Isogeometric Boundary Element Analysis of steady incompressible viscous flow, Part 2: 3-D problems (original) (raw)
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Isogeometric Boundary Element Analysis of steady incompressible viscous flow, Part 1: Plane problems
Computer Methods in Applied Mechanics and Engineering, 2017
A novel approach is presented to the Boundary Element analysis of steady incompressible flow. NURBS basis functions are used for describing the geometry of the problem and for approximating the unknowns. In addition, the arising volume integrals are treated differently to published work, that is, volumes are described by bounding NURBS curves instead of cells and a mapping is used. The advantage of our approach is that non-trivial boundary shapes can be described with very few parameters and that no generation of cells is required. For the solution of the non-linear equations both classical and modified Newton-Raphson methods are used. A comparison of the two methods is made on the classical example of a forced cavity flow, where accurate solutions are available in the literature. The results obtained agree well with published ones for moderate Reynolds numbers using both methods, but it is found that the latter requires a relaxation scheme and considerably more iterations to converge. Finally, it is shown on a practical example of an airfoil how more complex boundary shapes can be approximated with few parameters and a solution obtained with a small number of unknowns.
The boundary element method applied to incompressible viscous fluid flow
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An Integral equation formulation for steady flow of a viscous fluid is presented based on the boundary element method. The continuity, Navier-Stokes and energy equations are used for calculation of the flow field. The governing differential equations, in terms of primitive variables, are derived using velocity-pressure-temperature. The calculation of fundamental solutions and solutions tensor is showed. Applications to simple flow cases, such as the driven cavity, step, deep cavity and channel of multiple obstacles are presented. Convergence difficulties are indicated, which have limited the applications to flows of low Reynolds numbers.
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AIAA Journal, 1998
We present an innovative numerical approach for setting highly accurate nonlocal boundary conditions at the external computational boundaries when calculating three-dimensional compressible viscous ows over nite bodies. The approach is based on application of the special boundary operators analogous to Calderon's projections and the di erence potentials method by R y aben'kii; it extends our previous technique developed for the two-dimensional case. The new boundary conditions methodology has been successfully combined with the NASA-developed code TLNS3D and used for the analysis of wing-shaped con gurations in subsonic and transonic ow regimes. As demonstrated by the computational experiments, the improved external boundary conditions allow one to greatly reduce the size of the computational domain while still maintaining high accuracy of the numerical solution. Moreover, they may provide for a noticeable speedup of convergence of the multigrid iterations.
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Cavitation is one of the most important problems in hydrodynamic applications which causes a noticeable deterioration of the machine performance. Thus, accurate prediction of cavitation is very important in estimating the hydrodynamic performance of pumps, marine propellers and high speed hydrofoils. For this reason, substantial efforts have been taken by many researchers to develop capabilities to predict the extent of cavitation for various types of geometries. Several researchers have successfully analyzed the cavitation phenomenon and its application. They used different analytical and numerical methods to determine the shape and the size of the cavity and to know the velocities and/ or pressure along the boundaries. Due to the complexity of the cavitation problem and due to the difficulty to obtain analytical solutions many researchers prefer numerical methods. The most popular numerical technique for analyzing cavitation problems is the boundary element method (BEM). Boundary element method has gained its popularity from its simplified nature and many other reasons that will be encountered through the thesis. The boundary element method was used herein as a mathematical tool to solve the cavity flow around hydrofoils. Two major difficulties meet the researcher in this field of study. These difficulties are the determination of the cavity free surface and the potential at the leading point. In the present thesis, these difficulties are solved by a new suggested technique and it gives excellent results. Also, the new technique saves the time and effort needed by previous techniques. The algorithm is applied to solve the different symmetric hydrofoil NACA sections with studying the effect of three different parameters. Excellent agreement was obtained with the available existing results.
An ‘A Priori’ Model Reduction for Isogeometric Boundary Element Method
Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016), 2016
The aim of this work is to provide a promising way to accelerate the structural design procedure and overcome the burden of meshing. The concept of Isogeometric Analysis (IGA) has appeared in recent years and has become a powerful means to eliminate gaps between Computer Aided Design (CAD) and Computer Aided Engineering (CAE), giving a higher fidelity geometric description and better convergence properties of the solution. The Boundary Element Method (BEM) with IGA offers a better, more seamless integration since it uses a boundary representation for the analysis. However, the computational efficiency of IGABEM may be compromised by the dense and unsymmetrical matrix appearing in the calculation, and this motivates the present work. This study introduces an 'a priori' model reduction method in IGABEM analysis aiming to enhance efficiency. The problem is treated as a state evolution process. It proceeds by approximating the problem solution using the most appropriate set of approximation functions, which depend on Karhunen-Loève decomposition. Secondly, the model is re-analysed using a reduced basis, using the Krylov subspaces generated by the governing equation residual for enriching the approximation basis. Finally, the IGABEM calculation combined the model reduction strategy is proposed, which provides accurate and fast resolution of IGABEM problems compared with the traditional BEM solution. Moreover, the CPU time is drastically reduced. A simple numerical example illustrates the potential of this numerical technique.
The total pressure boundary element method (TPBEM) for steady viscous flow problems
Engineering Analysis with Boundary Elements, 1994
The application of the boundary element method to viscous flow problems is becoming increasingly important since it has offered great advantages in the solid mechanics field. In the present research work, the total pressure boundary element method (TPBEM) is presented. The method offers favorable features for convective diffusion steady flow problems. The flow kinematics is described by a standard velocity-vorticity boundary domain integral representation while the integral equation for the kinetics is expressed in terms of velocity, vorticity and the total pressure at the boundary. The present scheme allows the BEM to be used in an optimal manner. This is achieved by applying an implicit-explicit numerical procedure. The implicit system of equations is written only for the boundary unknowns, vorticity and total pressure, while the other interior unknowns are computed explicitly by a standard under-relaxation procedure. This indeed greatly reduces the CPU time needed to solve such a nonlinear viscous flow problem. Moreover, the computation of the total pressure at the boundary allows a direct determination of the aerodynamic coefficients for solid boundaries. Effectiveness and validity of the present TPBEM is illustrated by solving a standard benchmark driven cavity problem. Numerical results for Reynolds numbers 100 and 1000 show good agreement with other numerical solution and experimental data. The present method consumes almost half of the CPU time needed for a standard finite difference solution.