Keywords: finite element method, Delaunay method, virtual nodes, mixed interpolation, linear interpolation.

The finite element method has been successfully applied to solve a variety of engineering problems. This paper shows a method of generating the numerical model using the Delaunay method and how to apply the Delaunay method to a moving boundary problem. The Delaunay method is one of the techniques to generate the finite element mesh. Tetrahedral elements are generated with the node group which is set arbitrarily in the computational domains. A feature of Delaunay method is that a circumscribed sphere of a tetrahedron generated by the Delaunay method does not include nodes of other elements. The Delaunay method is useful and effective for re-meshing. When the re-meshing is performed, there is an immovable area. The mesh data for the immovable area is not input. Generally, the re-meshing area is much smaller than the immovable area. In this study, three kinds of meshes are prepared. In this research, a windmill is treated as the numerical model [1], and a moving boundary problem in the incompressible viscous fluid is analyzed. The Navier-Stokes equation is employed as the basic equation. As the spatial discretization, the finite element method using a mixed interpolation method is applied for the basic equation. The bubble function interpolation using the stabilized bubble function for the velocity and the linear interpolation for pressure are applied. For the temporal discretization, the Crank-Nicolson method is applied, and the fractional step method is applied to the incompressible Navier-Stokes equation. Velocity and pressure fields can be solved separately by the fractional method. After solving for velocity and pressure, the computational domain is updated. Using the updated mesh, the velocity must be interpolated because the positions of the nodes are different. Not only the mesh is updated but also the previous meshes are necessary for interpolation of velocities. Linear interpolation is applied to the interpolation of velocity. As conclusion, the three-dimensional mesh generation using the Delaunay method and flow analysis around a windmill using the finite element method has been presented. Arbitrary finite element meshes can be generated using the Delaunay method. The virtual nodes are effective for generating finite element meshes which have concave faces. The calculation time for generating meshes is reduced by using the technique presented for re-meshing. Flow analysis based on the incompressible Navier-Stokes equation can be analyzed using the finite element method. When re-meshing is applied, stable pressure and velocity can be obtained. The scheme presented is effective for flow analysis which has moving boundaries.

1

K. Harada, M. Kawahara, "An Analysis of Flow around a Propeller using Fictitious Domain Finite Element Method", Int. J. Energy, Environment and Economics, 2006.

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