A boundary and soil interface conforming unstructured local mesh refinement for geological structures
Introduction
Recent years have of course seen significant developments in hydrological and hydraulic modelling [1], [2], [3], [4]. Implicit in the latest model advances is the ability of numerical schemes to perform at high space–time resolution within numerical stability regimes (as determined by the Peclet and Courant number for example). One of the key areas requiring high-resolution meshing is the infiltration and pore water redistribution on unsaturated hillslopes. In this context it is advantageous not only to have high-resolution mesh representation, but also to employ adaptive meshing techniques to improve the accuracy of the solution in areas of high gradient and minimise computational time.
Adaptive mesh methods are widely regarded as reliable and efficient approaches to many problems of computational fluid dynamics [5], [6]. Normally adaptive mesh methods consist of using a mesh which has a variable density and is constructed according to a criterion based on the solution itself. In general there are two main strategies employed in adaptive mesh methods. The first one, named p-refinement, uses local high order approximation in the regions where the gradient of the solution is high. The second one, named h-refinement, maintains linear approximation but the element sizes are locally changed. For the sake of programming simplicity h-refinement has been widely adopted and there are three main approaches classified in the literature [6], [7]: r-refinement, local refinement and full remesh.
r-refinement consists of redistributing a fixed number of nodes over the domain, the structure and connectivity of the mesh remaining unaltered. The nodes are considered to be connected by springs whose stiffness depends on the solution. The equilibrium position of the spring system gives the ideal location of the nodes for the finite element analysis. Since the number of nodes is constant, the accuracy of the solution is restricted by the initial number of nodes. Calculation of the equilibrium position of the spring network is also a complicated exercise for unstructured meshes, due to the global dependence of the system.
Local refinement consists of adding new nodes in the regions where relatively large error occurs to subdivide the corresponding elements into smaller ones. This approach is simpler than the r-refinement and is more suitable for unstructured mesh generation. The technique usually used for the unstructured triangle mesh [8], [9] consists of dividing an initial triangle into two to four smaller elements by bisecting one to three edges of the element. However maintaining the high-quality element shape throughout the refined mesh is very difficult [9] to achieve using this technique. For this reason, alternative methods which decompose a triangle element by inserting a new point at its centroid or circumcircle centre are also widely used in the Delaunay triangulation process.
Full remesh is also a simple approach used with success in some studies [6]. Using this technique, the initial input data for the mesh generator is updated at every refining step and the mesh generator runs repeatedly. The generated mesh is therefore always independent from the previous one.
The biggest challenge for adaptive mesh generation in geological applications is creating a good quality grid while maintaining the conformation of soil interfaces and domain boundaries. When adaptive methods are adopted to solve an unsteady problem, the difficulty of the mesh generation increases greatly as the locally refined mesh is constrained by the geological structure of the soil.
This paper uses the Delaunay triangulation method to generate locally refined adaptive meshes. The critical objective of this paper is to develop a fully automated mesh adaptation code capable of interfacing with meshing codes that are typically available, but which nonetheless lack the automated adaptive meshing capability that we regard as necessary for hydrological applications. Specific treatments are proposed to deal with the soil interfaces and boundaries. The methods presented here make a fully automated adaptive mesh generation possible.
Section snippets
Delaunay triangulation and mesh density control
The computer code used for this study is based on EASYMESH, a 2D Delaunay mesh generator developed in C programming language by Niceno [10] and available for free on the internet. EASYMESH generates unstructured, Delaunay and constrained Delaunay triangulations. Although it already has some facilities to handle domains composed of more than one soil/rock type and to deal with local refining and coarsening of the mesh, EASYMESH cannot be readily used for the automatic mesh adaptation which needs
Adaptive mesh refinement
The techniques described above can be used quite easily to generate a static mesh as the length scales of the boundary points and the sources can be adjusted manually to achieve the result desired by the user. However, it is not suitable for solution-based refinement which requires an automated local refinement based on the solution of governing equations [19], especially when the domain consists of more than one soil/rock type. Consider, for instance, the situation shown in Fig. 7a, where six
Application
Fig. 15a shows a layered slope composed of different soils. There are three very thin parallel strata above which four other strata converge towards a point on the boundary. Fig. 15b shows an evenly distributed triangular mesh of the domain generated with an average element size of .
Let us now assume there are two source points indicating that local refinement is needed to make the local mesh size decreased to . These two source points do not necessarily reflect a real hydrological
Conclusions
In this paper, we have presented a local refinement technique usable in an hydrological context where the Delaunay triangulation is heavily constrained by the geology of the domain. A set of refinement and control criteria have been developed to allow the local refinement to be performed very close to the boundary and also to the soil interfaces. The technique performs well in a very complex situation involving soil layers of variable widths and shallow angles. The algorithm has been shown to
References (20)
- et al.
Aspects of adaptive mesh generation based on domain decomposition and Delaunay triangulation
Finite Elem. Anal. Des.
(1995) Efficient unstructured mesh generation by means of Delaunay triangulation and Bowyer–Watson algorithm
J. Comput. Phys.
(1993)A grid generation and flow solution method for the Euler equations on unstructured grids
J. Comput. Phys.
(1994)- et al.
Automatic adaptive finite element mesh generation over arbitrary two-dimensional domain using advancing front technique
Comput. Struct.
(1999) - et al.
Delaunay triangulation and 3D adaptive mesh generation
Finite Elem. Anal. Des.
(1997) Delaunay triangulation in computational fluid-dynamics
Comput. Math. Appl.
(1992)- et al.
Self-adaptive hierarchical finite-element solution of the one-dimensional unsaturated flow equation
Int. J. Numer. Meth. Fluids
(1990) - J.R. Lang, L.M. Abriola, A. Gamliel, Comparison of P, H, and R-version adaptive finite element solutions for...
- et al.
Moving finite-element model for one-dimensional infiltration in unsaturated soil
Water Resour. Res.
(1992) - et al.
A front-tracking numerical algorithm for liquid infiltration into nearly dry soils
Water Resour. Res.
(1999)
Cited by (2)
Four-triangles adaptive algorithms for RTIN terrain meshes
2009, Mathematical and Computer ModellingCitation Excerpt :By a dynamic environment we mean a computer application dealing with approximated meshes that eventually depend on time, a prescribed observed-terrain distance, a numerical solution within the mesh etc. Such applications include Interactive Visualization and Level of Details techniques [6,8,12], Real-time rendering [6], and Finite Element computations, see [3,5,18]. For instance, in landscape visualization, the accuracy needed in the various parts of a terrain depends on the distance from the viewpoint [6,8].
Unstructured surface mesh generation for topography using interpolating surface modeling
2009, Zhongguo Jixie Gongcheng/China Mechanical Engineering