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Finite-Volume Methods for Anisotropic Diffusion Problems on Skewed Meshes
Abstract
A direct numerical method for the anisotropic diffusion equation is presented and the balance-point method and adaptive method are also derived. All these methods are formulated and implemented in the in-house fluid flow solver GTEA. Two test cases, an isotropic problem and an anisotropic problem with exact solution on a skewed mesh, are chosen for comparison and validation. The error and computation time are illustrated. It is concluded that the direct method has the least computation time among all the diffusion problems and that all the methods meet the precision requirement in engineering computation even on skewed meshes.
... Zielinski and Voller [13] present a numerical approach to solution of space-fractional heat conduction equations. It is noteworthy that, in the context of the present work, Demirdžić [14] examined different discretization techniques of the diffusion term when using Cartesian, structured body-fitted and unstructured meshes; and Liu et al. [15] discussed relevant aspects on how distorted meshes affect computation of isotropic and anisotropic heat conduction. ...
... It is noteworthy that the EbFVM was able to handle a mesh with substantial element distortion with local and global errors somewhat smaller than those obtained by finite elements. The reader is referred to the work of Liu et al. [15], who presented a recent study focused solely on issues associated with mesh distortion and the FVM for heat conduction in isotropic and anisotropic materials. ...
... Nevertheless, the average mesh refinement ratio, r � 2:00185, measured by the average mesh size approximates Richardson's theoretical ratio (r ¼ 2). Therefore, the observed error order, p h 1 ;U , and corresponding error e R h 1 ;U for the global heat transfer coefficient are estimated from similar considerations which gave rise to Eq. (15). Table 4 shows the global heat transfer coefficient for meshes h 1 , h 2 , and h 3 and respective Richardson parameters. ...
This work describes the fundamentals of the element-based finite volume method for anisotropic heat conduction within the framework of the finite element space. Patch tests indicate no element inconsistencies or deficiencies when facing mesh distortion and poor aspect ratio. Convergence and accuracy assessments show that the method presents asymptomatic rate of convergence with discretization errors approaching a second-order scheme. Anisotropic heat conduction in a periodical solid lattice illustrates the application of the method. Application of an optimization technique demonstrates that the choice of a proper material orientation when manufacturing the solid lattice can increase the global heat transfer coefficient.
... However, the FVM based a classic two-point flux formula is inconsistent in anisotropic problems 10,11 . Multiple-point flux approximations 12,13,11 , e.g., generalized finite difference method, have to be employed. And the same as the FEM, the FVM also suffers from the mesh distortion problem and may have difficulties in analyzing problems with crack developments. ...
... Yet unlike the conventional FVMs, in which the gradient of temperature ∇ is interpolated on each subdomain boundary independently, in the present PG-FPM, ∇ is approximated on the Internal Points and is assumed constant in the entire subdomain. Consequently, the local discretization is consistent for both isotropic and anisotropic materials, whereas the two-point discretization scheme widely used in conventional FVMs is only consistent for isotropic problems 12,13 . The conservation laws are weakly satisfied by the IP Numerical Flux Corrections in the current PG-FPM-2. ...
Three kinds of Fragile Points Methods based on Petrov-Galerkin weak-forms (PG-FPMs) are proposed for analyzing heat conduction problems in nonhomogeneous anisotropic media. This is a follow-up of the previous study on the original FPM based on a symmetric Galerkin weak-form. The trial function is piecewise-continuous, written as local Taylor expansions at the Fragile Points. A modified Radial Basis Function-based Differential Quadrature (RBF-DQ) method is employed for establishing the local approximation. The Dirac delta function, Heaviside step function, and the local fundamental solution of the governing equation are alternatively used as test functions. Vanishing or pure contour integral formulation in subdomains or on local boundaries can be obtained. Extensive numerical examples in 2D and 3D are provided as validations. The collocation method (PG-FPM-1) is superior in transient analysis with arbitrary point distribution and domain partition. The finite volume method (PG-FPM-2) shows the best efficiency, saving 25% to 50% computational time comparing with the Galerkin FPM. The singular solution method (PG-FPM-3) is highly efficient in steady-state analysis. The anisotropy and nonhomogeneity give rise to no difficulties in all the methods. The proposed PG-FPM approaches represent an improvement to the original Galerkin FPM, as well as to other meshless methods in earlier literature.
... As mentioned above, numerous studies related to the discretization of the anisotropic diffusion term have appeared in the literature. Liu et al. [29] reported on two finite volume-based methods for the discretization of the diffusion term designated by the direct (a 13-point scheme) and the balance-point (a nine-point scheme) methods. Umansky et al. [30] discussed the construction of a finite-difference scheme for a strongly anisotropic diffusion equation on a misaligned grid and performed a quantitative assessment of the numerical error for several test cases. ...
In this paper, a fully implicit method for the discretization of the diffusion term is presented in the context of the cell-centered finite volume method. The newly developed fully implicit method is denoted by the modified implicit nonlinear diffusion (MIND) scheme. The method is used to solve several isotropic and anisotropic diffusion problems in two-dimensional domains covered with structured (quadrilateral elements) and unstructured (triangular elements) grid systems. The comparison of generated results with similar ones obtained using the semi-implicit scheme demonstrates the superior robustness and accuracy of the MIND scheme and its good convergence characteristics for all types of meshes.
Three kinds of Fragile Points Methods based on Petrov‐Galerkin weak‐forms (PGFPMs) are proposed for analyzing heat conduction problems in nonhomogeneous anisotropic media. This is a follow‐up of the previous study on the original FPM based on a symmetric Galerkin weak‐form. The trial function is piecewise continuous, written as local Taylor expansions at the Fragile Points. A modified Radial Basis Function‐based Differential Quadrature (RBF‐DQ) method is employed for establishing the local approximation. The Dirac delta function, Heaviside step function, and the local fundamental solution of the governing equation are alternatively used as test functions. Vanishing or pure contour integral formulation in subdomains or on local boundaries can be obtained. Extensive numerical examples in 2D and 3D are provided as validations. The collocation method (PG‐FPM‐1) is superior in transient analysis with arbitrary point distribution and domain partition. The finite volume method (PG‐FPM‐2) shows the best efficiency, saving 25% to 50% computational time comparing with the Galerkin FPM. The singular solution method (PG‐FPM‐3) is highly efficient in steady‐state analysis. The anisotropy and nonhomogeneity give rise to no difficulties in all the methods. The proposed PGFPM approaches represent an improvement to the original Galerkin FPM, as well as to other meshless methods in earlier literature.
Anisotropy problems are widely encountered in practical applications. In the present paper, a parallel and scalable multigrid (MG) method combined with the high-order compact difference scheme is employed to solve an anisotropy model equation on a rectangular domain. The present method is compared with the MG combined with the standard central difference scheme. Numerical results show that the MG method combined with the high-order compact difference scheme is more accurate and efficient than the MG with the standard central difference scheme for solving anisotropy elliptic problems. MG components (restriction, prolongation, relaxation, and cycling) and the corresponding parallelism characteristics are also discussed.
An unstructured finite-volume time-domain method (UFVTDM) is developed to solve two-dimensional transient heat conduction in multilayer functionally graded materials (FGMs). A four-node quadrilateral grid and a three-node triangular grid are employed to deal with mixed-grid problems. The accuracy of the method is improved by treating the quadrilateral grid as a bilinear element with consideration of both the linear term and the constant term. The improvement is validated to be vital to avoid violent numerical oscillation when applying the method to heat point-source problems. The accuracy and capability of the UFVTDM are validated by numerical tests.
The accuracy of the diamond scheme is experimentally investigated for anisotropic diffusion problems in two space dimensions. This finite volume formulation is cell-centered on unstructured triangulations and the numerical method approximates the cell averages of the solution by a suitable discretization of the flux balance at cell boundaries. The key ingredient that allows the method to achieve second-order accuracy is the reconstruction of vertex values from cell averages. For this purpose, we review several techniques from the literature and propose a new variant of the reconstruction algorithm that is based on linear Least Squares. Our formulation unifies the treatment of internal and boundary vertices and includes information from boundaries as linear constraints of the Least Squares minimization process. It turns out that this formulation is well-posed on those unstructured triangulations that satisfy a general regularity condition. The performance of the finite volume method with different algorithms for vertex reconstructions is examined on three benchmark problems having full Dirichlet, Dirichlet-Robin and Dirichlet–Neumann boundary conditions. Comparison of experimental results shows that an important improvement of the accuracy of the numerical solution is attained by using our Least Squares-based formulation. In particular, in the case of Dirichlet–Neumann boundary conditions and strongly anisotropic diffusions the good behavior of the method relies on the absence of locking phenomena that appear when other reconstruction techniques are used.
In this article, the weighted least-squares collocation method (WLSCM) is adopted to deal with two- and three-dimensional heat conduction problems in irregular domains. A radial basis function (RBF) is selected to construct the approximation function. To improve the accuracy and stability, some auxiliary points are increased within the domain of interest. Only inner nodes are used to construct the approximation function, and the equilibrium equations are satisfied not only at collocation points but also at auxiliary points, so the equations should be solved in a least-square sense. A 2-D case that has an analytical solution is simulated by the proposed method and the outcome verifies that the present method can obtain desired accuracy and efficiency. Then the current method is adopted to compute one 2-D and two 3-D cases of engineering heat conduction problems in irregular complex domains. The results show that the present method can deal effectively with the heat conduction problems of both 2-D and 3-D irregular domains.
This paper presents the relationships between some numerical methods suitable for a heterogeneous elliptic equation with application to reservoir simulation. The methods discussed are the classical mixed finite element method (MFEM), the control-volume mixed finite element method (CVMFEM), the support operators method (SOM), the enhanced cell-centered finite difference method (ECCFDM), and the multi-point flux-approximation (MPFA) control-volume method. These methods are all locally mass conservative, and handle general irregular grids with anisotropic and heterogeneous discontinuous permeability. In addition to this, the methods have a common weak continuity in the pressure across the edges, which in some cases corresponds to Lagrange multipliers. It seems that this last property is an essential common quality for these methods.
We develop a local flux mimetic finite difference method for second order elliptic equations with full tensor coefficients
on polyhedral meshes. To approximate the velocity (vector variable), the method uses two degrees of freedom per element edge
in two dimensions and n degrees of freedom per n-gonal mesh face in three dimensions. To approximate the pressure (scalar variable), the method uses one degree of freedom
per element. A specially chosen quadrature rule for the L
2-product of vector-functions allows for a local flux elimination and reduction of the method to a cell-centered finite difference
scheme for the pressure unknowns. Under certain assumptions, first-order convergence is proved for both variables and second-order
convergence is proved for the pressure. The assumptions are verified on simplicial meshes for a particular quadrature rule
that leads to a symmetric method. For general polyhedral meshes, non-symmetric methods are constructed based on quadrature
rules that are shown to satisfy some of the assumptions. Numerical results confirm the theory.
Meshless Local Petrov-Galerkin method (MLPG) is a truly meshless method, in which nodes are only used to construct the interpolation function and Gauss integration is applied on each local sub-domain. In this paper, numerical integration method of irregular domain is proposed. The numerical results show that the present method can deal with numerical integration of irregular domain efficiently and are in agreement with those of analytical solutions and the numerical results from FLUENT.
A new reconstruction algorithm is developed to obtain diffusion schemes
with cell-centered unknowns only. The main characteristic of the new
algorithm is the flexibility of stencils when the auxiliary unknowns are
reconstructed with cell-centered unknowns. The stencils are selected
depending on the mesh geometry and discontinuities of diffusion
coefficients. Moreover, an explicit expression is derived for
interpolating the auxiliary unknowns in terms of cell-centered unknowns,
and the auxiliary unknowns can be defined at any point on the edge. The
algorithm is applied to construct several new diffusion schemes, whose
effectiveness is illustrated by numerical experiments. For anisotropic
problems with or without discontinuities, nearly second order accuracy
is achieved on skewed meshes.
Natural convection and radiation heat transfer interaction commonly exists in engineering problems, and a numerical method for combined natural convection and radiation heat transfer is very important in practical engineering applications. In this article, the finite-volume method (FVM) for radiation is formulated and implemented in the fluid flow solver GTEA on hybrid grids. For comparison and validation, three test cases, an equilateral triangular enclosure and a square enclosure with/without baffles, are chosen. Then, natural convections in a cavity with/without baffles are simulated with the present FVM to take into account the radiation heat transfer effects. All the results obtained by the presented FVM agree very well with the exact solutions as well as results obtained by the zone method. Natural convection under low gravity is researched with Gr = 0.7 and 700, and the radiation effects on the temperature distribution are also studied with variation conduction-radiation numbers, Nr = 0.06, 0.1, and 0.15. It is found that the solutions are sensitive to the conduction-radiation number but do not change very much with the Grashof number.
A new reconstruction algorithm is proposed for constructing cell-centered diffusion schemes on distorted meshes. Its main feature is that edge unknowns are defined at certain balance points, the locations of which depend on the diffusion coefficient and the skewness of grid cells, so as to obtain a two-point reconstruction stencil. Implementing the new algorithm for the approximation of gradients, we extend the IDC (improved deferred correction) scheme, which was proposed by Traoré et al. [P. Traoré, Y. Ahipo, C. Louste, A robust and efficient finite volume scheme for the discretization of diffusive flux on extremely skewed meshes in complex geometries, J. Comput. Phys. 228 (2009) 5148–5159], to handle diffusion problems with discontinuous coefficients. Numerical results demonstrate the accuracy and efficiency of the extended scheme.
We introduce here a new finite volume scheme which was developed for the discretization of anisotropic diffusion problems; the originality of this scheme lies in the fact that we are able to prove its convergence under very weak assumptions on the discretization mesh. c 2007 Academie des sciences/ ´ Editions scientifiques et medicales Elsevier SAS Un nouveau schema volumes finis pour les probl ` emes de diffusion anisotrope :
The modern FGMs composite materials have a complex internal structure due to the fact that they consist of several different phases. In this paper, a sudden cooling process (thermal shock) at the upper side of FGM circular plates having discrete variation of the composite features was analysed. The samples were made of five ceramic layers (each of them 0.5 mm thick): purely Al2O3 layer and composite layers made of Al2O3 matrix and 5, 10, 15, 20 wt% content of ZrO2 (Fig. 1). The mechanical response of the plate is axisymmetric, whereas thermal properties are orthotropic. The non-stationary heat conduction equation was solved for arbitrary smooth or step variation of functions describing properties of the analysed material. The considered boundary conditions obey: perfect cooling and real cooling process of the material by introduction of heat transfer coefficient at the cooled surface. The Fourier–Kirchhoff equation for the cylindrical specimen is solved using two methods. The first one is a finite difference (FD) numerical code based on a generalized alternating direction implicit (ADI) method. In the second approach the considered problem was solved with the finite element method (FEM) of ABAQUS code. The heat transfer coefficient at the cooled surface was estimated with a linear monotonic model and with a linear model applying a step function. The solution gives the distribution of the temperature for different times. The theoretical results were compared with experimental ones [T. Sadowski, M. Boniecki, Z. Librant, A. Nakonieczny, Int. J. Heat Mass Transfer 50 (2007) 4461–4467]. Comparison between two numerical approaches was presented.
This paper establishes the convergence of a multi point flux approximation control volume method on rough quadrilateral grids. By rough grids we refer to a family of refined quadrilateral grids where the cells are not required to approach parallelograms in the asymptotic limit. In contrast to previous convergence results for these methods we consider here a version where the flux approximation is derived directly in the physical space, and not on a reference cell. As a consequence, less regular grids are allowed. However, the extra cost is that the symmetry of the method is lost.
A cell-centered finite volume method is presented for discretizing diffusion operator on general nonconforming meshes. The node values are accurately approximated using a new weighted interpolation formula, in which the calculation of the weight is adaptive to both geometric parameters and diffusion coefficients. It follows that an explicit expression, composed of cell-centered unknowns only, is obtained for the discretization of normal flux. Numerical results demonstrate that linear solutions are reproduced exactly on the nonconforming random grids, and that the convergence rate is close to second order for non-linear or discontinuous problems.
Advanced computational method for transient heat conduction analysis in continuously nonhomogeneous functionally graded materials (FGM) is proposed. The method is based on the local boundary integral equations with moving least square approximation of the temperature and heat flux. The initial-boundary value problem is solved by the Laplace transform technique. Both Papoulis and Stehfest algorithms are applied for the numerical Laplace inversion to obtain the time-dependent solutions. Numerical results are presented for a finite strip and a hollow cylinder with an exponential spatial variation of material parameters.
We propose a new cell-centered diffusion scheme on unstructured meshes. The main feature of this scheme lies in the introduction of two normal fluxes and two temperatures on each edge. A local variational formulation written for each corner cell provides the discretization of the normal fluxes. This discretization yields a linear relation between the normal fluxes and the temperatures defined on the two edges impinging on a node. The continuity of the normal fluxes written for each edge around a node leads to a linear system. Its resolution allows to eliminate locally the edge temperatures as function of the mean temperature in each cell. In this way, we obtain a small symmetric positive definite matrix located at each node. Finally, by summing all the nodal contributions one obtains a linear system satisfied by the cell-centered unknowns. This system is characterized by a symmetric positive definite matrix. We show numerical results for various test cases which exhibit the good behavior of this new scheme. It preserves the linear solutions on a triangular mesh. It reduces to a classical five-point scheme on rectangular grids. For non orthogonal quadrangular grids we obtain an accuracy which is almost second order on smooth meshes.
This paper presents a “simple” boundary element method for transient heat conduction in functionally graded materials, which leads to a boundary-only formulation without any domain discretization. For a broad range of functional material variation (quadratic, exponential and trigonometric) of thermal conductivity and specific heat, the non-homogeneous problem can be transformed into the standard homogeneous diffusion problem. A three-dimensional boundary element implementation, using the Laplace transform approach and the Galerkin approximation, is presented. The time dependence is restored by numerically inverting the Laplace transform by means of the Stehfest algorithm. A number of numerical examples demonstrate the efficiency of the method. The results of the test examples are in excellent agreement with analytical solutions and finite element simulation results.
We derive a cell-centered 2-D diffusion differencing scheme for arbitrary quadrilateral meshes inr-zgeometry using a local support-operators method. Our method is said to be local because it yields a sparse matrix representation for the diffusion equation, whereas the traditional support-operators method yields a dense matrix representation. The diffusion discretization scheme that we have developed offers several advantages relative to existing schemes. Most importantly, it offers second-order accuracy even on meshes that are not smooth, rigorously treats material discontinuities, and has a symmetric positive-definite coefficient matrix. The only disadvantage of the method is that it has both cell-center and face-center scalar unknowns as opposed to just cell-center scalar unknowns. Computational examples are given which demonstrate the accuracy and cost of the new scheme relative to existing schemes.
We investigate the convergence of a finite volume scheme for the approximation of diffusion operators on distorted meshes. The method was originally introduced by Hermeline [F. Hermeline, A finite volume method for the approximation of diffusion operators on distorted meshes, J. Comput. Phys. 160 (2000) 481–499], which has the advantage that highly distorted meshes can be used without the numerical results being altered. In this work, we prove that this method is of first-order accuracy on highly distorted meshes. The results are further extended to the diffusion problems with discontinuous coefficient and non-stationary diffusion problems. Numerical experiments are carried out to confirm the theoretical predications.
A new finite volume method is presented for discretizing general linear or nonlinear elliptic second-order partial-differential equations with mixed boundary conditions. The advantage of this method is that arbitrary distorted meshes can be used without the numerical results being altered. The resulting algorithm has more unknowns than standard methods like finite difference or finite element methods. However, the matrices that need to be inverted are positive definite, so the most powerful linear solvers can be applied. The method has been tested on a few elliptic and parabolic equations, either linear, as in the case of the standard heat diffusion equation, or nonlinear, as in the case of the radiation diffusion equation and the resistive diffusion equation with Hall term.
Numerical Simulation of Heat Conduction Problems on Irregular Domain by Using MLPG Method-Chinese Works
- Li Wu Xue-Hong
- Shen Zeng-Yao
- Sheng-Ping