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Workshop Ljubljana - Linz (popravek)

Date: 9. 7. 2018
Source: Numerical analysis seminar
Torek, 10.7.2018, od 10h do 12h, soba 3.06 na Jadranski 21
Dragi člani seminarja,

v sklopu bilateralnega projekta, ki ga imamo s kolegi iz Linza, bomo imeli v torek (in ne v sredo, kot je bilo pomotoma objavljeno), 10. julija manjši workshop, na katerem bodo svoje raziskovalne rezultate predstavili Thomas Takacs, Mario Kapl in Katharina Birner. Predavanja bodo od 10h-12h v predavalnici 3.06.

Povzetki predavanj so sledeči.

Thomas Takacs: Analysis-suitable G1 multi-patch parametrizations for isogeometric analysis

Abstract: Multi-patch spline parametrizations are often used in geometric design to represent geometrically complex domains of interest.  In isogeometric analysis (IGA), these representations are then used to perform numerical simulations  of  physical  processes  on  such  domains.   IGA  allows  for  discretizations  of  high  continuity  within single  patches.   This  is  necessary  e.g.   when  employing  a  standard  Galerkin  discretization  of  the  variational formulation  of  a  fourth  order  partial  differential  equation,  where C1 smoothness  is  needed.   In  this  talk  we discuss the imposition of C1 smoothness over planar multi-patch domains and its application to IGA.

We  deal  with  a  particular  class  of  multi-patch  spline  parametrizations  called  analysis-suitable G1 (AS-G1)  multi-patch  parametrizations,  introduced  in  [1].   This  class  of  parametrizations  has  to  satisfy  specific geometric continuity constraints, and is of importance since it allows to construct, on the multi-patch domain, C1 isogeometric spaces with optimal approximation properties.  In this context, on every patch individually, the isogeometric functions are B-splines composed with the inverse of the patch parametrization.

Such  AS-G1 multi-patch  parametrizations  are  suitable  for  modeling  complex  multi-patch  domains.   We present the theoretical foundations as well as basis constructions and demonstrate the dependence of the space on the multi-patch topology and on the patch parametrizations.  Moreover, we discuss constructions of AS-G1 multi-patch parametrizations for geometrically complex domains.
Mario Kapl: Solving the triharmonic equation over planar multi-patch domains using isogeometric analysis

Abstract: The triharmonic equation, a sixth order partial differential equation, is solved over bilinearly parameterized planar multi-patch domains by means of isogeometric analysis. This requires the use of a space of globally C2-smooth isogeometric functions as discretization space for the sixth order partial differential equation.

We present the construction of a C2-smooth isogeometric spline space which consists of three different types of functions called patch, edge and vertex functions corresponding to the single patches, edges and vertices of the multi-patch domain. The construction of the functions is simple, and is based on solving small systems of linear equations and/or on using simple explicit formulas.

In addition, all functions possess a small local support, and are well-conditioned. Several numerical examples demonstrate the potential of the constructed C2-smooth isogeometric spline space for solving the triharmonic equation.

Katharina Birner: C1-smooth geometrically continuous isogeometric functions on volumetric two-patch domains: dimension and basis

Abstract: One main advantage of isogeometric analysis is that it facilitates discretization spaces providing high order smoothness. However, when using multi-patch parameterizations of the computational domain, this needs special constructions to achieve smoothness across the interfaces between patches. In particular it is important to study the space of C1-smooth geometrically continuous functions on such domains. More precisely, it is of interest to investigate the dimension and to construct local bases for this space.

The space of C1-smooth geometrically continuous isogeometric splines on bilinear two- and multi-patch domains was recently studied, including explicit constructions of basis functions and numerical experiments indicating optimal approximation power. In this talk we extend these results to the trivariate case. In particular, we focus on two hexahedral volumetric domains given as general B-Spline maps with a suitable parameterization of the interface.

Lepo vabljeni!

Marjeta Knez