Wind power is an increasingly popular renewable energy. In the design process of the wind turbine blade, the accurate aerodynamic simulation is important. In the past, most of the wind turbine simulations were carried out with some low fidelity methods, such as the blade element momentum method [9]. Recently, with the rapid development of the supercomputers, high fidelity simulations based on 3D unsteady Navier-Stokes (N-S) equations become more popular. For example, Sorensen et al. studied the 3D wind turbine rotor using the Reynolds-Averaged Navier-Stokes (RANS) framework where a finite volume method and a semi-implicit method are used for the spatial and temporal discretization, respectively [17]. Bazilevs et al. investigated the aerodynamic of the NREL 5MW offshore baseline wind turbine rotor using large eddy simulation built with a deforming-spatial-domain/stabilized space-time

We propose and study a new parallel one-shot Lagrange--Newton--Krylov--Schwarz (LNKSz) algorithm for shape optimization problems constrained by steady incompressible Navier--Stokes equations discretized by finite element methods on unstructured moving meshes. Most existing algorithms for shape optimization problems iteratively solve the three components of the optimality system: the state equations for the constraints, the adjoint equations for the Lagrange multipliers, and the design equations for the shape parameters. Such approaches are relatively easy to implement, but generally are not easy to converge as they are basically nonlinear Gauss--Seidel algorithms with three large blocks. In this paper, we introduce a fully coupled, or the so-called one-shot, approach which solves the three components simultaneously. First, we introduce a moving mesh finite element method for the shape optimization

We characterize conformally flat spaces as the only compact self-dual manifolds which are U(1)-equivariantly and conformally decomposable into two complete self-dual Einstein manifolds with common conformal infinity. A geometric characterization of such conformally flat spaces is also given.

The notion of a generalized CRF-structure on a smooth manifold was recently introduced and studied by Vaisman (2008)[6]. An important class of generalized CRF-structures on an odd dimensional manifold M consists of CRF-structures having complementary frames of the form , where is a vector field and is a 1-form on M with ()= 1. It turns out that these kinds of CRF-structures give rise to a special class of what we called strong generalized contact structures in Poon and Wade [5]. More precisely, we show that to any CRF-structures with complementary frames of the form , there corresponds a canonical Lie bialgebroid. Finally, we explain the relationship between generalized contact structures and another generalization of the notion of a CauchyRiemann structure on a manifold.

Using deformations of singular twistor spaces, a generalisation of the connected sum construction appropriate for quaternionic manifolds is introduced. This is used to construct examples of quaternionic manifolds which have no quaternionic symmetries and leads to examples of quaternionic manifolds whose twistor spaces have arbitrary algebraic dimension.