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A four-field mixed formulation for incompressible finite elasticity

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Computer Methods in Applied Mechanics and engineering 2025年 444卷

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 ViennaA-1040 Austria HB Berrit Solhemsbackarna 73 SpångaSE-163 56 Sweden

In this work, we generalize the mass-conserving mixed stress (MCS) finite element method for Stokes equations (Gopalakrishnan et al., 2019), 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 experiments. © 2025 Elsevier B.V.

关键词： Galerkin methods

A DISCONTINUOUS GALERKIN METHOD BY PATCH RECONSTRUCTION FOR BIHARMONIC PROBLEM

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Journal of computational mathematics 2019年第4期37卷 524-540页

作者： Ruo Li Pingbing Ming Zhiyuan Sun Fanyi Yang Zhijian Yang CAPT LMAM and School of Mathematical Sciences Peking University Beijing 100871 China LSEC Institute of Computational Mathematics and Scientific/Engineering Computing AMSS Chinese Academy of Sciences Beijing 100190 China School of Mathematical Sciences University of Chinese Academy of Sciences No. 19A Yu-Quan Road Beijing 100049 China School of Mathematical Sciences Peking University Beijing 100871 China School of Mathematics and. Statistics Wuhan University Wuhan 430072 China

We propose a new discontinuous Galerkin method based on the least-squares patch reconstruction for the biharmonic problem. We prove the optimal error estimate of the proposed method. The two-dimensional and three-dimensional numerical examples are presented to confirm the accuracy and efficiency of the method with several boundary conditions and several types of polygon meshes and polyhedral meshes.

关键词： Least-squares problem Reconstructed basis function Discontinuous Galerkin method Biharmonic problem

Rational Framing Motions and Spatial Rational Pythagorean Hodograph Curves

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arXiv 2021年

作者： Kalkan, Bahar Scharler, Daniel F. Schröcker, Hans-Peter Šír, Zbyněk Van Yuzuncu Yil University Department of Mathematics Van65090 Turkey Universität Graz Institute of Mathematics and Scientific Computing Heinrichstr. 4 Graz8010 Austria Universität Innsbruck Department of Basic Sciences in Engineering Sciences Technikerstr. 13 Innsbruck6020 Austria Charles University Faculty of Mathematics and Physic Sokolovská 83 Prague186 75 Czech Republic

We propose a new method for constructing rational spatial Pythagorean Hodograph (PH) curves based on determining a suitable rational framing motion. While the spherical component of the framing motion is arbitrary, the translation part is determined be a modestly sized and nicely structured system of linear equations. Rather surprisingly, generic input data will only result in polynomial PH curves. We provide a complete characterization of all cases that admit truly rational (non-polynomial) solutions. Examples illustrate our ideas and relate them to existing *** Codes 65D17 Copyright © 2021, The Authors. All rights reserved.

关键词： Polynomials

Rubric-based holistic review: A promising route to equitable graduate admissions in physics

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Physical Review Physics Education Research 2022年第2期18卷 020140-020140页

作者： Nicholas T. Young K. Tollefson Remco G. T. Zegers Marcos D. Caballero Department of Physics and Astronomy Michigan State University East Lansing Michigan 48824 USA Department of Computational Mathematics Science and Engineering Michigan State University East Lansing Michigan 48824 USA National Superconducting Cyclotron Laboratory Michigan State University East Lansing Michigan 48824 USA Joint Institute for Nuclear Astrophysics Michigan State University East Lansing Michigan 48824 USA Center for Computing in Science Education and Department of Physics University of Oslo N-0316 Oslo Norway CREATE for STEM Institute Michigan State University East Lansing Michigan 48824 USA

As systematic inequities in higher education and society have been brought to the forefront, graduate programs are interested in increasing the diversity of their applicants and enrollees. Yet, structures in place to evaluate applicants may not support such aims. One potential solution to support those aims is rubric-based holistic review. Starting in 2018, our physics department implemented a rubric-based holistic review process for all applicants to our graduate program. The rubric assessed applicants on 18 metrics covering their grades, test scores, research experiences, noncognitive competencies, and fit with the program. We then compared faculty’s ratings of applicants by admission status, sex, and undergraduate program over a three-year period. We find that the rubric scores show statistically significant differences between admitted and nonadmitted students as hoped. We also find that differences in rubric scores based on sex or undergraduate program reflected known systematic inequities such as applicants from smaller and less prestigious undergraduate universities scoring lower on the physics GRE and women performing more volunteer work in academia. Our results then suggest rubric-based holistic review as a possible route to making graduate admissions in physics more equitable.

关键词： Diversity & inclusion Educational policy Graduate students

A plane wave study on the localized-extended transitions in the one-dimensional incommensurate systems

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arXiv 2020年

作者： Chen, Huajie Zhou, Aihui Zhou, Yuzhi School of Mathematical Sciences Beijing Normal University Beijing100875 China LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China Software Center for High Performance Numerical Simulation China Academy of Engineering Physics Beijing100088 China Institute of Applied Physics and Computational Mathematics Beijing100088 China

Based on our recently proposed plane wave framework, we theoretically study the localized-extended transition in the one dimensional incommensurate systems with cosine type of potentials, which are in close connection to many recent experiments in the ultracold atom and photonic crystal. We formulate a propagator based scattering picture for the transition at the ground state and single particle mobility edge, in which the deeper connection between the incommensurate potentials, eigenstate compositions and transition mechanism is revealed. We further show that there exists a upper limit of localization length for all localized eigenstates, leading to an fundamental difference to the Anderson localization. Numerical calculations are presented alongside the analysis to justify our statements. The theoretical analysis and numerical methods can also be generalized to systems in higher dimensions, with different potentials or beyond the single particle regime, which would benefit the future studies in the related fields. Copyright © 2020, The Authors. All rights reserved.

关键词： Elastic waves

AN OPTIMAL PIECEWISE CUBIC NONCONFORMING FINITE ELEMENT SCHEME FOR THE PLANAR BIHARMONIC EQUATION ON GENERAL TRIANGULATIONS

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arXiv 2019年

作者： Zhang, Shuo LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and System Sciences Chinese Academy of Sciences Beijing100190 China

This paper presents a nonconforming finite element scheme for the planar biharmonic equation which applis piecewise cubic polynomials (P3) and possesses O(h2) convergence rate in energy norm on general shape-regular triangulations. Both Dirichlet and Navier type boundary value problems are studied. The basis for the scheme is a piecewise cubic polynomial space, which can approximate the H4 functions with O(h2) accuracy in broken H2 norm. Besides, an equivalence (∇2h ·, ∇2h ·) = (∆h ·, ∆h ·), which is usually not true for nonconforming finite element spaces, is proved on the newly designed spaces. The finite element space does not correspond to a finite element defined with Ciarlet’s triple;however, a set of locally supported basis functions of the finite element space is still figured out. The notion of the finite element Stokes complex plays an important role in the analysis and also the construction of the basis functions. Copyright © 2019, The Authors. All rights reserved.

关键词： Finite element method

High-order semi-Lagrangian kinetic scheme for compressible turbulence

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Physical Review E 2021年第2期104卷 025301-025301页

作者： Dominik Wilde Andreas Krämer Dirk Reith Holger Foysi Department of Mechanical Engineering University of Siegen Paul-Bonatz-Straße 9-11 57076 Siegen-Weidenau Germany Institute of Technology Resource and Energy-efficient Engineering (TREE) Bonn-Rhein-Sieg University of Applied Sciences Grantham-Allee 20 53757 Sankt Augustin Germany Department of Mathematics and Computer Science Freie Universität Berlin Arnimallee 6 14195 Berlin Germany Fraunhofer Institute for Algorithms and Scientific Computing (SCAI) Schloss Birlinghoven 53754 Sankt Augustin Germany

Turbulent compressible flows are traditionally simulated using explicit time integrators applied to discretized versions of the Navier-Stokes equations. However, the associated Courant-Friedrichs-Lewy condition severely restricts the maximum time-step size. Exploiting the Lagrangian nature of the Boltzmann equation's material derivative, we now introduce a feasible three-dimensional semi-Lagrangian lattice Boltzmann method (SLLBM), which circumvents this restriction. While many lattice Boltzmann methods for compressible flows were restricted to two dimensions due to the enormous number of discrete velocities in three dimensions, the SLLBM uses only 45 discrete velocities. Based on compressible Taylor-Green vortex simulations we show that the new method accurately captures shocks or shocklets as well as turbulence in 3D without utilizing additional filtering or stabilizing techniques other than the filtering introduced by the interpolation, even when the time-step sizes are up to two orders of magnitude larger compared to simulations in the literature. Our new method therefore enables researchers to study compressible turbulent flows by a fully explicit scheme, whose range of admissible time-step sizes is dictated by physics rather than spatial discretization.

关键词： Compressible flows Kinetic theory Turbulence Lattice-Boltzmann methods Navier-Stokes equation

SWEMniCS: a software toolbox for modeling coastal ocean circulation, storm surges, inland, and compound flooding

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npj Natural Hazards

npj Natural Hazards 2024年第1期1卷 1-17页

作者： Clint Dawson Benjamin Pachev Jennifer Proft Eirik Valseth Mark Loveland Oden Institute for Computational Engineering and Sciences The University of Texas at Austin Austin TX USA The Department of Data Science The Norwegian University of Life Science Ås Norway Department of Scientific Computing and Numerical Analysis Simula Research Laboratory Oslo Norway Information and Technology Laboratory Engineering Research and Development Center US Army Corps of Engineers Vicksburg MS USA

Flooding from storm surges, rainfall-runoff, and their interaction into compounding events are major natural hazards in coastal regions. To assess risks of damages to life and properties alike, numerical models are needed to guide emergency responses and future assessments. Numerical models, such as ADCIRC have over many decades shown their usefulness in such assessments. However, these models have a high threshold in terms of new user engagement as development and compilation is not trivial for users trained in compiled programming languages. Here, we develop a new open-source finite element solver for the numerical simulation of flooding. The numerical solution of the underlying PDEs is developed using the finite element framework FEniCSx. The goal is a framework where new methods can be rapidly tested before time-consuming development into codes like ADCIRC. We validate the framework on several test cases, including large-scale computations in the Gulf of Mexico for Hurricane Ike (2008).

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An improved gradient method with approximately optimal stepsize based on conic model for unconstrained optimization

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arXiv 2019年

作者： Liu, Zexian Liu, Hongwei State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematics and Scientific/Engineering computing AMSS Chinese Academy of Sciences Beijing1000190 China School of Mathematics and Statistics Xidian University Xi’an710126 China

A new type of stepsize, which was recently introduced by Liu and Liu (Optimization, 67(3), 427-440, 2018), is called approximately optimal stepsize and is quit efficient for gradient method. Interestingly, all gradient methods can be regarded as gradient methods with approximately optimal stepsizes. In this paper, based on the work (Numer. Algorithms 78(1), 21-39, 2018), we present an improved gradient method with approximately optimal stepsize based on conic model for unconstrained optimization. If the objective function f is not close to a quadratic on the line segment between the current and latest iterates, we construct a conic model to generate approximately optimal stepsize for gradient method if the conic model can be used;otherwise, we construct some quadratic models to generate approximately optimal stepsizes for gradient method. The convergence of the proposed method is analyzed under suitable conditions. Numerical comparisons with some well-known conjugate gradient software packages such as CG_DESCENT (SIAM J. Optim. 16(1), 170-192, 2005) and CGOPT (SIAM J. Optim. 23(1), 296-320, 2013) indicate the proposed method is very promising. Copyright © 2019, The Authors. All rights reserved.

关键词： Gradient methods