NEČAS SEMINAR ON CONTINUUM MECHANICS

Hemostasis is responsible to stem blood loss after injury by platelet plug formation. Although being life essential, a major part of deaths in the western society is due to thrombotic events provoked by disorders of the hemostatic system. Therefore, a better understanding of the underlying mechanisms is needed.

The first talk presents the biological background and focuses on the kinetics (without taking flow into account). The mechanisms of platelet adhesion/aggregation and the chemical processes are explained in quite detail, the latter both from a cascade and from a cell-based view. Then, the ODE-model of Kenneth Mann serves to study feedback mechanisms, threshold behavior and impairement of thrombin production in the bleeding disorder hemophilia. These findings help to understand hemostatic tests.

place: UTIA, CAS, Pod Vodrenskou vezi 4, Praha

place: UTIA, CAS, Pod Vodrenskou vezi 4, Praha

place: UTIA, CAS, Pod Vodrenskou vezi 4, Praha

!!! Special memorial seminar, starts at 15:30 at the Building of the Faculty of Mathematics and Physics at Malostranska square 25. !!!

u_t - partial_x (u_x/|u_x|)=0 on R times (0,T).

Slides: 1 and 2, 3 and 4

Reference: K. R. Rajagopal, The elasticity of elasticity, Z. Angew. Math. Phys. 58 (2007) 309-317.

Full abstract

Contents of part I and II: Standard meshes 
1. A review of classical FE analysis and the difficulties for convection-diffusion problems
2. Stabilization methods
2.1 Residual based stabilization
2.2 Symmetric stabilization: CIP and LPS
3. Operator fitted methods

Contents of part III and IV: Layer-adapted meshes
1. Solution-decomposition and classification of meshes
2. Galerkin-FEM on layer-adapted meshes
3. Stabilization
4. Solution recovery

Many subcellular biological processes can be described in terms of diffusing and chemically reacting species. Several stochastic simulation algorithms (SSAs) suitable for the modelling of such reaction-diffusion processes will be analysed. The connections between SSAs and the deterministic models (based on reaction-diffusion partial differential equations (PDEs)) will be presented. I will consider chemical reactions both at a surface (e.g. a membrane with receptors) and in the bulk. I will show how the microscopic parameters should be chosen to achieve the correct macroscopic reaction rate. This choice is found to depend on which SSA is used.

The movement of unicellular organisms can also be viewed as a stochastic process - a biased random walk. Examples include chemotaxis of bacteria or amoeboid cells and in both cases, cells detect extracellular signals (attractants or repellents) and alter their behaviour accordingly. I will discuss the derivation of macroscopic PDEs (collective behaviour) from individual based models of unicellular organisms. I will also present modelling of animal groups with a focus on the behaviour of locusts. Systematic analysis of the experimental data reveals that individual locusts appear to increase the randomness of their movements in response to a loss of alignment by the group. I will show how properties of individual animal behaviour can be implemented in the self-propelled particle model to replicate the group-level dynamics seen in the experimental data.

Discretization of a continuous medium by the finite element method introduces dispersion error to numerical solutions of stress waves propagation. In the introduction, review is made of fundamental approaches used to derive the truncation error of the finite element method in a dynamic analysis, valid for elements with linear shape functions. In the second part, recent results accomplished by the authors are summarized, namely the extension of dispersion theory to quadratic finite elements, following the lines of reasoning introduced by Belytschko and Mullen for one-dimensional elements and those of Abboud and Pinsky, concerning the scalar Helmholtz equation. The main conclusions drawn may be stated as follows.

Reduced order modeling plays a central role in approximation of large-scale dynamical systems, and techniques based on Krylov subspaces are used in that area for decades, see, e.g. [2] and the recent monograph [1]. Krylov subspace methods can not only be viewed as tools for model reduction. Many of them by their nature represent model reductions based on matching moments. In order to explain the deep link between matching moments and Krylov subspace methods, we start with a modification of the classical Stieltjes moment problem, and show that its solution is given:

K. Rajagopal, A. Srinivasa, A thermodynamic frame work for rate type fluid models, Journal of Non-Newtonian Fluid Mechanics 80 (2000) 207-227.

M. Anand, K. R. Rajagopal, A shear-thinning viscoelastic fluid model for describing the flow of blood, International Journal of Cardiovascular Medicine and Science 4 (2) (2004) 59-68.

In the first part of the lectures we consider stationary reaction-diffusion equations of the type

while the second part is concerned with its time-dependent analogue

Both are equipped with homogeneous Dirichlet boundary conditions and, in case of the time-dependent problem, with appropriate initial conditions. The parameter ε >0 is small, while the reaction cofficient c is assumed to satisfy c ≥ γ² with some positive constant γ.

The efficiency of standard numerical methods deteriorates as the perturbation parameter ε approaches zero. This is because layers form. These are regions where the solution varies rapidely.

In the present talk, bounds for the Green’s function associated with the differential operators and are derived. These bounds are applied to obtain a posteriori error estimators in the maximum norm for difference schemes and for FEM. These estimators are robust with respect to the perturbation parameter. They can be applied to design adaptive mesh-movement algorithms that give numerical methods which converge uniformly with respect to the perturbation parameter ε.

Numerical results will be presented to illustrate the theoretical findings.

The lecture will be delivered also at the Seminar on Numerical mathematics on 1st April, 14:00, seminar room K3.

15:40 Prof. Josef Malek (Faculty of Math. and Phys., Charles University in Prague)
Introduction

15:55 Mgr. Martin Lanzendorfer (Institute of Computer Science, AS CR)
On the existence analysis of fluids whose viscosity depends on the pressure and the shear rate

16:25 MSc. Adrian Hirn (IWR, University of Heidelberg)
Finite element approximation of flows of fluids with shear rate and pressure dependent viscosity

16:45 MSc. Stefan Knauf (IWR, University of Heidelberg)
Numerical simulations for ball bearings containing fluids with pressure-dependent viscosity

In the literature one can find a variety of numerical methods appropriate for viscous compressible flow. However, there are very few results with reference to the convergence properties of these methods. For instance, it is an open problem whether any numerical method for the compressible Navier-Stokes equations (in more than one dimension) converges as discretization parameters tend to zero.

In this talk, I aim at discussing why proving convergence of numerical methods for viscous compressible flow is so hard. I will also present some recently developed methods that are provably convergent for compressible Stokes flow (neglecting convection). These methods solves half of the problems, but still seem inadequate for the problem with convection.

Glaciology is a physical discipline describing the flow and evolution of ice masses on the Earth. The evolution of grounded glaciers - ice-sheets - is primarily controlled by the climate forcing via surface temperature and precipitation/ablation rates and by the geothermal heat flux from the Earth’s interior. Given these inputs, the dynamics of ice-sheets on sufficiently long time and spatial scales may be viewed as a thermomechanically coupled free-surface flow of a non-Newtonian fluid subject to gravity forcing.

A large number of physical processes needs to be taken into account to provide a sufficiently realistic glacier model, some of which are still only poorly understood. Typical example is a generation and transport of meltwater in the ice matrix and its interaction with the glacier flow, that would require a mixture continuum thermodynamics rather than a single-component theory. Of the same importance are processes at the glacier base, such as the interaction of basal ice and meltwater with a sedimentary glacier bed, which strongly affects basal boundary conditions. For sufficiently low local temperatures, no-slip (“frozen bed”) Dirichlet boundary condition is applied, while in regions where basal temperature exceeds the local pressure-melting point, a rapid sliding occurs, governed by a Newton-type sliding law. As shown in recent numerical studies, such a type of thermally-conditioned boundary conditions may become a trigger of large-scale intrinsic instabilities of the whole ice-sheet, which is a phenomenon indicated by geological evidence. From a numerical point of view, glacier physics remains a challenging topic as even relatively crude and simplistic formulations of glacier flow problem remain rather intractable by standard numerical techniques such as finite-elements. Various approximations of the flow equations based on scaling analysis have been adopted, in order to reduce the huge computational demands.

I am going to discuss the issue of existence of stationary solutions to the Navier Stokes Equations for compressible, barotropic flow in a cylindrical domain. More precisely, we are interested in strong solutions in a vicinity of given laminar flows. I will show existence of such solution when the perturbed flow is a constant flow or a Poiseuille profile. In the latter case we need some assumptions on the viscosity, but these turn out natural and does not provide any serious limitations.

A main problem to face in the proof is the lack of compactness in the continuity equation. To overcome this diffuculty we can introduce a kind of Lagrangian coordinates. In the new system the continuity equation simplifies enabling us to apply the Banach Fixed Point Theorem.

Geofyzikalni studium vyvoje Zeme a dalsich teles slunecni soustavy je zalozeno na numerickem reseni rovnic popisujicich prenos tepla ve vysokoviskozni kapaline. Krome tradicni Bousinesqovy aproximace se dnes jiz standardne vyuziva take rozsirena Bousinesqova aproximace (zahrnujici disipacni a adiabaticke zahrivani a teplo uvolňujici se pri fazovych prechodech) a anelasticka aproximace (uvazujici narust hustoty s hloubkou). Jelikoz se jedna o velmi pomale procesy, lze v pohybove rovnici bezpecne zanedbat setrvacny clen. Vyznamnou komplikaci pri reseni rovnic zpravidla predstavuje viskozita, ktera se muze menit v rozmezi sesti i vice radu. Viskozita je obecne chapana jako funkce tlaku, teploty a napeti (a pripadne dalsich parametru, jako je napr. velikost zrna) a je casto zavedena tak, aby simulovala i dalsi typy deformaci, jako je krehke poruseni a plasticita. Dalsim specifikem geofyzikalnich modelu jsou fazove prechody, ať uz souvisejici s tavenim materialu nebo s jeho prechodem do vysokotlake faze s odlisnymi fyzikalnimi vlastnostmi. Chovani geofyzikalnich systemu je obvykle studovano ve dvourozmerne kartezske geometrii, stale castejsi jsou ale dnes i trojrozmerne sfericke modely. Nektere soucasne modely se snazi pracovat s volnou hranici a zahrnout deje, ktere souvisi s jeji deformaci (eroze, sedimentace). Z geofyzikalnich problemu resenych na katedre geofyziky MFF budou diskutovany tri priklady:
1) model litosfericke desky zanorujici se do plaste (priklad komplikovane reologie materialu s fazovymi prechody),
2) simulace geologickeho vyvoje Ceskeho masivu v prubehu variske orogeneze (model zahrnuje nekolik „nemisitelnych“ petrologickych fazi, krehke chovani svrchni kury a volnou erodujici hranici) a
3) generace podpovrchoveho oceanu na Saturnove mesici Enceladu v dusledku slapoveho zahrivani (trojrozmerny sfericky model, zahrnujici viskozni i viskoelastickou deformaci).

Literature:
S. Repin. A posteriori error estimates for PDE s, deGruyter, Berlin, 2008.
M. Fuchs and S. Repin. A Posteriori Error Estimates for the Approximations of the Stresses in the Hencky Plasticity Problem, Numer. Funct. Analysis and Optimization, 32(2011), 6, 610-640.
S. Repin and S. Sauter. Estimates of the modeling error for the Kirchhoff-Love plate model. C. R. Math. Acad. Sci. Paris 348 (2010), no. 17-18, 1039–1043.

A joint event as a 13th Colloquium Lecture of the School of Mathematics