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Computer big data modeling system based on finite element mathematical equation simulation

Publicado en línea: 15 Jul 2022
Volumen & Edición: AHEAD OF PRINT
Páginas: -
Recibido: 19 Feb 2022
Aceptado: 25 Apr 2022
Detalles de la revista
License
Formato
Revista
eISSN
2444-8656
Primera edición
01 Jan 2016
Calendario de la edición
2 veces al año
Idiomas
Inglés
Introduction

When high-speed moving bodies such as landslides, debris flows, and glaciers enter the water and impact water bodies in enclosed waters such as reservoirs, lakes, rivers, and bays, swells will be induced, and large-scale disasters will be produced. A representative example is a large-scale monolithic rock landslide that occurred on the bank of the Vaian Reservoir in Italy in 1963. The landslide body was 2 km long, 1.6 km wide, the landslide volume was 240 million m3, and the sliding speed was 15–30 m/s. When the landslide body breaks into the reservoir, a surge of up to 90–130m is generated, causing destructive damage to the reservoir bank and the market town downstream of the dam [1]. During the typhoon on July 6, 2011 in Japan, a large amount of debris flowed into the upper reaches of the Naban Igawa Pingguo Dam, causing swells exceeding 5m in height, causing damage to the dam gate and creating a flooded roof. At the same time, the Pingguo suspension bridge upstream of the dam was damaged. The Zipingpu Dam is a concrete face rockfill dam with a height of 158m and a storage capacity of 1.11 billion m3. It is only 17km away from the epicenter of the Wenchuan earthquake that occurred on May 12, 2008. The Wenchuan earthquake triggered the Zipingpu landslide. The height difference between the center of the landslide and the water surface of the reservoir was 700m. About 450,000 m3 of landslide rock mass rushed into the reservoir at high speed, generating swells as high as 25m, and about 70 lakeside fishermen were swarmed. Waves swept into the reservoir causing death, and 10 cars were swept into the lake [2].

The big data computer system is an important modern system supporting equipment. The traditional big data computer system is composed of special big data computer equipment. It receives the information of atmospheric static pressure, total atmospheric pressure, and total atmospheric temperature through the external interface, and then calculates it through the internal calculation, output absolute pressure altitude, relative pressure altitude, true airspeed, indicated airspeed, Mach number, atmospheric density, atmospheric density ratio, atmospheric static temperature and other related parameters required by peripherals. The traditional big data computer is cumbersome and takes up space. Wang Liangliang et al. [3] conducted in-depth research on the principle composition, architecture interface, flight parameters, etc. of the big data computer system, and carried out a model based on the latest model-based system engineering modeling tools Modeling, and taking full advantage of the advantages, a lot of simulation work has been done in the model stage. After the simulation is completed, the source code of the model is automatically generated, loaded into the new avionics display, and the actual test is carried out through the air pump test equipment. The model passed the test perfectly, successfully replaced the traditional big data computer system, and further realized the high integration of avionics display functions. Tian Lirong et al. [4] replaced big data computer equipment with models, reducing the weight of peripherals, which has important engineering practical significance for precision instruments whose weight is calculated in grams. Liu Wenxue et al. [5] implemented it based on the latest model-based systems engineering method. The model-based systems engineering method changed the traditional development model, and took the model as the core to separate the model from the platform, so that the model can be replicated on different platforms. It can effectively improve the development efficiency and reliability. Wang Xiaoyang et al. [6] compared the development process of modern technology and traditional technology. The model-based system engineering method has the following advantages: First, the code is automatically generated, and the generated code meets the D0178C-3 standard, eliminating the need for manual coding. hassle and possible errors, increasing efficiency. Secondly, the model is separated from the platform, which improves the reusability of the functional model and has strong portability.

In order to prevent surge disasters, a prediction method capable of establishing surge characteristics and magnitudes is required. The shallow water long wave equation is often used as the governing equation for swell. Because the displacement flow term is included in the shallow water long-wave equation, the stability of numerical calculation is very poor, and it is necessary to solve the stability problem of numerical calculation. In addition, the total coefficient matrix obtained by the finite element discretization is asymmetric. Therefore, in order to shorten the calculation time in large-scale surge analysis, it is necessary to develop a multi-core and multi-thread parallel solution method for linear equations. The classical process of computer modeling is shown in Figure 1 [7]. In this study, a sparse linear equation solver implemented on a shared memory machine provided by Intel MKL is used to develop a high-performance finite element analysis program for surge, and some theoretical solutions and experimental results are used to verify the developed program.

Figure 1

Computer big data modeling

Research Methods
Governing equation

The shallow water long-wave equations expressed in terms of vertical mean flow velocity include the continuity equation (1), and the motion equations (2) and (3) in the x and y directions: ηt+Ux+Vyzt=0 {{\partial \eta} \over {\partial t}} + {{\partial U} \over {\partial x}} + {{\partial V} \over {\partial y}} - {{\partial z} \over {\partial t}} = 0 Ut+x(uU)+y(vU)+ghηx+n2h73UU2+V2ε(2Ux2+2Uy2)=0 {{\partial U} \over {\partial t}} + {\partial \over {\partial x}}\left({uU} \right) + {\partial \over {\partial y}}\left({vU} \right) + gh{{\partial \eta} \over {\partial x}} + {{\partial {n^2}} \over {{h^{{7 \over 3}}}}}U\sqrt {{U^2} + {V^2}} - \varepsilon \left({{{{\partial ^2}U} \over {\partial {x^2}}} + {{{\partial ^2}U} \over {\partial {y^2}}}} \right) = 0 Vt+x(uV)+y(vV)+ghηy+gn2h73VU2+V2ε(2Vx2+2Vy2)=0 {{\partial V} \over {\partial t}} + {\partial \over {\partial x}}\left({uV} \right) + {\partial \over {\partial y}}\left({vV} \right) + gh{{\partial \eta} \over {\partial y}} + {{g{n^2}} \over {{h^{{7 \over 3}}}}}V\sqrt {{U^2} + {V^2}} - \varepsilon \left({{{{\partial ^2}V} \over {\partial {x^2}}} + {{{\partial ^2}V} \over {\partial {y^2}}}} \right) = 0

In the formula, η is the water surface elevation, z is the ground surface elevation, as shown in Figure 2, the water depth h=η z; the x-direction stream U=hu, the y-direction stream V=hv, g is the gravity acceleration, and n is Manning coefficient, ε is the water depth average eddy viscosity coefficient. In order to consider the change of the ground surface elevation caused by the landslide movement, which is different from the general shallow water long-wave equation, the water surface and the ground surface elevation are introduced in Equation (1).

Figure 2

Surge diagram

Direct integration of time

Considering the water surface elevation and the flux vectors in the x and y directions of the two states tn+1 and tn with a time interval of Δt, the time differential of the water surface elevation and the flux vectors in the x and y directions can be expressed as: ηt=ηn+1ηnΔt {{\partial \eta} \over {\partial t}} = {{{\eta ^{n + 1}} - {\eta ^n}} \over {\Delta t}} Ut=Un+1UnΔt {{\partial U} \over {\partial t}} = {{{U^{n + 1}} - {U^n}} \over {\Delta t}} Vt=Vn+1VnΔt {{\partial V} \over {\partial t}} = {{{V^{n + 1}} - {V^n}} \over {\Delta t}}

The water surface elevation at time tn+θh and the time differential of the beam vectors in the x and y directions at time tn+θu can be expressed as: ηn+θh=θhηn+1+(1θh)ηn {\eta ^{n + {\theta _h}}} = {\theta _h}{\eta ^{n + 1}} + \left({1 - {\theta _h}} \right){\eta ^n} Un+θu=θuUn+1+(1θu)Un {U^{n + {\theta _u}}} = {\theta _u}{U^{n + 1}} + \left({1 - {\theta _u}} \right){U^n} Vn+θu=θuVn+1+(1θu)Vn {V^{n + {\theta _u}}} = {\theta _u}{V^{n + 1}} + \left({1 - {\theta _u}} \right){V^n}

In the formula, θh and θu are time integration parameters, and the time integration is unconditionally stable when θh and θu ≥ 1/2. This paper takes θh =θu = 1/2, that is, the Crank-Nicolson method is used for time integration. Substitute equations (4)~(9) into equations (1)~(3), and we can get: [A11A12A13A21A220A310A33]{ηn+1Un+1Vn+1}={b1b2b3} \left[{\matrix{{{A_{11}}} \hfill & {{A_{12}}} \hfill & {{A_{13}}} \hfill \cr {{A_{21}}} \hfill & {{A_{22}}} \hfill & 0 \hfill \cr {{A_{31}}} \hfill & 0 \hfill & {{A_{33}}} \hfill \cr}} \right]\left\{{\matrix{{{\eta ^{n + 1}}} \hfill \cr {{U^{n + 1}}} \hfill \cr {{V^{n + 1}}} \hfill \cr}} \right\} = \left\{{\matrix{{{b_1}} \hfill \cr {{b_2}} \hfill \cr {{b_3}} \hfill \cr}} \right\}

Where: A11=1Δt(M+Ms)+θhKc {A_{11}} = {1 \over {\Delta t}}\left({M + {M_s}} \right) + {\theta _h}{K_c} A12=θu(B1+B1S) {A_{12}} = {\theta _u}\left({- {B_1} + {B_{1S}}} \right) A13=θu(B2+B2S) {A_{13}} = {\theta _u}\left({- {B_2} + {B_{2S}}} \right) A21=θh(G1+G1S) {A_{21}} = {\theta _h}\left({{G_1} + {G_{1S}}} \right) A22=1Δt(M+MS)+θu(A+AS+T+TS+KC+D) {A_{22}} = {1 \over {\Delta t}}\left({M + {M_S}} \right) + {\theta _u}\left({A + {A_S} + T + {T_S} + {K_C} + D} \right) A31=θh(G2+G2S) {A_{31}} = {\theta _h}\left({{G_2} + {G_{2S}}} \right) A33=A22 {A_{33}} = {A_{22}}

The flow velocities ũ and ṽ in the x and y directions are explicitly approximated and linearized using the Adams-Bashforth method with 2nd order accuracy. For the stability of numerical analysis, the time step size needs to satisfy the CFL condition, so the time step size of each time step may change. Using the idea of Adams-Bashforth method [8], the Adams-Bashforth method using unequal time steps in the program is: b1=Ub+1Δt(M+MS)ΔznKC(zn+θh+ηn) {b_1} = {U_b} + {1 \over {\Delta t}}\left({M + {M_S}} \right)\Delta {z^n} - {K_C}\left({{z^{n + {\theta _h}}} + {\eta ^n}} \right) b2=(11θh)A21ηn+[A22(A+AS+T+TS+KC+D)]Un {b_2} = \left({1 - {1 \over {{\theta _h}}}} \right){A_{21}}{\eta ^n} + \left[{{A_{22}} - \left({A + {A_S} + T + {T_S} + {K_C} + D} \right)} \right]{U^n} b3=(11θh)A31ηn+[A22(A+AS+T+TS+KC+D)]Vn {b_3} = \left({1 - {1 \over {{\theta _h}}}} \right){A_{31}}{\eta ^n} + \left[{{A_{22}} - \left({A + {A_S} + T + {T_S} + {K_C} + D} \right)} \right]{V^n} u˜=un+Δtn2Δtn1(unun1) \tilde u = {u_n} + {{\Delta {t_n}} \over {2\Delta {t_{n - 1}}}}\left({{u_n} - {u_{n - 1}}} \right)

In the formula, Δtn−1 and Δtn are the time step lengths of time steps n-1 and n, respectively. When the two time steps are equal, formula () can be simplified to the Adams-Bashforth method of equal time steps.

Multi-core and multi-threaded parallel solver PARDISO

The solution of the first-order simultaneous linear equation system Ax=b with the large-scale sparse matrix obtained by space and time discretization as the coefficient is one of the central tasks of the finite element analysis of the surge, so the high-speed and robust solution suitable for the sparse linear equation system is adopted device is very important [9]. The direct method can also solve the problem that the iterative method does not converge, especially if the matrix is symmetric positive timing, it can be solved if there is no numerical error. Because of these specialties, the direct method has been widely used. By considering the zero elements inside the one-dimensional storage of the variable bandwidth of the matrix, the direct solution method of the sparse linear equation system, which can further reduce the calculation amount and save memory, has now become the mainstream. The calculation steps of the direct method solvers of sparse linear equations including PARDISO generally include sequential calculation of four steps, such as reordering, symbol decomposition, LU decomposition, and forward and backward substitution. The calculation of each step is briefly described below.

Reordering is to use a suitable permutation matrix P, so that the elements filled in when LU decomposes PAPT are as few as possible. Filled elements here refer to elements that are zero before LU decomposition but non-zero after decomposition. Reordering methods include equal-scale reduction (minimum degree method, Reverse Cuthill-McKee method), triangulation (Markowitz method, Tewarson method), block (Stewart method, Nested Dissection method) and many other algorithms [10]. PARDISO uses the minimum order algorithm or the Nested Dissection algorithm in the METIS algorithm package for reordering. Symbol decomposition does not perform the decomposition calculation of specific elements, but only focuses on the distribution form of non-zero elements in the LU decomposition of matrix A, and finds the positions of non-zero elements after LU decomposition. From this, the amount of memory and calculation required for LU decomposition is calculated, and the memory required to save non-zero elements after LU decomposition is ensured, and the position of non-zero elements is recorded. In order to perform symbolic decomposition efficiently, the concept of column elimination tree is used to classify the problem as a path exploration problem of efficient graphs, thus making high-speed computation possible [11]. The actual LU factorization is performed with the amount of memory guaranteed during symbolic factorization. As the main LU decomposition method, refer to the right-looking algorithm on the right side of the update column, and the left-looking algorithm on the left side is widely known. The multi-core and multi-thread parallel solver PARDISO used in this program utilizes left-looking and right-looking algorithms to achieve efficient parallel computation.

The solution x can be obtained by forward substitution with the lower triangular matrix L decomposed by LU, and backward substitution with the upper triangular matrix U. If the first-order simultaneous linear equation system with the same coefficient matrix A is solved repeatedly (for example, when one loading step is calculated by the modified Newton-Raphson method), the solution can be solved by iteratively performing forward and backward substitution calculations. If the construction of the non-zero elements of the coefficient matrix is the same, then the calculation must be returned to the LU decomposition. Changes such as boundary conditions or analysis range then have to resynthesize the coefficient matrix and start the solution process from the reordering.

Another important concept in the solver is called a super node. The so-called super node is a set of columns in the upper triangular matrix L that are all non-zero, and each column has the same non-zero structure. For example, {1, 2}, {3, 4}, {5}, {6, 7, 8} are super nodes of order 2, 2, 1, and 3, respectively. After the left-looking algorithm is introduced into the super node, it can promote the partitioning, that is, the localization of data access, which can greatly improve the computing speed of the hierarchically structured memory computer. Assuming the zero elements of different non-zero constructs to be non-zero elements, generate super nodes, and in some cases more efficient super nodes can be obtained.

The multi-core and multi-thread parallel computing sparse linear equation solver PARDISO realizes parallelization in three levels: parallelization of tree elimination, parallelization of node level, and parallelization of data channel processing.

When PARDISO is used for surge finite element calculation, it needs to generate a list of the positions of the non-zero elements of the coefficient matrix A of the linear equation system. PARDISO uses a row-based storage method, that is, the deformed CSR (compressed sparse row) form, and the symmetric matrix only stores the upper half triangular elements. This method stores each non-zero data in row units. The storage of a sparse matrix A by PARDISO includes three arrays [12]:

values - non-zero elements of matrix A. The non-zero data of matrix A is mapped to the values array by the following columns and rowindex.

the column of the matrix where each element in columns-values is located.

rowindex - gives the position of each row's element in values.

Analysis of results
Data preprocessing and theoretical value calculation

Data preprocessing is responsible for preprocessing the initially obtained raw data into data files that can be easily parsed. The data preprocessing module in this system is divided into three processes: data batch decompression, data cleaning and preliminary analysis of data files. The batch decompression operation is because the original data exists in the form of multi-layer data compression packages, so it is necessary to realize the A large number of packets do batch deep decompression operations. The data parsing operation is to perform preliminary data parsing on the decompressed data files (xml format files and text files), and then obtain text data files stored in the form of key-value pairs. The data cleaning part is to perform corresponding operations after the data parsing process. Ensure the correctness and consistency of the data The final processed data will be stored in a specific directory for subsequent use. The data preprocessing function is provided by an independent PC-side application software. Here, we focus on the advantages of PC-side applications compared with web applications in performance and data processing efficiency, avoiding browser compatibility issues and network environment limitations in web applications. The influence of factors gives users a better immersive interactive experience.

Data transfer management is the “data porter” of the system, mainly responsible for the transfer and migration of system data. The data transmission management module includes two functions of data import and distributed storage. The data import function provides data to the user. The theoretical value calculation is based on reference materials and theoretical formulas, importing the data of major surge disasters in history into local files and relational databases. In the model system designed by text, the distributed storage function provides users with the function of migrating system data from relational database to h-base database of distributed storage, realizing distributed storage of large amounts of data, and calculating the surging generated by landslides. The high theoretical values are shown in Table 1:

Theoretical calculation of swell rise from historic landslides or partially underwater landslides

Date Location Landslide material landslide volume/×106m3 Death toll Theoretical value of surge rise/m
1756 Tjelle Granite gneiss 15 38 46.883
1792 Yuan Island Volcanic debris 500 >15000 10.282
1883 Krakatau Pyroxene -- 36000 35.426
1888 Ritter Island Basalt 5000 >100 20.732
1905 Disench.Bay Glacier ice 29 0 35.217
1934 Tafjord Gneiss 2~3 41 62.784
1936 Ravnefjell Gneiss 0.451 73 74.225
1958
1971 Yanahuin Lake Limestone 240 >2500 270.377
1980 M.St.Helens Rock 430 0 200.082
2008 Zipingpu R. Rock 0.45 >70 524.362
2010 513 Lake, Peru Glacier ice 0.5 1 30.683
Analysis of results

Using the developed software, the wave propagation with a water level difference of 0.8m in the 20m-long tank shown in Figure 3 was first calculated. This calculation example is the phenomenon of instantaneous collapse and discharge of a stationary dam. In order to compare with the theoretical solution of complete fluid, the average eddy viscosity coefficient of water depth ε=0 is assumed in the calculation, and the side wall of the water tank is assumed to be a slip condition. Figure 4 shows the theoretical solution of the water depth and flow velocity 1 s after the dam collapses.

Figure 3

Comparison of water depth results after 1s of swell caused by dam collapse

Figure 4

Comparison of flow velocity results after 1s of swell generated by dam collapse

Observing the curves and data in Figure 3 and Figure 4, it can be concluded by calculation that the normalized error of the flow rate obtained by the two is only between 5.2% and 6.8%, which is within the allowable error range specified by the D0178C-3 standard. It shows that the calculated results are basically consistent with the theoretical values, which verifies the correctness of the algorithm and model.

In order to confirm the calculation accuracy of the moving boundary problem, the trough is 10m long, the water level is 0.2m high on the left side of the center, and the instantaneous collapse of the dam in the anhydrous dry-bed trough is calculated on the right side. As in the previous example, in the calculation, it is assumed that the average water depth eddy viscosity coefficient ε=0, and the sidewall of the water tank is a slip condition. Through the comparison of the water depth calculation results and theoretical solutions after the dam collapsed for 1s. It can be seen that the calculation results and the theoretical solutions are also very consistent under the condition of dry-bed sink.

Conclusion

Prediction of swells generated when high-speed-moving landslides, debris flows, and glaciers impact water bodies in enclosed waters such as reservoirs, lakes, rivers, and bays is often a large-scale computational problem. In order to shorten the calculation time of large-scale surge analysis, this research uses PARDISO, a sparse linear equation solver implemented on a shared memory machine provided by Intel MKL, to develop a high-performance finite element analysis program for surge, and use some theoretical solutions and experimental results. The developed program is verified. The theoretical solution and numerical calculation results of vertical dam failure, as well as the comparison of laboratory experimental results and numerical calculation results, show that the developed high-performance finite element software for swell can quickly obtain reliable calculation results.

Figure 1

Computer big data modeling
Computer big data modeling

Figure 2

Surge diagram
Surge diagram

Figure 3

Comparison of water depth results after 1s of swell caused by dam collapse
Comparison of water depth results after 1s of swell caused by dam collapse

Figure 4

Comparison of flow velocity results after 1s of swell generated by dam collapse
Comparison of flow velocity results after 1s of swell generated by dam collapse

Theoretical calculation of swell rise from historic landslides or partially underwater landslides

Date Location Landslide material landslide volume/×106m3 Death toll Theoretical value of surge rise/m
1756 Tjelle Granite gneiss 15 38 46.883
1792 Yuan Island Volcanic debris 500 >15000 10.282
1883 Krakatau Pyroxene -- 36000 35.426
1888 Ritter Island Basalt 5000 >100 20.732
1905 Disench.Bay Glacier ice 29 0 35.217
1934 Tafjord Gneiss 2~3 41 62.784
1936 Ravnefjell Gneiss 0.451 73 74.225
1958
1971 Yanahuin Lake Limestone 240 >2500 270.377
1980 M.St.Helens Rock 430 0 200.082
2008 Zipingpu R. Rock 0.45 >70 524.362
2010 513 Lake, Peru Glacier ice 0.5 1 30.683

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