Corrected/Updated References

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1 K. Kashiyama, H. Ito, M. Behr and T. Tezduyar, "Massively Parallel Finite Element Strategies for Large-Scale Computation of Shallow Water Flows and Contaminant Transport", Extended Abstracts of the Second Japan-US Symposium on Finite Element Methods in Large-Scale Computational Fluid Dynamics, Tokyo, Japan (1994). Corrected/Updated References 1. T.E. Tezduyar, M. Behr, S. Mittal and A.A. Johnson, "Computation of Unsteady Incompressible Flows with the Stabilized Finite Element Methods: Space-Time Formulations, Iterative Strategies and Massively Parallel Implementations", New Methods in Transient Analysis, PVP-Vol.246/AMD-Vol.143, ASME, New York (1992) T.E. Tezduyar and T.J.R. Hughes, "Finite Element Formulations for Convection Dominated Flows with Particular Emphasis on the Compressible Euler Equations", AIAA Paper , Proceedings of AIAA 21st Aerospace Sciences Meeting, Reno, Nevada (1983). 5. K. Kashiyama, H. Ito, M. Behr and T. Tezduyar, "Three-step Explicit Finite Element Computation of Shallow Water Flows on a Massively Parallel Computer", International Journal for Numerical Methods in Fluids, 21 (1995) M. Behr and T.E. Tezduyar, "Finite Element Solution Strategies for Large-Scale Flow Simulations", Computer Methods in Applied Mechanics and Engineering, 112 (1994) 3-24.

2 To appear in the pre-conference proceedings of the Second US-Japan Symposium on Finite Element Methods in Large-Scale Computational Fluid Dynamics, Tokyo, Japan, March 14-16, 1993 MASSIVELY PARALLEL FINITE ELEMENT STRATEGIES FOR LARGE SCALE COMPUTATION OF SHALLOW WATER FLOWS AND CONTAMINANT TRANSPORT Kazuo Kashiyama 1, Hanae Ito 1, Marek Behr 2 and Tayfun Tezduyar 2 1 Chuo University, Tokyo, JAPAN 2 University of Minnesota, Minneapolis, MN, USA 1. INTRODUCTION Finite element computations of shallow water flows and contaminant transport can be applied to many practical problems: design of river, coastal and offshore structures, disaster prediction and other applications related to hydrodynamic, thermal and chemical transport behavior in oceans, lakes, and rivers. In this context, the finite element method is applicable to complicated water and land configurations. In practical computation of this type of problems, it is essential to use methods which are as efficient and fast as the available hardware allows. Also, in this type of problems, computations need to be carried out over long time durations to properly simulate and predict the phenomena of interest. In recent years, massively parallel finite element computations have been successfully applied to several large-scale compressible and incompressible flow problems, including those involving moving boundaries and interfaces and those in 3D [1]. These computations demonstrated the availability of a new level of finite element computational capability to solve practical flow problems. With the need for a high-performance computing environment to carry out simulations for practical problems in shallow water flows and contaminant transport, in this paper we present and employ a parallel explicit finite element method for computations based on unstructured grids. The finite element discretizations are based on a three-step explicit formulation both for the shallow water equations and the advection-diffusion equation governing the contaminant transport. In these discretizations, for numerical stabilization, we use selective lumping [2] for the shallow water equations and the streamline-upwind/petrov-galerkin (SUPG) technique [3] for the advectiondiffusion equation. Parallel implementation of these unstructured grid-based formulations are carried out on the Connection Machine CM-5. As an example, we carry out simulation of the effect of tidal waves on the Tokyo Bay and the spread of a pollutant injected into the Tokyo Bay. 2. GOVERNING EQUATIONS The governing equations of shallow water flo~s are oui Cb Tt + UjUi,j + g~,i + h+zui- AI(Ui,j+Uj,i),j + fi = 0, (1) ~~ + { (h+~)ui },i = 0, (2) where Ui is the mean horizontal velocity, ~is the water elevation, his the water depth, g is the gravitational acceleration, Cb is the coefficient of bottom friction, AI is the eddy

3 viscosity and fi is the Coriolis force. The Coriolis force can be given as: fi=-ku2, f2=ku1, in which k denotes the Coriolis acceleration. Transport of the contaminant, on the other hand, is governed by an advection-diffusion equation: ~~ + (<j>ui),i- K<j>,ii = 0, (3) where <1> is the concentration, Ui is the current velocity and K is the diffusion coefficient. 3. THREE-STEP EXPLICIT FINITE ELEMENT METHOD For the finite element spatial discretization of the governing equations, the selective lumping Galerkin and SUPG methods are used, respectively, for the shallow water and advection-diffusion equations. The three-node linear triangular element is employed for the spatial discretization For discretization in time, the three-step explicit time-integration scheme is employed [4]. This method contains the benefit of the conventional Taylor-Galerkin method. The numerical accuracy and stability of this scheme are di cus ed in reference [5]. Applying this procedure to the governing equations, the discretized equations in time can be obtained. 4. PARALLEL IMPLEMENTATION The data-parallel implementation has been carried out on the Connection Machine CM-5. For the implementation, two types of data arrays are used; element-level and equation-level [6]. The element-level arrays store the data with one element and its degrees of freedom associated with exactly one virtual proce or. On the other band, the equation-level arrays keep variables at the level of global equation system. The nodal data, coordinates, element level properties and element-level matrix and vector are tared in array of element-level type. The global increment variables are kept in an array of equation-level type. Communication operation from equation-level to element-level is called as a gather, while movement of the data from element-level to equation-level is called as a scatter. Both gather and. catter may be implemented efficiently on the Connection Machine computer. In order to ave the communication time the mesh partitioning [7] of Connection Machine Scientific Software Library (CMSSL) is used. The discretized element~level equation can be expressed as (4) where (Ma~L)e is the element-level lumped mass matrix, (x~)e is the element-level unknown vector and (f~)e is the element-level known vector. The computation of the element-level lumped mass matrix and the element-level known vector is performed on element level. Then, these values are assembled to nodal values by a scatter operation. The unknown variables x~ are solved by: and the introduction of the boundary conditions is taken care of at equation level. (5)

4 5. NUMERICAL EXAMPLE As a numerical example, simulation of tidal flow and contaminant transport in Tokyo Bay is carried out. Figure 1 shows the finite element idealization of Tokyo Bay which is partitioned for 128 processing nodes. The total number of elements and nodes are 56,893 and 30,105 respectively. This mesh is designed to keep the element Courant number is constant in the entire domain [8]. Figure 2 shows the computed current velocity distribution around Uraga-Suido. Figure 3 shows the contaminant spread. It can be seen that the contaminant is spread in accordance with the periodic oscillation due to tidal waves. The computational speed using 128 processing nodes is sec/step and 11.9 micsec/step/node. 6. CONCLUSION A three step explicit finite element solver applicable to unstructured meshes is successfully implemented on a massively parallel supercomputer. Present method can be usefully applied to large-scale computation of various shallow water flow problems. As an example, we have presented results from a simulation of tidal wave effects in the Tokyo Bay. The simulation also accounted for the spread of the contaminant due to tidal flows. ACKNOWLEDGMENT This research was supported by NSF under grant ASC Partial support for this work has also come from the ARO Contract Number DAAL03-89-C-0038 with the Army High Performance Computing Research Center at the University of Minnesota. REFERENCE [1] T.E. Tezduyar, M. Behr, S. Mittal and A.A. Johnson, Computation of unsteady incompressible flows with the stabilized finite element methods - Space-time formulations, iterative strategies and massively parallel implementations, New Methods in Transient Analysis, in P. Smolinski, W.K. Liu, G. Hulbert and K. Tamma (eds.), AMD-Vol.143, ASME, New York (1992) [2] M. Kawahara, H. Hirano, K. Tsubota and K. Inagaki, Selective lumping finite element method for shallow water flow, Int. J. Numer. Methods Fluids, Vol.2 (1982), [3] T.E. Tezduyar and T.J.R. Hughes, Finite element formulations for convection dominated flows with particular emphasis on the Euler equations, AIAA paper , Proceedings of AIAA 21st Aerospace Science Meeting, Reno, Nevada. (1983). [4] C.B.Jiang, M. Kawahara, K. Hatanaka and K. Kashiyama, A three-step finite element method for convection dominated incompressible flow, Comp. Fluid Dyn. Journal, Vol.l (1993), [5] K. Kashiyama, H. Ito, M. Behr and T.E. Tezduyar, Three step explicit finite element computation of tidal flow involving moving boundary on Connection machine, (in preparation) [6] M. Behr and T.E. Tezduyar, Finite element solution strategies for large-scale flow simulations, Comp. Meth. Appl. Mech. Eng., (in press). [7] Z. Johan, K.K. Mathur, K.K., S.L. Johnsson and T.J.R. Hughes, An efficient communication strategy for finite element methods on the connection machine CM-5 system, Thinking Machine Technical Report, Cambridge, MA, 1993 and Comp. Meth. Appl. Mech. Eng. (in press) [8] K. Kashiyama and T. Okada, Automatic mesh generation method for shallow water flow analysis, Int. J. Numer. Methods Fluids, Vol.l5, (1992)

5 Figure 1. Mesh partitioning for 128 processing nodes High Tide Figure 2. Computed tidal velocity distribution Low Tide Figure 3. Computed contaminant spread