Initially we applied the same data structure to grid generation, with further research, we found that the overall calculation efficiency of such a design is not high, and it is difficult to maintain. Finally, we designed different data structures based on the different characteristics of the two parts, in the grid generation, more consideration is given to flexibility and maintainability, a method to determine the flux function through the boundary of a triangular mesh is given. This method first obtains the two-dimensional Euler equations in the normal direction of the boundary of the control volume element, in the system of projection equations, the flux along the tangent to the boundary can be regarded as passive convection, therefore, the one-dimensional Riemann problem solver can be directly used to calculate the flux function passing through the boundary of the triangular control unit. This method under various flow conditions, including supersonic pipe flow and subsonic isentropic flow with strong shock waves, all have high precision and good stability.

#### Keywords

- Teaching process reform
- data structure
- grid generation

#### MSC 2010

- 68P15

“Data structure” is an important professional basic course in the computer professional curriculum system. The main content of data structure teaching is the design and implementation of the logical structure of the data, the storage structure and the algorithm of the core operation, it is an important theoretical and technical basis for computer programming. The training goal of the course is to require students to learn to analyze, study the characteristics of the data that is the object of computer processing, in order to select the appropriate logical structure, the storage structure and its corresponding algorithms require students to write programs with clear structure, correct and easy to read, comply with software engineering specifications. The learning effect of this course is not only related to the study of many subsequent professional courses, it is also related to the improvement of software design level and the training of professional quality. From the current teaching situation, because the course content is abstract and the dynamic storage structure is difficult to understand, the student feels unable to start the algorithm description, theory and practice cannot be combined well. For the study and practice of data structure, the idea of the algorithm is more meaningful than the specific implementation itself. Data structure is a course that requires careful consideration, there are many mature algorithms in this course, these algorithms are very classic, but some algorithms are for students, it's not easy to understand. Therefore, in teaching work, the main energy is still used in thinking, emphasize algorithmic ideas and ignore grammatical details. Because teachers need to pass a detailed anatomy of typical cases, what students get is not only the knowledge about this course, more is the ability to analyze and solve problems. The main reason most students are not interested in this course, i think this course has little practicality. Therefore, teachers should pay attention to practice when teaching this course, guide students to deal with daily problems in life cleverly, let students feel the connection between the course “Data Structure” and real life, in order to improve students’ interest in learning. For example, before introducing the concept of “tree”, let the students think about such a problem first, how to use a computer to store and manage family member information in a large family. Under the premise of students fully thinking, tell students that this question is a typical application of “trees”, then introduce the concept of “tree”, data extraction, storage methods, search and other operations, at this time, the students after the teacher's question, the interest in learning has improved a lot, then he will listen to the teacher's explanation attentively. After learning a knowledge point, ask some enlightening questions, guide students to actively explore the relationship between different knowledge points, master the knowledge learned. This not only increases the interaction between teachers and students, it also allows students to learn the content of the course from passive to active, aroused the subjective initiative of students in learning; At the same time, students’ thinking ability to analyze and solve problems has also been improved.

In the frontal approach, generally, the discrete line segments forming the boundary of the computational domain are used as the initial array of fronts, according to certain guidelines in this set, constantly select the corresponding edges to construct new triangle elements, and update the original front, until the front assembly is empty. From the perspective of the grid generation process, the algorithm given is also based on the requirements of the grid resolution, constantly constructing new triangular units, this is very similar to the frontal advancement method, but it is different from the front approach method, this algorithm is based on the effective triangle with the smallest radius, select the sides to construct the new triangle.

The concept of effective triangles is first given below: Consider the triangular element _{j}_{i}_{j}_{j}_{j}_{j}_{j}_{g}_{, j}, query the background grid, the expected value _{exp, j} of the radius of the circumcircle of the triangle unit at _{g}_{,j} can be obtained. Compare the value of _{j}_{exp, j}, if _{j}_{≤rexp, j}, then define _{j}_{j}_{j}_{j}_{j}_{j}_{j}

In order to evaluate the quality of grid cells, we introduce quality factor _{j}_{j}

Among them, _{j}_{j}_{j}_{j}_{j}_{j}_{j}_{j}_{j}_{j}_{j}_{j}

In order to facilitate the measurement of the overall quality of meshing, we introduced factors _{mean}_{joint} and _{min}, these factors are respectively defined as shown in formulas (2), (3) and (4).

Among them, _{j}_{mean}_{min} gives the quality factor of the worst-quality element in the meshing, _{joint} reflects the characteristics of the “bad” grid, when there are long and narrow elements whose quality is close to 0 in the meshing, then the value of _{joint} will be close to zero.

Two flexible and effective Delaunay grid internal node generation algorithms, Voronoi-vertex method and Voronoi-edge method, compare the calculation results of these two algorithms, it can be seen that the quality of the mesh obtained by the Voronoi-edge method is better than that of the Voronoi-vertex method.

Figure 1 compares the relevant parameters of these two grid divisions, it can be seen that, when generating a mesh for a simple area, the overall quality of the node generation algorithm we gave is slightly worse than the Voronoi-edge method. Figure 2 and Table 1 list the corresponding quality parameters and geometric parameters in these two cases, it can be seen that for a flow field with a complex boundary, the node generation algorithm based on the front propulsion method significantly improves the quality and arrangement of the grid cells near the boundary, and it can better guarantee the quality of the elements during meshing.

For the diffuser flow field, the distribution of quality circle q in the two grid divisions

The number of triangle elements whose prime factor q lies in different ranges | |||||||
---|---|---|---|---|---|---|---|

0.65–0.7 | 0.7–0.75 | 0.75–0.8 | 0.8–0.85 | 0.85–0.9 | 0.9–0.95 | 0.95–1 | |

Voronoi-edge method | 1 | 2 | 6 | 12 | 25 | 170 | 1481 |

Algorithm based on front propulsion method | 0 | 4 | 4 | 6 | 22 | 140 | 1535 |

When we compared different grid sizes, the geometric parameters and quality parameters of uniform meshing before and after using the node relaxation method. The node relaxation method is a kind of moving grid node position, in order to improve the overall quality of the grid. In two-dimensional meshing, for any node _{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}

Among them, n_{i} is the number of vertices of polygon _{i}_{i}_{,k} and _{i,k}_{i}

In actual calculations, the node relaxation method needs to be completed through multiple cycles, in each cycle, all internal nodes are moved to the center of gravity of their related Voronoi polygons.

The background grid method and the source function method are two more commonly used methods to control the grid resolution, the following are examples of non-uniform mesh division using these two methods.

(1) Use the background grid to control the grid division

Here are two examples of obtaining locally refined grids based on the background grid function. In the first example, we encrypt the local area near the boundary in the ring area; In the second example, we encrypt the local area near the diagonal of the rectangular area. The meshing process of these two calculation examples, as can be seen, the automatic grid generation algorithm we have given retains the characteristics of the front propulsion method. After the node relaxation method is used, there is generally no long and narrow cells. In the following examples, we all use the node relaxation method to smooth the final mesh.

(2) Use source function to control meshing

In an unstructured grid, you can also use the source function to control the resolution of the mesh, considering the quality of grid cells near the boundary, the extension length _{exp} at any point P in the calculation domain is defined as shown in formula (6).

Among them, _{exp, boundary} represents the density of discrete points on the boundary, the stretch length at the determined point P, _{j}_{j}_{j}_{j}_{j}_{exp} of the radius of the circumscribed circle at the point P according to _{exp}, as shown in formula (7).

In order to prove the versatility of algorithms and programs, we meshed areas with complex geometric shapes, this area is a complex connected domain, it contains 5 circles of equal size. The division is controlled by both the background grid and the source function, for any point P in the area to be divided, the background grid and the source function respectively define the corresponding expected stretch length:

The definition of

The definition of

Where _{j}

We apply the mesh generation algorithm to the unstructured meshing of the actual combustion chamber flow field. The corresponding geometric parameters and quality parameters are listed in Table 2. It can be seen that even for a flow field with a complex boundary, the given grid generation algorithm can also ensure the regular arrangement of grid cells near the boundary, and there is no long and narrow unit, when numerically calculating the flow field, this is particularly important.

Geometric parameters and quality parameters of unstructured grid division of the combustion chamber flow field

Flow field | Number of nodes | Number of sides | Number of triangles | _{mean} |
_{joint} |
_{min} |
---|---|---|---|---|---|---|

Flame tube | 3114 | 9113 | 6000 | 0.9770 | 0.9762 | 0.6264 |

Flow field behind the nozzle | 1395 | 3979 | 2584 | 0.9794 | 0.9783 | 0.6037 |

Annular combustion chamber | 584 | 1653 | 1069 | 0.9812 | 0.9806 | 0.7666 |

Shock wave solution

Let's first construct without considering the entropy condition, the weak solution of the nonlinear Riemann problem. Assuming passing through any point

Let
_{P}

Take the partial derivatives of both sides of equation (10) with respect to

Let

Regardless of the entropy condition, state _{l}_{1}(_{1}) by 1-shock wave, _{1}_{1}) can jump to another state set _{2}(_{1}, _{2}) through 2-shock wave, and so on, _{l}_{m}(_{m}_{m}

If ||_{l}–U_{r}_{r}_{1}, … _{m}_{r} = U_{m}_{1}, … _{m}

Assume that the conservation equations are truly nonlinear in the p characteristic region, that is, satisfy the conditions as shown in formula (13).

Among them, ∇ _{p}_{p}^{1}, … _{p}^{m}_{p}

Only ^{−} and ^{+} meet the conditions

There is a sudden jump from ^{−} to ^{+} in the p characteristic area. For any given initial conditions _{l}_{r}

The sparse wave solution is a self-similar continuous solution defined on a finite interval, which can be recorded in the following form as shown in formula (15).

Among them, _{1}) = _{l}_{2}) = _{r}

Let's determine the explicit expression of function _{1}_{2}

Substitute formulas (16) and (17) into the conservation equations and mark

It can be seen from equation (18) that the values of

Among them,
_{p}

In order to determine the explicit expression of function

Substituting equation (19) into equation (21), the expression of scale factor

From this, the relevant

Among them, _{1}) = _{l}_{2}) = _{r}_{p}_{1} and _{2}.

Let's discuss the conditions for the existence of sparse wave solutions between the initial states _{l}_{r}

Knowing from (18), the sparse wave solution _{l}_{r}_{l}_{r}_{l}_{r}_{p}_{l}_{r}

The grid generation process requires the grid division of the computational domain to form a graph, there are many ways to represent the graph. But the general method of graph representation, pay more attention to the connection relationship of the line segments between the nodes, in the process of adaptive mesh generation, the various geometric information of the triangle element is more important. So the representation of our graph is developed around a triangle. We use a list as a container for triangle units to facilitate the generation and deletion of triangles, at the same time, the triangle object establishes the adjacent relationship between the triangles through three pointers pointing to the adjacent triangles, in this way, the entire mesh can be determined. In order to improve the efficiency of the mesh generation algorithm, we also maintain some other geometric information such as edge Tables, it can be obtained by traversing the triangle unit, it is not essential information, but it is useful in the nodal relaxation method. In practice, such a representation method still satisfies the requirements of grid generation.

A new adaptive unstructured grid generation method that can be used in the numerical simulation of two-dimensional Euler equations is proposed. Based on the front propulsion method, this method is near triangular elements whose circumscribed circle radius does not meet the requirements of the background grid scale function, introduce new internal nodes of the grid, and then use Bowye algorithm to get the Delaunay grid division corresponding to the new point set. From the perspective of the grid generation process, this method retains the characteristics of the front propulsion method to a certain extent, so that the unit arrangement is relatively regular, and it is easy to ensure the quality of the cells near the boundary. On the other hand, since the connection relationship between nodes is determined according to the Delunay criterion, it is easy to realize the automation of grid generation.

#### For the diffuser flow field, the distribution of quality circle q in the two grid divisions

The number of triangle elements whose prime factor q lies in different ranges | |||||||
---|---|---|---|---|---|---|---|

0.65–0.7 | 0.7–0.75 | 0.75–0.8 | 0.8–0.85 | 0.85–0.9 | 0.9–0.95 | 0.95–1 | |

Voronoi-edge method | 1 | 2 | 6 | 12 | 25 | 170 | 1481 |

Algorithm based on front propulsion method | 0 | 4 | 4 | 6 | 22 | 140 | 1535 |

#### Geometric parameters and quality parameters of unstructured grid division of the combustion chamber flow field

Flow field | Number of nodes | Number of sides | Number of triangles | _{mean} |
_{joint} |
_{min} |
---|---|---|---|---|---|---|

Flame tube | 3114 | 9113 | 6000 | 0.9770 | 0.9762 | 0.6264 |

Flow field behind the nozzle | 1395 | 3979 | 2584 | 0.9794 | 0.9783 | 0.6037 |

Annular combustion chamber | 584 | 1653 | 1069 | 0.9812 | 0.9806 | 0.7666 |

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