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A Reissner-Mindlin plate formulation using symmetric Hu-Zhang elements via polytopal transformations

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arXiv 2023年

作者： Sky, Adam Neunteufel, Michael Hale, Jack S. Zilian, Andreas Institute of Computational Engineering and Sciences Department of Engineering Faculty of Science Technology and Medicine University of Luxembourg 6 Avenue de la Fonte Esch-sur-AlzetteL-4362 Luxembourg Institute of Analysis and Scientific Computing Technische Universität Wien Wiedner Hauptstr. 8-10 Wien1040 Austria

In this work we develop new finite element discretisations of the shear-deformable Reissner-Mindlin plate problem based on the Hellinger-Reissner principle of symmetric stresses. Specifically, we use conforming Hu-Zhang elements to discretise the bending moments in the space of symmetric square integrable fields with a square integrable divergence M ∈ HZ ⊂ Hsym(Div). The latter results in highly accurate approximations of the bending moments M and in the rotation field being in the discontinuous Lebesgue space φ ∈ [L2]2, such that the Kirchhoff-Love constraint can be satisfied for t → 0. In order to preserve optimal convergence rates across all variables for the case t → 0, we present an extension of the formulation using Raviart-Thomas elements for the shear stress q ∈ RT ⊂ H(div). We prove existence and uniqueness in the continuous setting and rely on exact complexes for inheritance of well-posedness in the discrete setting. This work introduces an efficient construction of the Hu-Zhang base functions on the reference element via the polytopal template methodology and Legendre polynomials, making it applicable to hp-FEM. The base functions on the reference element are then mapped to the physical element using novel polytopal transformations, which are suitable also for curved geometries. The robustness of the formulations and the construction of the Hu-Zhang element are tested for shearlocking, curved geometries and an L-shaped domain with a singularity in the bending moments M. Further, we compare the performance of the novel formulations with the primal-, MITC- and recently introduced TDNNS methods. © 2023, CC BY-NC-ND.

关键词： Mindlin plates

ON FEEDBACK STABILISATION FOR THE CAHN-HILLIARD EQUATION AND ITS NUMERICAL APPROXIMATION

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arXiv 2025年

作者： Egger, H. Fritz, M. Kunisch, K. Rodrigues, S.S. Johann Radon Institute for Computational and Applied Mathematics Linz Austria Institute for Numerical Mathematics Johannes Kepler University Linz Austria Institute for Mathematics and Scientific Computing Karl–Franzens University Graz Austria

We consider the stabilisation of solutions to the Cahn-Hilliard equation towards a given trajectory by means of a finite-dimensional static output feedback mechanism. Exponential stabilisation of the controlled state around the target trajectory is proven using careful energy estimates and a spectral condition which characterizes the strength of the feedback. The analysis is general enough to allow for pointwise and distributed measurements and actuation. The main results are derived via arguments that carry over to appropriate discretisation schemes which allows us to establish corresponding exponential stabilisation results also on the discrete level. The validity of our results and the importance of some of our assumptions are illustrated by numerical tests. Copyright © 2025, The Authors. All rights reserved.

关键词： Stabilization

A consistent diffuse-interface model for two-phase flow problems with rapid evaporation

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arXiv 2024年

作者： Schreter-Fleischhacker, Magdalena Munch, Peter Much, Nils Kronbichler, Martin Wall, Wolfgang A. Meier, Christoph Institute for Computational Mechanics Technical University of Munich Boltzmannstrasse 15 Garching85748 Germany Institute for High-Performance Scientific Computing University of Augsburg Universitätsstrase 12a Augsburg86159 Germany Department of Information Technology Uppsala University Box 337 Uppsala75105 Sweden Faculty of Mathematics Ruhr University Bochum Universitätsstrase 150 Bochum44780 Germany

We present accurate and mathematically consistent formulations of a diffuseinterface model for two-phase flow problems involving rapid evaporation. The model addresses challenges including discontinuities in the density field by several orders of magnitude, leading to high velocity and pressure jumps across the liquid-vapor interface, along with dynamically changing interface topologies. To this end, we integrate an incompressible Navier-Stokes solver combined with a conservative level-set formulation and a regularized, i.e., diffuse, representation of discontinuities into a matrix-free adaptive finite element framework. The achievements are three-fold: First, we propose mathematically consistent definitions for the level-set transport velocity in the diffuse interface region by extrapolating the velocity from the liquid or gas phase. They exhibit superior prediction accuracy for the evaporated mass and the resulting interface dynamics compared to a local velocity evaluation, especially for strongly curved interfaces. Second, we show that accurate prediction of the evaporation-induced pressure jump requires a consistent, namely a reciprocal, density interpolation across the interface, which satisfies local mass conservation. Third, the combination of diffuse interface models for evaporation with standard Stokes-type constitutive relations for viscous flows leads to significant pressure artifacts in the diffuse interface region. To mitigate these, we propose to introduce a correction term for such constitutive model types. Through selected analytical and numerical examples, the aforementioned properties are validated. The presented model promises new insights in simulation-based prediction of melt-vapor interactions in thermal multiphase flows such as in laser-based powder bed fusion of metals. Copyright © 2024, The Authors. All rights reserved.

关键词： Two phase flow

A FOUR-FIELD MIXED FORMULATION FOR INCOMPRESSIBLE FINITE ELASTICITY

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arXiv 2025年

作者： Fu, Guosheng Neunteufel, Michael Schöberl, Joachim Zdunek, Adam Department of Applied and Computational Mathematics and Statistics University of Notre Dame Notre DameIN46556 United States Fariborz Maseeh Department of Mathematics and Statistics Portland State University 1825 SW Broadway PortlandOR97201 United States Institute of Analysis and Scientific Computing TU Wien Wiedner Hauptstr. 8-10 WienA-1040 Austria HB Berrit Solhemsbackarna 73 Spånga SE-163 56 Sweden

In this work, we generalize the mass-conserving mixed stress (MCS) finite element method for Stokes equations [12], involving normal velocity and tangential-normal stress continuous fields, to incompressible finite elasticity. By means of the three-field Hu–Washizu principle, introducing the displacement gradient and 1st Piola–Kirchhoff stress tensor as additional fields, we circumvent the inversion of the constitutive law. We lift the arising distributional derivatives of the displacement gradient to a regular auxiliary displacement gradient field. Static condensation can be applied at the element level, providing a global pure displacement problem to be solved. We present a stabilization motivated by Hybrid Discontinuous Galerkin methods. A solving algorithm is discussed, which asserts the solvability of the arising linearized subproblems for problems with physically positive eigenvalues. The excellent performance of the proposed method is corroborated by several numerical *** Codes 74S05 (Primary) 74B20 (Secondary) Copyright © 2025, The Authors. All rights reserved.

关键词： Eigenvalues and eigenfunctions

Rfmnet: Robust Deep Functional Maps for Unsupervised Non-Rigid Shape Correspondence

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SSRN

SSRN 2023年

作者： Hu, Ling Li, Qinsong Liu, Shengjun Yan, Dong-Ming Xu, Haojun Liu, Xinru School of Mathematics and Statistics Hunan First Normal University China Institute of Engineering Modeling and Scientific Computing Central South University China Big Data Institute Central South University China National Laboratory of Pattern Recognition Chinese Academy of Sciences China

We introduce a novel strategy to compute the functional map in the deep functional map framework, which jointly considers training stability and more geometric shape features than previous works. We directly first produce a pointwise map by resorting to optimal transport and then convert it to an initial functional map. Such a mechanism mitigates the requirements for the descriptor and avoids the training instabilities resulting from the least square solver. Then, we successfully integrate a state-of-the-art geometric regularization for further optimizing the functional map. We show our novel computing functional map module brings more stable training even under encoding the functional map with high-frequency information and faster convergence speed. Finally, an unsupervised loss is presented for penalizing the correspondence distortion of Delta functions between shapes. Experimental results demonstrate our apparent superiority in correspondence quality and generalization across various shape discretizations and different datasets compared to the state-of-the-art learning methods. © 2023, The Authors. All rights reserved.

关键词： Delta functions

HOMOGENIZATION FOR POLYNOMIAL OPTIMIZATION WITH UNBOUNDED SETS

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arXiv 2021年

作者： Huang, Lei Nie, Jiawang Yuan, Ya-Xiang Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100190 China Department of Mathematics University of California 9500 Gilman Drive La Jolla San DiegoCA92093 United States Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China

This paper considers polynomial optimization with unbounded sets. We give a homogenization formulation and propose a hierarchy of Moment-SOS relaxations to solve it. Under the assumptions that the feasible set is closed at infinity and the ideal of homogenized equality constraining polynomials is real radical, we show that this hierarchy of Moment-SOS relaxations has finite convergence, if some optimality conditions (i.e., the linear independence constraint qualification, strict complementarity and second order sufficient conditions) hold at every minimizer, including the one at infinity. Moreover, we prove extended versions of Putinar-Vasilescu type Positivstellensatz for polynomials that are nonnegative on unbounded sets. The classical Moment-SOS hierarchy with denominators is also studied. In particular, we give a positive answer to a conjecture of Mai, Lasserre and Magron in their recent work. Polynomial optimization and Homogenization and Moment-SOS relaxations and Optimality conditions. Copyright © 2021, The Authors. All rights reserved.

关键词： Polynomials

A force-based gradient descent method for ab initio atomic structure relaxation

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arXiv 2022年

作者： Hu, Yukuan Gao, Xingyu Zhao, Yafan Liu, Xin Song, Haifeng State Key Laboratory of Scientific and Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China University of Chinese Academy of Sciences Beijing100049 China Laboratory of Computational Physics Institute of Applied Physics and Computational Mathematics Beijing100088 China CAEP Software Center for High Performance Numerical Simulation Beijing100088 China

Force-based algorithms for ab initio atomic structure relaxation, such as conjugate gradient methods, usually get stuck in the line minimization processes along search directions, where expensive ab initio calculations are triggered frequently to test trial positions before locating the next iterate. We present a force-based gradient descent method, WANBB, that circumvents the deficiency. At each iteration, WANBB enters the line minimization process with a trial stepsize capturing the local curvature of the energy surface. The exit is controlled by an unrestrictive criterion that tends to accept early trials. These two ingredients streamline the line minimization process in WANBB. The numerical simulations on nearly 80 systems with good universality demonstrate the considerable compression of WANBB on the cost for the unaccepted trials compared with conjugate gradient methods. We also observe across the board significant and universal speedups as well as the superior robustness of WANBB over several widely used methods. The latter point is theoretically established. The implementation of WANBB is pretty simple, in that no a priori physical knowledge is required and only two parameters are present without tuning. © 2022, CC BY.

关键词： Conjugate gradient method

Defining myocardial fiber bundle architecture in atrial digital twins

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Computers in Biology and Medicine 2025年 188卷 109774-109774页

作者： Piersanti, Roberto Bradley, Ryan Ali, Syed Yusuf Quarteroni, Alfio Dede’, Luca Trayanova, Natalia A. MOX - Laboratory of Modeling and Scientific Computing Dipartimento di Matematica Politecnico di Milano Milano Italy ADVANCE - Alliance for Cardiovascular Diagnostic and Treatment Innovation Johns Hopkins University Baltimore United States Research Computing Lehigh University BethlehemPA United States Department of Biomedical Engineering Johns Hopkins University Baltimore United States Mathematics Institute École Polytechnique Fédérale de Lausanne Lausanne Switzerland

A key component in developing atrial digital twins (ADT) - virtual representations of patients’ atria — is the accurate prescription of myocardial fibers which are essential for the tissue characterization. Due to the difficulty of reconstructing atrial fibers from medical imaging, a widely used strategy for fiber generation in ADT relies on mathematical models. Existing methodologies utilize semi-automatic approaches, are tailored to specific morphologies, and lack rigorous validation against imaging fiber data. In this study, we introduce a novel atrial Laplace–Dirichlet-Rule-Based Method (LDRBM) for prescribing highly detailed myofiber orientations and providing robust regional annotation in bi-atrial morphologies of any complexity. The robustness of our approach is verified in eight extremely detailed bi-atrial geometries, derived from a sub-millimeter Diffusion-Tensor-Magnetic-Resonance Imaging (DTMRI) human atrial fiber dataset. We validate the LDRBM by quantitatively recreating each of the DTMRI fiber architectures: a comprehensive comparison with DTMRI ground truth data is conducted, investigating differences between electrophysiology (EP) simulations provided by either LDRBM and DTMRI fibers. Finally, we demonstrate that the novel LDRBM outperforms current state-of-the-art (LDRBMs and Universal Atrial Coordinates) fiber models, confirming the exceptional accuracy of our methodology and the critical importance of incorporating detailed fiber orientations in EP simulations. Ultimately, this work represents a fundamental step towards the development of physics-based digital twins of the human atria, establishing a new standard for prescribing fibers in ADT. © 2025 The Authors

关键词： Magnetic resonance imaging

-matrix approximability of inverses of FEM matrices for the time-harmonic Maxwell equations

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Advances in computational mathematics 2022年第5期48卷 1-32页

作者： Faustmann, Markus Melenk, Jens Markus Parvizi, Maryam TU Wien Institute of Analysis and Scientific Computing Wien Austria Leibniz University Hannover Institute of Applied Mathematics Hannover Germany Cluster of Excellence PhoenixD (Photonics Optics and Engineering - Innovation Across Disciplines) Leibniz Universität Hannover Germany

The inverse of the stiffness matrix of the time-harmonic Maxwell equation with perfectly conducting boundary conditions is approximated in the blockwise low-rank format of $${\mathcal{H}}$$ -matrices. Under a technical assumption on the mesh, we prove that root exponential convergence in the block rank can be achieved, if the block structure conforms to a standard admissibility criterion.

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