Gilles Leborgne | Searcharxiv

arXiv2023

Inf-sup condition and locking: Understanding and circumventing. Stokes, Laplacian, bi-Laplacian, Kirchhoff--Love and Mindlin--Reissner locking type, boundary conditions

The inf-sup condition, also called the Ladyzhenskaya--Babu\v ska--Brezzi (LBB) condition, ensures the existence, uniqueness and well-posedness of a saddle point problem, relative to a partial differential equation. Discretization by the finite element method gives the discrete problem which must satisfy the discrete inf-sup condition. But, depending on the choice of finite elements, the discrete condition may fail. This paper attempts to explain why it fails from an engineer's perspective, and reviews current methods to work around this failure. The last part recalls the mathematical bases.

math.NA↗

arXiv2023

Objectivity in continuum mechanics, an introduction. Motions, Eulerian and Lagrangian variables and functions, deformation gradient, Lie derivatives, velocity-addition formula, Coriolis

In classical mechanics, there are two objectivities: The ``isometric objectivity'' which concerns the constitutive laws of materials once expressed in a reference frame, and the ``covariant objectivity'' which concerns the universal laws of physics required to be observer independent. Covariant objectivity is the main topic in this manuscript. We therefore first introduce motions and Eulerian variables and functions, then we deduce Lagrangian variables and functions, then define the function ``deformation gradient'', from which we deduce simple definitions of the Lie derivatives. Then we detail the construction of the velocity addition formula and recall the proof of the objective covariance of the Lie derivatives. And constituvive laws built from Lie derivatives are introduced.

physics.class-ph↗