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CONVERGENCE OF DISCONTINUOUS GALERKIN METHODS FOR QUASICONVEX AND RELAXED VARIATIONAL PROBLEMS

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arXiv 2025年

作者： Grekas, Georgios Koumatos, Konstantinos Makridakis, Charalambo Vikelis, Andreas Computer Electrical Mathematical Sciences and Engineering Division KAUST Saudi Arabia Department of Mathematics University of Sussex United Kingdom Institute of Applied & Computational Mathematics Foundation for Research & Technology-Hellas Greece Department of Mathematics and Applied Mathematics University of Crete Greece Faculty of Civil Engineering Czech Technical University of Prague Czech Republic

In this work, we establish that discontinuous Galerkin methods are capable of producing reliable approximations for a broad class of nonlinear variational problems. In particular, we demonstrate that these schemes provide essential flexibility by removing inter-element continuity while also guaranteeing convergent approximations in the quasiconvex case. Notably, quasiconvexity is the weakest form of convexity pertinent to elasticity. Furthermore, we show that in the non-convex case discrete minimisers converge to minimisers of the relaxed problem. In this case, the minimisation problem corresponds to the energy defined by the quasiconvex envelope of the original energy. Our approach covers all discontinuous Galerkin formulations known to converge for convex energies. This work addresses an open challenge in the vectorial calculus of variations: developing and rigorously justifying numerical schemes capable of reliably approximating nonlinear energy minimization problems with potentially singular solutions, which are frequently encountered in materials science. © 2025, CC BY.

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A deep neural network based surrogate model for damage identification in full-scale structures with incomplete noisy measurements

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Frontiers of Structural and Civil Engineering 2024年第3期18卷 393-410页

作者： Tram BUI-NGOC Duy-Khuong LY Tam T TRUONG Chanachai THONGCHOM T.NGUYEN-THOI Laboratory for Computational Mechanics Institute for Computational Science and Artificial IntelligenceVan Lang UniversityHo Chi Minh City 70000Vietnam Faculty of Mechanical-Electrical and Computer Engineering School of TechnologyVan Lang UniversityHo Chi Minh City 70000Vietnam Faculty of Civil Engineering School of TechnologyVan Lang UniversityHo Chi Minh City 70000Vietnam Department of Computer Science Aarhus UniversityAarhus 8000Denmark Thammasat University research unit in structural and foundation engineering Department of Civil EngineeringThammasat UniversityPathumthani 12120Thailand Laboratory for Applied and Industrial Mathematics Institute for Computational Science and Artificial IntelligenceVan Lang UniversityHo Chi Minh City 70000Vietnam Thammasat School of Engineering Thammasat UniversityPathumthani 12120Thailand

The paper introduces a novel approach for detecting structural damage in full-scale structures using surrogate models generated from incomplete modal data and deep neural networks(DNNs).A significant challenge in this field is the limited availability of measurement data for full-scale structures,which is addressed in this paper by generating data sets using a reduced finite element(FE)model constructed by SAP2000 software and the MATLAB programming *** surrogate models are trained using response data obtained from the monitored structure through a limited number of measurement *** proposed approach involves training a single surrogate model that can quickly predict the location and severity of damage for all potential *** achieve the most generalized surrogate model,the study explores different types of layers and hyperparameters of the training algorithm and employs state-of-the-art techniques to avoid overfitting and to accelerate the training *** approach’s effectiveness,efficiency,and applicability are demonstrated by two numerical *** study also verifies the robustness of the proposed approach on data sets with sparse and noisy measured ***,the proposed approach is a promising alternative to traditional approaches that rely on FE model updating and optimization algorithms,which can be computationally *** approach also shows potential for broader applications in structural damage detection.

关键词： vibration-based damage detection deep neural network full-scale structures finite element model updating noisy incomplete modal data

Sharp-interface limits for brittle fracture via the inverse-deformation formulation

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arXiv 2024年

作者： Healey, Timothy J. Paroni, Roberto Rosakis, Phoebus Department of Mathematics Cornell University IthacaNY14853 United States Department of Civil and Industrial Engineering University of Pisa Pisa56122 Italy Department of Mathematics and Applied Mathematics University of Crete Crete Heraklion70013 Greece Institute for Applied and Computational Mathematics Foundation of Research and Technology-Hellas Crete Heraklion70013 Greece

We derive sharp-interface models for one-dimensional brittle fracture via the inverse-deformation approach. Methods of Γ-convergence are employed to obtain the singular limits of previously proposed models. The latter feature a local, non-convex stored energy of inverse strain, augmented by small interfacial energy, formulated in terms of the inverse-strain gradient. They predict spontaneous fracture with exact crack-opening discontinuities, without the use of damage (phase) fields or pre-existing cracks;crack faces are endowed with a thin layer of surface energy. The models obtained herewith inherit the same properties, except that surface energy is now concentrated at the crack faces. Accordingly, we construct energy-minimizing configurations. For a composite bar with a breakable layer, our results predict a pattern of equally spaced cracks whose number is given as an increasing function of applied *** Codes 74R10 © 2024, CC BY.

关键词： Interfacial energy

A class of Discontinuous Galerkin methods for nonlinear variational problems

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arXiv 2023年

作者： Grekas, Georgios Koumatos, Konstantinos Makridakis, Charalambos Vikelis, Andreas Institute of Applied & Computational Mathematics Foundation for Research & Technology-Hellas Greece Aerospace Engineering and Mechanics University of Minnesota Minneapolis United States Department of Mathematics University of Sussex United Kingdom Department of Mathematics and Applied Mathematics University of Crete Greece Faculty of Mathematics University of Vienna Austria

In the context of Discontinuous Galerkin methods, we study approximations of nonlinear variational problems. We propose elementwise nonconforming finite element methods to discretize the continuous minimisation problem. Using Γ-convergence arguments we show that for convex energies the discrete minimisers converge to a minimiser of the continuous problem as the mesh parameter tends to zero, under the additional contribution of appropriately defined penalty terms at the level of the discrete energies. In addition, we provide error analysis in the case where the solution of the continuous minimization problem is smooth enough. Under appropriate assumptions we show local existence, uniqueness and corresponding optimal order error estimates for the discrete Euler-Lagrange equations of our scheme. We finally substantiate the feasibility of our methods by numerical examples. Copyright © 2023, The Authors. All rights reserved.

关键词： Finite element method

Theory of intermediate twinning and spontaneous polarization in the phase transformations of ferroelectric potassium sodium niobate

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arXiv 2023年

作者： Grekas, Georgios Pop-Ghe, Patricia-Lia Quandt, Eckhard James, Richard D. Aerospace Engineering and Mechanics University of Minnesota Minneapolis United States Institute of Applied and Computational Mathematics Foundation for Research and Technology-Hellas Heraklion Greece Inorganic Functional Materials Kiel University Kiel Germany

Potassium sodium niobate is considered a prominent material system as a substitute for toxic lead-containing ferroelectric materials. It exhibits first-order phase transformations and ferroelectricity with potential applications ranging from energy conversion to innovative cooling technologies, hereby addressing urgent societal challenges. However, a major obstacle in the application of potassium sodium niobate is its multi-scale heterogeneity and the lack of understanding of its phase transition pathway and microstructure. This can be seen from the findings of Pop-Ghe 2021 et al. [1] which also reveal the occurrence of intermediate twinning during the phase transition. Here we show that intermediate twinning is a consequence of energy minimization. We employ a geometrically nonlinear electroelastic energy function for potassium sodium niobate, including the cubic-tetragonal-orthorhombic transformations and ferroelectricity. The construction of the minimizers is based on compatibility conditions which ensure continuous deformations and pole-free interfaces. These minimizers agree with the experimental observations, including laminates between the tetragonal variants under the cubic to tetragonal transformation, crossing twins under the tetragonal to orthorhombic transformation, intermediate twinning and spontaneous polarization. This shows how the full nonlinear electroelastic model provides a powerful tool in understanding, exploring and tailoring the electromechanical properties of complex ferroelectric ceramics. Copyright © 2023, The Authors. All rights reserved.

关键词： Ferroelectric materials

Dynamical properties of neuromorphic Josephson junctions

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arXiv 2022年

作者： Chalkiadakis, Dimitrios Hizanidis, Johanne Department of Physics University of Crete Heraklion71003 Greece Institute of Applied and Computational Mathematics Foundation for Research and Technology-Hellas Heraklion70013 Greece

Neuromorphic computing exploits the dynamical analogy between many physical systems and neuron biophysics. Superconductor systems, in particular, are excellent candidates for neuromorphic devices due to their capacity to operate in great speeds and with low energy dissipation compared to their silicon counterparts. In this study we revisit a prior work on Josephson Junction-based "neurons" in order to identify the exact dynamical mechanisms underlying the system's neuron-like properties and reveal new complex behaviors which are relevant for neurocomputation and the design of superconducting neuromorphic devices. Our work lies at the intersection of superconducting physics and theoretical neuroscience, both viewed under a common framework, that of nonlinear dynamics theory. Copyright © 2022, The Authors. All rights reserved.

关键词： Neurons

Adapting Urbisphere’s Hyperspectral Library for Sentinel-2 Data: an Application to Heraklion’s Urban Landscape

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Adapting Urbisphere’s Hyperspectral Library for Sentinel-2 ...

IEEE International Symposium on Geoscience and Remote Sensing (IGARSS)

作者： Giannis Lantzanakis Dimitris Tsirantonakis Natalia Pynirtzi Dimitris Poursanidis Zina Mitraka Andreas Christen Sue Grimmond Joern Birkmann Nektarios Chrysoulakis Remote Sensing Lab Institute of Applied and Computational Mathematics Foundation for Research and Technology Hellas Heraklion Greece Environmental Meteorology Albert-Ludwigs-Universität Freiburg - Germany Urban Micromet University of Reading - UK Institute of Spatial and Regional Planning University of Stuttgart - Germany

ISBN: (数字)9798350360325

ISBN: (纸本)9798350360332

Mapping urban areas poses a significant challenge in Earth Observation. Accurate surface information is crucial for various urban planning and climate studies. The diverse use of artificial materials and mixed pixels complicates spectral analysis for multispectral and hyperspectral sensors, requiring higher resolution for identification. Moreover, existing hyperspectral libraries using spectroradiometers, lack the necessary variability required for effective Machine Learning model training. To address these gaps, the urbisphere urban hyperspectral library is under development, leveraging the spectroradiometer Spectral Evolution RS-3500. The library enables satellite image classification without the need for time-consuming on-site data collection. This study showcases the library’s potential to classify Sentinel-2 satellite images, utilizing an image from the city of Heraklion as a case study. The resulting material map was validated using a randomly generated independent dataset.

关键词： Training Spectroradiometers Accuracy Satellites Urban planning Machine learning Libraries

A POSTERIORI CERTIFICATION FOR PHYSICS-INFORMED NEURAL NETWORKS

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arXiv 2025年

作者： Ernst, Lewin Rekatsinas, Nikolaos Urban, Karsten Institute for Numerical Mathematics Ulm University Helmholtzstr. 20 Ulm89081 Germany Institute of Applied and Computational Mathematics Foundation of Research and Technology Nikolaou Plastira 100 Vassilika Vouton Crete HeraklionGR 70013 Greece

We propose rigorous lower and upper bounds for a PINN approximation to PDEs by efficiently computing the Riesz representations of suitable extension and restrictions of the PINN residual towards geometrically simpler domains, which are either embedded or enveloping the original domain. Error bounds are proven and detailed for elliptic as well as parabolic problems. Numerical experiments show the good quantitative behaviour of the derived upper and lower error *** Codes 35J20, 65M15, 68T07 Copyright © 2025, The Authors. All rights reserved.

关键词： Partial differential equations

The Fokas method for evolution partial differential equations

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Partial Differential Equations in applied mathematics 2025年 14卷

作者： Chatziafratis, A. Fokas, A.S. Kalimeris, K. Department of Mathematics and Statistics School of Pure and Applied Sciences University of Cyprus Cyprus Institute of Applied and Computational Mathematics Foundation for Research and Technology Crete Greece Department of Applied Mathematics and Theoretical Physics University of Cambridge United Kingdom Mathematics Research Center Academy of Athens Greece

In the late 1990s a novel methodology was introduced for solving boundary value problems for linear and integrable nonlinear PDEs. This new approach is known as the Unified Transform or the Fokas method. Here we discuss important developments regarding the implementation of this methodology to evolution PDEs. In particular, we analyse linear PDEs, including the heat equation, the Sobolev–Barenblatt pseudo-parabolic model, the Rubinshtein–Aifantis double-diffusion system, and the linearized Cahn–Hilliard model, as well as certain integrable nonlinear PDEs, like the nonlinear Schrödinger equation. Regarding the latter equation, emphasis is placed on the so-called linearizable boundary conditions, which remarkably, include the well-studied x-periodic initial value problem. © 2025 The Authors

关键词： Evolution PDEs Fokas method Integrable PDEs Unified transform