Trans. Tianjin Univ. 2012, 18: 90-96 DOI 10.1007/s12209-012-1714-9
Numerical Calculation of Channel Dredging Volume Using 3D Digital Stratum Model* MIAO Zhengjian(缪正建)，LI Mingchao(李明超)，ZHONG Denghua(钟登华) (State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, Tianjin 300072, China) © Tianjin University and Springer-Verlag Berlin Heidelberg 2012
Abstract： Prediction of channel dredging volume is critical for project cost estimation. However, many proposed approximate methods are not accurate. This paper presents a novel numerical method to accurately calculate the dredging volume using a 3D stratum model (DSM) and a channel surface model. First, the 3D DSM is constructed rapidly yet accurately from non-uniform rational B-splines (NURBS) surfaces through Boolean operation between a physical terrain model and a stratum surfaces model. Then, a parametric channel surface model is built from cross-section data and a channel center line using code implemented in the VC++ programming language. Finally, the volumes of different types of physical stratums can be calculated automatically and hierarchically to determine the dredging volume. Practical application shows that the DSM method is more precise and faster compared to the section method, and that the implementation of the developed software provides an interactive graphical user interface and visual presentation. Keywords： dredging volume; numerical calculation; digital stratum model; parametric modeling; surface integral
In dredging engineering, the quantity of sediment is an important factor that must be predicted to estimate the cost of the project. Many methods have been presented[1], such as the section method[2], contour method, square grid method[3], and digital elevation model (DEM) method[4]. Each of these methods has its individual features and applicable conditions and different accuracies. The section method is a traditional 2D approach to calculating dredging volume. It is more suitable for complex terrain with large fluctuations, especially for narrow and long terrain. The contour method is used to calculate the sediment quantity between two closed contour lines. It is suitable for terrain with a constant slope and closed contour lines, but its accuracy is not high. The square grid method is a common method with higher accuracy. However, the calculation result depends mainly on the size of the square grid. The smaller the grid is, the higher the precision is, the longer the computing time is. Therefore, it is more suitable for flat terrain. The DEM method, which is characterized by elevation, is an accurate method with higher precision and visual results, but it requires a large number of drilling and geophysical prospecting data. The first three methods mentioned above are ap-
proximate methods, producing a larger approximation error, while the DEM method applies a grid model, which requires large storage capacity. However, with the rapid development of computer modeling and 3D visualization, 3D engineering geological modeling[5,6] has been widely applied in many fields, including petroleum exploration, mining, tunnel excavation, and large-scale hydropower engineering. Because of the comparability between dredging engineering and hydropower engineering, a novel method is presented in this paper to calculate the required volume. This method is a numerical calculation method that is based mainly on 3D digital stratum model (DSM) and a channel surface model. Finally, with the surface integral method, the dredging volumes of different types of physical stratums can be calculated automatically and hierarchically. The 3D DSM requires less storage than other methods and the results show high precision and efficiency and are more visual. Therefore, it can meet the demands of dredging engineering.
1
Establishment of the 3D DSM
The accuracy of the 3D DSM depends greatly on the amount of the initial data available and their dispersion
Accepted date: 2011-11-02. *Supported by the Science Fund for Creative Research Groups of the National Natural Science Foundation of China (No. 51021004), National Natural Science Foundation of China(No. 50879056) and National Key Technologies R&D Program in the 12th Five-Year Plan of China(No. 2011BAB10B06). MIAO Zhengjian, born in 1986, male, doctorate student. Correspondence to MIAO Zhengjian, E-mail:
[email protected].
MIAO Zhengjian et al: Numerical Calculation of Channel Dredging Volume Using 3D Digital Stratum Model
over the study area. The general workflow of the 3D digital physical stratum modeling[7,8] is given in Fig.1. It includes three main modules: terrain modeling, stratum surface modeling, and physical stratum modeling. In the terrain modeling module[9], the triangulated irregular network (TIN) algorithm and NURBS technology are used to obtain the physical terrain model, which is the basis of the overall geologic solid model. In the stratum surface modeling module, with NURBS technology, stratum surfaces are built to describe geological objects. In the physical stratum modeling module, the results of the above two modules are seamlessly integrated by using Boolean operation. Following this integration, the corresponding 3D physical stratum model is built, and colors, texture mapping and the corresponding labels are added into the model. Finally, the 3D DSM can be output for geological analysis and application.
Fig.1
Workflow of 3D digital physical stratum modeling
1.1 Terrain modeling The digital terrain model (DTM) is a digital description with spatial location features and terrain attributes, which is the receptor of all the computing operations that establish the process of the whole model, and should meet the demands of small storage capacity, high precision and easy graphics computing operation. Therefore, NURBS technology is introduced to build a physical terrain model, enabling high-speed modeling based on a Tab.1
TIN model. Its specific modeling process is given as follows. (1) The original topography data is water depth data used in dredging engineering. CAD systems use text to express the elevation of the water depth point, and it is difficult to directly read text attributes when conducting the terrain surface modeling. Therefore, the water depth point text needs to be converted to a water depth point with elevation attributes. (2) Once the water depth point data has been generated, the TIN model can be created. The TIN model has high accuracy but large storage requirement, which is not conducive in the actual modeling operation. The model should, therefore, be as simple as possible and minimize the amount of stored data while still meeting the accuracy requirement. (3) After acquiring the TIN model, sample it to obtain section lines. (4) In the horizontal direction of the vertical section lines, draw up the network intersected with cutting lines to make up many quadrilateral grids. (5) Automatically generate the NURBS terrain surface on those grids. The points on the terrain surface may be unevenly distributed, and need to be adjusted to get a NURBS terrain surface with uniformly distributed points. (6) Draw a smaller outline body and use Boolean operation with the NURBS terrain surface model, and then construct the physical terrain model, which is also the receptor of the stratum modeling in the later operations. 1.2 Stratum surface modeling In dredging engineering, the geological stratum is mostly soils and rocks[10]. Due to the variety of soil data, such as drilling data and geological section data, the network lines fitting method[5] can be adopted to establish the soil NURBS surface models. According to the needs of the stratum modeling[6,11], the drilling data should be arranged in an Excel table, as shown in Tab.1 (only part of the table shown), and then read automatically into the modeling system to get points with the attributes of elevation. Meanwhile, the corresponding layers are added based on the stratum code.
Drilling data for stratum modeling
Drilling name
Stratum name
Stratum code
x coordinate/m
y coordinate/m
z coordinate/m
BZ1
Mucky silty clay
1-1
38 478 352.602
2 465 231.005
-9.05
BZ2
Mucky silty clay
1-1
38 479 674.515
2 465 231.005
-9.66
BZ3
Mucky silty clay
1-1
38 481 485.475
2 465 231.005
-9.47
BZ4
Silty clay
1-2
38 482 964.357
2 465 231.005
-15.64
—91—
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As the drilling data is limited, an interpolating fitting method is adopted in some areas, from which all the soil surface models may be established. Then, according to the sequence of the deposition of the soil, which is combined with the physical terrain model, the Boolean operation should be performed from top to bottom. Finally, the physical model for each soil layer will be obtained.
2
Parametric modeling of the channel
In dredging operation involving widening and deepening of a navigation channel, estimates of maintenance dredging quantities are needed, so a parametric modeling method of the channel is proposed adopting the VC++ programming language. Three parts are included (see Fig.2): input of initial data, creation of channel surface, and building of the dredged stratum model.
2.2 Interface design Fig.3 shows two cross-section lines and three kinds of soils for calculating the amount of sediment to be dredged. One line is the designed section, and the other is the calculated section, which considers overwidth and overdepth. In order to draw them automatically, the interface is designed based on VC++2005 (see Fig.4), containing three parts. Part A contains two design points (point A and point B) whose coordinates should be input first. The distance between point A and point B equals the designed width, which is calculated automatically in part B when clicking the “OK” button; the designed depth in Fig.3 is the z coordinate in part B. The third part has three important parameters: designed gradient, overwidth, and overdepth. When all the parameters are input and the “OK” button is clicked, these two cross-section lines will be drawn automatically by the modeling software.
Fig.3
Fig.2
Workflow of parametric modeling and calculation of channel
2.1 Initial data In dredging engineering construction, for the sake of maintaining the stabilization on both sides of the channel, the cross section is designed to be an inverse trapezoid, where its hemline length is the designed width and both sides are uptilted according to a designed gradient. Additionally, when the section is designed, the first stage is to obtain some parameters, such as the designed depth, designed width, channel center line, and the designed gradient; then, the overwidth and overdepth are determined according to the difference of the dredged soils; finally, the amount of sediment to be dredged is calculated. Therefore, the calculation values of overwidth and overdepth for various kinds of dredgers and the gradient values of different kinds of soils should be obtained first[12]. —92—
Schematic diagram of cross-section for calculating the sediment volume to be dredged
Fig.4
Interface of cross-section design
2.3 Channel modeling The channel always has different gradients along the center line. Thus, the cross-section lines should be obtained for all different stake numbers in construction. Then, all of the cross-section lines are lofted along the center line to build the channel excavation surface shown in Fig.5.
MIAO Zhengjian et al: Numerical Calculation of Channel Dredging Volume Using 3D Digital Stratum Model
Fig.5
3
Channel excavation surface model
Dredging volume calculation based on 3D DSM
direction, and N lj (u ) (0≤j≤n) is also a B-spline basis function of order l defined over the knot vector V={v0,…, vn+l| vj≤vj+1, j=0,…,(n+l-1)} in the v direction. For example, the recursive formula of N ik (u ) is given as 1 if ui u ui 1 1, N i (u ) otherwise 0, k u ui N ik 1 (u ) (2) N i (u ) u u i k 1 i ui k u k 1 N i 1 (u ) ui k ui 1
In Eq.(1), a closed spatial NURBS surface S(u, v) is provided, by which a closed spatial region is bound. According to the formula of Gauss and the principle of 3.1 Calculation principle of area and volume In the past, geometric calculation methods for 3D surface integral, the area A and volume V can be defined: A Su (u,v) Sv (u,v)dudv entities, including area and volume calculations, were S approximate methods, which produced a large approxi(3) S F (u,v)dudv mation error, e.g., the section method. The calculation V dΩ dxd ydz method presented here is based on 3D DSM, which is Ω built using the NURBS data structure. In order to obtain (4) S zdxd y S z (u,v) J dudv accurate results, the calculation of the NURBS curve and surface integral can be performed directly by utilizing the where the normal direction of surface S is outward; Su (u,v) and Sv (u,v) are the first partial derivative vectors formula of Gauss-Legendre[13]. Fig.6 demonstrates a NURBS surface based on one at one point of surface in the u and v directions; F (u, v) control point Pij (0≤i≤m, 0≤j≤n), which is controlled is a function calculating the length of the surface normal by (m+1) × (n+1) meshes. The NURBS surface is de- vector; S' is a closed bound domain where the region S is converted from the xoy coordinate system to the uov cofined as follows: m n ordinate system; and |J| is the Jacobi coordinate transforS (u, v) wij Pij Nik (u ) N lj (v) / mation[14]. i 0 j 0 Because the formulas of the area and volume of the m n k l NURBS surface contain double integrals, they can be wij N i (u ) N j (v) (1) i 0 j 0 calculated using the Gauss integral formula. The double normal Gauss-Legendre quadrature formula is: 1
1
1
1
Fig.6
NURBS surface(k＝l＝3)
In Eq.(1), wij is a weight factor corresponding to the control point Pij, 0≤i≤m, 0≤j≤n; the values of k and l are orders, generally k=l=3; N ik (u ) (0≤i≤m) is a Bspline basis function of order k defined over the knot vector U = {u0,…,um+k| ui≤ui+1, i=0,…, (m+k-1)} in the u
m
n
G (u ,v)dudv wi w j G (ai ,a j )
(5)
i 1 j 1
where wi and wj are the weights of the m and n orders of the Legendre orthogonal polynomial, while ai and aj are the zero points. If an integral interval is [ui, ui+1] and [vj, vj+1], before calculation, they should be transformed to the normal interval [-1, 1]. In general, the calculation principle of area and volume is stated as follows: S ( x(u , v ), y (u , v), z (u , v)) is any point on the NURBS surface S, and through it, we can calculate the normal vector. Meanwhile, the surface region should be divided into a series of rectangular regions with a discrete method, and for each small region transformed, the area and volume of which will be obtained by the Gauss-Legendre quadrature formula. Fi—93—
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nally, all the area and volume values must be combined sults are shown in Tab.4. to obtain the total area and volume in the whole region. Tab.2 Tunnel trench parameters 3.2 Model operation and quantity calculation Designed width/ Overwidth/ Overdepth/ Designed gradient So far, the channel excavation surface and 3D DSM m m m 42 1:3, 1:8 6.5 0.55 have been built based on the method of NURBS modeling. Following that, Boolean operation is performed on Tab.3 Coordinates of the control points the excavation surface and DSM for the excavation simuControl points x coordinate /m y coordinate /m z coordinate/m lation. As the stratums in the 3D DSM have been built P1 38 478 140.307 2 465 252.005 -19.86 independently and hierarchically, various types of physiP2 38 478 140.307 2 465 210.005 -19.86 cal excavated stratums can be obtained. Finally, the volP3 38 478 765.955 2 465 252.005 -38.388 umes of the physical stratums can be calculated autoP4 38 478 765.955 2 465 210.005 -38.388 matically using VC++ programming language to get the P5 38 480 977.515 2 465 252.005 -46.35 P6 dredging volume. 38 480 977.515 2 465 210.005 -46.35
4
P7
38 482 703.955
2 465 252.005
-40.135
P8
38 482 703.955
2 465 210.005
-40.135
P9
38 483 386.488
2 465 252.005
-19.86
P10
38 483 386.488
2 465 210.005
-19.86
Case study
The method presented here was applied to dredging volume calculations of the tunnel trench in the Hong Kong-Zhuhai-Macao bridge project with a total length of 5 249 m. A map of the area is shown in Fig.7, where the tunnel trench is displayed in a rectangular area. The data in the project are mainly topography data, drilling data and the geological section data. The topography data includes the plane position coordinates and water depth elevations. The plane position coordinates are obtained by DGPS technology and the water depth elevations are acquired by a multi-beam sounding system. However, due to high drilling costs, the amount of drilling data, which can describe the stratum structure accurately, is small. The geological section maps will be drawn based on these drilling data. Therefore, the initial data is limited and the accuracy of the stratum data is very low.
(a) Section 1
(b) Section 2
(c) Section 3
(d) Section 4
(e) Section 5
Fig.8 Fig.7
A map of the project area
The parameters of the tunnel trench model are given in Tab.2. There are several control points on the crosssection lines of the trench bottom, and their coordinates are shown in Tab.3. For comparison, we first use the section method to calculate the dredging volume. The five cross-sections of tunnel trench are shown in Fig.8. Then, according to the five cross-sections, the calculation re—94—
Five cross-sections of tunnel trench (bold lines representing the dredged surface, unit: m)
Tab.4 Calculation results of volume with the section method
1
Stake number/m K7+122
1 241.91
685.00
2 295 298.0
2
K7+807
5 459.69
1 726.44
12 181 018.3
3
K9+533.44
8 651.45
2 211.56
15 372 719.4
4
K11+745
5 250.70
626.00
2 057 305.2
5
K12+371
1 322.16
Sections
Total
Area/m2
Distance/m
Volume/m3
31 906 340.9
MIAO Zhengjian et al: Numerical Calculation of Channel Dredging Volume Using 3D Digital Stratum Model
From Tab.4, we can see the section method is an approximate method which calculates the area of each section and the distance between any two of them, then calculates each volume and finally adds them up to obtain the total volume. Besides, the section method cannot obtain the volume of each soil grade and thus, it is difficult to meet the requirement of dredging engineering. At the same time, using the digital stratum modeling method, the 3D DSM is built for the seabed dredging of the tunnel 7 282 m in length, 600 m in width, and 142 m in depth. It is a soil hierarchical model, each grade displayed by one color. We can then use the Boolean operation and rendering technique to output and save the data in graphic form to make the calculation results more visual, as shown in Fig.9. Based on the DSM, the volume of each soil grade is calculated respectively. The results are shown in Tab.5.
Tab.5
Soil grade
(b) 3D DSM dredged by the tunnel trench model(unit: m)
(c) 3D stratum model for the dredging volume (the soil classified by a small grade number is easier to excavate than that by a big one)
Fig.9
Boolean operation and calculation between 3D DSM and the tunnel trench model
Volume/m3 DSM method
Section method
2
13 092 637.5
——
3
5 634 135.7
——
4
4 357 643.2
——
5
4 548 443.5
——
6
3 698 232.4
—— ——
9
4 246.6
10
90 118.5
——
Total
31 425 457.4
31 906 340.9
Error
-1.5%
As the project is currently under construction, it is very difficult to compare the calculation results. But through the comparison with the section method, the method based on the 3D DSM shows higher computation efficiency and has better visual presentation. Therefore, it can provide a convenient tool for dredging volume calculation in dredging engineering.
5
(a) 3D DSM and tunnel trench model(unit: m)
Calculated volume of each soil grade by DSM and total calculated amount comparison
Conclusions
A novel numerical method is presented to calculate the dredging volume in a channel. It is mainly based on 3D DSM constructed using NURBS surfaces and channel surface model built adopting the parametric modeling method. Then Boolean operation is applied in the DSM to convert the results into a graphic display. Finally, by utilizing the method of surface integrals, the dredging volume is calculated automatically and hierarchically by the VC++ programming language. Practical engineering applications show that the method based on the 3D DSM, compared to the section method, shows higher precision, higher calculation efficiency and better visual presentation. However, the method is based on the 3D DSM, so the accuracy of results is dependent on the precision of the model. At the same time, the geological conditions in dredging engineering are very diverse and complex, and acquiring accurate data is very difficult. Therefore, 3D digital stratum modeling using NURBS is essential for volume calculations. As the method has high precision, calculation efficiency, and good visual results, it is an important and convenient supporting tool for volume calculation and can meet the requirements of many actual projects. In addition to its application to volume calculation for channels in dredging engineering, it can also be applied in —95—
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Vol.18 No.2 2012
other fields, such as railway and highway engineering, mining, tunnel excavation and petroleum exploration.
hanced NURBS modeling and visualization for large 3D
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