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ON POSITIVITY PRESERVATION OF HYBRID DISCONTINUOUS GALERKIN METHODS ON HYPERGRAPHS

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arXiv 2025年

作者： Knobloch, Petr Lederer, Philip L. Rupp, Andreas Department of Numerical Mathematics Faculty of Mathematics and Physics Charles University Prague Czech Republic Department of Mathematics University of Hamburg Germany Department of Mathematics Faculty of Mathematics and Computer Science Saarland University Saarbrücken Germany

Hybrid finite element methods, particularly hybridized discontinuous Galerkin (HDG) methods, are efficient numerical schemes for discretizing the diffusion equation, which encompasses two main physical principles: mass conservation and positivity preservation. While the former has been extensively analyzed in the literature, this paper investigates the latter. We state a theorem that guarantees the positivity of both the bulk and skeleton approximations to the primary unknown (concentration) and provide counterexamples for nonpositive discretizations. The theoretical findings are confirmed by numerical experiments. Copyright © 2025, The Authors. All rights reserved.

关键词： Finite element method

Sequential Solution Strategies for the Cahn-Hilliard-Biot Model

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Sequential Solution Strategies for the Cahn-Hilliard-Biot ...

European Conference on numerical mathematics and Advanced Applications, ENUMATH 2023

作者： Storvik, Erlend Riethmüller, Cedric Both, Jakub W. Radu, Florin Adrian Department of Computer Science Electrical Engineering and Mathematical Sciences Western Norway University of Applied Sciences Bergen Norway Institute of Applied Analysis and Numerical Simulation University of Stuttgart Stuttgart Germany Centre for Modeling of Coupled Subsurface Dynamics Department of Mathematics University of Bergen Bergen Norway

ISBN: (纸本)9783031861680

This paper presents a study of solution strategies for the Cahn-Hilliard-Biot equations, a complex mathematical model for understanding flow in deformable porous media with changing solid phases. Solving the Cahn-Hilliard-Biot system poses significant challenges due to its coupled, nonlinear and non-convex nature. We explore various solution algorithms, comparing monolithic and splitting strategies, focusing on both their computational efficiency and robustness. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2025.

关键词： Computational efficiency

Approximate stationarity in disjunctive optimization: concepts, qualification conditions, and application to MPCCs

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arXiv 2025年

作者： Käming, Isabella Mehlitz, Patrick Technische Universität Dresden Institute of Numerical Mathematics Dresden01062 Germany Philipps-Universität Marburg Department of Mathematics and Computer Science Marburg35032 Germany

In this paper, we are concerned with stationarity conditions and qualification conditions for optimization problems with disjunctive constraints. This class covers, among others, optimization problems with complementarity, vanishing, or switching constraints which are notoriously challenging due to their highly combinatorial structure. The focus of our study is twofold. First, we investigate approximate stationarity conditions and the associated strict constraint qualifications which can be used to infer stationarity of local minimizers. While such concepts are already known in the context of so-called Mordukhovich-stationarity, we introduce suitable extensions associated with strong stationarity. Second, a qualification condition is established which, based on an approximately Mordukhovich- or strongly stationary point, can be used to infer its Mordukhovich- or strong stationarity, respectively. In contrast to the aforementioned strict constraint qualifications, this condition depends on the involved sequences justifying approximate stationarity and, thus, is not a constraint qualification in the narrower sense. However, it is much easier to verify as it merely requires to check the (positive) linear independence of a certain family of gradients. In fact, it provides a characterization of stationarity and, thus, is weaker than these strict constraint qualifications. In order to illustrate the obtained findings, they are applied to optimization problems with complementarity constraints where they can be naturally extended to the well-known concepts of weak and *** Codes 49J53, 90C30, 90C33, 90C46 © 2025, CC BY-NC-ND.

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FIXED POINT RESULTS AND THE EKELAND VARIATIONAL PRINCIPLE IN VECTOR B-METRIC SPACES

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arXiv 2025年

作者： Precup, Radu Stan, Andrei Faculty of Mathematics and Computer Science Institute of Advanced Studies in Science and Technology Babeş-Bolyai University Cluj-Napoca400084 Romania Tiberiu Popoviciu Institute of Numerical Analysis Romanian Academy P.O. Box 68-1 Cluj-Napoca400110 Romania Department of Mathematics Babeş-Bolyai University Cluj-Napoca400084 Romania

In this paper, we extend the concept of b-metric spaces to the vectorial case, where the distance is vector-valued, and the constant in the triangle inequality axiom is replaced by a matrix. For such spaces, we establish results analogous to those in the b-metric setting: fixed-point theorems, stability results, and a variant of Ekeland’s variational principle. As a consequence, we also derive a variant of Caristi’s fixed-point theorem. © 2025, CC BY.

关键词： Vector spaces

Multiphysics simulations of microstructure influence on hysteresis and eddy current losses of electrical steel

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arXiv 2025年

作者： Kühn, Patrick Yang, Yangyiwei Chen, Guanyu Foster, Shanelle N. Egger, Herbert Xu, Bai-Xiang Mechanics of Functional Materials Division Institute of Materials Science Technische Universität Darmstadt Darmstadt64287 Germany Department of Electrical and Computer Engineering Michigan State University MI48824 United States Johann Radon Institute for Computational and Applied Mathematics Linz Austria Institute for Numerical Mathematics Johannes Kepler University Linz Austria

Improving efficiency of electrical machines requires fundamental knowledge on the mechanisms behind magnetic and eddy current losses of the magnetic core materials, with Fe-Si alloy as a prototype. These losses are intrinsically influenced by the microstructure of the materials. This necessitates physics-based, microstructure-informed multiscale simulations. In the present paper, we utilised micromagnetic simulations and computational homogenization methods to calculate the effective hysteresis and effective conductivities of Fe-Si electrical steels. To demonstrate the methodology, binder-jet printed electrical steel material samples with different microstructure were investigated. The microstructure samples were digitized based on both the descriptor-based synthetic reconstruction and SEM-image-based digitization. More samples were generated with varying microstructure features such as grain size and grain boundary phases. The micromagnetic simulations were then performed to investigate the magnetic hysteresis and hysteresis loss. The eddy current loss was also evaluated by using the effective conductivity through computational homogenization. By performing parameter research on a series of synthetic microstructures, effects of average grain size and grain boundary (GB) phase thickness on the hysteresis loss and eddy current loss were unveiled. An average grain size around 120 µm has the lowest hysteresis loss, although the eddy current loss increases with the grain size. Increasing GB-phase thickness helps reduce both losses. Results indicate the potential to decrease loss of magnetic core materials by microstructure optimization. © 2025, CC BY.

关键词： Silicon alloys

A Novel Generalized Conjugate Gradient Framework: Unlocking New Optimization Methods and Insights

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SSRN

SSRN 2025年

作者： Onuoha, Oluwaseun Biodun Bamigbola, Olabode Matthias Omole, Ezekiel Olaoluwa Moghrab, Issam A.R. Ibrahim, Sulaiman Mohammed Halilu, Abubakar Sani Raso, Mario Mathematical Sciences Department Adekunle Ajasin University Akungba Nigeria Mathematics Department University of Ilorin Nigeria Mathematics Programme Department of Physical Sciences Landmark University Omu-Aran Nigeria Information Systems and Technology Department Kuwait Technical College Kuwait Department of Computer Science College of Arts and Sciences University Central Asia Naryn Kyrgyzstan School of Quantitative Science Universiti Utara Malaysia Kedah Sintok06010 Malaysia Faculty of Education and Arts Sohar University Sohar311 Oman Department of Mathematics Sule Lamido University Kafin Hausa Nigeria Department of Computer Science Sapienza University of Rome Italy Numerical Optimization Research Group Bayero University Kano Nigeria

Conjugate Gradient (CG) methods are efficient techniques in addressing large-scale, unconstrained optimization problems because of their efficiency and minimal storage requirement. This paper introduces a new family of generalized CG methods by linearly combining terms from the numerators and denominators of ten existing CG methods to propose a novel update parameter. We derive two new methods from this general framework and conduct a comprehensive numerical simulation against established methods. numerical experiments on benchmark optimization problems show that one of the newly developed methods consistently surpasses conventional methods in both the speed of convergence and robustness. The second method competes favorably with existing methods. These findings indicate that the new class of generalized CG methods offers a promising direction for further research and practical applications in optimization techniques. © 2025, The Authors. All rights reserved.

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LINEAR MULTISTEP METHODS FOR VOLTERRA INTEGRAL AND INTEGRODIFFERENTIAL EQUATIONS

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mathematics OF COMPUTATION 1985年第172期45卷 439-461页

作者： VANDERHOUWEN, PJ RIELE, HJJT Department of Numerical Mathematics Centre for Mathematics and Computer Science Amsterdam Netherlands

Abstract: A general class of linear multistep methods is presented for numerically solving first- and second-kind Volterra integral equations, and Volterra integro-differential equations. These so-called VLM methods, which include the well-known direct quadrature methods, allow for a unified treatment of the problems of consistency and convergence, and have an analogue in linear multistep methods for ODEs, as treated in any textbook on computational methods in ordinary differential equations. General consistency and convergence results are presented (and proved in an Appendix), together with results of numerical experiments which support the theory.

关键词： numerical analysis Volterra integral and integro-differential equations linear multistep methods consistency convergence

A RIGOROUS COMPARISON OF THE EWALD METHOD AND THE FAST MULTIPOLE METHOD IN 2 DIMENSIONS

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computer PHYSICS COMMUNICATIONS 1995年第3期87卷 307-318页

作者： SOLVASON, D KOLAFA, J PETERSEN, HG PERRAM, JW Center for Mathematical Modelling and Numerical Simulation Department of Mathematics and Computer Science Odense University DK-5230 Odense M Denmark

The most efficient and proper standard method for simulating charged or dipolar systems is the Ewald method, which asymptotically scales as N-3/2 where N is the number of charges. However, recently the ''fast multipole method'' (FMM) which scales linearly with N has been developed. The break-even of the two methods (that is, the value of N below which Ewald is faster and above which FMM is faster) is very sensitive to the way the methods are optimized and implemented and to the required simulation accuracy. In this paper we use theoretical estimates and simulation results for the accuracies to carefully compare the two methods with respect to speed. We have developed and implemented highly efficient algorithms for both methods for a serial computer (a SPARCstation ELC) as well as a parallel computer (a T800 transputer based MEIKO computer). Breakevens in the range between N = 10000 and N = 30000 were found for reasonable values of the average accuracies found in our simulations. Furthermore, we illustrate how huge but rare single charge pair errors in the FMM inflate the error for some of the charges.

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Model of fractured medium and nondestructive control of composite materials

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Chinese Journal of Aeronautics 2024年第2期37卷 93-99页

作者： Vasily GOLUBEV Ilia NIKITIN Katerina BEKLEMYSHEVA Department of Computers Science and Numerical Mathematics Moscow Institute of Physics and TechnologyDolgoprudny 141700Russia Mathematical Modeling and Management Department of InformatizationInstitute of Computer Aided Design RASMoscow 123056Russia

Non-destructive testing of composites is an important issue in the modern aircraft *** are susceptible to the barely visible impact damage which can affect the residual strength of the material and occurs both during production and *** continuum model for describing the damaged zone is *** slip theory relations used for a continuous distribution of slip planes are *** the initial stage,the isotropic background model is *** model allows the material slippage along the fractures based on the Coulomb friction law with the small viscous *** this regime,the govern system of equations becomes *** overcome this difficulty,the explicit-implicit grid-characteristic scheme is *** standard ultrasound diagnostic procedure of damaged composite materials is successfully *** with the trivial free-surface fracture model,different reactions on the compression and stretch waves are *** approach provided an effective way for the simulation of complex dynamic behavior of damage zones.

关键词： Non-destructive testing Composite materials numerical modeling Continuum model Grid-characteristic method