Moukalled F., Mangani L., Darwish M. The finite volume method in computational fluid dynamics. An advanced introduction with OpenFOAM and Matlab (811443), страница 48
Текст из файла (страница 48)
A corrected averaging between C and F.c. Compare the two computed values with the exact gradient at point f.Fig. 9.23 Two elements in atwo dimensional plane withtheir centroids C and FFfCExercise 3Consider the mesh composed of equilateral triangular elements shown in Fig. 9.24.The coordinates of the mesh vertices are as shown. The temperatures at the cellcentroids are given byT ðx; y; zÞ ¼ 100 x2 þ y2 þ 1a.
Compute the gradient at 1 using the Green-Gauss method.b. Compute the gradient at 1 using the least squares approach with a limited stencil(i.e., nodes 2, 3 and 4) and with an extended stencil (i.e., nodes 2–13) used forthe reconstruction.c. Compare the least squares gradient with the limited and extended stencil and thegauss gradient to the exact gradient.3009Gradient Computation1121113112351610491/37080.51Fig. 9.24 A domain discretized using triangular elementsExercise 4Consider the uniform mesh shown in Fig.
9.25. The coordinates of the meshvertices are as shown. The temperatures at the cell nodes are given byT ðx; yÞ ¼ 100 x3 þ y3 þ xy þ 1compute the gradient at the centroid of cell 1 using the gauss gradient with andwithout face corrections and compare to the exact gradient given byrT ðx; yÞ ¼ 100 3x2 þ y i þ 100 3y2 þ x jFig. 9.25 Mesh system forExercise 4214213(0,0)9.7 Exercises301Exercise 5 (uFVM, OpenFOAM®)Using uFVM and then OpenFOAM® write a program thata. reads an OpenFOAM® mesh and sets a scalar field with values equal to/ðx; y; zÞ ¼ 10x2 y2 þ 3yb. computes the gradients using the second order least square method,c.
computes the gradients using the gauss gradient method with linearinterpolation,d. computes the gradients using the gauss gradient method with vertexinterpolation,e. and computes the root mean square error between each of the computed gradients and the exact gradient.Exercise 6 (uFVM, OpenFOAM®)Using uFVM and then OpenFOAM® write a program thata. reads an OpenFOAM® mesh and sets a scalar field with values equal to/ðx; y; zÞ ¼ 10x2 y2 þ 3xUse the exact gradients at the element nodes to compute the gradients at theinterior faces usingb.c.d.e.simple linear interpolation,corrected linear interpolation,and exact gradient formulation at the face centroids.Compute the root mean square error between each of the computed gradientsand the exact gradient.Exercise 7Starting with Eq.
(9.8) derive Eq. (9.9).Hint: rf rf 0 ¼ rf rC rcf 0 ¼ rf rf rFf 0 .Exercise 8 (OpenFOAM®)a. List all possible gradient type definitions available in OpenFOAM® by modifying the gradSchemes in the fvSchemes dictionary (Hint: just mistype ascheme, e.g., banana, and launch any solver or application, i.e., gradSchemes{default banana;}).b. Define in the dictionary file fvSchemes the option to evaluate the explicit gradient with the least square algorithm.c.
Use the Doxygen documentation [9] to analyze the member functioncorrectBoundaryConditions and explain its operations and aim.d. Define in the dictionary file fvSchemes the option to use a limited gradient (faceand cell) (Gauss limited 0.8;).3029Gradient ComputationReferences1. Ferziger JH, Peric ́M (2002) Computational Methods for Fluid Dynamics, 3rd edn. Springer,Berlin2. Soni B (1998) Hybrid techniques in computational fluid dynamics. Technical Report no.MSSU-COE-ERC-98-3, Engineering Research Center for Computational Field Simulation,Mississippi State University3.
Frink NT, Parikh P, Pirzadeh S (1991) Aerodynamic Analysis of Complex ConfigurationsUsing Unstructured Grids. AIAA 91-32924. Cabello J, Morgon K, Lohner R (1994) A Comparison of Higher Order Schemes Used in aFinite Volume Solver For Unstructured Grids. AIAA 94-2293. Presented at the 25th AIAAPlasmadynamics and Lasers Conference, Colorado Springs, CO5. Musaferija I, Gosman D (1997) Finite-Volume CFD procedure and adaptive error controlstrategy for grids of arbitrary topology.
J Comp Phys 138:766–7876. Barth TJ, Jespersen DC (1989) The design and application of upwind schemes on unstructuredmeshes. AIAA 89-03667. Ollivier-Gooch C, Van Altena M (2002) A high-order-accurate unstructured mesh finite-volumescheme for the advection–diffusion equation. J Comp Phys 181:729–7528. OpenFOAM (2015) Version 2.3.x.
http://www.openfoam.org9. OpenFOAM Doxygen (2015) Version 2.3.x. http://www.openfoam.org/docs/cpp/Chapter 10Solving the System of Algebraic EquationsAbstract The result of the discretization process is a system of linear equations ofthe form A/ ¼ b where the unknowns /, located at the centroids of the meshelements, are the sought after values. In this system, the coefficients of the unknownvariables constituting matrix A are the result of the linearization procedure and themesh geometry, while vector b contains all sources, constants, boundary conditions,and non-linearizable components. Techniques for solving linear systems of equationsare generally grouped into direct and iterative methods, with many sub-groups ineach category.
Since flow problems are highly non-linear, the coefficients resultingfrom their linearization process are generally solution dependent. For this reason andsince an accurate solution is not needed at each iteration, direct methods have beenrarely used in CFD applications. Iterative methods on the other hand have been morepopular because they are more suited for this type of applications requiring lowercomputational cost per iteration and lower memory.
The chapter starts by presentingfew direct methods applicable to structured and/or unstructured grids (Gauss elimination, LU factorization, Tridiagonal and Pentadiagonal matrix algorithms) to set theground for discussing the more widely used iterative methods in CFD applications.Then the performance and limitations of some of the basic iterative methods with andwithout preconditioning are reviewed. This include the Jacobi, Gauss-Siedel,Incomplete LU factorization, and the conjugate gradient methods. This is followedby an introduction to the multigrid method that is generally used in combination withiterative solvers to help addressing some of their important limitations.10.1 IntroductionThe starting point for any linear solver is the set of equations generated by thediscretization process, which are written mathematically asA/ ¼ b© Springer International Publishing Switzerland 2016F.
Moukalled et al., The Finite Volume Method in Computational Fluid Dynamics,Fluid Mechanics and Its Applications 113, DOI 10.1007/978-3-319-16874-6_10ð10:1Þ30330410Solving the System of Algebraic Equationswhere A is the matrix of coefficients of elements aij ; / the vector of unknownvariables /i , and b the vector of sources bi.
Using matrix numbering, the expandedform of Eq. (10.1) is given by2a116 a2166 ..4 .aN1a12a22...aN2......a1N1a2N1....... . . aNN13 2 32b13 /1a1N 6 / 7 6 b 727 6 276a2N 776 .. 7 6 .. 7.. 76 . 7 ¼ 6 . 77 6 7. 564 ... 5 4 ... 5aNNbN/Nð10:2ÞGenerally each row in the above matrix represents an equation defined over oneelement of the computational domain, and the non-zero coefficients are thoserelated to the neighbors of that element.
The coefficient aij measures the strength ofthe link between the value of /i at the centroid of the control volume and itsneighbors. As a cell is connected to only few neighbors, with their numberdepending on the connectivity of the elements in the discretized domain, many ofthe coefficients are zeros and the resulting A matrix is always sparse (i.e., thenon-zero coefficients are a very small fraction of the matrix).
If in addition themethod uses a structured grid system, the matrix A will be banded with all non-zeroelements aligned along few diagonals. Therefore methods for efficiently solvingsuch systems should exploit this characteristic.As mentioned above techniques for solving algebraic systems of equations arebroadly divided into two categories denoted by direct and iterative methods,respectively.
In a direct method, matrix A is inverted and the solution / is computed in one step as / ¼ A1 b. When the matrix A is large, computing its inverse iscomputationally very expensive requiring large memory. It is basically impracticalto apply direct linear solvers in CFD applications as they generally involve nonlinear systems of equations with their coefficients depending on the solutionnecessitating the use of an iterative process.On the other hand, with iterative algebraic solvers the solution algorithm isrepeatedly applied as many times as required until a pre-assigned level of convergence is reached without the need for a fully converged solution be attained atevery iteration.The presentation starts with few direct linear solvers applicable to structured andunstructured grid methods. This is followed by a description of solution algorithmsthat take advantage of the banded structure of the coefficient matrix in structuredgrid systems.
The main focus of the chapter is, however, on a specific class ofiterative linear algebraic solvers that has been identified to be generally very efficient and economical with the FVM, and has been exclusively implemented as thelinear solver in almost all finite volume-based codes.10.2Direct or Gauss Elimination Method30510.2 Direct or Gauss Elimination MethodEven though direct methods are not efficient at solving sparse systems of linearalgebraic equations due to their high computational cost, their discussion will pavethe way for introducing efficient iterative methods in the next section.
The simplestdirect method for finding solutions to the system of equations described byEq. (10.1) is the Gauss elimination technique, which will be described first. Thetransformation of the system into an equivalent upper triangular system, when usingthe Gauss elimination method, has motivated the development of the Lower-Upper(LU) triangulation method, which will also be presented. In this approach, matrixA is decomposed into the product of two matrices L and U with L being a lowertriangular matrix and U an upper triangular one.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
