arXiv subjects

Andreas Schröder

Publications and source records attributed to Andreas Schröder.

14 recordsLinked to original sources

arXiv2026

A posteriori error estimates for variational inequalities of the second kind in an abstract framework

In this paper, an abstract framework is introduced for the derivation of reliable a posteriori error estimators for variational inequalities of the second kind. Under certain conditions (local) efficiency of the error estimators can also be derived. The framework is based on the estimation of a residuum which results from an equivalent mixed formulation and from the replacement of the Lagrange multiplier by its (possibly post-processed) discrete counterpart. The approach is applied to an idealized frictional contact problem and a Mosolov type model problem in an $hp$-finite element discretization setup. Several numerical experiments demonstrate the broad applicability of the theoretical findings.

math.NA↗

arXiv2026

Wake interactions drive synchronized vortex merging in a hovering quadcopter

The most energetic coherent structure of a hovering full-scale quadcopter is associated with a self-organizing process in which the individual rotor vortices synchronize their frequencies while undergoing merging events, yielding a globally correlated structure. We identify and characterize this phenomenon by applying spectral modal and conditional analyses to assimilated three-dimensional velocity data acquired via Shake-The-Box Lagrangian particle tracking. The dataset captures a high-Re, turbulent flow further complicated by time-varying rotor speeds stemming from active flight control, low-frequency vehicle drift, finite spatio-temporal resolution, and measurement uncertainty. Most coherent structures recover established single-rotor features such as tip vortices and their subharmonic pairing. The globally synchronized vortex merging manifests as a spectral peak at an incommensurate frequency below the rotor band, which cannot be explained by single-rotor aerodynamics, subharmonic instabilities, or band-to-band triadic interactions. Instead, conditional averaging provides evidence of the aforementioned intermittent, distinctly non-subharmonic vortex-merging process involving all four rotor wakes. Establishing whether or not this phenomenon is observed across different flight conditions and configurations remains speculative; however, the consistent characterization of the globally synchronized vortex merging using complementary frequency- and time-domain analyses despite experimental complexities, in particular rotor speed variations, demonstrates its robustness.

physics.flu-dyn↗

arXiv2026

Lagrangian Curvature Statistics from Gaussian Subensembles in Turbulent Flows

A salient feature of fully turbulent flows far from onset is the intermittent occurrence of extreme fluctuations at small spatial and temporal scales. These have a qualitative and quantitative effect on the instantaneous curvature of a tracer particle trajectory as an intrinsically multi-scale observable. Here, we provide a complete statistical description of the curvature of tracer particle trajectories in turbulent flows that includes and quantifies intermittency effects. We derive an exact expression and a closed-form approximation for the curvature probability density function, both agree well with data obtained from laboratory experiments of different types of turbulent flows, and quantify the generic behavior of the system. The method can be extended to more complex systems such as plasma turbulence.

physics.flu-dyn↗

arXiv2026

hp-Finite Elements for Elastoplasticity

This article considers a model problem of elastoplasticity with linearly kinematic hardening and presents hp-finite element discretizations of two equivalent weak formulations each having their respective advantages. A mixed variational formulation is introduced to resolve the non-differentiablility of the so-called plasticity functional appearing in the weak formulation of the model problem as a variational inequality of the second kind. The discretization of the mixed formulation is then represented as a system of decoupled nonlinear equations which allows the application of an efficient semismooth Newton solver. Finally, an a priori and a posteriori error analysis is given.

math.NA↗

arXiv2025

A Semismooth Newton Solver and its application to an $hp$-FE Discretization in Elastoplasticity

In this paper, we consider a class of systems of nonlinear equations, which arise in discretized mixed formulations of problems in solid mechanics by $hp$-finite elements. We introduce a semismooth Newton solver for this specific class and prove its well-definedness and local convergence. Thereby, the analysis heavily relies on a special eigenvalue interplay of two matrices involved in the considered nonlinear system. Next, we apply the general results to an $hp$-finite element discretization of a problem in elastoplasticity, which can be formulated as a system of nonlinear equations of the above type by using biorthogonal basis functions. Finally, numerical examples demonstrate the applicability and robustness of the proposed semismooth Newton solver with respect to $h$ and $p$.

math.NA↗

arXiv2024

Error estimates for perturbed variational inequalities of the first kind

In this paper, we derive a priori error estimates for variational inequalities of the first kind in an abstract framework. This is done by combining the first Strang Lemma and the Falk Theorem. The main application consists in the derivation of a priori error estimates for Galerkin methods, in which "variational crimes" may perturb the underlying variational inequality. Different types of perturbations are incorporated into the abstract framework and discussed by various examples. For instance, the perturbation caused by an inexact quadrature is examined in detail for the Laplacian obstacle problem. For this problem, guaranteed rates for the approximation error resulting from the use of higher-order finite elements are derived. In numerical experiments, the influence of the number of quadrature points on the approximation error and on the quadrature-related error itself is studied for several discretization methods.

math.NA↗

arXiv2024

On an optimal AFEM for elastoplasticity

In this paper, optimal convergence for an adaptive finite element algorithm for elastoplasticity is considered. To this end, the proposed adaptive algorithm is established within the abstract framework of the axioms of adaptivity [Comput. Math. Appl., 67(6) (2014), 1195-1253], which provides a specific proceeding to prove the optimal convergence of the scheme. The proceeding is based on verifying four axioms, which ensure the optimal convergence. The verification is done by using results from [Numer. Math., 132(1) (2016), 131-154], which presents an alternative approach to optimality without explicitly relying on the axioms.

math.NA↗

arXiv2024

Mixed Finite Elements of Higher-Order in Elastoplasticity

In this paper a higher-order mixed finite element method for elastoplasticity with linear kinematic hardening is analyzed. Thereby, the non-differentiability of the involved plasticity functional is resolved by a Lagrange multiplier leading to a three field formulation. The finite element discretization is conforming in the displacement field and the plastic strain but potentially non-conforming in the Lagrange multiplier as its Frobenius norm is only constrained in a certain set of Gauss quadrature points. A discrete inf-sup condition with constant 1 and the well posedness of the discrete mixed problem are shown. Moreover, convergence and guaranteed convergence rates are proved with respect to the mesh size and the polynomial degree, which are optimal for the lowest order case. Numerical experiments underline the theoretical results.

math.NA↗

arXiv2024

A Posteriori Error Estimates for $hp$-FE Discretizations in Elastoplasticity

In this paper, a reliable a posteriori error estimator for a model problem of elastoplasticity with linear kinematic hardening is derived, which satisfies some (local) efficiency estimates. It is applicable to any discretization that is conforming with respect to the displacement field and the plastic strain. Furthermore, the paper presents $hp$-finite element discretizations relying on a variational inequality as well as on a mixed variational formulation and discusses their equivalence by using biorthogonal basis functions. Numerical experiments demonstrate the applicability of the theoretical findings and underline the potential of $h$- and $hp$-adaptive finite element discretizations for problems of elastoplasticity.

math.NA↗

arXiv2023

An $hp$-adaptive strategy based on locally predicted error reductions

We introduce a new $hp$-adaptive strategy for self-adjoint elliptic boundary value problems that does not rely on using classical a posteriori error estimators. Instead, our approach is based on a generally applicable prediction strategy for the reduction of the energy error that can be expressed in terms of local modifications of the degrees of freedom in the underlying discrete approximation space. The computations related to the proposed prediction strategy involve low-dimensional linear problems that are computationally inexpensive and highly parallelizable. The mathematical building blocks for this new concept are first developed on an abstract Hilbert space level, before they are employed within the specific context of $hp$-type finite element discretizations. For this particular framework, we discuss an explicit construction of $p$-enrichments and $hp$-refinements by means of an appropriate constraint coefficient technique that can be employed in any dimensions. The applicability and effectiveness of the resulting $hp$-adaptive strategy is illustrated with some $1$- and $2$-dimensional numerical examples.

math.NA↗

arXiv2023

Effects of anisotropy on the geometry of tracer particle trajectories in turbulent flows

Using curvature and torsion to describe Lagrangian trajectories gives a full description of these as well as an insight into small and large time scales as temporal derivatives up to order 3 are involved. One might expect that the statistics of these properties depend on the geometry of the flow. Therefore, we calculated curvature and torsion probability density functions (PDFs) of experimental Lagrangian trajectories processed using the Shake-the-Box algorithm of turbulent von Kármán flow, Rayleigh-Bénard convection and a zero-pressure-gradient turbulent boundary layer over a flat plate. The results for the von Kármán flow compare well with previous experimental results for the curvature PDF and numerical simulation of homogeneous and isotropic turbulence for the torsion PDF. Results for Rayleigh-Bénard convection agree with those obtained for Kármán flow, while results for the logarithmic layer within the boundary layer differ slightly, and we provide a potential explanation. To detect and quantify the effect of anisotropy either resulting from a mean flow or large-scale coherent motions on the geometry or tracer particle trajectories, we introduce the curvature vector. We connect its statistics with those of velocity fluctuations and demonstrate that strong large-scale motion in a given spatial direction results in meandering rather than helical trajectories.

physics.flu-dyn↗

arXiv2022

Race Driver Evaluation at a Driving Simulator using a physical Model and a Machine Learning Approach

Professional race drivers are still superior to automated systems at controlling a vehicle at its dynamic limit. Gaining insight into race drivers' vehicle handling process might lead to further development in the areas of automated driving systems. We present a method to study and evaluate race drivers on a driver-in-the-loop simulator by analysing tire grip potential exploitation. Given initial data from a simulator run, two optimiser based on physical models maximise the horizontal vehicle acceleration or the tire forces, respectively. An overall performance score, a vehicle-trajectory score and a handling score are introduced to evaluate drivers. Our method is thereby completely track independent and can be used from one single corner up to a large data set. We apply the proposed method to a motorsport data set containing over 1200 laps from seven professional race drivers and two amateur drivers whose lap times are 10-20% slower. The difference to the professional drivers comes mainly from their inferior handling skills and not their choice of driving line. A downside of the presented method for certain applications is an extensive computation time. Therefore, we propose a Long-short-term memory (LSTM) neural network to estimate the driver evaluation scores. We show that the neural network is accurate and robust with a root-mean-square error between 2-5% and can replace the optimisation based method. The time for processing the data set considered in this work is reduced from 68 hours to 12 seconds, making the neural network suitable for real-time application.

cs.LG↗

arXiv2021

Modeling Ice Friction for Vehicle Dynamics of a Bobsled with Application in Driver Evaluation and Driving Simulation

We provide an ice friction model for vehicle dynamics of a two-man bobsled which can be used for driver evaluation and in a driver-in-the-loop simulator. Longitudinal friction is modeled by combining experimental results with finite element simulations to yield a correlation between contact pressure and friction. To model lateral friction, we collect data from 44 bobsleigh runs using special sensors. Non-linear regression is used to fit a bob-specific one-track vehicle dynamics model to the data. It is applied in driving simulation and enables a novel method for bob driver evaluation. Bob drivers with various levels of experience are investigated. It shows that a similar performance of the top drivers results from different driving styles.

cs.HC↗

arXiv2012

Critical Parameter Values and Reconstruction Properties of Discrete Tomography: Application to Experimental Fluid Dynamics

We analyze representative ill-posed scenarios of tomographic PIV with a focus on conditions for unique volume reconstruction. Based on sparse random seedings of a region of interest with small particles, the corresponding systems of linear projection equations are probabilistically analyzed in order to determine (i) the ability of unique reconstruction in terms of the imaging geometry and the critical sparsity parameter, and (ii) sharpness of the transition to non-unique reconstruction with ghost particles when choosing the sparsity parameter improperly. The sparsity parameter directly relates to the seeding density used for PIV in experimental fluids dynamics that is chosen empirically to date. Our results provide a basic mathematical characterization of the PIV volume reconstruction problem that is an essential prerequisite for any algorithm used to actually compute the reconstruction. Moreover, we connect the sparse volume function reconstruction problem from few tomographic projections to major developments in compressed sensing.

math.NA↗