• Journal of the Chungcheong Mathematical Society
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• v.28 no.1
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• pp.53-63
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• 2015

In this paper, we define and study the Riemann-Stieltjes diamond-alpha integral on time scales. Many properties of this integral will be obtained. The Riemann-Stieltjes diamond-alpha integral contains the Riemann{Stieltjes integral and diamond-alpha integral as special cases.

• International Journal of Aeronautical and Space Sciences
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• v.18 no.4
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• pp.601-613
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• 2017

An adjoint-based error estimation and mesh adaptation study is conducted for two-dimensional viscous flows on unstructured hybrid meshes. The error in an integral output functional of interest is estimated by a dot product of the residual vector and adjoint variable vector. Regions for the mesh to be adapted are selected based on the amount of local error at each nodal point. Triangular cells in the adaptive regions are refined by regular refinement, and quadrangular cells near viscous walls are bisected accordingly. The present procedure is applied to single-element airfoils such as the RAE2822 at a transonic regime and a diamond-shaped airfoil at a supersonic regime. Then the 30P30N multi-element airfoil at a low subsonic regime with a high incidence angle (${\alpha}=21deg.$) is analyzed. The same level of prediction accuracy for lift and drag is achieved with much less mesh points than the uniform mesh refinement approach. The detailed procedure of the adjoint-based mesh refinement for the multi-element airfoil case show that the basic flow features around the airfoil should be resolved so that the adjoint method can accurately estimate an output error.