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The High-Order Variable-Coefficient Explicit-Implicit-Null Method for Diffusion and Dispersion Equations

应用数学与计算数学学报 2025年第1期7卷 115-150页

作者： Meiqi Tan Juan Cheng Chi-Wang Shu Graduate School China Academy of Engineering PhysicsBeijing 100088China Laboratory of Computational Physics Institute of Applied Physics and Computational MathematicsBeijing 100088China HEDPS Center for Applied Physics and TechnologyCollege of EngineeringPeking UniversityBeijing 100871China Division of Applied Mathematics Brown UniversityProvidenceRI 02912USA

For the high-order diffusion and dispersion equations,the general practice of the explicit-implicit-null(EIN)method is to add and subtract an appropriately large linear highest derivative term with a constant coefficient at one side of the equation,and then apply the standard implicit-explicit method to the equivalent *** call this approach the con-stant-coefficient EIN method in this paper and hereafter denote it by"CC-EIN".To reduce the error in the CC-EIN method,the variable-coefficient explicit-implicit-null(VC-EIN)method,which is obtained by adding and subtracting a linear highest derivative term with a variable coefficient,is proposed and studied in this *** with the local discon-tinuous Galerkin(LDG)spatial discretization,the VC-EIN method is shown to be uncon-ditionally stable and can achieve high order of accuracy for both one-dimensional and two-dimensional quasi-linear and nonlinear *** addition,although the computational cost slightly increases,the VC-EIN method can obtain more accurate results than the CC-EIN method,if the diffusion coefficient or the dispersion coefficient has a few high and narrow bumps and the bumps only account for a small part of the whole computational domain.

关键词： Diffusion equation Dispersion equation Stability Explicit-implicit-null(EIN)time discretization Local discontinuous Galerkin(LDG)method

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Rethinking materials simulations:Blending direct numerical simulations with neural operators

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npj Computational Materials 2024年第1期10卷 1767-1780页

作者： Vivek Oommen Khemraj Shukla Saaketh Desai Rémi Dingreville George Em Karniadakis School of Engineering Brown UniversityProvidenceRIUSA Division of Applied Mathematics Brown UniversityProvidenceRIUSA Center for Integrated Nanotechnologies Sandia National LaboratoriesAlbuquerqueNMUSA

Materials simulations based on direct numerical solvers are accurate but computationally expensive for predicting materials evolution across length-and time-scales,due to the complexity of the underlying evolution equations,the nature of multiscale spatiotemporal interactions,and the need to reach long-time *** develop a method that blends direct numerical solvers with neural operators to accelerate such *** methodology is based on the integration of a community numerical solver with a U-Net neural operator,enhanced by a temporal-conditioning mechanism to enable accurate extrapolation and efficient time-to-solution predictions of the *** demonstrate the effectiveness of this hybrid framework on simulations of microstructure evolution via the phase-field *** simulations exhibit high spatial gradients and the co-evolution of different material phases with simultaneous slow and fast materials *** establish accurate extrapolation of the coupled solver with large speed-up compared to DNS depending on the hybrid strategy *** methodology is generalizable to a broad range of materials simulations,from solid mechanics to fluid dynamics,geophysics,climate,and more.

关键词： dynamics method operators

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A High Order Positivity-Preserving Discontinuous Galerkin Remapping Method Based on a Moving Mesh Solver for ALE Simulation of the Compressible Fluid Flow

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Communications in Computational Physics 2023年第10期34卷 1177-1214页

作者： Xiaolu Gu Juan Cheng Chi-Wang Shu Graduate School China Academy of Engineering PhysicsBeijing 100088China Laboratory of Computational Physics Institute of Applied Physics and Computational MathematicsBeijing 100088China HEDPS Center for Applied Physics and Technologyand College of EngineeringPeking UniversityBeijing 100871China Division of Applied Mathematics Brown UniversityProvidenceRI 02912

The arbitrary Lagrangian-Eulerian(ALE)method is widely used in the field of compressible multi-material and multi-phase flow *** order to implement the indirect ALE approach for the simulation of compressible flow in the context of high order discontinuous Galerkin(DG)discretizations,we present a high order positivity-preserving DG remapping method based on a moving mesh solver in this *** remapping method is based on the ALE-DG method developed by Klingenberg et al.[17,18]to solve the trivial equation∂u/∂t=0 on a moving mesh,which is the old mesh before remapping at t=0 and is the new mesh after remapping at t=*** appropriate selection of the final pseudo-time T can always satisfy the relatively mild smoothness requirement(Lipschitz continuity)on the mesh movement velocity,which guarantees the high order accuracy of the remapping *** use a multi-resolution weighted essentially non-oscillatory(WENO)limiter which can keep the essentially non-oscillatory property near strong discontinuities while maintaining high order accuracy in smooth *** further employ an effective linear scaling limiter to preserve the positivity of the relevant physical variables without sacrificing conservation and the original high order *** experiments are provided to illustrate the high order accuracy,essentially non-oscillatory performance and positivity-preserving of our remapping *** addition,the performance of the ALE simulation based on the DG framework with our remapping algorithm is examined in one-and two-dimensional Euler equations.

关键词： Remapping discontinuous Galerkin method arbitrary Lagrangian-Eulerian high order accuracy multi-resolution WENO limiter positivity-preserving

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Preface: machine-learning approaches for computational mechanics

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applied mathematics and Mechanics(English Edition) 2023年第7期44卷 1035-1038页

作者： Z.LI Guohui HU Zhiliang WANG G.E.KARNIADAKIS Department of Mechanical Engineering Clemson UniversityClemsonSouth Carolina 29634U.S.A. Shanghai Institute of Applied Mathematics and Mechanics School of Mechanics and Engineering ScienceShanghai UniversityShanghai 200072China Division of Applied Mathematics Brown UniversityProvidenceRhode Island 02912U.S.A.

Machine-learning(ML) approaches have gained significant attention in recent years for their potential to revolutionize various scientific research fields,including computational mechanics,materials science,and biomedical *** the field of computational mechanics,various neural network models such as feedforward neural networks,convolutional neural networks.

关键词： neural networks engineering

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Lattice Isomorphism and L-homeomorphism

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Journal of Intelligent and Fuzzy Systems 2024年第4期46卷 9069-9082页

作者： Tomy, Navin Johnson, T.P. Department of Mathematics Cochin University of Science and Technology Kerala Cochin India Department of Mathematics Bharata Mata College Kerala Thrikkakara India Applied Sciences and Humanities Division School of Engineering Cochin University of Science and Technology Kerala Cochin India

This paper deals with lattice isomorphic L-topological spaces. We are concerned with a question: Under what conditions will a lattice isomorphic L-topological spaces be L-homeomorphic. We give contributions to this qu... 详细信息

关键词： Topology

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Bi-Orthogonal fPINN:A Physics-Informed Neural Network Method for Solving Time-Dependent Stochastic Fractional PDEs

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Communications in Computational Physics 2023年第9期34卷 1133-1176页

作者： Lei Ma Rongxin Li Fanhai Zeng Ling Guo George Em Karniadakis Department of Mathematics Shanghai Normal UniversityShanghai 200234China School of Mathematics Shandong UniversityShandong 250100China Division of Applied Mathematics Brown UniversityProvidenceRI 02906USAand School of EngineeringBrown UniversityProvidenceRI 02906USA

Fractional partial differential equations(FPDEs)can effectively represent anomalous transport and nonlocal ***,inherent uncertainties arise naturally in real applications due to random forcing or unknown material *** models considering nonlocal interactions with uncertainty quantification can be formulated as stochastic fractional partial differential equations(SFPDEs).There are many challenges in solving SFPDEs numerically,especially for long-time integration since such problems are high-dimensional and ***,we combine the bi-orthogonal(BO)method for representing stochastic processes with physicsinformed neural networks(PINNs)for solving partial differential equations to formulate the bi-orthogonal PINN method(BO-fPINN)for solving time-dependent ***,we introduce a deep neural network for the stochastic solution of the time-dependent SFPDEs,and include the BO constraints in the loss function following a weak *** automatic differentiation is not currently applicable to fractional derivatives,we employ discretization on a grid to compute the fractional derivatives of the neural network *** weak formulation loss function of the BO-fPINN method can overcome some drawbacks of the BO methods and thus can be used to solve SFPDEs with eigenvalue ***,the BO-fPINN method can be used for inverse SFPDEs with the same framework and same computational complexity as for forward *** demonstrate the effectiveness of the BO-fPINN method for different benchmark ***,we first consider an SFPDE with eigenvalue crossing and obtain good results while the original BO method *** then solve several forward and inverse problems governed by SFPDEs,including problems with noisy initial *** study the effect of the fractional order as well as the number of the BO modes on the accuracy of the BO-fPINN *** results demonstrate the flexibility and efficiency

关键词： Scientific machine learning uncertainty quantification stochastic fractional differential equations PINNs inverse problems

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Solid solution dependence of the deformation behavior in Mg-xZn(x=0,1,2 wt%)alloys:In-situ neutron diffraction and crystal plasticity modeling

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Journal of Magnesium and Alloys 2025年第2期13卷 823-838页

作者： Huai Wang Soo Yeol Lee You Sub Kim Huamiao Wang Wanchuck Woo Ke An School of Mechanical and Automotive Engineering Qingdao University of TechnologyQingdao 266520China Department of Materials Science and Engineering Chungnam National UniversityDaejeon 34134Republic of Korea Shanghai Institute of Applied Mathematics and Mechanics School of Mechanics and Engineering ScienceShanghai UniversityShanghai 200072China Neutron Science Division Korea Atomic Energy Research InstituteDaejeon 34057Republic of Korea Neutron Scattering Division Oak Ridge National LaboratoryOak Ridge 37831United States

The effects of solid solution on the deformation behavior of binary Mg-xZn(x=0,1,2 wt%)alloys featuring a designated texture that enables extension twinning under tension parallel to the basal pole in most grains,were investigated using in-situ neutron diffraction and the EVPSC-TDT *** diffraction was used to quantitatively track grain-level lattice strains and diffraction intensity changes(related to mechanical twinning)in differently oriented grains of each alloy during cyclic tensile/compressive *** measurements were accurately captured by the *** stress-strain curves of Mg-1 wt%Zn and Mg-2 wt%Zn alloys show as-expected solid solution strengthening from the addition of Zn compared to pure *** macroscopic yielding and hardening behaviors are explained by alternating slip and twinning modes as calculated by the *** solid solution's influence on individual deformation modes,including basal〈a〉slip,prismatic〈a〉slip,and extension twinning,was then quantitatively assessed in terms of activity,yielding behavior,and hardening response by combining neutron diffraction results with crystal plasticity *** Mg-1 wt%Zn alloy displays distinct yielding and hardening behavior due to solid solution softening of prismatic〈a〉***,the dependence of extension twinning,in terms of the twinning volume fraction,on Zn content exhibits opposite trends under tensile and compressive loadings.

关键词： Magnesium alloy Deformation behavior Solid solution Crystal plasticity modeling Neutron diffraction

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On Error-Based Step Size Control for Discontinuous Galerkin Methods for Compressible Fluid Dynamics

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应用数学与计算数学学报 2025年第1期7卷 3-39页

作者： Hendrik Ranocha Andrew R.Winters Hugo Guillermo Castro Lisandro Dalcin Michael Schlottke-Lakemper Gregor J.Gassner Matteo Parsani Applied Mathematics University of HamburgBundesstr.5520146 HamburgGermany Department of Mathematics Applied Mathematics Linköping UniversityLinköpingSweden Extreme Computing Research Center(ECRC) Computer Electrical and Mathematical Science and Engineering Division(CEMSE)King Abdullah University of Science and Technology(KAUST)Thuwal 23955-6900Saudi Arabia Applied and Computational Mathematics RWTH Aachen UniversityAachenGermany High-Performance Computing Center Stuttgart University of StuttgartStuttgartGermany Department of Mathematics and Computer Science University of CologneCologneGermany Center for Data and Simulation Science University of CologneCologneGermany Extreme Computing Research Center(ECRC) Computer Electrical and Mathematical Science and Engineering Division(CEMSE)King Abdullah University of Science and Technology(KAUST)Thuwal 23955-6900Saudi Arabia Physical Science and Engineering Division(PSE) King Abdullah University of Science and Technology(KAUST)Thuwal 23955-6900Saudi Arabia

We study a temporal step size control of explicit Runge-Kutta(RK)methods for com-pressible computational fluid dynamics(CFD),including the Navier-Stokes equations and hyperbolic systems of conservation laws such as the Euler *** demonstrate that error-based approaches are convenient in a wide range of applications and compare them to more classical step size control based on a Courant-Friedrichs-Lewy(CFL)*** numerical examples show that the error-based step size control is easy to use,robust,and efficient,e.g.,for(initial)transient periods,complex geometries,nonlinear shock captur-ing approaches,and schemes that use nonlinear entropy *** demonstrate these properties for problems ranging from well-understood academic test cases to industrially relevant large-scale computations with two disjoint code bases,the open source Julia pack-ages *** with *** and the C/Fortran code SSDC based on PETSc.

关键词： Explicit Runge-Kutta(RK)methods Step size control Compressible fluid dynamics Adaptivity in space and time Shock capturing

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An optimized Chebyshev–Halley type family of multiple solvers: Extensive analysis and applications

An optimized Chebyshev–Halley type family of multiple solve...

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作者： Rani, Litika Soleymani, Fazlollah Kansal, Munish Kumar Nashine, Hemant School of Mathematics Thapar Institute of Engineering and Technology Punjab India Zanjan Iran Mathematics Division School of Advanced Sciences and Languages VIT Bhopal University Bhopal India Department of Mathematics and Applied Mathematics University of Johannesburg Kingsway Campus Johannesburg South Africa

In this manuscript, we introduce a higher-order optimal family of Chebyshev–Halley type methods to solve a univariate nonlinear equation having multiple roots. The proposed scheme considers weight functions that are selected adequately to optimize the convergence order and demands only four functional evaluations at each iteration. An extensive convergence analysis is also provided that demonstrates the establishment of eighth-order convergence of the developed scheme. As a result, the efficiency index of our scheme is optimal according to the Kung–Traub conjecture. Numerical experiments are performed on some real-life as well as nonlinear academic problems. In addition, the dynamical study of iterative schemes reflects the good overview of their stability, convergence properties, and graphical aspects by drawing attraction basins in the complex plane. Numerical findings indicate that the proposed methods outperform the existing ones with similar characteristics. © 2022 John Wiley & Sons, Ltd.

关键词： Nonlinear equations

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Thin film chalcogenides and optical design for mid-infrared sensing applications

Thin film chalcogenides and optical design for mid-infrared ...

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CLEO: Science and Innovations in CLEO 2024, CLEO: S and I 2024 - Part of Conference on Lasers and Electro-Optics

作者： El-Kholy, Ibrahim Al-Alexis Khan, Amir Division of Physics Engineering Mathematics and Computer Science Optical Science Center for Applied Research Delaware State University DoverDE United States

ISBN: (纸本)9781957171395

Constituent elements of Ge-Se-S glasses were chosen with different compositions such as Ge2Se8, Ge2S8, Se2S8, and GeSe2S2 to study their optical properties for UV, visible, and Mid-IR range using different optical tools like UV-Vis, FTIR, and Raman spectroscopy for mid-infrared thin film applications. © Optica Publishing Group 2024

关键词： Fourier transform infrared spectroscopy

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