Article
Mathematics, Applied
Hailong Qiu
Summary: In this paper, an optimally accurate second-order time-stepping algorithm for nonstationary magneto-hydrodynamics equations is studied. The algorithm linearly treats the nonlinear terms in the momentum equations and magnetic equations, and adds stabilization terms for the discrete solutions of velocity, pressure, and magnetic field. It is derived that this algorithm is unconditionally stable and has optimally accurate error estimate. Numerical tests are conducted to validate the predicted convergence rate.
APPLIED NUMERICAL MATHEMATICS
(2023)
Article
Mathematics, Applied
Ruonan Cao, Nan Jiang, Huanhuan Yang
Summary: In this paper, three second-order, linear, unconditionally stable decoupling methods based on the Crank-Nicolson leap-frog time discretization are proposed for solving the ACNS phase field model. The ACNS system is decoupled using the artificial compression method and a splitting approach with an exponential scalar auxiliary variable. It is proven that all three algorithms are unconditionally long-time stable. Numerical examples are provided to validate the convergence rate, unconditional stability, and computational efficiency.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2023)
Article
Mathematics
Taohua Liu, Xiucao Yin, Yinghao Chen, Muzhou Hou
Summary: In this paper, a practical numerical method for solving a one-dimensional two-sided space-fractional diffusion equation with variable coefficients in a finite domain is investigated. The method is based on the classical Crank-Nicolson (CN) method combined with Richardson extrapolation, and it provides second-order exact numerical estimates in time and space. The unconditional stability and convergence of the method are also tested, and two numerical examples are presented and compared with the exact solution.
Article
Mathematics, Applied
Buyang Li, Shu Ma, Na Wang
Summary: This article discusses the numerical approximation of the two-dimensional nonstationary Navier-Stokes equations with H-1 initial data. By using special locally refined temporal stepsizes, it is proven that the linearly extrapolated Crank-Nicolson scheme, with the stabilized Taylor-Hood finite element method in space, can achieve second-order convergence in time and space. Numerical examples are provided to validate the theoretical analysis.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Yingwen Guo, Yinnian He
Summary: An efficient method is examined for the Oldroyd fluid, which reduces the nonlinear integro-differential equations to linear equations, significantly increasing computational efficiency. The approach uses finite element method in space and second-order Crank-Nicolson extrapolation in time. The method is unconditionally stable and convergent, with L-2 optimal error estimate and second-order accuracy. Numerical experiments demonstrate its accuracy, stability, and convergence.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
Xiaojuan Shen, Yunqing Huang, Xiaojing Dong
Summary: The Crank-Nicolson/Adams-Bashforth (CN/AB) algorithm is proposed for the nonstationary magnetohydro-dynamic (MHD) equations. The algorithm approximates the time derivative terms using a first-order Euler-backward scheme, an implicit second-order Crank-Nicolson scheme for linear terms, and an explicit Adams-Bashforth scheme for nonlinear terms. The finite element method is used for spatial discretization. Theoretical studies show that the algorithm is almost unconditionally stabilized. The optimal error estimate of the algorithm is obtained using a parabolic argument mathematical trick. Finally, numerical experiments are conducted to verify the theoretical results.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
Yang Li, Yanhong Bai, Minfu Feng
Summary: This paper presents a stabilized virtual element method for unsteady Navier-Stokes problems on polygonal meshes. The method uses equal-order virtual elements in space and the Crank-Nicolson scheme in time, and provides a fully discrete formula. By introducing local-projection type stabilizations, the method is able to avoid the discrete inf-sup condition and control spurious oscillations caused by high Reynolds numbers. The stability and error estimates for velocity and pressure are analyzed, and error estimates independent of the negative powers of the viscosity are derived. Numerical experiments are conducted to validate the theoretical results.
NUMERICAL ALGORITHMS
(2023)
Article
Mathematics, Applied
Yun-Bo Yang, Yao-Lin Jiang
Summary: This paper establishes unconditionally optimal error estimates for linearized second-order backward difference formula (BDF2) Galerkin finite element methods (FEMs) in describing the magnetic behavior in ferromagnetic materials using the Landau-Lifshitz equation. By splitting the error between exact and numerical solutions into temporal and spatial components, the study derives tau-independent spatial error and boundedness of the numerical solution in L-infinity norm. Optimal error estimates for r-th order FEMs (r = 1, 2) are obtained without restrictions on time step size, with numerical results confirming theoretical predictions and method efficiency in both two and three dimensional spaces.
APPLIED NUMERICAL MATHEMATICS
(2021)
Article
Mathematics, Applied
Cheng Wang, Jilu Wang, Steven M. Wise, Zeyu Xia, Liwei Xu
Summary: In this paper, a temporally second-order accurate numerical scheme for the Cahn-Hilliard-Magnetohydrodynamics system of equations is proposed and analyzed. The scheme utilizes a modified Crank-Nicolson-type approximation for time discretization and a mixed finite element method for spatial discretization. The modified Crank-Nicolson approximation allows for mass conservation and energy stability analysis. Error estimates are derived for the phase field, velocity, and magnetic fields, and numerical examples are presented to validate the proposed scheme's theoretical results.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Y. Rong, J. A. Fiordilino, F. Shi, Y. Cao
Summary: We study a modular Crank-Nicolson based Voigt regularization algorithm for the Navier-Stokes equations. The algorithm adds a minimally intrusive module that implements Voigt regularization and numerical dissipation, improving stability and accuracy in large-scale dynamics.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Yuan-Ming Wang, Zi-Yun Zheng
Summary: A new second-order difference approximation method for the Caputo fractional derivative is proposed, and its stability and convergence are rigorously proved through numerical experiments and theoretical analysis.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Mathematics, Applied
Archna Kumari, Vijay Kumar Kukreja
Summary: In this study, a robust septic Hermite collocation method (SHCM) is proposed to simulate the Kuramoto-Sivashinsky (KS) equation. The algorithm is demonstrated to be unconditionally stable using the von-Neumann approach. Convergence analysis shows second-order convergent in the temporal direction and sixth-order convergent in the spatial direction. The proposed technique outperforms other methods and matches well with the analytical solution.
APPLIED NUMERICAL MATHEMATICS
(2023)
Article
Engineering, Multidisciplinary
Xiaofeng Yang
Summary: This study presents a novel second-order time marching scheme for solving a highly coupled nonlinear two-phase fluid flow system with a full decoupling structure. The scheme introduces a nonlocal variable and an additional ordinary differential equation to maintain unconditional energy stability, and achieves high practical efficiency by solving several independent elliptic equations with constant coefficients at each time step. Various numerical simulations are conducted to demonstrate the stability and accuracy of the scheme.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2021)
Article
Mathematics, Applied
Wenjie Liu, Boying Wu
Summary: This study presents a fully discrete scheme for solving the two-dimensional second-order wave equation by discretizing space with the Legendre-Galerkin method and time with the Crank-Nicolson method. The fully discrete Crank-Nicolson Galerkin method is shown to have unconditional stability and optimal error estimates in both L-2 and H-1 norms, with numerical results confirming exponential convergence in space and second-order convergence in time, as well as discrete energy conservation and efficiency in long-time numerical calculations.
NUMERICAL ALGORITHMS
(2022)
Article
Mathematics, Applied
Xiong Liu, Wenming He
Summary: This paper investigates a multiscale homogenization theory for a second-order elliptic problem with rapidly oscillating periodic coefficients, proposing a new method for estimating the homogenization solution with weaker smoothness requirements than the classical theory.
JOURNAL OF FUNCTION SPACES
(2021)
Article
Computer Science, Interdisciplinary Applications
M. Mohebujjaman, L. G. Rebholz, T. Iliescu
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS
(2019)
Article
Engineering, Multidisciplinary
Nan Jiang, Changxin Qiu
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2019)
Article
Engineering, Multidisciplinary
Xiaoming He, Nan Jiang, Changxin Qiu
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING
(2020)
Article
Mathematics, Applied
Nan Jiang, Ying Li, Huanhuan Yang
Summary: This article presents a new second-order artificial compressibility ensemble method for simulating coupled surface-groundwater flow, which can decouple the system into two subphysics problems and use different time steps for higher computational efficiency. The algorithm reduces computational time by 96% compared to the nonensemble counterpart and is shown to be second-order convergent in numerical experiments.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2022)
Article
Mathematics, Applied
John Carter, Nan Jiang
Summary: This study presents a second order ensemble method for computing an ensemble of magnetohydrodynamics flows at small magnetic Reynolds number. The proposed algorithm requires solving only one linear system with multiple right-hands instead of multiple different linear systems, reducing computational cost and simulation time. Comprehensive stability and error analyses confirm conditional stability and second order time convergence, with numerical tests demonstrating the efficiency of the algorithm.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2022)
Article
Mathematics, Applied
Muhammad Mohebujjaman, Hongwei Wang, Leo G. Rebholz, Md. Abdullah Al Mahbub
Summary: This paper proposes an efficient algorithm for computing the ensemble average of incompressible MHD flows. The algorithm decouples Elsasser variables and allows for a shared coefficient matrix, resulting in improved computational efficiency. The stability and convergence of the algorithm are proven, and numerical tests are conducted to support the predicted convergence rates. The algorithm is then tested in lid-driven cavity and channel flow past a step problems to observe changes in physical behavior with increasing coupling number and deviation of uncertainties.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Mathematics, Applied
Nan Jiang, Changxin Qiu
Summary: This paper proposes an efficient algorithm for fast computation of a set of realizations of the stochastic Stokes-Darcy model. The algorithm only requires solving two linear systems with the same constant coefficient matrices, and stability and convergence analysis have been conducted. Numerical experiments support theoretical results and demonstrate the application of the method.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Biology
Hongwei Wang, Fernando G. Quintana, Yunlong Lu, Muhammad Mohebujjaman, Kanon Kamronnaher
Summary: This study conducted a detailed ordinal logistic regression analysis at a university in South Texas, USA, and found that this method is effective for analyzing categorical data. The analysis of survey results revealed that students who have sufficient whole grain food and exercise tend to have normal BMI, while BMI tends to exceed the normal range with increasing age.
Article
Energy & Fuels
Amzad Hossain, Md. Mamun Molla, Md. Kamrujjaman, Muhammad Mohebujjaman, Suvash C. Saha
Summary: In this study, the numerical investigation of magneto-hydrodynamic mixed convection flow and entropy formation of non-Newtonian Bingham fluid in a lid-driven wavy square cavity filled with nanofluid was conducted using the finite volume method (FVM). The results indicate that within the given parameter range, Reynolds number and nanoparticle volume fraction have a positive impact on the average Nusselt number (Nu over bar ), while Hartmann number and Bingham number have a negative impact on it. The entropy generation is also affected by these parameters.
Article
Engineering, Multidisciplinary
Muhammad Mohebujjaman, Syunichi Shiraiwa, Brian Labombard, John C. Wright, Kiran K. Uppalapati
Summary: A mathematical model for the charging simulation of non-insulated superconducting pancake solenoids is proposed and numerical solutions are obtained using various solvers. Scalability analysis shows that an iterative solver combination (FGMRES-GMRES) with the parallel Auxiliary Space Maxwell Solver (AMS) preconditioner outperforms a parallelized direct solver (MUMPS) even when there are two extremely different time scales in the system. Generally, the computational time of the iterative solver increases with the number of turns in the solenoids and/or the conductivity of the superconducting material.
ENGINEERING RESEARCH EXPRESS
(2023)
Article
Mathematics, Applied
Muhammad Mohebujjaman, Clarisa Buenrostro, Md. Kamrujjaman, Taufiquar Khan
Summary: We propose two novel fully discrete decoupled linearized algorithms for a nonlinearly coupled reaction-diffusion N-species competition model with harvesting or stocking effort. The algorithms are first and second order accurate in time and optimally accurate in space. We validate the convergence rates and efficacy of the algorithms through numerical experiments and synthetic data. Additionally, we study the effect of harvesting or stocking and diffusion parameters on the evolution of species population density numerically, and observe co-existence scenarios under optimal harvesting or stocking conditions.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematical & Computational Biology
Md Shahriar Mahmud, Md Kamrujjaman, Md Mashih Ibn Yasin Adan, Md Alamgir Hossain, Md Mizanur Rahman, Md Shahidul Islam, Muhammad Mohebujjaman, Md Mamun Molla
Summary: The study examined the impact of vaccination and vaccine efficacy rates during an ongoing pandemic with a mathematical model. It found that the Pfizer vaccine could control the pandemic situation in California by the end of 2023, but using vaccines with lower efficacy rates could extend the control period. Additionally, the pandemic in the whole U.S. is expected to be controlled by the end of 2026, but vaccination may need to continue until mid-2028 if using vaccines other than 100% effective ones or Pfizer.
INFECTIOUS DISEASE MODELLING
(2022)
Article
Mathematics, Applied
Nan Jiang, Huanhuan Yang
Summary: This report presents two second order, stabilized, scalar auxiliary variable (SAV) ensemble algorithms for fast computation of the Navier-Stokes flow ensembles. By utilizing the ensemble timestepping idea and a recently developed SAV approach, the proposed algorithms reduce computational cost, ensure long time stability, and improve accuracy for nonlinear flow ensembles.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Nan Jiang, Ying Li, Huanhuan Yang
Summary: The CNLFAC method is proposed for solving the Stokes-Darcy equations, achieving high stability and convergence without the need to solve a Poisson equation for pressure. It reduces computational time significantly and maintains robustness in numerical experiments.
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2021)
Article
Mathematics, Interdisciplinary Applications
M. Gunzburger, T. Iliescu, M. Mohebujjaman, M. Schneier
SIAM-ASA JOURNAL ON UNCERTAINTY QUANTIFICATION
(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)
