CN115455853A - Three-order compact reconstruction method based on finite volume method on unstructured grid - Google Patents

Three-order compact reconstruction method based on finite volume method on unstructured grid Download PDF

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CN115455853A
CN115455853A CN202211076411.XA CN202211076411A CN115455853A CN 115455853 A CN115455853 A CN 115455853A CN 202211076411 A CN202211076411 A CN 202211076411A CN 115455853 A CN115455853 A CN 115455853A
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张佳旺
李震
李�浩
琚亚平
张楚华
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Xian Jiaotong University
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Abstract

The invention discloses a three-order compact reconstruction method based on a finite volume method on an unstructured grid, which comprises the following steps: constructing an unstructured grid; reading coordinates of grid nodes in the unstructured grid and connection relations among grid units; calculating an integral average value of a reconstruction basis function in a grid unit according to the grid node coordinates; calculating an average flow field according to a finite volume method; calculating a reconstruction relation weight coefficient according to density distribution in a flow field; constructing a first derivative relation matrix and a second derivative relation matrix based on the weight coefficient, the integral average value of the reconstruction basis function, the connection relation among the grid units and the average flow field; and constructing a reconstruction polynomial second-order derivative term solving equation set, solving to obtain a reconstruction polynomial second-order derivative term, bringing the reconstruction polynomial second-order derivative term into the relation matrix of the first-order derivative and the second-order derivative, solving to obtain a reconstruction polynomial first-order derivative term, and finishing reconstruction.

Description

Three-order compact reconstruction method based on finite volume method on unstructured grid
Technical Field
The disclosure belongs to the technical field of computational fluid mechanics, and particularly relates to a three-order compact reconstruction method based on a finite volume method on an unstructured grid.
Background
Computational fluid mechanics is a method for obtaining an approximate numerical solution of a flow field by dispersing a fluid mechanics control equation set and determining solution conditions through a numerical method on a computer. The high-precision numerical value dispersion method has an important effect on improving the precision of numerical solution and further expanding the application range of computational fluid mechanics. Due to the advantages of processing complex geometric shapes, realizing flexible automatic generation and the like, the computational fluid mechanics method based on the unstructured grid has become a hot spot of current research. The finite volume method is a numerical method which is most widely applied in computational fluid mechanics due to the advantages of automatically meeting the flow conservation characteristic, the convection term windward characteristic and the like. However, the finite volume method widely applied to the unstructured grid at present only has second-order precision, and cannot meet the requirements of high-precision and high-resolution scenes such as pneumatic acoustics, vortex-dominated flow, direct numerical simulation of turbulence, large vortex simulation and the like, and the development of a high-order numerical method is an effective method for solving related problems. The following important challenges are faced in constructing a high-order finite volume method calculation method based on an unstructured grid:
unstructured grids do not have regular topological structures in three spatial dimensions, and a high-order method cannot be independently constructed in each dimension like structured grids. For the finite volume method, the core of the method for constructing high-precision numerical values on an unstructured grid is to construct a high-order reconstruction polynomial function of a reconstruction variable in a grid unit by using an integral average value of the reconstruction variable on a reconstruction template, wherein pioneering work is a k-order accurate reconstruction method, namely the integral average value of the high-order reconstruction function in the unit in the reconstruction template unit is equal to an average value obtained by calculation of the finite volume method, each coefficient of a k-order distribution function is solved by using a least square method, and in addition, a student popularizes a weighting basic non-oscillation reconstruction method on the structured grid to the unstructured grid. Because each grid unit can only provide a reconstruction condition for the conservation of the integral average value in the grid unit, and the number of undetermined coefficients of the reconstruction polynomial is rapidly increased along with the increase of dimensionality and order, the two reconstruction methods, namely a k-order accurate reconstruction method and a weighting basic flutter-free reconstruction method, have the problem of overlarge reconstruction template.
Disclosure of Invention
Aiming at the defects in the prior art, the invention aims to provide a three-order compact reconstruction method based on a finite volume method on an unstructured grid, which does not need to solve an implicit algebraic equation set in the reconstruction process, and has small calculated amount and high solving efficiency; in addition, boundary conditions are considered in the reconstruction process, and third-order reconstruction accuracy with the same boundary grid cells and internal grid cells is achieved.
In order to achieve the above object, the present disclosure provides the following technical solutions:
a three-order compact reconstruction method based on a finite volume method on an unstructured grid comprises the following steps:
s100: constructing an unstructured grid;
s200: reading coordinates of grid nodes in the unstructured grid and connection relations among grid units;
s300: calculating an integral average value of a reconstruction basis function in a grid unit according to the grid node coordinates;
s400: calculating an average flow field according to a finite volume method;
s500: calculating a reconstruction relation weight coefficient according to density distribution in a flow field;
s600: constructing a first derivative relation matrix and a second derivative relation matrix based on the weight coefficient, the integral average value of the reconstruction basis function, the connection relation among the grid units and the average flow field;
s700: and constructing a reconstruction polynomial second-order derivative term solving equation set, solving to obtain a reconstruction polynomial second-order derivative term, bringing the reconstruction polynomial second-order derivative term into the relation matrix of the first-order derivative and the second-order derivative, solving to obtain a reconstruction polynomial first-order derivative term, and finishing reconstruction.
Preferably, in step S300, the integrated average of the reconstruction basis functions in the grid cell is calculated by gaussian integration:
Figure BDA0003829284720000031
wherein omega i Is the volume of grid cell i, N is the number of Gaussian integration points, ω j Is the gaussian integral weight of the jth gaussian integral point,
Figure BDA0003829284720000032
for the ith reconstructed basis function of grid cell i,
Figure BDA0003829284720000033
to reconstruct basis functions
Figure BDA0003829284720000034
Integral mean, cor, within grid cell i i j Is the coordinate of the jth gaussian integration point in grid cell i.
Preferably, in step S500, the reconstruction relation weight coefficient is calculated by the following formula:
Figure BDA0003829284720000035
wherein,
Figure BDA0003829284720000036
is the weight of the adjacent cell nb of the grid cell i, nc is the number of cells adjacent to the i-face of the grid cell, ISS i j The smoothness ISS of each grid cell is the smoothness of the jth adjacent grid cell of grid cell i i Calculated by the following formula:
Figure BDA0003829284720000037
wherein ISS i To be the degree of smoothness of the grid cell i,
Figure BDA0003829284720000038
is the average density in the jth cell adjacent to grid cell i,
Figure BDA0003829284720000039
is the average density within grid cell i.
Preferably, in step S600, the first derivative and the second derivative relation matrix are expressed as follows:
Figure BDA00038292847200000310
wherein,
Figure BDA0003829284720000041
Figure BDA0003829284720000042
Figure BDA0003829284720000043
Figure BDA0003829284720000044
Figure BDA0003829284720000045
Figure BDA0003829284720000046
wherein,
Figure BDA0003829284720000047
the weight coefficients of the reconstructed relation in j cells adjacent to the grid cell i,
Figure BDA0003829284720000048
the mean value of the reconstructed variable in cell i calculated for the finite volume method,
Figure BDA0003829284720000049
average value, x, of reconstructed variables in j cells adjacent to cell i calculated by finite volume method i，j，c Is the abscissa, y, of the center of the jth grid cell adjacent to grid cell i i，j，c J is 1-Nc as the ordinate of the center of the j-th grid cell adjacent to grid cell i, nc is the number of grid cells adjacent to grid cell i, x i，c Is the abscissa, y, of the center of the grid cell i i，c Is the ordinate of the center of the grid cell i,
Figure BDA00038292847200000410
is the integral average of the function f within grid cell i.
In order to remember that,
Figure BDA00038292847200000411
is AuxMat i ，
Figure BDA00038292847200000412
Is AuxVec i 。
Preferably, in step S700, the equation system for solving the second derivative term of the reconstruction polynomial is expressed as follows:
Figure BDA0003829284720000051
wherein,
Figure BDA0003829284720000052
Figure BDA0003829284720000053
wherein,
Figure BDA0003829284720000054
is the kth row vector of the relationship matrix AuxMat of the jth cell adjacent to grid cell i,
Figure BDA0003829284720000055
is the kth row vector of the relationship matrix AuxMat of grid cell i,
Figure BDA0003829284720000056
the k-th component of the relationship matrix AuxVec for the j-th cell adjacent to grid cell i,
Figure BDA0003829284720000057
the k-th component, x, of the relationship matrix AuxVec for grid cell i i，j，c Is the abscissa, y, of the center of the jth cell adjacent to grid cell i i，j，c Is the ordinate, x, of the center of the jth cell adjacent to grid cell i i，c Is the abscissa, y, of the center of the grid cell i i，c Is the ordinate of the center of grid cell i. j ranges from 1 to Nc, and k ranges from 1 to 2.
The present disclosure further provides a third-order compact reconstruction apparatus based on the finite volume method on unstructured grid, including:
the unstructured grid generating module is used for generating unstructured grids;
the preprocessing module is used for reading node coordinates in the unstructured grid and the connection relation between grid units;
the integral calculation module is used for calculating an integral average value of the reconstruction basis function in the grid unit according to the grid node coordinates;
the finite volume method calculating module is used for calculating an average flow field according to a finite volume method;
the weight coefficient calculation module is used for calculating a weight coefficient of a reconstruction relation according to density distribution in a flow field;
the reconstruction relation matrix building module is used for building a first derivative and a second derivative relation matrix based on the weight coefficient, the integral average value of the reconstruction basis function, the connection relation among the grid units and an average flow field obtained by a finite volume method;
and the third-order reconstruction polynomial solving module is used for solving an equation set according to the constructed reconstruction polynomial second-order derivative term, solving to obtain a reconstruction polynomial second-order derivative term, and substituting the reconstruction polynomial second-order derivative term into a first-order derivative and second-order derivative relation matrix to solve to obtain a reconstruction polynomial first-order derivative term.
Preferably, the third-order reconstruction polynomial solving module includes:
the second order derivative item solving submodule is used for solving an algebraic equation system of the reconstruction polynomial second order derivative item to obtain a reconstruction polynomial second order derivative item;
and the first derivative item solving submodule is used for substituting the reconstructed polynomial second derivative item into the first derivative and second derivative item relation matrix to solve and obtain the reconstructed polynomial first derivative item.
Compared with the prior art, the beneficial effect that this disclosure brought does:
1. the problem of large templates of a high-order precision finite volume method on an unstructured grid is solved;
2. the explicit solution of the three-order reconstruction polynomial is realized, and compared with an implicit solution method, the calculation amount is small and the solution efficiency is high;
3. boundary conditions are considered in the reconstruction process, and full-field consistent third-order precision is achieved.
Drawings
FIG. 1 is a flow chart of a method provided by an embodiment of the present disclosure;
FIG. 2 is a schematic diagram of a two-dimensional unstructured grid;
FIG. 3 is a diagram showing the correspondence between the coordinates of Gaussian integration points in the actual unit and the coordinates of a reference unit;
FIG. 4 is a graph of the numerical accuracy results of the reconstruction method obtained by test function testing;
fig. 5 is a streamline distribution diagram of the square cavity drive flow Re =400 calculated by the method.
Detailed Description
Specific embodiments of the present disclosure will be described in detail below with reference to fig. 1 to 5. While specific embodiments of the disclosure are shown in the drawings, it should be understood that the disclosure may be embodied in various forms and should not be limited to the embodiments set forth herein. Rather, these embodiments are provided so that this disclosure will be thorough and complete, and will fully convey the scope of the disclosure to those skilled in the art.
It should be noted that certain terms are used throughout the description and claims to refer to particular components. As one skilled in the art will appreciate, various names may be used to refer to a component. This specification and claims do not intend to distinguish between components that differ in name but not function. In the following description and in the claims, the terms "include" and "comprise" are used in an open-ended fashion, and thus should be interpreted to mean "include, but not limited to. The description which follows is a preferred embodiment of the disclosure, but is made for the purpose of illustrating the general principles of the disclosure and not for the purpose of limiting the scope of the disclosure. The scope of the present disclosure is to be determined by the terms of the appended claims.
To facilitate an understanding of the embodiments of the present disclosure, the following detailed description is to be considered in conjunction with the accompanying drawings, and the drawings are not to be construed as limiting the embodiments of the present disclosure.
In one embodiment, as shown in fig. 1, a third-order compact reconstruction method based on finite volume on unstructured grid method includes the following steps:
s100: constructing an unstructured grid;
s200: reading coordinates of grid nodes in the unstructured grid and connection relations among grid units;
s300: calculating an integral average value of a reconstruction basis function in a grid unit according to the grid node coordinates;
s400: calculating an average flow field according to a finite volume method;
s500: calculating a weight coefficient of a reconstruction relation according to density distribution in a flow field;
s600: constructing a first derivative relation matrix and a second derivative relation matrix based on the weight coefficient, the integral average value of the reconstruction basis function, the connection relation among the grid units and the average flow field;
s700: and constructing a reconstruction polynomial second-order derivative term solving equation set, solving to obtain a reconstruction polynomial second-order derivative term, bringing the reconstruction polynomial second-order derivative term into the relation matrix of the first-order derivative and the second-order derivative, solving to obtain a reconstruction polynomial first-order derivative term, and finishing reconstruction.
The above embodiments constitute a complete technical solution of the present disclosure. Compared with the prior art, the technical scheme of the embodiment has the following advantages: 1. the problem of large templates of a high-order precision finite volume method on an unstructured grid is solved; 2. the explicit solution of the three-order reconstruction polynomial is realized, and compared with an implicit solution method, the calculation amount is small and the solution efficiency is high; 3. boundary conditions are considered in the reconstruction process, and full-field consistent third-order precision is achieved.
In another embodiment, in step S100, an unstructured grid, such as the unstructured grid shown in fig. 2, is generated by an unstructured grid generating program, and the grid includes two grid cells, five grid edges, one internal grid edge and four boundary grid edges, and four grid nodes.
In another embodiment, in step S200, grid node coordinates of the unstructured grid are read, and in the unstructured grid shown in fig. 2, grid node n is 1 -n 4 Are numbered 1-4 in sequence, and grid cell E 1 、E 2 Are numbered 1 and 2, respectively, and the grid boundary e 1 -e 5 The numbers of the two are 1 to 5 in sequence.
The coordinates of the grid nodes are:
n 1 ：(0，0)
n 2 ：(1，0)
n 3 ：(0，1)
n 4 ：(1，1)
the connection relationship of the grid cells includes:
internal grid edge:
2 3 1 2
boundary grid edge:
4 3 2 0
3 1 1 0
1 2 1 0
2 4 2 0
the unit connection relation rule is as follows: the first two numbers represent the numbers of two end points of the grid boundary, wherein the direction of the first point pointing to the second point is the direction of the grid boundary; the last two numbers represent the numbers of two grid cells connected by a grid boundary formed by the first two grid points, wherein the normal direction inside the grid boundary points to the grid cell represented by the third number, and the normal direction outside the grid boundary points to the grid cell represented by the fourth number. Note: the units outside the computation domain are denoted by 0.
In another embodiment, in step S300, according to the two-dimensional Taylor expansion:
Figure BDA0003829284720000091
wherein u is i (x, y) is a reconstruction polynomial function within grid cell i, (x) i，c ，y i，c ) Is the coordinate of the center of the grid cell i,
Figure BDA0003829284720000101
to reconstruct a polynomial u i The order derivatives at the center of the grid cell.
Taking integral average value of the reconstruction polynomial (1) in a grid unit i to obtain:
Figure BDA0003829284720000102
wherein omega i Is the volume of the grid cell i,
Figure BDA0003829284720000103
is the integral average of the reconstructed polynomial in grid cell i.
Substituting formula (1) into formula (2) yields:
Figure BDA0003829284720000104
wherein,
Figure BDA0003829284720000105
is the integrated average of the function f within grid cell i.
Substituting formula (3) into formula (1) for u in formula (1) i (x i，c ，y i，c ) The reconstruction polynomial is obtained as follows:
Figure BDA0003829284720000106
wherein,
Figure BDA0003829284720000107
is the integrated average of the reconstructed variable in grid cell i,
Figure BDA0003829284720000108
for the l-th undetermined coefficient of the reconstruction polynomial in grid cell i,
Figure BDA0003829284720000109
for the ith reconstruction basis function of the reconstruction polynomial in grid cell i,
Figure BDA00038292847200001010
is the integrated average of the basis functions in grid cell i,
Figure BDA0003829284720000111
is a zero mean basis function, x i，c Is the abscissa, y, of the center of the grid cell i i，c Is the ordinate of the center of the grid cell i,
Figure BDA0003829284720000112
is the integral average of the function f within grid cell i.
The reconstruction basis function in the grid unit has the specific form:
Figure BDA0003829284720000113
the integral average of the reconstructed basis functions in each grid cell is calculated by gaussian integration:
Figure BDA0003829284720000114
wherein omega i Is the volume of grid cell i, N is the number of Gaussian integration points, ω j Is the gaussian integral weight of the jth gaussian integral point,
Figure BDA0003829284720000115
for the ith reconstructed basis function of grid cell i,
Figure BDA0003829284720000116
is its integral mean, cor, within grid cell i i j Is the j-th Gaussian integration point coordinate in the grid unit i.
For the reference triangle composed of vertices 1' (0,0), 2' (1,0), 3' (0,1) given in fig. 3, the mapping relationship between the actual mesh cells and the reference cells is as follows:
Figure BDA0003829284720000117
wherein (x) 1 ，y 1 )，(x 2 ，y 2 )，(x 3 ，y 3 ) The physical coordinates of the three vertices 1,2,3 in the real cell, (x ', y') are the coordinates of the gaussian integration points a ', b', c ', d' in the reference cell, and (x, y) are the coordinates of the corresponding gaussian integration points a, b, c, d in the real cell.
The third-order gaussian integration point coordinates, i.e. the integration weights, in the reference unit are shown in table 1:
TABLE 1
Figure BDA0003829284720000121
In another embodiment, in step S400, the method of calculating the average flow field by using the finite volume method is a common method, and will not be described in detail, and the integral average value in the grid cell can be calculated by using the finite volume method
Figure BDA0003829284720000122
In another embodiment, in step S500, the reconstruction relation weight coefficient is calculated by the following formula:
Figure BDA0003829284720000123
wherein,
Figure BDA0003829284720000124
is the weight of the neighboring cell nb of the grid cell i, nc is the number of cells adjacent to the face of the grid cell i,
Figure BDA0003829284720000125
the smoothness ISS of each cell is the smoothness of the jth adjacent cell of the grid cell i i The calculation formula is as follows:
Figure BDA0003829284720000126
wherein ISS i To be the degree of smoothness of the unit i,
Figure BDA0003829284720000127
is the integrated average of the density in the jth cell adjacent to cell i,
Figure BDA0003829284720000128
is the integrated average of the density within cell i.
In another embodiment, in step S600, the integral average value of the cells adjacent to the i-plane of the grid cell is conserved, that is:
Figure BDA0003829284720000131
wherein nb is the number of the cell adjacent to the i-plane of the grid cell, Ω nb The volume of nb cells adjacent to the i-plane of the grid cell,
Figure BDA0003829284720000132
the integral average of the reconstructed variable in nb cells adjacent to the i-plane of the grid cell.
Based on the weight coefficients and the integrated average of the reconstructed basis functions, equation (10) may be further expressed as:
Figure BDA0003829284720000133
wherein,
Figure BDA0003829284720000134
is the integrated average of the reconstructed variable in grid cell i,
Figure BDA0003829284720000135
for the integrated average of the reconstructed variable in grid cell nb adjacent to grid cell i,
Figure BDA0003829284720000136
for the l-th undetermined coefficient of the reconstruction polynomial in grid cell i, (x) i，c ，y i，c ) Is the center coordinate of grid cell i, (x) nb，c ，y nb，c ) Is the center coordinate of the grid cell nb,
Figure BDA0003829284720000137
as the integrated average of the function f within grid cell i,
Figure BDA0003829284720000138
is the integral average of the function f within the grid cell nb.
Scanning all grid boundaries according to the connection relation among the grid units read in the step S200 to obtain an algebraic equation set constructed by a reconstructed variable integral mean conservation relation, wherein the algebraic equation set comprises:
Figure BDA0003829284720000139
wherein,
Figure BDA0003829284720000141
Figure BDA0003829284720000142
Figure BDA0003829284720000143
Figure BDA0003829284720000144
Figure BDA0003829284720000145
Figure BDA0003829284720000146
wherein,
Figure BDA0003829284720000147
the weight coefficients of the reconstructed relation in j cells adjacent to the grid cell i,
Figure BDA0003829284720000148
the mean value of the reconstructed variable in cell i calculated for the finite volume method,
Figure BDA0003829284720000149
average value, x, of reconstructed variables in j cells adjacent to cell i calculated by finite volume method i，j，c Is the abscissa, y, of the center of the jth grid cell adjacent to grid cell i i，j，c J is 1-Nc as the ordinate of the center of the j-th grid cell adjacent to grid cell i, nc is the number of grid cells adjacent to grid cell i, x i，c Is the abscissa, y, of the center of the grid cell i i，c Is the ordinate of the center of the grid cell i,
Figure BDA00038292847200001410
is the integral average of the function f within grid cell i.
For the boundary cell, the boundary condition also needs to be considered in the reconstruction process, and the general boundary condition is in the form as follows:
Figure BDA00038292847200001411
wherein u is i，b Is the value, c, of the reconstructed variable u at the boundary point b on the boundary cell i 1 ，c 2 ，c 3 The control parameter for the general boundary condition is determined according to the type of the boundary condition, for example, if a reconstruction variable U = U is specified on a boundary, c is determined in the reconstruction process of the cell i adjacent to the boundary 1 ＝0，c 2 ＝1，c 3 ＝U。
Since the reconstruction relation is provided through the boundary condition in the reconstruction process, the order of the reconstruction polynomial in the boundary unit does not need to be reduced, and therefore the third-order reconstruction precision with the same boundary unit and internal unit can be realized.
Based on the algebraic equation system constructed as above, the first and second derivatives relationship is expressed as follows:
Figure BDA0003829284720000151
wherein,
Figure BDA0003829284720000152
Figure BDA0003829284720000153
Figure BDA0003829284720000154
Figure BDA0003829284720000155
Figure BDA0003829284720000156
Figure BDA0003829284720000157
wherein,
Figure BDA0003829284720000158
the weight coefficients for the reconstruction relationship in the j cells adjacent to grid cell i,
Figure BDA0003829284720000159
the mean value of the reconstructed variable in cell i calculated for the finite volume method,
Figure BDA0003829284720000161
average value, x, of reconstructed variables in j cells adjacent to cell i calculated by finite volume method i，j，c Is the abscissa, y, of the center of the jth grid cell adjacent to grid cell i i，j，c Is a grid celli the vertical coordinate of the center of the j-th grid cell adjacent to the grid cell i, j is 1-Nc, nc is the number of grid cells adjacent to the grid cell i, and x i，c Is the abscissa, y, of the center of the grid cell i i，c Is the ordinate of the center of the grid cell i,
Figure BDA0003829284720000162
is the integral average of the function f within grid cell i.
In order to remember that,
Figure BDA0003829284720000163
the first and second derivative relationships may also be expressed as:
Figure BDA0003829284720000164
for the traditional high-order finite volume method, reconstruction is carried out only by utilizing the integral average value conservation relation (10) of the reconstruction variable in the reconstruction unit, for example, in the example, the two-dimensional third-order reconstruction polynomial (4) has five undetermined coefficients
Figure BDA0003829284720000165
At least five grid cells need to be applied with the conservation relation (10) of integral mean value of the reconstruction variable, but for a two-dimensional triangular grid, each grid cell only has three adjacent grid cells, so that the information of non-surface adjacent grid cells is needed in the reconstruction process, namely the reconstruction process of the traditional high-order finite volume method is non-compact. The present invention provides not only the conservation condition of the integral average value of equation (10) but also the conservation condition of the first derivative of the reconstruction polynomial in the face-adjacent cells, so that the third order reconstruction can be completed only by using the information of the face-adjacent cells, and the specific steps are as described in the following embodiments.
In another embodiment, in step S700, an algebraic equation system for solving the second-order derivative term is first constructed from the first-order derivative conservation relation of the reconstructed polynomial.
The first derivative of the reconstruction polynomial (4) for x is in the form:
Figure BDA0003829284720000171
the conservation relation (10) of the integral average of the analog reconstruction variables can obtain the conservation relation of the reconstruction polynomial to the first derivative of x:
Figure BDA0003829284720000172
namely:
Figure BDA0003829284720000173
wherein nb is a cell number adjacent to the mesh cell i,
Figure BDA0003829284720000174
for the jth undetermined coefficient of the reconstruction polynomial in unit i, (x) i，c ，y i，c ) Is the center coordinate of grid cell i, (x) nb，c ，y nb，c ) Is the center coordinate of the nb cell adjacent to grid cell i.
In conjunction with the first derivative and second derivative relationship matrix (15) obtained in step S600, equation (18) can be expressed as:
Figure BDA0003829284720000175
wherein,
Figure BDA0003829284720000176
reconstruct the jth row vector of the matrix AuxMat for cell i,
Figure BDA0003829284720000177
the jth component of the matrix AuxVec is reconstructed for cell i.
Due to the fact that
Figure BDA0003829284720000178
And
Figure BDA0003829284720000179
all the unknown, so the equation system formed by the formula (19) usually needs to solve all the units in the whole field simultaneously, namely implicit solution, the solution method has large calculation amount and low parallel solution efficiency, and in order to solve the problem, the reconstruction solution process is explicitly realized by the following method.
The second derivative term in equation (1) is approximated by a Taylor expansion, for example:
Figure BDA0003829284720000181
wherein,
Figure BDA0003829284720000182
for the second cross derivative of the reconstructed polynomial at the center of the grid cell i,
Figure BDA0003829284720000183
reconstructing the second cross derivative of the polynomial for the cell nb centers adjacent to grid cell i, (x) i，c ，y i，c ) Is the center coordinate of grid cell i, (x) nb，c ，y nb，c ) As the center coordinate of the nb cell adjacent to grid cell i, (x) nb，c -x i，c ) And (y) nb，c -y i，c ) Is the grid size h.
Namely:
Figure BDA0003829284720000184
similarly, for the other two second derivative terms:
Figure BDA0003829284720000185
substituting formula (21) and formula (22) into formula (1) yields:
Figure BDA0003829284720000186
in grid cell i (x-x) i，c )∝h，(y-y i，c ) Oc h, therefore, expanding equation (23) can be written as:
Figure BDA0003829284720000191
in order to remember that,
Figure BDA0003829284720000192
is a primitive polynomial u i Approximation of (x, y), equation (24) can be written as:
Figure BDA0003829284720000193
i.e. by
Figure BDA0003829284720000194
Approximation u i (x, y), then still with third order reconstruction accuracy. Therefore, it can be concluded that the coefficients of the highest order terms of the reconstruction polynomials in adjacent grids are made equal without affecting the reconstruction accuracy.
The highest order terms, i.e. second derivative terms, of neighboring reconstruction polynomials are made equal, i.e.,
Figure BDA0003829284720000195
equation (19) can then be:
Figure BDA0003829284720000196
similarly, a first derivative conservation relation of the reconstruction polynomial to y is obtained:
Figure BDA0003829284720000201
scanning all internal grid edges to obtain an algebraic equation system satisfied by a second derivative term:
Figure BDA0003829284720000202
wherein,
Figure BDA0003829284720000203
Figure BDA0003829284720000204
wherein,
Figure BDA0003829284720000205
is the kth row vector of the relationship matrix AuxMat of the jth cell adjacent to grid cell i,
Figure BDA0003829284720000206
is the kth row vector of the relationship matrix AuxMat of grid cell i,
Figure BDA0003829284720000207
the kth component of the relationship matrix AuxVec for the jth cell adjacent to grid cell i,
Figure BDA0003829284720000208
the k-th component, x, of the relationship matrix AuxVec for grid cell i i，j，c Is the abscissa, y, of the center of the jth cell adjacent to grid cell i i，j，c Is the ordinate, x, of the center of the jth cell adjacent to grid cell i i，c Is the abscissa, y, of the center of the grid cell i i，c Is the ordinate of the center of grid cell i. j ranges from 1 to Nc, and k ranges from 1 to 2.
Solving the algebraic equation set to obtain second derivative term
Figure BDA0003829284720000211
Substituting into the relation matrix of the first derivative and the second derivative obtained in step S600 to obtain the first derivative term of the reconstruction polynomial
Figure BDA0003829284720000212
And completing the reconstruction.
Next, the flow field calculation results based on the method of the present disclosure are given.
Dividing a 10 × 10, 14 × 14, 20 × 20, 28 × 28, 40 × 40 unstructured grid in a computational domain (0,1) × (0,1) and using an exponential function f 1 ＝x 2 e y As test function 1, with complex functions
Figure BDA0003829284720000213
As a test function 2, the reconstruction method is used for reconstructing the test function, the difference between a reconstructed variable value and an accurate value reconstructed at the edge center of a grid is used as a reconstruction error, and the method is defined according to the error:
Figure BDA0003829284720000214
where Error is the numerical Error, h is the grid size, and n is the precision order.
As can be seen from the above equation, the slope of the logError-high curve represents the accuracy of the numerical method. Fig. 4 is a result diagram of numerical accuracy of a reconstruction method obtained through a test function test, in fig. 4, an average slope of a logError-high curve reconstructed by the method of the present disclosure is 3.01, that is, the method of the present disclosure does have third-order accuracy, compared with a classical second-order center method, in fig. 4, an average slope of a logError-high curve reconstructed by a second-order center format is 2.00, that is, the second-order center format has second-order reconstruction accuracy, and for a test function 1, to achieve a reconstruction error of the reconstruction method of the present disclosure on a 10 × 10 grid, the second-order center method requires a grid number of 71 × 71, and for a test function 2, to achieve a reconstruction error of the reconstruction method of the present disclosure on a 10 × 10 grid, the second-order center method requires a grid number of 26 × 26, so the method of the present disclosure has higher accuracy compared with the classical second-order center method, and the grid number required for achieving the same calculation residual error is less.
The reconstruction method is utilized to carry out numerical calculation on the incompressible square cavity driving flow calculation example under the condition that the Reynolds number (Re) is 400, 80 multiplied by 80 unstructured grids are divided into a calculation domain (0,1) × (0,1), four edges of a square cavity are non-slip boundaries, the upper boundary is a driving edge, and the speed distribution in the x direction is set as: u (x, 1) =16x 2 (1-x) 2 The other boundary speeds are all 0. Fig. 5 is a distribution diagram of a square cavity driving flow Re =400 streamline calculated by the method, and it can be seen that a large vortex in the middle of a flow field and small vortices at two base angles can be clearly captured by applying the method of the present invention. Table 2 shows the comparison of the vortex center coordinates calculated by the method with the literature results, as follows:
TABLE 2
Figure BDA0003829284720000221
The calculation result of the spectrum type high-precision algorithm for the calculation example shows that the vortex center position of the middle large vortex is in the range of (0.565-0.578,0.616-0.629), the vortex center position of the left bottom angle small vortex is in the range of (0.038-0.048,0.038-0.043), and the vortex center position of the right bottom angle small vortex is in the range of (0.897-0.902,0.103-0.114). In summary, it can be seen that the present disclosure can better identify the shape and location of vortices in a separation flow.
In conclusion, the three examples with different complexities verify the accuracy and effectiveness of the method of the third-order compact reconstruction method based on the finite volume method on the unstructured grid, and have feasibility and necessity of popularization and application.
The technical solutions provided by the present disclosure are described in detail with reference to specific embodiments, and the description of the embodiments is only for assisting understanding of the core ideas of the present disclosure, and a person skilled in the art may change the specific implementation and application scope according to the ideas of the present disclosure. Accordingly, the description should not be construed as limiting the disclosure.

Claims (7)

1. A three-order compact reconstruction method based on a finite volume method on an unstructured grid comprises the following steps:
s100: constructing an unstructured grid;
s200: reading coordinates of grid nodes in the unstructured grid and connection relations among grid units;
s300: calculating an integral average value of a reconstruction basis function in a grid unit according to the grid node coordinates;
s400: calculating an average flow field according to a finite volume method;
s500: calculating a weight coefficient of a reconstruction relation according to density distribution in a flow field;
s600: constructing a first derivative relation matrix and a second derivative relation matrix based on the weight coefficient, the integral average value of the reconstruction basis function, the connection relation among the grid units and the average flow field;
s700: and constructing a reconstruction polynomial second-order derivative term solving equation set, solving to obtain a reconstruction polynomial second-order derivative term, bringing the reconstruction polynomial second-order derivative term into the relation matrix of the first-order derivative and the second-order derivative, solving to obtain a reconstruction polynomial first-order derivative term, and finishing reconstruction.
2. The method according to claim 1, wherein preferably, in step S300, the integrated average of the reconstruction basis functions in the grid cell is calculated by gaussian integration:
Figure FDA0003829284710000011
wherein omega i Is the volume of grid cell i, N is the number of Gaussian integration points, ω j Is the gaussian integral weight of the jth gaussian integral point,
Figure FDA0003829284710000012
for the ith reconstructed basis function of grid cell i,
Figure FDA0003829284710000013
to reconstruct basis functions
Figure FDA0003829284710000014
Integral mean, cor, within grid cell i i j Is the coordinate of the jth gaussian integration point in grid cell i.
3. The method according to claim 1, wherein in step S500, the reconstruction relation weight coefficient is calculated by:
Figure FDA0003829284710000021
wherein,
Figure FDA0003829284710000022
is the weight of the neighboring cell nb of the grid cell i, nc is the number of cells adjacent to the face of the grid cell i,
Figure FDA0003829284710000023
the smoothness ISS of each grid cell is the smoothness of the jth adjacent grid cell of grid cell i i Calculated by the following formula:
Figure FDA0003829284710000024
wherein ISS i As a grid celli the degree of smoothness of the film to be formed,
Figure FDA0003829284710000025
is the average density in the jth cell adjacent to grid cell i,
Figure FDA0003829284710000026
is the average density within grid cell i.
4. The method according to claim 1, wherein in step S600, the first derivative and second derivative relation matrices are represented as follows:
Figure FDA0003829284710000027
wherein,
Figure FDA0003829284710000028
Figure FDA0003829284710000029
Figure FDA00038292847100000210
Figure FDA0003829284710000031
Figure FDA0003829284710000032
Figure FDA0003829284710000033
wherein,
Figure FDA0003829284710000034
the weight coefficients of the reconstructed relation in j cells adjacent to the grid cell i,
Figure FDA0003829284710000035
the mean value of the reconstructed variable in cell i calculated for the finite volume method,
Figure FDA0003829284710000036
average value, x, of reconstruction variables in j cells adjacent to cell i calculated for finite volume method i，j，c Is the abscissa, y, of the center of the jth grid cell adjacent to grid cell i i，j，c For the ordinate of the center of the jth grid cell adjacent to grid cell i, j takes 1-Nc, nc being the number of grid cells adjacent to grid cell i, x i，c Is the abscissa, y, of the center of the grid cell i i，c Is the ordinate of the center of the grid cell i,
Figure FDA0003829284710000037
is the integral average of the function f in the grid cell i;
in order to remember that,
Figure FDA0003829284710000038
is AuxMat i ，
Figure FDA0003829284710000039
Is AuxVec i 。
5. The method of claim 1, wherein the solving of the equation set by the reconstruction polynomial second derivative term in step S700 is represented as follows:
Figure FDA00038292847100000310
wherein,
Figure FDA00038292847100000311
Figure FDA0003829284710000041
wherein,
Figure FDA0003829284710000042
is the kth row vector of the relationship matrix AuxMat of the jth cell adjacent to grid cell i,
Figure FDA0003829284710000043
for the kth row vector of the relation matrix AuxMat of grid cell i,
Figure FDA0003829284710000044
the k-th component of the relationship matrix AuxVec for the j-th cell adjacent to grid cell i,
Figure FDA0003829284710000045
the k-th component, x, of the relationship matrix AuxVec for grid cell i i，j，c Is the abscissa, y, of the center of the jth cell adjacent to grid cell i i，j，c Is the ordinate, x, of the center of the jth cell adjacent to grid cell i i，c Is the abscissa, y, of the center of the grid cell i i，c The value range of j is 1-Nc, and the value range of k is 1-2.
6. A three-order compact reconstruction device based on finite volume method on unstructured grid comprises:
the unstructured grid generating module is used for generating unstructured grids;
the preprocessing module is used for reading node coordinates in the unstructured grid and the connection relation between grid units;
the integral calculation module is used for calculating an integral average value of the reconstruction basis function in the grid unit according to the grid node coordinates;
the finite volume method calculating module is used for calculating an average flow field according to a finite volume method;
the weight coefficient calculation module is used for calculating a reconstruction relation weight coefficient according to density distribution in the flow field;
the reconstruction relation matrix building module is used for building a first derivative and a second derivative relation matrix based on the weight coefficient, the integral average value of the reconstruction basis function, the connection relation among the grid units and an average flow field obtained by a finite volume method;
and the third-order reconstruction polynomial solving module is used for solving an equation set according to the constructed reconstruction polynomial second-order derivative term, solving to obtain a reconstruction polynomial second-order derivative term, and substituting the reconstruction polynomial second-order derivative term into a first-order derivative and second-order derivative relation matrix to solve to obtain a reconstruction polynomial first-order derivative term.
7. The apparatus of claim 6, wherein the third order reconstruction polynomial solving module comprises:
the second order derivative item solving submodule is used for solving an algebraic equation system of the reconstruction polynomial second order derivative item to obtain a reconstruction polynomial second order derivative item;
and the first derivative item solving submodule is used for substituting the reconstructed polynomial second derivative item into the first derivative and second derivative item relation matrix to solve and obtain the reconstructed polynomial first derivative item.
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Cited By (1)

* Cited by examiner, † Cited by third party
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* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN116822312A (en) * 2023-08-30 2023-09-29 北京智芯微电子科技有限公司 Post-processing display method, system and medium for high-order unstructured grid unit
CN116822312B (en) * 2023-08-30 2024-01-26 北京智芯微电子科技有限公司 Post-processing display method, system and medium for high-order unstructured grid unit

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