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Abstract
Parametric Sensitivity Analysis (PSA) [1] can support the robustness and efficiency of shape optimisation as it the enables the designers to identify the subset of parameters which are most sensitive to the design’s physics variability. Once obtained, insensitive parameters can be excluded or fixed to reduce the dimensionality of the design space, thereby facilitating efficient exploration of the design space. For complex engineering problems, however, involving free shapes, such as ship hulls, PSA implementation often suffers from highcomputational cost especially when the design’s physics is computationally costly to evaluate.
In this work, we aim to tackle the above challenges by introducing and testing a computationally inexpensive PSA approach, in which, instead of design's physical properties geometric properties, such as moments, are used for evaluating parametric sensitivity. As physical properties rely strongly on design’s geometry and the evaluation of geometric properties is computationally inexpensive compared to the physical ones, we propose a geometrydriven approach to estimate parameters’ sensitivity. The selection of moments in our work is done based on two fundamental insights:geometric moments of a shape (i) are intrinsic properties of its underlying geometry, providing critical cues to aid its CAD [2] and (ii) provide a unifying medium between geometry of a design and its physical evaluation [3].
From a geometric point of view, these moments can provide the volume enclosed by the shape, its centre of mass,moments of inertia as well as higherorder moments. They also provide a geometric foundation for many physical analyses of design, such as structural analysis [4], meshless physical analysis [3], governing equations of motion [5], fluid simulations [6], hydrodynamic and hydrostatic stability [7], etc. More interestingly these moments also have been widely utilized to measure the extrinsic geometric similarity, which is utilized for object recognition [8] and shape retrieval [9] tasks.
To achieve the aforementioned objectives, we integrate moments with the Active Subspace Method (ASM) [10], which commences with the eigendecomposition of the original design space and partitions it into active and inactive subspaces. Afterwards, the active subspace is exploited to measure parametric sensitivities. The feasibility of our method is tested using as parent hull the US Navy Combatant DTMB 5415 ( http://www.simman2008.dk/5415/combatant.html) parameterised with 27 parameters – yielding a 27dimensional design space. ASM is then implemented using moments up to the 2nd order. The sensitivity results so obtained are then compared with ASM applied with the wave resistance coefficient (c_w) used as physical quantity. The results obtained from this study show a good correlation between the two approaches, which indicates that PSA performed with moments can ease the designer for efficient design parametrisation and can provide a good estimation of parameters' sensitivity on design physics.
REFERENCES
[1] S. Andrea, M. Ratto, T. Andres, F. Campolongo, J. Cariboni, D. Gatelli, M. Saisana, and S. Tarantola, Global sensitivity analysis: the primer, John Wiley & Sons, 2008.
[2] A. Krishnamurthy and S. McMains, “Accurate GPUaccelerated surface integrals for moment computation”, ComputerAided Design, Vol. 43, no. 10, pp. 12841295, (2011).
[3] A. Taber, G. Kumar, M. Freytag, and V. Shapiro, “A momentvector approach to interoperable analysis”, ComputerAided Design, Vol. 102, pp. 139147, (2018).
[4] B. W. Kim, S. Y. Hong, J. H. Kyoung, and S. K. Cho, “Evaluation of bending moments and shear forces at unit connections of very large floating structures using hydroelastic and rigid body analyses”, Ocean engineering, Vol. 34, no. 1112, pp. 16681679, (2007).
[5] Newman, John Nicholas. Marine hydrodynamics. The MIT press, 2018.
[6] P. Jin, B. Xie, and F. Xiao, “Multimoment finite volume method for incompressible flows on unstructured moving grids and its application to fluidrigid body interactions”, Computers & Structures, Vol. 221, pp. 91110, (2019).
[7] A. Biran, and R. L. Pulido, Ship hydrostatics and stability, ButterworthHeinemann, 2013.
[8] D.F. Atrevi, D. Vivet, F. Duculty, and B. Emile, “A very simple framework for 3D human poses estimation using a single 2D image: Comparison of geometric moments descriptors”, Pattern Recognition, Vol. 71, pp. 389401, (2017).
[9] L. Luciano, and A. B. Hamza, “A global geometric framework for 3D shape retrieval using deep learning”, Computers & Graphics, Vol. 79, pp. 1423, (2019).
[10] P.G. Constantine, Active subspaces: Emerging ideas for dimension reduction in parameter studies, Society for Industrial and Applied Mathematics, 2015.
In this work, we aim to tackle the above challenges by introducing and testing a computationally inexpensive PSA approach, in which, instead of design's physical properties geometric properties, such as moments, are used for evaluating parametric sensitivity. As physical properties rely strongly on design’s geometry and the evaluation of geometric properties is computationally inexpensive compared to the physical ones, we propose a geometrydriven approach to estimate parameters’ sensitivity. The selection of moments in our work is done based on two fundamental insights:geometric moments of a shape (i) are intrinsic properties of its underlying geometry, providing critical cues to aid its CAD [2] and (ii) provide a unifying medium between geometry of a design and its physical evaluation [3].
From a geometric point of view, these moments can provide the volume enclosed by the shape, its centre of mass,moments of inertia as well as higherorder moments. They also provide a geometric foundation for many physical analyses of design, such as structural analysis [4], meshless physical analysis [3], governing equations of motion [5], fluid simulations [6], hydrodynamic and hydrostatic stability [7], etc. More interestingly these moments also have been widely utilized to measure the extrinsic geometric similarity, which is utilized for object recognition [8] and shape retrieval [9] tasks.
To achieve the aforementioned objectives, we integrate moments with the Active Subspace Method (ASM) [10], which commences with the eigendecomposition of the original design space and partitions it into active and inactive subspaces. Afterwards, the active subspace is exploited to measure parametric sensitivities. The feasibility of our method is tested using as parent hull the US Navy Combatant DTMB 5415 ( http://www.simman2008.dk/5415/combatant.html) parameterised with 27 parameters – yielding a 27dimensional design space. ASM is then implemented using moments up to the 2nd order. The sensitivity results so obtained are then compared with ASM applied with the wave resistance coefficient (c_w) used as physical quantity. The results obtained from this study show a good correlation between the two approaches, which indicates that PSA performed with moments can ease the designer for efficient design parametrisation and can provide a good estimation of parameters' sensitivity on design physics.
REFERENCES
[1] S. Andrea, M. Ratto, T. Andres, F. Campolongo, J. Cariboni, D. Gatelli, M. Saisana, and S. Tarantola, Global sensitivity analysis: the primer, John Wiley & Sons, 2008.
[2] A. Krishnamurthy and S. McMains, “Accurate GPUaccelerated surface integrals for moment computation”, ComputerAided Design, Vol. 43, no. 10, pp. 12841295, (2011).
[3] A. Taber, G. Kumar, M. Freytag, and V. Shapiro, “A momentvector approach to interoperable analysis”, ComputerAided Design, Vol. 102, pp. 139147, (2018).
[4] B. W. Kim, S. Y. Hong, J. H. Kyoung, and S. K. Cho, “Evaluation of bending moments and shear forces at unit connections of very large floating structures using hydroelastic and rigid body analyses”, Ocean engineering, Vol. 34, no. 1112, pp. 16681679, (2007).
[5] Newman, John Nicholas. Marine hydrodynamics. The MIT press, 2018.
[6] P. Jin, B. Xie, and F. Xiao, “Multimoment finite volume method for incompressible flows on unstructured moving grids and its application to fluidrigid body interactions”, Computers & Structures, Vol. 221, pp. 91110, (2019).
[7] A. Biran, and R. L. Pulido, Ship hydrostatics and stability, ButterworthHeinemann, 2013.
[8] D.F. Atrevi, D. Vivet, F. Duculty, and B. Emile, “A very simple framework for 3D human poses estimation using a single 2D image: Comparison of geometric moments descriptors”, Pattern Recognition, Vol. 71, pp. 389401, (2017).
[9] L. Luciano, and A. B. Hamza, “A global geometric framework for 3D shape retrieval using deep learning”, Computers & Graphics, Vol. 79, pp. 1423, (2019).
[10] P.G. Constantine, Active subspaces: Emerging ideas for dimension reduction in parameter studies, Society for Industrial and Applied Mathematics, 2015.
Original language  English 

Title of host publication  IX Conference on Computational Methods in Marine Engineering 
Subtitle of host publication  Marine 2021 
Place of Publication  [Edinburgh] 
Number of pages  18 
Publication status  Published  3 Jun 2021 
Keywords
 parametric sensitivity
 freeform marine shapes
 marine engineering
 hydrodynamics
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Learning, Processing and Optimising Shapes (MCSAITNETN) (GRAPES)
European Commission  Horizon 2020
1/12/19 → 30/11/23
Project: Research