arXiv subjects

Karel Tůma

Publications and source records attributed to Karel Tůma.

15 recordsLinked to original sources

arXiv2026

A numerical benchmark for fluid--structure--contact interaction

We propose a two-dimensional benchmark for fluid-structure-contact interaction consisting of a deformable elastic disk falling under gravity within a viscous incompressible fluid and rebounding in the vicinity of the bottom wall. Solid deformability is essential, as rigid solids do not rebound in this framework. Besides this, the setting is deliberately kept simple to facilitate reproduction. The configuration is particularly challenging due to the well-known no-contact paradox, which can lead to a contactless rebound and forces numerical methods to resolve a vanishingly thin fluid layer in the near-contact region, making the dynamics highly sensitive to the spatial and temporal discretizations. In addition to no-slip boundary and interface conditions, a reduced porous modeling of surface roughness, either on the disk boundary or on the bottom wall, is also considered; this circumvents the no-contact paradox and enables genuine contact. An energy balance law is derived theoretically for all three cases. Eight numerical methodologies, developed by five research groups and spanning different model formulations, numerical methods, and codes (including both fitted and unfitted discretizations), are applied to the benchmark at several levels of spatial and temporal refinement. Quantities of interest of varying complexity are collected and compared, showing close agreement during the falling phase and increased sensitivity in the near-contact and rebound regimes. The setting and the results provide a suitable reference for the systematic assessment of fluid-structure-contact interaction solvers. The time histories of all quantities of interest for every approach and refinement level are provided as supplementary material.

math.NA↗

arXiv2026

A thermodynamically consistent Johnson-Segalman-Giesekus model: numerical simulation of the rod climbing effect

Viscoelastic rate-type fluids represent a popular class of non-Newtonian fluid models due to their ability to describe phenomena such as stress relaxation, non-linear creep, and normal stress differences. The presence of normal stress differences in a simple shear flow gives rise to forces acting in directions orthogonal to the primary flow direction. The rod climbing effect, i.e. the rise of a fluid along a rod rotating about its axis, is associated with this phenomenon. Within the class of viscoelastic rate-type fluids that includes the Oldroyd-B and Giesekus models with Gordon--Schowalter convected derivatives, we show -- by means of thermodynamical analysis and numerical simulations -- that a thermodynamically consistent variant of the Johnson--Segalman model captures experimental data exceedingly well and is therefore superior to other models in this class, including the standard Johnson--Segalman model, which is widely used in engineering applications but is shown here to be incompatible with the second law of thermodynamics. We release a robust and computationally efficient higher-order finite-element implementation as open-source software on GitHub. The implementation is based on an arbitrary Lagrangian--Eulerian (ALE) formulation of the governing equations and is developed using the Firedrake library.

physics.flu-dyn↗

arXiv2026

On bifurcations and traction forces on an obstacle in incompressible flow

A systematic numerical investigation of flow-regime transitions in the two-dimensional incompressible Navier-Stokes flow past a confined circular cylinder is presented. For a fixed benchmark geometry, we observe a clear empirical correspondence between qualitative changes in steady traction profiles, understood here as the pointwise force density given by the Cauchy stress tensor on the obstacle boundary, and bifurcations in the long-time behavior of the unsteady Navier-Stokes equations. The observed transitions include onset of time-periodic oscillations, the appearance of multiple steady solutions and loss of effective symmetry. The well-known planar Schäfer-Turek benchmark is considered for Reynolds numbers up to 500. Several numerical techniques are employed to compute steady solutions, boundary traction profiles, and linear stability spectra such as duality-based approach for traction evaluation, deflation methods for detecting multiple steady states, and both two- and three- dimensional linear stability analyzes. The results suggest that steady boundary traction profiles can serve as a sensitive diagnostic indicator of critical Reynolds numbers at which qualitative changes in flow dynamics occur. This suggests a computationally inexpensive, complementary approach for detecting flow-regime transitions within this benchmark configuration.

physics.flu-dyn↗

arXiv2025

In vivo evidence of blood flow slippage: failure of the no-slip boundary condition assumption

The assumption that blood adheres to vessel walls, the ``no-slip'' boundary condition, is an essential premise of cardiovascular fluid dynamics. Yet, whether it holds true \emph{in vivo} has not been established. Using 4D flow magnetic resonance imaging of the human thoracic aorta and modeling blood as a Navier--Stokes fluid, we quantify the velocity of blood at the wall. We find tangential wall velocities of about 30--80\% of the mean luminal velocity, providing clear evidence of blood slippage. To our knowledge, this is the first demonstration that the no-slip condition does not apply to blood flow \emph{in vivo}. This finding challenges a fundamental assumption in cardiovascular modeling and directly affects key blood flow characteristics such as pressure drop, vorticity, wall shear stress, and energy dissipation, all of which play important roles across a wide range of cardiovascular conditions.

physics.flu-dyn↗

arXiv2024

Fast construction of the discrete Green operator for a second order ordinary differential equation

We consider linear second order differential equation y''= f with zero Dirichlet boundary conditions. At the continuous level this problem is solvable using the Green function, and this technique has a counterpart on the discrete level. The discrete solution is represented via an application of a matrix -- the Green matrix -- to the discretised right-hand side, and we propose an algorithm for fast construction of the Green matrix. In particular, we discretise the original problem using the spectral collocation method based on the Chebyshev--Gauss--Lobatto points, and using the discrete cosine transformation we show that the corresponding Green matrix is fast to construct even for large number of collocation points/high polynomial degree. Furthermore, we show that the action of the discrete solution operator (Green matrix) to the corresponding right-hand side can be implemented in a matrix-free fashion.

math.NA↗

arXiv2023

Numerical approximation of a thermodynamically complete rate-type model for the elastic--perfectly plastic response

We analyse a numerical scheme for a system arising from a novel description of the standard elastic--perfectly plastic response. The elastic--perfectly plastic response is described via rate-type equations that do not make use of the standard elastic-plastic decomposition, and the model does not require the use of variational inequalities. Furthermore, the model naturally includes the evolution equation for temperature. We present a low order discretisation based on the finite element method. Under certain restrictions on the mesh we subsequently prove the existence of discrete solutions, and we discuss the stability properties of the numerical scheme. The analysis is supplemented with computational examples.

math.NA↗

arXiv2023

Indentation-induced martensitic transformation in SMAs: insights from phase-field simulations

Direct experimental characterization of indentation-induced martensitic microstructures in pseudoelastic shape memory alloys (SMAs) is not possible, and thus there is a lack of evidence and understanding regarding the microstructure pattern and related features. To fill this gap, in this work we employ the phase-field method to provide a detailed and systematic analysis of martensitic phase transformation during nanoindentation. A recently-developed finite-element-based computational model is used for this purpose, and a campaign of large-scale 3D simulations is carried out. First, the orientation-dependent indentation response in CuAlNi (a widely studied SMA) is examined. A detailed investigation of the predicted microstructures reveals several interesting features, some of them are consistent with theoretical predictions and some can be (to some extent) justified by experiments other than micro/nanoindentation. The results also highlight the key role of finite-deformation effects and elastic anisotropy of the phases on the model predictions. Next, a detailed study of indentation-induced martensitic transformation in NiTiPd (a potential low-hysteresis SMA) with varying Pd content is carried out. In terms of hysteresis, the results demonstrate the prevailing effect of the transformation volume change over phase compatibility in the conditions imposed by nanoindentation and emphasize on the dominant role of the interfacial energy at small scales. Results of such scope have not been reported so far.

physics.comp-ph↗

arXiv2023

A thermodynamic framework for non-isothermal phenomenological models of isotropic Mullins effect

The Mullins effect is a common name for a family of intriguing inelastic responses of various solid materials, in particular filled rubbers. Given the importance of the Mullins effect, there have been many attempts to develop mathematical models describing the effect. However, most of available models focus exclusively on the mechanical response, and are restricted to the idealised isothermal setting. We lift the restriction to isothermal processes, and we propose a full thermodynamic framework for a class of phenomenological models of the Mullins effect. In particular, we identify energy storage mechanisms (Helmholtz free energy) and entropy production mechanisms that on the level of stress--strain relation lead to the idealised Mullins effect or to the Mullins effect with permanent strain. The models constructed within the proposed framework can be used in the modelling of fully coupled thermo-mechanical processes, and the models are guaranteed to be consistent with the laws of thermodynamics.

cond-mat.soft↗

arXiv2021

A note on parametric resonance induced by a singular parameter modulation

We investigate the classical problem of motion of a mathematical pendulum with an oscillating pivot. This simple mechanical setting is frequently used as the prime example of a system exhibiting the parametric resonance phenomenon, which manifests itself by surprising stabilisation/destabilisation effects. In the classical case the pivot oscillations are described by a cosine wave, and the corresponding stability analysis requires one to investigate the behaviour of solutions to the Mathieu equation. This is not a straightforward procedure, and it does not lead to exact and simple analytical results expressed in terms of elementary functions. Consequently, the explanation of the parametric resonance phenomenon can be in this case obscured by the relatively involved technical calculations. We show that the stability analysis is much easier if one considers the pivot motion described by a non-smooth function -- a triangular or a nearly rectangular wave. The non-smooth pivot motion leads to the presence of singularities (Dirac distributions) in the corresponding Mathieu type equation, which seemingly further complicates the analysis. Fortunately, this is only a minor technical difficulty. Once the mathematical setting for the non-smooth forcing is settled down, the corresponding stability diagram is indeed straightforward to obtain, and the stability boundaries are, unlike in the classical case, given in terms of simple analytical formulae involving only elementary functions.

math.DS↗

arXiv2020

Temperature field and heat generation at the tip of a cutout in a viscoelastic solid body undergoing loading

Using the finite element method we quantitatively analyse temperature field evolution in a viscoelastic solid undergoing a loading--unloading process. In particular we investigate the temperature field inside a Kelvin--Voigt type viscoelastic body with a thin cutout. We find that the viscosity significantly contributes to the temperature field changes, and that the temperature field changes initiated by the loading--unloading process are strongly concentrated at the tip of the thin cutout. The predicted temperature field qualitatively corresponds to the temperature field observed in experiments focused on simultaneous heat and strain measurements at the crack tip inside materials such as the filled rubber.

cond-mat.soft↗

arXiv2020

Contactless rebound of elastic bodies in a viscous incompressible fluid

In this paper, we investigate the phenomenon of particle rebound in a viscous incompressible fluid environment. We focus on the important case of no-slip boundary conditions, for which it is by now classical that, under certain assumptions, collisions cannot occur in finite time. Motivated by the desire to understand this fascinating yet counterintuitive fluid-structure interaction, we introduce a reduced model which we study both analytically and numerically. In this simplified framework, we provide conditions which allow to prove that rebound is possible even in the absence of a topological contact. Our results lead to conjecture that a qualitative change in the shape of the solid is necessary for obtaining a physically meaningful rebound. We support the conjecture by also comparing numerical simulations performed for the reduced model with the finite element solutions obtained for the corresponding well-established PDE system.

math.AP↗

arXiv2019

Finite amplitude stability of internal steady flows of the Giesekus viscoelastic rate-type fluid

Using a Lyapunov type functional constructed on the basis of thermodynamical arguments we investigate the finite amplitude stability of internal steady flows of viscoelastic fluids described by the Giesekus model. Using the functional we derive bounds on the Reynolds and the Weissenberg number that guarantee the unconditional asymptotic stability of the corresponding steady internal flow, wherein the distance between the steady flow field and the perturbed flow field is measured with the help of the Bures--Wasserstein distance between positive definite matrices. The application of the theoretical results is documented in the finite amplitude stability analysis of Taylor--Couette flow.

physics.flu-dyn↗

arXiv2018

Motion of the vitreous humour in a deforming eye -- fluid-structure interaction between a nonlinear elastic solid and a nonlinear viscoleastic fluid

We study the motion of vitreous humour in a deforming eyeball. From the mechanical and computational perspective this is a task to solve a fluid-structure interaction problem between a complex viscoelastic fluid (vitreour humour) and a nonlinear elastic solid (sclera and lens). We propose a numerical methodology capable of handling the fluid-structure interaction problem, and we demonstrate its applicability via solving the corresponding governing equations in a realistic geometrical setting and for realistic parameter values. It is shown that the choice of the rheological model for the vitreous humour has a negligible influence on the overall flow pattern in the domain of interest, whilst it is has a significant impact on the mechanical stress distribution in the domain of interest.

physics.flu-dyn↗

arXiv2016

On thermodynamics of viscoelastic rate type fluids with temperature dependent material coefficients

We derive a class of thermodynamically consistent variants of Maxwell/Oldroyd-B type models for viscoelastic fluids. In particular, we study the models that allow one to consider temperature dependent material coefficients. This naturally calls for the formulation of a temperature evolution equation that would accompany the evolution equations for the mechanical quantities. The evolution equation for the temperature is explicitly formulated, and it is shown to be consistent with the laws of thermodynamics and the evolution equations for the mechanical quantities. The temperature evolution equation contains terms that are ignored or even not thought of in most of the works dealing with this class of fluids. The impact of the additional terms in the temperature evolution equation on the flow dynamics is documented by the solution of simple initial/boundary value problems.

physics.flu-dyn↗

arXiv2016

Colombeau algebra as a mathematical tool for investigating step load and step deformation of systems of nonlinear springs and dashpots

The response of mechanical systems composed of springs and dashpots to a step input is of eminent interest in the applications. If the system is formed by linear elements, then its response is governed by a system of linear ordinary differential equations, and the mathematical method of choice for the analysis of the response of such systems is the classical theory of distributions. However, if the system contains nonlinear elements, then the classical theory of distributions is of no use, since it is strictly limited to the linear setting. Consequently, a question arises whether it is even possible or reasonable to study the response of nonlinear systems to step inputs. The answer is positive. A mathematical theory that can handle the challenge is the so-called Colombeau algebra. Building on the abstract result by (Průša & Rajagopal 2016, Int. J. Non-Linear Mech) we show how to use the theory in the analysis of response of a simple nonlinear mass--spring--dashpot system.

math-ph↗