arXiv subjects

Tomáš Roubíček

Publications and source records attributed to Tomáš Roubíček.

At least 37 records · Page 2Linked to original sources

arXiv2021

Optimal control of flows of viscoelastic semi-compressible fluids

Semilinear parabolic systems with bi-linear nonlinearities cover a lot of applications and their optimal control leads to relatively simple optimality conditions. An example is the incompressible Navier-Stokes system for homogeneous fluids, which is however here modified towards a physically reasonable model of slightly (so-called ``semi'') compressible liquids rather than fully compressible gases. An optimal control problem optimizing also pressure on the boundary is considered and, in the simple variant, analysed as far as uniqueness of the control-to-state mapping and 1st-order optimality conditions in the 2-dimensional case and outlined in a nonsimple variant for the 3-dimensional case. Some other bi-linear parabolic systems as Cahn-Hilliard diffusion or magneto-hydrodynamics can be treated analogously.

math.AP↗

arXiv2021

Thermodynamically consistent model for poroelastic rocks towards tectonic and volcanic processes and earthquakes

A general-purpose model combining concepts from rational continuum mechanics, fracture and damage mechanics, plasticity, and poromechanics is devised in Eulerian coordinates, involving objective time derivatives. The model complies with mass, momentum, and energy conservation as well as entropy inequality and objectivity. It is devised to cover many diverse phenomena, specifically rupture of existing lithospheric faults, tectonic earthquakes, generation and propagation of seismic waves, birth of new tectonic faults, or volcanic activity, aseismic creep, folding of rocks, aging of rocks, long-distance saturated water transport and flow in poroelastic rocks, melting of rocks and formation of magma chambers, or solidification of magma.

physics.geo-ph↗

arXiv2021

A convective model for poro-elastodynamics with damage and fluid flow towards Earth lithosphere modelling

Devised towards geophysical applications for various processes in the lithosphere or the crust, a model of poro-elastodynamics with inelastic strains and other internal variables like damage (aging) and porosity as well as with diffusion of water is formulated fully in the Eulerian setting. Concepts of gradient of the total strain rate as well as the additive splitting of the total strain rate are used while eliminating the displacement from the formulation. It relies on that the elastic strain is small while only the inelastic and the total strains can be large. The energetics behind this model is derived and used for analysis as far as the existence of global weak energy-conserving solutions concerns. By this way, the model in [V. Lyakhovsky et al., Pure Appl. Geophys., 171:3099-3123, 2014] and [V. Lyakhovsky et al., Izvestiya, Physics of the Solid Earth, 43:13-23, 2007] is completed to make it mechanically consistent and amenable for analysis.

math.AP↗

arXiv2021

Fine metrizable convex relaxations of parabolic optimal control problems

Nonconvex optimal-control problems governed by evolution problems in infinite-dimensional spaces (as e.g. parabolic boundary-value problems) needs a continuous (and possibly also smooth) extension on some (preferably convex) compactification, called relaxation, to guarantee existence of their solutions and to facilitate analysis by relatively conventional tools. When the control is valued in some subsets of Lebesgue spaces, the usual extensions are either too coarse (allowing in fact only very restricted nonlinearities) or too fine (being nonmetrizable). To overcome these drawbacks, a compromising convex compactification is here devised, combining classical techniques for Young measures with Choquet theory. This is applied to parabolic optimal control problems as far as existence and optimality conditions concerns.

math.OC↗

arXiv2020

A note about hardening-free viscoelastic models in Maxwellian-type rheologies at large strains

Maxwellian-type rheological models of inelastic effects of creep type at large strains are revisited in relation to inelastic-strain gradient theories. In particular, we observe that a dependence of the stored-energy density on inelastic-strain gradients may lead to spurious hardening effects, preventing the model from accommodating large inelastic slips. The main result of this paper is an alternative inelastic model of creep type, where higher-order energy-contribution is provided by the gradients of the elastic strain and of the plastic strain rate, thus preventing the onset of spurious hardening under large slips. The combination of Kelvin-Voigt damping and Maxwellian creep results in a Jeffreys-type rheological model. Existence of weak solutions is proved via a Faedo-Galerkin approximation.

math.AP↗

arXiv2020

Coupled time discretisation of dynamic damage models at small strains

The dynamic damage model in viscoelastic materials in Kelvin-Voigt rheology is discretised by a scheme which is coupled, suppresses spurious numerical attenuation during vibrations, and has a variational structure with a convex potential for small time-steps. In addition, this discretisation is numerically stable and convergent for the time step going to zero. When combined with the FEM spatial discretisation, it leads to an implementable scheme and to that iterative solvers (e.g. the Newton-Raphson) used for the nonlinear algebraic systems at each time level have guaranteed global convergence. Models which are computationally used in some engineering simulations in a non-reliable way are thus stabilized and theoretically justified in this viscoelastic rheology. In particular, this model and algorithm can be used in a reliable way for a dynamic fracture in the usual phase-field approximation.

math.NA↗

arXiv2020

Staggered explicit-implicit time-discretization for elastodynamics with dissipative internal variables

An extension of the two-step staggered time discretization of linear elastodynamics in stress-velocity form to systems involving internal variables subjected to a possibly non-linear dissipative evolution is proposed. The original scheme is thus enhanced by another step for the internal variables which, in general, is implicit, although even this step might be explicit if no spatial gradients of the internal variables are involved. Using an abstract Banach-space formulation, a-priori estimates and convergence are proved under a CFL condition. The developed three-step scheme finds applications in various problems of continuum mechanics at small strain. Here, we consider in particular plasticity, viscoelasticity (creep), diffusion in poroelastic media, and damage.

math.NA↗

arXiv2020

From quasi-incompressible to semi-compressible fluids

A new concept of semi-compressible fluids is introduced for slightly compressible visco-elastic fluids (typically rather liquids than gasses) where mass density variations are negligible in some sense, while being directly controlled by pressure which is very small in comparison with the elastic bulk modulus. The physically consistent fully Eulerian models with specific dispersion of pressure-wave speed are devised. This contrasts to the so-called quasi-incompressible fluids which are described not physically consistently and, in fact, only approximate ideally incompressible ones in the limit. After surveying and modifying models for the quasi-incompressible fluids, we eventually devise some fully convective models complying with energy conservation and capturing phenomena as pressure-wave propagation with wave-length (and possibly also pressure) dependent velocity.

math.AP↗

arXiv2019

A monolithic model for phase-field fracture and waves in solid-fluid media towards earthquakes

Coupling of rupture processes in solids with waves also propagating in fluids is a prominent phenomenon arising during tectonic earthquakes. It is executed here in a single `monolithic' model which can asymptotically capture both damageable solids (rocks) and (visco-)elastic fluids (outer core or oceans). Both ruptures on pre-existing lithospheric faults and a birth of new faults in compact rocks are covered by this model, together with emission and propagation of seismic waves, including, e.g., reflection of S-waves and refraction of P-waves on the solid-fluid interfaces. A robust, energy conserving, and convergent staggered FEM discretisation is devised. Using a rather simplified variant of such models for rupture, three computational experiments documenting the applicability of this approach are presented. Some extensions of the model towards more realistic geophysical modelling are outlined, too.

physics.geo-ph↗

arXiv2019

A general thermodynamical model for adhesive frictional contacts between viscoelastic or poro-viscoelastic bodies at small strains

A general model covering a large variety of the so-called adhesive or cohesive, possibly also frictional, contact interfaces between visco-elastic bodies with inertia considered in a thermodynamical context is presented. A semi-implicit time discretisation which conserves energy, is numerically stable and convergent, and which advantageously decouples the system is devised. An extension to porous media with adhesive contact influenced by a diffusant concentration is devised, too.

math.AP↗

arXiv2019

Models of dynamic damage and phase-field fracture, and their various time discretisations

Several variants of models of damage in viscoelastic continua under small strains in the Kelvin-Voigt rheology are presented and analyzed by using the Galerkin method. The particular case, known as a phase-field fracture approximation of cracks, is discussed in detail. All these models are dynamic (i.e. involve inertia to model vibrations or waves possibly emitted during fast damage/fracture or induced by fast varying forcing) and consider viscosity which is also damageable. Then various options for time discretisation are devised. Eventually, extensions to more complex rheologies or a modification for large strains are briefly exposed, too.

math.AP↗

arXiv2019

Dynamic perfect plasticity and damage in viscoelastic solids

In this paper we analyze an isothermal and isotropic model for viscoelastic media combining linearized perfect plasticity (allowing for concentration of plastic strain and development of shear bands) and damage effects in a dynamic setting. The interplay between the viscoelastic rheology with inertia, elasto-plasticity, and unidirectional rate-dependent incomplete damage affecting both the elastic and viscous response, as well as the plastic yield stress, is rigorously characterized by showing existence of weak solutions to the constitutive and balance equations of the model. The analysis relies on the notions of plastic-strain measures and bounded-deformation displacements, on sophisticated time-regularity estimates to establish a duality between acceleration and velocity of the elastic displacement, on the theory of rate-independent processes for the energy conservation in the dynamical-plastic part, and on the proof of the strong convergence of the elastic strains. Existence of a suitably defined weak solutions is proved rather constructively by using a staggered two-step time discretization scheme.

math.AP↗

arXiv2019

Cahn-Hilliard equation with capillarity in actual deforming configurations

The diffusion driven by the gradient of the chemical potential (by the Fick/Darcy law) in deforming continua at large strains is formulated in the reference configuration with both the Fick/Darcy law and the capillarity gradient term considered at the actual configurations deforming in time. Static situations are analysed by the direct method. Evolution (dynamical) problems are treated by the Galerkin method, the actual capillarity giving rise to various new terms as e.g. the Korteweg-like stress and analytical difficulties related to them. Some other models (namely plasticity at small elastic strains or damage) with gradients at actual configuration allow for similar models and analysis.

math.AP↗

arXiv2019

Thermoviscoelasticity in Kelvin-Voigt rheology at large strains

The frame-indifferent thermodynamically-consistent model of thermoviscoelasticity at large strain is formulated in the reference configuration with using the concept of the second-grade nonsimple materials. We focus on physically correct viscous stresses that are frame indifferent under time-dependent rotations. Also elastic stresses are frame indifferent under rotations and respect positivity of the determinant of the deformation gradient. The heat transfer is governed by the Fourier law in the actual deformed configuration, which leads to a nontrivial description when pulled back into the reference configuration. Existence of weak solutions in the quasistatic setting, i.e.\ inertial forces are ignored, is shown by time discretization.

math.AP↗

arXiv2017

Seismic waves and earthquakes in a global monolithic model

The philosophy that a single "monolithic" model can "asymptotically" replace and couple in a simple elegant way several specialized models relevant on various Earth layers is presented and, in special situations, also rigorously justified. In particular, global seismicity and tectonics is coupled to capture e.g.\ (here by a simplified model) ruptures of lithospheric faults generating seismic waves which then propagate through the solid-like mantle and inner core both as shear (S) or pressure (P) waves, while S-waves are suppressed in the fluidic outer core and also in the oceans. The "monolithic-type" models have the capacity to describe all the mentioned features globally in a unified way together with corresponding interfacial conditions implicitly involved, only when scaling its parameters appropriately in different Earth's layers. Coupling of seismic waves with seismic sources due to tectonic events is thus an automatic side effect. The global ansatz is here based, rather for an illustration, only on a relatively simple Jeffreys' viscoelastic damageable material at small strains whose various scaling (limits) can lead to Boger's viscoelastic fluid or even to purely elastic (inviscid) fluid. Self-induced gravity field, Coriolis, centrifugal, and tidal forces are counted in our global model, as well. The rigorous mathematical analysis as far as the existence of solutions, convergence of the mentioned scalings, and energy conservation is briefly presented.

math.AP↗

arXiv2017

Variational methods for steady-state Darcy/Fick flow in swollen and poroelastic solids

Existence of steady states in elastic media at small strains with diffusion of a solvent or fluid due to Fick's or Darcy's laws is proved by combining usage of variational methods inspired from static situations with Schauder's fixed-point arguments. In the plain variant, the problem consists in the force equilibrium coupled with the continuity equation, and the underlying operator is non-potential and non-pseudomonotone so that conventional methods are not applicable. In advanced variants, electrically-charged multi-component flows through an electrically charged elastic solid are treated, employing critical points of the saddle-point type. Eventually, anisothermal variants involving heat-transfer equation are treated, too.

math.AP↗

arXiv2016

Quasistatic normal-compliance contact problem of visco-elastic bodies with Coulomb friction implemented by QP and SGBEM

The quasistatic normal-compliance contact problem of isotropic homogeneous linear visco-elastic bodies with Coulomb friction at small strains in Kelvin-Voigt rheology is considered. The discretization is made by a semi-implicit formula in time and the Symmetric Galerkin Boundary Element Method (SGBEM) in space, assuming that the ratio of the viscosity and elasticity moduli is a given relaxation-time coefficient. The obtained recursive minimization problem, formulated only on the contact boundary, has a nonsmooth cost function. If the normal compliance responds linearly and the 2D problems are considered, then the cost function is piecewise-quadratic, which after a certain transformation gets the quadratic programming (QP) structure. However, it would lead to second-order cone programming in 3D problems. Finally, several computational tests are presented and analysed, with additional discussion on numerical stability and convergence of the involved approximated Poincaré-Steklov operators.

math.NA↗

arXiv2015

Stress-driven solution to rate-independent elasto-plasticity with damage at small strains and its computer implementation

The quasistatic rate-independent damage combined with linearized plasticity with hardening at small strains is investigated. The fractional-step time discretisation is devised with the purpose to obtain a numerically efficient scheme converging possibly to a physically relevant stress-driven solutions, which however is to be verified a-posteriori by using a suitable integrated variant of the maximum-dissipation principle. Gradient theories both for damage and for plasticity are considered to make the scheme numerically stable with guaranteed convergence within the class of weak solutions. After finite-element approximation, this scheme is computationally implemented and illustrative 2-dimensional simulations are performed.

math.NA↗