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"ZOOM" SEMINAR
SCHOOL OF MECHANICAL ENGINEERING SEMINAR
Wednesday, March 3, 2021 at 14:00
Improved conditioning for non-conforming discretizations based on Nitsche's method
Shani Gabay
MSc Under the supervision of Prof. Isaac Harari

   Embedded methods are used to analyze complex problems involving complicated and evolving geometries, multi-phase flow interfaces, fluid–structure interaction and contact. The use of embedded methods gives rise to ill-conditioning in the presence of small "cut" elements. Thus, lead to convergence difficulties. Nitsche's formulation is a stabilized approach for enforcing surface constraints weakly and exhibits robust performance compared to other classical embedded approaches. Our goal is to improve the conditioning of Nitsche's method.
In previous work, Nitsche’s stabilization parameter was considered to be a significant source of ill-conditioning. Modifying Nitsche's formulation for interface problems by weighting averages of interface flux precludes very large values of the parameter in the presence of small domains or large contrasts in coefficients [1,2], although the ultimate effect on conditioning was not examined.  Our studies show that this approach by itself fails to alleviate severe ill conditioning.  Similar ideas for enforcing boundary conditions are not available.
Recent work [3] outlines two causes for ill-conditioning resulting from small cut elements: basis functions being small and higher-order shape functions becoming nearly linearly dependent. We claim these are in addition to the cause of small integration domain and high values of the stabilization parameter, as already mentioned. Our numerical studies show the degradation to be confined to the relatively small sub-matrices associated with the cut elements. Based on this finding, we propose an algebric elimination of the associated dof's, by the technique known as "static condensation", to overcome the ill-conditioning of the system. This simple and low-cost procedure has been tested and proved to be effective for all the examples of boundary and interface conditions presented in this work.
[1] Annavarapu, Chandrasekhar, Martin Hautefeuille, and John E. Dolbow. "A robust Nitsche’s formulation for interface problems." Computer Methods in Applied Mechanics and Engineering 225 (2012): 44-54.
[2] Jiang, Wen, Chandrasekhar Annavarapu, John E. Dolbow, and Isaac Harari. "A robust Nitsche's formulation for interface problems with spline‐based finite elements." International Journal for Numerical Methods in Engineering 104, no. 7 (2015): 676-696.
[3] de Prenter, Frits, et al. "Condition number analysis and preconditioning of the finite cell method." Computer Methods in Applied Mechanics and Engineering 316 (2017): 297-327.
Zoom Meeting
https://zoom.us/j/96584758181?pwd=WC9PMXdsYzJ3NFdEN2Q5ZUtOZEVjdz09

~~"ZOOM" SEMINAR
SCHOOL OF MECHANICAL ENGINEERING SEMINAR
Monday, December 14, 2020 at 14:00

Potential and non-potential effects in wave breaking

Dr. Anatoliy Khait
Centre for Mathematical Modelling and Flow Analysis
Manchester Metropolitan University, UK
a.khait@mmu.ac.uk

Wave breaking is sufficiently two-phase transient phenomenon occurring frequently in open seas and coastal surf zones. It is of significant importance in marine engineering due to its capability to produce extreme loadings that can severely damage or completely destroy coastal defense, offshore structure and marine vessel. Recent work1 suggests importance of wave breaking in formation of rogue waves. Wave breaking also plays an important role in the atmosphere-ocean system by facilitating and enhancing the physical, chemical and biological interactions across the air-sea interface.
Complexity of wave breaking phenomenon imposes significant restrictions on applicability of analytical analysis. Turbulent energy release and air-water mixing in the ocean boundary layer suggests that high fidelity numerical simulations and laboratory experiments are the only approaches producing reliable results2-6. In this seminar, attention will be paid to breaking wave trains that demonstrate spatiotemporal evolution significantly different from theoretical predictions based on potential approximations. We will discuss recently reported effect of nonlinear phase locking7-8 induced by the wave breaking. It will be shown that breaking-generated non-stationary sheared currents9 can be responsible for modification of waves’ dispersive properties leading to suppression of the wave train defocusing. Practical importance of the findings will be discussed, which includes the impact of wave breaking on rogue waves formation, as well as required corrections to ocean waves models.

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REFERENCES
1. M.L. McAllister, S. Draycott, T.A.A. Adcock, P.H. Taylor, T.S. van den Bremer, J. Fluid Mech., 860, (2019).
2. Z. Tian, M. Perlin, and W.J. Choi, J. Fluid Mech., 655, (2010).
3. Z. Tian, M. Perlin, and W.J. Choi, Phys. Fl., 24, (2012).
4. B.R. Seiffert, G. Ducrozet, Ocean Dynamics, 68, (2018).
5. C.C. Craciunescu, M. Christou, Appl. Ocean Res., 99, (2020).
6. A. Khait, L. Shemer, Phys. Fl., 30, (2018).
7. M. Derakhti, J.T. Kirby, J. Fluid Mech., 790, (2016).
8. A. Khait, Z. Ma, Submitted to J. Fluid Mech.
9. A. Khait, Z. Ma, L. Qian, W. Bai, Z. Lin, ISOPE 2020-TPC-0369.

About the presenter
Dr. Anatoliy Khait is a postdoctoral research associate at the Department of Computing and Mathematics at Manchester Metropolitan University. He earned his academic degrees from Ural Federal University (Russia). He attended intensive Master course at Genoa University (Italy) and Silesian University of Technology (Poland) funded by EU Tempus project. He has a substantial industrial experience working in a private design company as an engineer for hydraulic and pneumatic systems. Being a postdoctoral fellow under supervision of Prof. Lev Shemer at Tel Aviv University, Dr. Khait started research on nonlinear ocean waves dynamics combining theoretical, numerical and experimental approaches. His current research is focused on the problems of ocean aeration due to the wave breaking. He also continues research projects on various fluid dynamic and thermodynamic problems.