Condition aux limites de Neumann

Résumé

En mathématiques, une condition aux limites de Neumann (nommée d'après Carl Neumann) est imposée à une équation différentielle ou à une équation aux dérivées partielles lorsque l'on spécifie les valeurs des dérivées que la solution doit vérifier sur les frontières/limites du domaine.

Pour une équation différentielle, par exemple :

:y'' + y = 0~ la condition aux limites de Neumann sur l'intervalle [a, , b] s'exprime par : :y'(a) = \alpha \ \text{et} \ y'(b) = \beta où \alpha et \beta sont deux nombres donnés.

Pour une équation aux dérivées partielles, par exemple :

:\Delta y + y = 0~ où \Delta~ est le Laplacien (opérateur différentiel), la condition aux limites de Neumann sur un domaine \Omega \subset R^n s'exprime par : :\frac{\partial y}{\partial \overrightarrow{n}}(x) = f(x) \quad \forall x \in \partial\Omega où f est une fonction scalaire

Source officielle

https://fr.wikipedia.org/wiki/Condition_aux_limites_de_Neumann

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Regularity of solutions for a fourth-order elliptic system via Conservation law

Changyu Guo

In this paper, we obtain interior Holder continuity for solutions of the fourth-order elliptic system Delta(2)u = Delta(V center dot del u) + div(w del u) + W center dot del u formulated by Lamm and Riviere [Comm. Partial Differential Equations 33 (2008) 245-262]. Boundary continuity is also obtained under a standard Dirichlet or Navier boundary condition. We also use conservation law to establish a weak compactness result which generalizes a result of Riviere for the second-order problem.

2019

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In the paper we deal with shells with non-zero Gaussian curvature. We derive sharp Korn's first (linear geometric rigidity estimate) and second inequalities on that kind of shell for zero or periodic Dirichlet, Neumann, and Robin type boundary conditions. We prove that if the Gaussian curvature is positive, then the optimal constant in the first Korn inequality scales like h, and if the Gaussian curvature is negative, then the Korn constant scales like h(4/3), where h is the thickness of the shell. These results have a classical flavour in continuum mechanics, in particular shell theory. The Korn first inequalities are the linear version of the famous geometric rigidity estimate by Friesecke et al. for plates in Arch Ration Mech Anal 180(2): 183-236, 2006 (where they show that the Korn constant in the nonlinear Korn's first inequality scales like h(2)), extended to shells with nonzero curvature. We also recover the uniform Korn-Poincare inequality proven for "boundary-less" shells by Lewicka and Muller in Annales de l'Institute Henri Poincare (C) Non Linear Anal 28(3): 443-469, 2011 in the setting of our problem. The new estimates can also be applied to find the scaling law for the critical buckling load of the shell under in-plane loads as well as to derive energy scaling laws in the pre-buckled regime. The exponents 1 and 4/3 in the present work appear for the first time in any sharp geometric rigidity estimate.

Springer Verlag2017

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An accurate solution of the wave equation at a fluid-solid interface requires a correct implementation of the boundary condition. Boundary conditions at acousto-elastic interface require continuity of the normal component of particle velocity and traction, whereas the tangential components vanish. The main challenge is to model interface waves, namely the Scholte and leaky Rayleigh waves. In this study, we use a nodal discontinuous Galerkin (dG) finite-element method with the medium discretized using an unstructured uniform triangular meshes. The natural boundary conditions in the dG method is implemented by 1) using an explicit upwind numerical flux and 2) by using an implicit penalty flux and setting the modulus of rigidity of the acoustic medium to zero. The accuracy of these methods are evaluated by comparing the numerical solutions with analytical ones, with source and receiver at and away from the interface. The study shows that the solution obtained from the explicit and implicit boundary conditions produces the correct results. This is due to the fact that the stability of the dG scheme is determined the numerical flux, which also implements the boundary conditions by unifying the numerical solution at shared edges of the elements in an energy stable manner.

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