In this paper, we present a new model for heat transfer in compressible fluid flows. The model is derived from Hamilton’s principle of stationary action in Eulerian coordinates, in a setting where the entropy conservation is recovered as an Euler–Lagrange equation. A sufficient criterion for the hyperbolicity of the model is formulated. The governing equations are asymptotically consistent with the Euler equations for compressible heat conducting fluids, provided the addition of suitable relaxation terms. A study of the Rankine–Hugoniot conditions and Clausius–Duhem inequality is performed for a specific choice of the equation of state. In particular, this reveals that contact discontinuities cannot exist while expansion waves and compression fans are possible solutions to the governing equations. Evidence of these properties is provided on a set of numerical test cases.
{"title":"An Eulerian hyperbolic model for heat transfer derived via Hamilton’s principle: analytical and numerical study","authors":"Firas Dhaouadi, Sergey Gavrilyuk","doi":"10.1098/rspa.2023.0440","DOIUrl":"https://doi.org/10.1098/rspa.2023.0440","url":null,"abstract":"In this paper, we present a new model for heat transfer in compressible fluid flows. The model is derived from Hamilton’s principle of stationary action in Eulerian coordinates, in a setting where the entropy conservation is recovered as an Euler–Lagrange equation. A sufficient criterion for the hyperbolicity of the model is formulated. The governing equations are asymptotically consistent with the Euler equations for compressible heat conducting fluids, provided the addition of suitable relaxation terms. A study of the Rankine–Hugoniot conditions and Clausius–Duhem inequality is performed for a specific choice of the equation of state. In particular, this reveals that contact discontinuities cannot exist while expansion waves and compression fans are possible solutions to the governing equations. Evidence of these properties is provided on a set of numerical test cases.","PeriodicalId":20716,"journal":{"name":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","volume":"19 1","pages":""},"PeriodicalIF":3.5,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140025218","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Bifurcations organize the dynamics of many natural and engineered systems. They induce qualitative and quantitative changes to a system’s dynamics, which can have catastrophic consequences if ignored during design. In this paper, we propose a general computational method to control the local bifurcations of dynamical systems by optimizing design parameters. We define an objective functional that enforces the appearance of local bifurcation points at targeted locations or even encourages their disappearance. The methodology is an efficient alternative to bifurcation tracking techniques capable of handling many design parameters ( >102 ). The method is demonstrated on a Duffing oscillator featuring a hardening cubic nonlinearity and an autonomous van der Pol-Duffing oscillator coupled to a nonlinear tuned vibration absorber. The finite-element model of a clamped-free Euler–Bernoulli beam, coupled with a reduced-order modelling technique, is also used to show the extension to the shape optimization of more complicated structures. Results demonstrate that several local bifurcations of various types can be handled simultaneously by the bifurcation control framework, with both parameter and state target values.
{"title":"Multi-parametric optimization for controlling bifurcation structures","authors":"A. Mélot, E. Denimal, L. Renson","doi":"10.1098/rspa.2023.0505","DOIUrl":"https://doi.org/10.1098/rspa.2023.0505","url":null,"abstract":"Bifurcations organize the dynamics of many natural and engineered systems. They induce qualitative and quantitative changes to a system’s dynamics, which can have catastrophic consequences if ignored during design. In this paper, we propose a general computational method to control the local bifurcations of dynamical systems by optimizing design parameters. We define an objective functional that enforces the appearance of local bifurcation points at targeted locations or even encourages their disappearance. The methodology is an efficient alternative to bifurcation tracking techniques capable of handling many design parameters ( <jats:inline-formula> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>></mml:mo> </mml:mrow> <mml:msup> <mml:mn>10</mml:mn> <mml:mn>2</mml:mn> </mml:msup> </mml:math> </jats:inline-formula> ). The method is demonstrated on a Duffing oscillator featuring a hardening cubic nonlinearity and an autonomous van der Pol-Duffing oscillator coupled to a nonlinear tuned vibration absorber. The finite-element model of a clamped-free Euler–Bernoulli beam, coupled with a reduced-order modelling technique, is also used to show the extension to the shape optimization of more complicated structures. Results demonstrate that several local bifurcations of various types can be handled simultaneously by the bifurcation control framework, with both parameter and state target values.","PeriodicalId":20716,"journal":{"name":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","volume":"47 1","pages":""},"PeriodicalIF":3.5,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140025175","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Paolo Conti, Mengwu Guo, Andrea Manzoni, Attilio Frangi, Steven L. Brunton, J. Nathan Kutz
High-fidelity numerical simulations of partial differential equations (PDEs) given a restricted computational budget can significantly limit the number of parameter configurations considered and/or time window evaluated. Multi-fidelity surrogate modelling aims to leverage less accurate, lower-fidelity models that are computationally inexpensive in order to enhance predictive accuracy when high-fidelity data are scarce. However, low-fidelity models, while often displaying the qualitative solution behaviour, fail to accurately capture fine spatio-temporal and dynamic features of high-fidelity models. To address this shortcoming, we present a data-driven strategy that combines dimensionality reduction with multi-fidelity neural network surrogates. The key idea is to generate a spatial basis by applying proper orthogonal decomposition (POD) to high-fidelity solution snapshots, and approximate the dynamics of the reduced states—time-parameter-dependent expansion coefficients of the POD basis—using a multi-fidelity long short-term memory network. By mapping low-fidelity reduced states to their high-fidelity counterpart, the proposed reduced-order surrogate model enables the efficient recovery of full solution fields over time and parameter variations in a non-intrusive manner. The generality of this method is demonstrated by a collection of PDE problems where the low-fidelity model can be defined by coarser meshes and/or time stepping, as well as by misspecified physical features.
{"title":"Multi-fidelity reduced-order surrogate modelling","authors":"Paolo Conti, Mengwu Guo, Andrea Manzoni, Attilio Frangi, Steven L. Brunton, J. Nathan Kutz","doi":"10.1098/rspa.2023.0655","DOIUrl":"https://doi.org/10.1098/rspa.2023.0655","url":null,"abstract":"High-fidelity numerical simulations of partial differential equations (PDEs) given a restricted computational budget can significantly limit the number of parameter configurations considered and/or time window evaluated. Multi-fidelity surrogate modelling aims to leverage less accurate, lower-fidelity models that are computationally inexpensive in order to enhance predictive accuracy when high-fidelity data are scarce. However, low-fidelity models, while often displaying the qualitative solution behaviour, fail to accurately capture fine spatio-temporal and dynamic features of high-fidelity models. To address this shortcoming, we present a data-driven strategy that combines dimensionality reduction with multi-fidelity neural network surrogates. The key idea is to generate a spatial basis by applying proper orthogonal decomposition (POD) to high-fidelity solution snapshots, and approximate the dynamics of the reduced states—time-parameter-dependent expansion coefficients of the POD basis—using a multi-fidelity <jats:italic>long short-term memory</jats:italic> network. By mapping low-fidelity reduced states to their high-fidelity counterpart, the proposed reduced-order surrogate model enables the efficient recovery of full solution fields over time and parameter variations in a non-intrusive manner. The generality of this method is demonstrated by a collection of PDE problems where the low-fidelity model can be defined by coarser meshes and/or time stepping, as well as by misspecified physical features.","PeriodicalId":20716,"journal":{"name":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","volume":"264 1","pages":""},"PeriodicalIF":3.5,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140025177","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Josep Fontana-McNally, Eva Miranda, Daniel Peralta-Salas
We prove that the correspondence between Reeb and Beltrami vector fields presented in Etnyre & Ghrist (Etnyre, Ghrist 2000 Nonlinearity13 , 441–458 ( doi:10.1088/0951-7715/13/2/306 )) can be made equivariant whenever additional symmetries of the underlying geometric structures are considered. As a corollary of this correspondence, we show that energy levels above the maximum of the potential energy of mechanical Hamiltonian systems can be viewed as stationary fluid flows, though the metric is not prescribed. In particular, we showcase the emblematic example of the n -body problem and focus on the Kepler problem. We explicitly construct a compatible Riemannian metric that makes the Kepler problem of celestial mechanics a stationary fluid flow (of Beltrami type) on a suitable manifold, the Kepler–Euler flow .
{"title":"An equivariant Reeb–Beltrami correspondence and the Kepler–Euler flow","authors":"Josep Fontana-McNally, Eva Miranda, Daniel Peralta-Salas","doi":"10.1098/rspa.2023.0499","DOIUrl":"https://doi.org/10.1098/rspa.2023.0499","url":null,"abstract":"We prove that the correspondence between Reeb and Beltrami vector fields presented in Etnyre & Ghrist (Etnyre, Ghrist 2000 <jats:italic>Nonlinearity</jats:italic> <jats:bold>13</jats:bold> , 441–458 ( <jats:ext-link xmlns:xlink=\"http://www.w3.org/1999/xlink\" ext-link-type=\"uri\" xlink:href=\"http://dx.doi.org/10.1088/0951-7715/13/2/306\">doi:10.1088/0951-7715/13/2/306</jats:ext-link> )) can be made equivariant whenever additional symmetries of the underlying geometric structures are considered. As a corollary of this correspondence, we show that energy levels above the maximum of the potential energy of mechanical Hamiltonian systems can be viewed as stationary fluid flows, though the metric is not prescribed. In particular, we showcase the emblematic example of the <jats:inline-formula> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>n</mml:mi> </mml:math> </jats:inline-formula> -body problem and focus on the Kepler problem. We explicitly construct a compatible Riemannian metric that makes the Kepler problem of celestial mechanics a stationary fluid flow (of Beltrami type) on a suitable manifold, the <jats:italic>Kepler–Euler flow</jats:italic> .","PeriodicalId":20716,"journal":{"name":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","volume":"1 1","pages":""},"PeriodicalIF":3.5,"publicationDate":"2024-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140025113","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Motivated by observations of loop formation and elimination phenomena in elastic filaments subjected to torsion and axial end displacement, we develop a tension–torsion tester to study the slack–extension responses of filaments with varied initial twists. The experiments are conducted by initially twisting the filament by a specific degree and subsequently adjusting the axial end displacement. By continuously monitoring the correlation between torque and filament configuration, we can accurately determine the critical points associated with buckling, loop formation and loop elimination. The interconversion of link, twist and writhe is also tracked in the experiment. We establish a relation between torque and end displacement of filament that provides an insight into the conditions leading to instabilities. Three buckling criteria are also evaluated in the context of measurement data. Simulations on the slack and extension of the twisted filaments based on the Cosserat rod theory are performed, providing reliable predictions of the configuration evolution. Our results highlight the importance of twist and slack in torsional buckling processes, providing guidance for the control of systems with twisted slender structures.
{"title":"Toward a further understanding of the loop formation and elimination in twisted filament: experiments and validation","authors":"Jiongjiong Hu, Jiahui Teng, Lei Liu, Dabiao Liu","doi":"10.1098/rspa.2023.0719","DOIUrl":"https://doi.org/10.1098/rspa.2023.0719","url":null,"abstract":"Motivated by observations of loop formation and elimination phenomena in elastic filaments subjected to torsion and axial end displacement, we develop a tension–torsion tester to study the slack–extension responses of filaments with varied initial twists. The experiments are conducted by initially twisting the filament by a specific degree and subsequently adjusting the axial end displacement. By continuously monitoring the correlation between torque and filament configuration, we can accurately determine the critical points associated with buckling, loop formation and loop elimination. The interconversion of link, twist and writhe is also tracked in the experiment. We establish a relation between torque and end displacement of filament that provides an insight into the conditions leading to instabilities. Three buckling criteria are also evaluated in the context of measurement data. Simulations on the slack and extension of the twisted filaments based on the Cosserat rod theory are performed, providing reliable predictions of the configuration evolution. Our results highlight the importance of twist and slack in torsional buckling processes, providing guidance for the control of systems with twisted slender structures.","PeriodicalId":20716,"journal":{"name":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","volume":"171 1","pages":""},"PeriodicalIF":3.5,"publicationDate":"2024-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140025192","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A novel methodology for determining the singular point of an enveloped surface is put forward. Unlike some existing methods, the presented method starts directly from the equation of the enveloped surface instead of that of the generating surface, and it is thus called a direct method. The calculation for the normal vector of the enveloped surface is well simplified with the help of the moving frame approach, which makes the presented method feasible. The singularity condition equation is extracted by using the theory of linear algebra. For singular points with different properties, proper solving techniques are established, including resultant elimination and simple elimination. Applying the developed method, the undercutting characteristics of the Archimedes worm wheel are investigated from the perspective of spatial meshing. The numerical results demonstrate that the worm wheel generally has one undercutting limit line, whose trend is along the tooth width of the wheel. Locating on one side of the tooth surface and near the tooth root is a dangerous part of the worm wheel undercutting. The proposed method is beneficial for the development of gear meshing science.
{"title":"Direct method to determine singular point of enveloped surface and its application to worm wheel tooth surface","authors":"Jian Cui, Yaping Zhao, Qingxiang Meng, Gongfa Li","doi":"10.1098/rspa.2023.0369","DOIUrl":"https://doi.org/10.1098/rspa.2023.0369","url":null,"abstract":"A novel methodology for determining the singular point of an enveloped surface is put forward. Unlike some existing methods, the presented method starts directly from the equation of the enveloped surface instead of that of the generating surface, and it is thus called a direct method. The calculation for the normal vector of the enveloped surface is well simplified with the help of the moving frame approach, which makes the presented method feasible. The singularity condition equation is extracted by using the theory of linear algebra. For singular points with different properties, proper solving techniques are established, including resultant elimination and simple elimination. Applying the developed method, the undercutting characteristics of the Archimedes worm wheel are investigated from the perspective of spatial meshing. The numerical results demonstrate that the worm wheel generally has one undercutting limit line, whose trend is along the tooth width of the wheel. Locating on one side of the tooth surface and near the tooth root is a dangerous part of the worm wheel undercutting. The proposed method is beneficial for the development of gear meshing science.","PeriodicalId":20716,"journal":{"name":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","volume":"227 1","pages":""},"PeriodicalIF":3.5,"publicationDate":"2024-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140025397","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The buckling instability of cylindrical shells under axial compression has been one of the most renowned problems in structural engineering for several decades. Many pioneering works in the twentieth century have provided insights into understanding the shells’ infamous imperfection sensitivity and led to reliability-based designs. However, a recent surge in numerical studies of the snaking phenomenon explores the development of a localized stable post-buckling mode in axially compressed cylindrical shells. Hitherto, none of the experimental studies report on the evolution of azimuthal snaking. In this work, experimental studies are carried out with the objective of revealing the snaking phenomenon. The axial compression experiments are performed on 3D-printed shells made of thermoplastic polyurethane (TPU). The work’s novelty lies in the usage of TPU shells for slowing down the propagation of circumferential dimples and making it feasible to capture them using photogrammetry. Despite the match between the experimental and numerical mode shapes, the experiments reveal multiple routes for the snaking sequence. Furthermore, mode transitions such as reduction in circumferential wave number and transformation of symmetric mode into an asymmetric one are observed. These experimental results provide insights into the localized phenomenon of snaking and validate numerical solutions.
几十年来，圆柱形壳体在轴向压缩下的屈曲不稳定性一直是结构工程领域最著名的问题之一。二十世纪的许多开创性工作为了解壳体的缺陷敏感性提供了见解，并促成了基于可靠性的设计。然而，最近对蛇形现象的数值研究激增，探索了轴向压缩圆柱壳的局部稳定后屈曲模式的发展。迄今为止，还没有任何一项实验研究报告了方位蛇行的演变过程。在这项工作中，为了揭示蛇行现象，我们进行了实验研究。轴向压缩实验是在热塑性聚氨酯（TPU）制成的 3D 打印外壳上进行的。这项工作的新颖之处在于使用热塑性聚氨酯外壳来减缓圆周凹痕的传播，并使使用摄影测量捕捉凹痕成为可行。尽管实验和数值模式形状相吻合，但实验揭示了蛇形序列的多种路径。此外，还观察到模式转换，如圆周波数减少以及对称模式转变为不对称模式。这些实验结果提供了对局部蛇行现象的见解，并验证了数值解决方案。
{"title":"Experimental studies on snaking in 3D-printed cylindrical shells under axial compression using photogrammetry","authors":"V. Ravulapalli, G. Raju, V. Narayanamurthy","doi":"10.1098/rspa.2023.0631","DOIUrl":"https://doi.org/10.1098/rspa.2023.0631","url":null,"abstract":"The buckling instability of cylindrical shells under axial compression has been one of the most renowned problems in structural engineering for several decades. Many pioneering works in the twentieth century have provided insights into understanding the shells’ infamous imperfection sensitivity and led to reliability-based designs. However, a recent surge in numerical studies of the snaking phenomenon explores the development of a localized stable post-buckling mode in axially compressed cylindrical shells. Hitherto, none of the experimental studies report on the evolution of azimuthal snaking. In this work, experimental studies are carried out with the objective of revealing the snaking phenomenon. The axial compression experiments are performed on 3D-printed shells made of thermoplastic polyurethane (TPU). The work’s novelty lies in the usage of TPU shells for slowing down the propagation of circumferential dimples and making it feasible to capture them using photogrammetry. Despite the match between the experimental and numerical mode shapes, the experiments reveal multiple routes for the snaking sequence. Furthermore, mode transitions such as reduction in circumferential wave number and transformation of symmetric mode into an asymmetric one are observed. These experimental results provide insights into the localized phenomenon of snaking and validate numerical solutions.","PeriodicalId":20716,"journal":{"name":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","volume":"84 1","pages":""},"PeriodicalIF":3.5,"publicationDate":"2024-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140025034","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
M. Harrison, M. Riva, M. Mousavi Nezhad, A. Guadagnini
We derive analytical formulations relating the spatial covariance ( CY ) of (log-transformed) hydraulic conductivities to auto- and cross-covariances of porosity ( ϕ ) and representative soil particle sizes within the framework of the classical Terzaghi model. The latter provides an empirical relationship which is widely used to obtain conductivity estimates. We frame the study within recent stochastic approaches and conceptualize appropriate transformations of ϕ and representative soil particle size as Generalized Sub-Gaussian (GSG) spatially cross-correlated random processes. Consistency of the theoretical framework against sample distributions of ϕ and particle size is assessed through the analysis of field data. A perturbation-based approach yields workable expressions of CY upon truncating the otherwise exact analytical solution at given orders of approximations. Our analytical (truncated) log-conductivity covariance is in agreement with its Monte Carlo-based counterpart. A Global Sensitivity Analysis relying on classical Sobol indices quantifies the relative importance of all parameters embedded in the formulation of CY . We show that parameters driving the GSG nature of the distribution of (transformed) porosity are key to the main features of CY . We also document the relevance of properly capturing emergences of possible cross-correlations between ϕ and representative particle size to reconstruct conductivity fields.
在经典特尔扎吉模型的框架内，我们推导出了（对数变换）水力传导性的空间协方差（C Y）与孔隙度（j）和代表性土壤颗粒尺寸的自协方差和交叉协方差的相关分析公式。后者提供了一种经验关系，被广泛用于估算导流系数。我们将这项研究纳入最新的随机方法，并将 ϕ 和代表性土壤粒径的适当变换概念化为广义子高斯（GSG）空间交叉相关随机过程。通过分析实地数据，评估了理论框架与 ϕ 和粒径样本分布的一致性。采用基于扰动的方法，在给定近似阶数下截断原本精确的分析解，即可得到可行的 C Y 表达式。我们的分析（截断）对数电导协方差与基于蒙特卡洛的协方差一致。全局敏感性分析依赖于经典的索布尔指数，量化了 C Y 公式中所有参数的相对重要性。我们表明，驱动（转换）孔隙度分布 GSG 性质的参数是 C Y 主要特征的关键。我们还证明了正确捕捉 ϕ 与代表性粒度之间可能存在的交叉相关性对重建电导场的重要性。
{"title":"Estimation of auto-covariance of log hydraulic conductivity from Generalized Sub-Gaussian porosity and particle size random fields","authors":"M. Harrison, M. Riva, M. Mousavi Nezhad, A. Guadagnini","doi":"10.1098/rspa.2023.0476","DOIUrl":"https://doi.org/10.1098/rspa.2023.0476","url":null,"abstract":"We derive analytical formulations relating the spatial covariance ( <jats:inline-formula> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>Y</mml:mi> </mml:msub> </mml:math> </jats:inline-formula> ) of (log-transformed) hydraulic conductivities to auto- and cross-covariances of porosity ( <jats:inline-formula> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>ϕ</mml:mi> </mml:math> </jats:inline-formula> ) and representative soil particle sizes within the framework of the classical Terzaghi model. The latter provides an empirical relationship which is widely used to obtain conductivity estimates. We frame the study within recent stochastic approaches and conceptualize appropriate transformations of <jats:inline-formula> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>ϕ</mml:mi> </mml:math> </jats:inline-formula> and representative soil particle size as Generalized Sub-Gaussian (GSG) spatially cross-correlated random processes. Consistency of the theoretical framework against sample distributions of <jats:inline-formula> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>ϕ</mml:mi> </mml:math> </jats:inline-formula> and particle size is assessed through the analysis of field data. A perturbation-based approach yields workable expressions of <jats:inline-formula> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>Y</mml:mi> </mml:msub> </mml:math> </jats:inline-formula> upon truncating the otherwise exact analytical solution at given orders of approximations. Our analytical (truncated) log-conductivity covariance is in agreement with its Monte Carlo-based counterpart. A Global Sensitivity Analysis relying on classical Sobol indices quantifies the relative importance of all parameters embedded in the formulation of <jats:inline-formula> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>Y</mml:mi> </mml:msub> </mml:math> </jats:inline-formula> . We show that parameters driving the GSG nature of the distribution of (transformed) porosity are key to the main features of <jats:inline-formula> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>Y</mml:mi> </mml:msub> </mml:math> </jats:inline-formula> . We also document the relevance of properly capturing emergences of possible cross-correlations between <jats:inline-formula> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>ϕ</mml:mi> </mml:math> </jats:inline-formula> and representative particle size to reconstruct conductivity fields.","PeriodicalId":20716,"journal":{"name":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","volume":"5 1","pages":""},"PeriodicalIF":3.5,"publicationDate":"2024-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140025114","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The problem of thermal convection in a layer of viscous incompressible fluid is analysed. The heat flux law is taken to be one of Cattaneo type. The time derivative of the heat flux is allowed to be a material derivative, or a general objective derivative. It is shown that only one objective derivative leads to results consistent with what one expects in real life. This objective derivative leads to a Cattaneo–Christov theory, and the results for linear instability theory are in agreement with those for a material derivative. It is further shown that none of the theories allow a standard nonlinear, energy stability analysis. A further heat flux due to P.M. Mariano is added and then an analysis is performed for stationary convection, oscillatory convection, and fully nonlinear theory. For the material derivative case, the analysis proceeds and global nonlinear stability is achieved. For Cattaneo–Christov theory, it appears necessary to add a regularization term in the equation for the heat flux, and even then the analysis only works in two space dimensions, and is conditional upon the size of the initial data. For the three-dimensional situation, it is shown how a nonlinear stability analysis may be achieved with a Navier–Stokes–Voigt fluid rather than a Navier–Stokes one.
{"title":"Thermal convection with a Cattaneo heat flux model","authors":"M. Gentile, B. Straughan","doi":"10.1098/rspa.2023.0771","DOIUrl":"https://doi.org/10.1098/rspa.2023.0771","url":null,"abstract":"The problem of thermal convection in a layer of viscous incompressible fluid is analysed. The heat flux law is taken to be one of Cattaneo type. The time derivative of the heat flux is allowed to be a material derivative, or a general objective derivative. It is shown that only one objective derivative leads to results consistent with what one expects in real life. This objective derivative leads to a Cattaneo–Christov theory, and the results for linear instability theory are in agreement with those for a material derivative. It is further shown that none of the theories allow a standard nonlinear, energy stability analysis. A further heat flux due to P.M. Mariano is added and then an analysis is performed for stationary convection, oscillatory convection, and fully nonlinear theory. For the material derivative case, the analysis proceeds and global nonlinear stability is achieved. For Cattaneo–Christov theory, it appears necessary to add a regularization term in the equation for the heat flux, and even then the analysis only works in two space dimensions, and is conditional upon the size of the initial data. For the three-dimensional situation, it is shown how a nonlinear stability analysis may be achieved with a Navier–Stokes–Voigt fluid rather than a Navier–Stokes one.","PeriodicalId":20716,"journal":{"name":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","volume":"1 1","pages":""},"PeriodicalIF":3.5,"publicationDate":"2024-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140025045","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
I. I. Mullyadzhanov, A. S. Gudko, R. I. Mullyadzhanov, A. A. Gelash
We consider the model of the focusing one-dimensional nonlinear Schrödinger equation (fNLSE) in the presence of an unstable constant background, which exhibits coherent solitary wave structures—breathers. Within the inverse scattering transform (IST) method, we study the problem of the scattering data numerical computation for a broad class of breathers localized in space. Such a direct scattering transform (DST) procedure requires a numerical solution of the auxiliary Zakharov–Shabat system with boundary conditions corresponding to the background. To find the solution, we compute the transfer matrix using the second-order Boffetta–Osborne approach and recently developed high-order numerical schemes based on the Magnus expansion. To recover the scattering data of breathers, we derive analytical relations between the scattering coefficients and the transfer matrix elements. Then we construct localized single- and multi-breather solutions and verify the developed numerical approach by recovering the complete set of scattering data with the built-in accuracy providing the information about the amplitude, velocity, phase and position of each breather. To combine the conventional IST approach with the efficient dressing method for multi-breather solutions, we derive the exact relation between the parameters of breathers in these two frameworks.
我们考虑了存在不稳定恒定背景的聚焦一维非线性薛定谔方程（fNLSE）模型，该模型表现出相干孤波结构--呼吸器。在反向散射变换（IST）方法中，我们研究了在空间中定位的一大类呼吸器的散射数据数值计算问题。这种直接散射变换（DST）程序需要对带有与背景相对应的边界条件的辅助 Zakharov-Shabat 系统进行数值求解。为了求解，我们使用二阶 Boffetta-Osborne 方法和最近开发的基于马格努斯展开的高阶数值方案计算传递矩阵。为了恢复呼吸器的散射数据，我们推导出了散射系数与传递矩阵元素之间的分析关系。然后，我们构建了局部单呼吸器和多呼吸器解决方案，并通过提供每个呼吸器的振幅、速度、相位和位置信息的内置精度恢复整套散射数据来验证所开发的数值方法。为了将传统的 IST 方法与多呼吸器解的高效修整方法相结合，我们推导出了这两种框架中呼吸器参数之间的精确关系。
{"title":"Numerical direct scattering transform for breathers","authors":"I. I. Mullyadzhanov, A. S. Gudko, R. I. Mullyadzhanov, A. A. Gelash","doi":"10.1098/rspa.2023.0529","DOIUrl":"https://doi.org/10.1098/rspa.2023.0529","url":null,"abstract":"We consider the model of the focusing one-dimensional nonlinear Schrödinger equation (fNLSE) in the presence of an unstable constant background, which exhibits coherent solitary wave structures—breathers. Within the inverse scattering transform (IST) method, we study the problem of the scattering data numerical computation for a broad class of breathers localized in space. Such a direct scattering transform (DST) procedure requires a numerical solution of the auxiliary Zakharov–Shabat system with boundary conditions corresponding to the background. To find the solution, we compute the transfer matrix using the second-order Boffetta–Osborne approach and recently developed high-order numerical schemes based on the Magnus expansion. To recover the scattering data of breathers, we derive analytical relations between the scattering coefficients and the transfer matrix elements. Then we construct localized single- and multi-breather solutions and verify the developed numerical approach by recovering the complete set of scattering data with the built-in accuracy providing the information about the amplitude, velocity, phase and position of each breather. To combine the conventional IST approach with the efficient dressing method for multi-breather solutions, we derive the exact relation between the parameters of breathers in these two frameworks.","PeriodicalId":20716,"journal":{"name":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","volume":"47 1","pages":""},"PeriodicalIF":3.5,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140025130","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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