7 research outputs found
Riemann-Cartan Geometry of nonlinear dislocation mechanics
We present a geometric theory of nonlinear solids with distributed dislocations. In this theory the material manifold – where the body is stress free – is a Weitzenbock manifold, i.e. a manifold with a flat affine connection with torsion but vanishing non-metricity. Torsion of the material manifold is identified with the dislocation density tensor of nonlinear dislocation mechanics. Using Cartan’s moving frames we construct the material manifold for several examples of bodies with distributed dislocations. We also present non-trivial examples of zero-stress dislocation distributions. More importantly, in this geometric framework we are able to calculate the residual stress fields assuming that the nonlinear elastic body is incompressible. We derive the governing equations of nonlinear dislocation mechanics covariantly using balance of energy and its covariance
欠陥をもつ材料多様体の幾何学
要約のみTohoku University小谷元子論
Covariance-modulated optimal transport and gradient flows
We study a variant of the dynamical optimal transport problem in which the
energy to be minimised is modulated by the covariance matrix of the
distribution. Such transport metrics arise naturally in mean-field limits of
certain ensemble Kalman methods for solving inverse problems. We show that the
transport problem splits into two coupled minimization problems: one for the
evolution of mean and covariance of the interpolating curve and one for its
shape. The latter consists in minimising the usual Wasserstein length under the
constraint of maintaining fixed mean and covariance along the interpolation. We
analyse the geometry induced by this modulated transport distance on the space
of probabilities as well as the dynamics of the associated gradient flows.
Those show better convergence properties in comparison to the classical
Wasserstein metric in terms of exponential convergence rates independent of the
Gaussian target. On the level of the gradient flows a similar splitting into
the evolution of moments and shapes of the distribution can be observed.Comment: 84 pages, 4 figures. Comments are welcom