Tomáš Roubíček - Profile on Academia.edu (original) (raw)

Papers by Tomáš Roubíček

Applications of Mathematics

Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents ... more Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This document has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://dml.cz 35(1990) APLIKACE MATEMATIKY No. 5, 361-372

Mathematical Models and Methods in Applied Sciences, 2006

Damage of an elastic body undergoing large deformations by a "hard-device" loading poss... more Damage of an elastic body undergoing large deformations by a "hard-device" loading possibly combined with an impact (modeled by a unilateral frictionless contact) of another, ideally rigid body is formulated as an activated, rate-independent process. The damage is assumed to absorb a specific and prescribed amount of energy. A solution is defined by energetic principles of stability and balance of stored and dissipated energies with the work of external loading, realized here through displacement on a part of the boundary. Rigorous analysis by time discretization is performed.

International Journal of Plasticity, 2006

In this article, we develop a continuum mechanical model to simulate deformation and phase transf... more In this article, we develop a continuum mechanical model to simulate deformation and phase transformation processes in shape memory alloys. The model is based on a detailed description of the stored energy. Furthermore, the energy dissipation due to phase transformations is taken into account via the maximum-dissipation principle. The results from the 3D numerical simulations of stress induced transformations from the cubic to the tetragonal phase and martensitic variant reorientations in NiMnGa are compared with laboratory experiments on NiMnGa [001]-oriented single crystals.

Phase transformation in shape-memory alloys is known to cause electric resistivity variation that... more Phase transformation in shape-memory alloys is known to cause electric resistivity variation that, under electric current, may conversely influence Joule heat production and thus eventually the martensitic transformation itself. A ther-modynamically consistent general continuum-mechanical model at large strains is presented. In special cases, a proof of the existence of a weak solution is outlined, using a semidiscretization in time. Mathematics Subject Classification (2000). 35K55 · 74A15 · 74N10 · 80A17.

Existence of weak solutions is proved for a system of nonlinear parabolic equations/inequalities ... more Existence of weak solutions is proved for a system of nonlinear parabolic equations/inequalities describing evolution of magnetization, temperature, magnetic ¯eld, and electric ¯eld in electrically-conductive unsaturated ferromagnets. The system is derived from a recently-proposed thermodynamically-consistent continuum theory for the ferro/paramagnetic transition. Besides the standard viscous-like damping, dissipation due to eddy currents and domain-wall pinning is considered.

We derive a thermodynamically consistent general continuum-mechanical model describing mutually c... more We derive a thermodynamically consistent general continuum-mechanical model describing mutually coupled martensitic and ferro/paramagnetic phase transformations in electrically-conductive magnetostrictive materials such as NiMnGa. We use small-strain and eddy-current approximations, yet large velocities and electric current injected through the boundary are allowed. Fully nonlinear coupling of magneto-mechanical and thermal effects is considered. The existence of energy-preserving weak solutions is proved by showing convergence of time-discrete approximations constructed by a carefully designed semi-implicit regular-ized scheme.

A thermodynamically consistent mathematical model for hydrogen adsorption in metal hydrides is pr... more A thermodynamically consistent mathematical model for hydrogen adsorption in metal hydrides is proposed. Beside hydrogen diffusion, the model accounts for phase transformation accompanied by hysteresis, swelling, temperature and heat transfer, strain, and stress. We prove existence of solutions of the ensuing system of partial differential equations by a carefully-designed, semi-implicit approximation scheme. A generalization for a drift-diffusion of multi-component ionized " gas " is outlined, too.

We propose a thermodynamically consistent general-purpose model describing diffusion of a solute ... more We propose a thermodynamically consistent general-purpose model describing diffusion of a solute or a fluid in a solid undergoing possible phase transformations and damage, beside possible visco-inelastic processes. Also heat genera-tion/consumption/transfer is considered. Damage is modelled as rate-independent. The applications include metal-hydrogen systems with metal/hydride phase transformation, poroelastic rocks, structural and ferro/para-magnetic phase transformation , water and heat transport in concrete, and if diffusion is neglected, plasticity with damage and viscoelasticity, etc. For the ensuing system of partial differential equations and inclusions, we prove existence of solutions by a carefully devised semi-implicit approximation scheme of the fractional-step type. Mathematics Subject Classification. 35K55 · 35Q74 · 74A15 · 74R20 · 74N10 · 74F10 · 76S99 · 80A17 · 80A20.

We propose a continuum theory describing the evolution of magnetization and temperature in a rigi... more We propose a continuum theory describing the evolution of magnetization and temperature in a rigid magnetic body. The theory is based on a microforce balance, an energy balance, and an entropy imbalance. We advance the choice of a class of constitutive equations, consistent with the entropy imbalance, that appear appropriate to describe the phase transition taking place in a ferromagnet at the Curie point. By combining these constitutive equations with the balance laws, we formulate an initial-boundary value problem for the magnetization and temperature fields, and we prove existence of weak solutions.

The problem of the onset and propagation of an elastic-brittle delamination is considered. We stu... more The problem of the onset and propagation of an elastic-brittle delamination is considered. We study delamination processes for elastic bodies glued by an adhesive to each other or to a rigid outer boundary. The interfacial adhesive is assumed to store and also dissipate a specific amount of energy during the delamination process. Damage along the interfaces is taken into account by introducing an interface damage variable. The present approach is based on the so-called energetic-solution concept. After introducing an implicit time discretization and a spatial discretization along the boundaries, the boundary element method (BEM) is utilized to solve the pertinent recursive boundary-value problems arising at each time step and compute the stored elastic energy. The whole solution process is based on the global minimization of the sum of the elastic potential energy in the solids and adhesive layer, defined in terms of the displacements along boundaries and the damage parameter along th...

Computational Mechanics, 2013

The problem of quasistatic and rate-independent evolution of elastic-plastic-brittle delamination... more The problem of quasistatic and rate-independent evolution of elastic-plastic-brittle delamination at small strains is considered. Delamination processes for linear elastic bodies glued by an adhesive to each other or to a rigid outer surface are studied. The energy amounts dissipated in fracture Mode I (opening) and Mode II (shear) at an interface may be different. A concept of internal parameters is used here on the delaminating interfaces, involving a couple of scalar damage variable and a plastic tangential slip with kinematic-type hardening. The so-called energetic solution concept is employed. An inelastic process at an interface is devised in such a way that the dissipated energy depends only on the rates of internal parameters and therefore the model is associative. A fully implicit time discretization is combined with a spatial discretization of elastic bodies by the BEM to solve the delamination problem. The BEM is used in the solution of the respective boundary value problems, for each subdomain separately, to compute the corresponding total potential energy. Sample problems are analysed by a collocation BEM code to illustrate the capabilities of the numerical procedure developed.

ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 2013

An adhesive unilateral contact of elastic bodies with a small viscosity in the linear Kelvin-Voig... more An adhesive unilateral contact of elastic bodies with a small viscosity in the linear Kelvin-Voigt rheology at small strains is scrutinized. The flow-rule for debonding the adhesive is considered rate-independent and unidirectional, and inertia is neglected. The asymptotics for the viscosity approaching zero towards purely elastic material involves a certain defect-like measure recording in some sense natural additional energy dissipated in the bulk due to (vanishing) viscosity, which is demonstrated on particular 2-dimensional computational simulations based on a semi-implicit time discretisation and a spacial discretisation implemented by boundary-element method.

Discrete and Continuous Dynamical Systems - Series S, 2013

The quasistatic rate-independent evolution of a delamination in the so-called mixed mode, i.e. di... more The quasistatic rate-independent evolution of a delamination in the so-called mixed mode, i.e. distinguishing opening (Mode I) from shearing (Mode II), devised in , is rigorously analysed as far as existence of the so-called energetic solutions concerns. The model formulated at small strains uses a delamination parameter of Frémond's type combined with a concept of an interface plasticity, and is associative in the sense that the dissipative force driving the delamination has a potential which depends in a 1-homogeneous way only on rates of internal parameters. A sample numerical simulation documents that this model can really produce mixity-sensitive delamination.

Applied Mathematics & Optimization, 1986

An optimal-control problem of a variational inequality of the elliptic type is investigated. The ... more An optimal-control problem of a variational inequality of the elliptic type is investigated. The problem is approximated by a family of finite-dimensional problems and the convergence of the approximated optimal controls is shown. The finite-dimensional problems, being nonsmooth, are to be optimized by a bundle algorithm, which requires an element of Clarke's generalized gradient of the minimized function. A simple algorithm which yields this element is proposed. Some numerical experiments with a simple model problem have also been carried out.

Adaptive approximation algorithm for relaxed optimization problems

Quasistatic delamination models for Kirchhoff-Love plates

Zamm-zeitschrift Fur Angewandte Mathematik Und Mechanik, 2011

... Lorenzo Freddi1,∗ , Roberto Paroni2, Tomáš Roubıcek3,4, and Chiara Zanini1 1 Dipartimento di ... more ... Lorenzo Freddi1,∗ , Roberto Paroni2, Tomáš Roubıcek3,4, and Chiara Zanini1 1 Dipartimento di Matematica e Informatica, Universit`a di Udine, Via ... We further confine ourselves to Griffith-type delamination on such a prescribed surface which is positioned in a normal direction ...

Mathematical Methods in the Applied Sciences

Pod vodà arenskou vÄ eÄ zà 4; CZ-182 08 Praha 8;

Optimization, 1997

Optimal control problems with nonlinear equations usually do not have a solution, i.e. an optimal... more Optimal control problems with nonlinear equations usually do not have a solution, i.e. an optimal control. Nevertheless, if the cost functional is uniformly concave with respect to the state, the solution may exist. Using the Balder's technique based on a Young-measure relaxation, Bauer's extremal principle and investigation of extreme Young measures, the existence is demonstrated here for optimal control processes described by nonlinear integral equations.

Mathematical Methods in the Applied Sciences, 2002

Pod vodà arenskou vÄ eÄ zà 4; CZ-182 08 Praha 8;

A Rate-Independent Model for Inelastic Behavior of Shape-Memory Alloys

Multiscale Modeling & Simulation, 2003

We formulate a model describing rate-independent hysteretic re- sponse of shape-memory alloys und... more We formulate a model describing rate-independent hysteretic re- sponse of shape-memory alloys under slow external forcing. Under natural as- sumptions we prove that this model has solution. The microstructure is treated on a \mesoscopic" level, described by volume fractions of particular phases in terms of Young measures. The whole formulation is based on energetic function- als for energy storage and

Applications of Mathematics

Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents ... more Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This document has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://dml.cz 35(1990) APLIKACE MATEMATIKY No. 5, 361-372

Mathematical Models and Methods in Applied Sciences, 2006

Damage of an elastic body undergoing large deformations by a "hard-device" loading poss... more Damage of an elastic body undergoing large deformations by a "hard-device" loading possibly combined with an impact (modeled by a unilateral frictionless contact) of another, ideally rigid body is formulated as an activated, rate-independent process. The damage is assumed to absorb a specific and prescribed amount of energy. A solution is defined by energetic principles of stability and balance of stored and dissipated energies with the work of external loading, realized here through displacement on a part of the boundary. Rigorous analysis by time discretization is performed.

International Journal of Plasticity, 2006

In this article, we develop a continuum mechanical model to simulate deformation and phase transf... more In this article, we develop a continuum mechanical model to simulate deformation and phase transformation processes in shape memory alloys. The model is based on a detailed description of the stored energy. Furthermore, the energy dissipation due to phase transformations is taken into account via the maximum-dissipation principle. The results from the 3D numerical simulations of stress induced transformations from the cubic to the tetragonal phase and martensitic variant reorientations in NiMnGa are compared with laboratory experiments on NiMnGa [001]-oriented single crystals.

Phase transformation in shape-memory alloys is known to cause electric resistivity variation that... more Phase transformation in shape-memory alloys is known to cause electric resistivity variation that, under electric current, may conversely influence Joule heat production and thus eventually the martensitic transformation itself. A ther-modynamically consistent general continuum-mechanical model at large strains is presented. In special cases, a proof of the existence of a weak solution is outlined, using a semidiscretization in time. Mathematics Subject Classification (2000). 35K55 · 74A15 · 74N10 · 80A17.

Existence of weak solutions is proved for a system of nonlinear parabolic equations/inequalities ... more Existence of weak solutions is proved for a system of nonlinear parabolic equations/inequalities describing evolution of magnetization, temperature, magnetic ¯eld, and electric ¯eld in electrically-conductive unsaturated ferromagnets. The system is derived from a recently-proposed thermodynamically-consistent continuum theory for the ferro/paramagnetic transition. Besides the standard viscous-like damping, dissipation due to eddy currents and domain-wall pinning is considered.

We derive a thermodynamically consistent general continuum-mechanical model describing mutually c... more We derive a thermodynamically consistent general continuum-mechanical model describing mutually coupled martensitic and ferro/paramagnetic phase transformations in electrically-conductive magnetostrictive materials such as NiMnGa. We use small-strain and eddy-current approximations, yet large velocities and electric current injected through the boundary are allowed. Fully nonlinear coupling of magneto-mechanical and thermal effects is considered. The existence of energy-preserving weak solutions is proved by showing convergence of time-discrete approximations constructed by a carefully designed semi-implicit regular-ized scheme.

A thermodynamically consistent mathematical model for hydrogen adsorption in metal hydrides is pr... more A thermodynamically consistent mathematical model for hydrogen adsorption in metal hydrides is proposed. Beside hydrogen diffusion, the model accounts for phase transformation accompanied by hysteresis, swelling, temperature and heat transfer, strain, and stress. We prove existence of solutions of the ensuing system of partial differential equations by a carefully-designed, semi-implicit approximation scheme. A generalization for a drift-diffusion of multi-component ionized " gas " is outlined, too.

We propose a thermodynamically consistent general-purpose model describing diffusion of a solute ... more We propose a thermodynamically consistent general-purpose model describing diffusion of a solute or a fluid in a solid undergoing possible phase transformations and damage, beside possible visco-inelastic processes. Also heat genera-tion/consumption/transfer is considered. Damage is modelled as rate-independent. The applications include metal-hydrogen systems with metal/hydride phase transformation, poroelastic rocks, structural and ferro/para-magnetic phase transformation , water and heat transport in concrete, and if diffusion is neglected, plasticity with damage and viscoelasticity, etc. For the ensuing system of partial differential equations and inclusions, we prove existence of solutions by a carefully devised semi-implicit approximation scheme of the fractional-step type. Mathematics Subject Classification. 35K55 · 35Q74 · 74A15 · 74R20 · 74N10 · 74F10 · 76S99 · 80A17 · 80A20.

We propose a continuum theory describing the evolution of magnetization and temperature in a rigi... more We propose a continuum theory describing the evolution of magnetization and temperature in a rigid magnetic body. The theory is based on a microforce balance, an energy balance, and an entropy imbalance. We advance the choice of a class of constitutive equations, consistent with the entropy imbalance, that appear appropriate to describe the phase transition taking place in a ferromagnet at the Curie point. By combining these constitutive equations with the balance laws, we formulate an initial-boundary value problem for the magnetization and temperature fields, and we prove existence of weak solutions.

The problem of the onset and propagation of an elastic-brittle delamination is considered. We stu... more The problem of the onset and propagation of an elastic-brittle delamination is considered. We study delamination processes for elastic bodies glued by an adhesive to each other or to a rigid outer boundary. The interfacial adhesive is assumed to store and also dissipate a specific amount of energy during the delamination process. Damage along the interfaces is taken into account by introducing an interface damage variable. The present approach is based on the so-called energetic-solution concept. After introducing an implicit time discretization and a spatial discretization along the boundaries, the boundary element method (BEM) is utilized to solve the pertinent recursive boundary-value problems arising at each time step and compute the stored elastic energy. The whole solution process is based on the global minimization of the sum of the elastic potential energy in the solids and adhesive layer, defined in terms of the displacements along boundaries and the damage parameter along th...

Computational Mechanics, 2013

The problem of quasistatic and rate-independent evolution of elastic-plastic-brittle delamination... more The problem of quasistatic and rate-independent evolution of elastic-plastic-brittle delamination at small strains is considered. Delamination processes for linear elastic bodies glued by an adhesive to each other or to a rigid outer surface are studied. The energy amounts dissipated in fracture Mode I (opening) and Mode II (shear) at an interface may be different. A concept of internal parameters is used here on the delaminating interfaces, involving a couple of scalar damage variable and a plastic tangential slip with kinematic-type hardening. The so-called energetic solution concept is employed. An inelastic process at an interface is devised in such a way that the dissipated energy depends only on the rates of internal parameters and therefore the model is associative. A fully implicit time discretization is combined with a spatial discretization of elastic bodies by the BEM to solve the delamination problem. The BEM is used in the solution of the respective boundary value problems, for each subdomain separately, to compute the corresponding total potential energy. Sample problems are analysed by a collocation BEM code to illustrate the capabilities of the numerical procedure developed.

ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 2013

An adhesive unilateral contact of elastic bodies with a small viscosity in the linear Kelvin-Voig... more An adhesive unilateral contact of elastic bodies with a small viscosity in the linear Kelvin-Voigt rheology at small strains is scrutinized. The flow-rule for debonding the adhesive is considered rate-independent and unidirectional, and inertia is neglected. The asymptotics for the viscosity approaching zero towards purely elastic material involves a certain defect-like measure recording in some sense natural additional energy dissipated in the bulk due to (vanishing) viscosity, which is demonstrated on particular 2-dimensional computational simulations based on a semi-implicit time discretisation and a spacial discretisation implemented by boundary-element method.

Discrete and Continuous Dynamical Systems - Series S, 2013

The quasistatic rate-independent evolution of a delamination in the so-called mixed mode, i.e. di... more The quasistatic rate-independent evolution of a delamination in the so-called mixed mode, i.e. distinguishing opening (Mode I) from shearing (Mode II), devised in , is rigorously analysed as far as existence of the so-called energetic solutions concerns. The model formulated at small strains uses a delamination parameter of Frémond's type combined with a concept of an interface plasticity, and is associative in the sense that the dissipative force driving the delamination has a potential which depends in a 1-homogeneous way only on rates of internal parameters. A sample numerical simulation documents that this model can really produce mixity-sensitive delamination.

Applied Mathematics & Optimization, 1986

An optimal-control problem of a variational inequality of the elliptic type is investigated. The ... more An optimal-control problem of a variational inequality of the elliptic type is investigated. The problem is approximated by a family of finite-dimensional problems and the convergence of the approximated optimal controls is shown. The finite-dimensional problems, being nonsmooth, are to be optimized by a bundle algorithm, which requires an element of Clarke's generalized gradient of the minimized function. A simple algorithm which yields this element is proposed. Some numerical experiments with a simple model problem have also been carried out.

Adaptive approximation algorithm for relaxed optimization problems

Quasistatic delamination models for Kirchhoff-Love plates

Zamm-zeitschrift Fur Angewandte Mathematik Und Mechanik, 2011

... Lorenzo Freddi1,∗ , Roberto Paroni2, Tomáš Roubıcek3,4, and Chiara Zanini1 1 Dipartimento di ... more ... Lorenzo Freddi1,∗ , Roberto Paroni2, Tomáš Roubıcek3,4, and Chiara Zanini1 1 Dipartimento di Matematica e Informatica, Universit`a di Udine, Via ... We further confine ourselves to Griffith-type delamination on such a prescribed surface which is positioned in a normal direction ...

Mathematical Methods in the Applied Sciences

Pod vodà arenskou vÄ eÄ zà 4; CZ-182 08 Praha 8;

Optimization, 1997

Optimal control problems with nonlinear equations usually do not have a solution, i.e. an optimal... more Optimal control problems with nonlinear equations usually do not have a solution, i.e. an optimal control. Nevertheless, if the cost functional is uniformly concave with respect to the state, the solution may exist. Using the Balder's technique based on a Young-measure relaxation, Bauer's extremal principle and investigation of extreme Young measures, the existence is demonstrated here for optimal control processes described by nonlinear integral equations.

Mathematical Methods in the Applied Sciences, 2002

Pod vodà arenskou vÄ eÄ zà 4; CZ-182 08 Praha 8;

A Rate-Independent Model for Inelastic Behavior of Shape-Memory Alloys

Multiscale Modeling & Simulation, 2003

We formulate a model describing rate-independent hysteretic re- sponse of shape-memory alloys und... more We formulate a model describing rate-independent hysteretic re- sponse of shape-memory alloys under slow external forcing. Under natural as- sumptions we prove that this model has solution. The microstructure is treated on a \mesoscopic" level, described by volume fractions of particular phases in terms of Young measures. The whole formulation is based on energetic function- als for energy storage and