Development of a mixed control volume – Finite element method

SIAM Journal on Scientific Computing

We present a fourth-order accurate algorithm for solving Poisson's equation, the heat equation, and the advection-diffusion equation on a hierarchy of block-structured, adaptively refined grids. For spatial discretization, finite-volume stencils are derived for the divergence operator and Laplacian operator in the context of structured adaptive mesh refinement and a variety of boundary conditions; the resulting linear system is solved with a multigrid algorithm. For time integration, we couple the elliptic solver to a fourth-order accurate Runge-Kutta method, introduced by Kennedy and Carpenter [Appl. Numer. Math., 44 (2003), pp. 139-181], which enables us to treat the nonstiff advection term explicitly and the stiff diffusion term implicitly. We demonstrate the spatial and temporal accuracy by comparing results with analytical solutions. Because of the general formulation of the approach, the algorithm is easily extensible to more complex physical systems.

International Journal for Numerical Methods in Fluids, 2007

CFD modelling of 'real-life' thermo-fluid processes often requires solutions in complex three-dimensional geometries, which can result in meshes containing aspects that are badly distorted. Cell-centred finite volume methods (CC-FV), typical of most commercial CFD tools, are computationally efficient, but can lead to convergence problems on meshes that feature cells with highly non-orthogonal shapes. The control volume-finite element method (CVFE) uses a vertex-based approach and handles distorted meshes with relative ease, but is computationally expensive. A combined vertex-based-cell-centre technique (CFVM), detailed in this paper, allows solutions on distorted meshes where purely cell-centred solutions procedures fail. The method utilizes the ability of the vertex-based approach to resolve the flow field on a distorted mesh, enabling well established cell-centred physical models to be employed in the solution of other transported quantities. The vertex-based flow code is verified against a manufactured 3D solution and error norms are compared on meshes with various degrees of distortion. The CFVM method is validated with benchmark solutions for thermally driven flow and turbulent flow. Finally, the method is illustrated on three-dimensional turbulent flow over an aircraft wing on a distorted mesh where purely cell-centred techniques fail. The CFVM is relatively straightforward to embed within generic CC based CFD tools allowing it to be employed in a wide variety of processing applications. but also on the ability to solve on a mesh that matches the true geometry of the physical domain. Significant advances have been made in the development of numerical methods for the solution of flow on unstructured meshes. The most frequently used unstructured mesh discretization methods are the finite element method (FEM) [1] and the finite volume method (FV) . However, it is the Finite Volume principles, of applying conservation laws locally to control volumes, which lend themselves to easy physical interpretation (such as, fluxes, source terms and local conservation principles). This has led to the FV method being the preferred technique used in most of the commercially available CFD codes. Even successful finite element CFD codes, such as, FLITE3D [3], originally developed at the University of Wales Swansea and now extensively used within the aerospace industry , are formulated to ensure local conservation using a finite volume framework. There are a number of FV approaches, usually either vertex-centred, where the unknowns are defined at the mesh nodes, or cell-centred, where the unknowns are defined at the element centroid. The most widely used is the cell-centred (CC) approach, which is employed in many CFD codes (e.g. CFX [5], FLUENT [6], STAR-CD [7] and PHYSICA ). This technique is computationally efficient, on a highly orthogonal mesh, using simple approximations to discretize the terms in the transport equation, it has low memory requirements and fast simulation times. However, the method is not robust on a highly nonorthogonal mesh. Corrections have to be made to the usual discretization process to account for non-orthogonality in the mesh, see Section 3. On distorted grids these corrections are only first-order accurate . Local errors appear in the solution dependent on the extent of mesh distortion and the solution may diverge. Peric [10] investigated this effect on distorted meshes and proposed grid refinement and smoothing to reduce the error. Moulinec [11] investigated several different interpolation schemes to improve the derivatives at the cell faces on distorted meshes. Other authors proposed different interpolation schemes based on Taylor series expansions. Lehnhauser reported significantly improved accuracy with a multi-dimensional Taylor series expansion scheme, which ensured second-order accuracy, even on strongly distorted meshes. The problems encountered in discretizing the transport equations on distorted meshes are exacerbated when employing pressure-correction schemes in the solution of the incompressible Navier-Stokes equations. Addition of the non-orthogonal correction terms in the pressure-correction equation is expensive and complex and it is common practice to omit these terms . However, omission or simplification of these terms can introduce stability problems into the solution process and lead to difficulties with convergence on highly distorted meshes. Since in complex geometries it is often not possible to have a good quality mesh over the entire domain, many authors have sought to address this problem .

arXiv (Cornell University), 2016

Finite volume methods (FVMs) constitute a popular class of methods for the numerical simulation of fluid flows. Among the various components of these methods, the discretisation of the gradient operator has received less attention despite its fundamental importance with regards to the accuracy of the FVM. The most popular gradient schemes are the divergence theorem (DT) (or Green-Gauss) scheme, and the least-squares (LS) scheme. Both are widely believed to be secondorder accurate, but the present study shows that in fact the common variant of the DT gradient is second-order accurate only on structured meshes whereas it is zeroth-order accurate on general unstructured meshes, and the LS gradient is second-order and first-order accurate, respectively. This is explained through a theoretical analysis and is confirmed by numerical tests. The schemes are then used within a FVM to solve a simple diffusion equation on unstructured grids generated by several methods; the results reveal that the zeroth-order accuracy of the DT gradient is inherited by the FVM as a whole, and the discretisation error does not decrease with grid refinement. On the other hand, use of the LS gradient leads to second-order accurate results, as does the use of alternative, consistent, DT gradient schemes, including a new iterative scheme that makes the common DT gradient consistent at almost no extra cost. The numerical tests are performed using both an in-house code and the popular public domain PDE solver OpenFOAM.

Computational Mechanics, 1994

Recent developments in the application of a control-volume-based finite-element method, that has proven successful in solving incompressible flow problems, to the solution of compressible flow problems are presented. The finite Element Differential Scheme (FIELDS) is demonstrated to retain the pressure checker boarding problem for the case of Euler flow under certain conditions of flow. The source of this is investigated

Numerische Mathematik, 1990

The finite volume element method (FVE) is a discretization technique for partial differential equations. It uses a volume integral formulation of the problem with a finite partitioning set of volumes to discretize the equations, then restricts the admissible functions to a finite element space to discretize the solution. This paper develops discretization error estimates for general self-adjoint elliptic boundary value problems with FVE based on triangulations with linear finite element spaces and a general type of control volume.

2005

A new finite element method for the solution of the diffusion-advection equation is proposed. The method uses non-isoparametric exponentially-varying interpolation functions, based on exact, one-and two-dimensional solutions of the Laplace-transformed differential equation. Two eight-noded elements are developed and tested for convergence, stability, Peclet number limit, anisotropy, material heterogeneity, Dirichlet and Neumann boundary conditions and tolerance for mesh distortions. Their performance is compared to that of conventional, eight-and 12-noded polynomial elements.

2006

Two different approaches are proposed to enhance the efficiency of the numerical resolution of optimal control problems governed by a linear advection-diffusion equation. In the framework of the Galerkin-Finite Element (FE) method, we adopt a novel a posteriori error estimate of the discretization error on the cost functional; this estimate is used in the course of a numerical adaptive strategy for the generation of efficient grids for the resolution of the optimal control problem. Moreover, we propose to solve the control problem by adopting a reduced basis (RB) technique, hence ensuring rapid, reliable and repeated evaluations of inputoutput relationship. Our numerical tests show that by this technique a substantial saving of computational costs can be achieved.

2004

Finite volume methods are a class of discretization schemes that have proven highly successful in approximating the solution of a wide variety of conservation law systems. They are extensively used in fluid mechanics, meteorology, electromagnetics, semiconductor device simulation, models of biological processes and many other engineering areas governed by conservative systems that can be written in integral control volume form. This article reviews elements of the foundation and analysis of modern finite volume methods. The primary advantages of these methods are numerical robustness through the obtention of discrete maximum (minimum) principles, applicability on very general unstructured meshes, and the intrinsic local conservation properties of the resulting schemes. Throughout this article, specific attention is given to scalar nonlinear hyperbolic conservation laws and the development of high order accurate schemes for discretizing them. A key tool in the design and analysis of finite volume schemes suitable for discontinuity capturing is discrete maximum principle analysis. A number of building blocks used in the development of numerical schemes possessing local discrete maximum principles are reviewed in one and several space dimensions, e.g. monotone fluxes, TVD discretization, positive coefficient discretization, non-oscillatory reconstruction, slope limiters, etc. When available, theoretical results concerning a priori and a posteriori error estimates are given. Further advanced topics are then considered such as high order time integration, discretization of diffusion terms and the extension to systems of nonlinear conservation laws.

Journal of Aerospace Technology and Management, 2017

In this study conservation equations were implemented along the boundaries via ghost control-volume immersed boundary method. The control-volume finiteelement method was applied on a cartesian grid to simulate 2-D incompressible flow. In this approach, mass and momentum equations were conserved in the whole domain including boundary control volumes by introducing ghostcontrol volume concept. The Taylor problem was selected to validate the present method. Four different case studies of Taylor problem encompassing both inviscid and viscous flow conditions in ordinary and 45° rotated grid were used for more investigation. Comparisons were made between the results of the present method and those obtained from the exact solution. Results of the present method indicated accurate predictions of the velocity and pressure fields in midline, diagonal, and all boundaries. The agreement between the results of the present method and the exact solution was very good throughout the whole temporal domain. Furthermore, comparison of the rate of kinetic energy decay in viscous case showed same level of agreement between the results.

The discretisation of the gradient operator is an important aspect of finite volume methods that has not received as much attention as the discretisation of other terms of partial differential equations. The most popular gradient schemes are the divergence theorem (or Green-Gauss) scheme, and the least-squares scheme. Both schemes are generally believed to be second-order accurate, but the present study shows that in fact the divergence theorem gradient is second-order accurate only on structured meshes whereas it is zeroth-order accurate on general unstructured meshes, and the least-squares gradient is second-order and first-order accurate, respectively. This is explained through a theoretical analysis and is confirmed by numerical tests. Furthermore, the schemes are used within a finite volume method to solve a simple diffusion equation on unstructured grids generated by several methods; the results reveal that the zeroth-order accuracy of the divergence theorem gradient scheme is inherited by the finite volume method as a whole, and the discretisation error does not decrease with grid refinement. On the other hand, use of the least-squares gradient leads to second-order accurate results. These numerical tests are performed using both an in-house code and the popular public domain PDE solver OpenFOAM, which uses the divergence theorem gradient by default.