Optimal-order convergence in the $H^{1}$ norm is proved for an arbitrary Lagrangian–Eulerian (ALE) interface tracking finite element method (FEM) for the sharp interface model of two-phase Navier–Stokes flow without surface tension, using high-order curved evolving mesh. In this method, the interfacial mesh points move with the fluid’s velocity to track the sharp interface between two phases of the fluid, and the interior mesh points move according to a harmonic extension of the interface velocity. The error of the semidiscrete ALE interface tracking FEM is shown to be $O(h^{k})$ in the $L^infty (0, T; H^{1}(varOmega ))$ norm for the Taylor–Hood finite elements of degree $k geqslant 2$. This high-order convergence is achieved by utilizing the piecewise smoothness of the solution on each subdomain occupied by one phase of the fluid, relying on a low global regularity on the entire moving domain. Numerical experiments illustrate and complement the theoretical results.
{"title":"Optimal convergence of the arbitrary Lagrangian–Eulerian interface tracking method for two-phase Navier–Stokes flow without surface tension","authors":"Buyang Li, Shu Ma, Weifeng Qiu","doi":"10.1093/imanum/draf003","DOIUrl":"https://doi.org/10.1093/imanum/draf003","url":null,"abstract":"Optimal-order convergence in the $H^{1}$ norm is proved for an arbitrary Lagrangian–Eulerian (ALE) interface tracking finite element method (FEM) for the sharp interface model of two-phase Navier–Stokes flow without surface tension, using high-order curved evolving mesh. In this method, the interfacial mesh points move with the fluid’s velocity to track the sharp interface between two phases of the fluid, and the interior mesh points move according to a harmonic extension of the interface velocity. The error of the semidiscrete ALE interface tracking FEM is shown to be $O(h^{k})$ in the $L^infty (0, T; H^{1}(varOmega ))$ norm for the Taylor–Hood finite elements of degree $k geqslant 2$. This high-order convergence is achieved by utilizing the piecewise smoothness of the solution on each subdomain occupied by one phase of the fluid, relying on a low global regularity on the entire moving domain. Numerical experiments illustrate and complement the theoretical results.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"1 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143736499","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We construct and analyse finite element approximations of the Einstein tensor in dimension $N ge 3$. We focus on the setting where a smooth Riemannian metric tensor $g$ on a polyhedral domain $varOmega subset mathbb{R}^{N}$ has been approximated by a piecewise polynomial metric $g_{h}$ on a simplicial triangulation $mathcal{T}$ of $varOmega $ having maximum element diameter $h$. We assume that $g_{h}$ possesses single-valued tangential–tangential components on every codimension-$1$ simplex in $mathcal{T}$. Such a metric is not classically differentiable in general, but it turns out that one can still attribute meaning to its Einstein curvature in a distributional sense. We study the convergence of the distributional Einstein curvature of $g_{h}$ to the Einstein curvature of $g$ under refinement of the triangulation. We show that in the $H^{-2}(varOmega )$-norm this convergence takes place at a rate of $O(h^{r+1})$ when $g_{h}$ is an optimal-order interpolant of $g$ that is piecewise polynomial of degree $r ge 1$. We provide numerical evidence to support this claim. In the process of proving our convergence results we derive a few formulas for the evolution of certain geometric quantities under deformations of the metric.
我们构造并分析了维度$N ge $ 3的爱因斯坦张量的有限元近似。我们关注的是在多面体域$varOmega 子集$ mathbb{R}^{N}$上的光滑黎曼度量张量$g$被$varOmega $的简单三角剖分$mathcal{T}$上的分段多项式度量$g_{h}$所近似，其最大元径$h$。我们假设$g_{h}$在$mathcal{T}$的每一个余维上都具有单值切-切分量-$1$单纯形。一般来说，这样的度规不是经典可微的，但事实证明，人们仍然可以在分布意义上赋予它的爱因斯坦曲率意义。在三角剖分的细化下，研究了$g_{h}$的分布爱因斯坦曲率收敛于$g$的爱因斯坦曲率。我们证明了在$H^{-2}(varOmega)$范数中，当$g_{H}$是$g$的最优阶插值，且$g$是次$r ge $的分段多项式时，这种收敛以$O(H^{r+1})$的速率发生。我们提供了数字证据来支持这一说法。在证明收敛性结果的过程中，我们导出了若干几何量在度规变形下的演化公式。
{"title":"Finite element approximation of the Einstein tensor","authors":"Evan S Gawlik, Michael Neunteufel","doi":"10.1093/imanum/draf004","DOIUrl":"https://doi.org/10.1093/imanum/draf004","url":null,"abstract":"We construct and analyse finite element approximations of the Einstein tensor in dimension $N ge 3$. We focus on the setting where a smooth Riemannian metric tensor $g$ on a polyhedral domain $varOmega subset mathbb{R}^{N}$ has been approximated by a piecewise polynomial metric $g_{h}$ on a simplicial triangulation $mathcal{T}$ of $varOmega $ having maximum element diameter $h$. We assume that $g_{h}$ possesses single-valued tangential–tangential components on every codimension-$1$ simplex in $mathcal{T}$. Such a metric is not classically differentiable in general, but it turns out that one can still attribute meaning to its Einstein curvature in a distributional sense. We study the convergence of the distributional Einstein curvature of $g_{h}$ to the Einstein curvature of $g$ under refinement of the triangulation. We show that in the $H^{-2}(varOmega )$-norm this convergence takes place at a rate of $O(h^{r+1})$ when $g_{h}$ is an optimal-order interpolant of $g$ that is piecewise polynomial of degree $r ge 1$. We provide numerical evidence to support this claim. In the process of proving our convergence results we derive a few formulas for the evolution of certain geometric quantities under deformations of the metric.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"72 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143736497","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Complex Gaussian quadrature rules for oscillatory integral transforms have the advantage that they can achieve optimal asymptotic order. However, their existence for Hankel transform can only be guaranteed when the order of the transform belongs to $[0,1/2]$. In this paper we introduce a new family of Gaussian quadrature rules for Hankel transforms of integer order. We show that, if adding certain value and derivative information at the left endpoint, then complex generalized Gauss–Radau quadrature rules that guarantee existence can be constructed and their nodes and weights can be calculated from a half-size Gaussian quadrature rule with respect to the generalized Prudnikov weight function. Orthogonal polynomials that are closely related to such quadrature rules are investigated and their existence for even degrees is proved. Numerical experiments are presented to show the performance of the proposed rules.
{"title":"Complex generalized Gauss–Radau quadrature rules for Hankel transforms of integer order","authors":"Haiyong Wang, Menghan Wu","doi":"10.1093/imanum/draf007","DOIUrl":"https://doi.org/10.1093/imanum/draf007","url":null,"abstract":"Complex Gaussian quadrature rules for oscillatory integral transforms have the advantage that they can achieve optimal asymptotic order. However, their existence for Hankel transform can only be guaranteed when the order of the transform belongs to $[0,1/2]$. In this paper we introduce a new family of Gaussian quadrature rules for Hankel transforms of integer order. We show that, if adding certain value and derivative information at the left endpoint, then complex generalized Gauss–Radau quadrature rules that guarantee existence can be constructed and their nodes and weights can be calculated from a half-size Gaussian quadrature rule with respect to the generalized Prudnikov weight function. Orthogonal polynomials that are closely related to such quadrature rules are investigated and their existence for even degrees is proved. Numerical experiments are presented to show the performance of the proposed rules.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"61 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143677760","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the numerical approximation for a class of radial Dunkl processes corresponding to arbitrary (reduced) root systems in $mathbb{R}^{d}$. This class contains well-known processes such as Bessel processes, Dyson’s Brownian motions and square root of Wishart processes. We propose some semi-implicit and truncated Euler–Maruyama schemes for radial Dunkl processes and study their convergence rate with respect to the $L^{p}$-sup norm.
{"title":"Numerical schemes for radial Dunkl processes","authors":"Hoang-Long Ngo, Dai Taguchi","doi":"10.1093/imanum/draf005","DOIUrl":"https://doi.org/10.1093/imanum/draf005","url":null,"abstract":"We consider the numerical approximation for a class of radial Dunkl processes corresponding to arbitrary (reduced) root systems in $mathbb{R}^{d}$. This class contains well-known processes such as Bessel processes, Dyson’s Brownian motions and square root of Wishart processes. We propose some semi-implicit and truncated Euler–Maruyama schemes for radial Dunkl processes and study their convergence rate with respect to the $L^{p}$-sup norm.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"21 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143607954","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove rigorous a-posteriori error estimates for first-order finite-volume approximations of nonlinear systems of hyperbolic conservation laws in one spatial dimension. Our estimators rely on recent stability results by Bressan, Chiri and Shen, a new way to localize residuals and a novel method to compute negative-order norms of these local residuals. Computing negative-order norms becomes possible by suitably projecting test functions onto a finite dimensional space. Numerical experiments show that the error estimator converges with the rate predicted by a-priori error estimates.
{"title":"A-posteriori error estimates for systems of hyperbolic conservation laws via computing negative norms of local residuals","authors":"Jan Giesselmann, Aleksey Sikstel","doi":"10.1093/imanum/drae111","DOIUrl":"https://doi.org/10.1093/imanum/drae111","url":null,"abstract":"We prove rigorous a-posteriori error estimates for first-order finite-volume approximations of nonlinear systems of hyperbolic conservation laws in one spatial dimension. Our estimators rely on recent stability results by Bressan, Chiri and Shen, a new way to localize residuals and a novel method to compute negative-order norms of these local residuals. Computing negative-order norms becomes possible by suitably projecting test functions onto a finite dimensional space. Numerical experiments show that the error estimator converges with the rate predicted by a-priori error estimates.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"10 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143599894","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study a first-order system formulation of the (acoustic) wave equation and prove that the operator of this system is an isomorphism from an appropriately defined graph space to $L^{2}$. The results rely on well-posedness and stability of the weak and ultraweak formulation of the second-order wave equation. As an application, we define and analyze a space-time least-squares finite element method for solving the wave equation. Some numerical examples for one- and two-dimensional spatial domains are presented.
{"title":"Well-posedness of first-order acoustic wave equations and space-time finite element approximation","authors":"Thomas Führer, Roberto González, Michael Karkulik","doi":"10.1093/imanum/drae104","DOIUrl":"https://doi.org/10.1093/imanum/drae104","url":null,"abstract":"We study a first-order system formulation of the (acoustic) wave equation and prove that the operator of this system is an isomorphism from an appropriately defined graph space to $L^{2}$. The results rely on well-posedness and stability of the weak and ultraweak formulation of the second-order wave equation. As an application, we define and analyze a space-time least-squares finite element method for solving the wave equation. Some numerical examples for one- and two-dimensional spatial domains are presented.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"86 1 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143599893","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we analyze a nonconforming virtual element method to approximate the eigenfunctions and eigenvalues of the two dimensional Oseen eigenvalue problem. The spaces under consideration lead to a divergence-free method that is capable to capture properly the divergence at discrete level and the eigenvalues and eigenfunctions. Under the compact theory for operators, we prove convergence and error estimates for the method. By employing the theory of compact operators, we recover the double order of convergence of the spectrum. Finally, we present numerical tests to assess the performance of the proposed numerical scheme.
{"title":"A noncoforming virtual element approximation for the Oseen eigenvalue problem","authors":"Dibyendu Adak, Felipe Lepe, Gonzalo Rivera","doi":"10.1093/imanum/drae108","DOIUrl":"https://doi.org/10.1093/imanum/drae108","url":null,"abstract":"In this paper, we analyze a nonconforming virtual element method to approximate the eigenfunctions and eigenvalues of the two dimensional Oseen eigenvalue problem. The spaces under consideration lead to a divergence-free method that is capable to capture properly the divergence at discrete level and the eigenvalues and eigenfunctions. Under the compact theory for operators, we prove convergence and error estimates for the method. By employing the theory of compact operators, we recover the double order of convergence of the spectrum. Finally, we present numerical tests to assess the performance of the proposed numerical scheme.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"12 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143599895","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In Heimann, Lehrenfeld, and Preuß (2023, SIAM J. Sci. Comp., 45(2), B139–B165), new geometrically unfitted space–time Finite Element methods for partial differential equations posed on moving domains of higher-order accuracy in space and time have been introduced. For geometrically higher-order accuracy a parametric mapping on a background space–time tensor-product mesh has been used. In this paper, we concentrate on the geometrical accuracy of the approximation and derive rigorous bounds for the distance between the realized and an ideal mapping in different norms and derive results for the space–time regularity of the parametric mapping. These results are important and lay the ground for the error analysis of corresponding unfitted space–time finite element methods.
[2] [ei] Heimann, Lehrenfeld, and Preuß[2023]。介绍了求解高阶精度运动域上偏微分方程的新的几何非拟合时空有限元方法（p., 45(2), B139-B165）。为了提高几何精度，采用了背景时空张量积网格上的参数映射。本文主要讨论了这种近似的几何精度，推导了在不同范数下实现映射与理想映射之间的距离的严格界限，并推导了参数映射的时空正则性的结果。这些结果具有重要意义，为相应的非拟合时空有限元方法的误差分析奠定了基础。
{"title":"Geometry error analysis of a parametric mapping for higher order unfitted space–time methods","authors":"Fabian Heimann, Christoph Lehrenfeld","doi":"10.1093/imanum/drae098","DOIUrl":"https://doi.org/10.1093/imanum/drae098","url":null,"abstract":"In Heimann, Lehrenfeld, and Preuß (2023, SIAM J. Sci. Comp., 45(2), B139–B165), new geometrically unfitted space–time Finite Element methods for partial differential equations posed on moving domains of higher-order accuracy in space and time have been introduced. For geometrically higher-order accuracy a parametric mapping on a background space–time tensor-product mesh has been used. In this paper, we concentrate on the geometrical accuracy of the approximation and derive rigorous bounds for the distance between the realized and an ideal mapping in different norms and derive results for the space–time regularity of the parametric mapping. These results are important and lay the ground for the error analysis of corresponding unfitted space–time finite element methods.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"17 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143599897","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Francis R A Aznaran, Martina Bukač, Boris Muha, Abner J Salgado
We consider the interaction between a poroelastic structure, described using the Biot model in primal form, and a free-flowing fluid, modelled with the time-dependent incompressible Stokes equations. We propose a diffuse interface model in which a phase field function is used to write each integral in the weak formulation of the coupled problem on the entire domain containing both the Stokes and Biot regions. The phase field function continuously transitions from one to zero over a diffuse region of width $mathcal{O}(varepsilon)$ around the interface; this allows the weak forms to be integrated uniformly across the domain, and obviates tracking the subdomains or the interface between them. We prove convergence in weighted norms of a finite element discretization of the diffuse interface model to the continuous diffuse model; here the weight is a power of the distance to the diffuse interface. We, in turn, prove convergence of the continuous diffuse model to the standard, sharp interface, model. Numerical examples verify the proven error estimates, and illustrate application of the method to fluid flow through a complex network, describing blood circulation in the circle of Willis.
{"title":"Analysis and finite element approximation of a diffuse interface approach to the Stokes–Biot coupling","authors":"Francis R A Aznaran, Martina Bukač, Boris Muha, Abner J Salgado","doi":"10.1093/imanum/draf002","DOIUrl":"https://doi.org/10.1093/imanum/draf002","url":null,"abstract":"We consider the interaction between a poroelastic structure, described using the Biot model in primal form, and a free-flowing fluid, modelled with the time-dependent incompressible Stokes equations. We propose a diffuse interface model in which a phase field function is used to write each integral in the weak formulation of the coupled problem on the entire domain containing both the Stokes and Biot regions. The phase field function continuously transitions from one to zero over a diffuse region of width $mathcal{O}(varepsilon)$ around the interface; this allows the weak forms to be integrated uniformly across the domain, and obviates tracking the subdomains or the interface between them. We prove convergence in weighted norms of a finite element discretization of the diffuse interface model to the continuous diffuse model; here the weight is a power of the distance to the diffuse interface. We, in turn, prove convergence of the continuous diffuse model to the standard, sharp interface, model. Numerical examples verify the proven error estimates, and illustrate application of the method to fluid flow through a complex network, describing blood circulation in the circle of Willis.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"54 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143599896","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce the definition of tensorized block rational Krylov subspace and its relation with multivariate rational functions, extending the formulation of tensorized Krylov subspaces introduced in Kressner, D. & Tobler, C. (2010) Krylov subspace methods for linear systems with tensor product structure. SIAM J. Matrix Anal. Appl., 31,$1688$–$1714$. Moreover, we develop methods for the solution of tensor Sylvester equations with low multilinear or tensor train rank, based on projection onto a tensor block rational Krylov subspace. We provide a convergence analysis, some strategies for pole selection and techniques to efficiently compute the residual.
我们介绍了张量块有理克雷洛夫子空间的定义及其与多元有理函数的关系，扩展了 Kressner, D. & Tobler, C. (2010) Krylov subspace methods for linear systems with tensor product structure 中介绍的张量克雷洛夫子空间的表述。SIAM J. Matrix Anal.应用，31,$1688$-$1714$。此外，我们开发了基于投影到张量块有理 Krylov 子空间的低多线性或张量列车秩的张量 Sylvester 方程求解方法。我们提供了收敛性分析、极点选择的一些策略以及有效计算残差的技术。
{"title":"Tensorized block rational Krylov methods for tensor Sylvester equations","authors":"Angelo A Casulli","doi":"10.1093/imanum/draf001","DOIUrl":"https://doi.org/10.1093/imanum/draf001","url":null,"abstract":"We introduce the definition of tensorized block rational Krylov subspace and its relation with multivariate rational functions, extending the formulation of tensorized Krylov subspaces introduced in Kressner, D. & Tobler, C. (2010) Krylov subspace methods for linear systems with tensor product structure. SIAM J. Matrix Anal. Appl., 31,$1688$–$1714$. Moreover, we develop methods for the solution of tensor Sylvester equations with low multilinear or tensor train rank, based on projection onto a tensor block rational Krylov subspace. We provide a convergence analysis, some strategies for pole selection and techniques to efficiently compute the residual.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"10 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2025-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143546177","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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