Computers & Mathematics with Applications最新文献

英文中文

Robust iterative spectral algorithms for smooth solutions of time-fractional nonlinear diffusion problems and convergence analysis 时间分数非线性扩散问题平稳解的稳健迭代谱算法及收敛性分析

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2024-10-29 DOI: 10.1016/j.camwa.2024.10.015

Muhammad Usman , Muhammad Hamid , Dianchen Lu , Zhengdi Zhang , Wojciech Sumelka

Nonlinear time-fractional diffusion problems, a significant class of parabolic-type problems, appear in various diffusion phenomena that seem extensively in nature. Such physical problems arise in numerous fields, such as phase transition, filtration, biochemistry, and dynamics of biological groups. Because of its massive involvement, its accurate solutions have become a challenging task among researchers. In this framework, this article proposed two operational-based robust iterative spectral schemes for accurate solutions of the nonlinear time-fractional diffusion problems. Temporal and spatial variables are approximated using Vieta-Lucas polynomials, and derivative operators are approximated using novel operational matrices. The approximated solution, novel operational matrices, and uniform collection points convert the problem into a system of nonlinear equations. Here, two robust methods, namely Picard's iterative and Newton's, are incorporated to tackle a nonlinear system of equations. Some problems are considered in authenticating the present methods' accuracy, credibility, and reliability. An inclusive comparative study demonstrates that the proposed computational schemes are effective, accurate, and well-matched to find the numerical solutions to the problems mentioned above. The proposed methods improve the accuracy of numerical solutions from 27 % to 100 % when

M > 2

as compared to the existing results. The suggested methods' convergence, error bound, and stability are investigated theoretically and numerically.

非线性时分式扩散问题是抛物线型问题中的一个重要类别，广泛存在于自然界的各种扩散现象中。这类物理问题出现在相变、过滤、生物化学和生物群体动力学等众多领域。由于涉及面广，其精确求解成为研究人员面临的一项艰巨任务。在此框架下，本文提出了两种基于运算的鲁棒迭代谱方案，用于非线性时间-分数扩散问题的精确求解。时间和空间变量使用 Vieta-Lucas 多项式近似，导数算子使用新型运算矩阵近似。近似解、新型运算矩阵和统一收集点将问题转换为非线性方程组。在这里，两种稳健方法，即皮卡尔迭代法和牛顿法，被纳入到处理非线性方程组的方法中。在验证现有方法的准确性、可信度和可靠性时，考虑了一些问题。一项全面的比较研究表明，所提出的计算方案有效、准确，且与上述问题的数值解相匹配。与现有结果相比，当 M>2 时，建议的方法将数值解的精确度从 27% 提高到 100%。对建议方法的收敛性、误差范围和稳定性进行了理论和数值研究。

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引用次数: 0

Generalised adaptive cross approximation for convolution quadrature based boundary element formulation 基于卷积正交边界元素公式的广义自适应交叉逼近

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2024-10-28 DOI: 10.1016/j.camwa.2024.10.025

A.M. Haider , S. Rjasanow , M. Schanz

The acoustic wave equation is solved in time domain with a boundary element formulation. The time discretisation is performed with the generalised convolution quadrature method and for the spatial approximation standard lowest order elements are used. Collocation and Galerkin methods are applied. In the interest of increasing the efficiency of the boundary element method, a low-rank approximation such as the adaptive cross approximation (ACA) is carried out. We discuss a generalisation of the ACA to approximate a three-dimensional array of data, i.e., usual boundary element matrices at several complex frequencies. This method is used within the generalised convolution quadrature (gCQ) method to obtain a real time domain formulation. The behaviour of the proposed method is studied with three examples, a unit cube, a unit cube with a reentrant corner, and a unit ball. The properties of the method are preserved in the data sparse representation and a significant reduction in storage is obtained.

声波方程在时域中采用边界元素公式求解。时间离散采用广义卷积正交法，空间近似采用标准最低阶元素。采用了拼合和 Galerkin 方法。为了提高边界元方法的效率，我们采用了自适应交叉近似（ACA）等低阶近似方法。我们讨论了 ACA 的广义化，以近似三维数据阵列，即多个复频的常规边界元矩阵。这种方法被用于广义卷积正交（gCQ）方法中，以获得实时域公式。我们用三个例子研究了所提方法的性能，即一个单位立方体、一个带重入角的单位立方体和一个单位球。该方法的特性在数据稀疏表示中得以保留，并显著减少了存储量。

引用次数: 0

Innovative discretizations of PDEs: Towards an accurate representation of the reality PDEs 的创新离散化：实现对现实的准确呈现

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2024-10-28 DOI: 10.1016/j.camwa.2024.10.013

Fleurianne Bertrand, Daniele Boffi, Alexander Düster, Jean-Luc Guermond, Norbert Heuer, Jichun Li, Waldemar Rachowicz

引用次数: 0

A-priori and a-posteriori error estimates for discontinuous Galerkin method of the Maxwell eigenvalue problem 麦克斯韦特征值问题非连续伽勒金方法的先验和后验误差估计

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2024-10-24 DOI: 10.1016/j.camwa.2024.10.026

Jun Zhang , Zijiang Luo , Jiayu Han , Hu Chen

This paper is devoted to a-priori and a-posteriori error analysis of discontinuous Galerkin (DG) method for the Maxwell eigenvalue problem. The discrete compactness of DG space is proved so that the Babuška and Osborn spectral approximation theory can be applicable in the a-priori error analysis. Then we prove the optimal error estimates for DG eigenfunctions in mesh-dependent norm and DG eigenvalues. A special contribution of this work is to prove that the error in

L^{2}

-norm for smooth eigenfunctions is of higher order than that in mesh-dependent norm, so that the DG eigenvalues can approximate the true eigenvalues from upper. Another contribution of this work is to provide a-posteriori error analysis for the DG method. A reliable a-posteriori error estimator is analyzed. The upper bound property of DG eigenvalues and the robustness of adaptive methods are verified through numerical experiments.

本文致力于麦克斯韦特征值问题的非连续伽勒金（DG）方法的先验和后验误差分析。本文证明了 DG 空间的离散紧凑性，因此 Babuška 和 Osborn 光谱近似理论可用于先验误差分析。然后，我们证明了网格相关规范中 DG 特征函数和 DG 特征值的最优误差估计。这项工作的一个特殊贡献是证明了光滑特征函数的 L2 准则误差比网格相关准则误差的阶数更高，因此 DG 特征值可以从上逼近真实特征值。这项工作的另一个贡献是提供了 DG 方法的后验误差分析。分析了一个可靠的后验误差估计器。通过数值实验验证了 DG 特征值的上界特性和自适应方法的鲁棒性。

引用次数: 0

Fully parallel and pipelined sparse direct solver for large symmetric indefinite finite element problems 大型对称不定期有限元问题的全并行流水线稀疏直接求解器

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2024-10-24 DOI: 10.1016/j.camwa.2024.10.017

Yujie Wang , Shengquan Wang , Yong Cai , Guidong Wang , Guangyao Li

Sparse linear system solving is a primary computational cost in large-scale finite element analysis, and improving its performance is a key technological challenge in this field. Real-world engineering problems involve diverse materials, elements, and connectivity relationships, making it difficult for iterative methods to handle their global stiffness matrices. Direct methods, owing to their robustness, emerge as the preferred choice. In this paper, a novel block-based supernodal LDL^T numerical factorization method is introduced. The computational process is disassembled into distinct tasks, and the dependency relationships between these tasks are expressed via a directed acyclic graph to guide the calculation sequence. Based on this approach, a global task pool and local task stack are established to store task queues, enhancing data reuse and multicore collaboration efficiency. Additionally, an effective task dispatch and work-stealing mechanism is implemented to prevent performance degradation caused by load imbalances. Numerical experiments, including a publicly available matrix test set and real-world engineering finite element problems, are conducted to compare the parallel performances of the Pardiso, MUMPS, and proposed solver. The results illustrate that the proposed solver performs significantly better than the other solvers when handling various types of sparse matrices and diverse architectures of multicore processors.

稀疏线性系统求解是大规模有限元分析的主要计算成本，提高其性能是这一领域的关键技术挑战。现实世界的工程问题涉及多种材料、元素和连接关系，因此迭代法很难处理其全局刚度矩阵。直接方法因其稳健性而成为首选。本文介绍了一种新颖的基于块的超节点 LDLT 数值因式分解方法。计算过程被分解成不同的任务，这些任务之间的依赖关系通过有向无环图来表达，以指导计算顺序。基于这种方法，建立了全局任务池和本地任务栈来存储任务队列，从而提高了数据重用和多核协作效率。此外，还实施了有效的任务调度和抢工机制，以防止负载不平衡导致的性能下降。为了比较 Pardiso、MUMPS 和建议的求解器的并行性能，我们进行了包括公开矩阵测试集和实际工程有限元问题在内的数值实验。结果表明，在处理各种类型的稀疏矩阵和不同架构的多核处理器时，建议的求解器的性能明显优于其他求解器。

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引用次数: 0

A least-squares Fourier frame method for nonlocal diffusion models on arbitrary domains 任意域上非局部扩散模型的最小二乘傅立叶框架法

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2024-10-24 DOI: 10.1016/j.camwa.2024.10.024

Mengxia Shen , Haiyong Wang

We introduce a least-squares Fourier frame method for solving nonlocal diffusion models with Dirichlet volume constraint on arbitrary domains. The mathematical structure of a frame rather than a basis allows using a discrete least-squares approximation on irregular domains and imposing non-periodic boundary conditions. The method has inherited the one-dimensional integral expression of Fourier symbols of the nonlocal diffusion operator from Fourier spectral methods for any d spatial dimensions. High precision of its solution can be achieved via a direct solver such as pivoted QR decomposition even though the corresponding system is extremely ill-conditioned, due to the redundancy in the frame. The extension of AZ algorithm improves the complexity of solving the rectangular linear system to

O (N \log^{2} N)

for 1d problems and

O (N^{2} \log^{2} N)

for 2d problems, compared with

O (N^{3})

of the direct solvers, where N is the number of degrees of freedom. We present ample numerical experiments to show the flexibility, fast convergence and asymptotical compatibility of the proposed method.

我们介绍了一种最小二乘傅立叶框架方法，用于求解任意域上具有德里赫特体积约束的非局部扩散模型。框架而非基础的数学结构允许在不规则域上使用离散最小二乘近似，并施加非周期性边界条件。该方法继承了傅里叶谱方法中任意 d 空间维度非局部扩散算子傅里叶符号的一维积分表达式。由于框架中存在冗余，即使相应系统的条件极差，也可以通过直接求解器（如枢轴 QR 分解）实现高精度求解。与直接求解器的 O(N3)（其中 N 为自由度数）相比，AZ 算法的扩展提高了矩形线性系统的求解复杂度，1d 问题为 O(Nlog2N)，2d 问题为 O(N2log2N)。我们通过大量的数值实验证明了所提方法的灵活性、快速收敛性和渐进兼容性。

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引用次数: 0

A local projection stabilised HHO method for the Oseen problem 奥森问题的局部投影稳定 HHO 方法

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2024-10-24 DOI: 10.1016/j.camwa.2024.10.030

Gouranga Mallik , Rahul Biswas , Thirupathi Gudi

In this article, we consider a local projection stabilisation for a Hybrid High-Order (HHO) approximation of the Oseen problem. We prove an existence-uniqueness result under a stronger SUPG-like norm. We improve the stability and provide error estimation in stronger norm for convection dominated Oseen problem. We also derive an optimal order error estimate under the SUPG-like norm for equal-order polynomial discretisation of velocity and pressure spaces. Numerical experiments are performed to validate the theoretical results.

在本文中，我们考虑了奥森问题的混合高阶（HHO）近似的局部投影稳定问题。我们证明了更强 SUPG 类规范下的存在唯一性结果。我们提高了对流主导奥森问题的稳定性，并提供了更强规范下的误差估计。我们还为速度和压力空间的等阶多项式离散化推导出了 SUPG 类规范下的最优阶误差估计。我们进行了数值实验来验证理论结果。

引用次数: 0

Special Issue: Advanced COmputational Methods in ENgineering (ACOMEN 2022) 特刊：高级工程计算方法（ACOMEN 2022）

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2024-10-23 DOI: 10.1016/j.camwa.2024.10.012

Maarten Arnst, Christophe Geuzaine, Ludovic Noels, Jean-Philippe Ponthot

引用次数: 0

Accurate and parallel simulation of the anisotropic dendrite crystal growth by Lagrangian data assimilation with directional operator splitting 通过拉格朗日数据同化与方向算子拆分对各向异性树枝状晶体生长进行精确并行模拟

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2024-10-22 DOI: 10.1016/j.camwa.2024.10.020

Fenglian Zheng , Yan Wang , Xufeng Xiao

Dendritic crystal growth is a prevalent natural phenomenon that generates a crystalline structure resembling a tree during a phase transition. In practical computations, inaccuracies in model parameters and initial conditions can introduce observation errors, even leading to inaccurate results. To enhance the precision and efficiency of the numerical simulation, the data assimilation method with parallel simulation are considered in this study. Firstly, based on the phase-field dendritic crystal growth model, a Lagrangian data assimilation method, which adds Lagrange multiplier terms into the phase-field partial differential equations (PDEs), is presented to integrate the observed data of physical information to modify the numerical solution, thereby improving simulation accuracy. Secondly, to achieve efficient data assimilation, a parallel directional operator splitting method is presented to solve the modified data assimilation PDEs. Thirdly, in the section of numerical experiments, we investigate the validity of the method and assess the impact of various factors such as the Lagrange multiplier parameter, spatio-temporal sampling rate and parameter perturbation ratio on the effectiveness of data assimilation. The evaluation is conducted for two distinct problem categories: initial observation errors and model parameter errors. Experimental results demonstrate that our method can effectively assimilate experimental observations in simulations, thereby enhancing more accurate dendritic crystal growth processes.

树枝状晶体生长是一种普遍的自然现象，在相变过程中会产生类似树状的晶体结构。在实际计算中，模型参数和初始条件的不准确会带来观测误差，甚至导致结果不准确。为了提高数值模拟的精度和效率，本研究考虑采用数据同化方法进行并行模拟。首先，基于相场树枝状晶体生长模型，提出一种拉格朗日数据同化方法，在相场偏微分方程（PDEs）中加入拉格朗日乘数项，以整合物理信息的观测数据来修改数值解，从而提高模拟精度。其次，为了实现高效的数据同化，提出了一种并行定向算子拆分方法来求解修改后的数据同化偏微分方程。第三，在数值实验部分，我们研究了该方法的有效性，并评估了拉格朗日乘数参数、时空采样率和参数扰动比等各种因素对数据同化效果的影响。评估针对两个不同的问题类别：初始观测误差和模型参数误差。实验结果表明，我们的方法可以在模拟中有效地同化实验观测数据，从而提高树枝状晶体生长过程的准确性。

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引用次数: 0

Three-dimensional peridynamic modeling of damage and penetration in composite plates exposed to localized explosive blasts 复合材料板在局部爆炸冲击波作用下的损伤和穿透的三维围动力学建模

IF 2.9 2区数学 Q1 MATHEMATICS, APPLIED

Computers & Mathematics with Applications

Pub Date : 2024-10-22 DOI: 10.1016/j.camwa.2024.10.022

D.A. Abdoh

Localized failures in composites can lead to catastrophic consequences in some vehicles and systems, such as airplanes and submarines. The paper presents a new three-dimensional (3D) peridynamic model to explore the localized explosive effects on composite plates. The study utilizes the peridynamic method to simulate fractures and deformations in composite plates when exposed to localized explosive blasts. Accurate prediction of composites' performance during explosive blast events is crucial in the design process to avoid the deadly effects of their failures. This study highlights the following novelties: (1) We present, for the first time, a novel 3D mesh-free model to simulate the fracture and damage behavior of composite plates when exposed to explosive blasts; (2) The adopted numerical modeling technique enables highly efficient simulations of fractures and failures in composites when compared with other mesh-based numerical models; (3) We introduce a new mathematical framework to reflect the explosive pressure loads through different composite layers; (4) The study provides an accessible and efficient tool for engineers and researchers to enhance the design of composites in several industries instead relying on limited-access commercial software packages. In addition to the previous novelties, the paper presents a new parametric study that investigates the performance of several composite plates, such as titanium-aramid and aluminum-aramid composites, in explosive blast scenarios. Moreover, the roles of explosive mass and plate thickness in the failure mechanisms of composites are examined.

复合材料的局部失效可导致某些飞行器和系统（如飞机和潜艇）出现灾难性后果。本文介绍了一种新的三维（3D）周动模型，用于探索爆炸对复合材料板的局部影响。该研究利用周向动力学方法模拟复合材料板在受到局部爆炸冲击时的断裂和变形。在设计过程中，准确预测复合材料在爆炸冲击波事件中的性能对于避免其失效造成致命影响至关重要。本研究具有以下新颖之处：(1) 我们首次提出了一种新型三维无网格模型，用于模拟复合材料板在爆炸冲击波作用下的断裂和损伤行为；(2) 与其他基于网格的数值模型相比，所采用的数值建模技术能够高效模拟复合材料的断裂和损伤；(3) 我们引入了一个新的数学框架，以反映通过不同复合材料层的爆炸压力载荷；(4) 该研究为工程师和研究人员提供了一个易用、高效的工具，以提高多个行业的复合材料设计，而不是依赖于访问受限的商业软件包。除了之前的创新之外，本文还提出了一项新的参数研究，调查了几种复合材料板（如钛-芳纶和铝-芳纶复合材料）在爆炸冲击波情况下的性能。此外，还研究了爆炸质量和板材厚度在复合材料失效机制中的作用。

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