Prague, Czech Republic, 22nd–27th September 2024

The conference starts on 22nd September in the evening (registration/welcome drink), the first lecture takes place on Monday 23rd September, the last lecture takes place on Friday 27th September 2024 at noon.

- The schematic program is available here;
- The detailed scientific program can be downloaded here;
- Registration is possible during the whole week, starting on Sunday 22nd at 19:00 in the conference building;
- The informal opening ceremony takes place on Sunday 22nd at 19:15 in the conference building;
- The conference dinner will take place on Thursday evening at 19:00 at the Art Restaurant Mánes;
- On Wednesday afternoon, we plan to organise a guided tour "Mathematics in Prague". You will be able to indicate your interest in attending this during registration.

We consider the sharp interface limit of a Navier-Stokes/Allen-Cahn system, when a parameter $\varepsilon>0$ that is proportional to the thickness of the diffuse interface tends to zero, in a two dimensional bounded domain. In dependence on the mobility coefficient in the Allen-Cahn equation in dependence on $\varepsilon>0$ different limit systems or non-convergence can occur. In the case that the mobility vanishes as $\varepsilon$ tends to zero slower than quadratic we prove convergence of solutions to a smooth solution of a classical sharp interface model for well-prepared and sufficiently smooth initial data. The proof is based on a relative entropy method and the construction of sufficiently smooth solutions of a suitable perturbed sharp interface limit system.

This is a joint work with Julian Fischer and Maximilian Moser (ISTA Klosterneuburg, Austria).

Sharp interface limit of a Navier-Stokes/Allen-Cahn system with vanishing mobility (slides)The plate bending or Babuska paradox refers to the failure of convergence when a linear bending problem with simple support boundary conditions is approximated using polygonal domain approximations. We provide an explanation based on a variational viewpoint and identify sufficient conditions that avoid the paradox and which show that boundary conditions have to be suitably modified. We show that the paradox also matters in nonlinear thin-sheet folding problems and devise approximations that correctly converge to the original problem.

Babuska's paradox in linear and nonlinear bending theories (slides)To control instabilities in architected materials is a new challenge in which solid and structural mechanics become fully complementary. Homogenization techniques will be presented for periodic elastic structures subject to prestress, to give evidence to the emergence of material instabilities such as shear bands, occurring for both for compressive [1] and tensile [2] prestress. Moreover, the architecture of the analyzed structures leads to the emergence of multiple band gaps, flat bands, and Dirac cones [3]. The experience gained on structural flutter [4, 5] is exploited to implement a new concept, namely, the possibility of producing a Hopf bifurcation in a continuous medium. This possibility is proven through a rigorous application of Floquet-Bloch wave asymptotics, which yields an unsymmetric acoustic tensor governing the incremental dynamics of the effective material [6]. The latter represents the incremental response of a hypo-elastic solid, which does not follow from a strain potential and thus apparently breaks the wall of hyperelasticity, leading to non-Hermitian mechanics. The discovery of elastic materials capable of collecting or releasing energy in closed strain cycles through interactions with the environment introduces new micro and nano technologies and finds definite applications, for example, in the field of energy harvesting.

Acknowledgements: Financial support from ERC-ADG-2021-101052956-BEYOND is gratefully acknowledged.

References:

- Bordiga, G., Cabras, L., Piccolroaz, A., and Bigoni, D. “Dynamics of prestressed lattices: Homogenization, instabilities, and strain localization” J. Mech. Phys. Solids, 146, 104198 (2021).
- Bordiga G., Bigoni D. and A. Piccolroaz “Tensile material instabilities in elastic beam lattices lead to a closed stability domain”. Phil. Trans. Royal Soc. A, Vol. 380, 20210388, (2022).
- Cabras, L., Bigoni, D., and Piccolroaz, A. Dynamics of elastic lattices with sliding constraints. Proc. Royal Soc. A 480, 20230579, 2024.
- Rossi, M., Piccolroaz, A., Bigoni, D., “Fusion of two stable elastic structures resulting in an unstable system” J. Mech. Phys. Solids, 173, 105201 (2023).
- Bigoni, D., Dal Corso, F., Kirillov, O., Misseroni, D., Noselli, G., Piccolroaz, A. Flutter instability in solids and structures, with a view on biomechanics and metamaterials. Proc. Royal Soc. A 479, 20230523 (2023).
- Bordiga, G., Piccolroaz, A., and Bigoni, D., “A way to hypo-elastic artificial materials without a strain potential and displaying flutter instability” J. Mech. Phys. Solids, 158, 104665 (2022).

I will discuss a possible strategy to solve the Cauchy problem for nonlinear evolution PDEs by space-time convex optimization based on their weak formulation. One of the simplest example is the quadratic porous medium equation for which the Aronson-Benilan inequality is sharply used to prove that the strategy works for arbitrarily long time intervals. A similar result holds true for the Burgers equation. For the more challenging Euler equations, the concept of subsolution (in the sense of convex integration theory) plays a crucial role. Finally, I will mention how the Einstein equations in vacuum can be considered in that framework.

Solving initial value problems by space-time convex optimization (slides)Spinodal decomposition is a process where an initial homogeneous but unstable mixture sponta- neously separates into two or more stable phases with a distinctive arrangement termed spinodal structure. This process can be modeled with the Cahn-Hilliard equation [1] which is based on the definition of a chemical energy density depending on a variable (the phase field), represent- ing the smooth transition between the unstable initial mixture and the stable phases. Spinodal structures are characterized by a length scale that, if no competing processes are accounted for, coarsens over time until complete segregation of the stable phases. Recent studies suggest that for certain mixtures involving a solid matrix, the elastic parameters of the matrix govern the coarsening stage and can even arrest it, a phenomenon that is denoted as elastic microphase separation [2].

In this work [3], we propose a phase-field model that captures the main features of elastic microphase separation observed in [2]. We extend the Cahn-Hilliard free-energy functional [1] to include the elastic strain energy density as well as an additional coupling term. The model is first investigated in 1D and the results show that the mechanical deformation controls both the composition of the stable phases and the initial characteristic length of the spinodal structure. Moreover, we numerically show that the proposed coupling is able to predict the arrest of the coarsening phase at a length scale controlled by the model parameters. The formulation is then extended to the multi-dimensional setting and compared to experimental results. The numerical results show excellent agreement with the experimental evidence, especially in terms of initial and arrested pattern morphology.References:

- J. Cahn, and J. Hilliard, 1958, Free Energy of a Nonuniform System. I. Interfacial Free Energy. J. Chem. Phys. Vol 28, No 2.
- C. Fernández-Rico, S. Schreiber, H. Oudich, C. Lorenz, A. Sicher, T. Sai, V. Bauernfeind, S. Heyden, P. Carrara, L. De Lorenzis, Robert W. Style, and Eric R. Dufresne, 2023, Elastic microphase separation produces robust bicontinuous materials. Nat. Mater. pp. 1–7
- H. Oudich, P. Carrara, L. De Lorenzis, 2024, Phase-field modeling of elastic microphase separation, in prep.

Numerical methods for the simulation of transient systems with structure-preserving properties are known to exhibit greater accuracy and physical reliability, in particular over long durations. These schemes are often built on powerful geometric ideas for broad classes of problems, such as Hamiltonian or reversible systems. However, there remain difficulties in devising higher-order-in-time structure-preserving discretizations for nonlinear problems, and in conserving non-polynomial invariants.

In this work we propose a new, general framework for the construction of structure-preserving timesteppers via finite elements in time and the systematic introduction of auxiliary variables. The framework reduces to Gauss methods where those are structure-preserving, but extends to generate arbitrary-order structure-preserving schemes for nonlinear problems, and allows for the construction of schemes that conserve multiple higher-order invariants. We demonstrate the ideas by devising novel schemes that exactly conserve all known invariants of the Kepler and Kovalevskaya problems, arbitrary-order schemes for the compressible Navier–Stokes equations that conserve mass, momentum, and energy, and provably dissipate entropy, and multi-conservative schemes for the Benjamin-Bona-Mahony equation.

Designing conservative and accurately dissipative numerical integrators in time (slides)Necking is traditionally associated with plastic deformations and is commonly observed in the tension test of ductile materials. However, an increasing body of literature has demonstrated that elastic necking is also possible in soft materials due to multi-fields such as surface tension, residual stresses, and electric field. It is likely to become a more common phenomenon as more and more smart materials are synthesized (e.g. cellular materials and double-network hydrogels).

We report our recent research effort on characterising axisymmetric necking of a circular membrane under an all-round tension (or equivalently a square membrane under equibiaxial tension). We derive the bifurcation condition, conduct a weakly nonlinear analysis to obtain an amplitude equation in the form of a fourth-order ODE with quadratic nonlinearity and variable coefficients, and develop a 1D reduced model to describe the fully nonlinear regime where the necking front propagates from the center towards the edge. We shall explain the methodology in the purely mechanical case, but our initial motivation has come from applications associated with dielectric elastomer actuators under the combined action of mechanical stretching and an electric field. We wish to point out that in the axisymmetric case the governing equations has no translational invariance in the radial direction, and as a result it is not clear how a central manifold reduction can be applied to derive the amplitude equation in a rigorous manner (we used a perturbation approach). Thus, it is hoped that our talk will draw the attention of applied analysists at this multidisciplinary conference. Our amplitude equation mentioned above might also be of interest to theorists since although we were able to obtain a localised solution numerically, its solution properties await to be fully explored.

Acknowledgements: This work was conducted with Xiang Yu, Mi Wang and Lishuai Jin, and was supported by the National Natural Science Foundation of China (Grant No 12072224) and the Engineering and Physical Sciences Research Council, UK (Grant No EP/W007150/1).

References:

- Mi Wang, Lishuai Jin and Yibin Fu: Axisymmetric necking versus Treloar-Kearsley instability in a hyperelastic sheet under equibiaxial stretching, Math. Mech. Solids, 27 (2022), 1610-1631.
- Yibin Fu and Xiang Yu: Axisymmetric necking of a circular electrodes-coated dielectric membrane, Mechanics of Materials, 181 (2023), 104645.
- Xiang Yu and Yibin Fu: A 1D reduced model for the axisymmetric necking of a circular electrodes-coated dielectric membrane, to be submitted.

We construct a structure-preserving and thermodynamically consistent finite element method and time stepping scheme for heat conducting viscous fluids. The method is derived by discretizing a variational formulation for nonequilibrium thermodynamics that extends Hamilton’s principle for fluids to systems with irreversible processes. The resulting scheme preserves energy and mass balance to machine precision, while also ensuring compliance with the second law of thermodynamics, both at the spatially and temporally discrete levels. The method is shown to apply both with insulated and prescribed heat flux boundary conditions, as well as with prescribed temperature boundary conditions. The effectiveness of the scheme is demonstrated through its application to Rayleigh-Bénard convection.

Variational and thermodynamically consistent discretization for heat-conducting viscous fluids (slides)In this talk, I will focus on some models for the motion of rigid bodies immersed in a fluid. The different variants of this model have motivated many references in the past 20 years. Whatever the properties of the fluid (incompressible vs compressible, viscous vs inviscid, ...) a specific difficulty arises in the mathematical treatment because of collisions between the bodies or between one body and the container boundary. Such collisions are difficult to handle because they induce a geometrical singularity in the fluid domain and they require to introduce an advanced physical description of solid/solid contacts in the modeling. It is then mandatory to discuss the possibility of a collision and more generally to obtain an analytical description of the flow when solid bodies are close to contact. Such a description is also of high importance to the description of dense suspensions. In this talk, I will review results on this topic discussing the importance of the fluid and body properties and discuss related open problems.

Collisions in fluid/solid mixtures (slides)The mean curvature flow is a classical topic and has been studied extensively due to its applications in geometry. Because of the gradient flow structure, it is also relevant in Calculus of Variations and shape optimization, where also the volume preserving variants are important. I will discuss recent results on regularity and the long-time behavior of two globally well-defined weak solutions of physically relevant volume preserving geometric ﬂows: the volume preserving mean curvature and the Mullins-Sekerka flat ﬂow.

Regularity and asymptotical behavior of volume preserving geometric flows (slides)We present the state of the art in Multiscale Finite element methods, highlighting the many successes of these approaches but also their limitations. Our perspective is mathematical in nature and we address questions ranging from the links between the computational approaches and homogenization theory to software engineering questions. We explore some recent contributions that specifically aim at making the approaches more versatile, applicable to a wide class of different equations, and less intrusive in terms of software implementation when applied to practically relevant problems in the engineering sciences.

The talk is based upon a series of joint works with various collaborators, in particular Frederic Legoll, Alexei Lozinski, Rutger Biezemans, Amandine Boucart. The work is partially supported by the ONR and the EOARD.

Slides available on demand -- please email Claude Le Bris directly.

Mathematical models arising in science and engineering inherit several sources of uncertainties, such as model parameters, and initial or boundary conditions. To predict reliable results, deterministic models are insufficient, and more sophisticated methods are needed to analyze the influence of data uncertainties. In scientific computing the Monte Carlo method is typically used for uncertainty quantification. Despite its large popularity, the rigorous convergence analysis for compressible fluid flows was missing in general. In this talk, we will review our recent results obtained for the random compressible Euler and Navier-Stokes systems. We suppose that the initial and boundary data as well as model parameters, such as the viscosity coefficients, are random variables. Consequently, a solution of the PDE system will be a random process. The Monte Carlo methods is combined with a suitable deterministic discretization scheme, such as a finite volume method. We study both the statistical convergence rates as well as the approximation errors. The convergence for the deterministic Navier-Stokes or Euler system is realized via dissipative solutions. Assuming that the numerical solutions satisfy in probability suitable conditions leading to a global regular solution, we prove that the Monte Carlo finite volume method converges to a statistical strong solution. The convergence rates of the finite volume and statistical methods are discussed as well. Numerical experiments will illustrate theoretical results.

This research was supported by the German Science Foundation (DFG) under the grants TRR146 ”Multiscale Simulation Methods for Soft Matter Systems” and SPP 2410 ”Hyperbolic Balance Laws: Complexity, Scales and Randomness”.

Random compressible fluid flows (slides)Sea ice is one of the important components in global circulation models used for weather forecasting and especially for climate prediction. Sea ice is modeled as a 2D layer between the atmosphere and the ocean. While sea ice covers only the polar regions of the Earth, the sea ice component usually takes on the role of a coupler between the ocean and the atmosphere and is responsible for all energy transfer between these two phases. We focus on the dynamics of sea ice and provide an introduction to modeling it as a 2D fluid. Different rheologies are considered. The most established are approximations to a viscous plastic model, but recently different material types, e.g. brittle rheologies, are also considered. Finally, we describe the special requirements in terms of numerical discretization and implementation of such a sea ice model, which should fit well into the general framework of global climate models.

Vector-valued function spaces, their finite element sub-spaces, and relations between these spaces are well understood within the de Rham complex. The framework of differential forms and Hilbert complexes provides a unified framework for any space dimension. Various matrix-valued finite element spaces have been introduced and analyzed more or less independently. In this presentation we put these spaces into a so called 2-complex. We present applications in fluid dynamics, solid mechanics and relativity.

Distributional finite elements with applications for elasticity, fluids, and curvature (slides)In this talk I will describe several examples of kinetic equations for which the boundary conditions imply the exchange of some quantity (mass, momentum or energy) with the surroundings. A consequence of this is that the resulting solutions do not converge to an equilibrium solution, although in some cases the solutions can converge to steady states, but they can yield also periodic oscillations or in some cases solutions that can be described as self-similar solutions in some regions of the space of parameters. Specific examples of equations that will be described include the classical Boltzmann equation, the Smoluchowski coagulation equation, the Becker-Döring equation of the theory of nucleation and some of the kinetic equations arising in the so-called Wave Turbulence theory.

Kinetic equations describing open systems (slides)In this talk, we will explore the optimal approximation rates for the critical constants associated with fundamental inequalities in partial differential equations (PDE) analysis, specifically the Hardy and Sobolev inequalities. We will outline a systematic methodology to achieve sharp convergence rates.

Additionally, we will present several applications within the context of numerical approximation for parabolic evolution problems.

Qualitative numerics (slides)In order to reproduce, bacteria must remodel their cell wall at the division site. The division process is an interplay of the mass increase of Peptiglycan (PG), driven by the enzymatic activity of PG synthases, and the mechanical force exerted by the constricting Z-ring. We introduce a model that is able to reproduce correctly the shape of the division site during the constriction and septation phase of E. coli. The model represents mechanochemical coupling within the mathematical framework of morphoelasticity: It contains only two parameters, associated with volumetric growth and PG remodelling, that couple to the mechanical stress in the bacterial wall. Depending on the remodelling parameter, given by overall enzyme activity, different morphologies were recovered, corresponding either to mutant or wild type cells. In addition, a plausible range for the cell stiffness and turgor pressure was determined by comparing numerical simulations with data on cell lysis.

Stress-mediated growth determines E. coli division site morphogenesis (preprint)For the numerical simulations of nanoscale semiconductor devices, model equations based on quantum mechanical phenomena have to be employed. In this talk, I will focus on a macroscopic model, the so-called quantum hydrodynamics (QHD) system, that describes the time evolution of the charge and current electron densities in the device. For the one-dimensional model, I will present an existence result for global-in-time, weak solutions. Moreover, I will discuss the time-relaxation limit of the QHD system towards the so-called quantum drift-diffusion equation.

This is based on joint works with Pierangelo MArcati (GSSI) and Hao Zheng (Chinese Academy of Sciences).

The relaxation-time limit of the quantum hydrodynamics equations for semiconductors (slides)

In this talk, we explore the Hodge Laplacian in variable exponent spaces with differential forms on smooth manifolds. We present several results, including the Hodge decomposition in variable exponent spaces and a priori estimates. As an application, we derive Calderón-Zygmund estimates for variable exponent problems involving differential forms and discuss numerical approximations for nonlinear models with differential forms, which have applications in superconductivity.

This presentation is based on several works with Swarnendu Sil, Michail Surnachev, and Alex Kaltenbach.

Hodge decomposition in variable exponent spaces with applications to regularity theory (slides)We will focus on fundamental questions connected with the PDE systems describing the flow of (thermo-) viscoelastic fluids. Among these is the question of existence of solutions and their appropriate definition or the stability of the steady state solution.

Analysis of viscoelastic fluids (slides)We study boundary regularity for nonlinear systems depending on the symmetric gradient $\varepsilon u:=\tfrac{1}{2}(\nabla u + \nabla ^T u)$. An important example is given by the symmetric $p$-Laplace system $-\operatorname{div}(|\varepsilon u|^{p-2}\varepsilon u)=f$ for $1< p < \infty$, where our result is new in the degenerate case $p>2$. Furthermore, we consider more general systems of Orlicz growth.

Boundary regularity for systems with symmetric gradient (slides)Linear elasticity is a well-established and probably most used model in solid mechanics. Going back to Dal Maso, Negri And Percivale it can be rigorously derived via a $\Gamma$ Limit procedure from nonlinear models. As this procedure is purely variational, it does not directly translate to force balance models like elastodynamics. Nonetheless using an approximation procedure due to Benešová, Kampschulte and Schwarzacher, we explore the connection between the $\Gamma$ Limit and elastodynamics.

This is a joint work with Malte Kampschulte and Martin Kružík.

Linearization in elastodynamics (slides)The arbitrary Eulerian-Lagrangian (ALE) method is useful for solving fluid dynamics in a moving domain. Its use can be found also in the fluid-structure interaction problem. However, this powerful method cannot be used for large deformations. This issue can be solved by incorporating a remeshing strategy. We demonstrate that this technique can even solve the rebound of an elastic ball submerged in fluid.

By conditional regularity we mean that locally defined strong solutions of a given system of partial differential equations remain regular as long as their ``norm'' in specific function space remains bounded. In particular, we show that strong solutions of the compressible Navier-Stokes-Fourier system complemented with general Dirichlet boundary conditions remain regular as long as their amplitude remains bounded.

There are several applications of this result including

- Blow up criteria for strong solutions
- Convergence of bounded solutions to suitable numerical schemes
- Local regularity results in general function spaces

We present a mechanically consistent model for the numerical simulation of fluid-structure interactions (FSI) with contact [2]. The main novelty compared to previous works is the consideration of seepage through a porous layer of co-dimension one during contact. For the latter, a Darcy model is considered in a thin porous layer attached to a solid boundary in the limit of infinitesimal thickness. To obtain a stable and efficient numerical algorithm, we use a relaxation of the contact conditions and a weak imposition of the FSI coupling and contact conditions by means of a unified Nitsche approach [1]. To test the approach, we present a benchmark setting of a falling elastic ball including experimental results [3].

Joint work with E. Burman and M. A. Fernández.

References:

- E. Burman, M.A. Fernández, S. Frei: A Nitsche-based formulation for fluid-structure interactions with contact, ESAIM M2AN 54(2) 531-564 (2020)
- S. Frei, F.M. Gerosa, E. Burman, M.A. Fernández: A mechanically consistent model for fluid-structure interactions with contact including seepage, Comp Methods Appl Mech Eng 392, 114637 (2022)
- H. von Wahl, T. Richter, S. Frei, T. Hagemeier: Falling balls in a viscous fluid with contact: Comparing numerical simulations with experimental data, Phys Fluids 33, 033304 (2021)

We present a comprehensive mathematical framework for modeling biological materials in the human eye, such as the cornea, lens, and vitreous, within the context of continuum mechanics. These materials exhibit complex mechanical properties, including elasticity, viscoelasticity, and fluid-structure interactions. As part of this effort, we are developing a Virtual Eye to map the physiology and pathology of ocular tissues and to explore in silico therapies. We present the coupled PDE-systems used so far to understand the development of some meaningful disesases, discuss some adaptation to the standard analysis and show the 3D Finite Element simulations. The Virtual Eye provides new insights into the complex interactions within ocular tissues, allowing us to bridge the gap between theoretical understanding, pathology modeling, and computational therapy optimization.

Continuum mechanics and computational modeling of ocular tissues for in silico therapies (slides)By exploiting remarkable properties of the Crouzeix-Raviart and Raviart-Thomas finite elements, numerous works in recent years have been able to employ convex duality theory to derive error estimates for a diverse set of problems, including total variation minimisation, the p-Laplacian, the obstacle problem, elastoplastic torsion, among others. However, virtually all of the available results have been developed for scalar problems with homogeneous Dirichlet boundary conditions. This work extends the existing results in three directions, taking the incompressible Stokes and linear elasticity systems as prototypical examples: it considers vectorial as opposed to just scalar problems, it includes non-homogeneous mixed boundary conditions, as well as loads in the dual of the energy space.

Electro-energy-reaction-diffusion systems are thermodynamically consistent continuum models for reaction-diffusion processes that account for temperature and electrostatic effects in a way that total charge and energy are conserved. The question of the long-time asymptotic behaviour in electro-energy-reaction-diffusion systems and the characterisation of their equilibrium solutions leads to a maximisation problem of the entropy on the (generalised) manifold of states with fixed values for the linear charge and the convex nonlinear energy functional. In this talk, we discuss the existence, uniqueness, and regularity of solutions to this global optimisation problem. Physically relevant entropies being sublinear at infinity with respect to the internal energy density, the direct method leads us to extend the problem to the cone of finite measures. Our techniques are based on ideas from convex duality and Lagrange multiplier theory.

This is joint work with Michael Kniely and Alexander Mielke.

Most studies of patient-specific blood flow models prescribe a no-slip boundary condition at the walls. Although its implementation is straightforward, its validity at the blood-vessel wall interface is questionable. It has been suggested in (Nubar, 1971) for example, that a slip boundary condition can be considered for blood flow in certain situations. We will discuss some effects of prescribing the Navier-type slip boundary condition, which assumes a linear proportionality between the tangential part of the wall velocity and the shear stress using an additional parameter. Given some measured flow data, for example modern 4D-PC MRI image, we use variational data assimilation approach to estimate the Navier's slip parameter on the wall for real data.

Numerical investigation of blood flows with slip boundary conditions (slides)We propose models in nonlinear elasticity for nonsimple materials that include surface energy terms. Additionally, we also discuss living surface loads on the boundary. We establish corresponding linearized models and show their relationship to the original ones by means of $\Gamma$-convergence.

This is a joint work with E. Mainini (Genoa).

Linearization of finite elasticity with surface tension (slides)In a given domain we consider a generalized Newtonian incompressible heat conducting fluid with prescribed nonuniform temperature on the boundary of the domain and with the no-slip boundary conditions for the velocity. We study stability of equilibria if no external body forces are applied to the fluid. In dependence on the growth of the constitutively determined part of the Cauchy stress we identify different classes of proper solutions that converge to the equilibrium exponentially in a suitable metric. Consequently, the equilibrium is nonlinearly stable and attracts all weak solutions from these classes. We also show that these classes of solutions are nonempty.

Stability of steady states to generalized Navier-Stokes-Fourier system (slides)We study a nonlinear fluid-structure interaction problem between a viscoelastic beam and a compressible viscous fluid. The beam is immersed in the fluid which fills a two-dimensional rectangular domain with periodic boundary conditions in both directions, while both the beam and the fluid are under the effect of time-periodic forces. By using a decoupling approach, at least one time-periodic weak solution to this problem is constructed which has a bounded energy and a fixed prescribed mass. This is a joint work with Václav Mácha, Šárka Nečasová and Srdjan Trifunović.

On time-periodic solutions to an interaction problem between compressible viscous fluids and viscoelastic beams (slides)We consider the homogenization of stationary and evolutionary incompressible viscous non-Newtonian flows of Carreau-Yasuda type in domains perforated with a large number of periodically distributed small holes in $R^{3}$, where the mutual distance between the holes is measured by a small parameter $\epsilon>0$ and the size of the holes is $\epsilon^{\alpha}$ with $\alpha \in (1, \frac 32)$. The Darcy's law is recovered in the limit. Instead of using their restriction operator to derive the estimates of the pressure extension by duality, we use the Bogovskii type operator in perforated domains to deduce the uniform estimates of the pressure directly. Moreover, quantitative convergence rates are given.

This is a joint work with Florian Oschmann from Prague.

Qualitative/quantitative homogenization of some non-Newtonian flows in perforated domains (slides)We consider convex integral functional with random coefficients that feature a fast decay of corelations that we encode with a spectral gap inequality. We prove a quantitative two-scale expansion for the minimizers with optimal scaling and discuss applications to nonlinearly elastic laminates.

Quantitative stochastic homogenization of convex integral functionals (slides)In this short talk, we discuss a mathematical framework for rigorously justifying nonlinear constitutive relations between stress and linearized strain for elastic bodies. Each of these relations emerges as the leading-order relationship describing a one-parameter family of nonlinear constitutive relations between stress and nonlinear strain. The asymptotic parameter limits the strain range while allowing stresses to remain unconstrained. Our approach differs from the standard justification of classical linearized elasticity, which uses the displacement gradient as the asymptotic parameter within a fixed constitutive relation. The proposed framework lays the foundation for rigorously proving asymptotic convergence between solutions of boundary value problems determined by relations between stress and nonlinear strain and those governed by stress and linearized strain.

This talk is based on joint work with K. R. Rajagopal.

Towards mathematically justifying nonlinear constitutive relations between stress and linearized strain (slides)The essence of the phase-field (PF) method is that interfaces are treated as diffuse, which simplifies computational treatment. However, this comes at the cost that spatial discretization must be sufficiently fine to resolve the diffuse interfaces, leading to large-scale problems with associated high computational cost. There is thus a quest for PF formulations that allow a sharper representation of diffuse interfaces. In this talk, we present a recently developed hybrid diffuse-semisharp approach for microstructure evolution problems [1]. The approach combines a (diffuse) PF-like treatment of the interfacial energy with a (semisharp) treatment of weak discontinuities using the laminated element technique (LET) [2]. We show that the new approach (LET-PF) is more accurate than the conventional phase-field method so that a coarser mesh can be used to get results of similar accuracy, thus reducing the computational cost.

Joint work with J. Dobrzański.

References:

- J. Dobrzański, S. Stupkiewicz: Towards a sharper phase-field method: A hybrid diffuse-semisharp approach for microstructure evolution problems, Comp. Meth. Appl. Mech. Eng. 423, 116841 (2024)
- J. Dobrzański, K. Wojtacki, S. Stupkiewicz: Lamination-based efficient treatment of weak discontinuities for non-conforming finite element meshes, Comput. Struct. 291, 107209 (2024)

We are interested in the interaction of a viscous incompressible fluid with an elastic structure, where the structure is located on a part of the fluid boundary. It reacts to the surface forces induced by the fluid and deforms the reference domain to the moving domain. The fluid equations are coupled with the structure via the kinematic condition and the action-reaction principle on the interface. We study the 2D visco-elastic shell interacting with 3D Navier-Stokes equations. Especially in a general reference geometry (the shell deforms along the normal direction of the flexible boundary), we prove a counterpart of the classical Ladyzhenskaya-Prodi-Serrin condition yielding conditional regularity and uniqueness of a solution. This requires additionally the deformation of the shell is Lipschitz continuous. This is based on joint work with D. Breit (Clausthal), P. Mensah (Clausthal) and S. Schwarzacher (Uppsala).

Conditional regularity for an elastic shell interacting with the Navier-Stokes equation (slides)Several recent studies considered the high-friction limit for systems arising in fluid mechanics. Following this approach, we rigorously derive the nonlocal Cahn-Hilliard equation as a limit of the nonlocal Euler-Korteweg equation using the relative entropy method. Applying the recent result about relations between non-local and local Cahn-Hilliard, we also derive rigorously the large- friction nonlocal- to-local limit. The result is formulated for dissipative measure-valued solutions of the nonlocal Euler-Korteweg equation which are known to exist on arbitrary intervals of time. This approach provides a new method to derive equations not enjoying classical solutions via relative entropy method by introducing the nonlocal effect in the fluid equation. During the talk I will also discuss the high-friction limit of the Euler-Poisson system.

Cahn-Hillard and Keller-Segel systems as high-friction limits of gas dynamics (slides)We propose a linear monolithic finite element method for the approximation of an incompressible viscous fluid interacting with an elastic plate. We employ the full arbitrary Lagrangian–Eulerian (ALE) approach that works in the fixed computational domain. For time discretization, we employ the backward Euler method. For space discretization, we respectively use P1-bubble, P1, and P1 finite elements for the approximation of the fluid velocity, pressure, and structure displacement. We show that our method fulfills the geometrical conservation law and dissipates the total energy on the discrete level. Moreover, we prove the (optimal) linear convergence with respect to the sizes of the time step tau and the mesh h. We present numerical experiments involving a substantially deforming fluid domain that do validate our theoretical results. A comparison with a fully implicit (thus nonlinear) scheme indicates that our semi-implicit linear scheme is faster and as accurate as the fully implicit one.

This is a joint work with B. She and S. Schwarzacher.

The use of spectral trigonometric solvers in image-based computational micromechanics, first introduced by Moulinec and Suquet [1], has become widespread due to their computational efficiency, low memory requirements, and ease of implementation. In this talk, I will focus on the Finite Element (FE)-based reformulation of the original scheme, as pioneered by Schneider et al. [2] and Leuschner and Fritzen [3]. In particular, I will (1) provide a linear algebra perspective on their developments and (2) relate them to recent advances in Laplace preconditioning of elliptic partial differential equations. The adopted approach [4] involves preconditioning the periodic cell problem using a discrete Green's operator derived from a reference problem with constant coefficients. Theoretical analysis shows that the eigenvalues of the preconditioned matrix can be bounded from above and below by coefficients of the original and reference problems. This allows the system to be solved using the preconditioned conjugate gradient method in the number of iterations that are almost independent of the grid size. In terms of implementation, we take advantage of the fact that for generic arbitrary regular meshes, the system matrix of the reference problem exhibits a block-diagonal structure in the Fourier space and can be efficiently inverted using Fast Fourier Transform (FFT) techniques. As a result, the computational complexity of the scheme is dominated by the FFT, making it equivalent to that of spectral solvers. Unlike trigonometric spectral solvers, the proposed scheme works with arbitrary FE shape functions with local supports and does not exhibit the Fourier ringing phenomenon.

References:

- H. Moulinec, and P. Suquet, C R Acad Sci Paris, Vol. 318, pp. 1417--1423, 1994.
- M. Schneider, D. Merkert and M. Kabel, Int J Numer Methods Eng, Vol. 109, pp. 1461--1489, 2017.
- M. Leuschner and F. Fritzen, Comput Mech, Vol. 62, pp. 359--392, 2018.
- M. Ladecký et al., Appl Math Comput, Vol. 446, 127835, 2023.

Can fragmentation behaviour in brittle materials be tailored and/or controlled? Fragmentation (breaking into pieces) is important in applications ranging from high strain rate loading of armour plates and ballistic impacts to meteoritic collisions. Our work focuses on developing closed-form solutions to determine the minimum fragment size when a brittle functionally graded bar (1D) is subjected to a constant strain rate. We solve the second-order hyperbolic wave equation with variable coefficients using homotopy perturbation to determine the displacement fields. Further, the governing equation is recast into two first-order PDEs with the objective of decomposing the displacement field into its forward and backward components (Riemann Variables) that vary along the characteristic curves. By invoking geometric compatibility, stress continuity and using a cohesive law, we formulate the governing ODE represented in terms of the unknown cohesive displacements. This is solved using suitable stability conditions to determine the minimum fragment size alongside the critical time of fragmentation. Our work also demonstrates the role of material properties and functional gradation in the fragmentation response of brittle materials.

Dynamic fragmentation of functional graded brittle materials in 1d (slides)For an incompressible Newtonian fluid flowing around an obstacle, we are interested in the pointwise traction acting on it. To determine the local deformation of a solid obstacle, an accurate traction calculation is required. In addition to the classical approach that involves directly calculating the traction from the Cauchy stress tensor, we investigate the Poincaré-Steklov method, which is based on solving a dual problem and appears to provide more accurate results. Specifically, we show a better convergence rate for the latter method compared to the direct approach. The method is applied to the Turek benchmark, which considers a flow past a rigid cylinder. We also consider a rigid square prism as an obstacle to address non-smooth boundaries and singularities in the flow.

Flow around an obstacle: Various approaches to calculate pointwise traction (slides)In this talk, the numerical approximation of smart fluids is discussed. Smart fluids are characterised by the property that the power-law index is not a fixed constant, but variable dependent. The most common examples for smart fluids are electro-rheological fluids and chemically reacting fluids. More precisely, in this talk, a priori error estimates for a finite element approximation of the p(x)-Navier–Stokes equations, which is prototypical for the class of smart fluids, are derived assuming only appropriate fractional regularity properties of the velocity vector field and the kinematic pressure. Finally, numerical experiments are presented that confirm the quasi-optimality of the error decay rates.

Numerical methods for smart fluids (slides)The driving forces towards the study of fluid structure interaction have long come mainly from the side of fluids. As a result, many of the methods employed are inherited from there as well, with a strong focus on PDE-approaches such as fix-point theorems and Galerkin-approximations and a general danger of treating the solid simply as a "complicated" boundary condition. While these approaches were quite successful so far, they run into serious issues, when there is no convexity and no good linearization of the problem is possible. In contrast, by coming at the problem from the side of the solid, one tends to favor methods based on iterated minimization. While these methods take some extra work to be applied to the setting of fluids, it turns out that once this is done, they work extremely well also with the coupled problem. The aim of this talk is thus to give a short general overview about some of the main recent results involving variational techniques in the context of fluid structure interaction.

Variational aspects of fluid-structure interaction (slides)Viscoelastic rate-type fluid models are popular in many applications involving flows of fluid-like materials with complex micro-structure (e.g. biomaterials, geomaterials, synthetic rubbers). A well-developed mathematical theory for the most of these models is however missing. Our main purpose is to provide a complete proof of long-time and large-data existence of weak solutions to unsteady internal three-dimensional flows of the Giesekus model and its generalizations subject to a no-slip boundary condition.

On unsteady flows of viscoelastic fluids of Giesekus type (slides)In this talk we will review our work on implicitly constituted fluids with M. Bulíček and J. Málek. We will also present some recent results in numerical analysis for different boundary conditions obtained with A. Gazca Orozco, F. Gmeineder and T. Tscherpel.

On different constitutive relations and boundary conditions for fluids (slides)In recent years there has been an increasing activity on boundary value problems driven by a nonhomogeneous differential operator. In this talk, by using variational tools combined with suitable truncation and comparison techniques, we prove an existence and multiplicity result for the positive solutions (a bifurcation-type result) of Robin Boundary value problem which is global in the parameter. This is the joint work with Nikolaos S. Papageorgiou (Department of Mathematics, National Technical University, Zografou Campus, Athens 15780, Greece).

References:

- A. Ambrosetti, H. Brezis, G. Cerami: Combined effects of concave and convex nonlinearities in some elliptic problems. J.Funct. Anal. 122, 519-543 (1994).
- Y. Bai, N. S. Papageorgiou, S. Zeng: Multiplicity Results for Nonlinear Nonhomogeneous Robin Problems with Indefinite Potential Term. Results Math (2023) 78:134 doi:10.1007/s00025-023-01907-5.
- J. Garcia Azorero, I. Peral Alonso, Juan Manfredi: Sobolev versus H ̈older local minimizers and global multiplicity for some quasilinear elliptic equations. Communications in Contemporary Mathematics Vol. 02, No. 03, pp. 385-404 (2000).
- L. Gasinski, N. S. Papageorgiou: Exercises in Analysis. Part 2: Nonlinear Analysis. Springer, Cham (2016).
- L. Gasinski, N. S. Papageorgiou, Y. Zhang. Positive solutions for a class of nonlinear parametric Robin problems. Rend. Circ. Mat. Palermo, II. Ser (2023). https://doi.org/10.1007/s12215-023-00918-z.
- Z. Guo, Z. Zhang: W1,p versus C1 local minimizers and multiplicity results for quasilinear elliptic equations. Journal of Mathematical Analysis and Applications Volume 286, Issue 1, Pages 32-50 (2003).

In this talk, we study the manifold-valued Rudin-Osher-Fatemi (ROF) model, which involves an initial datum mapping from an open 2-dimensional domain to an open connected smooth n-dimensional Riemannian manifold. Firstly, we prove the existence and uniqueness of minimizers (subject to a curvature condition). Secondly, we present a series of regularity results on the associated PDE system of a relaxed functional with Neumann condition. Finally, we apply these results to the ROF model to obtain Lipschitz regularity of minimizers.

On the manifold-valued ROF model (slides)Magnetoelastic materials are smart materials characterised by a strong interplay between their mechanical and magnetic properties. Due to their capability to change shape in response to applied magnetic fields, they currently find use in many technological applications requiring a magnetomechanical transducer, e.g., actuators and sensors. In this talk, we consider the finite element approximation of a nonlinear system of PDEs modeling the dynamics of magnetization and displacement in magnetoelastic materials in the small strain regime. We present a fully discrete structure-preserving numerical scheme and discuss its analysis (well-posedness, discrete energy law, stability, unconditional convergence). This is joint work with Hywel Normington (University of Strathclyde).

Finite element methods for magnetoelastic materials (slides)The bending resistance of interfaces is often modeled by penalizing their curvature. This leads to a class of Sobolev-critical curvature functionals, including the Willmore energy and the Canham-Helfrich energy. One variational approach to handle the critical growth is to consider a weak class of surfaces given by (curvature) varifolds. In this talk, we will discuss a recent global regularity result for varifolds whose energy is below a sharp explicit threshold. This is based on joint work with C. Scharrer (Bonn).

Regularity of surfaces with nearly minimal bending (slides)We consider the Navier-Stokes-like system describing an incompressible fluid in a bounded Lipschitz domain or an infinite channel. It is coupled with the so-called dynamic slip boundary condition, i.e. the boundary condition contains the time derivative. We will summarize our results on the existence of finite-dimensional attractors for such a problem together with a specific upper bound of its fractal dimension.

On the Navier-Stokes like system with the dynamic slip boundary condition (slides)We present three novel classifications of the weak sequential (and strong) limits in $W^{1,p}$ of planar diffeomorphisms. We introduce a concept called QM condition which is a kind of separation property for pre-images of closed connected sets and show that $u$ satisfies this property exactly when it is the limit of Sobolev homeomorphisms. Further, we prove that $u \in W^{1,p}_{\operatorname{id}}((-1,1)^2, \mathbb{R}^2)$ is the limit of a sequence of homeomorphisms exactly when there are classically monotone mappings $g_{delta}:[-1,1]^2 \to \mathbb{R}^2$ and very small open sets $U_{delta}$ such that $g_{\delta} = u$ on $[-1,1]^2 \setminus U_{delta}$. Also, we introduce the so-called the three curve condition, which is in some sense reminiscent of the NCL condition introduced by Campbell Pratelli and Radici but for $u^{-1}$ instead of for $u$, and prove that a map is the $W^{1,p}$ limit of planar Sobolev homeomorphisms exactly when it satisfies this property. This improves on results by De Philippis and Pratelli answering a question of Iwaniec and Onninen.

hp-adaptive finite element methods allow to choose both the mesh size h and the polynomial degree p locally on every cell. Despite their excellent convergence properties, they are not widely used, possibly, due to their challenging implementation. Recent comprehensive support for hp-FEM in the deal.II library attempts to close this gap with hybrid multigrid preconditioning and matrix-free methods on parallel distributed systems. We investigate performance and scalability using realistic computations as test cases.

It is known that time-periodic hyperbolic problems lack the regularizing effect, but how much regularity of the data is needed to obtain a solution? The answer heavily depends on the geometry of the underlying domain. Criteria will be presented that describe admissible shapes of domains and quantify the regularity loss.

We study the bifurcation behaviour of viscoelastic fluid flows. In particular, we investigate steady states of Giesekus and FENE-CR fluid in a planar sudden expansion geometry with expansion ratio 1:4 where symmetry-breaking bifurcation occurs above critical Reynolds number. For bifurcation analysis, we use the deflated continuation method which combines the deflation techniques (find multiple solutions at fixed parameter values) with continuation methods (extend solution branches). During continuation and deflation, the left Cauchy--Green tensor associated with the elastic part of the fluid response loses positive definiteness. To preserve its positive definiteness, we solve reformulated equations for the matrix logarithm of the left Cauchy--Green tensor. The problem is solved by finite element method. In addition, we apply finite element stabilization techniques (DEVSS-TG, SUPG) used for viscoelastic flows. The numerical methods are implemented using Firedrake and Defcon library.

We consider the incompressible Navier-Stokes-Fourier system describing the motion of a non-Newtonian fluid in the two-dimensional bounded domain. In addition, the system is equipped with the entropy equation. We define the notion of a solution and prove its existence. We approach the problem by modifying techniques used in several papers studying the generalized NSF system and the entropy equation. Since we are treating the two-dimensional case as opposed to the more frequent 3D case, we are able to relax conditions on the initial data.