NEČAS SEMINAR ON CONTINUUM MECHANICS
organized by the Mathematical Institute of the Charles University
each Monday at 15:45
in the MFF UK building,
Sokolovská 83, lecture room K3, 2^{nd}
floor,
proceeds with the following presentations:

15:40  (UTIA AV CR + FSv CVUT Praha):  Boundary effects and weak lower semicontinuity for signed integral functionals on BV  Abstract: We characterize lower semicontinuity of integral functionals with respect to weak convergence in BV, including integrands whose negative part has linear growth. In addition, we allow for sequences without a fixed trace at the boundary. In this case, both the integrand and the shape of the boundary play a key role. This is made precise in our newly found condition, a quasisublinear growth from below at points of the boundary, which compensates for possible concentration effects generated by the sequence. Some applications to relaxation of variational problems with linear growth will be outlined. It is a joint work with B. Benesová
(Wurzburg) and S. Kromer (Cologne). 



15:40  (KM, FJFI CVUT Praha):  Modeling of moving curves by curvature driven flow and its application in discrete dislocation dynamics  Abstract: We investigate the numerical solution of the evolution law for the mean curvature flow of open or closed nonselfintersecting curves in a plane. The model schematically reads as normal velocity = (mean) curvature + force. We treat the motion law by means of the parametric method, resulting into a system of degenerate parabolic partial differential equations for the curve parametrization. Unlike other interfacecapturing methods, such as levelset method or phasefield method, the parametric approach allows us to treat the dynamics of open curves comfortably. The parametric equations are spatially discretized by means of the flowing finite volume method. To improve the quality of the numerical solution, we discuss the effect of artificial designed tangential terms redistributing the discretization points. In the second part of the lecture, we present the application of the curvature driven flow to a microscale modeling of basic mechanisms in discrete dislocation dynamics. 



15:40  Prof. Victor A. Kovtunenko (Inst. for Math. and Sci. Comp., Univ. of Graz, Austria, and Lavrent ev Inst. of Hydrodynamics, Novosibirsk, Russia):  On generalized PoissonNernstPlanck equations  Abstract: A strongly nonlinear system of PoissonNernstPlanck equations is considered. The diffusion laws are coupled with the Landau grand potential for entropy variables. The model describes electrokinetic phenomena on multiphase medium in physical, chemical, and biological sciences. The generalized model is supplemented by the mass balance, positivity and volume constraints, quasiFermi statistics depending on the pressure, and inhomogeneous Robin boundary conditions representing interfacial reactions .
In our research we aim at proper variational modelling, wellposedness properties and dynamic stability, as well as homogenization of the problem supported by rigorous estimate of the energy and entropy types. 



15:40  Dr. Koya Sakakibara (University of Tokyo):  Structurepreserving numerical scheme for the onephase HeleShaw problems by the method of fundamental solutions combined with the uniform distribution method  Abstract: In this talk, the onephase HeleShaw problems and their numerical scheme are considered. The onephase interior HeleShaw problem has curveshortening (CS), areapreserving (AP), and barycenterfixed (BF) properties. We construct numerical scheme which satisfies the above properties in a discrete sense. As a result, computing the normal velocities by the method of fundamental solutions and the tangential velocities by the uniform distribution method, a discrete version of CS, AP, and BFproperties are satisfied. The onephase exterior HeleShaw problem and onephase interior HeleShaw problem with sink/source points can also be treated. We also show some numerical results which exemplify the effectiveness of our scheme. 



15:40  (Inst. of Thermomechanics, Czech Acad. Sci.):  Constitutive model of NiTi SMA polycrystals: from experiments to simulations  Abstract: Mechanical response of polycrystals of NiTi shape memory alloys (SMA)
exhibits several interesting features, e.g. strong dependence on
temperature, loading mode or loading history. These effects result from
interplay of deformation mechanisms of various origin with the dominant
influence of martensitic phase transformation. In the talk, I will introduce
a constitutive model of textured NiTi SMA which allows for a realistic
description of the mechanical response under various loading conditions.
Particular attention has been paid to the description of martensite
reorientation, occurrence of intermediate phase and the localization effect.
Simulations demonstrating capabilities of the model both at the macro and
meso scale will be presented and compared to experimental data. 



15:40  (Abt. f. angew. Math., AlbertLudwigsUniversitat Freiburg):  Energy estimates, relaxation, and existence for strain gradient plasticity with cross hardening  Abstract: We consider a variational formulation of gradient elastoplasticity subject to a class of singleslip side conditions. Such side conditions typically render the associated boundaryvalue problems nonconvex. We first show that, for a large class of plastic deformations, a given singleslip
condition (specification of Burgers vectors and slip planes) can be relaxed
by introducing a microstructure through a twostage process of mollification
and lamination. This yields a relaxed side condition which only prescribes
slip planes and allows for arbitrary slip directions. This relaxed model
can be thought of as an aid to simulating macroscopic plastic behavior
without the need to resolve arbitrarily fine spatial scales. We then discuss
issues of existence of solutions for the relaxed model.
Finally, we apply this relaxed model to a specific system, in order to be
able to compare the analytical results with experiments. A rectangular
shear sample is clamped at each end, and is subjected to a prescribed
horizontal, modelled by an appropriate Dirichlet condition. We ask: how much
energy is required to impose such a shear, and how does the energy depend on
the aspect ratio of the sample? Assuming that just two slip systems are
active, we show that there is a critical aspect ratio, above which the
energy is strictly positive, and below which it is zero. Furthermore, in the
respective regimes determined by the aspect ratio, we prove energy scaling
bounds, expressed in terms of the amount of prescribed shear. 



15:40  (Faculty of Mathematics, University of Vienna, Austria):  Wulff Shape Emergence in Graphene  Abstract: In this talk the problem of understanding why particles selfassemble in macroscopic
clusters with overall polyhedral shape is investigated. At low temperature ground states
for a general finite number n of particles of suitable phenomenological energies possibly
accounting for two and threebody atomic interactions are shown to be connected subsets
of regular lattices L, such as the triangular and the hexagonal lattice. The hexagonal
lattice well represents the arrangement of carbon atoms in the graphene layers.
By means of a characterization of minimal configurations via a discrete isoperimetric
inequality, ground states will be seen to converge to the hexagonal Wulff shape as the
number n of particles tends to infinity. Furthermore, ground states are shown to be
given by hexagonal configurations with some extra particles at their boundary, and the
n3/4 scaling law for the deviation of ground states from their corresponding hexagonal
configurations is shown to hold. Precisely, the number of extra particles is carefully
estimated to be at most KL n3/4 + o(n3/4 ), where both the rate n3/4 and the explicitly
determined constant KL are proven to be sharp.
The new designed method allows to sharpen previous results [1, 5] for the triangular
setting [2] and allows to provide a first analytical evidence of the zigzagedge selectivity
and the emergence of the asymptotic Wulff shape for the hexagonal setting [3] in accor
dance with what is experimentally observed in the growth of graphene flakes [4]. Results
presented are in collaboration with Elisa Davoli and Ulisse Stefanelli (Vienna).
References
[1] Y. Au Yeung, G. Friesecke, and B. Schmidt, Minimizing atomic configurations of short range pair
potentials in two dimensions: crystallization in the Wulffshape, Calc. Var. Partial Differential Equations 44 (2012), 81100.
[2] E. Davoli, P. Piovano, and U. Stefanelli, Sharp n3/4 law for the minimizers of the edgeisoperimetric
problem on the triangular lattice, Submitted (2015).
[3] E. Davoli, P. Piovano, U. Stefanelli, Wulff shape emergence in graphene, Submitted (2015).
[4] Z. Luo, S. Kim, N. Kawamoto, A.M. Rappe, and A.T. Charlie Johnson, Growth mechanism of
hexagonalshape graphene flakes with zigzag edges, ACSNano 11 (2011), 19541960.
[5] B. Schmidt, Ground states of the 2D sticky disc model: fine properties and N 3/4 law for the deviation
from the asymptotic Wulffshape, J. Stat. Phys. 153 (2013), 727738. 



15:40  Dr. Diego Grandi (University of Vienna):  Modeling shape memory alloys at finite strains: solvability and linearization  Abstract: We discuss the macroscale modeling of shape memory alloys according to a finite strainversion of the SouzaAuricchio model. Assuming the isotropy of the hyperelastic stored energy functional, a convenient formulation in terms of the GreenStVenant transformation strain tensor can be established.
For the chosen rateindependent constitutive relation, coupled to a quasistatic elastic response and with an additional regularizing interfaceenergy contribution, the global existence of energetic solutions to the boundary value problem (i.e. variational evolution) is proven.
A similar finitestrain approach is applied to provide a model for the magnetoelastic evolution in magnetic shapememory materials.
The possibility of a rigorous linearization limit of the models at small strain is addressed within the framework of the evolutive variational convergence for rate independent processes. 



15:40  (University of Oxford):  Finite element approximation of nondivergence form PDEs  Abstract: Nondivergence form partial differential equations with discontinuous coefficients do not
generally possess a weak formulation, thus presenting an obstacle to their numerical solution
by classical finite element methods. Such equations arise in many applications from areas
such as probability and stochastic processes. These equations also arise as linearizations to
fully nonlinear PDEs, as obtained for instance from the use of iterative solution algorithms.
In such cases, it can rarely be expected that the coefficients of the operator be smooth or
even continuous. For example, in applications to HamiltonJacobiBellman equations, the
coefficients will usually be merely essentially bounded. In contrast to the study of divergence
form equations, it is usually not possible to define a notion of weak solution when the coefficients are nonsmooth. In the case of continuous but possibly nondifferentiable coefficients,
the CalderonZygmund theory of strong solutions establishes the wellposedness of the problem in sufficiently smooth domains. However, without additional hypotheses, wellposedness
is generally lost in the case of discontinuous coefficients. The aim of the lecture is to survey recent developments concerning the numerical approximation of such problems by finite
element methods.
The lecture is based on joint work with Iain Smears (INRIA Paris). 



15:40  (University of Oxford):  Numerical analysis of nonlocal CahnHilliard equations  Abstract: We discuss the numerical approximation of a class of nonlinear evolution problems that arise as L2 and H1 gradient flows for the ModicaMortola regularization of a certain functional on BV
involving the interfacial energy per unit length or unit area, the flat torus in Rd , and a nonnegative Fourier multiplier, that is continuous and symmetric, and which decays to zero at infinity.
Such functionals feature in mathematical models of patternformation in micromagnetics
and models of diblock copolymers. The resulting evolution equation is discretized by a Fourier
spectral method with respect to the spatial variables and a CrankNicolson or an implicit
midpoint scheme with respect to the temporal variable. We investigate the stability and
convergence properties of the proposed numerical schemes and illustrate the theoretical results
by numerical simulations.
The lecture is based on joint work with Christof Melcher (RWTH Aachen), Barbora Benesova (University of Wurzburg) and Nicolas Condette (Humboldt University, Berlin). 



15:40  (Inst. f. Mathematik, Technische Univ. Berlin):  Optimal control of some reactiondiffusion equations  Abstract: Results on the optimal control of the Nagumo equation and the
FitzHughNagumo system are surveyed. Special emphasis is laid on first and secondorder optimality conditions for sparse optimal controls. The secondorder analysis is applied to explain observed numerical stability with respect to certain perturbations. The theory is illustrated by various numerical examples. 

16:50  (Inst. f. Mathematik, Univ. Würzburg):  Existence of weak solutions for a model of magnetoelasticity  Abstract: In the talk, we will present a model for particles in
micromagnetic fluids. Such fluids have many technological applications. They
can not only be found in medical applications, but also in loud speakers and
shock absorbers.
We investigate micromagnetic material in the framework of complex fluids.
The system of PDEs to model the flow of the material is derived in a
continuum mechanical setting. We outline the process of modeling. Moreover,
we highlight the coupling between the elastic and the magnetic properties of
the material.
Restricting our scope to the two dimensional setting, we then prove
existence of solutions under the assumption of small initial data.
This is joint work with Carlos GarcíaCervera (Mathematics Department,
University of California, Santa Barbara, USA), Johannes Forster, Anja Schlömerkemper (Institute for Mathematics, University of
Würzburg, Germany), and Chun Liu (Department of Mathematics, Penn State University, University Park, USA). 



15:40  (Katedra matematiky, FJFI CVUT):  On modelling selforganisation in real systems  Abstract: Selforganisation in nature is widely recognised and is extensively
modelled. In some systems (e.g. Drosophila embryo) the spatial pattern
is not selforchestrated. On the other hand, Turing model of pattern
formation is capable of breaking symmetry without preexisting
positional information. This mechanism has driven numerous experimental
studies even in the context of developmental systems which suggest that
Turinglike morphogen interactions and patterns can occur in such
scenarios. However, a direct verification has remained elusive.
We start by introduction to the classical Turing instability. As the aim
is to reveal mechanism behind the observed pattern in nature, robustness
is required not only with respect to parameter sensitivity or the choice
of initial or boundary conditions but also with respect to the model
formulation itself. Only then are these models subjected to a detailed
mathematical analysis. We illustrate the essence of these ideas on the
reactiondiffusionadvection system, where we indicate that such a
system should be preferred from both physical and mathematical viewpoint
for selforganisation modelling. Particularly, we shall use the mixture
theory within extended irreversible thermodynamics to reveal what
evolution equations are relevant in real physical systems and can be
considered as small perturbations of reactiondiffusion equations and
mathematically analyse the possibility of the emergence of pattern in
RDA systems. Note that it is required to identify plausible extensions
of the Turing concept of selforganisation into more general cases. Such
extensions are not unambiguous but their discussion is beneficial even
for understanding the standart Turing model of spatial selforganisation. 



15:40  Dr. Jan Haskovec (King Abdullah Univ. of Sci. and Technology):  PDEbased modelling of biological network formation  Abstract: Motivated by recent papers describing rules for natural
network formation in discrete settings, we propose an
ellipticparabolic system of partial differential equations. The model
describes the pressure
field due to a Darcytype equation and the dynamics of the conductance
network under pressure force effects with a diffusion rate representing
randomness in the material structure. We prove the existence of global
weak solutions and of local mild solutions and study their long term
behavior. Moreover, we study the structure and stability properties of
steady states that play a central role to understand the pattern
capacity of the system. We show that patterns (network structures) occur
in the regime of small material randomness. Moreover, we present results
of systematic numerical simulations of the system that provide further
insights into the properties of the networktype solutions. 



15:40  Mgr. Michal Pavelka, PhD. (VSCHT, Praha):  A hierarchy of Poisson brackets  Abstract: Reversible part of evolution equations is often governed by a Poisson
bracket and energy. For example, Hamilton canonical equations are given by
the canonical Poisson bracket on cotangent bundle of classical mechanics and
by energy (a Hamiltonian). Similarly, Liouville equation is given by an
induced Liouville Poisson bracket. Reversible parts of more macroscopic
evolution equations, for example Boltzmann equation, NavierStokes equation,
equations for polymeric fluids or turbulent flows, are also generated by
Poisson brackets. The goal of this talk is to show how the more macroscopic
Poisson brackets are derived from the Liouville Poisson bracket in a
systematic way. 



15:40  Dr. Stanislav Parez (VSCHT, Praha):  Flow of granular materials: What can we say about landslides?  Abstract: Large landslides exhibit surprisingly long runout distances compared to a
rigid body sliding from the same slope, and the mechanism of this phenomena
has been studied for decades. Here we propose a scenario in which the
observed long runouts are explained via a granular flow, including its
spreading, but not including frictional weakening that has traditionally
been suggested to cause long runouts. Kinematics of the granular flow is
divided into center of mass motion and spreading due to flattening of the
flowing mass. We solve the center of mass motion analytically based on a
frictional law valid for granular flows, and find that the center of mass
runout is similar to that of a rigid body. Based on the shape of deposits
observed in experiments with collapsing granular columns and numerical
simulations of landslides, we estimate the effect of spreading and derive a
characteristic spreading length R_f ~ V^{1/3}. Spreading is shown to be an
important, often dominating, contribution
to the total runout distance. The combination of the predicted center of
mass runout and the spreading length gives the runout distance in a very
good match to natural landslides. 



15:40  (Lecole Polytechnique Montreal):  Modeling of flows of complex fluids by modeling their internal structure  Abstract: Fluids with internal structure that evolves in time on a scale that is comparable with the scale on which the macroscopic flow evolves exhibit
a complex flow behavior and are therefore called complex fluids. The
internal structure can be flowinduced (e.g. the structure emerging in
turbulent flows) or it can also be a mesoscopic or microscopic structure
of the fluids at rest (e.g. structure of macromolecules in polymeric fluids
or suspended particles in suspensions). I investigate consequences of the
requirement that the time evolution of complex fluids is compatible with
mechanics (i.e. it is a Hamiltonian time evolution) and compatible with
thermodynamics (it obeys the second law of thermodynamics). 

16:50  (Dept. of Civil Engr., IIT Madras):  Experimental Investigations on Asphalt Binders  What are the challenges?  Abstract: 90% of the highways and runways throughout the world use asphalt
binders for road construction. Such materials are byproducts of oil
refinery. They are a complicated mixture of thousands of hydrocarbons and
show diverse range of behavior as the temperature is varied. These material
exhibit transitory behavior from a viscoelastic fluid to viscoelastic solid
during service. In this talk, I present novel experimental techniques
designed to elicit the rich behavior of these materials. These include large
amplitude oscillatory shear, stress relaxation and creep and recovery. I
also show some interesting observations related to manifestation of normal
force and its relaxation. Such experimental data pose challenge while
modeling as most of the existing models cannot describe such behavior. 



15:40  (KNM MFF UK):  On a robust DG method for the solution of compressible Euler and NavierStokes equations  Abstract: The lecture presents numerical method for the numerical solution of
compressible flow, which is robust with respect to the Mach number and Reynolds numbers. It is based on the application of the discontinuous Galerkin method and allows the numerical simulation of compressible flow
with low Mach numbers up to an incompressible limit and high speed flow, in general in timedependent domains. The method is used for the solution of fluidstructure interaction problems. 



15:40  (Faculty of Math., Univ. Vienna):  Dynamic perfect plasticity as convex minimization  Abstract: We present a novel approximation of solutions to the equations of dynamic linearized perfect plasticity, based on a global variational
formulation of the problem by means of the WeightedInertia
DissipationEnergy (WIDE) approach. Solutions to the system of dynamic PrandtlReuss perfect plasticity are identified as limit of minimizers of parameterdependent energy functionals evaluated on trajectories (the WIDE functionals). Compactness is achieved by means of time
discretization, uniform energy estimate on minimizers of discretized
WIDEfunctionals, and passage to the limit in a parameterdependent
energy inequality. This is a joint work with Ulisse Stefanelli. 



15:40  (Math. Inst., Charles Uni.):  Minicourse of NonEquilibrium Thermodynamics. Part I.  Abstract: Nonequilibrium thermodynamics is the theory within which we should be able to derive evolution equations of any macroscopic or mesoscopic physical system. Although such a task is very important in modern physics and applied
mathematics, it seems to be still far away from current state of the art. The theory is still under construction.
The goal of this minicourse is to review some fundamental concepts and results of nonequilibrium thermodynamics, which we can use in mathematical
modeling, and to open discussion so that we find the most important open problems.
In the fist part, we will discuss the concepts of equilibrium,
nonequilibrium, levels of description, geometrization of thermodynamics, reversibility and irreversibility. 



15:40  (Math. Inst., Charles Uni.):  Minicourse of NonEquilibrium Thermodynamics. Part II.  Abstract: In the second part we will discuss mainly the reversible part of evolution equations. We will start with Liouville equation in the Hamiltonian form and we will pass to lower levels of description as kinetic theory or fluid mechanics. Perhaps we will have time to start talking about the irreversible evolution. 



15:40  (Univ. of California, Davis, & National Technical University of Athens):  Anisotropic Critical State Theory: Challenging a Paradigm in Granular Mechanics  Abstract: Consider the following thought experiment: load a granular specimen in triaxial compression till Critical State (CS) is reached, where shear deformation continues under fixed stress and zero volume change. At CS impose a rotation of stress Principal Axes (PA) keeping the stress principal values fixed. Will the sample continue being at CS or not?
The answer to this seemingly simple and of academic interest question can challenge the paradigm of Critical State Theory (CST) that defines failure and mechanical response of granular media in soil mechanics for more than half a century. The recently developed Anisotropic Critical State Theory (ACST) will be presented as a paradigm replacement for the classical CST. The main novel ingredient entering the new formulation is fabric, expressed in terms of a properly defined fabric tensor that evolves towards a unique norm CS value.
The presentation will be narrative providing stages of development and problems encountered and solved or still pending a solution. In the process constitutive models of soil plasticity will be presented within the framework of ACST, and the use of numerical/experimental techniques such as Discrete Element Method (DEM) and XRay tomography will be outlined. 



15:40  (Math. Inst., Charles Uni.):  Minicourse of NonEquilibrium Thermodynamics. Part III.  Abstract: We will derive the Poisson bracket of oneparticle kinetic theory, classical hydrodynamics and a theory of mixtures. We will then start discussing entropy. It will be introduced as the Shannon entropy at the Liouville level of description. Entropy of ideal gas at the levels of kinetic theory, classical hydrodynamics and thermodynamic equilibrium will be derived by the
principle of maximum entropy (MaxEnt). 



15:40  (Univ. Heidelberg, Inst. of Mathematik):  Selfsimilar lifting and persistent touchdown point solutions in the thinfilm equation  Abstract: In the talk I discuss the appearance of selfsimilar blowup solutions for thinfilm equations with different mobility
exponents. This is related to nonuniqueness phenomena for weak solution of the same equation. The proof is based on dynamical systems arguments. 



15:40  (Math. Inst., Charles Uni.):  Minicourse of NonEquilibrium Thermodynamics. Part IV.  Abstract: After having introduced energy, entropy and Poisson brackets on different levels of description, we will discuss various forms of irreversible evolution, in particular gradient dynamics (generated by dissipation potentials). We will show connection to the method of entropy production maximization and implicit constitutive relations. All so far brought up topics will be summarized in the formulation of the GENERIC framework, particular realizations and applications (solid mechanics, plasticity, chemical reactions, electrochemistry, diffusion, nonlocal phenomena) of which will dominate the following seminars. 



15:40  Mgr. Jan Stebel, PhD. (T. U. Liberec):  Shape optimization for Stokes problem with stickslip boundary conditions  Abstract: We consider the problem of finding an optimal shape of a domain
occupied by a viscous fluid. A part of the boundary represents a solid wall to which the fluid may or need not adhere depending on the magnitude of the shear stress. Such model describes e.g. hydrophobic
or microscale surfaces. The existence of an optimal shape is proved. Moreover, a regularizedpenalized problem is formulated and it is proved that its solutions converge to the solution of the original shape optimization problem. Finally we present an approximation of the problem and numerical results. This is a joint work with J. Haslinger and R.A.E. M{ a}kinen. 



15:40   ____________________________  

15:45  (MU AV CR + ZCU Plzen):  My life with J.N.  Abstract: The talk will deal with a rather private description of the long lasting collaboration with my teacher, colleague and friend, professor Jindrich Necas, and will mention some facts about common events (like seminars and workshops), about some of his students and about his results. 



15:40  (Universita di Roma `Tor Vergata):  A nonsmooth variant of the nonlinear diffusion equation  


Po semináři se podává opět čaj a káva.
Všichni zájemci jsou srdečně zváni. 
