Morten M. T. Wang

An element-by-element BICGSTAB iterative method for three-dimensional steady Navier-Stokes equations

Construction of a stabilized Galerkin upwind finite element model for steady and incompressible Navier-Stokes equations in three dimensions is the main theme of this study. In the time-independent context, the weighted residuals statement is kept biased in favor of the upstream flow direction by adding an artificial damping term of physical plausibility to the Galerkin framework. This upwind approach has significant advantage of seeking solutions free from cross-stream diffusion error. Finite element solutions have been found by mixed formulation, implemented in quadratic cubic elements which are characterized as possessing the so-called LBB (Ladyzhenskaya-Babuska-Brezzi) condition. An element-by-element BICGSTAB solution solver is intended to alleviate difficulties regarding the asymmetry and indefiniteness arising from the use of a mixed formulation for incompressible fluid flows. The developed three-dimensional finite element code is first rectified by solving a problem amenable to analytic solution. A well-known lid-driven cavity flow problem in a cubical cavity is also studied.

Implementation of a free boundary condition to Navier‐Stokes equations

Numerical simulation of a fluid flow involves the specification of boundary conditions along all or part of the boundary. Designs a means of handling outflow boundary conditions for the incompressible Navier‐Stokes equations. Addresses through‐flow problems involving the specification of outflow conditions at the synthetic boundary. This outflow boundary condition is applicable to a developing flow problem. The underlying objectives behind designing the boundary condition at the truncated boundary are three‐fold, namely: matching with Navier‐Stokes equations inside the domain; taking both non‐linear and diffusive contributions into account; and ensuring the discrete divergence‐free condition. In order to meet these requirements, follows the concept of a free boundary condition by taking the outflow nodal values of u, v and p as unknowns, which are coupled with the interior unknowns through the surface integrals in the momentum equations. The computed solutions can be legitimately regarded as solutions to conservation equations under consideration when both components of the surface traction vector approach zero. With the convergent property accommodated in the present mixed finite element analysis, the task remains to simply improve the accuracy. Demonstrates the capability of the proposed non‐linear outflow boundary conditions through several benchmark tests.

A monotone finite element method with test space of Legendre polynomials

This paper is concerned with the development of a multi-dimensional monotone scheme to deal with erroneous oscillations in regions where sharp gradients exist. The strategy behind the underlying finite element analysis is the accommodation of the M-matrix to the Petrov-Galerkin finite element model. An irreducible diagonal-dominated coefficient matrix is rendered through the use of exponential weighting functions. With a priori knowledge capable of leading to a Monotone matrix, the analysis model is well conditioned with the monotonicity-preserving property. In order to stress the effectiveness of test functions in resolving oscillations, we considered two classes of the convection-diffusion problem. As seen from the computed results, we can classify the proposed finite element model as legitimate for the problem free of boundary layer. Also, through the use of this model, we can capture the solution for the problem involving a high gradient. In this study, we are interested in a cost-effective method which ensures monotonicity irrespective of the value of the Peclet number throughout the entire domain. To gain access to these desired properties, it is tempting to bring in the Legendre polynomials and the characteristic information so that by virtue of the inherent orthogonal property the integral can be obtained exactly by two Gaussian integration points along each spatial direction while maintaining stability in the M-matrix satisfaction sense.

A monotone multidimensional upwind finite element method for advection-diffusion problems

We are interested in developing a multidimensional convective scheme that is capable of dealing with erroneous oscillations nearjumps. The strategy is based on the Petrov-Galerkin formulation, to which the underlying idea of the M matrix is added. The nature of the exponentially weighted upwind method is best illuminated by its matrix structure. We interpret the enhanced stability as being due to the attainable irreducible diagonal dominance. The accessible monotonicity condition enables us to construct a monotone stiffness matrix a priori, thereby laying the foundation for arriving at the manotonicity-preserving property. In order to show the merit of the proposed upwinding technique in resolving spurious oscillations generated by unresolved internal and boundary layers, we considered two classes of convection-diffusion problems. As seen from the computed results, we can attain an accurate finite-element solution for a problem free of boundary layer and can capture a high-gradient solution in the sharp layer.

Element-by-Element Parallel Computation of Incompressible Navier--Stokes Equations in Three Dimensions

Development of a stable finite element model for solving steady incompressible viscous fluid flows in three dimensions is the main theme of the present study. For stability reasons, weighting functions are designed in favor of field variables on the upstream side. For accuracy reasons, it is required that weighting functions be equipped with the streamline operator so that false diffusion errors can be largely suppressed. In the steady-state analysis of Navier--Stokes equations, we adopt the mixed formulation to preserve mass conservation on quadratic elements which accommodate the Ladyzhenskaya--Babuska--Brezzi (LBB) stability condition. To resolve difficulties arising from asymmetry and indefiniteness in the resulting large-size matrix equations, we abandon the elimination-like solution solver because the storage demand exceeds the ability of our hardware to solve for three-dimensional problems. A modern iteration solver, known as the biconjugate gradiant stabilized (BICGSTAB) solution solver, is thus implemented in an element-by-element fashion to effectively alleviate the problem. For performance reasons, the finite element code developed here should be implemented in a hardware environment which is suited to the use of an iterative solver. To this end, our analysis is implemented in shared memory parallel architectures, CRAY C-90 and J-90. We benchmark the parallel computing performance through a lid-driven cavity flow problem and a problem amenable to analytic solution.

A PETROV-GALERKIN FORMULATION FOR INCOMPRESSIBLE FLOW AT HIGH REYNOLDS NUMBER

SUMMARY A mixed finite element method was applied to solve a set of elliptic partial differential equations which corresponds to steady-state incompressible laminar flow. To obtain stable solutions at high Reynolds numbers, the Petrov-Galerkin finite element method was used to discretize the advective flux terms with a biquadratic velocity-bilinear pressure element. A priori knowledge of the M-matrix has been used as an underlying guide to enhance the solution stability. The main impetus and effort involve designing a test space of an exponential type. The test cases considered and the results obtained show that the proposed Petrov-Galerkin method is highly reliable and applicable to a wide range of flow conditions.

PRESSURE BOUNDARY CONDITIONS FOR A SEGREGATED APPROACH TO SOLVING INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

It has been well accepted that Diricklet and Neumann boundary conditions for Ike pressure Poisson equation give the same solution. The purpose of this article is to reveal that the above statement is computationally acceptable but is not theoretically correct. Analytic proof as well as computational evidences are presented through examples in support of our observation. In this work we address that the mixed finite-element formulation for solving incompressible Navier-Stokes equations in primitive variables is equivalent to the formulation that involves solving the pressure Poisson equation, subject to Neumann boundary conditions, Uerativety with the momentum equations provided the velocity field is classified as having divergence-free and conservative properties.

DISCUSSION OF NUMERICAL DEFICIENCY OF APPLYING A PARTIALLY WEIGHTED UPWIND FINITE-ELEMENT MODEL TO INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

Abstract The streamline upwind technique is extended to quadratic dements to analyze incompressible and viscous flow equations cast in the steady state. The biased part of the weighting Junctions is devised to achieve a nodally exact discretized one-dimensional equation, with an emphasis on grid nonuniformity. Two classes of upwinding finite-element models are considered. Our primary goal is to address the deficiency of the partially weighted finite-element model. Assessment is made of the stability and accuracy of the schemes devised. The integrity of the weighting functions chosen and the finite-element models considered is demonstrated analytically, and their performance is assessed systematically.

ON A COMPACT MIXED-ORDER FINITE ELEMENT FOR SOLVING THE THREE-DIMENSIONAL INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

SUMMARY Our work is an extension of the previously proposed multivariant element. We assign this refined element as a compact mixed-order element in the sense that use of this element offers a much smaller bandwidth. The analysis is implemented on quadratic hexahedral elements with a view to analysing a three-dimensional incompressible viscous flow problem using a method formulated within the mixed finite element context. The idea of constructing such a stable element is to bring the marker-and-cell (MAC) grid lay-out to the finite element context. This multivariant element can thus be classified as a discontinuous pressure element. We have several reasons for advocating the proposed multivariant element. The primary advantage gained is its ability to reduce the bandwidth of the matrix equation, as compared with its univariant counterparts, so that it can be effectively stored in a compressed row storage (CRS) format. The resulting matrix equation can be solved efficiently by a multifrontal solver owing to its reduced bandwidth. The coding is, however, complicated by the appearance of restricted degrees of freedom at mid-face nodes. Through analytic study this compact multivariant element has a marked advantage over the multivariant element of Gupta et al. in that both bandwidth and computation time have been drastically reduced. # 1997 by John Wiley & Sons, Ltd.

Multi‐dimensional monotone flux discretization scheme for convection dominated flows

Deals with the non‐stationary pure convection equation in two dimensions. An attribute of the method is that the advective fluxes are approximated by taking the flow orientations into consideration. The interfacial numerical fluxes are interpolated by virtue of the rational areas which depend on the corner velocity vectors. This leads to a discrete system containing dissipative artifacts in regions normal to the local streamline. Conducts two‐dimensional fundamental studies for the flux discretization developed. These analyses give insight into the order‐of‐accuracy, and the scheme stability. According to the underlying positivity definition, this explicit scheme is, furthermore, classified as conditionally monotonic. This scheme has been applied successfully to solve smooth, sharply varied, and discontinuous transport problems.