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AHPCRC Bulletin: Summer 1994 - Volume 4 Number 3

Finite Element Computation of Environmental Flow Problems

Marek Behr, AHPCRC

The problem of the environment affecting man-made structures and industrial complexes has been of concern to the engineering community for a long time. More recently, influence in the reverse direction, exemplified by industry-related changes in the environment itself, is also undergoing scrutiny. The first kind of problem is usually encountered in the design process preceding many types of construction. The second kind takes the form of impact studies, usually with the aim of quantifying and minimizing the changes in natural resources resulting from building, excavation, pollution, etc., before such projects begin. Together, these two kinds of problems become the main focus of environmental engineering.

One of the main research areas in this engineering field is the modeling of flows in oceans, bays, lakes, and rivers. These flows may be driven by tidal motion of the oceans, wind shear on the water surface, or the permanent river currents, and their characteristics vary greatly, depending on the scale of the observed phenomena. In the most common practical applications of these flow models, we seek to determine the water elevation and currents given a set of conditions at the water body's boundaries. The results allow us to predict, for example, how planned break-water structures, or channel excavation, will affect the flow patterns in a port or an estuary. Furthermore, given the existing elevation and currents, it is often desirable to foresee the transport of pollutants from an industrial site or a river and thus, the severity of its impact on the environment. Frequently, the search for elevation and currents, and transport analysis, proceed in a coupled fashion, with the transported chemical or thermal pollutant affecting the physical characteristics of the transporting fluid, and therefore its circulation patterns.

The numerical simulation of the tidal, riverine, and estuarine problems presents many challenges. Typically, there is a very wide range of length and time scales that need to be analyzed simultaneously in such computations. This problem is only partially alleviated by the use of Shallow Water Equations (SWE) approximation. The SWE are based on the assumption of negligible vertical velocity of the fluid and a hydrostatic pressure distribution. The full 3D Navier-Stokes equations are integrated over the water depth, leading to a set of 2D generalized advection-diffusion equations. Even with the SWE approximation, there is a great need to solve the moderate-to-large systems of equations very rapidly, as the number of time steps required in a typical environmental flow simulation is quite large. Therefore, there is a strong motivation to use the latest massively parallel computers in SWE simulations.

Tayfun Tezduyar, Professor of Aerospace Engineering and Mechanics and AHPCRC Interim Director, and this author, are involved in collaboration with two research groups, aiming to bring the power of AHPCRC high performance computing resources to finite element SWE modeling. Their initial cooperation involved Kazuo Kashiyama, Professor of Civil Engineering at Chuo University in Tokyo, Japan, and his team, and sought to carry out high-resolution computations of transport phenomena in the Tokyo Bay. More recent joint work has been undertaken with Charlie Berger from the US Army Waterways Experiment Station (WES) in Vicksburg, Mississippi. Here the AHPCRC researchers are using a massively parallel SWE implementation to simulate circulation in the Galveston Bay near Houston, Texas.

Figure 1. Contaminant transport in Tokyo Bay.
Kashiyama, who visited the AHPCRC for six months in 1993, is using a three-step explicit finite element formulation, both for the SWE and the advection-diffusion contaminant transport equation. Stabilization of the numerical scheme is achieved by the selective lumping technique for the SWE part of the problem and by streamline-upwind/Petrov-Galerkin (SUPG) technique for the transport part. The parallel implementation of this formulation, based on unstructured triangular grids, has been written for the AHPCRC CM-5 supercomputer, and utilizes some of the unique capabilities of the CM Scientific Software Library (CMSSL), such as recursive spectral bisection mesh partitioning and data distribution, which dramatically reduces communication between processing nodes and leads to faster computation rates. A series of increasingly fine finite element discretizations was used in the Tokyo Bay simulations, with the most recent mesh consisting of 207,799 elements. This resolution is sufficient to accurately represent many of the fine harbor features and small-scale depth variations present in the actual bay topography. The mesh is designed to keep the element Courant number constant in the entire domain, leading to high efficiency of the explicit code. At the open (rightmost) boundary, the sinusoidal tidal variation of the water elevation is specified with a period of 12 hours. Figure 1 shows, at 6.210 hour intervals, the spread of contaminant introduced at the beginning of the simulation at a single point. The computations were performed on a 256-node partition of the CM-5, with the mesh decomposed into 1024 partitions, with one mesh partition assigned to each vector execution unit of the machine.

Figure 2. Water elevation in Galveston Bay.
The Galveston Bay, on the Texas Gulf coast, has been the focus of investigations by the WES researchers, who aim to predict the impact of planned navigation channel changes on the ecosystem. The 40-foot-deep and 400-foot-wide navigational channel linking the Gulf of Mexico with the Port of Houston is considered for further expansion due to traffic demands. The resulting changes in circulation are studied with the aid of the RMA10-WES numerical code. With the hope of expanding this prediction capability, the AHPCRC researchers applied their massively parallel implementation of an implicit finite element code to the same problem. The code, previously used to simulate Tokyo Bay circulation patterns in concert with Kashiyama's studies, is based on the stabilized space-time formulation of the SWE model in its conservation form. The SUPG stabilization is used for all equation system components. The set of linear equations formed at each nonlinear iteration is solved with a GMRES update technique. Figure 2 shows the results from the initial computations of the Galveston Bay tidal flow on a 3682 element WES-generated mesh. The incident tidal wave, with a period of 12 hours, is again specified at the gulf boundary, to be replaced later by the actual measurements of the tidal wave elevation. Shown in Figure 2, at three hour intervals, is the water elevation, which has been magnified 105 times to bring it into the same order of magnitude as the horizontal dimensions. Further simulations will increase the resolution of the grid and incorporate relevant transport phenomena into the model.

The simulations described in this article demonstrate the applicability of the scalable computer architectures, such as the AHPCRC CM-5, to environmental engineering problems. With the potential of such parallel architectures to accommodate very large problems, the accuracy of numerical predictions in this field should increase considerably as more refined models and more complete sets of natural phenomena can be taken into account.