Article
Mathematics, Applied
Wei Yang, Xin Liu, Bin He, Yunqing Huang
Summary: In this paper, the a posteriori error estimator of the SDG method for variable coefficients time-harmonic Maxwell's equations is studied. Two a posteriori error estimators are proposed, one being the recovery-type estimator and the other being the residual-type estimator. The curl-recovery method for the staggered discontinuous Galerkin method (SDGM) is first proposed, and an asymptotically exact error estimator is constructed based on the super-convergence result of the postprocessed solution. The reliability and effectiveness of the residual-type a posteriori error estimator are also proved for variable coefficients time-harmonic Maxwell's equations. The efficiency and robustness of the proposed estimators are demonstrated through numerical experiments.
JOURNAL OF COMPUTATIONAL MATHEMATICS
(2023)
Article
Mathematics, Applied
Zhiqiang Cai, Jing Yang
Summary: In this paper, a class of discontinuous Galerkin finite element methods for advection-diffusion-reaction problems is presented, and a priori error estimates are established when the solution is only in H1+s (Omega) space with s is an element of(0, 1/2].
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
Yanjun Li, Hai Bi, Yidu Yang
Summary: In this paper, we studied the discontinuous Galerkin finite element method for the Steklov eigenvalue problem in inverse scattering. We presented complete error estimates including both a priori and a posteriori error estimators, and proved the reliability and efficiency of the a posteriori error estimators for eigenfunctions up to higher order terms. We also analyzed the reliability of estimators for eigenvalues, and conducted numerical experiments in an adaptive fashion to show the optimal convergence order of our method.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Huihui Cao, Yunqing Huang, Nianyu Yi
Summary: This paper investigates the adaptive direct discontinuous Galerkin method for second order elliptic equations in two dimensions, introducing a numerical flux with general weighted averages and proper weights for interface problems. In addition, it proposes a residual-type a posteriori error estimator and establishes global upper bounds and local lower bounds for errors in the DG norm. Several numerical examples are conducted to confirm the reliability and efficiency of the proposed error estimator and method.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
Yingying Xie, Ming Tang, Chunming Tang
Summary: In this paper, a weak Galerkin finite element method for solving the indefinite time-harmonic Maxwell equations is developed. By deducing an error equation, optimal a priori error estimates in both the energy norm and the L 2 norm are achieved. Numerical experiments are conducted to confirm the theoretical conclusions.
APPLIED MATHEMATICS AND COMPUTATION
(2022)
Article
Computer Science, Interdisciplinary Applications
Assyr Abdulle, Giacomo Rosilho de Souza
Summary: This paper introduces a local adaptive discontinuous Galerkin method for convection-diffusion-reaction equations. The method improves the accuracy of the solution by solving local elliptic problems in refined subdomains and provides an algorithm for the automatic identification of these subdomains. Numerical comparisons demonstrate the efficiency of the method.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
R. H. W. Hoppe
Summary: We consider an adaptive C0 Interior Penalty Discontinuous Galerkin (C0IPDG) approximation of the fourth order von Karman equations with homogeneous Dirichlet boundary conditions and an equilibrated a posteriori error estimator. The C0IPDG method can be derived from a six-field formulation of the finite element discretized von Karman equations. The equilibrated a posteriori error estimator provides an upper bound for the discretization error in the broken W 2,2 0 norm in terms of the associated primal and dual energy functionals. It requires the construction of equilibrated fluxes and equilibrated moment tensors which can be computed on local patches around interior nodal points of the triangulations. The relationship with a residual-type a posteriori error estimator is studied as well. Numerical results illustrate the performance of the suggested approach.
APPLIED NUMERICAL MATHEMATICS
(2023)
Article
Mathematics, Applied
Helmi Temimi
Summary: In this paper, a novel discontinuous Galerkin (DG) finite element method is developed to solve the Poisson's equation on Cartesian grids. The method first applies the standard DG method in the x-spatial variable, resulting in a system of ordinary differential equations (ODEs) in the y-variable. The method of line is then used to discretize the ODEs. The proposed fully DG scheme utilizes DG methods of different degrees in the x and y variables and achieves optimal convergence rate.
APPLIED NUMERICAL MATHEMATICS
(2023)
Article
Mathematics, Applied
Jing Wen, Jian Su, Yinnian He, Hongbin Chen
Summary: In this paper, semi-discrete and fully discrete schemes of the Stokes-Biot model are proposed and analyzed in detail. The existence and uniqueness of the semi-discrete scheme are proved, with a-priori error estimates derived. Numerical tests under matching and non-matching meshes validate the convergence analysis and support the theoretical results.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
Divay Garg, Kamana Porwal
Summary: In this article, the Dirichlet boundary control problem governed by the Poisson equation is studied, and symmetric discontinuous Galerkin finite element methods are designed and analyzed for its numerical approximation. The discrete optimality system is obtained by exploiting the symmetric property of the bilinear forms. By utilizing various intermediate problems, the optimal order convergence rates are obtained for the control in the energy and L-2 norms. Additionally, an a posteriori error estimator is derived using an auxiliary system of equations, which is proven to be reliable and efficient. Numerical experiments are conducted to validate the theoretical findings.
APPLIED NUMERICAL MATHEMATICS
(2023)
Article
Mathematics, Applied
Haitao Leng
Summary: In this paper, a hybridizable discontinuous Galerkin method with divergence-free and H(div)-conforming velocity field is proposed for the stationary incompressible Navier-Stokes equations. The pressure-robustness, which ensures that the a priori error estimates of the velocity are independent of the pressure error, is satisfied. Additionally, an efficient and reliable a posteriori error estimator is derived for the L-2 errors in the velocity gradient and pressure, under a smallness assumption. Numerical examples are provided to demonstrate the pressure-robustness and the performance of the obtained a posteriori error estimator.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Computer Science, Interdisciplinary Applications
Jiawei Sun, Chi-Wang Shu, Yulong Xing
Summary: This paper investigates one- and multi-dimensional stochastic Maxwell equations with additive noise and designs high order discontinuous Galerkin methods. The proposed methods satisfy the linear growth property of stochastic energy and preserve the multi-symplectic structure, with an optimal error estimate of the semi-discrete DG method analyzed.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
Z. Dong, L. Mascotto
Summary: This paper proves the effectiveness of hp-optimal error estimates for interior penalty discontinuous Galerkin methods (IPDG) in solving the biharmonic problem with homogeneous essential boundary conditions. The study considers both tensor product-type meshes in two and three dimensions, as well as triangular meshes in two dimensions. A key aspect of the analysis is the construction of a global H-2 piecewise polynomial approximants with hp-optimal approximation properties over the given meshes. The paper also discusses the hp-optimality of C-0-IPDG in two and three dimensions, as well as the stream formulation of the Stokes problem in two dimensions. Numerical experiments validate the theoretical predictions and show that p-suboptimality occurs in the presence of singular essential boundary conditions.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Engineering, Multidisciplinary
Liang Wang, Xinpeng Yuan, Chunguang Xiong
Summary: In this paper, we study the new isogeometric analysis penalty discontinuous Galerkin methods for convection problems on implicitly defined surfaces, which exhibit optimal convergence properties. Theoretical results are demonstrated through two numerical experiments.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2023)
Article
Mathematics, Applied
Haitao Leng, Yanping Chen
Summary: This paper investigates a hybridizable discontinuous Galerkin method for second order elliptic equations with Dirac measures. Theoretical analyses include a priori error estimate in L-2-norm under convex domain and quasi-uniform mesh assumption. Additionally, a posteriori error estimates for L-2-norm and W-1,W-p-seminorm errors are obtained through duality argument and Oswald interpolation, which are validated with numerical examples.
ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS
(2022)
Article
Mathematics, Applied
Huangxin Chen, Jingzhi Li, Weifeng Qiu
IMA JOURNAL OF NUMERICAL ANALYSIS
(2016)
Article
Chemistry, Physical
Zhijun Wang, Weifeng Qiu, Yong Yang, C. T. Liu
Article
Mathematics, Applied
Weifeng Qiu, Ke Shi
JOURNAL OF SCIENTIFIC COMPUTING
(2016)
Article
Mathematics, Applied
Bernardo Cockburn, Guosheng Fu, Weifeng Qiu
IMA JOURNAL OF NUMERICAL ANALYSIS
(2017)
Article
Mathematics, Applied
Weifeng Qiu, Ke Shi
IMA JOURNAL OF NUMERICAL ANALYSIS
(2016)
Article
Mathematics, Applied
Weifeng Qiu, Minglei Wang, Jiahao Zhang
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2016)
Article
Mathematics, Applied
Huangxin Chen, Weifeng Qiu, Ke Shi, Manuel Solano
JOURNAL OF SCIENTIFIC COMPUTING
(2017)
Article
Mathematics, Applied
Weifeng Qiu, Manuel Solano, Patrick Vega
JOURNAL OF SCIENTIFIC COMPUTING
(2016)
Article
Mathematics, Applied
Peipei Lu, Huangxin Chen, Weifeng Qiu
MATHEMATICS OF COMPUTATION
(2017)
Article
Mathematics, Applied
Aycil Cesmelioglu, Bernardo Cockburn, Weifeng Qiu
MATHEMATICS OF COMPUTATION
(2017)
Article
Mathematics, Applied
Huangxin Chen, Weifeng Qiu
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2017)
Article
Mathematics, Applied
Weifeng Qiu, Jiguang Shen, Ke Shi
MATHEMATICS OF COMPUTATION
(2018)
Article
Mathematics, Applied
Eric T. Chung, Weifeng Qiu
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2017)
Article
Mathematics, Applied
Guosheng Fu, Yanyi Jin, Weifeng Qiu
IMA JOURNAL OF NUMERICAL ANALYSIS
(2019)
Article
Engineering, Multidisciplinary
Akshay J. Thomas, Mateusz Jaszczuk, Eduardo Barocio, Gourab Ghosh, Ilias Bilionis, R. Byron Pipes
Summary: We propose a physics-guided transfer learning approach to predict the thermal conductivity of additively manufactured short-fiber reinforced polymers using micro-structural characteristics obtained from tensile tests. A Bayesian framework is developed to transfer the thermal conductivity properties across different extrusion deposition additive manufacturing systems. The experimental results demonstrate the effectiveness and reliability of our method in accounting for epistemic and aleatory uncertainties.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2024)
Article
Engineering, Multidisciplinary
Zhen Zhang, Zongren Zou, Ellen Kuhl, George Em Karniadakis
Summary: In this study, deep learning and artificial intelligence were used to discover a mathematical model for the progression of Alzheimer's disease. By analyzing longitudinal tau positron emission tomography data, a reaction-diffusion type partial differential equation for tau protein misfolding and spreading was discovered. The results showed different misfolding models for Alzheimer's and healthy control groups, indicating faster misfolding in Alzheimer's group. The study provides a foundation for early diagnosis and treatment of Alzheimer's disease and other misfolding-protein based neurodegenerative disorders using image-based technologies.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2024)
Article
Engineering, Multidisciplinary
Jonghyuk Baek, Jiun-Shyan Chen
Summary: This paper introduces an improved neural network-enhanced reproducing kernel particle method for modeling the localization of brittle fractures. By adding a neural network approximation to the background reproducing kernel approximation, the method allows for the automatic location and insertion of discontinuities in the function space, enhancing the modeling effectiveness. The proposed method uses an energy-based loss function for optimization and regularizes the approximation results through constraints on the spatial gradient of the parametric coordinates, ensuring convergence.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2024)
Article
Engineering, Multidisciplinary
Bodhinanda Chandra, Ryota Hashimoto, Shinnosuke Matsumi, Ken Kamrin, Kenichi Soga
Summary: This paper proposes new and robust stabilization strategies for accurately modeling incompressible fluid flow problems in the material point method (MPM). The proposed approach adopts a monolithic displacement-pressure formulation and integrates two stabilization strategies to ensure stability. The effectiveness of the proposed method is validated through benchmark cases and real-world scenarios involving violent free-surface fluid motion.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2024)
Article
Engineering, Multidisciplinary
Chao Peng, Alessandro Tasora, Dario Fusai, Dario Mangoni
Summary: This article discusses the importance of the tangent stiffness matrix of constraints in multibody systems and provides a general formulation based on quaternion parametrization. The article also presents the analytical expression of the tangent stiffness matrix derived through linearization. Examples demonstrate the positive effect of this additional stiffness term on static and eigenvalue analyses.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2024)
Article
Engineering, Multidisciplinary
Thibaut Vadcard, Fabrice Thouverez, Alain Batailly
Summary: This contribution presents a methodology for detecting isolated branches of periodic solutions to nonlinear mechanical equations. The method combines harmonic balance method-based solving procedure with the Melnikov energy principle. It is able to predict the location of isolated branches of solutions near families of autonomous periodic solutions. The relevance and accuracy of this methodology are demonstrated through academic and industrial applications.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2024)
Article
Engineering, Multidisciplinary
Weisheng Zhang, Yue Wang, Sung-Kie Youn, Xu Guo
Summary: This study proposes a sketch-guided topology optimization approach based on machine learning, which incorporates computer sketches as constraint functions to improve the efficiency of computer-aided structural design models and meet the design intention and requirements of designers.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2024)
Article
Engineering, Multidisciplinary
Leilei Chen, Zhongwang Wang, Haojie Lian, Yujing Ma, Zhuxuan Meng, Pei Li, Chensen Ding, Stephane P. A. Bordas
Summary: This paper presents a model order reduction method for electromagnetic boundary element analysis and extends it to computer-aided design integrated shape optimization of multi-frequency electromagnetic scattering problems. The proposed method utilizes a series expansion technique and the second-order Arnoldi procedure to reduce the order of original systems. It also employs the isogeometric boundary element method to ensure geometric exactness and avoid re-meshing during shape optimization. The Grey Wolf Optimization-Artificial Neural Network is used as a surrogate model for shape optimization, with radar cross section as the objective function.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2024)
Article
Engineering, Multidisciplinary
C. Pilloton, P. N. Sun, X. Zhang, A. Colagrossi
Summary: This paper investigates the smoothed particle hydrodynamics (SPH) simulations of violent sloshing flows and discusses the impact of volume conservation errors on the simulation results. Different techniques are used to directly measure the particles' volumes and stabilization terms are introduced to control the errors. Experimental comparisons demonstrate the effectiveness of the numerical techniques.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2024)
Article
Engineering, Multidisciplinary
Ye Lu, Weidong Zhu
Summary: This work presents a novel global digital image correlation (DIC) method based on a convolution finite element (C-FE) approximation. The C-FE based DIC provides highly smooth and accurate displacement and strain results with the same element size as the usual finite element (FE) based DIC. The proposed method's formulation and implementation, as well as the controlling parameters, have been discussed in detail. The C-FE method outperformed the FE method in all tested examples, demonstrating its potential for highly smooth, accurate, and robust DIC analysis.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2024)
Article
Engineering, Multidisciplinary
Mojtaba Ghasemi, Mohsen Zare, Amir Zahedi, Pavel Trojovsky, Laith Abualigah, Eva Trojovska
Summary: This paper introduces Lung performance-based optimization (LPO), a novel algorithm that draws inspiration from the efficient oxygen exchange in the lungs. Through experiments and comparisons with contemporary algorithms, LPO demonstrates its effectiveness in solving complex optimization problems and shows potential for a wide range of applications.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2024)
Article
Engineering, Multidisciplinary
Jingyu Hu, Yang Liu, Huixin Huang, Shutian Liu
Summary: In this study, a new topology optimization method is proposed for structures with embedded components, considering the tension/compression asymmetric interface stress constraint. The method optimizes the topology of the host structure and the layout of embedded components simultaneously, and a new interpolation model is developed to determine interface layers between the host structure and embedded components.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2024)
Article
Engineering, Multidisciplinary
Qiang Liu, Wei Zhu, Xiyu Jia, Feng Ma, Jun Wen, Yixiong Wu, Kuangqi Chen, Zhenhai Zhang, Shuang Wang
Summary: In this study, a multiscale and nonlinear turbulence characteristic extraction model using a graph neural network was designed. This model can directly compute turbulence data without resorting to simplified formulas. Experimental results demonstrate that the model has high computational performance in turbulence calculation.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2024)
Article
Engineering, Multidisciplinary
Jacinto Ulloa, Geert Degrande, Jose E. Andrade, Stijn Francois
Summary: This paper presents a multi-temporal formulation for simulating elastoplastic solids under cyclic loading. The proper generalized decomposition (PGD) is leveraged to decompose the displacements into multiple time scales, separating the spatial and intra-cyclic dependence from the inter-cyclic variation, thereby reducing computational burden.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2024)
Article
Engineering, Multidisciplinary
Utkarsh Utkarsh, Valentin Churavy, Yingbo Ma, Tim Besard, Prakitr Srisuma, Tim Gymnich, Adam R. Gerlach, Alan Edelman, George Barbastathis, Richard D. Braatz, Christopher Rackauckas
Summary: This article presents a high-performance vendor-agnostic method for massively parallel solving of ordinary and stochastic differential equations on GPUs. The method integrates with a popular differential equation solver library and achieves state-of-the-art performance compared to hand-optimized kernels.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2024)
