Generalized finite difference method for two-dimensional velocity-vorticity formulation and the assessment of its parallel efficiency

• Main Authors: Chang, Chien-Chung, 張建忠
• Other Authors: Fan, Chia-Ming
• Format: Others
• Language: zh-TW
• Published: 2015
• Online Access: http://ndltd.ncl.edu.tw/handle/xdb9qq

碩士 === 國立臺灣海洋大學 === 河海工程學系 === 103 === In this study, we use the generalized finite difference method (GFDM) to simulate the two-dimensional fluid fields of incompressible viscous fluid. In order to accelerate the computational speed, we parallelize our program on a multithreading computer which has shared memory architecture. To validate the feasibility, accuracy and the parallel efficiency of the meshless method proposed in this article, we provided three numerical examples. The velocity-vorticity formulation are solved in this study and it is another form of the standard Navier-Stokes equations. The problem of the pressure term and the pressure boundary condition in the primary-variable formulation of the Navier-Stokes equations can be overcame. In addition, the number of the unknown variables in the velocity-vorticity formulation is the same as the primary-variable formulation in two-dimensional fluid fields. Therefore, the velocity-vorticity formulation has great potential to be used in engineering applications. The methods, used for flow fields, usually require grids in the computational domain, such as the finite difference method and the finite volume method. Instead, we adopt the GFDM, a novel meshless method. It can get rid of mesh generation and numerical quadrature as well as construct a sparse matrix by introducing the localized concept. Moreover, we used the parallelizable non-symmetric sparse matrix direct solver to solve the resultant linear algebraic system. The GFDM is based on the moving-least-square method to approximate the derivatives at every node in computational domain by linear summation of nodal values. Therefore, the programming by using the GFDM is quite easy and convenient and can reduce the probability of ill-conditioned matrix. On the other hand, we use the implicit Euler method on the time quadrature, so we can adopt larger time step to acquire the stable result. To apply the above methods will result in non-linear algebraic equations. We solve it by the Newton’s method because its convergence is at least quadratic. In this paper, we analyze the two-dimensional flow fields of incompressible viscous fluid, described by the velocity-vorticity formulation, by using the GFDM, the implicit Euler method and the Newton’s method. To verify the accuracy and the stability of our model, we simulated some examples include cavity flow, rectangular cavity flow, and staggered double cavity flow. The results of these examples are very similar to previous work, so we can deduce that our model has the ability for accurately solving flow-field problems. We also examine the influence of the variables in our model, such as, the number of total nodes, the number of nodes in the sub-domain and the length of time steps. All these results and comparisons can verify the accuracy, consistency, and stability of our model. We also parallelized the developed model by OpenMP and test the efficiency on a shared memory computer. According to the results, our parallelized model provided a satisfying efficiency when four threads are used.