Reconstruction of a Steam Turbine Blade Using Piecewise Bernstein Polynomials and Transfinite Interpolation

Turbine blades are designed to achieve its maximum efficiency, but deformations caused by the exposition to extreme operating environments provokes reduction in the engine performance. Often, operators choose to repair a damaged blade instead of replacing it to save money, however, reconstructing its virtual model, commonly the first step in the repairing process, can be challenging due to the geometrical complexity of the blades, variability in deformations and the requirement to meet the dimensions specified by the manufacturer. This paper presents the reconstruction methodology of the clean virtual model of a steam turbine blade through numerical tools, as a previous step for regenerating a worn blade. First, few cross-sectional airfoil profiles are extracted from the damaged blade and are regenerated using Bernstein polynomials; then, using the previously obtained data, many more profiles are interpolated and stacked along the spanwise direction of the blade via Transfinite Interpolation in order to obtain a smooth and continuous representation of the reference blade. Final deviation between the reference and reconstructed model resulted in an average value of 1.5496 × 10−3 % and 9.685 × 10−5 % relative to the rotor diameter in the pressure and suction sides respectively, showing an accuracy that could be considered to be used in industrial applications or optimization.

The order of the best transfinite interpolation of functions with bounded laplacian with the help of harmonic splines on box partitions

We show that the error of the best transfinite interpolation of functions with bounded laplacian with the help of harmonic splines on box partitions comprising $$$N$$$ elements has the order $$$N^{-2}$$$ as $$$N \rightarrow \infty$$$.

A Hybrid Dynamic Mesh Generation Method for Multi-Block Structured Grid

In this paper, a hybrid dynamic mesh generation method for multi-block structured grid is presented based on inverse distance weighting (IDW) interpolation and transfinite interpolation (TFI). The major advantage of the algorithm is that it maintains the effectiveness of TFI, while possessing the ability to deal with multi-block structured grid from the IDW method. In this approach, dynamic mesh generation is made in two steps. At first, all domain vertexes with known deformation are selected as sample points and IDW interpolation is applied to get the grid deformation on domain edges. Then, an arc-length-based TFI is employed to efficiently calculate the grid deformation on block faces and inside each block. The present approach can be well applied to both two-dimensional (2D) and three-dimensional (3D) problems. The proposed method has been well-validated by several test cases. Numerical results show that dynamic meshes with high quality can be generated in an accurate and efficient manner.

Hybrid Method Combines Transfinite Interpolation with Series Expansion to Simulate the Anti-Plane Response of a Surface Irregularity

The responses to an incident plane SH wave on or near a surface irregularity which is embedded in an elastic half-plane are investigated. The surface irregularity represents a canyon, an alluvial valley or a hill. The wave function expansion method has been employed to solve surface irregularities, such as a semi-cylindrical canyon, a semi-cylindrical alluvial valley, or a semi-elliptical canyon and a semi-elliptical alluvial valley. These solutions to the scattering problem of SH wave can be used to test the accuracy of the other numerical methods. But solutions for surface irregularities with arbitrarily shapes cannot be found easily. A hybrid method combines the finite element method with series expansion is applied to solve scattering problems in this study. A subregion encloses the surface irregularity with a semi-circular auxiliary boundary can be meshed by the finite element method. By using the transfinite interpolation (TFI) produces excellent grid mesh on the subregion. The advantage of TFI is the flexibility to facilitate modeling of the subregion. On the other hand, the boundary data can be formulated by using a series representation with unknown coefficients. The Lamb's solution which satisfies the traction free condition and the radiation condition at infinity is implemented to be the basis function. The unknown coefficients can be obtained by satisfying the continuity conditions of the semi-circular auxiliary boundary between the subregion and the half-plane. The hybrid method that combines TFI with series expansion is successfully herein to solve the scattering problem by a surface irregularity. Numerical results in this study for special cases agree well with those in the published literatures. In this study, the steps and skills of hybrid method are described systematically and completely to solve the surface irregularity.

An Efficient Dynamic Mesh Generation Method for Complex Multi-Block Structured Grid

Aiming at a complex multi-block structured grid, an efficient dynamic mesh generation method is presented in this paper, which is based on radial basis functions (RBFs) and transfinite interpolation (TFI). When the object is moving, the multi-block structured grid would be changed. The fast mesh deformation is critical for numerical simulation. In this work, the dynamic mesh deformation is completed in two steps. At first, we select all block vertexes with known deformation as center points, and apply RBFs interpolation to get the grid deformation on block edges. Then, an arc-length-based TFI is employed to efficiently calculate the grid deformation on block faces and inside each block. The present approach can be well applied to both two-dimensional (2D) and three-dimensional (3D) problems. Numerical results show that the dynamic meshes for all test cases can be generated in an accurate and efficient manner.