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General finite-volume framework for saddle-point problems of various physics 各种物理鞍点问题的一般有限体积框架
IF 0.6 4区 数学 Q4 MATHEMATICS, APPLIED Pub Date : 2021-12-01 DOI: 10.1515/rnam-2021-0029
K. Terekhov
Abstract This article is dedicated to the general finite-volume framework used to discretize and solve saddle-point problems of various physics. The framework applies the Ostrogradsky–Gauss theorem to transform a divergent part of the partial differential equation into a surface integral, approximated by the summation of vector fluxes over interfaces. The interface vector fluxes are reconstructed using the harmonic averaging point concept resulting in the unique vector flux even in a heterogeneous anisotropic medium. The vector flux is modified with the consideration of eigenvalues in matrix coefficients at vector unknowns to address both the hyperbolic and saddle-point problems, causing nonphysical oscillations and an inf-sup stability issue. We apply the framework to several problems of various physics, namely incompressible elasticity problem, incompressible Navier–Stokes, Brinkman–Hazen–Dupuit–Darcy, Biot, and Maxwell equations and explain several nuances of the application. Finally, we test the framework on simple analytical solutions.
摘要本文研究了用于离散化和求解各种物理鞍点问题的一般有限体积框架。该框架应用Ostrogradsky-Gauss定理将偏微分方程的发散部分转化为表面积分,近似为界面上矢量通量的总和。利用谐波平均点的概念重构了界面矢量通量,即使在非均质各向异性介质中也能得到唯一的矢量通量。在矢量未知数处,考虑矩阵系数的特征值对矢量通量进行了修正,以解决双曲和鞍点问题,这些问题会引起非物理振荡和内支撑稳定性问题。我们将该框架应用于各种物理问题,即不可压缩弹性问题,不可压缩Navier-Stokes, Brinkman-Hazen-Dupuit-Darcy, Biot和Maxwell方程,并解释了应用中的几个细微差别。最后,我们在简单解析解上对框架进行了测试。
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引用次数: 2
An equilibrated a posteriori error estimator for an Interior Penalty Discontinuous Galerkin approximation of the p-Laplace problem p-拉普拉斯问题的内罚不连续伽辽金近似的平衡后验误差估计
IF 0.6 4区 数学 Q4 MATHEMATICS, APPLIED Pub Date : 2021-12-01 DOI: 10.1515/rnam-2021-0026
R. Hoppe, Youri Iliash
Abstract We are concerned with an Interior Penalty Discontinuous Galerkin (IPDG) approximation of the p-Laplace equation and an equilibrated a posteriori error estimator. The IPDG method can be derived from a discretization of the associated minimization problem involving appropriately defined reconstruction operators. The equilibrated a posteriori error estimator provides an upper bound for the discretization error in the broken W1,p norm and relies on the construction of an equilibrated flux in terms of a numerical flux function associated with the mixed formulation of the IPDG approximation. The relationship with a residual-type a posteriori error estimator is established as well. Numerical results illustrate the performance of both estimators.
摘要我们讨论了p-Laplace方程的内部惩罚间断Galerkin(IPDG)近似和平衡后验误差估计。IPDG方法可以从涉及适当定义的重建算子的相关最小化问题的离散化中导出。平衡后验误差估计器为破碎的W1,p范数中的离散化误差提供了上界,并且依赖于根据与IPDG近似的混合公式相关联的数值通量函数来构造平衡通量。建立了残差型后验误差估计器与残差型的关系。数值结果说明了这两种估计的性能。
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引用次数: 0
Stability analysis of functionals in variational data assimilation with respect to uncertainties of input data for a sea thermodynamics model 关于海洋热力学模型输入数据不确定性的变分数据同化中泛函的稳定性分析
IF 0.6 4区 数学 Q4 MATHEMATICS, APPLIED Pub Date : 2021-12-01 DOI: 10.1515/rnam-2021-0028
V. Shutyaev, E. Parmuzin, I. Gejadze
Abstract The problem of stability and sensitivity of functionals of the optimal solution of the variational data assimilation of sea surface temperature for the model of sea thermodynamics is considered. The variational data assimilation problem is formulated as an optimal control problem to find the initial state and the boundary heat flux. The sensitivity of the response functions as functionals of the optimal solution with respect to the observation data is studied. Computing the gradient of the response function reduces to the solution of a non-standard problem being a coupled system of direct and adjoint equations with mutually dependent initial and boundary values. The algorithm to compute the gradient of the response function is presented, based on the Hessian of the original cost functional. Stability analysis of the response function with respect to uncertainties of input data is given. Numerical examples are presented for the Black and Azov seas thermodynamics model.
摘要考虑了海洋热力学模型变分海面温度数据同化最优解的泛函的稳定性和灵敏度问题。将变分数据同化问题化为求初始状态和边界热通量的最优控制问题。研究了响应函数作为最优解的函数对观测数据的敏感性。计算响应函数的梯度可归结为一个非标准问题的解,即具有相互依赖的初始值和边值的直接方程和伴随方程的耦合系统。在原始代价泛函的Hessian基础上,提出了计算响应函数梯度的算法。给出了响应函数相对于输入数据不确定性的稳定性分析。给出了黑海和亚速海热力学模型的数值算例。
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引用次数: 2
Frontmatter Frontmatter
IF 0.6 4区 数学 Q4 MATHEMATICS, APPLIED Pub Date : 2021-11-01 DOI: 10.1515/rnam-2021-frontmatter5
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引用次数: 0
Numerical modelling of the transition of infected cells and virions between two lymph nodes in a stochastic model of HIV-1 infection 在HIV-1感染的随机模型中感染细胞和病毒粒子在两个淋巴结间转移的数值模拟
IF 0.6 4区 数学 Q4 MATHEMATICS, APPLIED Pub Date : 2021-11-01 DOI: 10.1515/rnam-2021-0024
N. Pertsev, V. Topchii, K. Loginov
Abstract The paper is focused on stochastic modelling of the process of transition of infected cells and virions of HIV-1 infection between two lymph nodes. The model is based on the following assumptions: (1) the duration of transition of infected cells and virions between two lymph nodes is set using a time-dependent function, (2) infected cells produce virions in the process of transition between two lymph nodes, (3) infected cells and virions may die when moving between two lymph nodes. The methods of the theory of branching random processes are used to study analytically the model variables. An algorithm for statistical modelling of the number of infected cells and virions in the second lymph node is presented. The results of computational experiments studying the distribution law of the number of virions produced by one infected cell depending on the duration of movement between two lymph nodes are presented.
摘要本文研究了HIV-1感染细胞和病毒粒子在两个淋巴结间转移过程的随机模型。该模型基于以下假设:(1)感染细胞和病毒粒子在两个淋巴结之间转移的持续时间使用时间依赖函数设定;(2)感染细胞在两个淋巴结之间转移的过程中产生病毒粒子;(3)感染细胞和病毒粒子在两个淋巴结之间移动时可能死亡。利用分支随机过程理论的方法对模型变量进行了解析研究。提出了一种用于统计第二淋巴结感染细胞和病毒粒子数量的算法。本文介绍了一个感染细胞产生的病毒粒子数量随两个淋巴结间运动时间的分布规律的计算实验结果。
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引用次数: 1
Numerical evaluation of the effectiveness of coronary revascularization 冠状动脉血运重建有效性的数值评估
IF 0.6 4区 数学 Q4 MATHEMATICS, APPLIED Pub Date : 2021-11-01 DOI: 10.1515/rnam-2021-0025
S. Simakov, T. Gamilov, F. Liang, D. Gognieva, Mariam K. Gappoeva, P. Kopylov
Abstract In the present paper we construct the model of coronary flow which utilizes the patient’s CT data on both large CA and perfusion. We reconstruct large coronary vessels based on the CT data and extrude possibly invisible vessels from the branches of the left coronary artery so that every terminal point supply every of standardized zone of myocardium. We apply a previously developed and validated 1D model of haemodynamics. Utilizing the patient-specific CTP data for modifying outflow terminal resistance at rest and under stress conditions is the major novelty of the present work. From our results it follows that acceptable values of fractional flow reserve (FFR) after the stenotic treatment may produce false negative output. Therefore healthy FFR values may be observed after ineffective revascularization. We also conclude that microvascular perfusion impairment plays significant role in correct computational estimation of haemodynamic indices before stenotic treatment. The advantage of the presented approach is the availability of transmural perfusion ratio assessment in everyday practice.
摘要在本文中,我们构建了冠状动脉血流模型,该模型利用了患者在大CA和灌注方面的CT数据。我们根据CT数据重建大的冠状动脉血管,并从左冠状动脉的分支中挤出可能不可见的血管,使每个终点都能提供心肌的每个标准区。我们应用了先前开发和验证的血液动力学1D模型。利用患者特异性CTP数据来修改静息和应激条件下的流出端阻力是本工作的主要新颖之处。根据我们的结果,狭窄治疗后血流储备分数(FFR)的可接受值可能会产生假阴性输出。因此,在无效的血运重建后可以观察到健康的FFR值。我们还得出结论,微血管灌注损伤在狭窄治疗前血流动力学指标的正确计算估计中起着重要作用。所提出的方法的优点是在日常实践中可以进行跨壁灌注比率评估。
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引用次数: 6
Herd immunity levels and multi-strain influenza epidemics in Russia: a modelling study 俄罗斯群体免疫水平和多株流感流行:一项模型研究
IF 0.6 4区 数学 Q4 MATHEMATICS, APPLIED Pub Date : 2021-11-01 DOI: 10.1515/rnam-2021-0023
V. Leonenko
Abstract In the present paper, we consider a compartmental epidemic model which simulates the co-circulation of three influenza strains, A(H1N1)pdm09, A(H3N2), and B, in a population with the history of exposure to these virus strains. A strain-specific incidence data for the model input was generated using long-term weekly ARI incidence and virologic testing data. The algorithm for model calibration was developed as a combination of simulated annealing and BFGS optimization methods. Two simulations were carried out, assuming the absence and the presence of protected individuals in the population, with 2017– 2018 and 2018–2019 epidemic seasons in Moscow as a case study. It was shown that strain-specific immune levels defined by virologic studies might be used in the model to obtain plausible incidence curves. However, different output parameter values, such as fractions of individuals exposed to particular virus strain in the previous epidemic season, can correspond to similar incidence trajectories, which complicates the assessment of herd immunity levels based on the model calibration. The results of the study will be used in the research of the interplay between the immunity formation dynamics and the circulation of influenza strains in Russian cities.
摘要在本文中,我们考虑了一个分区流行病模型,该模型模拟了三种流感毒株,a(H1N1)pdm09、a(H3N2)和B,在有接触这些病毒株历史的人群中的共同循环。使用长期每周ARI发病率和病毒学检测数据生成模型输入的菌株特异性发病率数据。模型校准算法是将模拟退火和BFGS优化方法相结合而开发的。假设人群中不存在和存在受保护的个体,进行了两次模拟,以莫斯科的2017-2018年和2018-2019年疫情季节为例进行了研究。研究表明,病毒学研究定义的菌株特异性免疫水平可用于该模型,以获得合理的发病率曲线。然而,不同的输出参数值,例如在上一个流行季节暴露于特定病毒株的个体的分数,可能对应于相似的发病轨迹,这使基于模型校准的群体免疫水平评估变得复杂。这项研究的结果将用于研究免疫形成动力学与俄罗斯城市流感毒株传播之间的相互作用。
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引用次数: 2
Two-scale haemodynamic modelling for patients with Fontan circulation Fontan循环患者的双尺度血流动力学模型
IF 0.6 4区 数学 Q4 MATHEMATICS, APPLIED Pub Date : 2021-11-01 DOI: 10.1515/rnam-2021-0022
T. Dobroserdova, Y. Vassilevski, S. Simakov, T. Gamilov, A. Svobodov, L. Yurpolskaya
Abstract Palliation of congenital single ventricle heart defects suggests multi-stage surgical interventions that divert blood flow from the inferior and superior vena cava directly to the right and left pulmonary arteries, skipping the right ventricle. Such system with cavopulmonary anastomoses and single left ventricle is called Fontan circulation, and the region of reconnection is called the total cavopulmonary connection (TCPC). Computational blood flow models allow clinicians to predict the results of the Fontan operation, to choose an optimal configuration of TCPC and thus to reduce negative postoperative consequences. We propose a two-scale (1D3D) haemodynamic model of systemic circulation for a patient who has underwent Fontan surgical operation. We use CT and 4D flow MRI data to personalize the model. The model is tuned to patient’s data and is able to represent measured time-averaged flow rates at the inlets and outlets of TCPC, as well as pressure in TCPC for the patient in horizontal position.We demonstrate that changing to quiescent standing position leads to other patterns of blood flow in regional (TCPC) and global haemodynamics. This confirms clinical data on exercise intolerance of Fontan patients.
摘要缓解先天性单心室心脏缺陷需要多阶段的手术干预,将下腔静脉和上腔静脉的血流直接转移到左右肺动脉,跳过右心室。这种具有腔肺吻合和单个左心室的系统被称为Fontan循环,重新连接的区域被称为全腔肺连接(TCPC)。计算血流模型使临床医生能够预测Fontan手术的结果,选择TCPC的最佳配置,从而减少术后的负面后果。我们为一名接受Fontan手术的患者提出了一个系统循环的双尺度(1D3D)血液动力学模型。我们使用CT和4D流MRI数据对模型进行个性化设置。该模型根据患者的数据进行了调整,能够表示TCPC入口和出口处测量的时间平均流速,以及处于水平位置的患者的TCPC中的压力。我们证明,改变为静止站立姿势会导致区域(TCPC)和全球血流动力学的其他模式。这证实了Fontan患者运动不耐受的临床数据。
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引用次数: 1
Frontiers in mathematical modelling of the lipid metabolism under normal conditions and its alterations in heart diseases 正常情况下脂质代谢的数学模型及其在心脏病中的变化的前沿
IF 0.6 4区 数学 Q4 MATHEMATICS, APPLIED Pub Date : 2021-11-01 DOI: 10.1515/rnam-2021-0021
G. Bocharov, D. Grebennikov, R. Savinkov
Abstract Pathophysiology of ischemic heart disease is a complex phenomenon determined by the interaction of multiple processes including the inflammatory, immunological, infectious, mechanical, biochemical and epigenetic ones. A predictive clinically relevant modelling of the entire trajectory of the human organism, from the initial alterations in lipid metabolism through to atherosclerotic plaque formation and finally to the pathologic state of the ischemic heart disease, is an open insufficiently explored problem. In the present review, we consider the existing mathematical frameworks which are used to describe, analyze and predict the dynamics of various processes related to cardiovascular diseases at the molecular, cellular, tissue, and holistic human organism level. The mechanistic, statistical and machine learning models are discussed in detail with special focus on the underlying assumptions and their clinical relevance. All together, they provide a solid computational platform for further expansion and tailoring for practical applications.
缺血性心脏病的病理生理是炎症、免疫、感染、机械、生化、表观遗传等多个过程共同作用的复杂现象。从最初的脂质代谢改变到动脉粥样硬化斑块形成,最后到缺血性心脏病的病理状态,人类机体整个轨迹的预测临床相关建模是一个尚未充分探索的开放性问题。在本综述中,我们考虑了现有的数学框架,用于描述,分析和预测在分子,细胞,组织和整体人体有机体水平上与心血管疾病相关的各种过程的动力学。详细讨论了机制、统计和机器学习模型,特别关注潜在的假设及其临床相关性。总之,它们为进一步扩展和裁剪实际应用提供了一个坚实的计算平台。
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引用次数: 0
A regularized isothermal phase-field model of two-phase solid–fluid mixture and its spatial dissipative discretization equations 固液两相混合物的正则化等温相场模型及其空间耗散离散化方程
IF 0.6 4区 数学 Q4 MATHEMATICS, APPLIED Pub Date : 2021-08-01 DOI: 10.1515/rnam-2021-0017
V. Balashov
Abstract The present paper is devoted to a model describing a two-phase isothermal mixture, in which one of the phases obeys solid-like (namely, elastic) rheology. A fully Eulerian description is considered. To describe the stress–strain behaviour of the solid phase the elastic energy term is added to the Helmholtz free energy. The term depends on Almansi strain tensor. In its turn, the strain tensor is defined as the solution of the corresponding evolutionary equation. Considered model belongs to the phase field family. Formally it describes two-component mixture and uses mass densities of the components as order parameters. A distinctive feature of the considered model is its preliminary regularization according to the quasi-hydrodynamic framework. The dissipativity in total energy is proved when periodic boundary conditions are imposed. A spatial dissipative semi-discrete (continuous in time and discrete in space) scheme based on staggered grids is suggested. The theoretical results remain valid in the absence of the regularization. The results of a numerical study in a 2D setting are presented.
摘要本文致力于描述两相等温混合物的模型,其中一相服从固体状(即弹性)流变。考虑一个完全欧拉的描述。为了描述固相的应力-应变行为,在亥姆霍兹自由能中加入弹性能项。这一项取决于Almansi应变张量。反过来,应变张量被定义为相应演化方程的解。所考虑的模型属于相场族。形式上,它描述双组分混合物,并使用组分的质量密度作为序参数。所考虑的模型的一个显著特征是它根据准流体动力框架进行了初步正则化。在周期边界条件下,证明了总能量的耗散率。提出了一种基于交错网格的空间耗散半离散(时间连续,空间离散)方案。在没有正则化的情况下,理论结果仍然有效。本文给出了二维环境下的数值研究结果。
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引用次数: 1
期刊
Russian Journal of Numerical Analysis and Mathematical Modelling
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