Prof. V. Garanzha (Russian Academy of Sciences, Moskau)
Thursday, September 14, 2017 - 14:00
Weierstraß-Institut
Mohrenstr. 39, 10117 Berlin, Raum: 405/406, 4. Etage
Seminar Numerische Mathematik
Theory of quasi-isometric (QI) mappings allows to reformulate in unified terms the intuitive idea of quasi-uniform meshes and presents natural multidimensional generalization of equidistribution principle. Suprisingly theory of QI mapping is not very rich. Rigorous results about existence of QI mappings are available only in 2d, see works by Reshetnyak, Bakelman, Bonk and Lang, Burago on parameterization of 2d manifolds of bounded curvature. In 2000 Garanzha suggested stored energy with guaranteed QI minimizer. Multidimensional "existence theorems" (Garanzha, 2005, 2010) essentially follow formulations of existence theorems of finite hyperelasticity due to John Ball: minimum of stored energy is attained on admssible mapping provided that at least one admissible mapping exists. It is not known how to find this initial admissible mapping. Another unsolved problem is lack of discrete convergence: one cannot prove that solution of discretized variational problem converges to exact minimizer. In order to construct QI mappings one has to solve continuation problem which minimizes certain evaluation of quasi-isometry constant. While providing rigorous foundations and best results in terms of distortion, this aproach is too complicated for solving engineering problems. Simple solution to this problem is based on introducing spatial weight distributions which premultiply density of stored energy (distortion measure) and sharply increase in the zones where small distortion is required. Strategies for devising these weight functions are discussed as applied to construction of smooth and orthogonal boundary layer meshes with precise stretching control, to construction of prismatic mesh layer using elastic springback technique, to surface flattening and parameterization and to construction of adaptive meshes.
submitted by lawrenz (marion.lawrenz@wias-berlin.de, 030 20372566)