Lie-symmetry group and modeling in non-isothermal fluid mechanics

Razafindralandy, Dina; Hamdouni, Aziz; Sayed, Nazir Al

doi:10.1016/j.physa.2012.05.063

Cited by 6 publications

(8 citation statements)

References 19 publications

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“…Lie symmetries and related ideas provide a set of powerful tools in the realm of mechanics of materials. While the Lie theory and related methods and ideas applicable to ODE/PDE models have been significantly extended to include nonlocal and approximate symmetries, methods of moving frames and co-frames, and various approaches to produce invariant (in some sense) numerical methods (e.g., [5,6,[64][65][66][67][68][69][70], it us usually the most basic and fundamental ideas and simplest Lie symmetry-based computations that yield most useful results for complex PDE models in nonlinear mechanics. In particular, symmetry ideas prove useful in many different ways to construct a constitutive law of a given material from experimental results, in terms of a functional relation between control variables and additional parameters.…”

Section: Discussion and Concluding Remarksmentioning

confidence: 99%

Symmetry analysis and equivalence transformations for the construction and reduction of constitutive models

Ganghoffer

Rahouadj

Cheviakov

2021

Adv. Model. and Simul. in Eng. Sci.

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A methodology based on Lie analysis is proposed to investigate the mechanical behavior of materials exhibiting experimental master curves. It is based on the idea that the mechanical response of materials is associated with hidden symmetries reflected in the form of the energy functional and the dissipation potential leading to constitutive laws written in the framework of the thermodynamics of irreversible processes. In constitutive modeling, symmetry analysis lets one formulate the response of a material in terms of so-called master curves, and construct rheological models based on a limited number of measurements. The application of symmetry methods leads to model reduction in a double sense: in treating large amounts number of measurements data to reduce them in a form exploitable for the construction of constitutive models, and by exploiting equivalence transformations extending point symmetries to efficiently reduce the number of significant parameters, and thus the computational cost of solving boundary value problems (BVPs). The symmetry framework and related conservation law analysis provide invariance properties of the constitutive models, allowing to predict the influence of a variation of the model parameters on the material response or on the solution of BVPs posed over spatial domains. The first part of the paper is devoted to the presentation of the general methodology proposed in this contribution. Examples of construction of rheological models based on experimental data are given for setting up a reduced model of the uniaxial creep and rupture behaviour of a Chrome-Molybdenum alloy (9Cr1Mo) at different temperatures and stress levels. Constitutive equations for creep and rupture master responses are identified for this alloy, and validated based on experimental data. Equivalence transformations are exemplified in the context of parameter reduction in fully nonlinear anisotropic fiber-reinforced elastic solids.

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Section: Discussion and Concluding Remarksmentioning

confidence: 99%

Symmetry analysis and equivalence transformations for the construction and reduction of constitutive models

Ganghoffer

Rahouadj

Cheviakov

2021

Adv. Model. and Simul. in Eng. Sci.

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“…In turbulence, a link between the symmetry group of the Navier-Stokes equations and Kolmogorov's −5/3 law has been exposed [105]. At the same time, scaling laws of various turbulent flows have been obtained in [106][107][108][109][110][111]. In these article, the classical scaling laws in the literature ( [112][113][114][115]) has also be obtained, but also new ones.…”

Section: Symmetry Group Of the Equations Of A Mechanical Problemmentioning

confidence: 99%

“…An approach developped in [120][121][122][123] enables to build a class of turbulence models which preserve the Lie symmetry group of the Navier-Stokes equations. The works have latter been extended to the anisothermal case [108,[124][125][126]. This approach will be recalled in Section 3.3.…”

Section: Symmetry Group Of the Equations Of A Mechanical Problemmentioning

confidence: 99%

“…An approach in [108,122,126] suggests to construct T s and h s such that the symmetry groups of equations ( 1), summarized in Table 2, are also symmetry groups of equations (1). With this approach, the resolved variables have the same properties as v , p and θ, regarding the consequences of symmetries (self-similar solutions, conservation laws, .…”

Section: Modellingmentioning

confidence: 99%

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Lie groups and continuum mechanics: where do we stand today?

de Saxcé,

Razafindralandy

2024

Comptes Rendus. Mécanique

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“…Symmetries play a fundamental role since they may encode exact model solutions (self-similar, vortex, shock solutions, …), conservation laws via Nother's theorem or physical principles (Galilean invariance, scale invariance, …) [55][56][57][58][59][60][61][62][63][64][65][66][67][68]. They have also been used for modelling purposes such as the establishment of wall laws in turbulent flows or the development of turbulence models [69][70][71][72][73][74][75][76]. It is then important that numerical schemes do not break symmetries if one wishes to reproduce numerically the cited model solutions and properties.…”

Section: Introductionmentioning

confidence: 99%

A review of some geometric integrators

Razafindralandy

Hamdouni

Chhay

2018

Adv. Model. and Simul. in Eng. Sci.

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Some of the most important geometric integrators for both ordinary and partial differential equations are reviewed and illustrated with examples in mechanics. The class of Hamiltonian differential systems is recalled and its symplectic structure is highlighted. The associated natural geometric integrators, known as symplectic integrators, are then presented. In particular, their ability to numerically reproduce first integrals with a bounded error over a long time interval is shown. The extension to partial differential Hamiltonian systems and to multisymplectic integrators is presented afterwards. Next, the class of Lagrangian systems is described. It is highlighted that the variational structure carries both the dynamics (Euler-Lagrange equations) and the conservation laws (Noether's theorem). Integrators preserving the variational structure are constructed by mimicking the calculus of variation at the discrete level. We show that this approach leads to numerical schemes which preserve exactly the energy of the system. After that, the Lie group of local symmetries of partial differential equations is recalled. A construction of Lie-symmetry-preserving numerical scheme is then exposed. This is done via the moving frame method. Applications to Burgers equation are shown. The last part is devoted to the Discrete Exterior Calculus, which is a structure-preserving integrator based on differential geometry and exterior calculus. The efficiency of the approach is demonstrated on fluid flow problems with a passive scalar advection.

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Lie-symmetry group and modeling in non-isothermal fluid mechanics

Cited by 6 publications

References 19 publications

Symmetry analysis and equivalence transformations for the construction and reduction of constitutive models

Symmetry analysis and equivalence transformations for the construction and reduction of constitutive models

Lie groups and continuum mechanics: where do we stand today?

A review of some geometric integrators

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