Least-squares solution of incompressible Navier-Stokes equations with the p-version of finite elements

A p-version of the least squares finite element method, based on the velocity-pressure-vorticity formulation, is developed for solving steady state incompressible viscous flow problems. The resulting system of symmetric and positive definite linear equations can be solved satisfactorily with the conjugate gradient method. In conjunction with the use of rapid operator application which avoids the formation of either element of global matrices, it is possible to achieve a highly compact and efficient solution scheme for the incompressible Navier-Stokes equations. Numerical results are presented for two-dimensional flow over a backward facing step. The effectiveness of simple outflow boundary conditions is also demonstrated

A 3-dimensional h-adaptive algorithm for species transport prediction

Development of 3-dimensional self-adaptive algorithms for unstructured finite element grids is recent to the solution of partial differential equations. An h-adaptive grid embedding method is employed to help solve the Navier-Stokes equations for fluid flow and scalar transport; This h-adaptive algorithm, in combination with the finite element solver, has been designed to solve simple to moderately complex 3-dimensional problems on high-end PC\u27s with at least 16 megabytes of ram, and more complex geometries on workstations and mainframes. The finite element solver is a one point Gauss-Legendre integration scheme which employs mass lumping, Cholesky skyline L-U decomposition, and Petrov-Galerkin upwinding; This thesis introduces and explains the Galerkin weighted residual finite element solution process with the use of the Laplace heat conduction equation. Development of the weak statements for the non-dimensional primitive variable Navier-Stokes equations is presented with a Poisson formulation for pressure. The explicit solution process of this Poisson formulation is described in detail. Various adaptive methods are presented with emphasis on grid embedDing Single element division or grid embedding allows for the use of the one point quadrature integration scheme used in the solution process. Finally the application of the adaptive process coupled with the finite element solver is applied to the solution of the Navier-Stokes equations along with the species transport equations

An Oseen Two-Level Stabilized Mixed Finite-Element Method for the 2D/3D Stationary Navier-Stokes Equations

We investigate an Oseen two-level stabilized finite-element method based on the local pressure projection for the 2D/3D steady Navier-Stokes equations by the lowest order conforming finite-element pairs (i.e., Q1−P0 and P1−P0). Firstly, in contrast to other stabilized methods, they are parameter free, no calculation of higher-order derivatives and edge-based data structures, implemented at the element level with minimal cost. In addition, the Oseen two-level stabilized method involves solving one small nonlinear Navier-Stokes problem on the coarse mesh with mesh size H, a large general Stokes equation on the fine mesh with mesh size h=O(H)2. The Oseen two-level stabilized finite-element method provides an approximate solution (uh,ph) with the convergence rate of the same order as the usual stabilized finite-element solutions, which involves solving a large Navier-Stokes problem on a fine mesh with mesh size h. Therefore, the method presented in this paper can save a large amount of computational time. Finally, numerical tests confirm the theoretical results. Conclusion can be drawn that the Oseen two-level stabilized finite-element method is simple and efficient for solving the 2D/3D steady Navier-Stokes equations

Preconditioning of a hybridized discontinuous Galerkin finite element method for the Stokes equations

We present optimal preconditioners for a recently introduced hybridized discontinuous Galerkin finite element discretization of the Stokes equations. Typical of hybridized discontinuous Galerkin methods, the method has degrees-of-freedom that can be eliminated locally (cell-wise), thereby significantly reducing the size of the global problem. Although the linear system becomes more complex to analyze after static condensation of these element degrees-of-freedom, the pressure Schur complement of the original and reduced problem are the same. Using this fact, we prove spectral equivalence of this Schur complement to two simple matrices, which is then used to formulate optimal preconditioners for the statically condensed problem. Numerical simulations in two and three spatial dimensions demonstrate the good performance of the proposed preconditioners

Stable finite element approximations of two-phase flow with soluble surfactant

A parametric finite element approximation of incompressible two-phase flow with soluble surfactants is presented. The Navier–Stokes equations are coupled to bulk and surfaces PDEs for the surfactant concentrations. At the interface adsorption, desorption and stress balances involving curvature effects and Marangoni forces have to be considered. A parametric finite element approximation for the advection of the interface, which maintains good mesh properties, is coupled to the evolving surface finite element method, which is used to discretize the surface PDE for the interface surfactant concentration. The resulting system is solved together with standard finite element approximations of the Navier–Stokes equations and of the bulk parabolic PDE for the surfactant concentration. Semidiscrete and fully discrete approximations are analyzed with respect to stability, conservation and existence/uniqueness issues. The approach is validated for simple test cases and for complex scenarios, including colliding drops in a shear flow, which are computed in two and three space dimensions

Penyusunan Model Elemen Hingga Persamaan Navier-Stokes 3 Dimensi Untuk Aliran Tak Mampat

ABSTRAK Fluid dynamics modeling is important for hydraulic structure design or any process which involes flow like transport of pollutant. A finite element model for 3-D unsteady and incompressible flow is developed. The fundamental equations are the momentum (Navier-Stokes) and the continuity equations. These equations are written in the primitive variables of velocity components and pressure. The numerical scheme employs time-splitting method. It consists of four stage solution procedures within a time step, i.e., Taylor-Galerkin convective approximation, viscous prediction, pressure correction, and velocity correction. The velocity-pressure solution based on the quadratic velocity and linear pressure interpolation. To show performance of the model, comparison to the analytical solution on simple cases and comparison to other researchers\u27s results was performed. The numerical simulations show that the numerical scheme based on Taylor-Galerkin pure convection equation is stable for Courant number S 0,39. The numerical scheme which involves convection and viscous terms is stable for certain range of Courant (Cr) and Peclet (Pe) number, for Peclet number 0 7 Courant number remains constant on 0,39. Keywords : Navier-Stokes equation, incompressible flow, finite element, Taylor-Galerki

A finite element method to solve the compressible Navier-Stokes equations in 3D with mesh enrichment procedure

A new computer code solving the compressible Navier-Stokes equations is described. The code solve the three-dimensional (3D), time-independant equations using the finite element method based on the P1-P1isoP2 element. Actually the numerical stability is ensured by a simple artificial viscosity method which will be improved in a future version. The results of numerical experiments for the flow around an ellipsoid will be presented in order to show the possibilities of the methods used. Finally, we discuss an efficient mesh enrichment procedure which can be used in 2 or 3 dimension

Analysis of an augmented mixed-FEM for the Navier-Stokes problem

In this paper we propose and analyze a new augmented mixed finite element method for the Navier-Stokes problem. Our approach is based on the introduction of a “nonlinearpseudostress” tensor linking the pseudostress tensor with the convective term, which leads to a mixed formulation with the nonlinear-pseudostress tensor and the velocity as the main unknowns of the system. Further variables of interest, such as the fluid pressure, the fluid vorticity and the fluid velocity gradient, can be easily approximated as a simple postprocess of the finite element solutions with the same rate of convergence. The resulting mixed formulation is augmented by introducing Galerkin least-squares type terms arising from the constitutive and equilibrium equations of the Navier-Stokes equations and from the Dirichlet boundary condition, which are multiplied by stabilization parameters that are chosen in such a way that the resulting continuous formulation becomes well-posed. Then, the classical Banach’s fixed point Theorem and Lax-Milgram’s Lemma are applied to prove well-posedness of the continuous problem. Similarly, we establish well-posedness and the corresponding Cea’s estimate of the associated Galerkin scheme considering any conforming finite element subspace for each unknown. In particular, the associated Galerkin scheme can be defined by employing Raviart-Thomas elements of degree k for the nonlinear-pseudostress tensor, and continuous piecewise polynomial elements of degree k + 1 for the velocity, which leads to an optimal convergent scheme. In addition, we provide two iterative methods to solve the corresponding nonlinear system of equations and analyze their convergence. Finally, several numerical results illustrating the good performance of the method are provided.Comisión Nacional de Investigación Científica y TecnológicaMinistry of Education, Youth and Sports of the Czech Republi