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Grants





Advanced flowfield analysis (GA1809628S)
from 01/01/2018
to 31/12/2020 investigator

Objectives:
The research deals with an advanced flowfield analysis, particularly for transitional and turbulent flows. Local vortex identification and more general classification of flow regions based on the velocity gradient are investigated. The velocity gradient is usually decomposed in strainrate tensor and vorticity tensor, consequently the identification and classification criteria are determined by the inner velocitygradient configuration. The impact of configuration is usually significant though hidden, and will be, including representative data sets, analyzed and described. Largescale 3D numerical experiments based on the solution of the NavierStokes equations (NSE) will be performed with the help of new effective methods (parallel domain decomposition (DD) with adaptive mesh refinement) while these methods will be further developed. Some suitable qualitative properties of the solutions to the NSE will be studied and described in detail, focusing on the regularity criteria with only one or two velocity components and one or several velocitygradient entries.



Advanced methods for flowfield analysis (GA1402067S)
from 01/01/2014
to 31/12/2016 investigator

Objectives:
The investigation aims at developing advanced methods for flowfield analysis particularly for transitional and turbulent flows. The work deals with decomposition techniques, mainly with decomposition of motion and vorticity. The vortexidentification criterion under development at present is associated with the corotation of line segments near a point and with the socalled residual vorticity obtained after the 'removal' of local shearing motion. The new quantity  the average corotation  is a vector, thus it provides a good starting point for developing both region and linetype vortexidentification methods, especially predictorcorrector type schemes. Further, the project aims to analyze the strainrate skeleton so as to draw a more complete picture of the flow. For testing purposes, largescale numerical experiments based on the solution of the NavierStokes equations (NSE) will be performed for selected 3D flow problems using finite element method on parallel supercomputers. Some qualitative properties of the solutions to the NSE will be studied and described in detail.



The finite element method for higher dimensional problems (IAA100190803)
from 01/01/2008
to 31/12/2012 investigator

The proposed project is a free continuation of the grant Finite element method for threedimensional problems IAA1019201, which terminated in 2006 and which was fulfilled with excellent results. The main goal of the new project will be a thorough mathematical and numerical analysis of the finite element method for solving partial differential equations in higher dimensional spaces. The necessity of solving such problems arises, e.g., in theory of relativity, statistical and particle physics, financial mathematics. In particular, we would like to deal with generation of simplicial meshes of polytopic domains. Further, we will investigate the existence and uniqueness of continuous and approximate solutions of problems that are often nonlinear. A special emphasize will be laid also on a priori and a posteriori error estimates, superconvergence, discrete maximum principle, stability of numerical schemes, etc.
