CN-115455853-B - Third-order compact reconstruction method based on finite volume method on unstructured grid

CN115455853BCN 115455853 BCN115455853 BCN 115455853BCN-115455853-B

Abstract

The invention discloses a third-order compact reconstruction method based on a finite volume method on unstructured grids, which comprises the steps of constructing unstructured grids, reading coordinates of grid nodes in the unstructured grids and connection relations among grid cells, calculating integral average values of reconstruction basis functions in the grid cells according to the grid node coordinates, calculating average flow fields according to the finite volume method, calculating reconstruction relation weight coefficients according to density distribution in the flow fields, constructing a first derivative relation matrix and a second derivative relation matrix based on the weight coefficients, the integral average values of the reconstruction basis functions, the connection relations among the grid cells and the average flow fields, constructing a reconstruction polynomial second derivative term solving equation set, solving to obtain a reconstruction polynomial second derivative term, bringing the reconstruction polynomial second derivative term into the first derivative relation matrix and the second derivative relation matrix, solving to obtain a reconstruction polynomial first derivative term, and completing reconstruction.

Inventors

ZHANG JIAWANG
LI ZHEN
LI HAO
JU YAPING
ZHANG CHUHUA

Assignees

西安交通大学
西安交通大学

Dates

Publication Date: 20260421
Application Date: 20220902
Priority Date: 20220902

Claims (6)

1. A three-order compact reconstruction method based on an unstructured grid finite volume method comprises the following steps: S100, constructing an unstructured grid; s200, reading the coordinates of grid nodes in the unstructured grid and the connection relation 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 relationship 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 grid units and the average flow field; S700, constructing a reconstruction polynomial second derivative term solving equation set, solving to obtain a reconstruction polynomial second derivative term, introducing the reconstruction polynomial second derivative term into the relation matrix of the first derivative and the second derivative, solving to obtain a reconstruction polynomial first derivative term, and completing reconstruction; in step S600, the first derivative and second derivative relation matrix is expressed as follows: , Wherein, the , , Wherein, the The weight coefficients of the relationship are reconstructed for j cells adjacent to grid cell i, The average value of the reconstructed variables in the unit i calculated for the finite volume method, The average value of the reconstruction variables in j cells adjacent to cell i calculated for the finite volume method, Is the abscissa of the center of the jth grid cell adjacent to grid cell i, Taking j as 1-Nc, nc being the number of grid cells adjacent to grid cell i, the ordinate of the center of the j-th grid cell adjacent to grid cell i, Is the abscissa of the center of the grid cell i, Is the ordinate of the center of the grid cell i, An integrated average value of the function f in the grid cell i; The number of the marks is recorded, and the number of the marks is recorded, Is that , Is that 。
2. The method according to claim 1, wherein in step S300, the integrated average value of the reconstructed basis functions within the grid cells is calculated by gaussian integration: , Wherein, the For the volume of grid cell i, N is Gao Siji points, The gaussian integral weight for the j-th gaussian integral point, The basis functions are reconstructed for the first of the grid cells i, Reconstructing basis functions The integrated average value within the grid cell i, Is the coordinates of the j-th gaussian integral point in grid cell i.
3. The method according to claim 1, wherein in step S500, the reconstruction relationship weight coefficient is calculated by: , Wherein, the The weight of the adjacent cell nb of the grid cell i, nc is the number of cells adjacent to the grid cell i face, For the smoothness of the j-th neighboring cell of grid cell i, the smoothness ISS i of each grid cell is calculated by: , Wherein ISS i is the smoothness of grid cell i, For the average density in the j-th cell adjacent to grid cell i, Is the average density within grid cell i.
4. The method according to claim 1, wherein in step S700, the system of reconstruction polynomial second derivative term solving equations is represented as follows: , Wherein, the , , Wherein, the Relationship matrix for the jth cell adjacent to grid cell i Is used to determine the vector of the (k) th row, Relationship matrix for grid cell i Is used to determine the vector of the (k) th row, Relationship matrix for the jth cell adjacent to grid cell i Is used to determine the (k) th component of the (c), Relationship matrix for grid cell i Is used to determine the (k) th component of the (c), Is the abscissa of the center of the j-th cell adjacent to grid cell i, Is the ordinate of the center of the j-th cell adjacent to grid cell i, Is the abscissa of the center of the grid cell i, The value range of j is 1-Nc, and the value range of k is 1-2 for the ordinate of the center of the grid cell i.
5. A third order compact reconstruction apparatus based on a finite volume method on an unstructured grid, comprising: An unstructured grid generation module for generating an unstructured grid; The preprocessing module is used for reading node coordinates in the unstructured grid and connection relations among grid cells; 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 calculation 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 relationship weight coefficient according to the density distribution in the flow field; the reconstruction relation matrix construction module is used for constructing a first derivative relation matrix and a second derivative relation matrix based on the weight coefficient, an integral average value of a reconstruction basis function, a connection relation among grid units and an average flow field obtained by a finite volume method; The third-order reconstruction polynomial solving module is used for solving the equation set according to the constructed reconstruction polynomial second derivative term, solving to obtain a reconstruction polynomial second derivative term, and carrying the reconstruction polynomial second derivative term into a first derivative and second derivative relation matrix to solve to obtain a reconstruction polynomial first derivative term; in the reconstruction relation matrix construction module, the first derivative relation matrix and the second derivative relation matrix are expressed as follows: , Wherein, the , , Wherein, the The weight coefficients of the relationship are reconstructed for j cells adjacent to grid cell i, The average value of the reconstructed variables in the unit i calculated for the finite volume method, The average value of the reconstruction variables in j cells adjacent to cell i calculated for the finite volume method, Is the abscissa of the center of the jth grid cell adjacent to grid cell i, Taking j as 1-Nc, nc being the number of grid cells adjacent to grid cell i, the ordinate of the center of the j-th grid cell adjacent to grid cell i, Is the abscissa of the center of the grid cell i, Is the ordinate of the center of the grid cell i, An integrated average value of the function f in the grid cell i; The number of the marks is recorded, and the number of the marks is recorded, Is that , Is that 。
6. The apparatus of claim 5, wherein the third order reconstruction polynomial solving module comprises: The second derivative term solving sub-module is used for solving algebraic equation sets of the reconstruction polynomial second derivative term to obtain the reconstruction polynomial second derivative term; And the first derivative term solving sub-module is used for bringing the second derivative term of the reconstruction polynomial into a relation matrix of the first derivative term and the second derivative term to obtain the first derivative term of the reconstruction polynomial.

Description

Third-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 The computational fluid dynamics is a method for obtaining a flow field approximate numerical solution by dispersing a fluid mechanics control equation set and a solution condition through a numerical method on a computer. The high-precision numerical discrete method has an important effect on improving the precision of numerical solution and further expanding the application range of computational fluid dynamics. Unstructured grids have become a hotspot of current research due to the advantages of being able to handle complex geometries, enabling flexible automatic generation, etc. The finite volume method is the most widely applied numerical method in computational fluid dynamics because of the advantages of automatically meeting the flow conservation characteristic, the windward characteristic of the convection term and the like. However, the limited volume method widely applied to unstructured grids at present has only second-order precision, and cannot meet the requirements of high-precision high-resolution scenes such as aerodynamic acoustics, vortex dominant flow, turbulence direct numerical simulation, large vortex simulation and the like, and the development of the high-order numerical method is an effective method for solving related problems. Constructing a high-order finite volume method computing method based on unstructured grids faces the following important challenges: Unstructured grids do not have a regular topology in three spatial dimensions, and cannot construct higher-order methods in each dimension alone like structured grids. For the finite volume method, the core of the high-precision numerical method is to construct a reconstruction variable higher-order reconstruction polynomial function in a grid unit by utilizing an integral average value of a reconstruction variable on a reconstruction template, wherein the initial work is a k-order accurate reconstruction method, namely the integral average value of the higher-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 obtained by utilizing a least square method, and in addition, a learner extends the weighted basic concussion-free reconstruction method on the structured grid to the unstructured grid. Because each grid cell can only provide one reconstruction condition for conservation of integral average value in the grid cell, and the number of the coefficient to be determined of the reconstruction polynomial increases rapidly along with the increase of dimension and order, the two reconstruction methods, namely a k-order accurate reconstruction method and a weighted basically concussion-free reconstruction method, have the problem of overlarge reconstruction templates. Disclosure of Invention Aiming at the defects in the prior art, the purpose of the present disclosure is to provide a three-order compact reconstruction method based on a finite volume method on an unstructured grid, wherein the method does not need to solve an implicit algebraic equation set in the reconstruction process, has small calculation amount and high solving efficiency, and in addition, considers boundary conditions in the reconstruction process, thereby realizing the consistent three-order reconstruction precision of boundary grid cells and internal grid cells. In order to achieve the above object, the present disclosure provides the following technical solutions: A three-order compact reconstruction method based on an unstructured grid finite volume method comprises the following steps: S100, constructing an unstructured grid; s200, reading the coordinates of grid nodes in the unstructured grid and the connection relation 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 relationship 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 grid units and the average flow field; and S700, constructing a reconstruction polynomial second derivative term solving equation set, solving to obtain a reconstruction polynomial second derivative term, introducing the reconstruction polynomial second derivative term into the relation matrix of