Journal of Computational Physics最新文献_第8页

Preventing mass loss in the standard level set method: New insights from variational analyses 防止标准水平集方法中的质量损失：变分分析的新见解

IF 3.8 2区物理与天体物理 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Computational Physics

Pub Date : 2024-10-11 DOI: 10.1016/j.jcp.2024.113495

Kaustubh Khedkar , Amirreza Charchi Mamaghani , Pieter Ghysels , Neelesh A. Patankar , Amneet Pal Singh Bhalla

For decades, the computational multiphase flow community has grappled with mass loss in the level set method. Numerous solutions have been proposed, from fixing the reinitialization step to combining the level set method with other conservative schemes. However, our work reveals a more fundamental culprit: the smooth Heaviside and delta functions inherent to the standard formulation. Even if reinitialization is done exactly, i.e., the zero contour interface remains stationary, the use of smooth functions lead to violation of mass conservation. We propose a novel approach using variational analysis to incorporate a mass conservation constraint. This introduces a Lagrange multiplier that enforces overall mass balance. Notably, as the delta function sharpens, i.e., approaches the Dirac delta limit, the Lagrange multiplier approaches zero. However, the exact Lagrange multiplier method disrupts the signed distance property of the level set function. This motivates us to develop an approximate version of the Lagrange multiplier that preserves both overall mass and signed distance property of the level set function. Our framework even recovers existing mass-conserving level set methods, revealing some inconsistencies in prior analyses. We extend this approach to three-phase flows for fluid-structure interaction (FSI) simulations. We present variational equations in both immersed and non-immersed forms, demonstrating the convergence of the former formulation to the latter when the body delta function sharpens. Rigorous test problems confirm that the FSI dynamics produced by our simple, easy-to-implement immersed formulation with the approximate Lagrange multiplier method are accurate and match state-of-the-art solvers.

几十年来，计算多相流界一直在努力解决水平集方法中的质量损失问题。从固定重新初始化步骤到将水平集方法与其他保守方案相结合，已经提出了许多解决方案。然而，我们的工作揭示了一个更根本的罪魁祸首：标准公式固有的平滑海维塞函数和德尔塔函数。即使重新初始化精确完成，即零轮廓界面保持静止，光滑函数的使用也会导致违反质量守恒。我们提出了一种使用变分分析的新方法，将质量守恒约束纳入其中。这就引入了一个拉格朗日乘数，强制实现整体质量平衡。值得注意的是，随着三角函数的尖锐化，即接近狄拉克三角极限，拉格朗日乘数趋近于零。然而，精确的拉格朗日乘法破坏了水平集函数的符号距离特性。这促使我们开发一种近似版本的拉格朗日乘法器，它既能保持总体质量，又能保持水平集函数的符号距离特性。我们的框架甚至恢复了现有的质量保证水平集方法，揭示了之前分析中的一些不一致之处。我们将这种方法扩展到流固耦合（FSI）模拟的三相流。我们提出了沉浸式和非沉浸式两种形式的变分方程，证明了当体三角函数变得尖锐时，前者的公式会向后者收敛。严格的测试问题证实，我们简单易用的浸没式公式与近似拉格朗日乘法器方法所产生的 FSI 动力学是精确的，与最先进的求解器不相上下。

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引用次数: 0

A DSMC-CFD coupling method using surrogate modelling for low-speed rarefied gas flows 低速稀薄气体流的代用建模 DSMC-CFD 耦合方法

IF 3.8 2区物理与天体物理 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Computational Physics

Pub Date : 2024-10-11 DOI: 10.1016/j.jcp.2024.113500

Giorgos Tatsios , Arun K. Chinnappan , Arshad Kamal , Nikos Vasileiadis , Stephanie Y. Docherty , Craig White , Livio Gibelli , Matthew K. Borg , James R. Kermode , Duncan A. Lockerby

A new Micro-Macro-Surrogate (MMS) hybrid method is presented that couples the Direct Simulation Monte Carlo (DSMC) method with Computational Fluid Dynamics (CFD) to simulate low-speed rarefied gas flows. The proposed MMS method incorporates surrogate modelling instead of direct coupling of DSMC data with the CFD, addressing the limitations CFD has in accurately modelling rarefied gas flows, the computational cost of DSMC for low-speed and multiscale flows, as well as the pitfalls of noise in conventional direct coupling approaches. The surrogate models, trained on the DSMC data using Bayesian inference, provide noise-free and accurate corrections to the CFD simulation enabling it to capture the non-continuum physics. The MMS hybrid approach is validated by simulating low-speed, steady-state, force-driven rarefied gas flows in a canonical 1D parallel-plate system, where corrections to the boundary conditions and stress tensor are considered and shows excellent agreement with DSMC benchmark results. A comparison with the typical domain decomposition DSMC-CFD hybrid method is also presented, to demonstrate the advantages of noise-avoidance in the proposed approach. The method also inherently captures the uncertainty arising from micro-model fluctuations, allowing for the quantification of noise-related uncertainty in the predictions. The proposed MMS method demonstrates the potential to enable multiscale simulations where CFD is inaccurate and DSMC is prohibitively expensive.

本文介绍了一种新的微观-宏观-代用（MMS）混合方法，它将直接模拟蒙特卡罗（DSMC）方法与计算流体动力学（CFD）相结合，模拟低速稀薄气体流动。所提出的 MMS 方法采用了代用模型，而不是将 DSMC 数据与 CFD 直接耦合，从而解决了 CFD 在精确模拟稀薄气体流动方面的局限性、DSMC 在低速和多尺度流动方面的计算成本以及传统直接耦合方法中的噪声隐患。使用贝叶斯推理方法在 DSMC 数据上训练的代用模型为 CFD 模拟提供了无噪声的精确修正，使其能够捕捉到非连续的物理特性。MMS 混合方法通过模拟典型一维平行板系统中的低速、稳态、力驱动稀薄气体流进行了验证，其中考虑了对边界条件和应力张量的修正，并显示出与 DSMC 基准结果的极佳一致性。此外，还与典型的域分解 DSMC-CFD 混合方法进行了比较，以展示所提议方法在避免噪声方面的优势。该方法还能捕捉微观模型波动带来的不确定性，从而量化预测中与噪声相关的不确定性。拟议的 MMS 方法展示了在 CFD 不准确、DSMC 昂贵的情况下进行多尺度模拟的潜力。

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引用次数: 0

A performant energy-conserving particle reweighting method for Particle-in-Cell simulations 用于 "单元内粒子 "模拟的高性能能量守恒粒子再加权方法

IF 3.8 2区物理与天体物理 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Computational Physics

Pub Date : 2024-10-11 DOI: 10.1016/j.jcp.2024.113454

Jeremiah J. Boerner , Taylor Hall , Russell Hooper , Matthew T. Bettencourt , Matthew M. Hopkins , Anne M. Grillet , Jose L. Pacheco

A new particle-based reweighting method is developed and demonstrated in the Aleph Particle-in-Cell with Direct Simulation Monte Carlo (PIC-DSMC) program. Novel splitting and merging algorithms ensure that modified particles maintain physically consistent positions and velocities. This method allows a single reweighting simulation to efficiently model plasma evolution over orders of magnitude variation in density, while accurately preserving energy distribution functions (EDFs). Demonstrations on electrostatic sheath and collisional rate dynamics show that reweighting simulations achieve accuracy comparable to fixed weight simulations with substantial computational time savings. This highly performant reweighting method is recommended for modeling plasma applications that require accurate resolution of EDFs or exhibit significant density variations in time or space.

在 Aleph Particle-in-Cell with Direct Simulation Monte Carlo (PIC-DSMC) 程序中开发并演示了一种新的基于粒子的重新加权方法。新颖的分割和合并算法确保修改后的粒子保持物理上一致的位置和速度。通过这种方法，只需进行一次重新加权模拟，就能有效地模拟等离子体在密度数量级变化过程中的演变，同时准确地保留能量分布函数（EDF）。静电鞘和碰撞率动力学演示表明，重新加权模拟的精确度可与固定权重模拟相媲美，同时大大节省了计算时间。对于需要精确解析 EDF 或在时间或空间上表现出显著密度变化的等离子体应用建模，推荐使用这种高性能的重新加权方法。

引用次数: 0

Multiscale mixed methods with improved accuracy: The role of oversampling and smoothing 提高精度的多尺度混合方法：超采样和平滑的作用

IF 3.8 2区物理与天体物理 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Computational Physics

Pub Date : 2024-10-10 DOI: 10.1016/j.jcp.2024.113490

Dilong Zhou , Rafael T. Guiraldello , Felipe Pereira

Multiscale mixed methods based on non-overlapping domain decompositions can efficiently handle the solution of significant subsurface flow problems in very heterogeneous formations of interest to the industry, especially when implemented on multi-core supercomputers. Efficiency in obtaining numerical solutions is dictated by the choice of interface spaces that are selected: the smaller the dimension of these spaces, the better, in the sense that fewer multiscale basis functions need to be computed, and smaller interface linear systems need to be solved. Thus, in solving large computational problems, it is desirable to work with piecewise constant or linear polynomials for interface spaces. However, for these choices of interface spaces, it is well known that the flux accuracy is of the order of

10^{- 1}

.

This study is dedicated to advancing an efficient and accurate multiscale mixed method aimed at addressing industry-relevant problems. A distinctive feature of our approach involves subdomains with overlapping regions, a departure from conventional methods. We take advantage of the overlapping decomposition to introduce a computationally highly efficient smoothing step designed to rectify small-scale errors inherent in the multiscale solution. The effectiveness of the proposed solver, which maintains a computational cost very close to its predecessors, is demonstrated through a series of numerical studies. Notably, for scenarios involving modestly sized overlapping regions and employing just a few smoothing steps, a substantial enhancement of two orders of magnitude in flux accuracy is achieved with the new approach.

基于非重叠域分解的多尺度混合方法可以有效地解决工业界感兴趣的异质地层中的重要地下流动问题，尤其是在多核超级计算机上实施时。获取数值解的效率取决于界面空间的选择：这些空间的维度越小越好，因为需要计算的多尺度基函数越少，需要求解的界面线性系统越小。因此，在解决大型计算问题时，界面空间最好使用片断常数或线性多项式。然而，众所周知，对于这些界面空间的选择，通量精度为 10-1 量级。本研究致力于推进一种高效、精确的多尺度混合方法，旨在解决工业相关问题。与传统方法不同的是，我们的方法涉及具有重叠区域的子域。我们利用重叠分解引入了计算效率极高的平滑步骤，旨在纠正多尺度求解中固有的小尺度误差。通过一系列数值研究，我们证明了所提出的求解器的有效性，其计算成本与前代求解器非常接近。值得注意的是，对于涉及大小适中的重叠区域的情况，只需采用几个平滑步骤，新方法就能将通量精度大幅提高两个数量级。

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引用次数: 0

Improved physics-informed neural networks for the reinterpreted discrete fracture model 用于重新解释离散断裂模型的改进型物理信息神经网络

IF 3.8 2区物理与天体物理 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Computational Physics

Pub Date : 2024-10-10 DOI: 10.1016/j.jcp.2024.113491

Chao Wang , Hui Guo , Xia Yan , Zhang-Lei Shi , Yang Yang

This paper is the first attempt to apply improved-physics-informed neural networks (I-PINNs) to simulate fluid flow in fractured porous media based on the reinterpreted discrete fracture model (RDFM). The RDFM, first introduced by Xu and Yang, is a hybrid-dimensional model where Dirac-delta functions are used to characterize fractures and superposed with the permeability tensor. In this paper, we apply the physical information neural networks (PINNs) to RDFM. Different from the traditional PINNs where the PDE residual was used as the loss function, we adopt the finite element discretization of RDFM to build the loss function, avoiding the large gradient problem and difficulties in automatic differentiation. This new method is named as the improved PINNs (I-PINNs). Moreover, we combine the RDFM with incompressible miscible displacement in porous media. The bound-preserving technique of the I-PINNs is proposed and applied to the coupled system mentioned above, keeping the numerical concentration to be between 0 and 1. It is worth noting that one of the advantages of I-PINNs compared to PINNs is that it can better capture the pressure gradient at the fractures. Compared with traditional finite element methods for flow equations, I-PINNs do not request the inversion of the stiffness matrix. In addition, different from the traditional bound-preserving technique for contaminant transportation, I-PINNs preserve the physical bounds without taking a limited time step. Several numerical experiments are given to verify the feasibility and accuracy of the I-PINNs.

本文首次尝试基于重新解释的离散裂缝模型（RDFM），应用改进的物理信息神经网络（I-PINNs）来模拟裂缝多孔介质中的流体流动。RDFM 由 Xu 和 Yang 首次提出，是一种混维模型，其中 Dirac-delta 函数用于描述裂缝特征，并与渗透率张量叠加。本文将物理信息神经网络（PINN）应用于 RDFM。与传统 PINNs 使用 PDE 残差作为损失函数不同，我们采用 RDFM 的有限元离散化来建立损失函数，避免了大梯度问题和自动微分的困难。这种新方法被命名为改进 PINNs（I-PINNs）。此外，我们还将 RDFM 与多孔介质中的不可压缩混杂位移相结合。值得注意的是，与 PINNs 相比，I-PINNs 的优势之一是能更好地捕捉裂缝处的压力梯度。与处理流动方程的传统有限元方法相比，I-PINN 无需对刚度矩阵进行反演。此外，与传统的污染物输送边界保留技术不同，I-PINNs 保留了物理边界，而无需采取有限的时间步长。本文给出了几个数值实验来验证 I-PINNs 的可行性和准确性。

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引用次数: 0

Adaptive hyperbolic-cross-space mapped Jacobi method on unbounded domains with applications to solving multidimensional spatiotemporal integrodifferential equations 无界域上的自适应双曲跨空间映射雅可比法及其在求解多维时空整微分方程中的应用

IF 3.8 2区物理与天体物理 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Computational Physics

Pub Date : 2024-10-10 DOI: 10.1016/j.jcp.2024.113492

Yunhong Deng , Sihong Shao , Alex Mogilner , Mingtao Xia

In this paper, we develop a new adaptive hyperbolic-cross-space mapped Jacobi (AHMJ) method for solving multidimensional spatiotemporal integrodifferential equations in unbounded domains. By devising adaptive techniques for sparse mapped Jacobi spectral expansions defined in a hyperbolic cross space, our proposed AHMJ method can efficiently solve various spatiotemporal integrodifferential equations such as the anomalous diffusion model with reduced numbers of basis functions. Our analysis of the AHMJ method gives a uniform upper error bound for solving a class of spatiotemporal integrodifferential equations, leading to effective error control.

本文开发了一种新的自适应双曲交叉空间映射雅可比（AHMJ）方法，用于求解无界域中的多维时空整微分方程。通过设计定义在双曲交叉空间中的稀疏映射雅可比谱展开的自适应技术，我们提出的 AHMJ 方法可以高效地求解各种时空整微分方程，如减少基函数数量的反常扩散模型。我们对 AHMJ 方法的分析给出了求解一类时空微分方程的统一误差上限，从而实现了有效的误差控制。

引用次数: 0

The energy-diminishing weak Galerkin finite element method for the computation of ground state and excited states in Bose-Einstein condensates 计算玻色-爱因斯坦凝聚态基态和激发态的能量递减弱伽勒金有限元法

IF 3.8 2区物理与天体物理 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Computational Physics

Pub Date : 2024-10-10 DOI: 10.1016/j.jcp.2024.113497

Lin Yang , Xiang-Gui Li , Wei Yan , Ran Zhang

In this paper, we employ the weak Galerkin (WG) finite element method and the imaginary time method to compute both the ground state and the excited states in Bose-Einstein condensate (BEC) which is governed by the Gross-Pitaevskii equation (GPE). First, we use the imaginary time method for GPE to get the nonlinear parabolic partial differential equation. Subsequently, we apply the WG method to spatially discretize the parabolic equation. This yields a semi-discrete scheme, in which an energy function is explicitly defined. For the case

β ⩾ 0

, we demonstrate that the energy is diminishing with respect to time t at each time step. Applying the backward Euler scheme for temporal discretization yields a fully discrete scheme. For the case

β = 0

, we provide a mathematical justification, establishing the convergence analysis for the numerical solution of the ground state. Moreover, based on the theory of solving eigenvalue problems using the WG method, we present the error estimates between the ground state and its numerical solution under the

H^{1}

and

L^{2}

norms. Numerical experiments are provided to illustrate the effectiveness of the proposed schemes. Moreover, the results indicate that our method also can compute the first excited state, achieving optimal convergence orders.

本文采用弱 Galerkin（WG）有限元法和虚时间法计算受格罗斯-皮塔耶夫斯基方程（GPE）支配的玻色-爱因斯坦凝聚态（BEC）的基态和激发态。首先，我们使用虚时间法求解 GPE 的非线性抛物线偏微分方程。随后，我们采用 WG 方法对抛物方程进行空间离散化。这样就得到了一个半离散方案，其中明确定义了能量函数。对于 β⩾0 的情况，我们证明能量在每个时间步相对于时间 t 是递减的。应用后向欧拉方案进行时间离散化，可得到一个完全离散的方案。对于 β=0 的情况，我们提供了数学理由，建立了基态数值解的收敛分析。此外，基于使用 WG 方法求解特征值问题的理论，我们提出了 H1 和 L2 规范下基态与其数值解之间的误差估计。我们还提供了数值实验来说明所提方案的有效性。此外，结果表明我们的方法也能计算第一激发态，并达到最佳收敛阶数。

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引用次数: 0

Error analysis of kernel/GP methods for nonlinear and parametric PDEs 非线性和参数 PDE 的核/GP 方法的误差分析

IF 3.8 2区物理与天体物理 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Computational Physics

Pub Date : 2024-10-10 DOI: 10.1016/j.jcp.2024.113488

Pau Batlle , Yifan Chen , Bamdad Hosseini , Houman Owhadi , Andrew M. Stuart

We introduce a priori Sobolev-space error estimates for the solution of arbitrary nonlinear, and possibly parametric, PDEs that are defined in the strong sense, using Gaussian process and kernel based methods. The primary assumptions are: (1) a continuous embedding of the reproducing kernel Hilbert space of the kernel into a Sobolev space of sufficient regularity; and (2) the stability of the differential operator and the solution map of the PDE between corresponding Sobolev spaces. The proof is articulated around Sobolev norm error estimates for kernel interpolants and relies on the minimizing norm property of the solution. The error estimates demonstrate dimension-benign convergence rates if the solution space of the PDE is smooth enough. We illustrate these points with applications to high-dimensional nonlinear elliptic PDEs and parametric PDEs. Although some recent machine learning methods have been presented as breaking the curse of dimensionality in solving high-dimensional PDEs, our analysis suggests a more nuanced picture: there is a trade-off between the regularity of the solution and the presence of the curse of dimensionality. Therefore, our results are in line with the understanding that the curse is absent when the solution is regular enough.

我们使用基于高斯过程和核的方法，为任意非线性、可能是参数的、在强意义上定义的 PDEs 的求解引入先验 Sobolev 空间误差估计。主要假设有(1) 将核的再现核希尔伯特空间连续嵌入到具有充分正则性的索博廖夫空间；以及 (2) 微分算子和相应索博廖夫空间之间的 PDE 解映射具有稳定性。证明围绕核内插的 Sobolev 规范误差估计展开，并依赖于解的最小化规范特性。如果 PDE 的解空间足够平滑，误差估计值就会显示出维度良性收敛率。我们将这些观点应用于高维非线性椭圆 PDE 和参数 PDE。虽然最近的一些机器学习方法被认为在求解高维 PDE 时打破了维度诅咒，但我们的分析表明了一个更微妙的情况：在解的规则性和维度诅咒的存在之间存在权衡。因此，我们的结果符合这样一种理解：当解足够规则时，诅咒就不存在。

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引用次数: 0

High-order schemes of exponential time differencing for stiff systems with nondiagonal linear part 具有非对角线性部分的刚性系统的指数时差高阶方案

IF 3.8 2区物理与天体物理 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Computational Physics

Pub Date : 2024-10-10 DOI: 10.1016/j.jcp.2024.113493

Evelina V. Permyakova , Denis S. Goldobin

Exponential time differencing methods are a power tool for high-performance numerical simulation of computationally challenging problems in condensed matter physics, fluid dynamics, chemical and biological physics, where mathematical models often possess fast oscillating or decaying modes—in other words, are stiff systems. Practical implementation of these methods for the systems with nondiagonal linear part of equations is exacerbated by infeasibility of an analytical calculation of the exponential of a nondiagonal linear operator; in this case, the coefficients of the exponential time differencing scheme cannot be calculated analytically. We suggest an approach, where these coefficients are numerically calculated with auxiliary problems. We rewrite the high-order Runge–Kutta type schemes in terms of the solutions to these auxiliary problems and practically examine the accuracy and computational performance of these methods for a heterogeneous Cahn–Hilliard equation, a sixth-order spatial derivative equation governing pattern formation in the presence of an additional conservation law, and a Fokker–Planck equation governing macroscopic dynamics of a network of neurons.

指数时差法是对凝聚态物理、流体动力学、化学和生物物理中具有计算挑战性的问题进行高性能数值模拟的有力工具，这些问题的数学模型通常具有快速振荡或衰减模式--换句话说，是刚性系统。由于无法对非对角线性算子的指数进行分析计算，这些方法在非对角线性方程系统中的实际应用变得更加困难；在这种情况下，无法对指数时差方案的系数进行分析计算。我们建议采用一种方法，利用辅助问题对这些系数进行数值计算。我们根据这些辅助问题的解重写了高阶 Runge-Kutta 类型方案，并实际检验了这些方法在异质 Cahn-Hilliard 方程、存在额外守恒定律的支配模式形成的六阶空间导数方程以及支配神经元网络宏观动力学的福克-普朗克方程中的精度和计算性能。

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引用次数: 0

The method of fundamental solutions for multi-particle Stokes flows: Application to a ring-like array of spheres 多粒子斯托克斯流的基本解法：对环状球阵列的应用

IF 3.8 2区物理与天体物理 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Computational Physics

Pub Date : 2024-10-10 DOI: 10.1016/j.jcp.2024.113487

Josiah J.P. Jordan, Duncan A. Lockerby

A method is presented for calculating Stokes flow around multiple particles of arbitrary shape. It uses the Method of Fundamental Solutions (MFS) applied to single particles, combined with an iterative scheme to resolve the many particle-particle hydrodynamic interactions; an approach that is reminiscent of the Method of Reflections. The attractive features of the proposed method are inherited from the MFS — simplicity and accuracy — while providing orders of magnitude computational speed-up for large particle systems. The method is verified through a series of test cases, including those involving strong lubrication forces and non-spherical particles. Unlike applications of the Method of Reflections reported in the literature, the iterative scheme we propose (a block Gauss-Seidel approach to solving a particle-particle interaction matrix) converges for all the cases we consider, for both resistance and mobility problems. The scheme is applied to the study of Stokes flow around ring-like arrays of spheres. We show that the relationship between globally applied velocity or force to the response of individual spheres can be described by just 5 coefficients (or 9 in total) for any given configuration. The results indicate that for 7–10 spheres in sedimentation, there exists a certain spacing that produces steady-state translation of the ring, independent of its orientation. For large numbers of spheres, slender-body theory can be applied to the problem, providing remarkably close agreement to the numerical results over a wide range of parameters.

本文介绍了一种计算任意形状的多个粒子周围斯托克斯流的方法。该方法采用了适用于单个粒子的基本解法（MFS），并结合迭代方案来解决许多粒子间的流体动力学相互作用；这种方法让人想起反射法（Method of Reflections）。所提出的方法继承了 MFS 的诱人特点--简单、准确，同时为大型粒子系统提供了数量级的计算速度。通过一系列测试案例，包括涉及强润滑力和非球形粒子的案例，对该方法进行了验证。与文献中报道的反射法应用不同，我们提出的迭代方案（求解粒子-粒子相互作用矩阵的分块高斯-赛德尔方法）在我们考虑的所有情况下，无论是阻力问题还是流动性问题，都能收敛。我们将该方案应用于研究环状球阵列周围的斯托克斯流。我们发现，对于任何给定配置，全局施加的速度或力与单个球体响应之间的关系只需 5 个系数（或总共 9 个系数）即可描述。结果表明，对于沉积中的 7-10 个球体，存在一定的间距，可以产生与环的方向无关的稳态平移。对于大量的球体，细长体理论可应用于该问题，在广泛的参数范围内与数值结果非常接近。

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引用次数: 0