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We address the computational challenges encountered in turbocharger turbine and exhaust manifold flow analysis. The core computational method is the Space–Time Variational Multiscale (ST-VMS) method, and the other key methods are the ST Isogeometric Analysis (ST-IGA), ST Slip Interface (ST-SI) method, ST/NURBS Mesh Update Method (STNMUM), and a general-purpose NURBS mesh generation method for complex geometries. The ST framework, in a general context, provides higher-order accuracy. The VMS feature of the ST-VMS addresses the computational challenges associated with the multiscale nature of the unsteady flow in the manifold and turbine, and the moving-mesh feature of the ST framework enables high-resolution computation near the rotor surface. The ST-SI enables moving-mesh computation of the spinning rotor. The mesh covering the rotor spins with it, and the SI between the spinning mesh and the rest of the mesh accurately connects the two sides of the solution. The ST-IGA enables more accurate representation of the turbine and manifold geometries and increased accuracy in the flow solution. The STNMUM enables exact representation of the mesh rotation. The general-purpose NURBS mesh generation method makes it easier to deal with the complex geometries we have here. An SI also provides mesh generation flexibility in a general context by accurately connecting the two sides of the solution computed over nonmatching meshes. That is enabling us to use nonmatching NURBS meshes here. Stabilization parameters and element length definitions play a significant role in the ST-VMS and ST-SI. For the ST-VMS, we use the stabilization parameters introduced recently, and for the ST-SI, the element length definition we are introducing here. The model we actually compute with includes the exhaust gas purifier, which makes the turbine outflow conditions more realistic. We compute the flow for a full intake/exhaust cycle, which is much longer than the turbine rotation cycle because of high rotation speeds, and the long duration required is an additional computational challenge. The computation demonstrates that the methods we use here are very effective in this class of challenging flow analyses.

The Space–Time Computational Analysis (STCA) with key components that include the ST Variational Multiscale (ST-VMS) method and ST Isogeometric Analysis (ST-IGA) is being increasingly used in fluid mechanics computations with complex geometries. In such computations, the ST-VMS serves as the core method, complemented by the ST-IGA, and sometimes by additional key components, such as the ST Slip Interface (ST-SI) method. To make the ST-IGA use, and in a wider context the IGA use, even more practical in fluid mechanics computations, NURBS volume mesh generation needs to be easier and as automated as possible. To that end, we present a general-purpose NURBS mesh generation method. The method is based on multi-block structured mesh generation with existing techniques, projection of that mesh to a NURBS mesh made of patches that correspond to the blocks, and recovery of the original model surfaces to the extent they are suitable for accurate and robust fluid mechanics computations. It is expected to retain the refinement distribution and element quality of the multi-block structured mesh that we start with. The flexibility of discretization with the general-purpose mesh generation is supplemented with the ST-SI method, which allows, without loss of accuracy, C − 1 continuity between NURBS patches and thus removes the matching requirement between the patches. We present a test computation for a turbocharger turbine and exhaust manifold, which demonstrates that the general-purpose mesh generation method proposed makes the IGA use in fluid mechanics computations even more practical.

Heart valve flow computation requires accurate representation of boundary layers near moving solid surfaces, including the valve leaflet surfaces, even when the leaflets come into contact. It also requires dealing with a high level of geometric complexity. We address these computational challenges with a Space–Time (ST) method developed by integrating three special ST methods in the framework of the ST Variational Multiscale (ST-VMS) method. The special methods are the ST Slip Interface (ST-SI) and ST Topology Change (ST-TC) methods and ST Isogeometric Analysis (ST-IGA). The computations are for a realistic aortic-valve model with prescribed valve leaflet motion and actual contact between the leaflets. The ST-VMS method functions as a moving-mesh method, which maintains high-resolution boundary layer representation near the solid surfaces, including leaflet surfaces. The ST-TC method was introduced for moving-mesh computation of flow problems with TC, such as contact between the leaflets of a heart valve. It deals with the contact while maintaining high-resolution representation near the leaflet surfaces. The ST-SI method was originally introduced to have high-resolution representation of the boundary layers near spinning solid surfaces. The mesh covering a spinning solid surface spins with it, and the SI between the spinning mesh and the rest of the mesh accurately connects the two sides. In the context of heart valves, the SI connects the sectors of meshes containing the leaflets, enabling a more effective mesh moving. In that context, integration of the ST-SI and ST-TC methods enables high-resolution representation even when the contact is between leaflets that are covered by meshes with SI. It also enables dealing with contact location change or contact and sliding on the SI. By integrating the ST-IGA with the ST-SI and ST-TC methods, in addition to having a more accurate representation of the surfaces and increased accuracy in the flow solution, the element density in the narrow spaces near the contact areas is kept at a reasonable level. Furthermore, because the flow representation in the contact area has a wider support in IGA, the flow computation method becomes more robust. The computations we present for an aortic-valve model with two different modes of prescribed leaflet motion show the effectiveness of the ST-SI-TC-IGA method.

We focus on computational analysis of flow-driven string dynamics. The objective is to understand how the strings carried by a fluid interact with the solid surfaces present and get stuck on or around those surfaces. Our target application is turbomachinery, such as understanding how strings get stuck on or around the blades of a fan. The components of the method we developed for this purpose are the Space–Time Variational Multiscale (ST-VMS) and ST Slip Interface (ST-SI) methods for the fluid dynamics, and a one-way-dependence model and the Isogeometric Analysis (IGA) for the string dynamics. The ST-VMS method is the core computational technology and it also has the features of a turbulence model. The ST-SI method allows in a consistent fashion slip at the interface between the mesh covering a spinning solid surface and the mesh covering the rest of the domain, and with this, we maintain high-resolution representation of the boundary layers near spinning solid surfaces such as fan blades. With the one-way-dependence model, we compute the influence of the flow on the string dynamics, while avoiding the formidable task of computing the influence of the string on the flow, which we expect to be small. The IGA for the string dynamics gives us not only a higher-order method and smoothness in the structure shape, but also smoothness in the fluid dynamics forces calculated on the string. To demonstrate how the method can be used in computational analysis of flow-driven string dynamics, we present the pilot computations we carried out, for a duct with cylindrical obstacles and for a ventilating fan.

We focus on turbocharger computational flow analysis with a method that possesses higher accuracy in spatial and temporal representations. In the method we have developed for this purpose, we use a combination of (i) the Space–Time Variational Multiscale (ST-VMS) method, which is a stabilized formulation that also serves as a turbulence model, (ii) the ST Slip Interface (ST-SI) method, which maintains high-resolution representation of the boundary layers near spinning solid surfaces by allowing in a consistent fashion slip at the interface between the mesh covering a spinning surface and the mesh covering the rest of the domain, and (iii) the Isogeometric Analysis (IGA), where we use NURBS basis functions in space and time. The basis functions are spatially higher-order in all representations, and temporally higher-order in representation of the solid-surface and mesh motions. The ST nature of the method gives us higher-order accuracy in the flow solver, and when combined with temporally higher-order basis functions, a more accurate representation of the surface motion, and a mesh motion consistent with that. The spatially higher-order basis functions give us again higher-order accuracy in the flow solver, a more accurate, in some parts exact, representation of the surface geometry, and better representation in evaluating the second-order spatial derivatives. Using NURBS basis functions with a complex geometry is not trivial, however, once we generate the mesh, the computational efficiency is substantially increased. We focus on the turbine part of a turbocharger, but our method can also be applied to the compressor part and thus can be extended to the full turbocharger.

We present a method for structural and fluid mechanics computations of ram-air parachutes. A ram-air parachute is a parafoil inflated by the airflow through the inlets at the leading edge. It has better control and gliding capability than round parachutes. Reliable analysis of ram-air parachutes requires accurate representation of the parafoil geometry, fabric porosity and the complex, multiscale flow behavior involved in this class of problems. The key components of the method are (i) the Space–Time Variational Multiscale (ST-VMS) method, (ii) the version of the ST Slip Interface (ST-SI) method where the SI is between a thin porous structure and the fluid on its two sides, (iii) the ST Isogeometric Analysis (ST-IGA), and (iv) special-purpose NURBS mesh generation techniques for the parachute structure and the flow field inside and outside the parafoil. The ST-VMS method is a stabilized formulation that also serves as a turbulence model and can deal effectively with the complex, multiscale flow behavior. With the ST-SI version for porosity modeling, we deal with the fabric porosity in a fashion consistent with how we deal with the standard SIs and how we enforce the Dirichlet boundary conditions weakly. The ST-IGA, with NURBS basis functions in space, gives us, with relatively few number of unknowns, accurate representation of the parafoil geometry and increased accuracy in the flow solution. The special-purpose mesh generation techniques enable NURBS representation of the structure and fluid domains with significant geometric complexity. The test computations we present are for building a starting parachute shape and a starting flow field associated with that parachute shape, which are the first two key steps in fluid–structure interaction analysis. The computations demonstrate the effectiveness of the method in this class of problems.

We present a flow analysis of a wave-energy air turbine, commonly known as Wells turbine. The focus here is on the computational method used in the analysis, based on the Streamline-Upwind/Petrov-Galerkin (SUPG) and Pressure-Stabilizing/Petrov-Galerkin (PSPG) stabilizations and the Discontinuity-Capturing Directional Dissipation (DCDD). The SUPG/PSPG stabilization is used rather widely and successfully. The DCDD, first introduced to complement the SUPG/PSPG method in computations of incompressible flows in the presence of sharp solution gradients, was also shown to perform well in standard turbulent-flow test computations when compared to the Smagorinsky Large Eddy Simulation (LES) model. The results obtained in our computational analysis of the Wells turbine here compare favorably to the available experimental data, and this demonstrates that the DCDD method performs well also in turbomachinery flows.

The Space–Time Variational Multiscale (ST-VMS) method was introduced to function as a moving-mesh method. It is the VMS version of the Deforming-Spatial-Domain/Stabilized ST (DSD/SST) method. It has reasonably good turbulence modeling features and serves as a core computational method. The ST Slip Interface (ST-SI) method was introduced to addresses the challenge involved in high-resolution representation of the boundary layers near spinning solid surfaces. The mesh covering a spinning solid surface spins with it and thus maintains the high-resolution representation near it. The ST-TC method was introduced for moving-mesh computation of flow problems with topology changes, such as contact between solid surfaces. It deals with the TC while maintaining high-resolution boundary layer representation near solid surfaces. The “ST-SI-TC” method we introduce here integrates the ST-SI and ST-TC methods in the ST-VMS framework. It enables accurate flow analysis when we have a spinning solid surface that is in contact with a solid surface. We present two test computations with the ST-SI-TC method, and they are both with models of flow around a rotating tire with road contact and prescribed deformation, one with a 2D model, and one with a 3D model.

We present an overview of the special computer modeling techniques we have developed recently for flapping-wing aerodynamics of a locust. The wing motion and deformation data is from an actual locust, extracted from high-speed, multi-camera video recordings of the locust in a wind tunnel. The special techniques have been developed around our core computational technique, which is the Deforming-Spatial-Domain/Stabilized Space–Time (DSD/SST) formulation. Here we use the version of the DSD/SST formulation derived in conjunction with the variational multiscale (VMS) method, and this version is called “DSD/SST-VMST.” The special techniques are based on using, in the space–time flow computations, NURBS basis functions for the temporal representation of the motion and deformation of the locust wings. Temporal NURBS basis functions are used also in representation of the motion of the volume meshes computed and in remeshing. In this special-issue paper, we present a condensed version of the material from [1] , concentrating on the flapping-motion modeling and computations, and also a temporal-order study from [2] .

Comment is provided on Dutta and Ray (2004) to identify the source of the two computational-analysis pictures included in the article.

The objective of this paper is to show that use of the element-vector-based definition of stabilization parameters, introduced in [T.E. Tezduyar, Computation of moving boundaries and interfaces and stabilization parameters, Int. J. Numer. Methods Fluids 43 (2003) 555–575; T.E. Tezduyar, Y. Osawa, Finite element stabilization parameters computed from element matrices and vectors, Comput. Methods Appl. Mech. Engrg. 190 (2000) 411–430], circumvents the well-known instability associated with conventional stabilized formulations at small time steps. We describe formulations for linear advection–diffusion and incompressible Navier–Stokes equations and test them on three benchmark problems: advection of an L-shaped discontinuity, laminar flow in a square domain at low Reynolds number, and turbulent channel flow at friction-velocity Reynolds number of 395.

The Sequentially-Coupled Arterial Fluid–Structure Interaction (SCAFSI) technique is one of the special techniques developed recently by the Team for Advanced Flow Simulation and Modeling (T☆AFSM) for FSI modeling of blood flow and arterial dynamics. The SCAFSI technique, which was introduced as an approximate FSI approach in arterial fluid mechanics, is based on the assumption that the arterial deformation during a cardiac cycle is driven mostly by the blood pressure. In the SCAFSI, first we compute a “reference” arterial deformation as a function of time, driven only by the blood pressure profile of the cardiac cycle. Then we compute a sequence of updates involving mesh motion, fluid dynamics calculations, and recomputing the arterial deformation. Although the SCAFSI technique was developed and tested in conjunction with the stabilized space–time FSI (SSTFSI) technique, it can also be used in conjunction with other FSI modeling techniques categorized as moving-mesh methods. The SSTFSI technique is based on the Deforming-Spatial-Domain/Stabilized Space–Time (DSD/SST) formulation and includes the enhancements introduced recently by the T☆AFSM. The arterial structures can be modeled with the membrane or continuum elements, both of which are geometrically nonlinear, and the continuum element can be made of linearly-elastic or hyperelastic material (Mooney–Rivlin or Fung). Here we provide an overview of the SCAFSI technique and present a number of test computations for abdominal aortic and cerebral aneurysms, where the arterial geometries used in the computations are close approximations to the patient-specific image-based data.

We provide an overview of the finite element methods we developed for fluid dynamics problems. We focus on stabilized formulations and moving boundaries and interfaces. The stabilized formulations are the streamline-upwind/Petrov–Galerkin (SUPG) formulations for compressible and incompressible flows and the pressure-stabilizing/Petrov–Galerkin (PSPG) formulation for incompressible flows. These are supplemented with the discontinuity-capturing directional dissipation (DCDD) for incompressible flows and the shock-capturing terms for compressible flows. Determination of the stabilization and shock-capturing parameters used in these formulations is highlighted. Moving boundaries and interfaces include free surfaces, two-fluid interfaces, fluid–object and fluid–structure interactions, and moving mechanical components. The methods developed for this class of problems can be classified into two main categories: interface-tracking and interface-capturing techniques. The interface-tracking techniques are based on the deforming-spatial-domain/stabilized space–time (DSD/SST) formulation, where the mesh moves to track the interface. The interface-capturing techniques were developed for two-fluid flows. They are based on the stabilized formulation, over typically non-moving meshes, of both the flow equations and an advection equation. The advection equation governs the time-evolution of an interface function marking the interface location. We also describe some of the additional methods and ideas we introduced to increase the scope and accuracy of these two classes of techniques. Among them is the enhanced-discretization interface-capturing technique (EDICT), which was developed to increase the accuracy in capturing the interface. Also among them is the mixed interface-tracking/interface-capturing technique (MITICT), which was introduced for problems that involve both interfaces that can be accurately tracked with a moving-mesh method and interfaces that call for an interface-capturing technique.

As a sequel to “Finite elements in fluids: stabilized formulations and moving boundaries and interfaces” [Tezduyar TE. Finite elements in fluids: stabilized formulations and moving boundaries and interfaces. Comput Fluids, in press, doi:10.1016/j.compfluid.2005.02.011 .], in this article we provide an overview of the special methods and enhanced solution techniques we developed in conjunction with the methods described in the accompanying paper. The methods and ideas highlighted here were introduced to increase the scope and accuracy of the stabilized formulations and interface-tracking and interface-capturing techniques highlighted in the accompanying paper. They include special methods for fluid–object interactions, for flows involving objects in fast, linear or rotational relative motion, and for two-fluid flows. They also include enhanced solutions techniques, where we have enhancement in spatial discretization, enhancement in time discretization, and enhancement in iterative solution of non-linear and linear equation systems.

In computation of flow problems with fluid–solid interfaces, an interface-tracking technique, where the fluid mesh moves to track the interface, would allow us to have full control of the resolution of the fluid mesh in the boundary layers. With an interface-capturing technique (or an interface locator technique in the more general case), on the other hand, independent of how accurately the interface geometry is represented, the resolution of the fluid mesh in the boundary layer will be limited by the resolution of the fluid mesh at the interface. In computation of flow problems with fluid–fluid interfaces where the interface is too complex or unsteady to track while keeping the remeshing frequency under control, interface-capturing techniques, with enhanced-discretization as needed, could be used as more flexible alternatives. Sometimes we may need to solve flow problems with both fluid–solid interfaces and complex or unsteady fluid–fluid interfaces. The Mixed Interface-Tracking/Interface-Capturing Technique (MITICT) was introduced for computation of flow problems that involve both interfaces that can be accurately tracked with a moving mesh method and interfaces that are too complex or unsteady to be tracked and therefore require an interface-capturing technique. As the interface-tracking technique, we use the Deforming-Spatial-Domain/Stabilized Space–Time (DSD/SST) formulation. The interface-capturing technique rides on this, and is based on solving over a moving mesh, in addition to the Navier–Stokes equations, the advection equation governing the time-evolution of the interface function. For the computations reported in this paper, as interface-capturing technique we are using one of the versions of the Edge-Tracked Interface Locator Technique (ETILT).

Recent reports indicate that the rupture risk for cerebral aneurysms is less than the risk of surgical complications. Being able to predict the rupture of aneurysms would help making better-informed decisions and avoiding unnecessary surgical operations. The wall shear stress is known to play an important role in vascular diseases. We carry out computational fluid–structure interaction analyses to investigate the influence of the arterial-wall deformation on the hemodynamic factors, including the wall shear stress distribution. The results show various patterns of this influence, depending very much on the arterial geometry.

In recent years we introduced a number of enhancements to the space–time techniques we developed for computer modeling of Fluid–Structure Interaction (FSI) problems. These enhancements, which include more sophisticated fluid–structure coupling and improved mesh generation, are enabling us to address more of the computational challenges involved. Our objective here is to demonstrate the robustness of these techniques in FSI modeling of parachutes involving complex designs. As a numerical example, we have selected a conceptual parachute design with geometric complexities resembling those seen in some of the advanced parachute designs proposed and tested in recent times. We describe our FSI modeling techniques and how we compute the descent and glide performance of this conceptual parachute design. � 2005 Elsevier Ltd. All rights reserved.

We provide an overview of some of the interface-tracking and interface-capturing techniques we developed for finite element computation of flow problems with moving boundaries and interfaces. This category of flow problems includes fluid–particle, fluid–object and fluid–structure interactions; free-surface and two-fluid flows; and flows with moving mechanical components. Both classes of techniques are based on stabilized formulations. The interface-tracking techniques are based on the deforming-spatial-domain/stabilized space–time (DSD/SST) formulation, where the mesh moves to track the interface. The interface-capturing techniques, developed primarily for free-surface and two-fluid interface flows, are formulated typically over non-moving meshes, using an advection equation in addition to the flow equations. The advection equation governs the evolution of an interface function that marks the location of the interface. We also highlight some of the methods we developed to increase the scope and accuracy of these two classes of techniques.

We describe the space–time finite element techniques we developed for computation of fluid–structure interaction (FSI) problems. Among these techniques are the deforming-spatial-domain/stabilized space–time (DSD/SST) formulation and its special version, and the mesh update methods, including the solid-extension mesh moving technique (SEMMT). Also among these techniques are the block-iterative, quasi-direct and direct coupling methods for the solution of the fully discretized, coupled fluid and structural mechanics equations. We present some test computations for the mesh moving techniques described. We also present numerical examples where the fluid is governed by the Navier– Stokes equations of incompressible flows and the structure is governed by the membrane and cable equations. Overall, we demonstrate that the techniques we have developed have increased the scope and accuracy of the methods used in computation of FSI problems. � 2005 Elsevier B.V. All rights reserved.

The streamline-upwind/Petrov–Galerkin (SUPG) formulation is one of the most widely used stabilized methods in finite element computation of compressible flows. It includes a stabilization parameter that is known as “τ”. Typically the SUPG formulation is used in combination with a shock-capturing term that provides additional stability near the shock fronts. The definition of the shock-capturing term includes a shock-capturing parameter. In this paper, we describe, for the finite element formulation of compressible flows based on conservation variables, new ways for determining the τ and the shock-capturing parameter. The new definitions for the shock-capturing parameter are much simpler than the one based on the entropy variables, involve less operations in calculating the shock-capturing term, and yield better shock quality in the test computations.

We provide an overview of the role of parallel finite element computations in fluid mechanics. We emphasize the class of flow problems involving moving boundaries and interfaces. Some of the computationally most challenging flow problems, such as fluid–object and fluid–structure interactions as well as free-surface and two-fluid flows, belong to this class. In the development of the methods targeting this class of problems, the computational challenges involved need to be addressed in such a way that 3D computation of complex applications can be carried out efficiently on parallel computers. This requirement has to be one of the key factors in designing various components of the overall solution approach, such as solution techniques for the discretized equations and mesh update methods for handling the changes in the spatial domain occupied by the fluid. This overview includes a description of how the computational challenges are addressed and how the computational schemes can be enhanced with special preconditioning techniques.

Hemodynamic factors such as the wall shear stress are believed to affect a number of cardiovascular diseases including atherosclerosis and aneurysm. Since resolving phenomena in a living human body is currently beyond the capabilities of in vivo measurement techniques, computer modeling is expected to play an important role in gaining a better understanding of the relationship between the cardiovascular diseases and the hemodynamic factors. We have developed a computer modeling technique for cardiovascular hemodynamic simulations. With this modeling technique, patient-specific 3D geometry of an artery can be analyzed. We take into account some of the important factors in human body for the purpose of demonstrating in vivo situations in a virtual world. The interaction between the blood flow and the deformation of the arterial walls is a factor that we are specifically focusing on. For such fluid–structure interactions, we have developed a computer modeling tool based on the deforming-spatial-domain/stabilized space–time (DSD/SST) formulation. This simulation tool is applied to a patient-specific model under pulsatile blood flow conditions. The simulations show that the flow behavior with compliant arterial walls is different from what we see with rigid arterial walls. Consequently, the distribution of the wall shear stress on the compliant arterial walls is significantly different from that on the rigid arterial walls. We deduce that the compliance of the arterial walls needs to be taken into account in cardiovascular hemodynamic simulations, and the computer modeling tool we have developed can be effective in investigation of cardiovascular diseases.

Two preconditioners are presented for equation systems of strongly coupled fluid–structure interaction computations where the structure is modeled by shell elements. These preconditioners fall into the general category of incomplete LU factorization. The two differ mainly in whether the coefficient matrix is factorized node by node or variable-by-variable. In the variable-wise preconditioner, a modified Schur complement system for pressure is solved approximately with a few iterations using a special preconditioner. The efficiencies of the two preconditioners are compared for different finite element formulations of the fluid mechanics part, including formulations with SUPG and PSPG stabilizations.

In computation of fluid–structure interactions involving large displacements, we use a mesh update method composed of mesh moving and remeshing-as-needed. For problems with complex geometries, we need automatic mesh moving techniques that reduce the need for remeshing. We also would like that these mesh moving techniques allow us to control mesh resolution near the fluid–structure interfaces so that we can represent the boundary layers more accurately. In the mesh moving techniques we designed, the motion of the nodes is governed by the equations of elasticity, and mesh deformation is handled selectively based on element sizes and deformation modes. This is helping us reduce the frequency of remeshing. With the solid-extension mesh moving technique presented in this paper, we are also able to limit mesh distortion in thin layers of elements placed near fluid–structure interfaces.

We investigate, for linear and higher-order elements, various ways of calculating the advective limit of the stabilization parameter (“ τ ”) used in the streamline-upwind/Petrov–Galerkin (SUPG) formulation of flow problems. In the context of a pure advection test problem, we compare the “UGN-based”, element-matrix-based, and element-node-based calculations of the advective limit of the τ . Our investigation shows that the performances of the “UGN-based” and element-matrix-based τ definitions are comparable, with the element-matrix-based definition yielding somewhat lower τ values. We also show that for both definitions, as the polynomial orders increase, the τ values decrease, as they should.

The enhanced discretization and solution techniques are among the advanced computational methods we rely on in simulation and modeling of complex flow problems, including those with moving boundaries and interfaces. The set of enhanced discretization and solution techniques includes those based on enhancement in spatial discretization, enhancement in time discretization, and enhancement in iterative solution of nonlinear and linear equation systems. The enhanced-approximation linear solution technique (EALST) was introduced to increase the performance of the iterative technique used in solution of the linear equation systems when some parts of the computational domain may offer more of a challenge for the iterative method than the others. The EALST can be used for computations based on semi-discrete or space–time formulations.

The enhanced-discretization space–time technique (EDSTT) was developed for the purpose of being able to, in the context of a space–time formulation, enhance the time-discretization in regions of the fluid domain requiring smaller time steps. Such requirements are often encountered in time-accurate computations of fluid–structure interactions, where the time-step size required by the structural dynamics part is smaller, and carrying out the entire computation with that time-step size would be too inefficient for the fluid dynamics part. In the EDSTT-single-mesh (EDSTT-SM) approach, a single space–time mesh, unstructured both in space and time, would be used to enhance the time-discretization in regions requiring smaller time steps. In the EDSTT-multi-mesh (EDSTT-MM) approach, we complement the space–time concept of the deforming-spatial-domain/stabilized space–time (DSD/SST) formulation with the multi-mesh concept of the enhanced-discretization interface-capturing technique (EDICT). In applications to fluid–structure interactions, the structural dynamics modeling is based on a single space–time mesh and the fluid dynamics modeling is based on two space–time meshes. The structural dynamics interface nodes in the space–time domain also belong to the second fluid mesh, which accommodates the time-step requirement of the structural dynamics. We apply the EDSTT-SM and EDSTT-MM approaches to a number of test problems to demonstrate how these methods work and why they would be desirable to use in time-accurate computations.

In this paper we describe a computational technique for simulation of the fluid–structure interactions of a parachute crossing the far wake of an aircraft. This technique relies on using the long-wake flow data already computed, in our case, with the Multi-Domain Method (MDM) we developed earlier. The fluid–structure interaction computations are carried out over a domain enclosing the parachute and moving with the payload. This domain functions as one of the subdomains of the MDM designed specifically for the parachute fluid–structure interactions considered here. The boundary conditions for this subdomain are extracted from the long-wake flow data, at locations corresponding to the positions of those boundaries in the subdomain over which the wake flow data were computed. The Navier–Stokes equations of incompressible flows, governing the fluid dynamics, are solved with the Deforming-Spatial-Domain/Stabilized Space–Time (DSD/SST) formulation, which can handle changes in the spatial domain occupied by the fluid. This formulation is coupled to the finite element formulation used for solving the membrane equations governing the structural mechanics of the parachute canopy and the equations governing the mechanics of the suspension lines. The numerical example included demonstrates how the technique described here, functioning as a component of the MDM, enables us to simulate the fluid–structure interactions of a parachute crossing an aircraft wake.

The dynamics of parachutes involves complex interaction between the parachute structure and the surrounding flow field. Accurate representation of parachute systems requires treatment of the problem as a fluid–structure interaction (FSI). In this paper we present the numerical simulations we performed for the purpose of comparison to a series of cross-parachute wind tunnel experiments. The FSI model consists of a 3-D fluid dynamics (FD) solver based on the Deforming-Spatial-Domain/Stabilized Space–Time (DSD/SST) procedure, a structural dynamics (SD) solver, and a method of coupling the two solvers. These FSI simulations include the prediction of the coupled FD and SD behavior, drag histories, flow fields, structural behavior, and equilibrium geometries for the structure. Comparisons between the numerical results and the wind tunnel data are conducted for three cross-parachute models and at three different wind tunnel flow speeds.

In this paper, an enhanced finite element formulation for unsteady incompressible flows with moving interfaces is presented. The weak form of the Navier–Stokes equations, written using a generalized streamline operator technique, is coupled with the movement of the interface between two immiscible fluids defined through an independent moving mesh. The position of the interface is updated using a Lagrangian formulation. In this framework, a global mass conservation corrector algorithm and an enhanced element integration technique are proposed to improve accuracy. The method is applied to a number of test problems with moving interfaces.

We present the multi-domain method (MDM) for computation of unsteady flow past a cargo aircraft and around a parachute crossing the aircraft's far wake. The base computational methods used here are the stabilized semi-discrete and space-time finite element formulations developed earlier. In the MDM, the computational domain is divided into an ordered sequence of overlapping subdomains. The flow field computed over Subdomain-1, which contains the aircraft, supplies the inflow boundary conditions for Subdomain-2, which is used for computing the long-wake flow. Subdomain-3 contains the parachute, and moves across Subdomain-2. The boundary conditions for Subdomain-3 are extracted from the flow field computed over Subdomain-2, at locations corresponding to the positions of the boundaries of Subdomain-3 as it crosses Subdomain-2. The computation over Subdomain-1, which contains a complex but fixed object, is based on a general-purpose implementation of the semi-discrete formulation. The computation over Subdomain-2, which contains no objects, is based on a special-purpose implementation that exploits the simplicity of the mesh to increase the computational speed. The computation over Subdomain-3, which contains a complex and moving object, is based on a general-purpose implementation of the space-time formulation. With a numerical example, we show that different methods can be brought together in the context of the MDM to address the computational challenges involved in the aerodynamics of a parachute crossing the far wake of an aircraft.

In this paper we present a 3D computational technique for simulation of complex, real-world flow problems with fast-rotating mechanical components. This technique is based on the Deformable-Spatial-Domain/Stabilized Space-Time (DSD/SST) formulation, Shear-Slip Mesh Update Method (SSMUM), and an efficient parallel implementation for distributed-memory parallel computing platforms. The DSD/SST formulation was developed earlier for flow problems with moving boundaries and interfaces, including flows with moving mechanical components. The DSD/SST formulation requires, as a companion method, an effective mesh update strategy, especially in complex flow problems. The SSMUM was developed to meet the mesh update requirements in simulation of flow problems with fast translations, and recently, with a new version of SSMUM, fast rotations. As an example of the class of challenging simulations that can be carried out by this technique, we present computation of flow around a helicopter with its rotor in motion.

We present computational methods for 3D simulation of fluid–object interactions in spatially periodic flows. These methods include a stabilized space-time finite element formulation for incompressible flows with spatial periodicity, automatic mesh generation and update techniques for fluid–object mixtures with spatial periodicity, and parallel implementations. The methods can be applied to uni-periodic (i.e., periodic in one direction), bi-periodic, or tri-periodic flows. The methods are described here in the context of tri-periodic flows with fluid–object interactions, and are applied to the simulation of sedimentation of particles in a fluid. We present several case studies where the results obtained provide notable insight into the behavior of fluid–particle mixtures during sedimentation.

We propose new ways of computing the stabilization parameters used in the stabilized finite element methods such as the streamline-upwind/Petrov–Galerkin (SUPG) and pressure-stabilizing/Petrov–Galerkin (PSPG) formulations. The parameters are computed based on the element-level matrices and vectors, which automatically take into account the local length scales, advection field and the Reynolds number. We describe how we compute these parameters in the context of first a time-dependent advection–diffusion equation and then the Navier–Stokes equations of unsteady incompressible flows.

We present a parallel finite element computational method for 3D simulation of fluid–structure interactions (FSI) in parachute systems. The flow solver is based on a stabilized finite element formulation applicable to problems involving moving boundaries and governed by the Navier–Stokes equations of incompressible flows. The structural dynamics (SD) solver is based on the total Lagrangian description of motion, with cable and membrane elements. The nonlinear equation system is solved iteratively, with a segregated treatment of the fluid and SD equations. The large linear equation systems that need to be solved at every nonlinear iteration are also solved iteratively. The parallel implementation is accomplished using a message-passing programming environment. As a test case, the method is applied to computation of the equilibrium configuration of an anchored ram-air parachute placed in an air stream.

We present a parallel computational strategy for carrying out 3-D simulations of parachute fluid–structure interaction (FSI), and apply this strategy to a round parachute. The strategy uses a stabilized space-time finite element formulation for the fluid dynamics (FD), and a finite element formulation derived from the principle of virtual work for the structural dynamics (SD). The fluid–structure coupling is implemented over compatible surface meshes in the SD and FD meshes. Large deformations of the structure are handled in the FD mesh by using an automatic mesh moving scheme with remeshing as needed.

We present the stabilized-finite-element/interface-capturing (SFE/IC) method developed for parallel computation of unsteady flow problems with two-fluid interfaces and free surfaces. The SFE/IC method involves stabilized formulations, an interface-sharpening technique, and the enforcement of global mass conservation for each fluid. The SFE/IC method has been efficiently implemented on the CRAY T3E parallel supercomputer. A number of 2D test problems are presented to demonstrate how the SFE/IC method works and the accuracy it attains. We also show how the SFE/IC method can be very effectively applied to 3D simulation of challenging flow problems, such as two-fluid interfaces in a centrifuge tube and operational stability of a partially filled tanker truck driving over a bump.

We present the 3D implementation and applications of the enhanced-discretization interface-capturing technique (EDICT) in computation of unsteady flows with two-fluid interfaces. In such computations, EDICT can be used as a very effective method, which combines the flexibility and efficiency of interface-capturing techniques with the accuracy provided by enhanced discretization at the interfaces. A stabilized finite element interface-capturing technique is used as the base formulation to solve, over a typically non-moving mesh, the Navier–Stokes equations and an advection equation governing the interface function. To increase the accuracy in modeling the interfaces, we use finite element functions with multiple components at and near the interfaces, with each component coming from a different level of mesh refinement. With its parallel implementation on advanced high-performance computing platforms such as the CRAY T3E, EDICT is a powerful tool for the simulation of a complex, 3D unsteady flow problems with two fluid-interfaces, including free surfaces.

A fluid–object interaction model for an interior ballistics problem is presented. The fluid is a compressible gas and is modeled using the Deformable-Spatial-Domain/Stabilized-Space-Time (DSD/SST) formulation. The objects can move axially within the model domain, and their motion is determined by the fluid pressure forces and collisions with other objects and rigid boundaries. The model is implemented assuming axisymmetry of the geometry and the flow field. The fluid mesh is composed of structured regions of quadrilateral elements and unstructured regions of triangular elements. The structured elements are used near the surface of the objects in order to better resolve the boundary layer, while the unstructured elements are used elsewhere in the domain. As the objects move, the mesh deformation needed to accommodate these motions takes place only in the unstructured parts of the mesh. Application to an interior ballistics problem is presented and discussed.

The Shear-Slip Mesh Update Method, designed to handle certain classes of flow problems with moving boundaries and interfaces, is presented. Specifically, we focus on problems with large but regular boundary displacements, such as straight-line translation or rotation. These motions are accommodated by using a thin layer of deforming space—time elements, together with limited remeshing without any projection at space—time slab interfaces. As examples, 2D flow around two counter-rotating squares and 3D flow past a propeller are presented.

We present a new, multi-domain parallel computational method for simulation of unsteady flows involving a primary object, a long wake region and, possibly, a secondary object affected by the wake flow. The method is based on the stabilized finite element formulation of the time-dependent Navier—Stokes equations of incompressible flows. In the multi-domain computational method the entire simulation domain is divided into an ordered sequence of overlapping subdomains. The flow data computed over the leading subdomain is used for specifying the inflow boundary conditions for the next subdomain. The subdomain corresponding to the wake would not involve any objects, hence the mesh constructed over this domain would be structured. A special-purpose finite element implementation for structured meshes is used for the wake domain to achieve much higher computational speeds compared to a general-purpose implementation. We present verification studies for the multi-domain method and special-purpose implementation, followed by two numerical examples. The first example is the wake behavior behind a circular cylinder. The second one is the aerodynamic effect of tip vortices released from a leading wing on a trailing wing placed in the far wake.

This paper provides an overview of some of the CFD methods developed by the Team for Advanced Flow Simulation and Modeling (T*AFSM) [http://www.mems.rice.edu/TAFSM/]. The paper also provides many examples of three-dimensional flow simulations carried out with these CFD methods and advanced parallel supercomputers. The methods and tools described in this paper include: stabilized finite element formulations; formulations for flows with moving boundaries and interfaces; mesh update methods; iterative solution techniques for large nonlinear equation systems; and parallel implementation of these methods. Our target is to be able to address effectively certain classes of flow simulation problems. These include: unsteady flows with interfaces; fluid–object interactions; fluid–structure interactions; airdrop systems; aerodynamics of complex shapes; and contaminant dispersion.

A unified approach to computing compressible and incompressible flows is proposed. The governing equation for pressure is selected based on the local Mach number. In the incompressible limit the divergence-free constraint on velocity field determines the pressure, while it is the equation of state that governs the pressure solution for the compressible flows. Stabilized finite element formulations, based on the space-time and semi-discrete methods, with the ‘augmented’ conservation variables are employed. The ‘augmented’ conservation variables consist of the usual conservation variables and pressure as an additional variable. The formulation is applied to various test problems involving steady and unsteady flows over a large range of Mach and Reynolds numbers.

We present the Enhanced-Discretization Interface-Capturing Technique (EDICT) for computation of unsteady flow problems with interfaces, such as two-fluid and free-surface flows. In EDICT, we solve, over a non-moving mesh, the Navier-Stokes equations together with an advection equation governing the evolution of an interface function with two distinct values identifying the two fluids. The starting point for the spatial discretization of these equations are the stabilized finite element formulations which possess good stability and accuracy properties. To increase the accuracy in modeling the interfaces, we use finite element functions corresponding to enhanced discretization at and near the interface. These functions are designed to have multiple components, with each component coming from a different level of mesh refinement over the same computational domain. The primary component of the functions for velocity and pressure comes from the base mesh called Mesh-1. A subset of the elements in Mesh-1 are identified to be at or near the interface, and depending on where the interface is, this subset could change from one time level to another. A Mesh-2 is constructed by patching together the second-level meshes generated over this subset of elements, and the second component of the functions for velocity and pressure comes from Mesh-2. For the interface function, we have a third component coming from a Mesh-3 which is constructed by patching together the third-level meshes generated over a subset of elements in Mesh-2. With parallel computation of the test problems presented here, we demonstrate that the EDICT can be used very effectively to increase the accuracy of the base finite element formulations.

We present parallel computation of 3D, unsteady, incompressible flow past a sphere. The Navier-Stokes equations of incompressible flows are solved using a stabilized finite element formulation. Equal-order interpolation functions are used for velocity and pressure. The second-order accurate time-marching within the solution process is carried out in an implicit fashion. The coupled, nonlinear equations generated at each time step are solved using an element-vector-based iteration technique. The computed value of the primary frequency associated with vortex shedding is in close agreement with experimental measurements. The computation was performed on the Thinking Machines CM-5.

In this paper we describe parallel computational methods for 3D simulation of the dynamics and fluid dynamics of a parafoil with prescribed, time-dependent shape changes. The mathematical model is based on the time-dependent, 3D Navier-Stokes equations governing the incompressible flow around the parafoil and Newton's law of motion governing the dynamics of the parafoil, with the aerodynamic forces acting on the parafoil calculated from the flow field. The computational methods developed for these 3D simulations include a stabilized space-time finite element formulation to accommodate for the shape changes, special mesh generation and mesh moving strategies developed for this purpose, iterative solution techniques for the large, coupled nonlinear equation systems involved, and parallel implementation of all these methods on scalable computing systems such as the Thinking Machines CM-5. As an example, we report 3D simulation of a flare maneuver in which the parafoil velocity is reduced by pulling down the flaps. This simulation requires solution of over 3.6 million coupled, nonlinear equations at every time step of the simulation.

In this paper, shared-memory parallel implementations of a finite element formulation for unsteady interior flows with fluid-structure interactions are presented. The parallel computing platforms targeted are the CRAY C90, the Silicon Graphics (SGI) ONYX and the SGI Power Challenge. The formulation is based on the stabilized space-time finite element method developed earlier for a more general class of flow problems involving moving boundaries and interfaces. The specific test problem used in the performance evaluations involves fluid-structure interactions between a barotropic working fluid and one of the two pistons surrounding this fluid. We demonstrate that advanced formulations applicable to complex problems can be implemented in a parallel computing environment without resulting in a significant distraction from the scientific objectives of solving such complex problems.

In this article we present parallel 3D finite element computation of unsteady incompressible flows around circular cylinders. We employ stabilized finite element formulations to solve the Navier-Stokes equations on a thinking machine CM-5 supercomputer. The time integration is based on an implicit method, and the coupled, nonlinear equations generated every time step are solved iteratively, with an element-vector based evaluation technique. This strategy enables us to carry out these computations with millions of coupled, nonlinear equations, and thus resolve the flow features in great detail. At Reynolds number 300 and 800, our results indicate strong 3D features arising from the instability of the columnar vortices forming the Karman street. At Re = 10 000 we employ a large eddy simulation (LES) turbulence model.

A high performance computing research tool has been developed for 3D simulation of fluid-particle interactions with the number of particles reaching 100. The tool is based on a stabilized space-time finite element formulation for moving boundaries and interfaces and parallel computing. Other components of this tool include: fast automatic mesh generation with structured layers of elements around the particles and unstructured meshes elsewhere; an automatic mesh moving method combined with remeshing as needed; accurate and efficient projection of the solution between the old and new meshes after each remesh; surface mesh refinement as two spheres or a sphere and the tube wall get close; and multi-platform computing. We apply this tool to the simulation of two cases involving 101 spheres falling in a liquid-filled tube. In both cases the initial distribution of the spheres in the tube is random. In the first simulation the size of the spheres is also random, whereas in the second case it is uniform. We demonstrate that the tool developed can be used for simulation of this class of problems with computing durations kept at acceptable levels.

A new 3D finite element flow simulation capability for fluid-particle interactions is presented and applied to study time-dependent behavior of multiple spheres falling in a liquid-fluid tube. This capability is based on the flow simulation strategies such as stabilized space-time formulation for moving boundaries and interfaces, automatic mesh generation with structured layers of elements around the spheres, automatic mesh moving with remesh only as needed, and the implementation of these strategies on massively parallel computing platforms. Several cases of multiple spheres falling in a liquid-filled tube are studied, with the number of spheres ranging from two to five. In all cases, depending on the number of spheres and their initial arrangement, a stable state is eventually reached with all spheres arranged in a pattern corresponding to that stable state, and with all of them falling with the same terminal velocity.