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Preface： theory, methods, and applications of mesoscopic modeling

Applied Mathematics and Mechanics(english edition) 2018年第1期39卷 1-2页

作者： Z. LI guohui HU g.e. karniadakis Division of Applied Mathematics Brown UniversityProvidenceRhode Island 02912U.S.A. Shanghai Institute of Applied Mathematics and Mechanics Shanghai UniversityShanghai 200072China

With increasing attention to complex fluids and soft matter, we have witnessed a fast- growing research in mesoscopic modeling and simulation in the past decades. The development of mesoscopic methods offers many potential opportunities as well as challenges in modeling of complex materials for diverse applications. Despite significant prog.e.s in the past decade, mesoscopic methods are still under development. New formulation in the models, novel theo- retical interpretations, and innovative numerical algorithms often appear in literature. These mesoscopic methods have been already applied to a large number of problems, including poly- mer and colloidal suspensions, multiphase fluids, biological materials, and blood rheology. New applications of mesoscopic modeling in different areas are still emerging.

关键词： theory methods mesoscopic modeling

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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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efficient and Accurate Numerical Methods Using the Accelerated Spectral Deferred Correction for Solving Fractional Differential equations

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Numerical Mathematics(Theory,Methods and Applications) 2022年第4期15卷 876-902页

作者： Xuejuan Chen Zhiping Mao george em karniadakis School of Science Jimei UniversityXiamen 361021China School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific ComputingXiamen UniversityXiamenChina Division of Applied Mathematics Brown UniversityProvidenceRI 02912USA

We develop an efficient and accurate spectral deferred correction(SDC)method for fractional differential equations(FDes)by extending the algorithm in[14]for classical ordinary differential equations(ODes).Specifically,we discretize the resulted Picard integral equation by the SDC method and accelerate the convergence of the SDC iteration by using the generalized minimal residual algorithm(gMReS).We first derive the correction matrix of the SDC method for FDes and analyze the convergence region of the SDC *** then present several numerical examples for stiff and non-stiff FDes including fractional linear and nonlinear ODes as well as fractional phase field models,demonstrating that the accelerated SDC method is much more efficient than the original SDC method,especially for stiff ***,we resolve the issue of low accuracy arising from the singularity of the solutions by using a geometric mesh,leading to highly accurate solutions compared to uniform mesh solutions at almost the same computational ***,for long-time integration of FDes,using the geometric mesh leads to g.e.t computational savings as the total number of deg.e.s of freedom required is relatively small.

关键词： Stiff problem generalized minimal residual geometric mesh refinement long time evolution fractional phase field models

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Learning two-phase microstructure evolution using neural operators and autoencoder architectures

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npj Computational Materials 2022年第1期8卷 1804-1816页

作者： Vivek Oommen Khemraj Shukla Somdatta goswami Rémi Ding.e.ille george em karniadakis School of engineering Brown UniversityProvidenceRIUSA Division of Applied Mathematics Brown UniversityProvidenceRIUSA Center for Integrated Nanotechnolog.e. Sandia National LaboratoriesAlbuquerqueNM87185USA

Phase-field modeling is an effective but computationally expensive method for capturing the mesoscale morphological and microstructure evolution in ***,fast and generalizable surrogate models are needed to alleviate the cost of computationally taxing processes such as in optimization and design of *** intrinsic discontinuous nature of the physical phenomena incurred by the presence of sharp phase boundaries makes the training of the surrogate model *** develop a framework that integrates a convolutional autoencoder architecture with a deep neural operator(DeepONet)to learn the dynamic evolution of a two-phase mixture and accelerate time-to-solution in predicting the microstructure *** utilize the convolutional autoencoder to provide a compact representation of the microstructure data in a low-dimensional latent *** DeepONet is trained in the latent space,it can be used to replace the high-fidelity phase-field numerical solver in interpolation tasks or to accelerate the numerical solver in extrapolation tasks.

关键词： materials phase operators

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