Lectures on FEM and applications

Venue: TIFR Center for Applicable Mathematics
Date : 20 and 21 July, 2015

Numerical Zoom for localized Multi-Scale Problems

Olivier Pironneau
Universite Pierre et Marie Curie-Paris 6
UMR 7598 Laboratoire Jacques-Louis Lions
Paris, F-75005 France

We shall present a numerical method similar to the zooming technique that every body use to see a graphic detail. The natural framework is the Schwarz domain decomposition method on non matching grid for which we shall present.

• A necessary and sufficient condition for convergence
• Alternatives similar to the Hilbert Decomposition Method and to Mortar or Arlequin approximations

The safety assessment of a nuclear waste repository underground in a clay layer is by nature a multiscale problem for which, in principle it is not possible to obtain a numerical solution without extensive computer resources. However if the solution is needed only in some small restricted region of space then the multiscale decomposition described earlier can be applied with much less computer memory and cpu time than a full solution of the problem with a resolution of all scales.

Slides 1, Slides 2

Fluid-structure interaction analyses for blood flow related to aortic aneurysms

Hiroshi Suito
Graduate School of Environmental and Life Science, Okayama University
3-1-1, Tsushima-naka, Kita-ku, Okayama, 700-8530, Japan
Email: suito@okayama-u.ac.jp
URL: http://www.ems.okayama-u.ac.jp/suito/index-e.html

In this talk, fluid-structure interaction analyses for blood flow in the thoracic aorta related to aortic aneurysms are presented. Thoracic aortic aneurysm is one of the life-threatening diseases, which slowly grows with advancing age of the patients and may be at a risk of rupture. Many papers have reported the risk of rupture increases with size of the aneurysms; however, natural history of development of aneurysms has not been fully understood. There are so many parameters characterizing the blood flow, for example, geometric, kinematic and physiologic parameters. In this study, we draw attention to geometrical characteristics of the blood vessels. Difference of the geometry of the blood vessels brings about a difference of the flow characteristics, which causes different distributions of wall shear stresses. A number of patient-specific models of the aorta as constructed from CT scans are considered in this study. We compute the flow field with the variational multiscale version of the Deforming-Spatial-Domain/Stabilized Space-Time method (DSD/SST-VMST) [1,2]. Fluid-structure interaction is handled with the Sequentially-Coupled Arterial FSI (SCAFSI) technique [3].

REFERENCES

1. T.E. Tezduyar, "Stabilized finite element formulations for incompressible flow computations", Advances in Applied Mechanics, Vol. 28, pp. 1-44 (1992).
2. K. Takizawa and T.E. Tezduyar, "Multiscale space-time fluid-structure interaction techniques", Computational Mechanics, Vol. 248, No. 3, pp. 247-267 (2011).
3. T.E. Tezduyar, K. Takizawa, C. Moorman, S. Wright and J. Christopher, "Multiscale Sequentially-Coupled Arterial FSI Technique", Computational Mechanics, Vol. 46, pp. 17-29 (2010).

Slides

Nitsche's method for Incompressible Flows

Roland Becker
Universite du Pays de l'Adour
Math. Dept., Pau, France

We propose a generalization of Nitsche' classical method for the Poisson problem with Dirichlet boundary conditions to the incompressible Euler and Navier-Stokes equations. The main idea is to achieve an automatic weighting of convective and diffusive contributions in a variational formulation, allowing to deal with all P\'eclet numbers.

Slides

Coupled simulation for atmospheric and water current models for lakes and ponds

Hiroshi Suito
Graduate School of Environmental and Life Science, Okayama University
3-1-1, Tsushima-naka, Kita-ku, Okayama, 700-8530, Japan
Email: suito@okayama-u.ac.jp
URL: http://www.ems.okayama-u.ac.jp/suito/index-e.html

In this talk, computational simulations for flows in closed water areas such as lakes, ponds and inland seas are presented. Flow simulation of water currents is a useful tool for ascertaining the flow behavior. Since the advection and diffusion of contaminants in lakes are much affected by water flows, accurate information of flow behavior is important for estimating the pollution and/or eutrophication and for decision making for better planning and controls. Various forces such as incoming flows, temperature/salinity distributions and wind effects drive flows in lakes. In shallow lakes without big inflowing rivers, flows are driven mainly by the wind. Wind blowing near the water surface gives a certain amount of shear stress to the water body, which introduces flows of the water. However, the water flow does not always occur parallel to the wind because of various factors such as the lake shapes or depth distributions. Furthermore, wind fields are not homogeneous because obstacles such as mountains, hills, buildings, and forests around the targeted lake can affect the airflow above the lake. Therefore, coupled simulations between wind fields in the air and water current fields in the lake are highly recommended.

Slides