maximin distance
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AbstractThis work proposes a framework for projection-based model order reduction (MOR) of computational models aiming at a mechanical analysis of abdominal aortic aneurysms (AAAs). The underlying full-order model (FOM) is patient-specific, stationary and nonlinear. The quantities of interest are the von Mises stress and the von Mises strain field in the AAA wall, which result from loading the structure to the level of diastolic blood pressure at a fixed, imaged geometry (prestressing stage) and subsequent loading to the level of systolic blood pressure with associated deformation of the structure (deformation stage). Prestressing is performed with the modified updated Lagrangian formulation (MULF) approach. The proposed framework aims at a reduction of the computational cost in a many-query context resulting from model uncertainties in two material and one geometric parameter. We apply projection-based MOR to the MULF prestressing stage, which has not been presented to date. Additionally, we propose a reduced-order basis construction technique combining the concept of subspace angles and greedy maximin distance sampling. To further achieve computational speedup, the reduced-order model (ROM) is equipped with the energy-conserving mesh sampling and weighting hyper reduction method. Accuracy of the ROM is numerically tested in terms of the quantities of interest within given bounds of the parameter domain and performance of the proposed ROM in the many-query context is demonstrated by comparing ROM and FOM statistics built from Monte Carlo sampling for three different patient-specific AAAs.

Metamodels provide an efficient means for the approximation of response surfaces of systems, particularly for resource-intensive experiment designs. It is oftentimes the case that interest is focused on a specific region of the parameter space. We propose an efficient recipe for the local approximation of response surfaces using Polynomial Chaos techniques. For systems embedded in high-dimensional settings, a basis-adapted spectral representation is exploited locally for dimension reduction. The proposed approach comprises an initial heuristic global solution for parameter space exploration using an approximate global Polynomial Chaos metamodel, followed by a local design being refined through an active learning scheme. The problem of turbulent flow around a symmetric airfoil is considered. Statistical estimators based on the local, active, basis-adapted approach show less bias and faster convergence as compared to the estimators from a global solution. References B. J. Bichon, M. S. Eldred, L. P. Swiler, S. Mahadevan, and J. M. McFarland. Efficient global reliability analysis for nonlinear implicit performance functions. AIAA J. 46(10):2459–2468, 2008. doi: 10.2514/1.34321. G. E. P. Box and N. R. Draper. Empirical Model-Building and Response Surfaces. Wiley, 1987. V. Dubourg, B. Sudret, and F. Deheeger. Metamodel-based importance sampling for structural reliability analysis. Prob. Eng. Mech. 33:47–57, 2013. doi: 10.1016/j.probengmech.2013.02.002. R. G. Ghanem and P. D. Spanos. Stochastic finite element: A spectral approach. Dover, 1991. doi: 10.1007/978-1-4612-3094-6. Z. G. Ghauch. Leveraging adapted polynomial chaos metamodels for real-time Bayesian updating. J. Verif. Valid. Uncert. 4(4):041003, 2020. doi: 10.1115/1.4045693. Z. G. Ghauch, V. Aitharaju, W. R. Rodgers, P. Pasupuleti, A. Dereims, and R. G. Ghanem. Integrated stochastic analysis of fiber composites manufacturing using adapted polynomial chaos expansions. Compos. Part A: Appl. Sci. 118:179–193, 2019. doi: 10.1016/j.compositesa.2018.12.029. M. E. Johnson, L. M. Moore, and D. Ylvisaker. Minimax and maximin distance designs. J. Stat. Plan. Infer. 26(2):131–148, 1990. doi: 10.1016/0378-3758(90)90122-B. A. Notin, N. Gayton, J. L. Dulong, M. Lemaire, P. Villon, and H. Jaffal. RPCM: A strategy to perform reliability analysis using polynomial chaos and resampling. Euro. J. Comput. Mech. 19(8):795–830, 2010. doi: 10.3166/ejcm.19.795-830. OpenCFD. OpenFOAM User’s Guide. 2019. https://www.openfoam.com/documentation/user-guide. V. Picheny, D. Ginsbourger, O. Roustant, R. T Haftka, and N.-H. Kim. Adaptive designs of experiments for accurate approximation of target regions. J. Mech. Design. 132(7):071008, 2010. doi: 10.1115/1.4001873. C. Thimmisetty, P. Tsilifis, and R. Ghanem. Homogeneous chaos basis adaptation for design optimization under uncertainty: Application to the oil well placement problem. AI EDAM 31(3):265–276, 2017. doi: 10.1017/S0890060417000166. R. Tipireddy and R. Ghanem. Basis adaptation in homogeneous chaos spaces. J. Comput. Phys. 259:304–317, 2014. doi: 10.1016/j.jcp.2013.12.009. P. Tsilifis and R. G. Ghanem. Reduced Wiener chaos representation of random fields via basis adaptation and projection. J. Comput. Phys. 341:102–120, 2017. doi: 10.1016/j.jcp.2017.04.009. Turbulence Modeling Resource. NASA Langley Research Center. Washington, DC, 2018. http://turbmodels.larc.nasa.gov/.

Abstract One attractive class of space-filling designs for computer experiments is that of maximin distance designs. Algorithmic search for such designs is commonly used but this method becomes ineffective for large problems. Theoretical construction of maximin distance designs is challenging; some results have been obtained recently, often by employing highly specialized techniques. This paper presents an easy-to-use method for constructing maximin distance designs. The method is versatile as it is applicable for any distance measure. Our basic idea is to construct large designs from small designs and the method is effective because the quality of large designs is guaranteed by that of small designs, as evaluated by the maximin distance criterion.