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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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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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Data-driven geometric parameter optimization for PD-GMRES

arXiv

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arXiv 2025年

作者： Duvenbeck, Lennart Riethmüller, Cedric Rohde, Christian Institute of Applied Analysis and Numerical Simulation University of Stuttgart Stuttgart70569 Germany

Restarted GMRES is a robust and widely used iterative solver for linear systems. The control of the restart parameter is a key task to accelerate convergence and to prevent the well-known stagnation phenomenon. We focus on the Proportional-Derivative GMRES (PD-GMRES), which has been derived using control-theoretic ideas in [Cuevas Núñez, Schaerer, and Bhaya (2018)] as a versatile method for modifying the restart parameter. Several variants of a quadtree-based geometric optimization approach are proposed to find a best choice of PD-GMRES parameters. We show that the optimized PD-GMRES performs well across a large number of matrix types and we observe superior performance as compared to major other GMRES-based iterative solvers. Moreover, we propose an extension of the PD-GMRES algorithm to further improve performance by controlling the range of values for the restart parameter. Copyright © 2025, The Authors. All rights reserved.

关键词： Linear systems

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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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A hyperbolic model for two-layer thin film flow with a perfectly soluble anti-surfactant

arXiv

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arXiv 2025年

作者： Barthwal, Rahul Rohde, Christian Institute of Applied Analysis and Numerical Simulation University of Stuttgart Pffafenwaldring 57 Stuttgart Germany

We consider the motion of a two-phase thin film that consists of two immiscible viscous fluids and is endowed with an anti-surfactant solute. The presence of such solute particles induces variations of the surface tension and interfacial stress and can drive a Marangoni-type flow. We first analyze a lubrication limit and derive one-dimensional evolution equations of film heights and solute concentrations. Then, under the assumption that the capillarity and diffusion effects are negligible and the solute is perfectly soluble, we obtain a conservative first-order system in terms of film heights and concentration gradients. This reduced system is found to be strictly hyperbolic for a certain set of states and to admit an entire class of entropy/entropy-flux pairs. We also provide a strictly convex entropy for the hyperbolic system. Thus, the well-posedness for the Cauchy problem is given. Moreover, the system is almost a Temple-class system which allows to provide explicit solutions of the Riemann problem. The paper concludes with some numerical experiments using a Godunov-type finite volume method which relies on the exact Riemann *** Codes 35L40, 35L45, 35L65, 76A20, 78M35 © 2025, CC BY.

关键词： Finite volume method

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Robust and Efficient Preconditioners for Stokes–Darcy Problems 10th

Robust and Efficient Preconditioners for Stokes–Darcy Pro...

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

作者： Strohbeck, Paula Riethmüller, Cedric Göddeke, Dominik Rybak, Iryna Institute of Applied Analysis and Numerical Simulation University of Stuttgart Pfaffenwaldring 57 Stuttgart70569 Germany Stuttgart Centre for Simulation Science University of Stuttgart Allmandring 5b Stuttgart70569 Germany

ISBN: (纸本)9783031408632

Coupled systems of porous media and free flow can be modelled by the Stokes equations in the free-flow domain, Darcy’s law in the porous medium, and an appropriate set of coupling conditions on the fluid–porous interface. Discretisation of the coupled Stokes–Darcy problem leads to a large, sparse, ill-conditioned, and nonsymmetric linear system. We discretise the system using the MAC scheme, i.e., the finite volume method on staggered grids. To accelerate convergence of the GMRES method, efficient preconditioners are needed. We propose a block diagonal, a block triangular and a constraint preconditioner for the Stokes–Darcy problem with the classical set of coupling conditions based on the Beavers–Joseph condition and the generalised coupling conditions which were developed for arbitrary flow to the interface. We show the robustness and efficiency of the proposed preconditioners in numerical experiments. © 2023, The Author(s), under exclusive license to Springer Nature Switzerland AG.

关键词： Finite volume method

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A RIGOROUS FORMULATION OF DENSITY FUNCTIONAL THEORY FOR SPINLESS ELECTRONS IN ONE DIMENSION

arXiv

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arXiv 2025年

作者： Corso, Thiago Carvalho Institute of Applied Analysis and Numerical Simulation University of Stuttgart Pfaffenwaldring 57 Stuttgart70569 Germany

MSC Codes Primary: 35J10 Secondary:81Q05, 81V74, 46N50In this paper, we present a completely rigorous formulation of Kohn-Sham density functional theory for spinless electrons living in one dimensional space. More precisely, we consider Schrödinger operators of the form (formula presented) acting on (formula presented) where the external and interaction potentials v and w belong to a suitable class of distributions. In this setting, we obtain a complete characterization of the set of pure-state v-representable densities on the interval. Then, we prove a Hohenberg-Kohn theorem that applies to the class of distributional potentials studied here. Lastly, we establish the differentiability of the exchange-correlation functional and therefore the existence of a unique exchange-correlation potential. We then combine these results to provide a rigorous formulation of the Kohn-Sham scheme. In particular, these results show that the Kohn-Sham scheme is rigorously exact in this setting. © 2025, CC BY-NC-SA.

关键词： Schrodinger equation

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