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We propose a numerical scheme based on the principles of IsogeometricAnalysis (IgA) for a geometrical pattern formation induced evolution ofmanifolds. The development is modelled by the use of the Gray-Scott equationsfor pattern formation in combination with an equation for the displacement ofthe manifold. The method forms an alternative to the classical finite-elementmethod. Our method is based on partitioning the initially spherical geometryinto six patches, which are mapped onto the six faces of a cube. Majoradvantages of the new formalism are the reconstruction of the manifold based onbicubic spline-functions used for the computation of the concentrations as wellas the evolution of the mapping operator. These features give a smoothrepresentation of the manifold which, in turn, may lead to more realisticresults. The method successfully reproduces the smooth but complicatedgeometrical patterns found on the surface of human brains.

Isogeometric Analysis of the Gray-Scott Reaction-Diffusion Equations for  Pattern Formation on Evolving Surfaces and Applications to Human Gyrification