arXiv subjects

Sebastian Schwarzacher

Publications and source records attributed to Sebastian Schwarzacher.

At least 19 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

Range Failure, Resolvent Growth, and Resonance in Partially Dissipative Systems

We introduce several notions of resonance for systems exhibiting weak or partial dissipation: $\dot u=Au+f(t)$, with $A$ the generator of a strongly stable semigroup $S(t)$ on a Hilbert space $H$. We refer to resonance as the phenomenon where a time-periodic $f$ may yield an unbounded $u$, classically occurring when $A$ has imaginary eigenvalues. In infinite dimensions, however, unbounded growth may depend on topological choices, and several sorts of``resonance" may occur. Classically, existence of $T$-periodic $u$ under $T$-periodic $f$ is equivalent to a range condition: $\mathcal{R}(I-S(T))=H$. Consequently, its failure permits unbounded solution growth and we elucidate that connection through properties of $S(t)$. We describe a hierarchy of range failures, and provide an associated resonance taxonomy. The growth rate of ~$||(A-λI)^{-1}||_H$ on ~$i\R$ dictates a regularity gap between $u$ and $f$, and, for rapid growth in $λ$, that gap can be infinite. In said case, we demonstrate implications for periodic solvability, and a mechanism for constructing smooth resonant forces. The theory developed here is motivated by (and demonstrated for) a hyperbolic-parabolic system, where the latter component provides the only system dissipation. Such dynamics are a simplification of fluid-structure phenomena, which demonstrate strong stability but do not support unconditional periodic well-posedness. Though point-spectral resonance is ruled out, we note that $\mathcal R(I-S(T))\neq H$. Our main result here shows that, for a carefully constructed geometry, a classical heat-wave system possesses periods $T$ which yield infinite derivative loss and, via supporting results, yield resonance with smooth forcing.

math.AP↗

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↗

arXiv2026

Well-posedness theorems in fluid-structure interaction: perfectly elastic shells

In this work, we consider the interaction of a 3D incompressible fluid with a 2D flexible shell that occupies (a part of) the boundary of the fluid domain. We assume that the shell is perfectly elastic while the fluid is governed by the Navier--Stokes equations. Consequently, damping within the coupled system comes entirely from the parabolic fluid subsystem. Our main result is the construction of a local-in-time unique strong solution to the system of PDEs. Standard techniques from the literature do not apply here. They are restricted to visco-elastic structures, where the corresponding solid phase is parabolic. Our construction relies on a different method built upon a new estimate for the acceleration of the system. In the case of a 2D viscous incompressible fluid interacting with a 1D perfectly elastic shell we can extend the local solution globally in time (until a possible self-intersection of the shell).

math.AP↗

arXiv2026

Fluid-Structure interactions with Navier- and full-slip boundary conditions

We show the existence of weak solutions to the fluid-structure interaction problem of a largely deforming viscoelastic bulk solid with a viscous fluid governed by the incompressible Navier-Stokes equations. In contrast to previous works, the fluid is allowed to slip along the solid boundary; namely, the so called Navier-slip boundary conditions are considered. Such boundary conditions naturally involve the time-changing outer normal of the fluid domain. Hence, their dependence on the varying geometry is one degree higher than in the previously considered no-slip case, which makes it necessary to adjust the concept of weak coupled solutions. Two classes of test functions are introduced: test functions that are continuous over the fluid-solid domain, and fluid-only test functions with nonzero tangential component at the boundary. The weak equations are established until the point of contact, and moreover, compatibility with the strong formulation is shown.

math.AP↗

arXiv2026

Thermal effects in fluid structure interactions

In this article we consider two different heat conducting fluids each modelled by the incompressible Navier-Stokes-Fourier system separated by a non-linear elastic Koiter shell. The motion of the shell changes the domain of definition of the two separated fluids. For this setting we show the existence of a weak solution. The heat capacity of the shell is given energetically. It allows to consider transmission laws ranging from insulation to superconductivity.We follow a variational approach for fluid-structure interactions. To include temperature a novel two step minimization scheme is used to produce an approximation. The weak solutions are energetically closed and include a strictly positive temperature.

math.AP↗

arXiv2025

Time-Periodic Solutions for Hyperbolic-Parabolic Systems

Time-periodic weak solutions for a coupled hyperbolic-parabolic system are obtained. A linear heat and wave equation are considered on two respective $d$-dimensional spatial domains that share a common $(d-1)$-dimensional interface $Γ$. The system is only partially damped, leading to an indeterminate case for existing theory (Galdi et al., 2014). We construct periodic solutions by obtaining novel a priori estimates for the coupled system, reconstructing the total energy via the interface $Γ$. As a byproduct, geometric constraints manifest on the wave domain which are reminiscent of classical boundary control conditions for wave stabilizability. We note a ``loss" of regularity between the forcing and solution which is greater than that associated with the heat-wave Cauchy problem. However, we consider a broader class of spatial domains and mitigate this regularity loss by trading time and space differentiations, a feature unique to the periodic setting. This seems to be the first constructive result addressing existence and uniqueness of periodic solutions in the heat-wave context, where no dissipation is present in the wave interior. Our results speak to the open problem of the (non-)emergence of resonance in complex systems, and are readily generalizable to related systems and certain nonlinear cases.

math.AP↗

arXiv2025

The Stokes problem with Navier boundary conditions in irregular domains

We consider the steady Stokes equations supplemented with Navier boundary conditions including a non-negative friction coefficient. We prove maximal regularity estimates (including the prominent spaces $W^{1,p}$ and $W^{2,p}$ for $1<p<\infty$ for the velocity field) in bounded domains of minimal regularity. Interestingly, exactly one derivative more is required for the local boundary charts compared to the case of no-slip boundary conditions. We demonstrate the sharpness of our results by a propos examples.

math.AP↗

arXiv2024

Regularity estimates of a fluid-free surface evolution

In this work the evolution of a fluid droplet in vacuum is considered. This means that the surface tension and the fluid forces are in equilibrium at the free boundary. The fluid is governed by the incompressible quasi-steady Stokes equation. We present higher order energy estimates for this setting in the planar case. In particular bounds of the curvature and its tangential derivative combined with the second and third spacial derivatives of the fluid velocity as respective dissipation. These estimates are shown to hold until the point of a topological degeneracy. They provide quantitative bounds, that depend on specific properties of the initial geometry only. The work contrasts previous approaches, which are based on the use of local coordinates and instead performs all estimates in an Eulerian setting. Indeed, the estimates provided here are geometrically intrinsic and collapse only once these intrinsic qualities break.

math.AP↗

arXiv2024

Stability and convergence of in time approximations of hyperbolic elastodynamics via stepwise minimization

We study step-wise time approximations of non-linear hyperbolic initial value problems. The technique used here is a generalization of the minimizing movements method, using two time-scales: one for velocity, the other (potentially much larger) for acceleration. The main applications are from elastodynamics namely so-called generalized solids, undergoing large deformations. The evolution follows an underlying variational structure exploited by step-wise minimisation. We show for a large family of (elastic) energies that the introduced scheme is stable; allowing for non-linearities of highest order. If the highest order can assumed to be linear, we show that the limit solutions are regular and that the minimizing movements scheme converges with optimal linear rate. Thus this work extends numerical time-step minimization methods to the realm of hyperbolic problems.

math.NA↗

arXiv2024

Existence of strong solutions for a perfect elastic beam interacting with Navier-Stokes equations

A perfectly elastic beam is situated on top of a two dimensional fluid canister. The beam is deforming in accordance to an interaction with a Navier-Stokes fluid. Hence a hyperbolic equation is coupled to the Navier-Stokes equation. The coupling is partially of geometric nature, as the geometry of the fluid domain is changing in accordance to the motion of the beam. Here the existence of a unique strong solution for large initial data and all times up to geometric degeneracy is shown. For that an a-priori estimate on the time-derivative of the coupled solution is introduced. For the Navier-Stokes part it is a borderline estimate in the spirit of Ladyzhenskaya applied directly to the in-time differentiated system.

math.AP↗

arXiv2024

Time-periodic weak solutions for the interaction of an incompressible fluid with a linear Koiter type shell under dynamic pressure boundary conditions

In many occurrences of fluid-structure interaction time-periodic motions are observed. We consider the interaction between a fluid driven by the three dimensional Navier-Stokes equation and a two dimensional linearized elastic Koiter shell situated at the boundary. The fluid-domain is a part of the solution and as such changing in time periodically. On a steady part of the boundary we allow for the physically relevant case of dynamic pressure boundary values, prominent to model inflow/outflow. We provide the existence of at least one weak time-periodic solution for given periodic external forces that are not too large. For that we introduce new approximation techniques and a-priori estimates.

math.AP↗

arXiv2023

Ladyzhenskaya-Prodi-Serrin condition for fluid-structure interaction systems

We consider the interaction of a viscous incompressible fluid with a flexible shell in three space dimensions. The fluid is described by the three-dimensional incompressible Navier--Stokes equations in a domain that is changing in accordance with the motion of the structure. The displacement of the latter evolves along a visco-elastic shell equation. Both are coupled through kinematic boundary conditions and the balance of forces. We prove a counterpart of the classical Ladyzhenskaya-Prodi-Serrin condition yielding conditional regularity and uniqueness of a solution. Our result is a consequence of the following three ingredients which might be of independent interest: {\bf (i)} the existence of local strong solutions, {\bf (ii)} an acceleration estimate (under the Serrin assumption) ultimately controlling the second-order energy norm, and {\bf (iii)} a weak-strong uniqueness theorem. The first point, and to some extent, the last point were previously known for the case of elastic plates, which means that the relaxed state is flat. We extend these results to the case of visco-elastic shells, which means that more general reference geometries are considered such as cylinders or spheres. The second point, i.e. the acceleration estimate for three-dimensional fluids is new even in the case of plates.

math.AP↗

arXiv2023

Construction of a right inverse for the divergence in non-cylindrical time dependent domains

We construct a stable right inverse for the divergence operator in non-cylindrical domains in space-time. The domains are assumed to be Hölder regular in space and evolve continuously in time. The inverse operator is of Bogovskij type, meaning that it attains zero boundary values. We provide estimates in Sobolev spaces of positive and negative order with respect to both time and space variables. The regularity estimates on the operator depend on the assumed Hölder regularity of the domain. The results can naturally be connected to the known theory for Lipschitz domains. As an application, we prove refined pressure estimates for weak and very weak solutions to Navier--Stokes equations in time dependent domains.

math.AP↗

arXiv2023

Stability and error estimates of a linear numerical scheme approximating nonlinear fluid-structure interactions

In this paper, we propose a linear and monolithic finite element method for the approximation of an incompressible viscous fluid interacting with an elastic and deforming plate. We use the arbitrary Lagrangian-Eulerian (ALE) approach that works in the reference domain, meaning that no re-meshing is needed during the numerical simulation. 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 $τ$ 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, at least in stable configurations.

math.NA↗

arXiv2022

Existence and regularity of weak solutions for a fluid interacting with a non-linear shell in three dimensions

We study the unsteady incompressible Navier-Stokes equations in three dimensions interacting with a non-linear flexible shell of Koiter type. This leads to a coupled system of non-linear PDEs where the moving part of the boundary is an unknown of the problem. The known existence theory for weak solutions is extended to non-linear Koiter shell models. We introduce a-priori estimates that reveal higher regularity of the shell displacement beyond energy estimates. These are essential for non-linear Koiter shell models, since such shell models are non-convex (w.r.t.\ terms of highest order). The estimates are obtained by introducing new analytical tools that allow to exploit dissipative effects of the fluid for the (non-dissipative) solid. The regularity result depends on the geometric constitution alone and is independent of the approximation procedure; hence it holds for arbitrary weak solutions. The developed tools are further used to introduce a generalized Aubin-Lions type compactness result suitable for fluid-structure interactions.

math.AP↗

arXiv2022

Unrestricted deformations of thin elastic structures interacting with fluids

In this paper we discuss the motion of a beam in interaction with fluids. We allow the beam to move freely in all coordinate directions. We consider the case of a beam situated in between two different fluids as well as the case where the beam is attached only to one fluid. In both cases the fluid-domain is time changing. The fluid is governed by the incompressible Navier-Stokes equations. The beam is elastic and governed by a hyperbolic partial differential equation. In order to allow for large deformations the elastic potential of the beam is non-quadratic and naturally possesses a non-convex state space. We derive the existence of weak-solutions up to the point of a potential collision.

math.AP↗

arXiv2022

Time-periodic weak solutions for an incompressible Newtonian fluid interacting with an elastic plate

Under the action of a time-periodic external forces we prove the existence of at least one time-periodic weak solution for the interaction between a three-dimensional incompressible fluid, governed by the Navier- Stokes equation and a two dimensional elastic plate. The challenge is that the Eulerian domain for the fluid changes in time and is a part of the solution. We introduce a two fixed-point methodology: First we construct a time-periodic solutions for a given variable time-periodic geometry. Then in a second step a (set-valued) fixed point is performed w.r.t.\ the geometry of the domain. The existence relies on newly developed a-priori estimates applicable for both coupled and uncoupled variable geometries. Due to the expected weak regularity of the solutions such Eulerian estimates are unavoidable. Note in particular, that only the fluid is assumed to be dissipative. But the here produced a-priori estimates show that its possible to exploit the dissipative effects of the fluid also for the solid deformation. The existence of periodic solutions for a given geometry is valid for arbitrary large data. The existence of periodic coupled solutions to the fluid-structure interaction is valid for all data that excludes a self-intersection a-priori.

math.AP↗