Article
Mathematics, Applied
Kamalesh Kumar, P. Pramod Chakravarthy, Higinio Ramos, Jesus Vigo-Aguiar
Summary: This article presents a stable finite difference approach for numerical approximation of singularly perturbed differential-difference equations (SPDDEs). The proposed scheme is oscillation-free and more accurate than conventional methods on a uniform mesh. Error estimates show linear convergence of the scheme in space and time variables. The results are extrapolated using Richardson extrapolation technique for better approximations. Numerical examples from literature validate the theory and demonstrate good performance of the proposed method.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
Alan Demlow, Sebastian Franz, Natalia Kopteva
Summary: In this paper, we prove residual-type a posteriori error estimates in the maximum norm for a linear scalar elliptic convection-diffusion problem that may be singularly perturbed. Similar to the error analysis in the energy norm, a dual norm of the convective derivative of the error must be added to the natural energy norm in order for the natural residual estimator to be reliable and efficient. We also define a mesh-dependent weighted seminorm of the convective error in the maximum-norm setting. These estimates are proved under the assumption that certain natural estimates hold for the Green's function for the problem at hand. Numerical experiments confirm the effectiveness of our estimators in capturing the maximum-norm error behavior for singularly perturbed problems.
IMA JOURNAL OF NUMERICAL ANALYSIS
(2023)
Article
Mathematics, Applied
Yao Cheng, Yanjie Mei, Hans-Gorg Roos
Summary: In this paper, we analyze the error of both the semi-discretization and the full discretization of a time-dependent convection-diffusion problem. By using the local discontinuous Galerkin method and the implicit theta-scheme, we obtain error estimates with respect to space and time and discuss improved estimates and the use of a discontinuous Galerkin discretization in time.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Mathematics, Applied
Andres Contreras, Xavier Lamy
Summary: In this study, we establish small energy Holder bounds for the minimization problem of anisotropic energies in liquid crystal models. By analyzing the decay of energy and the regularity of harmonic maps, we demonstrate the local uniform convergence and boundary convergence of the minimization solution u, resolving the previously open problem.
Article
Mathematics, Applied
Pedro Toniol Cardin
Summary: This paper provides a geometric analysis of relaxation oscillations in the context of planar fast-slow systems with a discontinuous right-hand side. The conditions for the existence of a stable crossing limit cycle and the convergence of the cycle to a crossing closed singular trajectory are given. The regularization of the crossing relaxation oscillator and the existence of a relaxation oscillation in the regularized vector field are studied. The results are demonstrated with examples including a model of an arch bridge with nonlinear viscous damping.
Article
Mathematics, Applied
Natalia Kopteva, Richard Rankin
Summary: The symmetric interior penalty discontinuous Galerkin method and its weighted averages version are applicable on shape-regular nonconforming meshes for solving singularly perturbed semilinear reaction-diffusion equations. Residual-type a posteriori error estimates in maximum norm are given, with error cons...
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2023)
Article
Mathematics, Applied
Jianwei Zhou, Huiyuan Li, Zhimin Zhang
Summary: In this paper, the authors investigate a posteriori error estimates of the Galerkin spectral methods for second-order equations. They propose a simple type of error estimator based on the expansion coefficients of known quantities such as the right-hand term. The authors show that the decay rate of the high frequency coefficients of the right-hand term serves as an ideal a posteriori error estimator. They also establish a posteriori error estimates on the Galerkin spectral method applied to the singular perturbation problem, where the efficiency is given by the approximation errors of the weighted L-2-projection of the right-hand function and the reliability is determined by the truncation errors of the right-hand function together with the low frequency coefficients.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Zhongdi Cen, Jian Huang, Aimin Xu
Summary: This paper considers a system of singularly perturbed initial value problems with weak constrained conditions on the coefficients. The system is transformed into a system of first-order singularly perturbed problems with integral terms, and a hybrid difference method is utilized to approximate the transformed system. A posteriori error analysis for the discretization scheme on an arbitrary mesh is presented, and a solution-adaptive algorithm based on a posteriori error estimation is devised. Numerical experiments show the uniform convergence behavior of second-order for the scheme.
Article
Mathematics, Applied
Sebastian Franz
Summary: The discretisation of H-div-functions on rectangular meshes involves three families of finite elements, namely Raviart-Thomas, Brezzi-Douglas-Marini, and Arnold-Boffi-Falk elements. In order to prove convergence of numerical methods using these elements, sharp interpolation error estimates are important, especially in an anisotropic setting.
APPLIED MATHEMATICS LETTERS
(2021)
Article
Mathematics, Applied
Jian Huang, Zhongdi Cen, Aimin Xu
Summary: This paper presents a hybrid difference method for approximating a parameterized singular perturbation problem and introduces an improved a posteriori error estimation for the difference scheme. A solution-adaptive algorithm based on the error estimation is designed, leading to second-order uniform convergence according to numerical experiments.
APPLIED MATHEMATICS LETTERS
(2021)
Article
Computer Science, Interdisciplinary Applications
Aditya Kaushik, Vijayant Kumar, Manju Sharma, Nitika Sharma
Summary: This paper introduces a modified graded mesh for solving singularly perturbed reaction-diffusion problems using a recursive generation method. The numerical solution is based on finite element method with polynomials of degree at least p. The parameter uniform convergence of optimal order in epsilon-weighted energy norm is proven. Test examples and comparative analysis with other adaptive meshes validate the effectiveness of the proposed method.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2021)
Article
Mathematics, Applied
S. Chandra Sekhara Rao, Abhay Kumar Chaturvedi
Summary: This study proposes a finite difference method for solving a system of singularly perturbed reaction-diffusion equations, discretized using central difference scheme in space and backward difference scheme in time on a Shishkin mesh. The convergence analysis shows almost second-order parameter-uniform convergence in space and first-order convergence in time. Numerical experiments are conducted to confirm the efficiency of the method.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Article
Mathematics, Applied
Xiaoqi Ma, Jin Zhang
Summary: In this study, we investigate the supercloseness property of the nonsymmetric interior penalty Galerkin (NIPG) method on a Bakhvalov-type mesh for a singularly perturbed convection diffusion problem. We propose a new composite interpolation method that combines Gaul3 Radau projection outside the layer and Gaul3 Lobatto projection inside the layer. By selecting appropriate penalty parameters at different mesh points, we obtain the supercloseness of k + 21th order (k >= 1) in an energy norm.
APPLIED MATHEMATICS LETTERS
(2023)
Article
Mathematics, Applied
Satpal Singh, Renu Choudhary, Devendra Kumar
Summary: We propose a spline-based numerical technique for solving two-parameter singularly perturbed problems with discontinuous convection and source terms. The scheme discretizes the problem in the temporal direction using the Crank-Nicolson formula, and in the spatial direction, trigonometric B-spline basis functions are employed. The presence of perturbation parameters and discontinuous convection/source terms leads to the formation of interior and boundary layers in the solution, and our main objective is to accurately resolve these layers and develop a uniformly convergent scheme. The proposed method initially shows nearly first and second order convergence in the spatial and temporal directions respectively, and the accuracy in the spatial direction is further improved using Richardson extrapolation technique, as demonstrated by two numerical examples.
COMPUTATIONAL & APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Ludvig Of Klinteberg, Chiara Sorgentone, Anna-Karin Tornberg
Summary: This paper presents a method for estimating quadrature errors in the evaluation of layer potentials defined over smooth curved surfaces. The results are highly accurate, with low computational cost, and can be practically applied.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Mathematics, Applied
Fabien Bechet, Norbert Siedow, Dominique Lochegnies
FINITE ELEMENTS IN ANALYSIS AND DESIGN
(2015)
Article
Thermodynamics
Norbert Siedow, Dominique Lochegnies, Fabien Bechet, Philippe Moreau, Hiroshi Wakatsuki, Nobuhiro Inoue
INTERNATIONAL JOURNAL OF HEAT AND MASS TRANSFER
(2015)
Article
Materials Science, Ceramics
Kossiga Agboka, Fabien Bechet, Norbert Siedow, Dominique Lochegnies
INTERNATIONAL JOURNAL OF APPLIED GLASS SCIENCE
(2018)
Article
Computer Science, Interdisciplinary Applications
Arnaud Lejeune, Fabien Bechet, Hakim Boudaoud, Norman Mathieu, Michel Potier-Ferry
ADVANCES IN ENGINEERING SOFTWARE
(2012)
Article
Computer Science, Software Engineering
C. De Souza, F. Bechet, D. Leguillon, E. Sanchez-Palencia
BIT NUMERICAL MATHEMATICS
(2008)
Article
Mechanics
Fabien Bechet, Arnaud Lejeune, Michel Potier-Ferry
COMPTES RENDUS MECANIQUE
(2010)
Article
Mathematics, Interdisciplinary Applications
F. Bechet, E. Sanchez-Palencia, O. Millet
COMPUTATIONAL MECHANICS
(2009)
Article
Mechanics
F. Bechet, O. Millet, E. Sanchez-Palencia
INTERNATIONAL JOURNAL OF SOLIDS AND STRUCTURES
(2009)
Article
Mechanics
F. Bechet, O. Millet, E. Sanchez-Palencia
INTERNATIONAL JOURNAL OF SOLIDS AND STRUCTURES
(2010)
Article
Mechanics
F. Bechet
EUROPEAN JOURNAL OF MECHANICS A-SOLIDS
(2019)
Article
Mathematics, Interdisciplinary Applications
F. Bechet, E. Sanchez-Palencia, O. Millet
COMPUTATIONAL MECHANICS
(2008)
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)