NEČAS SEMINAR ON CONTINUUM MECHANICS
organized by the Mathematical Institute of the Charles University
each Monday at 15:40
in the MFF UK building,
Sokolovská 83, lecture room K1, 2^{nd}
floor,
proceeds with the following presentations:

15:40  prof. Ing. Tomas Roubicek, DrSc. (Charles Univ. & Acad. Sci.):  Adhesive contact of viscoelastic bodies and defect measures arising by vanishing viscosity  Abstract: An adhesive unilateral contact of elastic bodies with a small
viscosity in the linear KelvinVoigt rheology at small strains
is scrutinized. The flowrule for debonding the adhesive is
considered rateindependent and unidirectional. The asymptotics
for the viscosity or for external loading speed approaching zero
is proved in some special cases, in particular when inertia is
neglected or when delamination is in Mode II (pure shear).
The solutions thus obtained involve certain defectlike measures
recording in some sense natural additional energy dissipated in
the bulk due to (vanishing) viscosity. Very typically (and perhaps surprisingly), these measures show to be nontrivial. Reflecting also the conventional engineering concept, the delamination is thus
driven rather by stress than energy. An explicit example
leading to a nontrivial defect measure is given. Computation
simulations based on algorithm based on a semiimplicit
discretisation in time (launched by C.G.Panagiotopoulos)
evaluating numerically such measures in 2dimensional situations will be presented, too. 



15:40  RNDr. Petr Salac, CSc. (TU Liberec):  Optimalizace chlazeni razniku pri lisovani skla  Abstract: Prednaska je venovana vyuziti tvarove optimalizace pri reseni technologickych problemu pri lisovani velke sklenene produkce. Pro dosazeni vysoke kvality povrchu vylisku je potreba, aby v okamziku separace lisovaciho nastroje a vylisku byla povrchova teplota nastroje konstantni predem zvolene hodnoty. Regulace teploty razniku je provadena vnitrnim chlazenim proudici vodou v dutine razniku. Nejprve je provadena optimalizace zmenou tvaru chladici dutiny a nasledne regulaci rychlosti proudici vody uzitim tzv. regulacniho proudoveho telesa. Za stavovou ulohu bereme variacni formulaci rovnice energie s uvazovanym potencialnim proudenim chladici vody. Ucelovy funkcional je definovan ve tvaru druhe mocniny Lr2 normy z rozdilu mezi predepsanou konstantou a teplotou na vnejsim povrchu razniku. Je dokazana existence a jednoznacnost reseni stavove ulohy a existence reseni obou uloh optimalniho navrhu. Rovnez budou prezentovany pomerne uspokojive vysledky laboratornich experimentu. 



15:40  Lu Yong (Universite Paris Diderot  Paris 7):  Highfrequency limit of the MaxwellLandauLifshitz system in the diffractive optics regime: the application of normal form method  Abstract: We study semilinear MaxwellLandauLifshitz systems in one
space dimension. For highly oscillatory and prepared initial data, we construct WKB approximate solutions over long times O(1/epsilon).
The leading terms of the WKB solutions solve cubic Schrodinger
equations. We show that the nonlinear normal form method of Joly,
Metivier and Rauch applies to this context. This implies that the
Schrodinger approximation stays close to the exact solution of
MaxwellLandauLifshitz over diffractive times. 



15:40  (Weierstrass Institut fur angewandte Analysis und Stochastik, Berlin):  Mathematical and numerical modeling of coupled processes in electrochemical devices  Abstract: Electrochemical devices have manifold applications in technical systems. In particular, the growing share of fluctuating renewable sources in the supply of energy calls for the ability to store electrical energy in significantly larger quantities as before. In this context, electrochemical storage methods based on secondary
batteries, fuel cells, redox flow cells and electrolysis cells are in the focus of significant new research efforts, which include modeling from the molecular scale up to the system scale.
At the macroscale of a single electrochemical cell, coupled nonlinear systems of PDEs describe tightly coupled flow, transport, reactions and electric field. In the talk, we review elements of mathematical models of this kind, and discuss mathematical and numerical challenges connected with their investigation.
In this context, we discuss the advantages and challenges of the
implicit Euler, Voronoi box based finite volume method which allows to
derive a framework for the numerical implementation of mathematical
models based on reactiondiffusionconvection systems. Particular
advantages of the method are unconditional stability, positivity,
discrete maximum principle, local and global mass conservation, and
efficient ways to solve stationary and time dependent cases. It
relies on the ability to create Delaunay meshes conforming to interior
and exterior boundaries. We mention challenges in connection with the resolution of boundary layers and handling of anisotropies.
We present results of cooperation with electrochemical groups concerning the modeling of direct methanol fuel cells and thin layer flow cells. 



15:40  Petr Puncochar (AON Benfield):  Mathematical Modeling inside Catastrophe Models  Abstract: Serious number of fatalities and financial losses caused by
natural disasters in recent decades affected significantly the
insurance/reinsurance industry and led to rapid development of models that quantify financial loss occurring on insured portfolios. Impact Forecasting is Aon Benfield (market leading reinsurance broker) catastrophe (CAT) model development center whose primary aim is to develop these tools. Apart of exposure and vulnerability component, the very complex CAT models contain
hazard part that covers the modeling of extreme behavior of natural
phenomena, such as wind storm, flood or earthquake. The flood model
development with a particular focus on hydrodynamic flood extent modeling using numerical models will be described in detail as well as probabilistic modeling that helps to determine the correct probability of various synthetic events. Finally the loss estimation and other CAT model components are shown to see all components in context. 



15:40  (Centrum materialoveho vyzkumu, Fak. chemicka, VUT Brno):  Thermodynamics and kinetics of chemical reactions  Abstract: In chemistry, thermodynamics and kinetics are traditionally considered to be independent disciplines. Thermodynamics is said to give criteria on direction or feasibility of a reaction and to give no information on its rate; this is the domain of chemical kinetics. However, they meet in
equilibrium of chemical reactions, at least. Different thermodynamic and
kinetic descriptions of common equilibrium have been combined to find
some ``thermodynamic restrictions on kinetics``. Validity and correctness of
these results, which are in fact derived on the basis of equilibrium
thermodynamics, are limited to ideal systems. Nonequilibrium
thermodynamics points to closer contacts between chemical thermodynamics
and kinetics and even enables to formulate a framework for design of
chemical rate equations and rate networks which are thus consistent with
thermodynamics. Then also (thermodynamic) restrictions on rate constants
(coefficients, better speaking), well known to chemists, follow. Further,
the role of ``dependent reactions``, which are also sometimes looked for and
discussed in chemical kinetics, in description of reaction rates is
naturally clarified. The thermodynamic methodology will be illustrated on
simple example of reacting mixture of three isomers. 



15:40  Prof.Dr. Anja Schloemerkemper (Univ. Wuerzburg):  NonLaminate Microstructures in Two Kinds of MonoclinicI Martensite  Abstract: The most common shape memory alloys are monoclinicI martensite. We study their zero energy states and have two surprising results: First, there is a fivedimensional continuum in which the energy minimising microstructures are T3s, i.e. in finiterank laminates. To our knowledge, this is the first real material in which T3s occur. We discuss some of the consequences of this discovery. Second, there are in fact two types of monoclinicI martensite, which di ffer by their convex polytope structure but not by
their symmetry properties. It happens that all known materials belong to one of the two types. We explore whether materials belonging to the other type would have superior properties since they have different zeroenergy states. Our analysis uses algebraic methods, in particular the theory of convex polytopes. This is joint work with I.V. Chenchiah. 



15:40  (New technologies research center, Univ. of West Bohemia):  Thermal Analysis of Brakes and Frictionally Excited Thermoelastic Instability  Abstract: V prispevku bude ukazan inzenyrsky pristup k problematice brzd z hlediska matematickeho modelovani a vypoctu. Pozornost bude venovana nekterym aktualnim problemum ohrevu brzd, predevsim vsak jejich nachylnosti ke vzniku termoelasticke nestability vyvolane trenim. 



15:40  (University of Warsaw, Inst. of Appl. Math. & Mech. + Charles Univ., Math. Inst.):  An introduction to finite strain gradient crystal plasticity  Abstract: The aim is to briefly present finite strain gradient crystal plasticity. We derive the kinematic equations form the Kroner
decomposition and give basic thermodynamic description of the evolution of the Kirchhoff stress. In the second part of the talk we present an approach such that material is treated as a highly viscous, incompressible, anisotropic fluid which flows through an adjustable crystal lattice. Moreover implicit constitutive relation between slip rate and resolved shear stress is covered.
As a motivation of our numerical experiments we consider the Equal Channel Angular Extrusion. The experiment, in which a
specimen under presence of a high pressure together with large shear strains, exhibits significant changes in the internal
structure. The results of simulations by mixed finite element, are presented. This is joint work with Jan Kratochvil, Josef Malek, Martin Kruzik and Jaroslav Hron. 



15:40  C.Bertoglio, R.Chabiniok, J.Tintera (Technical University of Munich, Germany; King s College London, UK; Inst. Clinical & Experimental Medicine, CR):  Biophysical modeling of cardiovascular system in clinical setup  Abstract: Biophysical modeling coupled with clinical data has the potential to extract some additional metrics which is not directly visible in the data and could be used for more accurate diagnosis and understanding of disease progress. This datamodel coupling relies on a good balance between the types of data, model complexity and data assimilation techniques.
During this seminar, we are going to address all these three aspects and we will demonstrate them with an example of modeling of
various parts of the cardiovascular system. 



15:40   On Thursday May 2, 2pm5pm, at Refectory at Mala Strana, there will be the Necas Center Inauguration and Colloquium talk of Endre Suli (University of Oxford).  Abstract: The event organized by the dean of the faculty and vicedean for mathematics will take place at Refectory at Mala Strana faculty building. 



15:40  Petr Paus (Katedra aplikovane matematiky FJFI CVUT):  Numerical analysis of the critical crossslip annihilation distance and the cyclic saturation stress in copper and nickel single crystals  Abstract: An interpretation of the experimentally determined critical distance of
the screw dislocation annihilation in persistent slip bands [1] is still
an open question. We attempt to analyze this problem using discrete
dislocation dynamics simulations. Glide dislocations are represented by
parametrically described curves. The model is based on the numerical
solution of the dislocation motion law belonging to the class of
curvature driven curve dynamics. We focus on the simulation of the
crossslip of two dislocation curves of the opposite signs where each
evolves in a different primary slip plane in a channel of a persistent
slip band. The dislocations move under their mutual interaction, the
line tension and an applied stress forming a screw dislocation dipole. A
crossslip leads to annihilation of the dipolar parts. In the changed
topology each dislocation evolves in two slip planes and the plane where
crossslip occurred. The goal of our work is to determine the conditions
under which the crossslip occurs and the saturation stress required.
The simulation of the dislocation evolution and merging is performed by
improved parametric approach and numerical stability is enhanced by the
tangential redistribution of the discretization points. The critical
annihilation distance and the saturation stress determined by the
simulations are close to the experimental values. 



15:40  Prof. Eduardo Casas (Universidad de Cantabria, Santander, Spain):  Elliptic Control Problems in Measure Spaces with Sparse Solutions  Abstract: In the control of distributed parameter systems, those formulated by partial
differential equations, usually we cannot put control devices at every point of
the domain. Actually, we are allowed to use small regions to put the controllers.
Then, the big issue is which region is the most convenient to localize them. Of
course, we have to determine the power of the controllers as well. These controls
are called sparse because they are not zero only in a small region of the domain.
In the last few years, some researchers have focused their investigation in this
direction. First, it was observed that the use of the L1 norm of the control in the
cost functional leads to the sparsity of the solution. Of course, this introduces
some mathematical difficulties in the problem due to the lack of differentiability
of this functional. However, despite this difficulty, a lot of progress has been
done and the numerical computations show the interest and applicability of this
Taking a further step in this direction, we find that many times it is even
desirable to put the controllers only in a zero Lebesgue measure set (along a line
or on a surface). These controllers cannot be identified with functions, they are
measures. This is the starting point of a new type of control problems where
the controls are Borel measures. Adding the norm of the measure to the cost
functional, we obtain optimal controls having the desired sparsity property.
This talk deals with the analysis of optimal control problems in measure
spaces, which are known to promote sparse solutions. The semilinear elliptic
case is considered. First and second order optimality conditions are derived and
some of the structural properties of their solutions, in particular sparsity, are
discussed. Some numerical results are also presented. 



15:40  Prof.RNDr.Josef Malek, DSc and Mgr. Josef Zabensky (MUUK):  On Darcy, Forchheimer, and Brinkman models for flows through porous media and their generalizations  Abstract: We introduce several models connected with the classical models of
Darcy, Forchheimer, and Brinkman and
present two results concerning the existence of weak solutions and their
properties. 



15:40  (Geological Survey of Israel, Jerusalem):  Continuum Damage Mechanics for Brittle Rocks and Geophysical Applications  Abstract: We present a basic concept of the continuum damage rheology model based on thermodynamic principles and fundamental observations of rock deformation. Fundamental nonlinear aspects of rock deformation, such as microcrack and flaw nucleation, development of process zones at rupture tips, and branching from the main rupture plane are of crucial importance for evolutionary selforganization of faults at various spatiotemporal domains. 3D numerical simulations reproduce the main features of a quasistatic fault evolution at various scales from laboratory sample testing to processes associated with hydraulic borehole stimulation and regional lithospheric models. A recently developed theoretical model for continuum damagebreakage mechanics allows detailed description of the dynamic rupture process. This new approach combines and extends previous results of a continuum damage model that accounts for distributed cracks using a scalar damage parameter, and a continuum breakage model that measures the relative distance of a given grain size distribution of a granular phase to the ultimate distribution with a breakage parameter. Several features of the model including development of wide damage zones and its localization into a narrow slip zone with transition from slow to rapid dynamic slip are illustrated using numerical simulations. 



10:00  PLACE EXCEPTIONALLY AT MATH. INST. Zitna 25, P2. ____ Dr. Radek Erban (University of Oxford):  Hybrid Modelling of Reaction, Diffusion and Taxis Processes in Biology  Abstract: I will discuss methods for spatiotemporal modelling in cellular and molecular biology. Three classes of models will be considered:
(i) microscopic (molecularbased, individualbased) models which are
based on the simulation of trajectories of individual molecules
and their localized interactions (for example, reactions);
(ii) mesoscopic (latticebased) models which divide the computational
domain into a finite number of compartments and simulate the time
evolution of the numbers of molecules in each compartment; and
(iii) macroscopic (deterministic) models which are written in terms of
reactiondiffusionadvection partial differential equations (PDEs)
for spatially varying concentrations.
In the first part of my talk, I will discuss connections between the
modelling frameworks (i)(iii). I will consider chemical reactions both at
a surface and in the bulk. In the second part of my talk, I will present
hybrid (multiscale) algorithms which use models with a different level of
detail in different parts of the computational domain. The main goal of
this multiscale methodology is to use a detailed modelling approach in
localized regions of particular interest (in which accuracy and
microscopic detail is important) and a less detailed model in other
regions in which accuracy may be traded for simulation efficiency. I will
also discuss hybrid modelling of chemotaxis where an individualbased
model of cells is coupled with PDEs for extracellular chemical signals. 



15:40  prof. RNDr. Ales Pultr, DrSc. (KAM MFF UK):  Bezbodovy pristup k pojmu prostoru  



15:40  (Necas Center for Mathematical Modeling and Mathematical Institute of the Charles University):  On implicitly constituted fluids and implicitly constituted interactions of a fluid with a solid boundary  Abstract: In the analysis of weak solutions relevant to evolutionary
flows of incompressible fluids with nonconstant viscosity or with
nonlinear constitutive equation, it is in general an open question whether
a globally integrable pressure exists if the flows are subject to noslip
boundary conditions. Here we overcome this deficiency by considering
threshold boundary conditions stating that the fluid adheres to the
boundary until certain critical value for the wall shear stress is reached.
Once the wall shear stress exceeds this critical value, the fluid slips.
The main ingredient in our approach is to look at this type of
activated, stickslip, boundary condition as an implicit constitutive
equation on the boundary.
We present key steps in the proof of the longtime and largedata
existence of weak solutions, with integrable pressure,
to unsteady internal threedimensional flows of the Bingham (and
NavierStokes) fluids subject to such threshold slip boundary conditions.
This is a joint work with Miroslav Bulicek. We also mention
several essential generalizations.
This particular result will be put in a general scope linking modeling
in continuum physics with computer simulations via the PDE analysis of
initial and boundary value problems and numerical analysis of
finitedimensional solution algorithms. 



15:40  Dr. Giordano Tierra (MFF UK):  Numerical approximations for the CahnHilliard equation and some related models  Abstract: The diffuse interface theory, which was originally developed as methodology for modeling and approximating solidliquid phase transitions in which the effects of surface tension and nonequilibrium thermodynamic behavior may be
important at the surface. The diffuse interface model describes the interface by a mixing energy represented as a layer
of small thickness. This idea can be traced to van der Waals , and is the foundation for the phasefield theory for
phase transition and critical phenomena. Thus, the structure of the interface is determined by molecular forces; the
tendencies for mixing and demixing are balanced through the nonlocal mixing energy. The method uses an auxiliary
function (socalled phasefield function) to localize the phases, assuming distinct values in the bulk phases (for instance
1 in a phase and 1 in the other one) away from the interfacial regions over which the phase function varies smoothly.
The CahnHilliard model describes the complicated phase separation and coarsening phenomena in the mixture of different fluids, solid or gas where only two different concentration
phases can exist stably.
During the seminar, different numerical schemes to approximate the CahnHilliard model will be presented, showing
the advantage and disadvantages of each scheme. In particular, the focus will be on the study of the constraints on the
physical and discrete parameters that can appear to assure the energystability, unique solvability and, in the case of
nonlinear schemes, the convergence of Newton s method to the nonlinear schemes.
Moreover, an adaptive time stepping algorithm will be presented. This algorithm is based on the numerical
dissipation introduced in the discrete energy law in each time step. The behavior of the schemes and the effectiveness
of the adapttime algorithm will be compared through several computational experiments.
In the second part of the seminar, several physically motivated models such as liquid crystals,
vesicle membranes, twophase fluids (with same and different densities) and mechanical behavior of biofilms will be introduced. The key point is to try to extend the ideas presented for the CahnHilliard equation to preserve the properties of the original
models while the numerical schemes are efficient in time. Finally, some numerical simulations for these models will be
presented to show the effectiveness of the proposed numerical schemes. 



15:40  (Institut fur Angewandte Analysis und Numerische Simulation, Universitat Stuttgart):  Coupling porous medium and free flow systems: Mathematical modeling.  Abstract: Fluid flows and species transport in coupled porous medium and free flow systems appear in a wide spectrum of environmental settings (evaporation from the soil influenced by the wind, overland
flow interactions with groundwater aquifers, groundwater pollution), industrial applications (filtration, insulation, drying, fuel cells) and biological processes (flows in blood vessels and biological tissues, transport of drugs and nutrients).
The governing equations of these two systems have been widely investigated (the free flow is usually
modeled by the Stokes or NavierStokes equations and the porous medium is typically assumed to satisfy
a variation of the Darcy law), but a challenge arises in describing the transition between the free flow and
porous medium flow regimes. Modeling the coupling of the free flow and porous medium systems can
be done through the sharp interface approach by imposing the appropriate set of interface conditions
at the boundary between the flow domains or by considering a transition zone between two flow regions
and developing a transition region model.
The lecture is focused on modeling strategies for the coupled systems. We consider mathematical
models for the free flow and porous medium domains and different sets of interface conditions at the fluidporous interface. We start with the singlefluidphase stationary problem, consider different coupling techniques and gradually complicate the problem by considering nonstationary flows, adding a
second fluid phase to the porous medium and taking into account species and energy transport. 



15:40  .  Seminar se nekona  statni svatek  



15:40  (Institute of Mathematics AS CR):  Maximal dissipation and well posedness for models of inviscid fluids  Abstract: We discuss the principle of maximal dissipation introduced in 1974 by C.M.Dafermos in the light of recent results obtained by the method of convex integration for the compressible Euler system. We show examples of nonuniqueness in the class of admissible entropy solutions and inspecting the method of construction we infer that all violate the principle of maximal dissipation. We therefore conjecture that the latter should be retained as a criterion of well posedness, at least for certain problems in fluid mechanics. 



15:40  (MU UK):  On the boundary regularity of the weak solutions to the magneto hydrodynamics system  Abstract: Magneto hydrodynamics system describe the motion of a charged fluid in the magnetic field. We investigate a sufficient conditions of local regularity of suitable weak solutions to this system near the smooth boundary. Our goal is to generalize the known CaffarelliKohnNirenberg theorem. 



15:40  (Dept. de Ing. Matem., Universidad de Concepcion, Chile):  ATTENTION: Location, date and time change! 19.11. at 14:15 at the Institute of Computer Science A finite element method for a threedimensional fluidsolid interaction problem  Abstract: We introduce and analyze a new finite element method for a
threedimensional fluidsolid interaction problem. The media are
governed by the acoustic and elastodynamic equations in timeharmonic regime, and the transmission conditions are given by the equilibrium of forces and the equality of the corresponding normal displacements. We employ a dualmixed variational formulation in the solid, in which the Cauchy stress tensor and the rotation are the only unknowns, and maintain
the usual primal formulation in the fluid. The main novelty of our
method, with respect to previous approaches for a 2D version of this problem, consists of the introduction of the first transmission condition as part of the definition of the space to which the stress of the solid and the pressure of the fluid belong. As a consequence, and since the second transmission condition becomes natural, no Lagrange multipliers on the coupling boundary are needed, which certainly leads to a much simpler variational formulation. We show that a suitable decomposition of the space of stresses and pressures allows the application of the BabuskaBrezzi theory and the Fredholm alternative for concluding the solvability of the whole coupled problem. The unknowns of the fluid and the solid are then approximated, respectively, by Lagrange and ArnoldFalkWinther finite element subspaces of order 1, which yields a conforming Galerkin scheme. In this way, the
stability and convergence of the discrete method relies on a stable
decomposition of the finite element space used to approximate the stress and the pressure variables, and also on a classical result on projection methods for Fredholm operators of index zero.
The lecture of Professor Gabriel Gatica, originally scheduled on November 18, 2013 is due to speakers late arrival to Prague rescheduled to November 19 at 14:15. The lecture will be organized as a joint event of the Seminar of Computational Methods
and the Necas Seminar on Continuum Mechanics.




15:40   The seminar is cancelled because of the workshop of the MORE project at Chateau Liblice.  



15:40  (MU AV CR):  Efficient solution methods for modelling of flows around insect wings.  Abstract: We deal with a pressurecorrection method for solving unsteady
incompressible flows. In this approach, five subsequent equations are solved within each time step. These correspond to three scalar
convectiondiffusion problems, one for each component of velocity, a
pure Neumann problem for the correction of pressure, and a problem of the L2 projection for pressure update. We present a comparative study of several parallel preconditioners and Krylov subspace methods from the PETSc library and investigate their suitability for solving the arising linear systems after discretizing by the finite element method. The target application are largescale simulations of flows around wings of insects. This is a joint work with Fehmi Cirak. 



15:40  (Institut f. Angew. Mathematik, Univ. Heidelberg):  A Fully Eulerian Formulation for FluidStructure Interactions  Abstract: This presentation is about a monolithic variational formulation for
fluidstructure interaction problems. Instead of the standard approach  where the fluidproblem is transformed into an artificial coordinate system  the Arbitrary Lagrangian Eulerian coordinates  we state both subproblems, fluid and solid in the Eulerian coordinate system. By this approach, we avoid the introduction of artificial coordinates, that usually
give rise to problems, when very large deformation, motion or contact of the structure takes place.
As a Eulerian approach our method involves some specific difficulties: First, it is a fixedmesh interfacecapturing scheme, where the interface between solid and fluid is freely moving and must be captured by the discretization. Here, we introduce the Initial Point Set technique as an alternative to LevelSets. Second, as the type of equation changes within
the domain from fluid to solid, the Eulerian approach is an interface problem with a solution, that lacks regularity across the interface. To deal with this weak discontinuities, we introduce a locally modified finite element approach.
Numerical results will demonstrate the potential of the new Fully Eulerian approach to deal with problems involving large deformation, motion and contact. 



15:40  (Department of Earth Sciences, ETH Zurich Institute fuer Geophysik):  The dynamics and evolution of terrestrial planets in our solar system and beyond  Abstract: Convection of the rocky mantle is the key process that drives the interior evolution and surface tectonics of terrestrial planets Earth, Venus, Mars and Mercury, yet these planets are quite different. Mantle convection in Earth causes plate tectonics, controls heat loss from the metallic core (which generates the magnetic field) and drives longterm volatile cycling between the atmosphere/ocean and interior. Plate tectonics is thus a key process, yet exactly how plate tectonics arises is still quite uncertain; other terrestrial planets like Venus and Mars instead have a stagnant lithosphere like a single plate covering the entire planet. Here, numerical modelling of the interior dynamics and thermochemical evolution of Venus, Mars and Earth is presented, to develop a unified framework that can be applied to predicting the dynamics and possible habitability of terrestrial planets around other stars (superEarths as well as smaller planets), of which astronomers have so far found ~10s. 


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