Construction of meshes for the finite volume method. Finite volume method. Dividing the volume of a cube into finite volumes

Numerical solution of problems of mathematical physics is one of the main methods for studying real phenomena. The combined use of computational and physical experiments in the analysis of any phenomenon allows, on the one hand, to reduce the number of expensive experimental measurements, and on the other hand, to verify and improve mathematical models.

As the speed of computing systems increases, new demands are placed on numerical methods for solving problems of mathematical physics. The development and improvement of modern methods for discretizing conservation laws, which provide the ability to simulate ever new classes of problems and obtain significantly better results when solving known ones, is an important area of ​​research.

Modern computational algorithms should provide the most accurate description of areas with complex geometry. This is possible using non-orthogonal and unstructured meshes. Compared to arbitrary non-orthogonal meshes, for unstructured simplicial meshes (triangulation in the two-dimensional case and partition into tetrahedrons in the three-dimensional case), local condensations are easier to implement (for example, behind a back step, in a zone of sudden narrowing, in the vicinity of the attachment point), and also, if necessary , adaptation of the computational grid depending on the behavior of the solution. Thus, even when discretizing conservation laws in geometrically simple domains that can be accurately represented by a collection of rectangular elements, unstructured simplicial meshes have a number of advantages. Despite the obvious advantages of unstructured grids for approximating arbitrary regions and the possibility of automatically constructing simplicial partitions, they have practically not been used in computational fluid dynamics, and only in the last 15 years have they become increasingly popular. According to the testimony of B. Stoufflett et al., the reason for this is the sharply increasing calculation time when moving to unstructured approaches. The fact is that the position of non-zero elements in the matrices of discrete analogues depends on the contiguity of grid nodes and arbitrarily, the matrices are stored using universal formats and data structures. The operations of multiplying a sparse matrix by a vector and incomplete factorization become much more “expensive”. At the same time, systems of computational fluid dynamics equations are interconnected nonlinear systems of equations, the implicit solution schemes of which have a multi-level iterative nature, so that at each of the “global” iterations it is necessary to solve several systems of linear algebraic equations. It was with the advent of powerful computing systems, as well as thanks to the development of adaptive and multigrid methods, that it became possible to use unstructured grids and corresponding spatial discretization schemes for modeling hydro-gasdynamic processes.

The most common discretization method in the unstructured case is the finite element method (FEM). Let us note such advantages of the method as the preservation of the symmetric nature of the self-adjoint part of differential operators in their discrete analogues (this is achieved by a special choice of the space of test functions coinciding with the space of trial functions), the possibility of increasing the accuracy of approximation by increasing the degree of interpolation polynomials of the local solution representation (the so-called p and h-p versions of FEM, ), natural consideration of boundary conditions of the second and third kind. The finite element method has an established technological basis, in particular

Methods for approximating internal products under the assumption of a piecewise polynomial representation of the solution and parameters of a boundary value problem, namely: the use of basis decomposition of the corresponding finite element space, classes of integral formulas that make it possible to accurately integrate arbitrary products of basis functions over partition elements and edges (faces) of elements,

Standard interpolation apparatus.

The technologies of the method make it possible to simply and uniformly construct discrete analogues of initial boundary value problems, with various types of boundary conditions under the assumption of a certain degree of smoothness of the solution and piecewise polynomial behavior of the coefficients of the equations and boundary conditions , .

In a number of applications, such as modeling supersonic and transonic gas flows and calculations using shallow water models, the local conservatism of the schemes used to discretize conservation laws is very important. The finite element method does not allow one to trace with satisfactory accuracy the features of emerging discontinuous solutions, and the traditional approach to solving such problems is the finite volume method. When discretizing a system of conservation laws by the finite volume method, the computational domain is approximated by a set of open finite volumes, then the researcher takes a “step back”, moving to the integral form of the original system of equations; Using the Ostrogradsky-Gauss formula, we move from volume integration to boundary integral, so that the method of approximating flows through the faces of finite volumes completely determines the computational scheme. According to S. Patankar's monograph, "for most researchers working in the field of hydrodynamics and heat transfer, the finite element method still seems shrouded in mystery. The variational formulation and even the Galerkin method do not lend themselves to simple physical interpretation." At the same time, finite volume schemes have a certain physical meaning of the balance of fluxes and source terms in each of the finite volumes that approximate the computational domain, which makes the finite volume method more attractive. The “simplicity” of MKO is one of the reasons for the lack of a general technological basis for the method.

So, the advantages of the classical version of the MKO (finite volume/finite difference method, FVDM) include the local conservatism of discrete schemes, greater simplicity and clarity, and the possibility of naturally taking into account boundary conditions of the second kind. In addition, in the case of solving problems with a predominance of convection, the implementation of counterflow schemes is simplified, since the flows through the faces of finite volumes are both analyzed and approximated quantities.

Attempts to systematize finite-volume approximations led to a partial combination of FEM technologies and the principle of integration over finite volumes; the earliest of them go back to the work of B. R. Baliga, K. Prakash and S. Patankar and are known as CVFEM methods (control-volume-based finite element methods), hereinafter referred to as finite volume/finite element methods (FVM/FE). The authors of the method pursued the goal of constructing conservative schemes of the finite volume method, using one of the main advantages of FEM - the ability to approximate complex geometries using unstructured meshes. The profile functions in this class of methods are “of an auxiliary nature”; the belonging of the solution to finite element spaces is not emphasized. Barycentric sets are used as a dual partition.

For the first time, the problem of the lack of universal technological principles of the finite volume/finite difference method (MKO/KR, FVDM) is discussed in the work of Z. Kaya “On the finite volume/element method”. The author draws the reader's attention to the "unsystematic nature of the finite volume/finite difference method"; When approximating systems of conservation laws by the finite volume/finite difference method, approximations of various classes can be used within the same work, which significantly complicates the analysis of the convergence of such schemes. A solution to this problem is proposed - the joint use of the ideas of the finite element method (searching for a solution in some finite element space and using the piecewise polynomial behavior of the solution to calculate flows) and the integral form of conservation laws. Thus, finite volume/element methods (FME/E, “box methods”, FVE) arose in an attempt to create “more systematic finite volume technologies”. The absence of general technological principles of finite volume/finite difference methods is also noted in the works of Ya. JI. Guryeva and V.P. Ilyin.

Finite volume/finite element methods (FVE) and finite volume/finite element methods (CVFEM) use consistent finite element spaces of functions linear on simplexes and belong to the class of cell-vertex finite volume schemes, Fig. 1, a.

A number of computational fluid dynamics schemes (modeling of viscous incompressible flows) use inconsistent finite element spaces, in particular, the Crousey-Raviard space linear on elements continuous at the centers of the edges of test functions. Finite volume methods using inconsistent finite element spaces were proposed by S. Choi and D. Kwak, studied in a number of works by other authors (the so-called subvolume methods, covolume method) and are schemes with the calculation of unknowns at the centers of edges (

The most common schemes for solving gas dynamics problems and modeling anthropogenic disasters using shallow water equations are cell-centered finite volume schemes, Fig. 1, f. Their popularity is due to the fact that in the case of calculating unknowns in centroids, most gas dynamics schemes (S.K. Godunov schemes, TVD schemes) can be transferred to unstructured grids without fundamental technological changes. o a in

Fig.1. Location of calculation points in relation to the FE grid nodes.

This work primarily considers classes of finite volume methods with the calculation of unknowns at triangulation nodes (MKO/E, MKO/FE) and edge centers (subvolume methods); in the future we will also say “finite volume methods using finite element spaces.” These classes of methods, according to a number of studies (, ), for convection-diffusion problems provide better approximations to the solution than methods with the calculation of unknowns at the centers of cells. One of the main reasons is that for the methods listed above, the continuity of the first derivatives of test functions on the elements of the dual mesh is preserved.

An effective approach to solving problems with a predominance of convection is the use of the Galerkin method with symmetric test functions for the self-adjoint part of differential operators and upstream MCO schemes for their asymmetric part, the so-called. mixed finite element/volume methods (FEM/O, MEV, mixed element/volume method).

The dissertation work is devoted, in particular, to improving the technologies of the finite volume method for the indicated classes of methods (MKO/E, MKO/CE, FEM/O, subvolume methods). At the moment, these methods do not have established technologies for taking into account the piecewise polynomial behavior of the solution, source terms and transfer coefficients. We can list the following reasons for the imperfection of the apparatus for exact integration of polynomials in finite volume methods using finite element spaces:

1. Unlike the finite element method, the finite volume method does not have a p-version, since with the introduction of additional nodes and several types of dual meshes, the local conservatism of a number of variables of the system of conservation laws in relation to “alien” finite volumes is violated. Thus, approximations are limited to lower order finite element spaces.

2. Compared to the finite element method, finite volume methods are characterized by greater freedom in choosing the spaces of test functions, which in this case turn out to be related to the location of the points of calculation of the unknowns in relation to the discretization nodes (schemes with the location of the unknowns at nodes, midpoints of edges, centroids simplexes) and the method of constructing a dual grid (using barycentric, orthocentric, circumcentric sets). Combined with the ability to use collocated or staggered grids, this gives the full variety of existing MCM schemes in each application.

For MCM methods for discretizing conservation laws that use finite element spaces, a careful selection of these spaces for solution, coefficients of equations and source terms partly loses its meaning if the method does not have developed means of taking into account piecewise polynomial representations, in particular, the apparatus for exact integration of polynomials over elements of the dual mesh, subdomains of elements and segments of border edges. As a consequence, the results of calculations using the constructed schemes should be considered from the point of view of the effects of numerical integration, taking into account various methods of their implementation; it becomes significantly more difficult to compare research results with the works of other authors, etc.

So, this work is devoted to the revision of existing MCM/FEM technologies for constructing discrete analogues of convection-diffusion type problems.

The technology for taking into account the piecewise polynomial representation of the solution, the coefficients of the equation and those included in the boundary conditions, as well as source terms in finite volume methods using finite element spaces must satisfy the following requirements:

1) allow arbitrary combinations of polynomial representations of coefficients and solutions on partition elements, as well as increasing the degree of interpolation polynomials of the local solution representation;

2) use unified principles of approximation when calculating the contributions of elements corresponding to various terms of the equation (diffusion, convective, reaction terms, source terms), as well as contributions from edges that approximate parts of boundaries with different types of boundary conditions specified on them;

3) allow a homogeneous generalization to the three-dimensional case;

4) take into account the experience of well-developed finite element technologies, in particular, the use of basis expansion of finite element spaces and the advantages of accurately integrating piecewise polynomial representations of the solution and transfer coefficients;

5) provide a unified technological basis for mixed FEM/O approximations that use two sets of test functions - finite volume and finite element - to approximate one equation;

6) the principles of technology should remain unchanged during the transition from the use of consistent finite element spaces (finite volume/finite element methods with calculation of unknowns at nodes) to the use of inconsistent finite elements (methods with calculation of unknowns at the centers of triangulation edges);

7) the technology can be used to approximate various classes of physical problems.

Of the existing technologies of finite volume methods using finite element spaces (finite volume/element methods (FVE), finite volume/finite element methods (CVFEM), subvolume methods, mixed volume/element methods (MEV)), none one does not meet the above requirements. Thus, the creation of new technologies for these classes of methods using simplicial partitions and barycentric sets as dual ones seems to be a relevant research topic.

In the case of a significant predominance of convection, comparison of various MCO discretization schemes, as well as comparison of calculations using the finite element method and the finite volume method, actually comes down to a comparison of the corresponding upwind schemes.

The most studied and often used in the unstructured case are upwind schemes of the class of finite volume methods with the calculation of variables at the centers of cells. Despite the fact that the edges of the partition elements are not parallel to the coordinate axes, these schemes in most cases are of a one-dimensional nature, since they come down to solving the problem of the decay of a discontinuity on the lines connecting the centroids of simplexes. Calculations using such schemes do not reproduce the multidimensional structure of the flow and have excessive numerical diffusion. To construct upstream second-order approximation schemes, a significant expansion of the template is necessary, which in the unstructured case leads to a significant complication of the corresponding data structures.

Upstream schemes for schemes with the calculation of unknowns at triangulation nodes and the midpoints of its edges are currently few in number (see). In some cases, the upstream approximation principle comes down to using one value of a scalar substance - at a simplex node lying upstream, or two weighted values ​​- at the ends of a simplex edge lying upstream. Only one of the known schemes, the FLO (Flow Oriented Upwind Scheme), developed by K. Prakash and S. Patankar, takes advantage of the calculation of unknowns at the nodes - the ability to construct asymmetric profile functions. But calculations using this scheme are considered unsatisfactory, since the scheme does not have the property of positivity, and iterative processes often diverge.

Estimating the numerical diffusion introduced by the use of upwind schemes on simplicial grids is a problem in its own right. Existing works in this direction, providing theoretical estimates of convergence characteristics, are limited to a variety of schemes for calculating variables in the centers of cells. Therefore, estimating the convergence rate of upwind MCO/FE schemes using a series of numerical experiments is of particular importance.

So, the construction and comparative analysis of upwind MCO/FE schemes on unstructured grids is a current research topic.

The goal of the work is to develop computational technologies for finite volume methods using finite element spaces to approximate problems of convective-diffusion type. To achieve this goal, the following research objectives were formulated:

1) improvement of technologies for discretizing systems of conservation laws using the finite volume/finite element method on simplicial grids, using barycentric partitions as dual partitions;

2) development of technologies for approximation of problems of convective-diffusion type with significant first derivatives; construction, implementation and comparative analysis of upstream schemes on unstructured grids, in particular, conducting computational experiments to assess the order of approximation of the proposed and most accurate known schemes, as well as comparison of the characteristics of upstream schemes based on MCE/FE and FEM;

3) creation, based on the developed technologies, of software packages that make it possible to adequately simulate viscous incompressible flows of liquids and gases in geometrically complex areas, in stationary and non-stationary cases.

Research methods. Methods of computational mathematics. Comparative analysis of technologies for exact integration of polynomials in finite element methods, finite volumes/elements, distributed residuals. Experimental assessment of the convergence rate of upwind flow schemes for problems with an analytical solution. Calculations on a set of condensed finite element partitions, followed by analysis of convergence to experimental data.

The scientific novelty of the work is as follows:

1. A new technology is proposed for taking into account the piecewise polynomial representation of the solution, transfer coefficients and source terms when discretizing initial-boundary value problems using the methods of finite volumes/elements, finite volumes/finite elements and subvolumes. The technology is based on the use of basis expansion of finite element spaces in terms of barycentric simplicial coordinates, with further exact integration of their monomials. For the MKO/FE, MKO/E schemes with the calculation of variables at triangulation nodes, three classes of formulas for the exact integration of monomials of barycentric coordinates are proposed: over segments of the dual mesh in an element, over barycentric subregions and segments of boundary edges. For subvolume methods using inconsistent finite element spaces, it is proposed to use the principle of exact integration of basis functions and the corresponding integral formulas are obtained.

2. A method is proposed for constructing upwind MCO/FE schemes on simplicial grids, based on separate approximation of mass fluxes and scalar substance values ​​on segments of the dual grid. The concepts of a local matrix of weight coefficients of a countercurrent scheme, internal to the elements of the schemes, and local positivity of the schemes are introduced. A counterflow scheme of exponential class has been proposed, and its analogue has been constructed for the MKO with the calculation of unknown simplexes in barycenters.

3. Experimental estimates of the convergence rate of the upwind scheme with weighing of mass flows and the proposed exponential class scheme were obtained. Using solutions of the boundary layer type, the stability of the constructed schemes was analyzed and compared with upstream FEM schemes.

4. Using the proposed approximation technologies for convection-diffusion type problems, a set of programs was created for modeling viscous incompressible flows in natural velocity-pressure variables and a number of computational experiments were carried out confirming the effectiveness of the constructed schemes.

Structure and scope of the dissertation. The dissertation consists of an introduction, four chapters, a conclusion, a list of references, an appendix and contains 173 pages, including 10 tables and 51 figures. The bibliography contains 117 titles.

Conclusion

This work is devoted to the development of computational technologies for finite volume methods on simplicial grids, using finite element spaces and barycentric partitions as dual ones for approximating problems of convective-diffusion type! The work obtained the following main results for defense:

1. A new technology is proposed for taking into account the piecewise polynomial representation of the solution, transfer coefficients and source terms when discretizing initial-boundary value problems using the methods of finite volumes/elements, finite volumes/finite elements and subvolumes. The technology is based on the use of basis expansion of finite element spaces in terms of barycentric simplicial coordinates, with further exact integration of their monomials. For the MKO/FE, MKO/E schemes with the calculation of variables at triangulation nodes, three classes of formulas for the exact integration of monomials of barycentric coordinates are proposed: over segments of the dual mesh in an element, over barycentric subregions and segments of boundary edges. For subvolume methods using inconsistent finite element spaces, it is proposed to use the principle of exact integration of basis functions and the corresponding integral formulas are obtained.

2. A method is proposed for constructing upwind MCO/FE schemes on simplicial grids, based on separate approximation of mass fluxes and scalar substance values ​​on segments of the dual grid. The concepts of a local matrix of weight coefficients of an upstream scheme, internal to the elements of the schemes, and local positivity of the schemes are introduced. A counterflow scheme of exponential class has been proposed, and its analogue has been constructed for the MKO with the calculation of unknown simplexes in barycenters.

3. Experimental estimates of the convergence rate of the upwind scheme with weighing of mass flows and the proposed exponential class scheme were obtained. Using solutions of the boundary layer type, the stability of the constructed schemes was analyzed and compared with upstream FEM schemes. It is shown that the completed MCO/FE schemes make it possible to track the features of boundary layer solutions much more accurately than the schemes of the Petrov-Galerkin method with asymmetric basis functions (Legendre polynomials), finite element schemes of Rice and Schnipke, as well as combined finite element schemes of a higher order of approximation developed by T. Sheu, S. Wang, and S. Tsai.

4. Using the proposed approximation schemes for convective-diffusion type problems, a set of programs was created for modeling viscous incompressible flows in natural velocity-pressure variables, on combined grids, using interpolation polynomials of pressure and velocity of the same order; A number of computational experiments were carried out to confirm the effectiveness of the constructed circuits.

5. For a reference flow in a channel behind a backward step, the interaction of the input effect and the effect of using upstream approximations is shown for the first time.

So, the technology proposed in the work for discretizing initial-boundary value problems using the finite element/finite volume method on simplicial grids is an effective way to approximate systems of conservation laws, the developed upwind schemes have good convergence characteristics, and the use of discretization methods for a system of Navier-Stokes equations with the same order interpolation for the components of the velocity-pressure vector allows one to obtain results that are in good agreement with experimental data. Classes of finite volume/finite element methods on simplicial meshes, the technological basis of which is the exact integration of monomials of barycentric coordinates, are effective methods for modeling viscous incompressible flows in areas with complex boundary geometry.

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