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MethodsX. 2020; vii: 101061.

Method for generating high-quality tetrahedral meshes of geological models past utilizing CGAL

Received 2022 Jul 3; Accustomed 2022 Sep vii.

Abstract

High-quality computational meshes are crucial in the analysis of displacements and stabilities of stone and soil masses. In this newspaper, we present a method for generating high-quality tetrahedral meshes of geological models to exist used in stability analyses of rock and soil masses. The method is implemented by utilizing the Computational Geometry Algorithms Library (CGAL). The input is a geological model consisting of triangulated surfaces, and the output is a loftier-quality tetrahedral mesh of the geological model. To demonstrate the effectiveness of the presented method, we utilise it to generate a serial of computational meshes of geological model, and we then analyse the stabilities of the stone and soil slopes on the basis of the generated tetrahedral mesh models. The applications demonstrate the effectiveness and practicability of the nowadays method.

• A method for generating high-quality tetrahedral meshes of geological models is presented.
• We evaluate the quality of the tetrahedral mesh of geological model using iv metrics.
• 3 applications demonstrate the effectiveness and practicability of the presented method.

Keywords: Computational geometry, Geological model, Mesh generation, Tetrahedral mesh, Algorithm, CGAL

Graphical abstract

Image, graphical abstract

Specifications table

Subject area Area	Geotechnical Technology
More specific subject area	Mesh Generation and Numerical Simulation
Method name	GeoTet
Proper name and reference of original method	N/A
Resource availability	N/A

Method details

In this department, we will draw the implementation details of presented method for generating high-quality tetrahedral meshes of geological models.

Presented method for generating loftier-quality tetrahedral meshes of geological models

The presented method is devoted to tetrahedral mesh generation of a geological model, see Fig. 1. The domain representing the geological model is typically composed of several strata, each one must be the matrimony of at least one connected component of the complement of the 2nd triangulated surface. Subdomain indices belonging to the corresponding subdomain must be divers and the orientation of each surface patch should exist adamant. Before the meshing process, the input pairs of the subdomain should exist constructed according to the orientation and the subdomain indices, which represent the subdomains on both sides of the triangulated surfaces.

Fig. 1

Workflow of the presented method.

The meshing process will preserve the features of the geological model. The mesh generator is customized to output the meshes that cater to the user'south needs as much as possible, such as the mesh size and user-customized quality criteria in Fig. 2. The output meshes are composed of tetrahedral meshes, which gauge each input domain feature: subdomains, boundary triangulated surfaces or input domain features with dimensions of 0 or i.

Illustration of the process of generating the tetrahedral mesh of a geological model. (a) All the surfaces must be triangulated and guaranteed the geometric and topological consistency. (b) The domain is divided several subdomain, each subdomain is identified by a subdomain index. (c) Each triangulated surfaces is orientated. We employ any point located in subdomain to determine the positive or negative orientation. (d) The geological model is meshed with tetrahedrons, which can be used as the computational model in numerical analysis.

Step 1: Inputting a geological model consisting of triangulated surfaces

The geological model to be meshed is enclosed by a collection of triangulated surfaces, forming a circuitous as in Fig. 2(a). Each triangle of the triangulated surface is called a facet. In that location are two constraints on triangulated surfaces. Commencement, triangulated surfaces of the complex cannot self-intersect. 2nd, ii triangulated surfaces of the complex are either disjoint or share a subset of their edge edges. Fig. 3(a) shows invalid input. The intersection line cannot guarantee topological consistency. At that place are many isolated points. In this method, isolated points are forbidden. Fig. iii(b) shows valid input. Thus, an important precondition for the meshing algorithm to work correctly is that any two surface patches cannot intersect. If two surfaces intersect, the intersection edges must exist part of the input surfaces and take 1-dimensional features.

(a) The two triangulated surfaces are not valid inputs since the topological consistency is not promised. (b) The two triangulated surfaces are valid inputs. Geometric and topological consistency are guaranteed.

Step 2: Determining the orientation of each triangulated surface

The orientation of each triangulated surface must be identified. Each triangulated surface is oriented and has two sides. Each triangulated surface has a positive orientation and a negative orientation. The orientation of each surface is the aforementioned every bit that of all its facets. The positive side is the union of the positive side of each triangulated facet, usually called the "outside" of the surface. The negative side is the other side of the surface. We can utilize whatsoever betoken located in the subdomain to determine the orientation of the triangulated surfaces in Fig. 2(c).

The office CGAL::orientation (p, q, r, s) in CGAL can exist used to determine the orientation of a plane composed of the betoken p, q, r, which is equanimous of a triangle in the surface patch. Point s is located on the left side or right side of this triangle. Therefore, taking the three points of facets of triangulated surfaces and a point located in the subdomain as the function argument, we can determine the orientation.

Step iii: Amalgam the input pairs of subdomains

In a geological model, each subdomain represents the corresponding stratum. The subdomains have indices of integer blazon, and the exterior of the mesh domain is associated with the subdomain index 0. Each subdomain alphabetize ranging from 1 to the number of subdomains should be divers. It is also of import that for each triangulated surface, the subdomain indices on both sides of the surface are known. The input pairs of subdomains are used to combine the subdomain indices of both sides of each triangulated surface. One subdomain index is on the positively oriented "side" of the triangulated surface. The other subdomain index is on the negative side. The interface of sub-domains is supposed to be provided and sub-domains are sharing input interfaces with dissimilar subdomain indices. Besides, nosotros also marker respective subdomain indices for the interface. For each triangulated surface, there is a corresponding input pair of subdomain indices. The order of the constructed input pair of triangulated surfaces should be consistent with the order in which the triangulated surfaces are saved. These input pairs of subdomain indices associating the subdomain with triangulated surfaces can guarantee right mesh generation for each subdomain.

Step iv: Generating tetrahedral meshes

To achieve the user's needs with respect to the size of the mesh elements and the accuracy of the purlieus approximation, we practical some criteria implemented by CGAL. The criteria can drive the Delaunay refinement process and mainly contain the jail cell size and facet size. The cell size controls the size of the tetrahedral mesh and the facet size controls the size of the surface facets. The mesh generation process is a Delaunay refinement process followed past an optimization. The optimization phase contains four optimization processes: Optimal Delaunay Triangulation (ODT) smoothing [1], a Lloyd smoothing [two], a perturber and an exuder [3]. It should be executed as follows: an ODT smoother, a Lloyd smoother, a sliver perturber, and a sliver exuder. ODT is optimal Delaunay triangulation. The procedure of each application can be activated based on the needs of the user. Later the optimization stage, the average mesh quality will be improved. When the plan finishes, it volition generate high-quality tetrahedral meshes, such as that in in Fig. 2(d), and will output a mesh format file containing the betoken, triangle and tetrahedral mesh information.

In full general, the surface mesh is typically stock-still during the volume mesh generation. When the required volume size is less than half the size of the surface meshes, the method will refine the surface meshes for the generation of the required volume sizes by the package. When the required book size is greater than half the size of the surface meshes and less than the size of surface meshes, the original surface meshes are fixed, and the refinement is ignored. And the output volition be loftier-quality mesh. Thus, we recommend that the size of the required volume and the surface meshes are as consequent as possible. When the required book size is greater than the size of the surface meshes, the method will generate sick-shape meshes which can influence the result of numerical simulation. Thus, we recommend that the required volume size should less than or equal to surface meshes size.

The conformity of surface mesh ways that at that place is no isolated point. The method will automatically guarantee the conformity in the refinement procedure so that the user does not need to do additional settings. Only if the initial surface meshes is conformal, the method will generate meshes with overall conformity.

Method validation

Applications of the Presented Method

To demonstrate the effectiveness of the presented method, we employ it to generate a serial of computational meshes of geological model. FLAC^3D [4] is oftentimes used for numerical simulation in geotechnical applied science applications. We as well developed an application interface to catechumen the mesh format file generated by CGAL to a FLAC^3D format file. We so analysed the stabilities of the stone and soil slopes on the ground of the generated tetrahedral mesh models.

The measurement metrics of the element quality include (i) the qualitytest1, (2) the qualitytest2, (3) the aspect ratio, and (4) the orthogonality. Qualitytest1 and qualitytest2 are the measurements of the volume over the edge length and skew, see Eqs. (1) and (2), respectively.

where B represents the volume over the border length, V represents tetrahedron book and 50 represents shortest edge length.

$T_{s g e w} = (V_{i d eastward a l} - V) / V_{i d e a l}$

(2)

where T_skew represents the skew, V represents tetrahedron volume and 5_platonic is the volume co-ordinate to the radius of the circumscribed ball.

The attribute ratio of tetrahedral mesh is the ratio of the longest edge length to the shortest border length in Eq. (three).

$A_{r a t i o} = L_{50 o n k due east southward t} / {Fifty}_{s h o r t due east s t}$

(3)

where A_ratio represents the aspect ratio, Fifty_longest represents longest edge length and Fifty_shortest represents shortest edge length.

The ortho skew quality for meshes is computed using the face normal vector, and the vector from the tetrahedron centroid to each of the faces. Fig. iv illustrates the vectors used to decide the ortho skew quality for a mesh.

Fig. 4

Vectors used to compute the ortho skew quality of a tetrahedron.

The ortho skew for a tetrahedron is computed equally the maximum of the following quantities computed for each face i in Eq. (iv).

$O_{r a t i o} = \frac{\underset{A_{i}}{\to} . \underset{F_{i}}{\to}}{\underset{| A_{i} |}{\to} . \underset{| F_{i} |}{\to}}$

(4)

where $\vec{A_{i}}$ is the face up normal vector and $\vec{F_{i}}$ is a vector from the centroid of the tetrahedron to the centroid of that face up.

Result

Application Case 1

This model is enclosed by 21 triangulated surfaces and contains eight strata. There are two thin layers and a mistake in this model. For this kind of geological model, different mesh sizes for different subdomains based on the user's needs tin can generally amend fulfil the engineering requirements. Nosotros provide different mesh size parameters. Later the meshing process, we obtain a series of tetrahedral meshes. We so import these meshes into FLAC^3D.

Fig. 5 shows the mesh generated past the presented method. The mesh size for thin layers and faults is prepare to 4 and the size of the meshes of other strata is set up to ten. Finally, the model is divided into 62,994 nodes, and 312,542 tetrahedral meshes. The boundary condition is the aforementioned as that in the above application cases. The Mohr-Coulomb model was used to simulate stress and strain under a self-weight state. The maximum displacement is approximately 0.1 m (Fig. 6). A mesh quality analysis based on the value distribution of the meshes is shown in Fig. 13. Approximately 68% of all the meshes have values in the range from 0.5 to 1.0 and the proportion of values less than 0.2 is extremely low in Fig. 7(a). The distribution of mesh quality by and large has a acme at the value at 0.6 in Fig. vii(b) and the proportion of meshes with values greater than 0.5 is over 65% in Fig. 7(c). Shine distributions are also visible in Fig. 7(d). In general, the distribution of mesh quality generally has a elevation at a value from 0.6 to 0.eight. The majority of meshes have a quality value between 0.five and 1.0. More than 60% of the meshes have a value greater than 0.iv. Thus, all the tetrahedral meshes are able to satisfy the requirements of numerical simulations. The results also bear witness that these computational meshes can be used for numerical simulation in other kinds of commercial software.

The input geological model and generated computational mesh in application case 1.

Numerically calculated results in application instance ane.

Statistics of the quality of computational mesh in awarding case i.

Statistics of the quality of computational mesh in application instance iii.

Application Case 2

The geological model represents a slope located in Shanxi Province in China. The computational mesh comprises 54496 vertices, 70970 triangles, and 259652 tetrahedral meshes (Fig. viii). The 10-centrality side and Y-axis side were adopted to constrain the X-direction and Y-management displacement respectively. The bottom boundary of the model fixed the Z-management deportation, and the pinnacle boundary is the free boundary. The computational results show the displacement contour and the maximum principal stress contour under the self-weight state (Fig. 9).

The input geological model and generated computational mesh in application example 2.

Numerically calculated results of application case 2.

Mesh quality analysis is performed based on the value distribution of the meshes in Fig. ten. In Fig. 10 (a) and 10 (d), the distribution of the mesh quality is polish. Approximately 65% of all the tetrahedral meshes have values in the range from 0.5 to 1.0. Less than 5% of the mesh has a value less than 0.2. In Fig. 10 (b) and 10 (c), the distribution still has a tiptop at a value between 0.half dozen and 0.8, which indicates a lower proportion of distorted meshes. Therefore, all the computational tetrahedral meshes are able to satisfy the requirements of numerical simulations (Fig. xi).

Statistics of the quality of computational mesh in application instance ii.

The input geological model and generated computational mesh in application case 3.

Application Instance 3

This gradient contains a large deformation, developing cracks and bulging, and is taken as a typical example for stability analysis. The computational meshes contain 99338 vertices, 203098 triangles, and 365631 tetrahedral meshes. The boundary status is the same as that in application case 1. After the numerical simulation, the deportation and the maximum principal stress contour under the self-weight land are shown in Fig. 12. The maximum deportation is approximately 0.vii m. Fig. thirteen provides the quality distribution of the tetrahedral meshes. A mesh quality analysis based on the value distribution of the meshes is shown in Fig. 13. Approximately 55% of all the meshes accept values in the range from 0.5 to one.0 and the proportion of values less than 0.ane is extremely low (Fig. 13(a)). The distribution of mesh quality has a peak at a value between 0.half-dozen and 0.viii and the proportion of meshes with a good attribute ratio is over sixty% in Fig. 13(c). In general, the majority of meshes had a high-quality value betwixt 0.5 and 1.0, and smooth distributions are also visible in Fig. 13, which indicates the high quality in the model. Thus, all the tetrahedral meshes are able to satisfy the requirements of numerical simulations.

Numerically calculated results of awarding case three.

Discussion

Effectiveness of the presented method

Mesh generation of a geological model can be achieved by the presented method. Currently, this method tin generate high-quality tetrahedral meshes of geological models consisting of triangulated surfaces. These tetrahedral meshes encounter the mesh criteria by equally much as possible. The output meshes are reasonably well represented by geological features.

Examples demonstrate that the method is applied. These meshes tin be used in other simulation software. We import these computational meshes into FLAC^3D and prepare boundary weather condition for the model. The simulation results are obtained from FLAC3D which is frequently used for numerical simulation; and the results of the calculation testify that these tetrahedral meshes of the geological model tin can exist used in stability analysis of rock and soil masses. Thus, the simulation results are also reliable.

Measuring the mesh quality is of import for this method. For the dissimilar mesh generation methods, the measurement metrics of the chemical element quality are similar, which ordinarily include the aspect ratio and the orthogonality. Besides, we also add the metrics of the volume over the border length and skew. These quantities are mainly used to make up one's mind the quality based on the degree and edge length of elements and to identify degenerate elements. In general, the majority of meshes had a high-quality value and smooth distributions are also visible from the statistic of mesh quality metrics, which indicates the high quality in the model.

Advantages and shortcomings

The method of utilizing CGAL is easy to reproduce. Moreover, compared with other methods for mesh generation, this method focuses but on mesh generation for geological models. Thus, it is piece of cake to implement.

At that place is a procedure of optimizing mesh in the procedure of mesh generation for the improvement of the boilerplate quality of the mesh. The method can eliminate some distortion meshes effectively and tin can improve the average quality of the mesh after optimization. There are four inherent optimization functions in CGAL. The method can telephone call its optimization functions to improve the overall mesh quality in this process. The functions not simply focus on remove some badly shaped local tetrahedral meshes but also endeavour to improve the overall mesh quality. In this procedure, some purlieus edges typically demand to exist preserved in the input domain. The optimization process is automatically executed and the final output is optimized meshes.

Tetrahedral meshes with different mesh sizes tin be generated in different subdomains, only at that place is likewise a shortcoming in these tetrahedral meshes. In each subdomain, adaptive computational meshes cannot be generated. That is the mesh size is near the same, which increases the number of triangular and mesh vertices. Such largescale tetrahedral meshes volition lead to an increased calculation time and a decreased computational efficiency of the model.

Outlook and Future Work

In the time to come, we plan to advise a much more adaptive mesh generation method for very circuitous geological models. The mesh generation time volition likewise exist significantly reduced with that method. Co-ordinate to the features of the geometric model, quite adaptive high-quality tetrahedral mesh generation and smoothing [five,half dozen] of the geological model volition be implemented.

Conclusion

We take presented a method for generating high-quality tetrahedral meshes of geological models to be used in the stability analysis of rock and soil masses. The input is a geological model consisting of triangulated surfaces, and the output is a high-quality tetrahedral mesh of the geological model past utilizing CGAL. To demonstrate the effectiveness of the method, we implement it to generate a series of computational meshes in geological model, and we analysed the stabilities of the stone and soil slopes on the basis of the generated tetrahedral mesh models. The applications demonstrated the effectiveness and practicability of the presented method.

Additional information:

Introduction

Numerical simulation has become an indispensable means to solve scientific problems and conduct scientific research in many fields. High-quality meshes are key to the success of all numerical simulations [7,8]. High-quality meshes are normally a balance between quality and calculation time, and depression-quality, extremely thin or distorted elements often forbid the convergence of numerical calculations and increase belittling errors. Among the many automatic mesh generation algorithms, Delaunay triangulation [ix,10] has been widely developed and applied due to the loftier quality of the generated mesh and easy implementation of the algorithm.

Diverse commercial software programs have been adult and applied to numeral calculations in recent years [11], [12], [13], [14]. For example, HyperMesh can generate all the mesh types required in finite element calculations and ANSA is similar to HyperMesh. Information technology is widely used in the automotive field. The reward of these programs lies in the generation of surface meshes. ANSYS ICEM CFD is a total-featured CFD mesh generation tool. Information technology supports not only hexahedral meshes only also tetrahedral meshes, pentagonal meshes, and triangular prisms, which are sufficient for mesh generation of complex geometric model. It is more than focused on generating fluid meshes. GridPro is suitable for geometric models from which structural meshes are easy to generate, just the software cannot generate unstructured meshes. Pointwise produces extremely high-quality structured and unstructured meshes.

Several open source mesh generators have also been developed, such as CGAL, TetGen, Ani3D, MFEM, NETGEN, and Stellar. These packages are widely used in diverse areas that require geometric computation. TetGen is very popular for generating three-dimensional tetrahedral meshes and Delaunay triangulations [15]. Ani3D can also generate unstructured tetrahedral meshes and tin can exist used to generate quasi-optimal meshes [16]. MFEM was designed for finite element methods [17]. NETGEN generated meshes based on abstract rules [18]. Stellar is often used to optimize tetrahedral meshes to produce high quality tetrahedral meshes [xix].

In the field of geotechnical engineering science, geological models can be divided into computational tetrahedral, hexahedral or hybrid meshes [xx,21]. Much work has focused on hexahedron-dominant mesh generation methods [22]. Meng [23] presented a novel link-Delaunay-triangulation method to achieve geometric and topological consistency. Indirect methods [24,25] were proposed to convert Delaunay triangular meshes into quadrilateral meshes by combining adjacent pairs of triangles. Constrained triangulated surfaces can be used to mesh a three-dimensional geological model past using a series of tetrahedral meshes [26], [27], [28].

Commercial software can generate high-quality meshes, merely these programs are expensive. Compared with the above open source packages, CGAL is based on Delaunay refinement. Thus, there are remeshed surfaces in the model. CGAL also provides two global optimistic functions and two local optimistic functions to better the average quality of the mesh. Therefore, method for mesh generation based on CGAL is easily implement.

Background: the CGAL library

The Computational Geometry Algorithms Library (CGAL) [9] is an open source C++ graphics algorithm library that provides underlying half-border data structures. It is a very convenient and efficient program for iteratively traversing points, edges and faces. CGAL provides information structures and algorithms related to computational geometry, such as Voronoi diagrams, triangulations, Boolean operations on polygons and polyhedrons, curve finishing and its applications, mesh generation, geometry processing, and shape analysis.

Because of its open up source and portable characteristics, it has already been widely used in computer-aided design and modelling, calculator graphics, geographic data systems, numerical methods, medical imaging, and other fields that crave geometric ciphering.

CGAL mainly comprises iii components: kernels, computational geometry basic data structures and algorithms and non-geometric tools. CGAL is a header-just C++ template library and is based on a generic programming paradigm of concepts and models. Each of the main functions in CGAL exists as a class template. The input and output functions in each class are overloaded. Therefore, different data types can exist easily and flexibly handled with overloaded functions.

The presented method utilizing CGAL, is committed to the generation of isotropic simplified meshes for discretized 3D domains constructed past a series of triangulated surfaces. The 3D infinite to be meshed is chosen the domain, and it is required that it exist divisional and defined by a collection of initial triangulated surfaces. The domain may exist subdivided into several subdomains. Some points and edges may demand to be approximated in the mesh; these points and edges are defined as indispensable 1-dimensional features and 0-dimensional features respectively. In addition, 3-dimensional features and 2-dimensional features are defined as subdomains and boundary triangulated surfaces, respectively.

Annunciation of Competing Involvement

The Authors confirm that at that place are no conflicts of interest.

Acknowledgements

This research was jointly supported by the National Natural Science Foundation of China (Grant Nos. 11602235 and 41772326), and the Fundamental Research Funds for China Central Universities (2652018091, 2652018107 and 2652018109). The authors would like to thank the editor and reviewers for their contributions to the paper.

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