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A Multiscale Method for Two-Component,Two-Phase Flow with a Neural Network Surrogate

Communications on applied Mathematics and Computation 2024年第4期6卷 2265-2294页

作者： Jim Magiera Christian Rohde Institute of Applied Analysis and Numerical Simulation University of StuttgartPfaffenwaldring 5770569StuttgartGermany

Understanding the dynamics of phase boundaries in fluids requires quantitative knowledge about the microscale processes at the *** consider the sharp-interface motion of the compressible two-component flow and propose a heterogeneous multiscale method(HMM)to describe the flow fields *** multiscale approach combines a hyperbolic system of balance laws on the continuum scale with molecular-dynamics(MD)simulations on the microscale ***,the multiscale approach is necessary to compute the interface dynamics because there is—at present—no closed continuum-scale *** basic HMM relies on a moving-mesh finite-volume method and has been introduced recently for the compressible one-component flow with phase transitions by Magiera and Rohde in(J Comput Phys 469:111551,2022).To overcome the numerical complexity of the MD microscale model,a deep neural network is employed as an efficient surrogate *** entire approach is finally applied to simulate droplet dynamics for argon-methane mixtures in several space *** our knowledge,such compressible two-phase dynamics accounting for microscale phase-change transfer rates have not yet been computed.

关键词： Phase transition Hyperbolic balance laws for multi-component fluids Multiscale modeling Moving-mesh methods Deep neural networks

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Stokes–Brinkman–Darcy Models for Coupled Free-Flow and Porous-Medium Systems 10th

Stokes–Brinkman–Darcy Models for Coupled Free-Flow and P...

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10th International Symposium on Finite Volumes for Complex Applications, FVCA10 2023

作者： Ruan, Linheng Rybak, Iryna Institute of Applied Analysis and Numerical Simulation University of Stuttgart Pfaffenwaldring 57 Stuttgart70569 Germany

ISBN: (纸本)9783031408632

Coupled systems involving free flow and porous medium have gained significant attention in recent years due to their prevalence in environment and industry. Most of the coupling approaches are suitable only for flows parallel to the fluid–porous interface, and a generalization of the coupling concept is required. In this work, we consider a thin transition region between the free-flow and porous-medium domains, which stores and transports mass, momentum, and energy. The flow system of interest is incompressible and single-phase. The model comprises the Stokes equations in the free-flow domain, the Brinkman equations in the transition region, and Darcy’s law in the porous medium. These models are coupled through suitable interface conditions. numerical simulation results for the coupled full-dimensional Stokes–Brinkman–Darcy model are provided. A dimensionally reduced formulation for the coupled model is proposed in the case of a thin transition region. This model consists of the averaged Brinkman equations of co-dimension one, which are coupled to the full-dimensional Stokes and Darcy’s equations in the free flow and porous medium, respectively. © 2023, The Author(s), under exclusive license to Springer Nature Switzerland AG.

关键词： Porous materials

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Mathematical Challenges for the Theory of Hyperbolic Balance Laws in Fluid Mechanics: Complexity, Scales, Randomness

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Jahresbericht der Deutschen Mathematiker-Vereinigung 2024年第4期126卷 283-311页

作者： Lukáčová-Medvid’ová, Mária Rohde, Christian Institute of Mathematics Johannes Gutenberg-University Mainz Staudingerweg 9 Mainz 55128 Germany Institute for Applied Analysis and Numerical Simulation University of Stuttgart Pfaffenwaldring 57 Stuttgart 70569 Germany

Understanding the dynamics of hyperbolic balance laws is of paramount interest in the realm of fluid mechanics. Nevertheless, fundamental questions on the analysis and the numerics for distinctive hyperbolic features related to turbulent flow motion remain vastly open. Recent progress on the mathematical side reveals novel routes to face these concerns. This includes findings about the failure of the entropy principle to ensure uniqueness, the use of structure-preserving concepts in high-order numerical methods, and the advent of tailored probabilistic approaches. Whereas each of these three directions on hyperbolic modelling are of completely different origin they are all linked to small- or subscale features in the solutions which are either enhanced or depleted by the hyperbolic nonlinearity. Thus, any progress in the field might contribute to a deeper understanding of turbulent flow motion on the basis of the continuum-scale mathematical models. We present an overview on the mathematical state-of-the-art in the field and relate it to the scientific work in the DFG Priority Research Programme 2410. As such, the survey is not necessarily targeting at readers with comprehensive knowledge on hyperbolic balance laws but instead aims at a general audience of reseachers which are interested to gain an overview on the field and associated challenges in fluid mechanics. © The Author(s) 2024.

关键词： Hyperbolic balance laws Shock wave theory Stochastic modelling and uncertainty quantification Structure- and asymptotic-preserving numerics

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Data, Information, Knowledge, Wisdom, and Explainable Artificial Intelligence

Data, Information, Knowledge, Wisdom, and Explainable Artif...

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International Workshop on Designing the Conceptual Landscape for a XAIR Validation Infrastructure, DCLXVI 2024

作者： Schembera, Björn Institute of Applied Analysis and Numerical Simulation University of Stuttgart Allmandring 5b Stuttgart 70569 Germany

ISBN: (纸本)9783031892738

The data – information – knowledge – wisdom (DIKW)-hierarchy has become a standard framework for examining relationships among data, information, knowledge. This article explores the integration of DIKW with the FAIR and XAIR principles. These principles drive essential transformation processes within the DIKW model: the FAIR principles support the transition from data to information, while XAIR principles enable information to become actionable knowledge. Wisdom is only created particularly through ethical and scientific practices that transcend purely technical considerations. At each level, these transformations reflect distinct epistemic values: FAIR fosters technical knowledge about data, XAIR enables scientific knowledge, and ethics and scientific practices contribute to broader societal knowledge. AI technology plays a dual role, making research machine-readable and interpretable while simultaneously enhancing transformation processes through AI-driven techniques. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2025.

关键词： Artificial Intelligence Dark Data Data FAIR Information Knowledge Research Data Management Wisdom XAIR

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Online adaptive surrogates for the value function of high-dimensional nonlinear optimal control problems ⁎

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IFAC-PapersOnLine 2025年第1期59卷 331-336页

作者： Tobias Ehring Bernard Haasdonk Institute of Applied Analysis and Numerical Simulation (IANS) University of Stuttgart 70569 Stuttgart Germany

We introduce a strategy that generates an adaptive surrogate of the value function of high-dimensional nonlinear optimal control problems. It exploits the relevant operating domain online on which the resulting surrogate satisfies the Hamilton–Jacobi–Bellman (HJB) equation up to a given threshold. The approximate value function is based on Hermite kernel regression, where the data stems from open-loop control of reduced-order optimal control problems. As a measure of accuracy, the full-order HJB residual, known as the Bellman error, is used to determine whether the current Hermite kernel surrogate is sufficient or further training is required. In addition, the reduced-order model can also be improved using the full-order data if the same HJB-based error indicator suggests that the current reduced system is not accurate enough. numerical experiments support the effectiveness of the new scheme.

关键词： Feedback control surrogate modeling model order reduction kernel methods online adaptive learning

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Relativistic and electron-correlation effects in static dipole polarizabilities for main-group elements

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Physical Review A 2024年第4期110卷 042805页

作者： YingXing Cheng Numerical Mathematics for High Performance Computing Institute of Applied Analysis and Numerical Simulation University of Stuttgart Pfaffenwaldring 57 Stuttgart 70569 Germany

In this study, I compute the static dipole polarizability of main-group elements using the finite-field method combined with relativistic coupled-cluster and configuration-interaction simulations. The computational results closely align with the values recommended in the 2018 Table of static dipole polarizabilities of neutral elements [Mol. Phys. 117, 1200 (2019)]. Additionally, I investigate the influence of relativistic effects and electron correlation on atomic dipole polarizabilities. Specifically, three types of relativistic effects impacting dipole polarizabilities are studied: scalar-relativistic, spin-orbit coupling, and fully relativistic Dirac-Coulomb effects. The results indicate that scalar-relativistic effects are predominant for atoms in groups 1 and 2, with minimal influence from spin-orbit coupling effects. Conversely, for elements in groups 13–18, scalar-relativistic effects are less significant, while spin-orbit coupling significantly affects elements starting from the fourth row in groups 13 and 14 and from the fifth row in groups 15–18. In each category of relativistic effects, the impact of electron correlation is evaluated. The results show that electron correlation significantly influences dipole polarizability calculations, particularly for atoms from groups 1, 2, 13, and 14, but is less significant for atoms from groups 15–18. This study provides a comprehensive and consistent data set of dipole polarizabilities and contributes to a systematic understanding of the roles of relativistic and electron-correlation effects in atomic dipole polarizabilities, serving as a valuable reference for future research.

关键词： Electron correlation calculations for atoms & ions Electronic structure of atoms & molecules Relativistic & quantum electrodynamic effects in atoms, molecules,& ions Spin-orbit coupling

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Stochastic Runge-Kutta–Munthe-Kaas Methods in the Modelling of Perturbed Rigid Bodies

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Advances in applied Mathematics and Mechanics 2022年第2期14卷 528-538页

作者： Michelle Muniz Matthias Ehrhardt Michael Günther Renate Winkler Institute of Mathematical Modelling Analysis and Computational Mathematics(IMACM)Applied Mathematics and Numerical AnalysisBergische Universität WuppertalGaußstraße 2042119 WuppertalGermany

In this paper we present how nonlinear stochastic Itˆo differential equations arising in the modelling of perturbed rigid bodies can be solved numerically in such a way that the solution evolves on the correct *** this end,we formulate an approach based on Runge-Kutta–Munthe-Kaas(RKMK)schemes for ordinary differ-ential equations on ***,we provide a proof of the mean-square convergence of this stochastic version of the RKMK schemes applied to the rigid body problem and illustrate the effectiveness of our proposed schemes by demonstrating the structure preservation of the stochastic RKMK schemes in contrast to the stochastic Runge-Kutta methods.

关键词： Stochastic Runge-Kutta method Runge-Kutta–Munthe-Kaas scheme nonlinear Itˆo SDEs rigid body problem

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A posteriori single- and multi-goal error control and adaptivity for partial differential equations

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Advances in applied Mechanics 2024年 59卷 19-108页

作者： Endtmayer, Bernhard Langer, Ulrich Richter, Thomas Schafelner, Andreas Wick, Thomas Leibniz University Hannover Institute of Applied Mathematics Germany Leibniz University Hannover Cluster of Excellence PhoenixD Germany Johannes Kepler University Linz Institute of Numerical Mathematics Austria Austrian Academy of Sciences RICAM Linz Austria Otto von Guericke University Magdeburg Institute of Analysis and Numerics Germany

ISBN: (纸本)9780443294501

This chapter reviews goal-oriented a posteriori error control, adaptivity and solver control for finite element approximations to boundary and initial-boundary value problems for stationary and non-stationary partial differential equations, respectively. In particular, coupled field problems with different physics may require simultaneously the accurate evaluation of several quantities of interest, which is achieved with multi-goal oriented error control. Sensitivity measures are obtained by solving an adjoint problem. Error localization is achieved with the help of a partition-of-unity. We also review and extend theoretical results for efficiency and reliability by employing a saturation assumption. The resulting adaptive algorithms allow to balance discretization and non-linear iteration errors, and are demonstrated for four applications: Poisson's problem, non-linear elliptic boundary value problems, stationary incompressible Navier-Stokes equations, and regularized parabolic p-Laplace initial-boundary value problems. Therein, different finite element discretizations in two different software libraries are utilized, which are partially accompanied with open-source implementations on GitHub. © 2024

关键词： Navier Stokes equations

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Interaction of defects in disordered materials – Insights from a nonlinear decomposition of the nonaffine displacement field

Interaction of defects in disordered materials – Insights f...

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93rd Annual Meeting of the International Association of applied Mathematics and Mechanics, GAMM 2023

作者： Focks, Tobias Bamer, Franz Wu, Zhao Markert, Bernd Stamm, Benjamin Institute of General Mechanics RWTH Aachen University Aachen Germany Institute of Applied Analysis and Numerical Simulation University of Stuttgart Stuttgart Germany

In this paper, we link the unique mechanical properties to structural aspects using a splitting method in order to disentangle nonlinear interactions of Stone– Wales defects as fundamental perturbations in a crystalline environment. We investigate the spatial features of the displacement field achieved by this splitting method and couple it to macroscopic material properties. The nonlinear interaction broadens the understanding of material failure at the onset of disorder and may also play a role in the emergence of the Boson peak. © 2023 The Authors. Proceedings in applied Mathematics and Mechanics published by Wiley-VCH GmbH.

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Dynamic fracture with a continuum-kinematics-based peridynamic and a phase-field approach

Dynamic fracture with a continuum-kinematics-based peridynam...

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92nd Annual Meeting of the International Association of applied Mathematics and Mechanics, GAMM 2022

作者： Friebertshäuser, Kai Thomas, Marita Tornquist, Sven Weinberg, Kerstin Wieners, Christian University of Siegen Siegen Germany Weierstrass Institute for Applied Analysis and Stochastics Berlin Germany FU Berlin Germany Institute of Applied and Numerical Mathematics KIT Karlsruhe Germany

The notion of dynamic fracture with continuum-kinematics-based peridynamics is presented in this work. A geometrically precise version of peridynamics called continuum-kinematics-based peridynamics adds surface-or volume-based interactions to the traditional peridynamic bonds, accurately capturing the finite deformation kinematics. The point families produced from the horizon of the material points are used to construct the surfaces and volumes taken into account for these non-local interactions. In continuum kinematics-based peridynamics, the traditional bond-stretch damage technique is insufficient for fracture. Due to the loss of strength in the internal force densities of the material points, it is now extended to the surface-and volume-based interactions by new failure factors. numerical examples demonstrate that the proposed approach effectively manages crack propagation, impact damage, and spontaneous crack initiation under dynamic loading circumstances with large deformations. When the results are compared to phase-field calculations, there is a remarkable agreement concerning the damage patterns. © 2023 The Authors. Proceedings in applied Mathematics & Mechanics published by Wiley-VCH GmbH.

关键词： Kinematics

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