Engineering Analysis with Boundary Elements最新文献_第5页

Wave attenuation on a floating rigid dock by multiple surface-piercing vertical thin perforated barriers 多个表面穿孔垂直薄壁屏障对浮动刚性船坞的波浪衰减作用

IF 4.2 2区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Engineering Analysis with Boundary Elements

Pub Date : 2024-10-14 DOI: 10.1016/j.enganabound.2024.105985

Dipankar Paul , Harekrushna Behera

The scattering of gravity waves interacting with an array of multiple surface-piercing thin porous barriers is explored based on the hypothesis of linearized potential flow for finite water depth. The barriers are assumed to be stationed at a finite distance from each other and on the lee side of the dock. Also, it is hypothesized that the waves passing through the porous barriers follow Darcy’s law. The wave properties such as reflection and transmission coefficients, dissipation of wave energy, and horizontal wave force on the floating rigid dock are studied to check the effectiveness of different numbers of barriers and length (

H

) of barriers, their porosity, the spacing between barriers, and the distance between the last barrier and the floating rigid dock. It has been witnessed that more surface piercing barriers are obviously helpful in relieving the force due to the wave interaction with the floating rigid dock. It is noticed that implementing four perforated barriers not only reduces reflection by around 70% but also enhances wave energy dissipation by 90%, with equal size of barrier lengths being the most effective. The porous barriers are more conducive to alleviating the wave force than the rigid barriers. Additionally, it is observed that there is zero reflection when the barriers’ length is set at

H / h = 0.7

and set porosity at 1.212 (where

h

is the water depth). On the other hand, the critical incidence angle

31.81 °

for reflection is also noticed with barriers of length

H / h = 0.6

. Further, the expansion of the normalized spacing between the structures helps reflect and transmit the waves along with the dissipation of wave energy to display a periodic pattern. The free surface elevation plots certainly help to fortify the claim of having multiple barriers as a tool to mitigate the wave force on the floating dock.

根据有限水深线性化势流假设，探讨了重力波与多个表面穿透薄多孔壁垒阵列相互作用的散射问题。假定屏障之间的距离有限，且位于码头的靠岸一侧。此外，还假设穿过多孔屏障的波浪遵循达西定律。研究了波浪的特性，如反射系数和透射系数、波能耗散以及浮动硬质码头上的水平波力，以检验不同数量的屏障和屏障长度（H）、屏障的孔隙率、屏障之间的间距以及最后一个屏障与浮动硬质码头之间的距离的有效性。结果表明，更多的表面穿孔屏障明显有助于缓解波浪与浮动刚性码头相互作用时产生的力。我们注意到，设置四个穿孔屏障不仅能减少约 70% 的反射，还能增强 90% 的波浪能量消散，而屏障长度大小相等的效果最好。与刚性屏障相比，多孔屏障更有利于减轻波力。此外，当屏障长度设定为 H/h=0.7，孔隙率设定为 1.212（h 为水深）时，可以观察到零反射。另一方面，当屏障长度为 H/h=0.6 时，反射的临界入射角为 31.81°。此外，结构间归一化间距的扩大有助于波浪的反射和传输，以及波浪能量的消散，从而显示出周期性模式。自由表面高程图无疑有助于加强将多重屏障作为减轻浮动码头上波浪力的工具的说法。

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引用次数: 0

An improved radial basis reproducing kernel particle method for geometrically nonlinear problem analysis of SMAs 用于 SMA 几何非线性问题分析的改进型径向基重现核粒子法

IF 4.2 2区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Engineering Analysis with Boundary Elements

Pub Date : 2024-10-11 DOI: 10.1016/j.enganabound.2024.105990

Yijie Zhang , Gaofeng Wei , Tengda Liu , Ming Song , Shasha Zhou

In this paper, the radial basis function (RBF) without shaped parameter is utilized in the radial basis reproducing kernel particle method (RRKPM), and an improved radial basis reproducing kernel particle method (IRRKPM) is proposed. Compared with traditional RKPM, the IRRKPM effectively reduces the impact of different kernel functions on calculation precision, and is further employed to examine geometrically nonlinear problems associated with shape memory alloys (SMAs). The displacement boundary condition is enforced via the penalty function method, while the Galerkin integration method in its weak form, along with the total Lagrangian (TL) approach, is utilized to derive the geometrically nonlinear equations for SMAs within the IRRKPM framework. The equilibrium equations are then solved using the Newton Raphson (N-R) iterative method. The impact of the different penalty factor and the radius control parameter of influence domain on errors is analyzed, the computational precision of the IRRKPM is compared with the RRKPM, and the computational stability is evaluated. Finally, the suitability of the IRRKPM for the analysis of geometrically nonlinearity problems in SMAs are confirmed through specific numerical examples.

本文在径向基重现核粒子法（RRKPM）中使用了无整形参数的径向基函数（RBF），并提出了改进的径向基重现核粒子法（IRRKPM）。与传统的 RKPM 相比，IRRKPM 有效地减少了不同核函数对计算精度的影响，并进一步用于研究与形状记忆合金 (SMA) 相关的几何非线性问题。在 IRRKPM 框架内，通过惩罚函数法强制执行位移边界条件，同时利用弱形式的 Galerkin 积分法和总拉格朗日 (TL) 方法推导出 SMA 的几何非线性方程。然后使用牛顿-拉斐尔森（N-R）迭代法求解平衡方程。分析了不同的惩罚因子和影响域半径控制参数对误差的影响，比较了 IRRKPM 与 RRKPM 的计算精度，并评估了计算稳定性。最后，通过具体的数值示例证实了 IRRKPM 适用于分析 SMA 中的几何非线性问题。

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引用次数: 0

Fully coupled dynamic hydraulic fracturing of saturated porous media based on the numerical manifold method 基于数值流形法的饱和多孔介质全耦合动态水力压裂技术

IF 4.2 2区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Engineering Analysis with Boundary Elements

Pub Date : 2024-10-10 DOI: 10.1016/j.enganabound.2024.105987

Tao Wan , Hong Zheng , Wenan Wu , Shanyong Wang , Shuaixing Zhao , Zibo Fan

Accurate and efficient simulation of dynamic hydraulic fracturing of the saturated porous media has always been a pivotal topic in the oil and gas extraction. Leveraging the Numerical Manifold Method (NMM) and its inherent cutting technique, this paper proposes a fully coupled hydraulic fracturing model based on the u-p format, which incorporates the overall momentum balance and continuity conditions in both porous media and fractures. NMM approximations and the Newmark implicit algorithm are employed respectively to discretize the spatial and time domains, and the resulting system is solved based on the Newton-Raphson method. By imposing flow boundary conditions on the fracture surfaces, the present model accounts for fluid loss without introducing extra filtration coefficients. Using the Mohr-Coulomb-based LT criterion and the maximum circumferential stress criterion to determine whether crack propagation has occurred and crack propagation direction respectively, the present model is capable of simulating initiation and development of multiple cracks under hydraulic stimulations. Through modeling the KGD hydraulic fracturing, hydraulic fracturing of a pre-cracked cubic specimen and fracture interference phenomena during expansion of multiple fracture ports of a single injection well, accuracy and effectiveness of the model are validated.

准确高效地模拟饱和多孔介质的动态水力压裂一直是油气开采领域的关键课题。本文利用数值歧管法（NMM）及其固有的切割技术，提出了一种基于 u-p 格式的全耦合水力压裂模型，该模型包含了多孔介质和裂缝的整体动量平衡和连续性条件。分别采用 NMM 近似和 Newmark 隐式算法对空间域和时间域进行离散，并基于 Newton-Raphson 方法对所得到的系统进行求解。通过在裂缝表面施加流动边界条件，本模型在不引入额外过滤系数的情况下考虑了流体损失。本模型采用基于莫尔-库仑的 LT 准则和最大圆周应力准则，分别确定裂缝是否扩展和裂缝扩展方向，能够模拟水力刺激下多条裂缝的产生和发展。通过模拟 KGD 水力压裂、预裂立方体试样的水力压裂以及单注水井多裂口扩张过程中的裂缝干涉现象，验证了该模型的准确性和有效性。

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引用次数: 0

An improved high-precision polyhedron SBFEM with combinatorial interpolation strategies 采用组合插值策略的改进型高精度多面体 SBFEM

IF 4.2 2区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Engineering Analysis with Boundary Elements

Pub Date : 2024-10-08 DOI: 10.1016/j.enganabound.2024.105991

Xiupeng Nie , Degao Zou , Kai Chen , Guoyang Yi , Xianjing Kong

Computational accuracy and solution efficiency are crucial indicators for evaluating the performance of finite element numerical algorithms, and the corresponding improvements in these areas are the motivation for the development of computational science. In this paper, a flexible and high-precision polyhedron algorithm is proposed in conjunction with SBFEM and the triangle/quadrilateral interpolation functions. The main content can be summarized as follows: The construction equations for a mixed-order polyhedron element are derived, meanwhile, a generalized procedural framework is designed for multi-performance applications, including automatic elements type identification, coupled solutions, and dynamic storage, and the integrated development is completed based on self-developed software GEODYNA. The correctness, convergence and practicability of the proposed method are demonstrated through several examples in different dimensions. The results show that the proposed method obtains results with an error of <3 % compared to theoretical solutions and the fine numerical solutions, while significantly reducing computational costs; Besides, the proposed method overcomes the limitations of conventional methods on mesh shape and can be seamlessly integrated with octree algorithms, which offers a powerful technical means for the efficient and high-precision analyses of bending deformation and stress concentration problems. It is foreseeable that a good application potential in high-precision structural analysis would be revealed.

计算精度和求解效率是评价有限元数值算法性能的重要指标，这些方面的相应改进是计算科学发展的动力。本文结合 SBFEM 和三角形/四边形插值函数，提出了一种灵活的高精度多面体算法。主要内容可概括如下：推导了混合阶多面体元素的构造方程，同时设计了多性能应用的通用程序框架，包括元素类型自动识别、耦合求解和动态存储，并基于自主开发的软件 GEODYNA 完成了集成开发。通过几个不同维度的实例，证明了所提方法的正确性、收敛性和实用性。结果表明，与理论解法和精细数值解法相比，所提方法得到的结果误差仅为 3%，同时显著降低了计算成本；此外，所提方法克服了传统方法在网格形状上的局限性，可与八叉树算法无缝集成，为高效、高精度地分析弯曲变形和应力集中问题提供了强有力的技术手段。可以预见，该方法在高精度结构分析中将有很好的应用前景。

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引用次数: 0

The multiple scattering effect of elastic waves propagation in an inhomogeneous medium 弹性波在非均质介质中传播的多重散射效应

IF 4.2 2区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Engineering Analysis with Boundary Elements

Pub Date : 2024-10-08 DOI: 10.1016/j.enganabound.2024.105983

Lingyan Shen, Keyan Li , Yonggui Liu, Xiaofei Ji, Boyang Zhang, Zhibin Lin

Unraveling the elastic wave propagating in an inhomogeneous medium is critical from both scientific and engineering viewpoints. Here, we propose and validate a double defects model based on unified mechanics framework to study the multiple scattering effect induced by the interaction between elastic waves and defects. The governing equations describing the dispersion and attenuation in frequency space are derived. In order to describe the multiple scattering effect, the Green's function method is employed together with the discrete boundary element method to establish the relation of macroscopic defect density and microscopic defect structure. The results show that the multiple scattering effect originates from the interaction between adjacent defects, and the limit of the multiple scattering (strong interaction) is approximately 6 times the characteristic length of the defect, namely the affected area of a single defect. Due to the stronger interaction, wave velocity decays more seriously for higher defects density than those in the lower density, and there exists no strong coupling between multi-scattering effect and multi-scale effect. The present work provides an efficient way to understand the multi-scattering effect of elastic waves in an inhomogeneous medium.

从科学和工程的角度来看，揭示在非均质介质中传播的弹性波至关重要。在此，我们提出并验证了基于统一力学框架的双缺陷模型，以研究弹性波与缺陷之间的相互作用引起的多重散射效应。推导出了描述频率空间中的频散和衰减的支配方程。为了描述多重散射效应，采用了格林函数法和离散边界元法来建立宏观缺陷密度和微观缺陷结构的关系。结果表明，多重散射效应源于相邻缺陷之间的相互作用，多重散射的极限（强相互作用）约为缺陷特征长度的 6 倍，即单个缺陷的影响区域。由于强相互作用，高密度缺陷的波速衰减比低密度缺陷的波速衰减更严重，多重散射效应与多尺度效应之间不存在强耦合。本研究为理解非均质介质中弹性波的多重散射效应提供了一种有效的方法。

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引用次数: 0

Machine learning applied to estate pricing for residential rentals in dynamic urban markets—The case of São Paulo city 将机器学习应用于动态城市市场住宅租赁的房地产定价--圣保罗市的案例

IF 4.2 2区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Engineering Analysis with Boundary Elements

Pub Date : 2024-10-08 DOI: 10.1016/j.enganabound.2024.105988

Wesley F. Maia , Sergio A. David

This study conducts a comprehensive investigation into real estate rental pricing in São Paulo city, employing an innovative approach that combines advanced machine learning techniques with geospatial and natural language processing (NLP) analyses. The research analyzed a robust dataset comprising 47,243 rental listings, gathered through web scraping techniques. Following a rigorous data cleaning and preprocessing procedure, the study focused on 35,486 instances, incorporating a variety of variables that go beyond conventional metrics, including textual descriptions and geographic information, enriching the analysis and market understanding. Several regression models were implemented and compared, including linear approaches, Support Vector Machines, and ensemble methods such as Gradient Boosting, LightGBM, and XGBoost. The Blending model, which integrates multiple modeling techniques, stood out as the most accurate, achieving a Root Mean Squared Logarithmic Error (RMSLE) of 0.2923 on the test set. This result emphasizes the superiority of hybrid modeling strategies in complex pricing tasks. The findings of this study have significant practical implications. They provide landlords and tenants with a powerful data-driven tool for informed decision-making, reflecting the nuances and complexity of São Paulo’s real estate market. The practical implementation of the model in an interactive web application not only demonstrates its utility in the real-world scenario but also serves as a model for future applications in real estate analysis. This work contributes to mitigating the waste of time and energy when it comes to searching for and pricing residential rentals in a large city, through the use of machine learning that shows its power and potential in accurately estimating rental prices in dynamic urban markets, allowing that more assertive and economical decisions can be taken within a social-sustainable-technological perspective.

本研究采用创新方法，将先进的机器学习技术与地理空间和自然语言处理（NLP）分析相结合，对圣保罗市的房地产租赁定价进行了全面调查。该研究分析了一个强大的数据集，其中包括通过网络刮擦技术收集的 47,243 份租房信息。经过严格的数据清理和预处理程序后，研究重点放在 35,486 个实例上，并纳入了各种超出传统指标的变量，包括文本描述和地理信息，从而丰富了分析和市场理解。实施并比较了几种回归模型，包括线性方法、支持向量机以及梯度提升、LightGBM 和 XGBoost 等集合方法。整合了多种建模技术的混合模型最为准确，在测试集上的均方根对数误差 (RMSLE) 为 0.2923。这一结果凸显了混合建模策略在复杂定价任务中的优越性。这项研究的结果具有重要的现实意义。它们为业主和租户提供了一个强大的数据驱动工具，用于做出明智的决策，反映了圣保罗房地产市场的细微差别和复杂性。该模型在交互式网络应用程序中的实际应用不仅证明了其在现实世界中的实用性，还为未来的房地产分析应用树立了典范。这项工作通过使用机器学习，展示了其在动态城市市场中准确估算租金价格的能力和潜力，有助于减少在大城市中搜索和定价住宅租金时的时间和精力浪费，从而可以从社会可持续发展技术的角度做出更加果断和经济的决策。

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引用次数: 0

The reduced variational multiscale element free Galerkin method for three-dimensional steady incompressible Stokes and Navier–Stokes equations 三维稳定不可压缩斯托克斯方程和纳维-斯托克斯方程的简化变分多尺度元素自由伽勒金方法

IF 4.2 2区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Engineering Analysis with Boundary Elements

Pub Date : 2024-10-07 DOI: 10.1016/j.enganabound.2024.105984

Yujie Fan , Xiaohua Zhang

In the paper, we extend the reduced variational multiscale element free Galerkin (RVMEFG) method to solve the three-dimensional steady incompressible Stokes and Navier–Stokes equations. This method is simplified based on the three-dimensional variational multiscale element free method (VMEFG), which the standard Galerkin discretization is used to momentum conservation equation and the variational multiscale method is used to mass conservation equation. The RVMEFG method inherits the advantages of the VMEFG method, which allows equal linear basis approximation of both velocity and pressure and avoids the Ladyzhenskaya–Babuška–Breezi (LBB) condition. Meanwhile, it can naturally generate the stabilization matrix. Furthermore, compared to the VMEFG method, the RVMEFG method can save computational cost. To verify the numerical stability, computational accuracy and efficiency of the method, five numerical problems are tested, and compared with the exact solution, VMEFG solution. It is shown that the RVMEFG method can avoid numerical oscillations and have higher computational efficiency which can save computational time. Meanwhile, it can guarantee the numerical accuracy for the three-dimensional steady incompressible Stokes and Navier–Stokes problems.

本文扩展了简化变分多尺度无元素 Galerkin（RVMEFG）方法，用于求解三维稳定不可压缩斯托克斯方程和纳维-斯托克斯方程。该方法在三维变分多尺度无元素方法（VMEFG）的基础上进行了简化，动量守恒方程采用标准 Galerkin 离散法，质量守恒方程采用变分多尺度方法。RVMEFG 方法继承了 VMEFG 方法的优点，允许对速度和压力进行等线性基近似，避免了 Ladyzhenskaya-Babuška-Breezi（LBB）条件。同时，它还能自然生成稳定矩阵。此外，与 VMEFG 方法相比，RVMEFG 方法可以节省计算成本。为了验证该方法的数值稳定性、计算精度和效率，对五个数值问题进行了测试，并与精确解、VMEFG 解进行了比较。结果表明，RVMEFG 方法可以避免数值振荡，具有更高的计算效率，可以节省计算时间。同时，它能保证三维稳定不可压缩斯托克斯和纳维-斯托克斯问题的数值精度。

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引用次数: 0

A new isogeometric boundary element method to analyze two-dimensional potential and elasticity problems 分析二维势能和弹性问题的新等几何边界元方法

IF 4.2 2区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Engineering Analysis with Boundary Elements

Pub Date : 2024-10-07 DOI: 10.1016/j.enganabound.2024.105986

Jiaxing Chen, Lei Wang, Jiawei Xiang

To reduce the impact of inhomogeneous boundary conditions on computational accuracy and the impact of redundant geometry refinement on computational efficiency in traditional isogeometric boundary element methods (IGABEM), a new IGABEM that introduces geometry-independent field approximation (GIFT) is proposed and applied to 2D potential and elasticity problems. Unlike in IGABEM, element nodes are located on the boundary, which facilitates the precise application of boundary conditions on nodes to improve computational accuracy. The constructed element features multiple inner nodes, allowing for the use of fewer elements in solving problems, which is beneficial for reducing redundancy and improving the computational efficiency of IGABEM. First, the geometrically precise BEM element is constructed using non-uniform rational B-splines (NURBS) to describe the geometry, while the field is approximated using B-spline interpolation and a transformation matrix. Second, the calculation formats of problems are derived by using parameter mapping. Third, the calculation of variables at boundary points is performed on the element through the relationship between variables. The subsequent processing is like the BEM. Finally, several numerical examples are discussed. It can be concluded from examples that the proposed method can obtain a high-precision solution and reduce computation costs.

为了减少非均质边界条件对计算精度的影响以及冗余几何细化对传统等几何边界元方法（IGABEM）计算效率的影响，我们提出了一种引入几何无关场近似（GIFT）的新型等几何边界元方法，并将其应用于二维势能和弹性问题。与 IGABEM 不同的是，元素节点位于边界上，这有利于在节点上精确应用边界条件，从而提高计算精度。构建的元素具有多个内部节点，在解决问题时可以使用较少的元素，这有利于减少冗余和提高 IGABEM 的计算效率。首先，使用非均匀有理 B-样条曲线（NURBS）构建几何精确的 BEM 元素来描述几何，同时使用 B-样条插值和变换矩阵来近似场。其次，通过参数映射得出问题的计算格式。第三，通过变量之间的关系在元素上对边界点的变量进行计算。随后的处理过程与 BEM 类似。最后，讨论了几个数值示例。从实例中可以得出结论，所提出的方法可以获得高精度的解，并降低计算成本。

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引用次数: 0

A boundary-type algorithm based on the discrete ordinates for neutron transport equation 基于中子输运方程离散序数的边界型算法

IF 4.2 2区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Engineering Analysis with Boundary Elements

Pub Date : 2024-10-05 DOI: 10.1016/j.enganabound.2024.105981

YanTing Cheng , Mei Huang , XiaoPing Ouyang , Hiroshi Matsuda , Chihiro Morita

The spatial distribution of neutron flux in the core of a nuclear reactor plays a crucial role in ensuring nuclear safety. It can be determined by solving the neutron transport equation, where computational resource consumption is gradually receiving more attention, especially in problems with multiple reflective boundary conditions. It is common practice to assume initial boundary values in one angular direction to obtain a global solution, and then use reflective boundary conditions to solve for the other directions, iterating until convergence. However, this approach has the drawback of requiring the global solution, and each iteration necessitates storing the neutron flux values, consuming time and storage space. In order to address this issue, this paper proposes a precise and efficient boundary-type method, the Half Boundary Method (HBM). This method establishes relationships between adjacent node values through integration and mathematical deduction, ultimately obtaining the relationship among any node and the boundary values in any direction. As a result, iteration is only required for boundary values, reducing the iteration amount and thus saving time and storage space. Using the discrete ordinate (S_N) method for angular discretization and HBM for spatial discretization, this paper solved the steady-state neutron transport equation for two-dimensional systems modeled with Cartesian geometry. The method is validated using several benchmark problems, including the 2D ISSA benchmark, the 2D mono-group k-eigenvalue problem and BWR rod bundle test problem. All of these problems demonstrate good results and effectively reduced computational time.

核反应堆堆芯中子通量的空间分布对确保核安全起着至关重要的作用。它可以通过求解中子输运方程来确定，其中的计算资源消耗逐渐受到越来越多的关注，尤其是在具有多重反射边界条件的问题中。通常的做法是在一个角度方向上假设初始边界值以获得全局解，然后使用反射边界条件求解其他方向，反复进行直到收敛。然而，这种方法的缺点是需要全局解，而且每次迭代都需要存储中子通量值，耗费时间和存储空间。为了解决这个问题，本文提出了一种精确高效的边界型方法--半边界法（HBM）。该方法通过积分和数学推导建立相邻节点值之间的关系，最终得到任意节点与任意方向边界值之间的关系。因此，只需要对边界值进行迭代，减少了迭代次数，从而节省了时间和存储空间。本文使用离散序数法（SN）进行角度离散化，使用 HBM 进行空间离散化，求解了以笛卡尔几何建模的二维系统的稳态中子输运方程。该方法通过几个基准问题进行了验证，包括二维 ISSA 基准问题、二维单组 k 特征值问题和 BWR 杆束测试问题。所有这些问题都显示了良好的结果，并有效地减少了计算时间。

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引用次数: 0

Localized space-time Trefftz method for diffusion equations in complex domains 复杂域中扩散方程的局部时空特雷弗兹法

IF 4.2 2区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Engineering Analysis with Boundary Elements

Pub Date : 2024-10-05 DOI: 10.1016/j.enganabound.2024.105977

Li-Dan Hong , Weichung Yeih , Cheng-Yu Ku , Yan Su

This paper introduces an advanced localized space-time Trefftz method tackling boundary value predicaments within complex two-dimensional domains governed by diffusion equations. In contrast to the widespread space-time collocation Trefftz method, which typically produces dense and ill-conditioned matrices, the proposed strategy employs a localized collocation scheme to remove these constraints. In particular, this is beneficial in multi-connected configurations or when dealing with significant variations in field values. To the best of our knowledge, this is the first space-time collocation Trefftz method adaptation, which is referred to as the localized space-time Trefftz method in this paper. The latter combines the space-time collocation Trefftz method principles, which allows to eliminate the need for mesh and numerical quadrature in its application. The localized space-time Trefftz method represents each interior node expressed as a linear blend of its immediate neighbors, while the space-time collocation Trefftz method applies numerical techniques within distinct subdomains. A sparse system of linear algebraic equations with internal points satisfying the governing equation, and boundary points satisfying the boundary conditions, allows to obtain numerical solutions using matrix systems. The localized space-time Trefftz method retains the easy-to-use properties and mesh-free structure of the space-time collocation Trefftz method, and it mitigates its ill-conditioning characteristics. Due to the localization principle and the consideration of overlapping subdomains, the solutions in the proposed localized space-time Trefftz method are more simply and compactly expressed compared with those in the space-time collocation Trefftz method, especially when dealing with multiply-connected domains. Numerical examples for simply-connected and multiply-connected domains are then provided to demonstrate the high precision and simplicity of the proposed localized space-time Trefftz method. The obtained results show that the latter has very high accuracy in solving two-dimensional diffusion problems. Compared with the traditional space-time collocation Trefftz method, the proposed mesh-free strategy yields solutions with higher precision while significantly reducing the instability.

本文介绍了一种先进的局部时空特雷弗茨方法，用于解决由扩散方程控制的复杂二维域中的边界值难题。广泛使用的时空配准 Trefftz 方法通常会产生密集且条件不佳的矩阵，与之相比，本文提出的策略采用了局部配准方案来消除这些限制。特别是在多连接配置或处理场值的显著变化时，这种方法更有优势。据我们所知，这是首次对时空配准特雷弗茨方法进行调整，本文将其称为局部时空特雷弗茨方法。后者结合了时空配准 Trefftz 方法的原理，因此在应用中无需网格和数值正交。局部时空特里夫兹法将每个内部节点表示为其近邻的线性混合，而时空配位特里夫兹法则在不同的子域内应用数值技术。稀疏线性代数方程系统的内部点满足控制方程，边界点满足边界条件，这样就可以利用矩阵系统获得数值解。局部时空特雷弗兹方法保留了时空配位特雷弗兹方法的易用性和无网格结构，并减轻了其条件不佳的特点。由于采用了局部化原理并考虑了重叠子域，与时空配准特里夫兹方法相比，所提出的局部时空特里夫兹方法的解表达得更简单、更紧凑，尤其是在处理多连接域时。然后，提供了简单连接域和多重连接域的数值示例，以证明所提出的局部时空特雷弗茨方法的高精度和简便性。结果表明，后者在解决二维扩散问题时具有非常高的精度。与传统的时空配位 Trefftz 方法相比，所提出的无网格策略能得到精度更高的解，同时大大降低了不稳定性。

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引用次数: 0