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Port-Hamiltonian Fluid-Structure Interaction Modeling and Structure-Preserving Model Order Reduction of a Classical Guitar

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arXiv 2022年

作者： Rettberg, Johannes Wittwar, Dominik Buchfink, Patrick Brauchler, Alexander Ziegler, Pascal Fehr, Jörg Haasdonk, Bernard Institute of Engineering and Computational Mechanics University of Stuttgart Pfaffenwaldring 9 Stuttgart70569 Germany Institute of Applied Analysis and Numerical Simulation University of Stuttgart Pfaffenwaldring 57 Stuttgart70569 Germany

A fluid-structure interaction model in a port-Hamiltonian representation is derived for a classical guitar. We combine the laws of continuum mechanics for solids and fluids within a unified port-Hamiltonian (pH) modeling approach by adapting the discretized equations on second-order level in order to obtain a damped multi-physics model. The high-dimensionality of the resulting system is reduced by model order reduction. The article focuses on pH-systems in different state transformations, a variety of basis generation techniques as well as structure-preserving model order reduction approaches that are independent from the projection basis. As main contribution a thorough comparison of these method combinations is conducted. In contrast to typical frequency-based simulations in acoustics, transient time simulations of the system are presented. The approach is embedded into a straightforward workflow of sophisticated commercial software modeling and flexible in-house software for multi-physics coupling and model order reduction. © 2022, CC BY-NC-SA.

关键词： Fluid structure interaction

Improved a posteriori Error Bounds for Reduced port-Hamiltonian Systems

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arXiv 2023年

作者： Rettberg, Johannes Wittwar, Dominik Buchfink, Patrick Herkert, Robin Fehr, Jörg Haasdonk, Bernard Institute of Engineering and Computational Mechanics University of Stuttgart Pfaffenwaldring 9 Stuttgart70569 Germany Institute of Applied Analysis and Numerical Simulation University of Stuttgart Pfaffenwaldring 57 Stuttgart70569 Germany

Projection-based model order reduction of dynamical systems usually introduces an error between the high-fidelity model and its counterpart of lower dimension. This unknown error can be bounded by residual-based methods, which are typically known to be highly pessimistic in the sense of largely overestimating the true error. This work applies two improved error bounding techniques, namely (a) a hierarchical error bound and (b) an error bound based on an auxiliary linear problem, to the case of port-Hamiltonian systems. The approaches rely on a second approximation of (a) the dynamical system and (b) the error system. In this paper, these methods are for the first time adapted to port-Hamiltonian systems by exploiting their structure. The mathematical relationship between the two methods is discussed both, theoretically and numerically. The effectiveness of the described methods is demonstrated using a challenging three-dimensional port-Hamiltonian model of a classical guitar with fluid-structure *** Codes 65L70, 34C20 Copyright © 2023, The Authors. All rights reserved.

关键词： Dynamical systems

Ontologies for Models and Algorithms in applied Mathematics and Related Disciplines

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arXiv 2023年

作者： Schembera, Björn Wübbeling, Frank Kleikamp, Hendrik Biedinger, Christine Fiedler, Jochen Reidelbach, Marco Shehu, Aurela Schmidt, Burkhard Koprucki, Thomas Iglezakis, Dorothea Göddeke, Dominik Institute of Applied Analysis and Numerical Simulation University of Stuttgart Germany Institute of Applied Mathematics: Analysis and Numerics University of Münster Germany Fraunhofer Institute for Industrial Mathematics Kaiserslautern Germany Zuse Institute Berlin Germany Weierstrass Institute for Applied Analysis and Stochastics Berlin Germany University Library University of Stuttgart Germany University of Stuttgart Germany

In applied mathematics and related disciplines, the modeling-simulation-optimization workflow is a prominent scheme, with mathematical models and numerical algorithms playing a crucial role. For these types of mathematical research data, the Mathematical Research Data Initiative has developed, merged and implemented ontologies and knowledge graphs. This contributes to making mathematical research data FAIR by introducing semantic technology and documenting the mathematical foundations accordingly. Using the concrete example of microfracture analysis of porous media, it is shown how the knowledge of the underlying mathematical model and the corresponding numerical algorithms for its solution can be represented by the ontologies. © 2023, CC BY.

关键词： Optimization algorithms

A linear domain decomposition method for two-phase flow in porous media

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arXiv 2017年

作者： Seus, David Radu, Florin A. Rohde, Christian University of Stuttgart Institute of Applied Analysis and Numerical Simulation Pfaffenwaldring 57 Stuttgart70569 Germany University of Bergen Department of Mathematics Postboks 7803 Bergen5020 Norway

This article is a follow up of our submitted paper [11] in which a decomposition of the Richards equation along two soil layers was discussed. A decomposed problem was formulated and a decoupling and linearisation technique was presented to solve the problem in each time step in a fixed point type iteration. This article extends these ideas to the case of two-phase in porous media and the convergence of the proposed domain decomposition method is rigorously shown. Copyright © 2017, The Authors. All rights reserved.

关键词： Domain decomposition methods

Randomized Symplectic Model Order Reduction for Hamiltonian Systems

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arXiv 2023年

作者： Herkert, R. Buchfink, P. Haasdonk, B. Rettberg, J. Fehr, J. Institute of Applied Analysis and Numerical Simulation University of Stuttgart Pfaffenwaldring 57 Stuttgart70569 Germany Institute of Engineering and Computational Mechanics University of Stuttgart Pfaffenwaldring 9 Stuttgart70569 Germany

simulations of large scale dynamical systems in multi-query or real-time contexts require efficient surrogate modelling techniques, as e.g. achieved via Model Order Reduction (MOR). Recently, symplectic methods like the complex singular value decomposition (cSVD) or the SVD-like decomposition have been developed for preserving Hamiltonian structure during MOR. In the current contribution, we show how symplectic structure preserving basis generation can be made more efficient with randomized matrix factorizations. We present a randomized complex SVD (rcSVD) algorithm and a randomized SVD-like (rSVD-like) decomposition. We demonstrate the efficiency of the approaches with numerical experiments on high dimensional systems. Copyright © 2023, The Authors. All rights reserved.

关键词： Singular value decomposition

hp-multilevel monte carlo methods for uncertainty quantification of compressible flows

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arXiv 2018年

作者： Beck, Andrea Dürrwachter, Jakob Kuhn, Thomas Meyer, Fabian Munz, Claus-Dieter Rohde, Christian Institute of Aerodynamics and Gas Dynamics University of Stuttgart Pfaffenwaldring 21 Stuttgart70569 Germany Institute of Applied Analysis and Numerical Simulation University of Stuttgart Pfaffenwaldring 57 Stuttgart70569 Germany

We propose a novel hp-Multilevel Monte Carlo method for the quantification of uncertainties in compressible flows using the Discontinuous Galerkin method as deterministic solver. The multilevel approach exploits hierarchies of uniformly refined meshes jointly with an increasing polynomial degree for the ansatz space. It allows for a very large range of resolutions in the physical space and thus an efficient decrease of the statistical error. We prove that the overall complexity of the hp-Multilevel Monte Carlo method to compute the mean field with prescribed accuracy is of quadratic order with respect to the accuracy. We also propose a novel and simple approach to estimate a lower confidence bound for the optimal number of samples per level, which helps to prevent overshooting the optimal number of samples. The method is in particular adapted to the needs of high-performance computing. Our theoretical results are verified by numerical experiments for the two dimensional compressible Navier-Stokes equations. In particular we consider a cavity flow problem from computational acoustics, demonstrating that the method is suitable to handle complex engineering *** Codes 35R60, 65C05, 65M60 Copyright © 2018, The Authors. All rights reserved.

关键词： Monte Carlo methods

Towards a Knowledge Graph for Models and Algorithms in applied Mathematics

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arXiv 2024年

作者： Schembera, Björn Wübbeling, Frank Kleikamp, Hendrik Schmidt, Burkhard Shehu, Aurela Reidelbach, Marco Biedinger, Christine Fiedler, Jochen Koprucki, Thomas Iglezakis, Dorothea Göddeke, Dominik Institute of Applied Analysis and Numerical Simulation University of Stuttgart Germany Institute of Applied Mathematics: Analysis and Numerics University of Münster Germany Weierstrass Institute for Applied Analysis and Stochastics Berlin Germany Mathematics of Complex Systems Zuse Institute Berlin Germany Fraunhofer Institute for Industrial Mathematics Kaiserslautern Germany University Library University of Stuttgart Germany University of Stuttgart Germany

Mathematical models and algorithms are an essential part of mathematical research data, as they are epistemically grounding numerical data. In order to represent models and algorithms as well as their relationship semantically to make this research data FAIR, two previously distinct ontologies were merged and extended, becoming a living knowledge graph. The link between the two ontologies is established by introducing computational tasks, as they occur in modeling, corresponding to algorithmic tasks. Moreover, controlled vocabularies are incorporated and a new class, distinguishing base quantities from specific use case quantities, was introduced. Also, both models and algorithms can now be enriched with metadata. Subject-specific metadata is particularly relevant here, such as the symmetry of a matrix or the linearity of a mathematical model. This is the only way to express specific workflows with concrete models and algorithms, as the feasible solution algorithm can only be determined if the mathematical properties of a model are known. We demonstrate this using two examples from different application areas of applied mathematics. In addition, we have already integrated over 250 research assets from applied mathematics into our knowledge graph. © 2024, CC BY.

关键词： Knowledge graph

Error control for statistical solutions

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arXiv 2019年

作者： Giesselmann, Jan Meyer, Fabian Rohde, Christian Department of Mathematics TU Darmstadt Dolivostraße 15 Darmstadt64293 Germany Institute of Applied Analysis and Numerical Simulation University of Stuttgart Pfaffenwaldring 57 Stuttgart70569 Germany

Statistical solutions have recently been introduced as a an alternative solution framework for hyperbolic systems of conservation laws. In this work we derive a novel a posteriori error estimate in the Wasserstein distance between dissipative statistical solutions and numerical approximations, which rely on so-called regularized empirical measures. The error estimator can be split into deterministic parts which correspond to spatio-temporal approximation errors and a stochastic part which reflects the stochastic error. We provide numerical experiments which examine the scaling properties of the residuals and verify their *** Codes Primary: 35L65, 65M15, Secondary: 65M60, 65M700 Copyright © 2019, The Authors. All rights reserved.

关键词： Galerkin methods

A Surrogate-Assisted Uncertainty-Aware Bayesian Validation Framework and its Application to Coupling Free Flow and Porous-Medium Flow

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arXiv 2021年

作者： Mohammadi, Farid Eggenweiler, Elissa Flemisch, Bernd Oladyshkin, Sergey Rybak, Iryna Schneider, Martin Weishaupt, Kilian Institute for Modelling Hydraulic and Environmental Systems University of Stuttgart Pfaffenwaldring 61 Stuttgart70569 Germany Institute of Applied Analysis and Numerical Simulation University of Stuttgart Pfaffenwaldring 57 Stuttgart70569 Germany

Existing model validation studies in geoscience often disregard or partly account for uncertain-ties in observations, model choices, and input parameters. In this work, we develop a statistical framework that incorporates a probabilistic modeling technique using a fully Bayesian approach to perform a quantitative uncertainty-aware validation. A Bayesian perspective on a validation task yields an optimal bias-variance trade-off against the reference data. It provides an integrative metric for model validation that incorporates parameter and conceptual uncertainty. Additionally, a surrogate modeling technique, namely Bayesian Sparse Polynomial Chaos Expansion, is employed to accelerate the computationally demanding Bayesian calibration and validation. We apply this validation framework to perform a comparative evaluation of models for coupling a free flow with a porous-medium flow. The correct choice of interface conditions and proper model parameters for such coupled flow systems is crucial for physically consistent modeling and accurate numerical simulations of applications. We develop a benchmark scenario that uses the Stokes equations to describe the free flow and considers different models for the porous-medium compartment and the coupling at the fluid-porous interface. These models include a porous-medium model using Darcy's law at the representative elementary volume scale with classical or generalized interface conditions and a pore-network model with its related coupling approach. We study the coupled flow problems' behaviors considering a benchmark case, where a pore-scale resolved model provides the reference solution. With the suggested framework, we perform sensitivity analysis, quantify the parametric uncertainties, demonstrate each model's predictive capabilities, and make a probabilistic model *** Codes 65C20, 62F15, 65C05, 35Q35, 76D07, 76M12, 76S05 Copyright © 2021, The Authors. All rights reserved.

关键词： Sensitivity analysis