In this paper, the boundary layer equations (abbreviation BLE) for exterior flow around an obstacle are established using semi-geodesic coordinate system (S-coordinate) based on the curved two dimensional surface of t...In this paper, the boundary layer equations (abbreviation BLE) for exterior flow around an obstacle are established using semi-geodesic coordinate system (S-coordinate) based on the curved two dimensional surface of the obstacle. BLE are nonlinear partial differential equations on unknown normal viscous stress tensor and pressure on the obstacle and the existence of solution of BLE is proved. In addition a dimensional split method for dimensional three Navier-Stokes equations is established by applying several 2D-3C partial differential equations on two dimensional manifolds to approach 3D Navier-Stokes equations. The examples for the exterior flow around spheroid and ellipsoid are presents here.展开更多
In this work, the two-dimensional convective Brinkman-Forchheimer equa- tions are considered. The well-posedness for the variational problem and its mixed finite element approximation is established, and the error est...In this work, the two-dimensional convective Brinkman-Forchheimer equa- tions are considered. The well-posedness for the variational problem and its mixed finite element approximation is established, and the error estimates based on the conforming approximation are obtained. For the computation, a one-step Newton (or semi-Newton) iteration algorithm initialized using a fixed-point iteration is proposed. Finally, numerical experiments using a Taylor-Hood mixed element built on a structured or unstructured triangular mesh are implemented. The numerical results obtained using the algorithm are compared with the analytic data, and are shown to be in very good agreement. Moreover, the lid-driven problem at Reynolds numbers of 100 and 400 is considered and analyzed.展开更多
The incompatible numerical manifold method （INMM） is based on the finite cover approximation theory, which provides a unified framework for problems dealing with continuum and discontinuities. The incompatible numer...The incompatible numerical manifold method （INMM） is based on the finite cover approximation theory, which provides a unified framework for problems dealing with continuum and discontinuities. The incompatible numerical manifold method employs two cover systems as follows. The mathematical cover system provides the nodes for forming finite covers of the solution domain and the weighted functions, and the physical cover system describes geometry of the domain and the discontinuous surfaces therein. In INMM, the mathematical finite cover approximation theory is used to model cracks that lead to interior discontinuities in the process of displacement. Therefore, the discontinuity is treated mathematically instead of empirically by the existing methods. However, one cover of a node is divided into two irregular sub-covers when the INMM is used to model the discontinuity. As a result, the method sometimes causes numerical errors at the tip of a crack. To improve the precision of the INMM, the analytical solution is used at the tip of a crack, and thus the cover displacement functions are extended with higher precision and computational efficiency. Some numerical examples are given.展开更多
A decoupling method with different subdomain time steps for the non-stationary Navier- Stokes/Darcy model is formulated and analyzed. The method has asynchronous time steps, which adopt small time steps in the fluid r...A decoupling method with different subdomain time steps for the non-stationary Navier- Stokes/Darcy model is formulated and analyzed. The method has asynchronous time steps, which adopt small time steps in the fluid region and large time steps in the porous region. It saves relatively large amount of CPU time. Stability and convergence of the method are proved. The numerical results are presented to illustrate the features of the proposed method.展开更多
In this paper, a new operator splitting scheme is introduced for the numerical solution of the incompressible Navier-Stokes equations. Under some mild regularity assumptions on the PDE solution, the stability of the s...In this paper, a new operator splitting scheme is introduced for the numerical solution of the incompressible Navier-Stokes equations. Under some mild regularity assumptions on the PDE solution, the stability of the scheme is presented, and error estimates for the velocity and the pressure of the proposed operator splitting scheme are given.展开更多
In this study,the dynamics of elliptic membrane shell model has been proposed and discussed numerically for the first time.Firstly,we show that the solution of this model exists and is unique.Secondly,we consider spat...In this study,the dynamics of elliptic membrane shell model has been proposed and discussed numerically for the first time.Firstly,we show that the solution of this model exists and is unique.Secondly,we consider spatial and time discretizations of the time-dependent elliptic membrane shell by finite element method and Newmark scheme,respectively.Then,the corresponding existence,uniqueness,stability,convergence and a priori error estimate are given.Finally,we present numerical results involving a portion of an ellipsoidal shell and a portion of a spherical shell to verify the efficiency and convergence of the numerical scheme.展开更多
文摘In this paper, the boundary layer equations (abbreviation BLE) for exterior flow around an obstacle are established using semi-geodesic coordinate system (S-coordinate) based on the curved two dimensional surface of the obstacle. BLE are nonlinear partial differential equations on unknown normal viscous stress tensor and pressure on the obstacle and the existence of solution of BLE is proved. In addition a dimensional split method for dimensional three Navier-Stokes equations is established by applying several 2D-3C partial differential equations on two dimensional manifolds to approach 3D Navier-Stokes equations. The examples for the exterior flow around spheroid and ellipsoid are presents here.
基金supported by the National Natural Science Foundation of China(Nos.11461068,11362021,and 11401511)the Doctoral Foundation of Xinjiang Uygur Autonomous Region of China(No.BS110101)
文摘In this work, the two-dimensional convective Brinkman-Forchheimer equa- tions are considered. The well-posedness for the variational problem and its mixed finite element approximation is established, and the error estimates based on the conforming approximation are obtained. For the computation, a one-step Newton (or semi-Newton) iteration algorithm initialized using a fixed-point iteration is proposed. Finally, numerical experiments using a Taylor-Hood mixed element built on a structured or unstructured triangular mesh are implemented. The numerical results obtained using the algorithm are compared with the analytic data, and are shown to be in very good agreement. Moreover, the lid-driven problem at Reynolds numbers of 100 and 400 is considered and analyzed.
基金supported by the Natural Science Foundation of Shandong Province for Excellent Young and Middle-aged Scientist (2007BS04045 and 2008BS04009)the Natural Science Foundation of Shandong Province(Y2006B24 and Y2008A 11)
文摘The incompatible numerical manifold method （INMM） is based on the finite cover approximation theory, which provides a unified framework for problems dealing with continuum and discontinuities. The incompatible numerical manifold method employs two cover systems as follows. The mathematical cover system provides the nodes for forming finite covers of the solution domain and the weighted functions, and the physical cover system describes geometry of the domain and the discontinuous surfaces therein. In INMM, the mathematical finite cover approximation theory is used to model cracks that lead to interior discontinuities in the process of displacement. Therefore, the discontinuity is treated mathematically instead of empirically by the existing methods. However, one cover of a node is divided into two irregular sub-covers when the INMM is used to model the discontinuity. As a result, the method sometimes causes numerical errors at the tip of a crack. To improve the precision of the INMM, the analytical solution is used at the tip of a crack, and thus the cover displacement functions are extended with higher precision and computational efficiency. Some numerical examples are given.
基金The authors would like to thank the referees for helpful comments and suggestions, which lead to substantial improvements of the presentation.The work was Supported by the National Nature Foundation of China （NO.11401422）, the Provincial Natural Science Foundation of Shanxi （NO.2015011001, 2014011005-4）,China Postdoctoral Science Foundation funded project （No.2015M582336）
文摘A decoupling method with different subdomain time steps for the non-stationary Navier- Stokes/Darcy model is formulated and analyzed. The method has asynchronous time steps, which adopt small time steps in the fluid region and large time steps in the porous region. It saves relatively large amount of CPU time. Stability and convergence of the method are proved. The numerical results are presented to illustrate the features of the proposed method.
文摘In this paper, a new operator splitting scheme is introduced for the numerical solution of the incompressible Navier-Stokes equations. Under some mild regularity assumptions on the PDE solution, the stability of the scheme is presented, and error estimates for the velocity and the pressure of the proposed operator splitting scheme are given.
基金Acknowledgements This work was supported in part by the National Natural Science Foundation of China （Grant Nos. 11401422, 11172194）, the Provincial Soft Science Foundation of Shaanxi Province （No. 2014041007）, and Provincial Science Foundation of Shanxi （Nos. 2014011005, 2015011001）.
基金supported by the National Natural Science Foundation of China(NSFC 11971379,11571275,11572244)by the Natural Science Foundation of Shaanxi Province(2018JM1014).
文摘In this study,the dynamics of elliptic membrane shell model has been proposed and discussed numerically for the first time.Firstly,we show that the solution of this model exists and is unique.Secondly,we consider spatial and time discretizations of the time-dependent elliptic membrane shell by finite element method and Newmark scheme,respectively.Then,the corresponding existence,uniqueness,stability,convergence and a priori error estimate are given.Finally,we present numerical results involving a portion of an ellipsoidal shell and a portion of a spherical shell to verify the efficiency and convergence of the numerical scheme.
