Tuesday, 7 May 2019, 2 p.m. (sharp),

dr Mauro Mariani, HSE, Mosca,

at the conference room of Aula Beltrami, Dipartimento di Matematica in Pavia, will give a lecture titled:

TBA

and

at 3 p.m. (sharp),

Pascal Weinmüller, JKU, Linz

at the conference room of IMATI CNR in Pavia, will give a lecture titled:

Construction of approximate C1 bases on two-patch domains in isogeometric analysis

and

at 4 p.m. (sharp),

Cy Maor (University of Toronto)

at the conference room of IMATI CNR in Pavia, will give a lecture titled:

Elasticity and curvature: the elastic energy of non-Euclidean thin bodies

and

Wednesday 8 May, at 3 p.m. (sharp),

Jinchao Xu, Penn State University,

at the conference room of IMATI CNR in Pavia, will give a lecture titled:

Extended Galerkin Method

as part of the Applied Mathematics Seminar (IMATI-CNR e Dipartimento di Matematica, Pavia).

At the end a refreshment will be organized.

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Abstracts.

Mauro Mariani: TBA

Pascal Weinmüller: One key element of isogeometric analysis is that it allows high order smoothness within one patch. However, this is not the case on so-called multi-patch geometries where the global continuity is C0. Therefore, for C1 isogeometric functions, a special construction for the basis is needed. Such spaces are of interest when solving numerically fourth-order PDE problems, such as the biharmonic equation, using an isogeometric Galerkin method.

With the construction of so called analysis-suitable G1 (in short, AS-G1) parameterizations it is possible, under certain additional assumptions, to have C1 isogeometric spaces with optimal approximation properties. These geometries satisfy certain constraints along the interfaces.

The problem is that most complex geometries are not AS-G1 geometries. Therefore we define basis functions for isogeometric spaces by enforcing approximate C1 conditions. For this reason the defined function spaces are not exactly C1 but only approximately. We study the convergence behaviour and define function spaces that behave optimally under h-refinement, by locally introducing functions of higher polynomial degree and/or lower regularity. For the numerical tests we focus on the influence of a single, non-trivial interface within a two-patch domain.

Cy Maor: Non-Euclidean, or incompatible elasticity, is an elastic theory for bodies that do not have a reference (stress-free) configuration. It applies to many systems, in which the elastic body undergoes plastic deformations or inhomogeneous growth (e.g. plants, self-assembled molecules). Mathematically, it is a question of finding the "most isometric" immersion of a Riemannian manifold (M,g) into Euclidean space of the same dimension, by minimizing an appropriate energy functional.

Much of the research in non-Euclidean elasticity is concerned with elastic bodies that have one or more slender dimensions (such as leaves), and finding appropriate dimensionally-reduced models for them. In this talk I will give an introduction to non-Euclidean elasticity, and then focus on thin bodies and present some recent results and open problems on the relations between their elastic behavior and their curvature.

Based on joint work with Asaf Shachar.

Jinchao Xu: In this talk, I will present a general framework, known as extended Galerkin method, for the derivation and analysis of many different types of finite element methods (including various discontinuous Galerkin methods).

For second order elliptic equation, this framework employs 4 different discretization variables, $u_h, p_h, \hat u_h$ and $\hat p_h$, where $u_h$ and $p_h$ are for approximation of $u$ and $p=\grad u$ inside each element, and $\hat u_h$ and $\hat p_h$ are for approximation of $u$ and $p\cdot n$ on the boundary of each element. The resulting 4-field discretization is proved to satisfy inf-sup conditions that are uniform with respect to all discretization and penalization parameters. As a result, most existing finite element and discontinuous Galerkin methods can be derived and analyzed using this general theory by making appropriate choices of discretization spaces and penalization parameters.