ADDITIONAL PERFORMANCE ASPECTS OF MARCHING METHODS FOR ELLIPTIC EQUATIONS

According to the transformation relationships between the Cartesian coordinates and the general curvilinear coordinates, the governing equations of the model are derived as the forms in the general curvilinear coordinates from those in the Cartesian coordinates. In the model, the contravariant velocities are adopted as the independent variables in non-orthogonal grids. The momentum equations keep strongly conservative forms and the boundary conditions can be given easily. The model used a staggered grid arrangement. The discrete equations are solved using the SIMPLIC algorithms. The numerical model has been validated against the bifurcated flow of which the diversion angle is 30 degree. Compared with the measured values, the numerical shallow water model is shown to be capable of simulating the water domains with irregular boundaries.

RETRACTED: Quartic spline collocation methods for elliptic partial differential equations

ABSTRACTThis article presents a new element-based finite-volume discretization approach for the solution of incompressible flow problems on co-located grids. The proposed method, called the method of proper closure equations (MPCE), employs a proper set of physically relevant equations to constrain the velocity and pressure at integration points. These equations provide a proper coupling between the nodal values of pressure and velocity. The final algebraic equations are not segregated in this study and are solved in a fully coupled manner. To show the applicability and performance of the method, it is tested on several steady two-dimensional laminar-flow benchmark cases. The results indicate that the method simulates the fluid flow in complex geometries and on nonorthogonal computational grids accurately. Also, it is shown that the method is robust in the sense that it does not require severe underrelaxation even at relatively high Reynolds numbers. In each test case, the required underrelaxation parameter, the number of iterations, and the corresponding CPU time are reported.

This textbook covers fundamental and advanced concepts of computational fluid dynamics, a powerful and essential tool for fluid flow analysis. It discusses various governing equations used in the field, their derivations, and the physical and mathematical significance of partial differential equations and the boundary conditions. It covers fundamental concepts of finite difference and finite volume methods for diffusion, convection-diffusion problems both for cartesian and non-orthogonal grids. The solution of algebraic equations arising due to finite difference and finite volume discretization are highlighted using direct and iterative methods. Pedagogical features including solved problems and unsolved exercises are interspersed throughout the text for better understanding. The textbook is primarily written for senior undergraduate and graduate students in the field of mechanical engineering and aerospace engineering, for a course on computational fluid dynamics and heat transfer. The textbook will be accompanied by teaching resources including a solution manual for the instructors. Written clearly and with sufficient foundational background to strengthen fundamental knowledge of the topic. Offers a detailed discussion of both finite difference and finite volume methods. Discusses various higher-order bounded convective schemes, TVD discretisation schemes based on the flux limiter essential for a general purpose CFD computation. Discusses algorithms connected with pressure-linked equations for incompressible flow. Covers turbulence modelling like k-ε, k-ω, SST k-ω, Reynolds Stress Transport models. A separate chapter on best practice guidelines is included to help CFD practitioners.

In this paper, a nonorthogonal locally 1-D finite difference time-domain (LOD-NFDTD) method is presented for electromagnetic (EM) scattering involving curvilinear coordinate system. Formulations of scattered field and convolutional perfectly matched layer absorbing boundary condition in generalized nonorthogonal grids for LOD-NFDTD are also presented. The nonorthogonal grids are used to fully mesh the computational domain which leads to efficient computation. Moreover, the proposed technique requires fewer arithmetic operations than the alternating direction implicit NFDTD method leading to a reduction of CPU time. The numerical dispersion of the proposed method as a function of Courant-friedrich-lewy number (CFLN) is also presented. Computational results for EM scattering from 2-D conducting, dielectric, and coated cylinders as well as overfilled dielectric and bent cavity structures are presented.

An efficient strategy for the solution of N-S Equations using collocated, non-orthogonal grids is presented. The governing equations have been discretized in the physical plane itself without co-ordinate transformation, thereby retaining the lucidity of the basic finite volume method. The non-orthogonal terms and QUICK type corrections for the convective terms in the momentum equations are treated explicitly, while the other terms are taken in implicit form. In the pressure correction equation, the non-orthogonal terms have been dropped altogether. The discretized equations have been solved by the preconditioned conjugate gradient square method. The specific combination of above steps has resulted in better convergence properties as compared to those of existing algorithms, even for highly skewed grids. The scheme has been validated against benchmark solutions such as lid-driven flow in square and skewed cavities and experi mental results of flow over a single cylinder. Its applicability has also been illustrated for flow through a bank of staggered cylinders, with anti-symmetric inlet and outlet boundary conditions. Copyright © 1999 John Wiley & Sons, Ltd.

The laminar flow of viscoplastic fluids inside a lid-driven skewed cavity has been investigated using a numerical scheme based on finite volume method considering Bingham model. Viscoplasticity is characterized by a yield stress, below which the materials behave as solids, and above which they deform and flow according to different constitutive relations. The governing two dimensional unsteady incompressible Navier-Stokes equations were initially non-dimensionalized using appropriate transformation. Then the dimensionless form of these equations is transformed to curvilinear coordinates to simulate complex geometry. The transformed equations are then discretized with appropriate boundary conditions to deal with the non-orthogonal grids. The code is first validated against the existing benchmark results for two-dimensional lid driven square cavity problem considering both Newtonian and non-Newtonian fluids. Then the code is applied to the skewed cavity problem involving non-Newtonian fluid which can be described by the Bingham model. The constitutive equation is regularized as proposed by Papanastasiou [1]. Moreover, grid independence test has been performed for a skewed cavity for different values of Bingham numbers. Reynolds number and Bingham number are two important parameters which can describe the flow behavior of Bingham fluid in the skewed cavity. In this research, the skewness of the geometry has been changed by changing the skew angle. The consequent numerical results are presented in terms of the velocity and streamlines for the different values of Bingham numbers having a different angle of a skewed cavity.

An explicit time-marching procedure to calculate transonic turbine or compressor cascade flow has been developed. The integral unsteady Euler equations are approximated by a conservative finite volume discretization using split space operators in a predictor-corrector scheme. Results are obtained with the physical flow field geometry. At solid walls streamline differentiation is used to update boundary values in a nonorthogonal grid which is automatically generated and sectionally aligned to the boundary. Near the leading and trailing edges and at captured shock waves the computation is stabilized by a switched numerical Shuman-type filter. To minimize computing time the code is vector-optimized for use on a CRAY-1/S computer. In this paper particular details of the calculation procedure are discussed and sample calculations are presented.

This paper deals with the Hessian discretisation method (HDM) for fourth-order semi-linear elliptic equations with a trilinear non-linearity. The HDM provides a generic framework for the convergence analysis of several numerical methods, such as the conforming and nonconforming finite element methods (ncFEMs) and methods based on gradient recovery (GR) operators. The Adini ncFEM and GR method, a specific scheme that is based on cheap, local reconstructions of higher-order derivatives from piecewise linear functions, are analysed for the first time for fourth-order semi-linear elliptic equations with trilinear non-linearity. Four properties, namely, the coercivity, consistency, limit-conformity and compactness, enable the convergence analysis in HDM framework that does not require any regularity of the exact solution. Two important problems in applications, namely, the Navier–Stokes equations in stream function vorticity formulation and the von Karman equations of plate bending, are discussed. Results of numerical experiments are presented for the Morley and Adini ncFEMs, and the GR method.

A new higher order finite element method for elliptic partial differential equations on a stationary smooth surface $\Gamma$ is introduced and analyzed. We assume that $\Gamma$ is characterized as the zero level of a level set function $\phi$ and only a finite element approximation $\phi_h$ (of degree $k \geq 1$) of $\phi$ is known. For the discretization of the partial differential equation, finite elements (of degree $m\geq 1$) on a piecewise linear approximation of $\Gamma$ are used. The discretization is lifted to $\Gamma_h$, which denotes the zero level of $\phi_h$, using a quasi-orthogonal coordinate system that is constructed by applying a gradient recovery technique to $\phi_h$. A complete discretization error analysis is presented in which the error is split into a geometric error, a quadrature error, and a finite element approximation error. The main result is a $H^1(\Gamma)$-error bound of the form $c(h^m + h^{k+1})$. Results of numerical experiments illustrate the higher order convergence of t...

In this paper, a modified intrinsic extended finite element method (XFEM) for one-dimensional and two-dimensional elliptic equations with discontinuous coefficients and interfaces is proposed. We improve the intrinsic XFEM by changing the shape functions of the critical nodes. The improved shape functions can be used to catch the discontinuous information near interfaces. In addition, we modify the Gauss integration in special elements cut by interfaces. Numerical experiments are presented to verify the feasibility and superiority of the modified intrinsic XFEM compared with the standard FEM and extrinsic XFEM for this type of problem. Results also show that the modified intrinsic XFEM can generate an approximate solution whose error is $$O(h^2)$$ in an $$\hbox {L}^2$$ -norm and O(h) in an energy norm if the Q1 element is used.

In this paper, we propose a BDDC preconditioner for the mortar-type rotated finite element method for second order elliptic partial differential equations with piecewise but discontinuous coefficients. We construct an auxiliary discrete space and build our algorithm on an equivalent auxiliary problem, and we present the BDDC preconditioner based on this constructed discrete space. Meanwhile, in the framework of the standard additive Schwarz methods, we describe this method by a complete variational form. We show that our method has a quasi-optimal convergence behavior, i.e., the condition number of the preconditioned problem is independent of the jumps of the coefficients, and depends only logarithmically on the ratio between the subdomain size and the mesh size. Numerical experiments are presented to confirm our theoretical analysis. MSC:65N55, 65N30.

The authors are continuing to develop electromagnetic plasma simulation techniques applicable to geometrically complex domains discretized by a non-orthogonal, unstructured, mixed-polyhedral mesh. Maxwell`s curl equations are advanced in time using the recently developed discrete surface integral (DSI) method. This algorithm preserves, in the absence of particles, the electric and magnetic field divergence constraints and is a direct generalization of the standard staggered-grid finite-difference method. The plasma is represented by particles which move through the unstructured grid obeying all of the normal particle-in-cell approximations. They show details of the DSI field algorithm and of the method for tracking and weighting particles on the distorted mesh elements. Several numerical examples will be presented which demonstrate the present capabilities to integrate the DSI and PIC algorithms. Preliminary calculations which include particle field emission and wave boundary conditions will be presented.

We propose an efficient multigrid algorithm for solving anisotropic elliptic difference equations. The algorithm is based on using Chebyshev’s explicit iterations at smoothing stages and in solving coarse-grid equations. We have developed a procedure for adapting smoothers to anisotropy and present examples, which show that adaptation improves the efficiency of the multigrid method and scalability of the parallel code.

The paper presents the numerical solution of line contact thermal elastohydrodynamic lubrication (EHL) with bio-based lubricant. The model comprises Reynolds equation, film thickness, load balance and energy equations with appropriate boundary conditions by incorporating viscosity–pressure–temperature and density–pressure–temperature relations. Second-order finite difference scheme is used for the discretised form and their equation. The multigrid method with full approximation scheme is used to solve the Reynolds equation along with multilevel multi-integration method for film thickness equation. The pressure, film thickness and temperature distributions for two rolling velocities and various loads with a bio-based lubricant are presented in detail. The present findings yield a reduction in the minimum film thickness for high speed. Details of pressure spike as a function of relevant parameters are given. The results are compared with earlier findings based on different methods.