5.1.2009
15:40
Dr. Atsushi Suzuki
(Faculty of Mathematics, Kyushu University, Japan):
Preconditioned conjugate gradient solver for the Stokes equations
Abstract: The Stokes equations are fundamental equations of incompressible fluid. Discretized problem of the Stokes equations by finite elements consists of an indefinite matrix. We use conjugate gradient (CG) method with preconditioners to utilize the property of symmetry of the matrix. Since the matrix is not positive definite, there is a possibility of break down of the algorithm. However, if we do not encounter the break down, we can obtain approximate solution in the CG method. We can also employ indefinite matrix as a preconditioner. This simple algorithm helps to develop large-scale 3-D computational codes. Understanding of the CG procedure as variational problems in Krylov subspaces and results of numerical computation will be shown.
17:10
Doc. Ing. Dr. Miroslav Rozloznik
(Institute of Computer Sciences, Czech Academy of Sciences, Prague):
Saddle point problems and the conjugate gradient method with indefinite preconditioning
Abstract: In this talk we consider the solution of saddle-point problems that arise in many application areas such as computational fluid dynamics, electromagnetism, optimization and nonlinear programming. Particular attention has been paid recently to the iterative solution of these systems and to various preconditioning techniques. Several structure-dependent schemes have been proposed and analyzed. Indeed, the block pattern of saddle-point systems enables to take into account not only simple preconditioning strategies and scalings, but also preconditioners with a particular block structure. Here we analyze the null-space projection (or constraint) indefinite preconditioner. Since it was shown that the behavior of most of nonsymmetric Krylov subspace methods can be in this case related to the convergence of preconditioned conjugate gradient method (PCG) we study in detail its theoretical properties and propose simple procedures for correcting its possible misconvergence. The numerical behavior of the scheme is discussed and the maximum attainable accuracy of the approximate solution computed in finite precision arithmetic is estimated. This contribution is a joint work with V. Simoncini. M. Rozloznik and V. Simoncini, Krylov subspace methods for saddle point problems with indefinite preconditioning, SIAM J. Matrix. Anal. Appl.(2002), Vol. 24, No. 2, pp. 368--391.
12.1.2009
15:40
Mgr. Vit Prusa, Ph.D.
(Mathematical institute of Charles University):
Stability of fluid flow - recent developments
Abstract: The lecture will briefly introduce some basic notions and classical approaches used in theory of stability of fluid flow, and subsequently it will summarize some well-known fundamental results. After the introductory part, some weaknesses of the presented approaches will be pointed out, and the rest of the lecture will be devoted to some recent approaches to stability, namely to so called transient growth theory and self-sustaining processes theory. These fruitful approaches were developed on an engineering (read as non-rigorous ) level, and are probably still lacking a rigorous mathematical background, therefore the aim of the second part of the lecture is to attract attention to these approaches and possibly stimulate rigorous investigation of these approaches.
17:20
(Institute of Thermomechanics, Czech Academy of Sciences):
Stability of viscous flow-Thermodynamic point of view
Abstract: Thermodynamics of open systems offers a new concept for description of real material objects. The basic ideas come out the time irreversibility of processes and stability of states. The II. Law of Thermodynamics can be interpreted as an evolution law of all material systems, which are in interaction with their surroundings. The most important quantity is the entropy, which is defined by the balance law of entropy. The production of the entropy gives information about the processes into the systems. The convexity of the thermodynamic potentials (e.g. entropy, total enthalpy etc.) inform us about the stability of the system states. Under the appropriate outer conditions the fluctuations can to drive the systems to an instability. The consequence is the creation or decay of dissipative structures. When the new dissipative structure appears, the system is going further from the thermodynamic equilibrium to the new stable state. However, if the dissipative structure disappears the systems tends to the thermodynamic equilibrium without any transport processes. This approach is applied to the derivation of thermodynamic stability criteria for Blasius, Poisseulle and Couette flows . Moreover, the fluid flow stability enhancement by temperature gradient is discussed for heated Coanda flow. All theoretical conclusions are compared with physical experiments.
2.3.2009
15:40
(MFF UK Praha):
Delamination problems
Abstract: Contact problems of elastic solids with activated, rate-independent delamination on the contact boundary (also called debonding or adhesive contacts) will be addressed. Models at large or small strains will be presented first in a quasistatic formulation based on the concept of energetic solutions. In particular, a brittle delamination model (i.e. only either completely rigidly glued contact or fully debonded contact allowed) is exposed and shown to be a limit of a delamination with elastically responding glue that can be thus understood as a regularization which is also suitable to numerical implementation. Further extension to dynamical contact problems in visco-elastic solids and its possible thermodynamics will be briefly outlined, too. The talk will include also results from joint works with M.Kocvara, A.Mielke, L.Scardia, and C.Zanini.
17:20
Mgr. Libor Pavlicek
(MFF UK Praha):
Uvod do Di Perna-Lionsovy teorie
Abstract: Pracovni seminar NCMM.
9.3.2009
15:40
dr. Giuseppe Tomassetti
(Univ. Roma II Tor Vergata):
Theories of shearable beams and plates as gamma limit of three-dimensional micropolar elasticity
Abstract: In this talk I will outline some recent developments on the mathematical justification of theories of shearable rods and plates. In particular, I will discuss what happens when micropolar elasticity is adopted as parent theory from which the plate theory is deduced. In particular, I will show that for beams and plates quite different scalings are in order.
17:20
Mgr. Libor Pavlicek
(MFF UK):
Uvod do Di Perna-Lionsovy teorie II.
Abstract: Student seminar of the Necas Center
16.3.2009
15:40
Dr. Joerg Wolf
(University of Magdeburg):
On the pressure of the Navier-Stokes equations from different points of view
Abstract: The Navier-Stokes equations and related system describe an incompressible viscous fluid, where the volume is preserved. Therefore the pressure p has to be introduced as an additional unknown quantity. From the theory of the non-stationary Stokes system using the well-known properties of the Stokes operator under suitable conditions the pressure for the Navier-Stokes equation can be introduced. However, in general this method does not work, for example if the viscosity ν is non-constant. In this first lecture will introduce different methods for introducing a global and local pressure in general situations.
17:20
Mgr. Libor Pavlicek
(MFF UK Praha):
Uvod do Di Perna-Lionsovy teorie III.
Abstract: Student seminar of the Necas Center
23.3.2009
15:40
Prof.Dr. Soeren Bartels
(Institute for Numerical Simulation, University of Bonn):
Approximation of Harmonic Maps and Wave Maps
Abstract: Partial differential equations with a nonlinear pointwise constraint defined through a manifold occur in a variety of applications: The magnetization of a ferromagnet can be described by a unit length vector field and the orientation of the rod-like molecules that constitute a liquid crystal is often modeled by a vector field that attains its values in the real projective plane thus respecting the head-to-tail symmetry of the molecules. Other applications arise in geometric modeling, quantum mechanics, and general relativity. Simple examples reveal that it is impossible to satisfy pointwise constraints exactly by lowest order finite elements. For two model problems we discuss the practical realization of the constraint, the efficient solution of the resulting nonlinear systems of equations, and weak accumulation of approximations at exact solutions.
17:20
Prof. Piotr Mucha
(University of Warsaw):
Analysis of equations arising from the theory of crystal growth
Abstract: I would like to introduce some simple models of equations which determine evolution of crystals. They base on singular and degenerated parabolic systems. A model problem can be represented by the equation of the total variation

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

Analysis of this type of equations requires a new approach to define such quantity as smooth function , compositions of multi-functions. Then we will be able to obtain solutions so good that we will call them almost classical . Although in the system there appears formally a product of two Dirac deltas.
18:30
Mgr. Miroslav Bulicek, Ph.D.
(MUUK):
Partial regularity for Navier-Stokes equations by De Giorgi s method
Abstract: Student seminar of the Necas center.
30.3.2009
15:40
Prof. Antonin Novotny
(University of Toulon):
Singular limits in the thermodynamics of viscous fluids - part I/IV
17:20
Prof. Antonin Novotny
(University of Toulon):
Singular limits in the thermodynamics of viscous fluids - part II/IV
18:30
Mgr. Miroslav Bulicek, Ph.D.
(Mathematical Institute of the Charles University):
Partial regularity for Navier-Stokes equations by De Giorgi s method - part II
Abstract: Student seminar of the Necas center.
6.4.2009
15:40
Prof. Maurizio Grasselli
(Politecnico di Milano):
Asymptotic behavior of dissipative evolution equations - Part I/IV
Abstract: Semigroups of operators, trajectories and orbits, phase space, equilibria, invariant sets, omega-limit sets. bounded absorbing sets, attracting sets, compactness and asymptotic compactness, global attractors and their properties.

Slides: 1 and 2, 3 and 4

17:20
Prof. Antonin Novotny
(University of Toulon):
Singular limits in the thermodynamics of viscous fluids - part IV/IV
18:30
Mgr. Miroslav Bulicek, Ph.D.
(Mathematical Institute of the Charles University):
Partial regularity for Navier-Stokes equations by De Giorgi s method - part III
Abstract: Student seminar of the Necas center.
13.4.2009
15:40
seminar cancelled
Abstract: Easter Monday
20.4.2009
15:40
dr. Zdenek Fiala
(UTAM AV CR):
Geometrical interpretation of the logarithmic strain and the Zaremba-Jaumann time derivative
Abstract: After a brief review of some concepts in computational solid mechanics (objective time derivative, logarithmic strain, logarithmic time derivative), we adress the finite deformations from the viewpoint of natural geometry of the set of all symmetric, positive-definite real matrices $Sym^+$ to find a geometrical interpretation of the Zaremba-Jaumann time derivative and the logarithmic strain. Since the right Cauchy-Green deformation tensors $mathbf{C}=mathbf{F}^ ext{T}mathbf{F}$ are symmetric, positive-definite matrices, a deformation process is represented by a time-parameterized trajectory $mathbf{C}_t: o Sym^+$. As an open convex cone in the ambient space of all symmetric matrices $sym$, which is an inner product vector space, the set $Sym^+$ inherits this way its Riemannian geometry to become a constant negatively-curved manifold. From the viewpoint of continuum mechanics, the Riemannian metric naturaly enters $Sym^+$ via the stress power. We can then employ the tools of differential geometry to find out that the Zaremba-Jaumann derivative is related to geometrically consistent linearization via the covariant derivative, and the logarithmic strain to geodesics in $Sym^+$, which prove to be matrix exponentials. Adopting this interpretation we can consider initially-deformed and undeformed states in a unified way, resulting in particular to generalization of the logarithmic strain, and in general to some universal conclusions concerning the incremental approach. Moreover, within this interpretation we geometrically distinguish stress, deformation, and rate-of-deformation tensors.
17:20
Dr. Riccarda Rossi
(Univ. Brescia, Italy):
A variational approach to doubly nonlinear and rate-independent evolutions
Abstract: A wide class of dissipative phenomena can be modelled by doubly nonlinear abstract evolution equations in Banach spaces, featuring a convex dissipation functional and an energy functional. When the dissipation functional is positively homogeneous of degree 1, the related doubly nonlinear equation models rate-independent evolutions, arising in several branches of continuum mechanics, often in connection with hysteresis problems. In many relevant applications, the driving energy functional may be nonsmooth and nonconvex. Hence, the resulting rate-independent evolutions have jumps, which require additional modelling. Consequently, a new approach has been proposed, based on the analysis of such systems as limit of suitable viscous evolutions (given by doubly nonlinear equations with a superlinear dissipation functional). Combining variational and reparametrization techniques, in collaboration with Alexander Mielke and Giuseppe Savare we have developed the notion of parametrized rate-independent evolution , which shall be presented in this talk.
18:30
Dr. Petr Kaplicky
(MFF UK Praha):
On W^{1,p} estimates for elliptic equations in divergence form
Abstract: We refer beautiful method of proving L^p theory for large class of elliptic PDE s from article L. A. Caffarelli, I. Peral: On W^{1,p} estimates for elliptic equations in divergence form. Comm. Pure Appl. Math. 51 (1998), no. 1, 1--21.
27.4.2009
16:00
Prof. RNDr. Jan Kratochvil, DrSc.
(FSv CVUT Praha):
Modeling of microstructure formation in materials exposed to severe plastic deformation
Abstract: It is well documented that severe plastic deformation produces ultrafine grained materials with extraordinary mechanical properties. Very high strength and relatively good ductility is attributed to their fine microstructure. One of methods suitable for experimental and theoretical studies of the microstructure evolution during severe plastic deformation is the high pressure torsion (HPT).
The proposed crystal plasticity model outlines a possible mechanism of ultrafine structure formation as observed in HPT experiments. The HPT deformation can be interpreted as a plastic flow though the adjustable crystal lattice. In the deformation process the lattice is settled to provide the energetically most effective plastic flow, analogously as a riverbed adjusts to a water flow. HPT deformation as seen in the radial direction can be roughly described as a plastic flow through a channel with crystal lattice structure. The flow is exposed to the boundary conditions: zero velocity at the bottom and a prescribed velocity at the upper surface. The model reveals rotations of slip systems caused by the imposed shear strain. An axial compression and a shear stress of twist govern this process. Local variations in the crystal lattice orientation are responsible for the microstructure fragmentation. The accompanied continuous reconstruction of the deformation substructure is probably the main reason of the observed strengthening.
17:30
Prof. Kumbakonam Rajagopal
(Texas A&M University):
The elasticity of elasticity
Abstract: In this note we assert that the usual interpretation of what one means by elasticity is much too insular and illustrates our thesis by introducing implicit constitutive theories that can describe the non-dissipative response of solids. There is another important aspect to the introduction of such an implicit approach to the non-dissipative response of solids, the development of a hierarchy of approximations wherein, while the strains are infinitesimal the relationship between the stress and the linearized strain is non-linear. Such approximations would not be logically consistent within the context of explicit theories of Cauchy elasticity or Green elasticity that are currently popular.

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

4.5.2009
15:40
Prof. Gerhard Huisken
(Universitat Tubingen, Germany):
Mean curvature flow with surgery
Abstract: A family of hypersurfaces is said to move by mean curvature if at every point the speed in normal direction is given by the mean curvature of the surface. This quasilinear parabolic system decreases area and tends to uniformize the shape of the evolving surfaces. It has a similar analytic behavior as the Ricci flow of Riemannian metrics, that was recently used by Hamiltoon and Perelman for the proof of the Poincare conjecture. The lecture explains recent work by Huisken and Sinestrari on mean curvature flow with surgeries that can extend the flow through singularities in a controlled way. In particular it is shown that a certain class of hypersurfaces called 2-convex exhibits the same structure of singularities and surgeries as 3-dimensional Ricci flow.
17:20
Prof. Roger Lewandowski
(University Rennes 1):
Regularity and uniqueness results for the k-u system
Abstract: We first introduce the turbulence model that couples the mean velocity and the mean pressure of a flow field with its turbulent kinetic energy thru eddy viscosities. After the recall of the classical existence result, we show a regularity and uniqueness result in the steady-state case.
18:30
Dr. Petr Kaplicky
(MFF UK Praha):
On W^{1,p} estimates for elliptic equations in divergence form - part II
Abstract: We refer beautiful method of proving L^p theory for large class of elliptic PDE s from article L. A. Caffarelli, I. Peral: On W^{1,p} estimates for elliptic equations in divergence form. Comm. Pure Appl. Math. 51 (1998), no. 1, 1--21.
11.5.2009
15:40
Antonio Andre Novotny
(National Laboratory for Scientific Computing, Rio de Janeiro):
Topological Sensitivity Analysis
Abstract: The topological asymptotic analysis provides the sensitivity of a given shape functional with respect to an infinitesimal singular domain perturbation. Therefore, this sensitivity can be naturally used as a descent direction in an optimization algorithm. The concept of topological derivative is an extension of the classical notion of derivative. It has been rigorously introduced in 1999 by Sokolowski & Zochowski. Since then, the notion of topological derivative has proved extremely useful in the treatment of a wide range of problems in mechanics, optimization, inverse analysis and image processing and has become a subject of intensive research. In this seminar it will be presented some recent developments and also some applications of the topological derivative in image processing, inverse problems, topology optimization design of load bering structures and, finally, in the synthesis and optimal design of microstructures to meet a specified macroscopic behavior.
18.5.2009
15:40
Prof. A. M. Saendig
(University of Stuttgart):
Regularity results for nonlinear convection-diffusion problems
Abstract: The knowledge of analytical existence and regularity results for solutions of nonlinear nonstationary convection-diffusion equations is important for their numerical treatment. In particular, error estimates for Discontinous Galerkin Methods demand a high classical regularity of the solution [2]. This regularity is not guaranteed, if the domain is nonsmooth or if the boundary conditions change. Starting from weak solutions in anisotropic Sobolev spaces L2(I,V) [3], we analyse their regularity for Dirichlet boundary conditions in polygonal domains using the theory of semigroups in Lp [4]. Problems with changing boundary conditions can be handled analogously. Essential is, that we benefit from the introduction of Sobolev spaces with attached asymptotics in the linear stationary case [1], which reflects also the behavior of the solutions to nonlinear nonstationary convection-diffusion problems near corner points of a polygon under certain conditions.

Full abstract

5.10.2009
15:40
(IAM, University of Freiburg, Germany):
Finite element analysis of the p-Stokes system
Abstract: We derive apriori estimates for the finite element solutions of the p-Stokes system. The estimates for the velocity are optimal and presented in terms of quasi-norms. We also prove convergence of the pressure.
12.10.2009
15:40
(MUUK, Charles University, Prague):
On compressible and incompressible generalized Newtonian fluids
17:20
(IAM University of Heidelberg, Germany):
Modeling Vapor Transport in Cold Dry Soil
Abstract: We consider vapor transport in cold dry soil with high temperature fluctuations. Such systems occur in permafrost soil in the northwest of Tibet. A model is set up consisting of two temperature fields, air density, vapor saturation and water content as free variables. In contrast to most numerical models the assumption on constant full vapor saturation is dropped. Instead, the air phase allows for over- and undersaturation. Mathematical solvability results for suitable assumptions on the coefficients and boundary conditions will be demonstrated.
19.10.2009
15:40
(TU Dresden):
Finite element methods for convection-diffusion problems Part I
Abstract: Convection-diffusion equations are a fundamental subproblem for models in various applications. Typically, diffusion is less significant than convection: on a windy day the smoke from a chimney moves in the direction of the wind and the influence of the diffusion is small. Finite element discretizations of convection-diffusion equations are not trivial because stability problems arise and layer phenomena lead to difficulties with the interpolation error. In part I of our lecture we discuss several finite element techniques on standard meshes. Because on standard meshes it is difficult to resolve layer structures we analyze in part II the use of layer-adapted meshes.
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
Parts I and II will form the contents of 4 lectures delivered on 19 and 26 October in the lecture hall K1 at 15:40 (Seminar on Continuum Mechanics) and on 22 and 29 October in the lecture hall K3 at 14:00 (Seminar on Numerical Mathematics). The parts III and IV will be the subject of the course which will be delivered during 1 - 14 March 2010.
17:20
Mgr. R. Chabiniok
(INRIA Rocquencourt, France):
Biomechanical model of the heart function: validation and clinical applications
Abstract: First we describe an approach that we propose to model the electromechanical behavior of the heart. The modeling of the heart tissue is based on an electrically-activated contraction law formulated via multiscale considerations and is consistent with various key physiological and thermomechanical requirements. Then we present several steps of the model validation and some clinical applications, in particular:
1. Validation in 1D using experimental data of cardiac muscle fiber contraction, and in 3D using clinical data or data from specially designed experiments.
2. Physiological simulations of the infarcted heart with possible study of the heart remodeling after the infarction.
3. Clinical application on modeling of cardiac resynchronization therapy (CRT).
26.10.2009
15:40
Prof. Hans-Goerg Roos
(TU Dresden):
Finite element methods for convection-diffusion problems Part II
2.11.2009
15:40
Prof. Joga I. Rao
(Department of Mechanical Engineering, New Jersey Institute of Technology, USA):
MODELING THE THERMO-MECHANICS OF SHAPE MEMORY POLYMERS (part 1)
Abstract: Shape memory polymers have the ability to retain a temporary shape, which can be reset to the original shape with the use of a suitable trigger, typically an increase in temperature or exposure to light. These temporary shapes can be very complex and the deformations involved large. These materials are finding use in a large variety of important applications; hence the need to model their behavior. The first talk will provide an overview of the different types of shape memory polymers, in particular crystallizable or glassy shape memory polymers and light activated polymers. In crystallizable shape memory polymers the temporary shape is fixed by a crystalline phase, while return to the original shape is due to the melting of this crystalline phase. In glassy polymers the temporary shape is fixed by the formation of a glassy phase and the return to the original shape is initiated by heating above the glass transition temperature. For light activated shape memory polymers exposure of the polymer to UV light at a specific frequency initiates the formation of cross-links that are responsible for the temporary shape. Exposure to UV light of a different frequency is responsible for cleavage of these cross-links and return to the original shape. In this talk we will discuss the underlying mechanisms and our approach to formulating constitutive equations to model the thermo-mechanical behavior of these polymers. The modeling is done within the framework of natural configurations utilizing a full thermodynamic approach. The application of the models developed to different boundary value problems of interest will be discussed.
17:20
Prof. Joga I. Rao
(Department of Mechanical Engineering, New Jersey Institute of Technology, USA):
MODELING THE THERMO-MECHANICS OF SHAPE MEMORY POLYMERS (part 2)
Abstract: The second talk will focus on the details of the model for crystallizable shape memory polymers and its relation to models developed for the other kinds of shape memory polymers. Particularly we will discuss each aspect of the model. Initially these materials are elastomers. On cooling, crystallization is initiated and the rate at which crystallization takes place is related to the thermodynamics. Natural configurations of this newly formed crystalline are prescribed based on experimental data. An important aspect of these materials that is overlooked in models developed is the anisotropic response of the material after the onset of crystallization. The anisotropy in the behavior is directly included in the model and depends on the deformation in the original material at the onset of crystallization. Finally the reverse transition from a semi-crystalline material back to an elastomer is included in a similar manner. The predictions of the model are compared with experimental data available. A variety of boundary value problems are solved using the model developed. The model has also been included into a large deformation finite element code by creating a user defined subroutine, which is then used to solve complex boundary value problems.
9.11.2009
15:40
(Institute of Mathematics, Academy of Sciences, Praha):
Propagation of acoustic waves on general unbounded domains with applications to incompressible limits of the Navier-Stokes system
Abstract: We discuss the problem of propagation of acoustic waves and their oscillations in the so-called incompressible limit for the Navier-Stokes system. We derive Lighthill's acoustic analogy and study its basic properties. Finally, a necessary and sufficient condition will be given for the acoustic waves to decay locally to zero.
16.11.2009
15:40
Prof. Dr. Yasushi Taniuchi
(Department of Mathematics, TU Darmstadt):
On the uniqueness of almost periodic-in-time solutions to the Navier-Stokes equations in unbounded domains
Abstract: We present a uniqueness theorem for almost periodic-in-time solutions to the Navier-Stokes equations in $3$-dimensional exterior domains $Omega$. It is known that there exists a small almost periodic-in-time solution in $C(R;L^{3}_w(Omega))$ to the Navier--Stokes equations for a small almost periodic-in-time force. Here $L^n_w(Omega)$ denotes weak $L^n$ space. Thus far, with respect to the uniqueness of almost periodic-in-time solutions to the Navier--Stokes equations in exterior domain, roughly speaking, it has been only known that a small almost periodic-in-time solution in $BC(R;L^{3}_w)$ is unique within the class of solutions which have sufficiently small $L^{infty}( L^{3}_w)$-norm, i.e., that if $u$ and $v$ are $L^3_w$-solutions for the same force $f$, and if both of them are small, then $u=v$. In this talk, we will show that a small almost periodic-in-time solution in $BC(R;L^{3}_wcap L^{6,2})$ is unique within the class of all almost periodic-in-time solutions in $BC(R;L^{3}_wcap L^{6,2})$, i.e., we will show that if $u$ and $v$ are almost periodic-in-time solutions in $BC(R;L^{3}_wcap L^{6,2})$ for the same force $f$, and if one of them are small, then $u=v$.
23.11.2009
15:40
(Mathematical Institute, University of Oxford):
Stochastic and Multiscale Modelling in Biology I
Abstract: Some recent advances in the development and analysis of methods for stochastic and multiscale modelling of biological systems will be presented. The biological examples will cover problems ranging over different length and time scales, including processes on the molecular level (e.g. genes and proteins), cellular level (e.g. cell motility) and population level (e.g. social insect behaviour).

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.

17:20
(Mathematical Institute, University of Oxford):
Stochastic and Multiscale Modelling in Biology II
Abstract: This is a continuation of my talk Stochastic and Multiscale Modelling in Biology I . I will discuss further the challenges of mathematical modelling of biological systems mentioned in my first talk. In all model systems considered, I will discuss connections and differences between deterministic models (mean-field ODEs/PDEs) and stochastic simulation algorithms. I will also present multiscale modelling approaches linking models with a different-level of detail together.
30.11.2009
15:40
Prof. Louisette Priester
(Universite Paris 11, France):
Stress relaxation at grain boundaries: Models and experiments. Consequences on material plasticity
Abstract: Grain boundaries (and more generally interfaces) play a fundamental role in most properties of crystalline materials, and particularly in their mechanical behaviour. The study of the elementary processes that occur at grain boundaries is necessary to understand the plastic deformation of the material and constitutes a prerequisite to control, then to improve its global properties. To approach intergranular phenomena, we first have to know the equilibrium structure of grain boundaries at three levels that are strongly linked: geometrical, mechanical and atomic. Then we must describe their defects, mainly their linear defects that are responsible for their mechanical behaviour. The interfacial strains and stresses are generally approached in terms of discrete dislocations, however, as a preliminary, we will briefly evoke how the continuum theory of defects may apply to grain boundary. During plastic deformation, interactions between lattice dislocations and grain boundaries necessary occur giving rise to intergranular dislocations. The stresses associated to these dislocations must be relieved in order that the deformation may go on. The main object of this seminar is to describe the interaction and the relaxation phenomena at grain boundaries. We will first present the existing models, then we will see how they are confirmed (or invalidate) by some simulations and transmission electron microscopy observations. It will be proved that all the reactions (entrance of lattice dislocations in grain boundaries and relaxation processes) depend on the fine grain boundary structure: then, the answer to the deformation differs from one grain boundary to the other. The latter statement involves that the role of different grain boundaries in the global deformation of the material must also differ. In the last part of the seminar, we will try to see how the macroscospic plasticity may be affected by the microscopic phenomena. This implies to relate the individual behaviours to the collective ones (grain boundary network) that depend, in particular, on the grain boundary distribution. These efforts constitute a first step towards a grain boundary engineering, a concept that is still in its infancy.
7.12.2009
15:40
(Institute of Thermomechanics, ASCR):
Accuracy and numerical stability of finite element solutions to wave propagation problems
Abstract:

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.

  • i) A spurious optical branch in the spectrum existed.
  • ii) The associated modes possessed infinite phase velocity, finite group velocity and strongly focused polarization.
  • iii) It was further shown, in terms of dispersion curves, that the quadratic elements had much more favourable properties than the linear ones. This may, however, not be true of diagonalized (lumped) mass matrices.

    Indeed, a detailed study of the mass matrix lumping schemes for higher order elements reveals substantial deterioration of accuracy due to increased dispersion manifested by the deviation of numerical velocities from the continuum ones. Moreover, a characteristic pattern of nodal mass distribution for each method strongly influences the stability limit in explicit integration algorithms. The central difference method is analysed as a typical representative, employing both the derived dispersion curves to establish the critical time step as well as its simple estimate offered by the Fried theorem, which imposes bounds on the system eigenvalues. Further, an attempt is made to improve efficiency of lumping procedures; to this end, a variable parameter, x, is defined whose role is to distribute total mass between the elements corner and midside nodes. Based on that, dispersion analysis is carried out for varying x as well as the critical Courant number computed. For example, it is shown in terms of dispersion curves and stability theory that the Hinton-Rock-Zienkiewicz (HRZ) mass ratio is far from optimum and, on the contrary, the most accurate travelling wave-train representation is surprisingly obtained when 92% of total mass is coalesced into four midside nodes, whereas only the 8% share is placed to the corner nodes.

    The talk contains two numerical examples. In the first example, an infinite 2D space loaded by a point-wise source is considered to test the spatial finite element discretization by the eight node serendipity elements. The loading frequency gradually changes from zero to high values to mimic the dispersion response to a broad loading spectrum. With the second example, the analytical solution to the longitudinal impact of two cylindrical bars as in the split-Hopkinson pressure bar test, derived by Vales et al., is employed to gauge dispersion for a contact-impact problem defined by the serendipity elements. Both examples results show superior agreement with the developed theory.



  • Po semináři se podává opět čaj a káva.
    Všichni zájemci jsou srdečně zváni.
    [tea]