by support vector machines and the use of kernel PCA.
The second approach, called Local AS or LAS [11] , relies on the construction of local linear models exploiting a clustering of the data. This is done using a particular distance metric induced by the global active subspace. In this way the clusters align transversally to the global AS and are able to capture local variations along the global inactive direction. Below we present an illustrative example where we consider a bidimensional function expressed by the difference of two quartic terms. On the right the classical density-based k-means clustering compared to k-medoids with the supervised distance metric described above. The four different sufficient summary plots highlight how we are able to localize the approximation error and to find optimal rotations of smaller regions of the domain. We are thus able to unveil low-dimensional structures of the function of interest. This approach is very versatile and allows to impose additional criteria for the cluster division, such as the optimal AS dimension for each cluster or a prescribed approximation accuracy.
Multi-fidelity data fusion
Multi-fidelity models are of great importance due to their capability of fusing information coming from different simulations and/or sensors. Usually, such methods rely on different computational grids or on simplified physical models in order to build a hierarchy of fidelities.
Instead, we introduce a low dimensional bias in a chain of Gaussian processes so we can fight the curse of dimensionality affecting engineering quantities of interest, especially for many-query applications. We seek a gradient-based reduction of the parameter space through linear active subspaces or a nonlinear transformation of the input space. Then we build a low-fidelity response surface based on such reduction, thus enabling multi-fidelity Gaussian process regression without the need of running new simulations. This has a great potential in the data scarcity regime affecting many engineering applications, as we have shown in the automotive field in collaboration with the Innovation Center Europe of Volkswagen AG [12]. We depicted the framework below where a low-fidelity model is built through AS starting from the same high-fidelity data.
Optimization of cuise ship hulls
The shipbuilding industry is facing a radical change towards solutions with a smaller environmental impact. This goal can be achieved in many ways, such as low emissions engines, optimized shape designs with lower wave resistance and noise generation, or by reducing the metal raw materials used during the manufacturing process, for example. This is particularly important also considering the supply chain disruptions caused by the recent pandemic.
Here, we focus on the amount of steel needed to build modern passenger ship hulls by presenting a structural optimization pipeline based on model order reduction and parameter space reduction [13]. This approach can be used in many different engineering fields, due to its modularity and non-intrusiveness. We consider the cruise ship built by Fincantieri SpA. We created different Python modules around the commercial code already used within the company.
Following the figure above, we start from the parametrized structural model, where we can modify the thickness of a prescribed set of metal plates in different regions, and we collect a database of solution snapshots. Then we exploit POD with interpolation and the multi-fidelity approach presented above to increase the precision of the modal coefficients predictions. We can thus predict in real-time the stress and buckling usage factor fields, and the quantity of interest during the Bayesian optimization phase. After an optimum is found we can validate it using the high-fidelity solver and add it to the solutions database in order to make the ROMs model accurate in that specific neighborhood, and optimization, until no new points are found.
In the figure below, instead, the parameterization and deformation of propeller blades in order to reduce the noise and decrease the impact on aquatic wildlife. This was possible thanks to the shape morphing software packages BladeX and PyGeM [14] developed within SISSA mathLab and freely available online.
ARGOS webserver and open source scientific software libraries
ARGOS is a computational web server under development to allow real time computing, thanks to an ERC PoC (Proof of Concept) to valorise the results, methodologies and software libraries developed during the ERC project AROMA-CFD, by overcoming several methodological barriers and limitations.
Within this framework, ARGOS (http://argos.sissa.it) represents the dissemination tool of numerical simulation to a vast audience of companies and institutions, through simple web interfaces by providing graphical and ready-to-use applications involving Reduced Order Modelling. This fast computational web-server is also perfectly integrable in several emerging fields and technologies like digital twins and data-driven modeling, in need of important computing technologies, able to combine computational capabilities with data assimilation and analytics, but also in fields where augmented intelligence of machines — thanks to machine learning and internet of things — is a growing need. ARGOS can be applied in other several applications, like the mechatronic field, where real time computing is emerging, from 3D printing to additive manufacturing. The use of real time web computing facilities in medical problems is even more challenging due to physiological and morphological data to be assimilated and recast in the reduced computational model, whose web interface would allow to export numerical simulation in hospitals, even on small portable devices (a dedicated section for real time computing in cardiovascular modelling ATLAS is available at http://argos.sissa.it/atlas).
http://mathlab.sissa.it/cse-software
Acknowledgements
This article also appears inside ECCOMAS Newsletter June 2022.
We acknowledge the European Commission H2020 RISE ARIA (Accurate ROMs for Industrial Applications) project, the industrial Ph.D. grant sponsored by Fincantieri S.p.A. (IRONTH Project), the project SHip OPtimization with Reduced Order Methods (SH.OP. ROMs) carried out in the context of the IRISS initiative by SMACT Competence Center, and the European Union Funding for Research and Innovation — Horizon 2020 Program — in the framework of European Research Council Executive Agency: H2020 ERC CoG 2015 AROMA-CFD project 681447 “Advanced Reduced Order Methods with Applications in Computational Fluid Dynamics” P.I. Professor Gianluigi Rozza.
References
[1] G. Rozza, M. Hess, G. Stabile, M. Tezzele, and F. Ballarin. Basic Ideas and Tools for Projection- Based Model Reduction of Parametric Partial Differential Equations. In P. Benner, S. Grivet-Talocia, A. Quarteroni, G. Rozza, W. H. A. Schilders, and L. M. Silveira, editors, Model Order Reduction, volume 2, chapter 1, pages 1–47. De Gruyter, Berlin, Boston, 2020.
[2] M. Tezzele, N. Demo, A. Mola, and G. Rozza. An integrated data-driven computational pipeline with model order reduction for industrial and applied mathematics. In M. Gunther and W. Schilders, editors, Novel Mathematics Inspired by Industrial Challenges, n. 38 in Mathematics in Industry. Springer International Publishing, 2022.
[3] P.G. Constantine. Active subspaces: Emerging ideas for dimension reduction in parameter studies. SIAM Spotlights, 2015.
[4] M. Gadalla, M. Cianferra, M. Tezzele, G. Stabile, A. Mola, and G. Rozza. On the comparison of LES data-driven reduced order approaches for hydroacoustic analysis. Computers & Fluids, 216:104819, 2021.
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[7] N. Demo, M. Tezzele, A. Mola, and G. Rozza. Hull Shape Design Optimization with Parameter Space and Model Reductions, and Self-Learning Mesh Morphing. Journal of Marine Science and Engineering, 9(2):185, 2021.
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[9] F. Romor, M. Tezzele, and G. Rozza. ATHENA: Advanced Techniques for High dimensional parameter spaces to Enhance Numerical Analysis. Software Impacts, 10:100133, 2021.
[10] F. Romor, M. Tezzele, A. Lario, and G. Rozza. Kernel-based Active Subspaces with application to CFD parametric problems using Discontinuous Galerkin method. arXiv preprint arXiv:2008.12083, 2020.
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[12] F. Romor, M. Tezzele, M. Mrosek, C. Othmer, and G. Rozza. Multi-fidelity data fusion through parameter space reduction with applications to automotive engineering. arXiv preprint arXiv:2110.14396, 2021.
[13] M. Tezzele, L. Fabris, M. Sidari, M. Sicchiero, and G. Rozza. A multi-fidelity approach coupling parameter space reduction and non-intrusive POD with application to structural optimization of passenger ship hulls. Submitted, 2022.
[14] M. Tezzele, N. Demo, A. Mola, and G. Rozza. PyGeM: Python Geometrical Morphing. Software Impacts, 7:100047, 2021.
