Accurate Visualization of Pathlines in an Unstructured Finite-Volume Hydrological Model

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1 1 Accurate Visualization of Pathlines in an Unstructured Finite-Volume Hydrological Model Adrienne Dunham Dept. of Computer Science Seattle Pacific University Seattle, WA, USA Rachael Luhr Dept. of Computer Science Montana State University Bozeman, MT, USA SPONSOR Clemente Izurieta Dept. of Computer Science, Montana State University Bozeman, MT, USA SPONSOR Geoff Poole Dept. of Land Resources and Environmental Sciences Montana State University Bozeman, MT, USA Abstract With improved technology, there has been a lot of interest and growth in the field of computational fluid dynamics and visualizations using stream objects. However, the most prevalent visualization software in use has put very tight constraints on the hydrological model in order to function correctly. In this paper, we describe progress towards implementing a method for tracing pathlines in a 3-dimensional unstructured finite-volume mesh. We found a recently published algorithm that coincides with our numerical solution to the groundwater flow equations. We hope to integrate the algorithm with an open source visualization software, VisIt, and to make it publicly available. Our final intent is to integrate VisIt with an ecological modeling framework, Network Exchange Objects, under development at Montana State University. Being able to improve the accuracy of finite-volume groundwater flow models would provide valuable visual information to hydrologists analyzing this complex data. Keywords pathline; particle tracking; unstructured mesh; finite volume; groundwater flow I. INTRODUCTION A. Problem and Importance The past few decades have rapidly expounded upon the computational approaches to visualizing fluid flows. Hydrologists are very interested in visualizing water movement through natural models. This information can be used to predict events like floods and water shortages as well as track particle transport. Visualizing stream objects plays an integral role in the comprehension and analysis of these complex systems. Various methods have recently been developed and extended to solve increasingly complex models, such as unstructured meshes. We explore developments for visualizing stream objects, compatible in a 3-dimensional, transient, unstructured, finite-volume groundwater flow system, based on output from an ecological modeling framework. This paper is organized as follows: In section 2 we provide the background for understanding hydrological models. Section 3 explains the methods employed to visualize these types of models. Our findings are described in Section 4, and Section 5 discusses future work and its unique application with emerging hydrological modeling techniques. II. BACKGROUND A. Hydrology Background and Terms This research advanced an ongoing collaboration between the Department of Computer Science and the Department of Land Resources and Environmental Sciences [5] at Montana State University. The focus on visualizing hydrology data is consistent with computational fluid dynamics. Specifically, we concentrated on representing groundwater flow, by using sampled data from the Nyack Floodplain of the Middle Fork of the Flathead River. The derivation of the groundwater flow equation is outside of the scope of this paper; however, it is important to note that this equation includes partial differentials. To solve such would provide the deterministic mathematical model for groundwater flow, meaning that a response (such as a pathline) can be predetermined based on a mathematical understanding of the process [7]. The most common way to visualize fluid flow is by means of particle tracking. In order to compute and visualize the movement of particles, a velocity field must be available for the model. The fluid velocity is the numerical solution for the set of two equations: Darcy s law and the pressure equation [3]. Darcy s law, published in 1856, is an equation for flow through a porous medium [4]. Once we have this velocity field, we can start with visualizing the water movement. B. Data Visualization Concepts In this paper we focus on how to trace and visualize pathlines on a 3-dimensional unstructured mesh. A massless

2 2 particle seeded at a particular point follows a line defined as a streamline or pathline. Pathlines are closely related to streamlines, curved lines everywhere tangent to the instantaneous velocity [20, 3, 14]. In a static velocity vector field, streamlines and pathlines coincide. However, since most natural data is transient, or time-dependent, we are interested in visualizing pathlines. A pathline traces the trajectory of a particle through the domain as the flow changes in time [20, 3, 14]. Many of our resources on particle tracking primarily or exclusively discuss streamlines. In most cases, the extension of such algorithms to include time variance is a simple factor. There is a plethora of research regarding particle tracking and the generation of streamlines. The complexity arises when using an unstructured mesh, or grid, rather than a structured one. Traditionally, structured meshes have been widely used, as they are much simpler to analyze. In three dimensions, they consist of a grid of vertices, comprised of cells ordered along the x, y, and z axes. However, unstructured meshes can more accurately represent complex natural geometries [25]. Unstructured meshes consist of unorganized polyhedra. Typically, these include tetrahedra or hexahedra, although a fully unstructured mesh is not limited by such. One common unstructured mesh is generated from a set of vertices using the Delaunay triangulation algorithm, extended to create a 3- dimensional tetrahedralization. The Voronoi tessellation, related to the Delaunay triangulation, is another common method that creates a set of polyhedral cells around vertices. Vertices can either specify the centers of cells or the corners. Special attention is given to no-flow boundaries of the system when particle tracking. The definition of boundaries plays an essential role in the accurate representation of fluid flow. Typically, these boundaries are represented by a cell face with flux value of zero. Cells with permeability of zero also behave as no-flow boundaries. Cells through which no fluid flows are called inactive cells. C. Existing Developments in Hydrology Modeling Hydrological visualization software uses output from corresponding models. In 1984, development of these hydrological models began with Pollock [16], who created a method to simulate groundwater flow. This method was implemented in MODFLOW, the United States Geological Survey finite-difference groundwater flow model. MODFLOW is currently one of the most used numerical models for groundwater flow. Five years later MODPATH was released to calculate pathlines in finite-difference structured meshes. MODPATH is a particle tracking postprocessing package that computes pathlines based on output from MODFLOW. Since then, MODFLOW has been used often and has become the standard for modeling groundwater flow for those models with structured grids and finitedifference methods. However, there has been growing demand to improve upon Pollock s method so it will work for different and more complex models. There have been many extensions and improvements to MODFLOW since the late 1980s. It has been extended to work with irregular and partially unstructured meshes [3, 18]. However, MODFLOW was primarily developed to work with finite-difference models. We have a finite-volume model, and therefore Pollock s algorithm is not sufficient. The difference between these types of models will be further discussed in Section 3. Network Exchange Objects (NEO) is a simulation framework that is currently under development at Montana State University [5]. NEO facilitates the development of simulation models that describe the behavior of complex ecological and hydrological systems, including groundwater systems. It is our hope in the future to integrate NEO with visualization software. III. METHODS A. Choosing Visualization Software To find suitable data visualization software, our biggest constraints were finding a program that is free and open source. Even given these limitations, there are many programs from which to choose. We limited our search to programs that dealt specifically with computational fluid dynamics. This search resulted in three potential programs: OpenDX, OpenFOAM, and VisIt. OpenDX was the most familiar of the possible programs. It was released by IBM in 1991 and, for its time, exceptionally innovative in data visualization [13]. Since its creation, OpenDX was used consistently to visualize and analyze complex scientific data. Because of this, it comprises several desirable functions such as a graphical user interface and streamline module. However, since the project became open source, it has not had a strong online developer community and the most recent update to the website was in The latest release was written in entirely in the C programming language and is very hard to follow and understand. After downloading it and attempting to get it to run, we encountered many issues. The most recent release is only compatible with Windows 2000/XP/2003 and therefore, was not able to run on our Windows 7 machines. The second potential data visualization program for our purposes was OpenFOAM, which is used to solve complex fluid flows. Released in 2004, OpenFOAM holds the advantage in that is it tailored specifically to computational fluid dynamics [12]. At first glance, it seemed like it may have worked for our problem. Unfortunately, it could only be run on the Linux operating system. This was not ideal because not all of our available machines had Linux installed on them; therefore, we decided to pursue other, more compatible options that required less set up. VisIt is a visualization tool developed at Lawrence Livermore National Laboratory (LLNL) in 2002 [23]. It can be run on both Unix and Windows machines to generate visualizations. VisIt provides extensive functionality for many types of visualizations. Useful features include its powerful user interface, support of both 2- and 3-dimensional structured and unstructured meshes, and number of modifiable scalar and vector variables. An active developer community provides rapid support, maintaining the software and updating the release every couple of months. There is also a VisIt user community wiki and forum where tutorials are uploaded and

3 3 questions are discussed and answered. We downloaded the current version and had no issues getting it up and running. Because of these advantages, we chose to use VisIt as our main visualization software. B. VisIt Upon choosing a program with which to work, we acquired a Microsoft Access database from the Department of Land Resources and Environmental Sciences with preliminary data about the Nyack Floodplain of the Middle Fork of the Flathead River, collected over the course of 6 months from the beginning of March to the end of August For ease of use, we converted this into several tab delimited text files; still, this data would have to undergo a transformation to be readable by VisIt. VisIt accepts several types of file formats, the most extensive of which are vtk and silo [23]. We downloaded, compiled, and linked the libraries of these two, and were able to create simple C++ programs that wrote out each file type. VTK (Visualization ToolKit) creates an ASCII-based file format [24]. It has a relatively low learning curve, and held the advantage of readability from a text editor. However, VisIt provides more documentation for the silo file format, which is capable of storing more complex information. The silo format was developed at LLNL, like VisIt, and therefore conforms to many of the requirements for visualizing data in the VisIt software, including storing 3D unstructured meshes and various scalar and vector variables [22]. In addition, silo files may be stored in a database, which is especially advantageous when working with time-varying data. Along with the silo download came silex, a complementary program for viewing the contents of silo files. Although for our data, vtk would have sufficed, we chose to utilize silo for the aforementioned reasons. We then wrote a modularized C++ program that read in text files containing data about the 3D vectors, mesh polygons, and velocities for each day, and wrote out a database of 184 silo files. From VisIt, we could draw the mesh and view the velocities at each node (Figure 1). The visualization could be animated as it varied over time, iterating through each day in the silo database. This was helpful in comprehending how the groundwater velocities fluctuate between seasons. For plotting stream objects, VisIt allows several advanced options. Each can be seeded in a variety of ways including a discrete point list, line, plane, and box around the entire data set. Streamlines are plotted using the velocity field of only one time slice at a time. An additional option allows for plotting pathlines, using the current time slice as a starting time. Fig. 2. Streamlines (dark blue) and pathlines (light blue) seeded at each velocity vector. Fig. 3. An exaggerated z axis (x100) shows 300 streamlines moving between layers of the model, colored according to speed. Fig. 1. The tetrahedralized mesh with velocities at each node, scaled and colored according to magnitude (red being highest and blue lowest). Fig ,000 pathlines randomly seeded throughout the entire system domain, colored according to speed.

4 4 Fig. 5. Unstructured cells in the topographical mesh. Although VisIt is easy to use and provides much functionality, we still had to overcome a few obstacles to get our data to visualize correctly. We first took the data and plotted it in VisIt as something called a "pointmesh" which visualizes each vector as a point in 3D space. However, there is no way to interpolate the velocity field of a pointmesh, as there is no specified volume contained within the domain. VisIt provides a resample operator, which transposes each point to a regular grid, through which velocity may be interpolated. The resampling solution is not ideal as it approximates the data and fails to directly address unstructured meshes. Instead, we ran the nodes data through a program called TetGen, which created a 3D tetrahedralization of the data using the Delenauy algorithm [21]. After converting the output from TetGen to silo, VisIt was able to interpolate the velocity of the subsequent tetrahedrons. However, the boundaries of the domain were not observed, as TetGen generated a convex mesh around the entire domain (see Figures 1 4). We acquired system boundary information in the form of a shapefile, which is a file format also supported by VisIt. This data accurately represents the 2-dimensionsioanl topological boundaries of the system and divides the domain into polygons (see Figure 5). Those polygons bordering the complex boundaries often contained hundreds of edges. Rather than fully unstructured meshes, VisIt only supports 3D unstructured meshes comprised of any combination of tetrahedrons, pyramids, triangular prisms, and hexahedrons [23]. Because VisIt supports only this limited number of polygon types in their unstructured meshes, it automatically triangulated each polygon. From there we exported the database in silo format and used a C++ program to give each polygon z values and extend them into 3D triangular prisms, through which streamlines and pathlines could be computed (see Figure 6). Fig ,000 streamlines randomly seeded in the topographical mesh. Working with VisIt also revealed some limitations of the software. Because VisIt is a general-purpose visualization tool, it does not accommodate all specific hydrology models. Nonetheless, this software is very powerful and provides great potential for future development in its field. C. Numerical Analysis There are two methods for solving the groundwater flow equations and integrating stream objects. The first of these is an analytical solution, which is capable of solving for the exact solution. Pollock s method is a semi-analytical one, in which interpolating the velocity is a simple function computed within each grid cell [16, 3]. However, the analytical method assumes highly and perhaps unrealistically idealized parameters and boundaries [7]. This works well for structured meshes; however with unstructured meshes, heterogeneous geologic structures, an analytical solution may not exist. Field data is complex and while an analytical solution could be obtained by the simplification of its underlying structure (such as the transformation of an unstructured mesh into a structured one), this would introduce an unnecessary margin of error [7]. In this case, numerical methods provide a means of a finite, approximate solution using discretized variables such as time and space [7]. One common numerical solver is Runge-Kutta [3, 25]. In data visualization, more complicated interpolation methods use Euler s method or higher-order Runge-Kutta methods [3]. VisIt provides the Dormand-Prince, Forward Euler, Leapfrog, Adams-Bashforth, and Runge-Kutta 4 th order streamline and pathline integrators. However, these methods solve for ordinary differential equations, whereas a typical groundwater model solves for partial differentials. Common numerical approximations of the groundwater flow partial differentials include finite-difference, finite-element, mixedfinite element, and finite-volume methods [3, 25]. Also, with an unstructured mesh, it is necessary to interpolate the velocity to create a continuous velocity field. A

5 5 continuous velocity field provides velocity values at any given point in the mesh. This is essential since unstructured meshes do not have an organized order or structure to their cells. Only the mixed finite element method contains explicit velocity interpolation [25]. There are numerous velocity interpolation methods for other types of numerical solutions and meshes, for which the velocity may not be directly available. These include Pollock s method [16], extensions thereof [2, 15, 6], Matringe s method [8, 9], the corner velocity interpolation method [3], the extended corner velocity interpolation method [18], and Painter s method [14]. Since the trend toward unstructured meshes, most research explores particle tracking algorithms for finite-difference and finite-element grids; little attention has been given to particle tracking on a finite-volume grid, one of the constraints of our example data [7, 14]. Existing groundwater flow codes developed according to the finite-volume method hold a wide user community and are well-worth accommodating [14]. Both Zhang and Painter explore methods of tracing streamlines on unstructured finite-volume grids. These numerical methods have both been tested against their analytical counterparts to determine low margins of error. D. Zhang s Study Though various methods have been proposed to deal with unstructured grids, most assume grids composed entirely of tetrahedrons and quadrilaterals. Zhang explores and expands upon existing solutions to the problem of n-polygons (n < 4). The finite-volume method can be subdivided into three different types of data: cell-centered, point-distributed, and control volume finite element [25]. We have control volume finite element data. The n-polygons problem occurs in particular grid generations, such as this. Two generic approaches are proposed for solving such a problem: extending existing algorithms, and dividing the cells into sub triangles and sub quadrilaterals. The first was attempted by Rasmussen, by extending Haegland s corner velocity interpolation method [18]. The second was undertaken by Cordes and Kinzelbach [2] and Prevost et al. [15], resulting in higher-order quadrilateral refinement, lowerorder quadrilateral refinement, and triangular refinement methods. Using an extensive series of tests, Zhang et al. affirms the accuracy of higher-order quadrilateral refinement and locally conservative triangular refinement methods. E. Painter s Method Painter s method is unique because it does not resort to subdividing cells and it explicitly accounts for tracing pathlines in case of a transient model. This method first reconstructs cell-centered velocities, and applies a smoothing procedure to account for discontinuities between cells at their faces. The boundary constrained equation forces no-flow conditions on the cells to be honored, given that in such a case the cell s node is placed on the boundary. Barycentric, or cellcentered, interpolation can then be used to interpolate the velocity at any given point. Finally, a particle s position may be calculated and tracked. IV. FUTURE WORK A. Implementing an Integration Method We have been in contact with the Los Alamos National Laboratory to gain access to the source code that implements Painter s method. This code is in the release process and should be available to the general public shortly. Given code to compute pathlines on a control volume unstructured mesh, integration and implementation with a visualization program should be straightforward. VisIt s streamline/pathline infrastructure is modular, allowing extensions of its integrators. The underlying Parallel Integral Curve System (PICS) has 5 access points: how to interpolate a vector field, how to do an advection step, how to parallelize the work, how to analyze a particle, and where to place seeds [1]. Each is maintained separately. We would need to integrate with the second access point, known as the solver, which requests the velocity at different locations and decides where a particle should advance. This would involve creating a new derived class from the base class, avtivpsolver. It may also be necessary to write a new function for the first access point, velocity interpolation, because even though VisIt currently interpolates unstructured meshes, Painter s method takes preliminary steps to reconstruct velocities. The final challenge would be getting the user interface to display the new solver. If our effort is successful, this algorithm will be readily available to use with this free, open source visualization software. This addition would be very helpful to the field of computational fluid dynamics that deals with finite-volume groundwater models. We expect that many will benefit from this implementation. B. Integrating with NEO Once we have a new, working integration method implemented within VisIt, we can work on integrating NEO with VisIt to produce accurate visualizations of groundwater models from the Department of Land Resources and Environmental Sciences at MSU. In order to perform this integration seamlessly, we have two different options. The first option is that the output would be taken from NEO, ran through a program to be formatted into a file version that VisIt can understand (either silo or vtk file format), and then ran through VisIt to produce the actual visualizations. The second option is to build a database plug-in that connects NEO and VisIt. This may be the best option because it would allow for the ability to output data directly from the NEO database and input the data into VisIt with minimal formatting changes. However, more research needs to be done in order to decide which option is the best to use. Being able to visualize the output from NEO is very important to the hydrologists we work with because, through visualizations, they can analyze the data and see inconsistencies, or lack thereof, in their data that may not have been apparent by looking at the NEO output alone. Therefore, this step is essential in continuing this research.

6 6 ACKNOWLEDGEMENTS The authors would like to thank Drs. Clem Izurieta and Geoff Poole for providing input and continual guidance. Additional thanks goes to the VisIt Users Group, who were patient and answered all of our questions in a very timely manner. This project was supported by the National Science Foundation through the Sustainable Networks Research Experience for Undergraduates program at Montana State University. [25] Zhang, Yanbin, M. J. King, and Akhil Datta-Gupta. "Robust streamline tracing using inter-cell fluxes in locally refined and unstructured grids." Water Resources Research 48.6 (2012). REFERENCES [1] Childs, Hank. Re: New Streamline Integrator. to Adrienne Dunham. 23 July [2] Cordes, Christian, and Wolfgang Kinzelbach. "Continuous groundwater velocity fields and path lines in linear, bilinear, and trilinear finite elements." Water Resources Research (1992): [3] Hægland, Håkon, et al. "Improved streamlines and time-of-flight for streamline simulation on irregular grids." Advances in Water Resources 30.4 (2007): [4] Hornberger, George M. Elements of Physical Hydrology. Baltimore: Johns Hopkins UP, Print. [5] Izurieta, Poole, et al. Development and Application of a Simulation Environment (NEO) for Integrating Empirical and Computational Investigations of System-Level Complexity. In Information Science and Applications (ICISA), 2012 International Conference on (pp. 1-6). IEEE [6] Jimenez, Eduardo, et al. "Spatial error and convergence in streamline simulation." SPE Reservoir Simulation Symposium [7] Konikow, L. F. "Use of numerical models to simulate groundwater flow and transport." US Geological Survey (1996). [8] Mahrous, Karim, et al. "Topological segmentation in three-dimensional vector fields." Visualization and Computer Graphics, IEEE Transactions on 10.2 (2004): [9] Matringe, Sébastien F., Ruben Juanes, and Hamdi A. Tchelepi. "Robust streamline tracing for the simulation of porous media flow on general triangular and quadrilateral grids." Journal of Computational Physics (2006): [10] Matringe, Sebastien, Ruben Juanes, and Hamdi Tchelepi. "Tracing streamlines on unstructured grids from finite volume discretizations." SPE Journal 13.4 (2008): [11] MODFLOW and Related Programs [online] 2013, [12] OpenFOAM [online] 2013, [13] Open Visualization Data Explorer [online] 2013, [14] Painter, Scott L., C. W. Gable, and S. Kelkar. "Pathline tracing on fully unstructured control-volume grids." Computational Geosciences 16.4 (2012): [15] Prevost, Mathieu, Michael Edwards, and Martin Blunt. "Streamline tracing on curvilinear structured and unstructured grids." Spe Journal 7.2 (2002): [16] Pollock, David W. "Semianalytical computation of path lines for finitedifference models." Ground Water 26.6 (1988): [17] Pollock, D.W., 2012, User Guide for MODPATH Version 6 A Particle-Tracking Model for MODFLOW: U.S. Geological Survey Techniques and Methods 6 A41, 58 p. [18] Rasmussen, A. F. "Streamline tracing on irregular geometries." 12th European Conference on the Mathematics of Oil Recovery [19] Silo [online] 2013, [20] Telea, Alexandru C. Data Visualization: Principles and Practice. Wellesley, MA: K Peters, Print. [21] TetGen: A Quality Tetrahedral Mesh Generator [online] 2013, [22] VisIt Users Group [online] 2013, [23] VisIt Visualization Tool [online] 2013, [24] VTK - The Visualization Toolkit [online] 2013,