arXiv subjects

Stefan Frei

Publications and source records attributed to Stefan Frei.

At least 19 recordsLinked to original sources

arXiv2026

Finite-Dimensional MOR-Based RHC for Steering 2D Navier-Stokes Equations to Desired Trajectories

This paper investigates the local exponential stabilization of the two-dimensional Navier--Stokes equations to a given reference trajectory by means of receding horizon control (RHC). The control is realized as a linear combination of finitely many actuators, represented by indicator functions supported on subsets of a prescribed control subdomain. We establish local exponential stabilizability and suboptimality for the resulting RHC scheme. Numerical experiments for two flow configurations of increasing complexity illustrate the theoretical findings and assess the practical performance of the method. In addition, we propose a model-order-reduced RHC approach based on proper orthogonal decomposition, which significantly reduces the computational cost while maintaining favorable closed-loop stabilization performance in the numerical experiments.

math.OC↗

arXiv2026

Numerical Analysis of a Cut Finite Element Approach for Fully Eulerian Fluid-Structure Interaction with Fixed Interface

This work develops and analyzes a variational-monolithic unfitted finite element formulation of a linear fluid-structure interaction problem in Eulerian coordinates with a fixed interface. The overall discretization is based on a backward Euler scheme in time and finite elements in space. For the spatial discretization we employ a cut finite element method on a mesh consisting of quadrilateral elements. We use a first-order in time formulation of the elasticity equations, inf-sup stable finite elements in the fluid part and Nitsche's method to incorporate the coupling conditions. Ghost penalty terms guarantee the robustness of the approach independently of the way the interface cuts the finite element mesh. The main objective is to establish stability and a priori error estimates. We prove optimal-order error estimates in space and time and substantiate them with numerical tests.

math.NA↗

arXiv2026

Analysis and numerical simulation of a spatio-temporal Ricker-type model for the control of Aedes aegypti mosquitoes with Sterile Insect Techniques

Sterile Insect Technique (SIT) is widely regarded as a promising, environmentally friendly and chemical-free strategy for the prevention and control of dengue and other vector-borne diseases. In this paper, we develop and analyze a spatio-temporal reaction-diffusion model describing the dynamics of three mosquito subpopulations involved in SIT-based biological control of Aedes aegypti mosquitoes. Our sex-structured model explicitly incorporates fertile females together with fertile and sterile males that compete for mating. Its key features include spatial mosquito dispersal and the incorporation of spatially heterogeneous external releases of sterile individuals. We establish the existence and uniqueness of global, non negative, and bounded solutions, guaranteeing the mathematical well-posedness and biological consistency of the system. A fully discrete numerical scheme based on the finite element method and an implicit-explicit time-stepping scheme is proposed and analyzed. Numerical simulations confirm the presence of a critical release-size threshold governing eradication versus persistence at a stable equilibrium with reduced total population size, in agreement with the underlying ODE dynamics. Moreover, the spatial structure of the model allows us to analyze the impact of spatial distributions, heterogeneous releases, and periodic impulsive control strategies, providing insight into the optimal spatial and temporal deployment of SIT-based interventions.

math.NA↗

arXiv2025

Numerical Simulations of Fully Eulerian Fluid-Structure Contact Interaction using a Ghost-Penalty Cut Finite Element Approach

In this work, we develop a cut-based unfitted finite element formulation for solving nonlinear, nonstationary fluid-structure interaction with contact in Eulerian coordinates. In the Eulerian description fluid flow modeled by the incompressible Navier-Stokes equations remains in Eulerian coordinates, while elastic solids are transformed from Lagrangian coordinates into the Eulerian system. A monolithic description is adopted. For the spatial discretization, we employ an unfitted finite element method with ghost penalties based on inf-sup stable finite elements. To handle contact, we use a relaxation of the contact condition in combination with a unified Nitsche approach that takes care implicitly of the switch between fluid-structure interaction and contact conditions. The temporal discretization is based on a backward Euler scheme with implicit extensions of solutions at the previous time step. The nonlinear system is solved with a semi-smooth Newton's method with line search. Our formulation, discretization and implementation are substantiated with an elastic falling ball that comes into contact with the bottom boundary, constituting a challenging state-of-the-art benchmark.

math.NA↗

arXiv2024

A non-intrusive neural-network based BFGS algorithm for parameter estimation in non-stationary elasticity

We present a non-intrusive gradient and a non-intrusive BFGS algorithm for parameter estimation problems in non-stationary elasticity. To avoid multiple (and potentially expensive) solutions of the underlying partial differential equation (PDE), we approximate the PDE solver by a neural network within the algorithms. The network is trained offline for a given set of parameters. The algorithms are applied to an unsteady linear-elastic contact problem; their convergence and approximation properties are investigated numerically.

math.NA↗

arXiv2024

Attached and separated rotating flow over a finite height ridge

This paper discusses the effect of rotation on the boundary layer in high Reynolds number flow over a ridge using a numerical method based on stabilised finite elements that captures steady solutions up to Reynolds number of order $10^6$. The results are validated against boundary layer computations in shallow flows and for deep flows against experimental observations reported in Machicoane et al. (Phys. Rev. Fluids, 2018). In all cases considered the boundary layer remains attached, even at large Reynolds numbers, provided the Rossby number of the flow is sufficiently small. At any fixed Rossby number the flow detaches at sufficiently high Reynolds number to form a steady recirculating region in the lee of the ridge. At even higher Reynolds numbers no steady flow is found. This disappearance of steady solutions closely reproduces the transition to unsteadiness seen in the laboratory.

physics.flu-dyn↗

arXiv2024

Modeling and numerical simulation of fully Eulerian fluid-structure interaction using cut finite elements

We present a monolithic finite element formulation for (nonlinear) fluid-structure interaction in Eulerian coordinates. For the discretization we employ an unfitted finite element method based on inf-sup stable finite elements. So-called ghost penalty terms are used to guarantee the robustness of the approach independently of the way the interface cuts the finite element mesh. The resulting system is solved in a monolithic fashion using Newton's method. Our developments are tested on a numerical example with fixed interface.

math.NA↗

arXiv2023

A locally modified second-order finite element method for interface problems and its implementation in 2 dimensions

The locally modified finite element method, which is introduced in [Frei, Richter: SINUM 52(2014), p. 2315-2334], is a simple fitted finite element method that is able to resolve weak discontinuities in interface problems. The method is based on a fixed structured coarse mesh, which is then refined into sub-elements to resolve an interior interface. In this work, we extend the locally modified finite element method {in two space dimensions} to second order using an isoparametric approach in the interface elements. Thereby we need to take care that the resulting curved edges do not lead to degenerate sub-elements. We prove optimal a priori error estimates in the $L^2$-norm and in a discrete energy norm. Finally, we present numerical examples to substantiate the theoretical findings.

math.NA↗

arXiv2023

Analysis of an implicitly extended Crank-Nicolson scheme for the heat equation on a time-dependent domain

We consider a time-stepping scheme of Crank-Nicolson type for the heat equation on a moving domain in Eulerian coordinates. As the spatial domain varies between subsequent time steps, an extension of the solution from the previous time step is required. Following Lehrenfeld \& Olskanskii [ESAIM: M2AN, 53(2):\,585-614, 2019], we apply an implicit extension based on so-called ghost-penalty terms. For spatial discretisation, a cut finite element method is used. We derive a complete a priori error analysis in space and time, which shows in particular second-order convergence in time under a parabolic CFL condition. Finally, we present numerical results in two and three space dimensions that confirm the analytical estimates, even for much larger time steps.

math.NA↗

arXiv2023

Towards parallel time-stepping for the numerical simulation of atherosclerotic plaque growth

The numerical simulation of atherosclerotic plaque growth is computationally prohibitive, since it involves a complex cardiovascular fluid-structure interaction (FSI) problem with a characteristic time scale of milliseconds to seconds, as well as a plaque growth process governed by reaction-diffusion equations, which takes place over several months. In this work we combine a temporal homogenization approach, which separates the problem in computationally expensive FSI problems on a micro scale and a reaction-diffusion problem on the macro scale, with parallel time-stepping algorithms. It has been found in the literature that parallel time-stepping algorithms do not perform well when applied directly to the FSI problem. To circumvent this problem, a parareal algorithm is applied on the macro-scale reaction-diffusion problem instead of the micro-scale FSI problem. We investigate modifications in the coarse propagator of the parareal algorithm, in order to further reduce the number of costly micro problems to be solved. The approaches are tested in detailed numerical investigations based on serial simulations.

math.NA↗

arXiv2022

Efficient coarse correction for parallel time-stepping in plaque growth simulations

In order to make the numerical simulation of atherosclerotic plaque growth feasible, a temporal homogenization approach is employed. The resulting macro-scale problem for the plaque growth can be further accelerated by using parallel time integration schemes, such as the parareal algorithm. However, the parallel scalability is dominated by the computational cost of the coarse propagator. Therefore, in this paper, an interpolation-based coarse propagator, which uses growth values from previously computed micro-scale problems, is introduced. For a simple model problem, it is shown that this approach reduces both the computational work for a single parareal iteration as well as the required number of parareal iterations.

math.NA↗

arXiv2021

On temporal homogenization in the numerical simulation of atherosclerotic plaque growth

A temporal homogenization approach for the numerical simulation of atherosclerotic plaque growth is extended to fully coupled fluid-structure interaction (FSI) simulations. The numerical results indicate that the two-scale approach yields significantly different results compared to a simple heuristic averaging, where only stationary long-scale FSI problems are solved, confirming the importance of incorporating stress variations on small time-scales. In the homogenization approach, a periodic fine-scale problem, which is periodic with respect to the heart beat, has to be solved for each long-scale time step. Even if no exact initial conditions are available, periodicity can be achieved within only 2-3 heart beats by simple time-stepping.

math.NA↗

arXiv2021

A mechanically consistent model for fluid-structure interactions with contact including seepage

We present a new approach for the mechanically consistent modelling and simulation of fluid-structure interactions with contact. The fundamental idea consists of combining a relaxed contact formulation with the modelling of seepage through a porous layer of co-dimension 1 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. In combination with a relaxation of the contact conditions the computational model is both mechanically consistent and simple to implement. We analyse the approach in detailed numerical studies with both thick- and thin-walled solids, within a fully Eulerian and an immersed approach for the fluid-structure interaction and using fitted and unfitted finite element discretisations.

math.NA↗

arXiv2020

Eulerian time-stepping schemes for the non-stationary Stokes equations on time-dependent domains

This article is concerned with the discretisation of the Stokes equations on time-dependent domains in an Eulerian coordinate framework. Our work can be seen as an extension of a recent paper by Lehrenfeld & Olshanskii [ESAIM: M2AN, 53(2):585-614, 2019], where BDF-type time-stepping schemes are studied for a parabolic equation on moving domains. For space discretisation, a geometrically unfitted finite element discretisation is applied in combination with Nitsche's method to impose boundary conditions. Physically undefined values of the solution at previous time-steps are extended implicitly by means of so-called ghost penalty stabilisations. We derive a complete a priori error analysis of the discretisation error in space and time, including optimal $L^2(L^2)$-norm error bounds for the velocities. Finally, the theoretical results are substantiated with numerical examples.

math.NA↗

arXiv2020

Falling balls in a viscous fluid with contact: Comparing numerical simulations with experimental data

We evaluate a number of different finite element approaches for fluid-structure (contact) interaction problems against data from physical experiments. For this we take the data from experiments by Hagemeier [Mendeley Data, doi: 10.17632/mf27c92nc3.1]. This consists of trajectories of single particles falling through a highly viscous fluid and rebounding off the bottom fluid tank wall. The resulting flow is in the transitional regime between creeping and turbulent flows. This type of configuration is particularly challenging for numerical methods due to the large change of the fluid domain and the contact between the wall and particle. In the numerical simulations we consider both rigid body and linear elasticity models for the falling particles. In the first case, we compare results obtained with the well established Arbitrary Lagrangian Eulerian (ALE) approach and a moving domain CutFEM method together with a simple and common approach for contact avoidance. For the full fluid-structure interaction (FSI) problem with contact, we use a fully Eulerian approach in combination with a unified FSI-contact treatment using Nitsche's method. For higher computational efficiency we use the geometrical symmetry of the experimental set up to reformulate the FSI system into two spatial dimensions. Finally, we show full three dimensional ALE computations to study the effects of small perturbations in the initial state of the particle to investigate deviations from a perfectly vertical fall observed in the experiment. The methods are implemented in open-source finite element libraries and the results are made freely available to aide reproducibility.

physics.flu-dyn↗

arXiv2020

Weak imposition of Signorini boundary conditions on the boundary element method

We derive and analyse a boundary element formulation for boundary conditions involving inequalities. In particular, we focus on Signorini contact conditions. The Calderón projector is used for the system matrix and boundary conditions are weakly imposed using a particular variational boundary operator designed using techniques from augmented Lagrangian methods. We present a complete numerical a priori error analysis and present some numerical examples to illustrate the theory.

math.NA↗

arXiv2020

Efficient approximation of flow problems with multiple scales in time

In this article we address flow problems that carry a multiscale character in time. In particular we consider the Navier-Stokes flow in a channel on a fast scale that influences the movement of the boundary which undergoes a deformation on a slow scale in time. We derive an averaging scheme that is of first order with respect to the ratio of time-scales $ε$. In order to cope with the problem of unknown initial data for the fast scale problem, we assume near-periodicity in time. Moreover, we construct a second-order accurate time discretisation scheme and derive a complete error analysis for a corresponding simplified ODE system. The resulting multiscale scheme does not ask for the continuous simulation of the fast scale variable and shows powerful speed-ups up to 1:10000 compared to a resolved simulation. Finally, we present some numerical examples for the full Navier-Stokes system to illustrate the convergence and performance of the approach.

math.NA↗

arXiv2019

3D-2D Stokes-Darcy coupling for the modelling of seepage with an application to fluid-structure interaction with contact

In this note we introduce a mixed dimensional Stokes-Darcy coupling where a $d$ dimensional Stokes' flow is coupled to a Darcy model on the $d-1$ dimensional boundary of the domain. The porous layer introduces tangential creeping flow along the boundary and allows for the modelling of boundary flow due to surface roughness. This leads to a new model of flow in fracture networks with reservoirs in an impenetrable bulk matrix. Exploiting this modelling capability, we then formulate a fluid-structure interaction method with contact, where the porous layer allows for mechanically consistent contact and release. Physical seepage in the contact zone due to rough surfaces is modelled by the porous layer. Some numerical examples are reported, both on the Stokes'-Darcy coupling alone and on the fluid-structure interaction with contact in the porous boundary layer.

math.NA↗