1. bookAHEAD OF PRINT
Détails du magazine
License
Format
Magazine
eISSN
2444-8656
Première parution
01 Jan 2016
Périodicité
2 fois par an
Langues
Anglais
access type Accès libre

Data structure simulation for the reform of the teaching process of university computer courses

Publié en ligne: 15 Jul 2022
Volume & Edition: AHEAD OF PRINT
Pages: -
Reçu: 10 Feb 2022
Accepté: 31 Mar 2022
Détails du magazine
License
Format
Magazine
eISSN
2444-8656
Première parution
01 Jan 2016
Périodicité
2 fois par an
Langues
Anglais
Abstract

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

MSC 2010

Introduction

“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.

Method
The generation algorithm of internal nodes of Delaunay grid based on the front propulsion method

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 Tj in the existing meshing Di, the circumcircle of Tj is denoted by Bj, the radius of Bj is rj. The center of gravity of Tj is denoted by Pg, j, query the background grid, the expected value rexp, j of the radius of the circumcircle of the triangle unit at Pg,j can be obtained. Compare the value of rj with the value of rexp, j, if rjrexp, j, then define Tj to be an acceptable triangle; Otherwise, Tj is defined as an unacceptable triangle. Unacceptable triangles can be divided into valid triangles and preliminary triangles, the specific definition is as follows: For unacceptable triangle Tj, if among the three sides of Tj, at least one edge is a boundary, or in the triangle adjacent to Tj, at least one triangle is an acceptable triangle, then Tj is defined as a valid triangle, otherwise, Tj is defined as a preliminary triangle.

Evaluation Criteria for the Quality of Triangular Elements

In order to evaluate the quality of grid cells, we introduce quality factor q to the definition of quality factor qj of triangular element Tj as shown in formula (1). qj=2γjrj {{\rm{q}}_j} = 2{{{\gamma _j}} \over {{r_j}}}

Among them, γj is the radius of the inscribed circle of Tj, rj is the radius of the circumcircle of Tj. The quality factor qj is a scalar variable between 0 and 1, if the quality factor qj of Tj is the maximum value of 1, then Tj is an equilateral triangle; If the quality factor qj of Tj is less than 0.02 (y;/r;<0.01), then Tj is defined as a long and narrow triangle, at this time, Tj is seriously degraded.

In order to facilitate the measurement of the overall quality of meshing, we introduced factors Qmean, Qjoint and Qmin, these factors are respectively defined as shown in formulas (2), (3) and (4). Qmean=1Nj=1nqj {Q_{mean}} = {1 \over N}\sum\limits_{j = 1}^n {{q_j}} 1Qjoint=1Nj=1n1qj {1 \over {{Q_{jo\,{\mathop{\rm int}}}}}} = {1 \over N}\sum\limits_{j = 1}^n {{1 \over {{q_j}}}} Qmin=min1jNqj {Q_{\min}} = \mathop {min\,}\limits_{1 \le j \le N} {q_j}

Among them, qj is the quality factor of each unit in the grid division, N is the total number of units. From the definition of these factors, it can be seen that, Qmean describes the average quality of the grid, Qmin gives the quality factor of the worst-quality element in the meshing, Qjoint 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 Qjoint will be close to zero.

Results and analysis
Comparison of meshing results

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.

Figure 1

Comparison of two Delaunay internal node generation algorithms for a circular computing city

Figure 2

For the diffuser flow field, the comparison of two Delaunay internal node generation algorithms

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
Node relaxation method

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 Pi, the nodes that have an adjacent relationship with Pi constitute the Voronoi polygon Zi, move Pi to the center of gravity of Zi, the quality of the triangle unit with Pi as the vertex can be optimized. The calculation formula of the barycentric coordinates xi and yi of Zi is shown in formula (5). xi=1nik=1nixi,k,yi=1nik=1niyi,k {x_i} = {1 \over {{n_i}}}\sum\limits_{k = 1}^{{n_i}} {{x_{i,k}},\,{y_i} = {1 \over {{n_i}}}} \,\sum\limits_{k = 1}^{{n_i}} {{y_{i,k}}}

Among them, ni is the number of vertices of polygon Zi, xi,k and yi,k are the vertex coordinates of Zi.

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.

Non-uniform grid

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 lexp at any point P in the calculation domain is defined as shown in formula (6). lexp=min(lexp,boundary,AjeBj|SjP|)j=1,M {l_{\exp}} = \,\min \left({{l_{\exp,\,boundary}},\,{A_j}{e^{{B_j}\left| {{S_j} - P} \right|}}} \right)\,j = 1,\,M

Among them, lexp, boundary represents the density of discrete points on the boundary, the stretch length at the determined point P, Aj and Bj respectively represent source Sj, j = 1, the amplification factor and attenuation factor of M, SjP represents the distance from point P to source Sj. Taking the radius of the circle circumscribed by the triangle as the characteristic scale of the unit, therefore, it is necessary to obtain the expected value rexp of the radius of the circumscribed circle at the point P according to lexp, as shown in formula (7). rexp=13lexp {r_{\exp}} = {1 \over {\sqrt 3}}{l_{\exp}}

Non-uniform meshing of complex geometric regions

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: lexpB l_{\exp}^B , lexpS l_{\exp}^S .

The definition of lexpB l_{\exp}^B is: The centers of the surrounding four circles define a rectangular area P¯2 {\bar P_2} to any point P in the computational domain, if PP¯2 P \in {\bar P_2} , then the stretch length lexpB l_{\exp}^B at point P is 0.01; If PK and the stretched length lexpB l_{\exp}^B at point P is 0.05.

The definition of lexpS l_{\exp}^S is shown in formula (8). lexp,soure=AeB|OjP|,j=1,2,3,4,5 {l_{\exp,\,soure}}\, = A{e^{B\left| {{O_j} - P} \right|}},\,j = 1,2,3,4,5

Where Oj is the position of the center of the circle, A is 0.024, and B is 6.931473. In the process of meshing, the stretch length at point P is taken as shown in formula (9). lexp=min(lexpB,lexpS) {l_{\exp}} = \min \left({l_{\exp}^B,\,l_{\exp}^S} \right)

Grid division of the combustion chamber flow field

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 Qmean Qjoint Qmin
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
Solutions for solving nonlinear Riemann problems of hyperbolic conservation equations

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 U^Rm \hat U \in {R^m} , there is a family of characteristic lines composed of m curves, the discontinuity formed by U^ \hat U and the points on the characteristic line satisfies the Rankine-Hugoniot condition.

Let U˜P(ξ;U^) {\tilde U_P}\left({\xi ;\hat U} \right) represent a characteristic line in the characteristic curve family passing through U^ \hat U , U˜P(0;U^)=U^ {\tilde U_P}\left({0;\hat U} \right) = \hat U , the propagation speed of U˜P(ξ;U^) {\tilde U_P}\left({\xi ;\hat U} \right) is sP(ξ;U^) {s_P}\left({\xi ;\hat U} \right) . Both U˜P(ξ;U^) {\tilde U_P}\left({\xi ;\hat U} \right) and sP(ξ;U^) {s_P}\left({\xi ;\hat U} \right) are expressions with one parameter, can be abbreviated as U˜P(ξ) {\tilde U_P}\left(\xi \right) and sP(ξ) respectively. The discontinuity formed by U˜P(ξ;U^) {\tilde U_P}\left({\xi ;\hat U} \right) and U^ \hat U satisfies the Rankine-Hugoniot condition as shown in formula (10). F(U˜P(ξ))F(U^)=sp(ξ)(U˜p(U˜p(ξ)U^)),p=1,,m F\left({{{\tilde U}_P}\left(\xi \right)} \right) - F\left({\hat U} \right) = {s_p}\left(\xi \right)\left({{{\tilde U}_{\rm{p}}}\left({{{\tilde U}_{\rm{p}}}\left(\xi \right)\hat U} \right)} \right),\,p = 1,\, \ldots,\,m

Take the partial derivatives of both sides of equation (10) with respect to ξ, and set ξ = 0, there is equation (11). F(U^)U˜P(0)=sp(0)U˜P(0).P=1,,m F\left({\hat U} \right)\tilde U_P^{'}\left(0 \right) = {s_p}\left(0 \right)\tilde U_P^{'}\left(0 \right).\,P = 1,\, \ldots,\,m

Let λp(U^) {\lambda _p}\left({\hat U} \right) represent the characteristic value of F(U^) F\left({\hat U} \right) , r¯p(U^) {\bar r_p}\left({\hat U} \right) represents the corresponding right eigenvector, knowing from (11), U˜P(0)r¯p(U^) \tilde U_P^{'}\left(0 \right)\infty {\bar r_p}\left({\hat U} \right) , sp(0)=λp(U^) {s_p}\left(0 \right) = {\lambda _p}\left({\hat U} \right) .

Regardless of the entropy condition, state Ul, it can jump to state set U1(ξ1) by 1-shock wave, U1(ξ1) can jump to another state set U2(ξ1, ξ2) through 2-shock wave, and so on, Ul can jump to state set Um(ξm, … ξm) after m channel shock wave, and satisfy the condition as shown in formula (12). Umξp|ξ1,.ξm=0r¯p(Ul),p=1,2,,m {{\partial {U_m}} \over {\partial {\xi _p}}}\left| {_{{\xi _1}, \ldots.{\xi _m} = 0}\infty {{\bar r}_p}\left({{U_l}} \right),\,p = \,1,2, \ldots,m} \right.

If ||Ul–Ur|| is small enough for any state Ur, the parameter ξ1, … ξm that satisfies Ur = Um (ξ1, … ξm) is uniquely determined, therefore, when the picking conditions are not considered, the weak solution of the nonlinear Riemann problem is composed of m shock waves.

Sparse wave solution

Assume that the conservation equations are truly nonlinear in the p characteristic region, that is, satisfy the conditions as shown in formula (13). λp(U)rp(U)0,U \nabla {\lambda _p}\left(U \right) \cdot {\vec r_p}\left(U \right) \ne 0,\,\forall U

Among them, ∇ λp (U) = (∂λp / λu1, … ∂λp / λum) is the gradient of λp(U). Use s to denote the propagation speed of the shock wave, and the entropy condition can be expressed as shown in formula (14).

Only U and U+ meet the conditions λp(U)>s>λp(U+) {\lambda _p}\left({{U^ -}} \right) > s > {\lambda _p}\left({{U^ +}} \right)

There is a sudden jump from U to U+ in the p characteristic area. For any given initial conditions Ul and Ur of the Riemann problem, the corresponding shock wave solution may not satisfy the entropy condition, it does not exist in physics, so it is necessary to construct a sparse solution U(x, t) of Riemann's problem.

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). U(x,t)={W(x/tUrUl)ξ1t<xxξ2txξ1t<ξ2t U\left({x,t} \right) = \left\{{W\left({\mathop {x/t}\limits_{{U_r}}^{{U_l}}} \right){\xi _1}t < \mathop x\limits_{x \ge {\xi _2}t}^{x \le {\xi _1}t} < {\xi _2}t} \right.

Among them, W(x / t) is a smooth vector function, W(ξ1) = Ul, W(ξ2) = Ur.

Let's determine the explicit expression of function W(x / t). According to the definition of function W(x / t), when ξ1t < 2t, the partial derivatives of U(x, t) with respect to t and with respect to x are as shown in formulas (16) and (17), respectively. Ut(x,t)=xt2W(x/t) {U_t}\left({x,t} \right) = - {x \over {{t^2}}}W'\left({x/t} \right) Ux(x,t)=1tW(x/t) {U_x}\left({x,t} \right) = {1 \over t}W'\left({x/t} \right)

Substitute formulas (16) and (17) into the conservation equations and mark ξ = x / t to obtain formula (18). F(W(ξ))W(ξ)=ξW(ξ) F'\left({W\left(\xi \right)} \right)W'\left(\xi \right) = \xi W'\left(\xi \right)

It can be seen from equation (18) that the values of W′ (ξ) and ξ need to meet the conditions shown in equation (19). W(ξ)=α(ξ)r¯p(W(ξ)),ξR W'\left(\xi \right) = \alpha \left(\xi \right){\bar r_p}\left({W\left(\xi \right)} \right),\xi \in R ξ=λp(W(ξ)) \xi = {\lambda _p}\left({W\left(\xi \right)} \right)

Among them, rp(W(ξ)) {\vec r_p}\left({W\left(\xi \right)} \right) represents a certain feature vector of F′(W(ξ)), and λp(W(ξ)) represents the corresponding feature value.

In order to determine the explicit expression of function W(ξ), take the partial derivative of ξ on both sides of equation (20) to obtain equation (21). 1=λp(W(ξ))W(ξ) 1 = \nabla {\lambda _p}\left({W\left(\xi \right)} \right)\, \cdot W'\left(\xi \right)

Substituting equation (19) into equation (21), the expression of scale factor α(ξ) can be obtained, as shown in equation (22). α(ξ)=1λp(W(ξ))rp(W(ξ)) \alpha \left(\xi \right) = {1 \over {\nabla {\lambda _p}\left({W\left(\xi \right)} \right) \cdot {{\vec r}_p}\left({W\left(\xi \right)} \right)}}

From this, the relevant W (ξ) partial differential equations can be obtained as shown in formula (23). W(ξ)=rp(W(ξ))λp(W(ξ))rp(W(ξ)),ξ1ξξ2 W'\left(\xi \right) = {{{{\vec r}_p}\left({W\left(\xi \right)} \right)} \over {\nabla {\lambda _p}\left({W\left(\xi \right)} \right) \cdot {{\vec r}_p}\left({W\left(\xi \right)} \right)}},\,{\xi _1}\, \le \xi \le {\xi _2}

Among them, W(ξ1) = Ul, W(ξ2) = Ur needs to point out that, in order to ensure that the value of the denominator in (23) is finite, λp must change monotonously between ξ1 and ξ2.

Let's discuss the conditions for the existence of sparse wave solutions between the initial states Ul and Ur. First introduce the concept of integral curve, for vector rp(U) {\vec r_p}\left(U \right) , if the tangent line at any point U on a curve is parallel to rp(U) {\vec r_p}\left(U \right) , it is said that this curve is the integral curve of rp(U) {\vec r_p}\left(U \right) .

Knowing from (18), the sparse wave solution W(ξ) must change along the integral curve of the eigenvector rp(W(ξ)) {\vec r_p}\left({W\left(\xi \right)} \right) , therefore, one of the necessary conditions for a sparse wave solution between Ul and Ur is: States Ul and Ur are on the same integral curve. Another necessary condition for the existence of sparse wave solutions between Ul and Ur is: The system of conservation equations is truly nonlinear in the p characteristic region, in order to ensure that A changes monotonously along the entire integral curve, so that the function is single-valued. Because on the integral curve, the state quantity can only move in the direction in which λp(W(ξ)) increases, therefore, the states Ul and Ur also need to meet the conditions shown in formula (24). λp(Ul)<λp(Ur) {\lambda _p}\left({{U_l}} \right) < {\lambda _p}\left({{U_r}} \right)

Data structure of the grid generation process

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.

Conclusions

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.

Figure 1

Comparison of two Delaunay internal node generation algorithms for a circular computing city
Comparison of two Delaunay internal node generation algorithms for a circular computing city

Figure 2

For the diffuser flow field, the comparison of two Delaunay internal node generation algorithms
For the diffuser flow field, the comparison of two Delaunay internal node generation algorithms

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 Qmean Qjoint Qmin
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

Zhang Y, Li H, Clark J D. Experimental simulation of mathematical learning process based on ‘chunk-objective’[J]. Applied Mathematics and Nonlinear Sciences, 2020, 5(2):425–434. ZhangY LiH ClarkJ D Experimental simulation of mathematical learning process based on ‘chunk-objective’[J] Applied Mathematics and Nonlinear Sciences 2020 5 2 425 434 10.2478/amns.2020.2.00061 Search in Google Scholar

Zhao Y. Analysis of Trade Effect in Post-Tpp Era: Based on Gravity Model and Gtap Model[J]. Applied Mathematics and Nonlinear Sciences, 2020, 5(1):61–70. ZhaoY Analysis of Trade Effect in Post-Tpp Era: Based on Gravity Model and Gtap Model[J] Applied Mathematics and Nonlinear Sciences 2020 5 1 61 70 10.2478/amns.2020.1.00007 Search in Google Scholar

Darrah N J, Hadley D E, Packel L, et al. A Simulation Center Geriatric Teaching Experience in Interprofessional Communication[J]. Journal of the American Geriatrics Society, 2016, 64(6):1355. DarrahN J HadleyD E PackelL A Simulation Center Geriatric Teaching Experience in Interprofessional Communication[J] Journal of the American Geriatrics Society 2016 64 6 1355 10.1111/jgs.1415627321620 Search in Google Scholar

Adam H, Pau A, Cesare C, et al. BIGNASim: a NoSQL database structure and analysis portal for nucleic acids simulation data[J]. Nucleic Acids Research, 2016(D1):D272–D278. AdamH PauA CesareC BIGNASim: a NoSQL database structure and analysis portal for nucleic acids simulation data[J] Nucleic Acids Research 2016 D1 D272 D278 10.1093/nar/gkv1301470291326612862 Search in Google Scholar

Wanli L I, Qie X, Shenming F U, et al. Simulation of quasi-linear mesoscale convective systems in northern China: Lightning activities and storm structure[J]. Advances in Atmospheric Sciences, 2016, 33(001):85–100. WanliL I QieX ShenmingF U Simulation of quasi-linear mesoscale convective systems in northern China: Lightning activities and storm structure[J] Advances in Atmospheric Sciences 2016 33 001 85 100 10.1007/s00376-015-4170-3 Search in Google Scholar

E Chacón, Tarazona P, Bresme F. A computer simulation approach to quantify the true area and true area compressibility modulus of biological membranes[J]. The Journal of Chemical Physics, 2015, 143(3):034706. ChacónE TarazonaP BresmeF A computer simulation approach to quantify the true area and true area compressibility modulus of biological membranes[J] The Journal of Chemical Physics 2015 143 3 034706 10.1063/1.492693826203041 Search in Google Scholar

Parker W S. Computer Simulation, Measurement, and Data Assimilation [J]. The British Journal for the Philosophy of Science, 2017, 68(1):273–304. ParkerW S Computer Simulation, Measurement, and Data Assimilation [J] The British Journal for the Philosophy of Science 2017 68 1 273 304 10.1093/bjps/axv037 Search in Google Scholar

Medvedev P V, Soldatov M A, Shapovalov V V, et al. Analysis of the Local Atomic Structure of the MIL-88à Metal–Organic Framework by Computer Simulation Using XANES Data[J]. JETP Letters, 2018, 108(5):318–325. MedvedevP V SoldatovM A ShapovalovV V Analysis of the Local Atomic Structure of the MIL-88a Metal–Organic Framework by Computer Simulation Using XANES Data[J] JETP Letters 2018 108 5 318 325 10.1134/S0021364018170083 Search in Google Scholar

Chen X, P Zillé, Shao L, et al. Optical flow for incompressible turbulence motion estimation[J]. Experiments in Fluids, 2015, 56(1):8. ChenX ZilléP ShaoL Optical flow for incompressible turbulence motion estimation[J] Experiments in Fluids 2015 56 1 8 10.1007/s00348-014-1874-6 Search in Google Scholar

Avakian A, Gellmann R, Ricoeur A. Nonlinear modeling and finite element simulation of magnetoelectric coupling and residual stress in multiferroic composites[J]. Acta Mechanica, 2015, 226(8):2789–2806. AvakianA GellmannR RicoeurA Nonlinear modeling and finite element simulation of magnetoelectric coupling and residual stress in multiferroic composites[J] Acta Mechanica 2015 226 8 2789 2806 10.1007/s00707-015-1336-0 Search in Google Scholar

Articles recommandés par Trend MD

Planifiez votre conférence à distance avec Sciendo