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Flow simulation and high performance computing T. Tezduyar, S. Aliabadi, M. Behr, A. Johnson, V. Kalro, M. litke Computational Mechanics 18 (1996) 397-4\z!- Springer-Verlag r996 Abstract Flow simulation

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Flow simulation and high performance computing T. Tezduyar, S. Aliabadi, M. Behr, A. Johnson, V. Kalro, M. litke Computational Mechanics 18 (1996) 397-4\z!- Springer-Verlag r996 Abstract Flow simulation is a computational tool for exploring 1 science and technology involving flow applications. It can lntroduction provide cost-effective alternatives or complements to laboratory In recent years, the power and diversity of flow simulation as a experiments, field tests and prototyping. Flow simulation relies tool for exploring science and technology involving flow applications have reached new levels. This offers a promising future heavily on high performance computing (HPC). We view HPC as having two major components. One is advanced algorithms for providing cost-effective alternatives or complements to capable of accurately simulating complex, real-world problems. laboratory experiments, field tests and prototyping. This new The other is advanced computer hardware and networking with power of flow simulation is based mainly on recent developments in high performance computing (HPC). Both of what we sufficient power, memory and bandwidth to execute those simulations. While HPC enables flow simulation, flow simulation see as the two major components of HPC have contributed to motivates development of novel HPC techniques. This paper this. These components are: advanced algorithms capable of focuses on demonstrating that flow simulation has come a long accurately simulating complex, real-world problems; and way and is being applied to many complex, real-world problems advanced computer hardware and networking with sufficient in different fields ofengineering and applied sciences, particularly in aerospacengineering and applied fluid mechanics. is also true that there is a two-way exchange between flow simu- power, memory and bandwidth to execute those simulations. It Flow simulation has come a long way because HPC has come lation and HPC; while HPC enables flow simulation, flow simulation motivates development of novel HPC techniques. In this a long way, This paper also provides a brief review of some of the recently-developed HPC methods and tools that has played paper, we show that flow simulation has come a long way and a major role in bringing flow simulation where it is today. A to a point of being a practical tool for complex, real-world number of 3D flow simulations are presented in this paper as problems in different fields ofengineering and applied sciences, examples of the level of computational capability reached with in our case, aerospacengineering and applied fluid mechanics. In this paper we also briefly review some of the recently- recent HPC methods and hardware. These examples are, flow around a fighter aircraft, flow around two trains passing in a developed HPC methods and tools that have played a major tunnel, large ram-air parachutes, flow over hydraulic structures, role in making flow simulation a powerful tool that it is today. contaminant dispersion in a model subway station, airflow In this review and in providing examples, we include some past an automobile, multiple spheres falling in a liquid-filled material extracted from other articles written by these authors tube, and dynamics of a paratrooper jumping from a cargo and their group, particularly a recent article by Tezduyar et al. aircraft. ( 1 ee6). We consider applications involving compressible as well as incompressible flows, mostly with time-dependent behavior, and mostlywith moving boundaries and interfaces. The governing equations for this class of problems are given in Section 2. The flow simulation capabilities developed recently to address Communicated by S. N. Atluri, 13 May 1996 this type ofapplications cover a large class offlow regimes T. Tezduyar, S. Aliabadi, M. Behr, A. ohnson, V. Kalro, M. Litke, and patterns: flows involving complex geometries (Tezduyar Aerospace Engineering and Mechanics, Army HPC Research Center, et al. (1994), Johnson and Tezduyar (1996a)), flows with free University of Minnesota, 1100 Washington Avenue South, Minneapolis, surfaces (Soulaimani et al. (1991), Behr and Tezduyar (1994)), MN 55415, USA tidal flows (Kashiyama et ai. (1995)), twojiquid interfaces (Johnson and Tezduyar (1994), Ray et al. (1996)), fluid-particle Correspondence to: T. Tezduyar interactions (fohnson and Tezduyar (1995), johnson and Sponsored by ARO, ARPA, NASA-JSC, and by the Army High T ezduyar ( I 996b ), fl uid-structure interactions (Mittal and Performance Computing Research Center under the auspices of the Tezduyar (1995), Ray et al. (1996)), 2D aeroelasticity (Farhat Department of the Army, Army Research Laboratory cooperative et al. (1995)) and moving mechanical components (Ray et al. agreement number DAAH / contract number (1996)). We also see a significant amount of activity in HPCbased flow simulation of applications in chemical engineering DAAH04-95-C The content does not necessarily reflect the position or the policy of the Government, and no official endorsement and materials science (see for example Salinger et al. (1994), should be inferred. CRAY C90 time and support for the second author was provided in part by the Minnesota Supercomputer Institute. Xiao et al. (1995)). Because of the geometric complexities involved in a typical, Dedicated to the 1Oth anniversarv of Computational Mechanics real-world problem, especially if combined with the presence 397 of moving boundaries and interfaces, the finite element method One of the most notable of these groups is the one. led by with unstructured meshes becomes one of the best strategies for Shephard (see Shephard and Georges (1991) and Ozturan et al. flow simulation, and in fact sometimes the only practical choice. (1994)). This group is also particularly interested in research The advances of the last two decades in stabilized finite element on parallel mesh generation as well as mesh adaptation. For formulations have made this method an even more credible the class of flow simulations we are interested in. we needed and powerful approach in flow computations. The most notable a 3D automatic mesh generation capability which allows stabilization techniques are, streamline-upwind/petrov- structured layers of elements near solid surfaces and unstructured meshes elsewhere in the domain, and we developed Galerkin (SUPG) formulation for incompressible flows (Brooks and Hughes (1982)), SUPG formulation for compressible flows a mesh generation tool (fohnson and Tezduyar (1996a)) (Tezduyar and Hughes (1983), Le Beau and Tezduyar (1991), which satisfies this and some other requirements, and yet Aliabadi et al. (1993)), Galerkin/least-squares (GLS) formu- continues to evolve to meet the special needs of our simulations as such needs arise. The mesh generation issues are 398 lation (Hughes et al. (1989), Hansbo and Szepessy (1990)), and pressure-stabilizing/petrov-galerkin (PSPG) formulation for covered in more detail in Section 4. incompressible flows (Tezduyar(1991)). The stabilization prevents the numerical oscillations and instabilities that might boundaries and interfaces, the changes in the shape ofthe spatial In our parallel computations of flows problems with moving be encountered when the flow involves high Reynolds and/or domain are handled by updating the mesh by using the combination of moving the mesh and remeshing (i.e., generating a Mach numbers and strong shocks and sharp boundary layers. The stabilization also permits the use of equal-order interpolation functions for velocity and pressure and other unknowns, tion becomes too high (Tezduyar et al. (1992d), Johnson and new set of elements and grid points) when the mesh distor- which is a significant advantage in parallel implementations. Tezdtyar (1995)). Moving the mesh is accomplished by special In all cases, the stabilization is accomplished still in the mesh moving techniques designed for specific problems and context of a weighted residual formulation and without by an automatic mesh moving algorithm for more general cases. introducing excessive numerical dissipation, and that is the This automatic mesh moving is based on the motion of the main attractive point about this class of methods. A brief grid points being governed by the equations of elasticity, with review ofthese stabilization techniques is given in Section 3, the boundary conditions imposed by the motion of the fluid together with the specific stabilized finite element formulations employed in carrying out the flow simulations reported were used earlier by other researchers, e.g. Lynch (1982). boundaries and interfaces. Similar mesh moving techniques in this paper. Mesh moving issues were also addressed by Farhat and Lanteri When we need to solve problems with changing spatial domains, such as those encountered in flows with moving bound- springs. The methods for 3D mesh update are covered in Sec- (199a) by using a pseudo-structural model with fictitious aries and interfaces, depending on the circumstances, one can tion 5. chose between fixed grid and moving grid methods. As the most The large coupled, nonlinear equation systems that need to notable moving grid methods, we can list the moving finite elements, arbitrary Lagrangian-Eulerian, and the space-time for- every pseudo-time step in a steady-state computation) are be solved at every time step in a time-accurate computation (or mulations. The application of the stabilized space-time finite solved using iterative methods (Behr et al. (1993), Kalro and element formulation to flows with moving boundaries and interfaces was introduced first in the context of incompressible methods, are: a) formation of the residual vector of the linear Tezduyar (1995)). The main components of these iterative flows in Tezduyar et al. (1992c), and soon after that in the equation system (that needs to be solved at every iteration of context of compressible flows in Aliabadi and Tezduyar (1993). a Newton-Raphson sequence used in solving the coupled, nonlinear equations); b) designing a preconditioning matrix which The stabilized space-time formulation was in fact used earlier by other researchers to solve problems with fixed spatial cost-effectively and reasonably approximates the Jacobian domains (Hughes and Hulbert (1988), Hansbo and Szepessy matrix in each Newton-Raphson iteration; and c) updating the (1990), Shakib (1988)), however we do not see any major solution vector in an optimal way based on the residual and advantage in using this relatively costly method for problems solution increment vectors of the linear equation system. In all with just fixed domains. When the flow problem does not our parallel computations, we form the residual vector of the involve any moving boundaries and interfaces, we find it equally linear equation systems by using, depending on the size of satisfactory, and a lot less costly, to use the PSPG formulation which is basically a reduced version of the GLS method, is free from even element-level matrices. Such matrix-free the problem, the element-level matrices or a method which but which is still a weighted residual method. Section 3 also methods were successfully used also by other researchers in includes the description of the stabilized space-time formulations used in carrying out the flow simulations reported in this ohan et al. (1995)). For preconditioning matrices, we are the context ofparallel computations (Johan et al. (1991), paper. currently using simple ones, such as diagonal or nodal-blockdiagonal preconditioners (Shakib et al. (1989)), in our produc- Mesh generation continues to be an important issue and sometimes a serious bottleneck in large-scale computations. tion computations. We are also experimenting with more The 3D finite element meshes needed can be generated, depending on the complexity of the problem geometry, by (CEBE) and mixed CEBE and cluster companion (CC) pre- sophisticated ones, such as the clustered element-by-element using either special mesh generators designed for specific conditioners (Liou and Tezduyar (1992), Tezduyar et al. problems, or an automatic mesh generator. We believe that (1992b)). To update the solution vector, we typically use the still a significant amount of research and development effort GMRES technique (Saad and Schultz (1986)). This is a very needs to go into 3D automatic mesh generation, and there are common update technique for iterative computations involving some research groups who are doing that rather successfully. non-symmetric matrices. The parallel computations can be carried out on shared or distributed memory systems, or sometimes a combination of the 2.1 Compressible flows two in the context multi-platform computing or by connecting The Navier-Stokes equations of compressible flows can be several identical shared memory systems together. Parallel written in the following vector form: implementations on shared memory systems (e.g., a 20-processor SGI ONYX or a 12-processor SGI Power Challenge) are 6U 'F, AE, based on the fact that the processors use the same central - + -;r:o ct cxi Oxi memory, and therefore this parallel computing paradigm is relatively ono, Vre(o,T), (1) simple to adapt to. For distributed memory systems (e.g., where U : ( p, pu, pu, p4' pe) is the vector of conservation a 512-processor Thinking Machines CM-5 or a 512-processor variables, and F, and E, are, respectively, the Euler and viscous CRAY T3D), the implementations could be based on dataparallel flux vectors defined as or message-passing paradigms, and this requires more elaborate work. The user-friendly Connection Machine software ,,- libraries made available by a group of researchers at Thinking \ Machines (lohnsson and Mathur (1989), Mathur and I u,pu, i lohnsson F,:l u,pur* 6,rp (1992)), encouraged a number of researchers to implement their l, finite element flow simulation formulations on the Connection \ ''ou' + 5''P I Machine systems (fohan etal. (1992),Tezduyar et al. (1992d)). \ u,(pe+p) Our parallel computations on the Connection Machine systems \2) for flow problems were reported first in Tezduyar et al. (1992d) 0 \ for 2D and axisymmetric simulations, and soon after that in ltl,, i Tezduyar et ai. (1992a) and Tezduyar et al. ( 1993) for 3D simulations. More 3D simulations were reported in Behr and Tezduyar Ei: ttl,, I (3) trl;, - - li (1994), Mittal and Tezdtyar (1994), Tezduyar et al. (1994), qt+ lt)i1,u*f Aliabadi and Tezduyar (1995), Mittal and Tezdtyar (1995), ohnson and Tezduyar (1995) and Kalro et al. (1996). The 3D Here simulations we carried out on the T3D were reported first [T],, are the components of the Newtonian viscous stress tensor: in Tezduyar et al. (1995). Some of the other notable parallel computing activities for flow simulations were reported by Farhat and his coworkers, for example in Farhat et al. (1993) and Farhat and Lanteri (1994) for 2D viscous flows, and in T:2pe(u), (4) Farhat et al. (1995) for 2D aeroelasticity. The parallel computing studies by Shephard and his coworkers (Ozturan et al. (1994)) emphasize automatic mesh generation and mesh adaptivity. where p is the dynamic viscosity and e is the strain rate tensor, and q,are the components ofthe heat flux vector. The equation of state is modeled with the ideal gas assumption. Equation (1) can also be written in the following form: Currently, the majority of our parallel computations are carried out on the AHPCRC's 896-node CM-5 and the Minnesota aaaau 'U A/ AU\ Supercomputer Center's 512-node T3D. Some of our smaller. *A,- ' -. lk,,^ l:0 on Q, Vre (0,7), (5) ct 1xi t x,\ ' t'x1 / scale parallel computations are performed on the AHPCRC's 20-processor SGI Onyx. The applications that will be presented where in this paper are: supersonic flow around fighter airplane; flow around two trains passing each other in a tunnel; steady-state decent ofa large ram air parachute; longitudinal dynamics and if, A,: flare maneuver ofa large ram air parachute; flow past the spillway ru-., (6) of the Olmsted Dam; contaminant dispersion in a model subway station; airflow past an automobile; multiple spheres AU falling in a liquid-filled tube; and dynamics of a paratrooper K,,. :E, cx, (7) jumping from a cargo plane. Appropriate sets ofboundary and initial conditions are 2 assumed to accompany Eq. (5). Governing equations The flow simulations are based on the solution of timedependent Navier-Stokes equations of compressible and 2.2 lncompressible flows incompressible flows. In stating those equations here, {), and The Navier-Stokes equations of incompressible flows can be (0, T ) will denote the space and time domains, where { is written in the following vector form: the boundary of {),. ln general, the spatial domain may change with respecto time, and the subscript f indicates such timedependence. The symbols p(x,t),u(x, f),p(x, f) and e(x, /) representhe density, velocity, pressure and the total energy, res- \dr / p(* *u.vu - r)- v.o: o on Qt v/ (0, T), (8) pectively. The external forces (e.g. gravity) are represented by f(x, r). V.u:0 on!), V/e (0,7), (e) 399 4oo where p is assumed to be constant, and 6: -pi * T. For the compressible and incompressible flow applications later discussed in Section 8, the space-time formulation (10) begins with the weak form of the governing equations being written over the associated space-time domain of the problem, This equation set is completed by an appropriate set ofboundary conditions and an initial condition consisting ofa divergence-free velocity field specified over the entire domain: by dividing this domain into a sequence of space-time slabs Q,, where Q, is the slice of the space-time domain between the time levels f, and /,*,. The integrations involved in the weak form are then performed over Q,. The finite element interpolation u(x,0) : u,, V.uo : 0 on Oo. (ll) functions used are continuous in space but discontinuous across time levels. To reflect this situation, we use the notation (.) 2.3 and ('),* to denote the function values at f, as approached from Turbulence model below and above respectively. Each space-time slab Q, is decomposed into space-time elements Qi, where e :7,2,...,(n 1)u Most computational grids which are in practical use today are not able to resolve the flow features well enough to fully capture The subscript n used with n , is to account for the general case turbulence effects. Acknowledging this restriction, significant in which the number of space-tim elements may change from effort is spent in the computational community to devise and one space-time slab to other. In our computations, we use firstorder improve turbulence modeling. Several reviews of the work in polynomials as interpolation functions. this area have been published, see e.g. Speziale ( 1991 ). A notable attempt has been made recently to integrate the concepts of 3.1 turbulence modeling and stabilization techniques (Hughes Compressible flows (1995), Hughes and Stewart (1996)). In the finite element formulation of compressible flows, for each In our high-reynolds number flow computations, we employ slab Q,, we first define appropriate finite-dimensional function a simple Smagorinsky turbulence model (Smagorinsky(1963), spaces 9l and {l corresponding to the trial solutions and Kato and Ikegawa (1991)). In this approach, the kinematic weighting functions, respectively. The stabilized space-time viscosity rr is augmented by an eddy viscosity: formulation of (5) can then be written as follows: given (U'),-, frnd Uh e 9lsuch that Y Wh e{!: v e 1, * (ch)' (2t(u):e (u)) 1/2, (r2) /?t)u,aui'\ /ewh\/,aui\ where C is the model constant and h is the element length. Unless I w'.1 -+Ai= cxi/ ldq+ {- l.{ Ki,= ldq stated otherwise, the constant C is set to a, \.r a,\.xi/\'cxj/ 3 + Stabilized finite element formulations J (wh),*.((uft),* - Q\; ) da The SUPG formulation for incompressible flows was introduced in Hughes and Brooks (197

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