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The methodology previously proposed by the authors to solve particle-laden turbulent flows through a multiscale approach is extended by introducing a continuous function for thedispersed phaseconcentration. The proposed continuous model is especially useful for studying the motion of particle streams in which gravitational and inertial effects cause the particles to deviate from a simple trajectory following the surrounding flow, as would be the case for the limit of very small, massless particles. The results show an excellent comparison between the solutions obtained using the continuous model and simulations evaluating the forces on each particle individually. Distinct advantages of the continuous approach are a much lower computational overhead, a better load balance and ease ofparallelization. The multiscale methodology proposed may be used in the area ofmodeling and simulationof airborne infectious diseases.

An overview of high-fidelity modeling of pathogen propagation, transmission and mitigation in the built environment is given. In order to derive the required physical and numerical models, the current understanding of pathogen, and in particular virus transmission and mitigation is summarized. The ordinary and partial differential equations that describe the flow, the particles and possibly the UV radiation loads in rooms or HVAC ducts are presented, as well as proper numerical methods to solve them in an expedient way. Thereafter, the motion of pedestrians, as well as proper ways to couple computational fluid dynamics and computational crowd dynamics to enable high-fidelity pathogen transmission and infection simulations is treated. The present review shows that high-fidelity simulations of pathogen propagation, transmission and mitigation in the built environment have reached a high degree of sophistication, offering a quantum leap in accuracy from simpler probabilistic models. This is particularly the case when considering the propagation of pathogens via aerosols in the presence of moving pedestrians.

Computational fluid dynamics (CFD) models of the carotid artery are constructed from contrast-enhanced magnetic resonance angiography (MRA) using a deformable model and a surface-merging algorithm. Physiologic flow conditions are obtained from cine phase-contrast MRA at two slice locations below and above the carotid bifurcation. The methodology was tested on image data from a rigid flow-through phantom of a carotid artery with 65% degree stenosis. Predicted flow patterns are in good agreement with MR flow measurements at intermediate slice locations. Our results show that flow in a rigid flow-through phantom of the carotid bifurcation with stenosis can be simulated accurately with CFD. The methodology was then tested on flow and anatomical data from a normal human subject. The sum of the instantaneous flows measured at the internal and external carotids differs from that at the common carotid, indicating that wall compliance must be modeled. Coupled fluid-structure calculations were able to reproduce the significant dampening of the velocity waveform observed between different slices along the common carotid artery. Visualizations of the blood flow in a compliant model of the carotid bifurcation were produced. A comparison between compliant and rigid models shows significant differences in the time-dependent wall shear stress at selected locations. Our results confirm that image-based CFD techniques can be applied to the modeling of hemodynamics in compliant carotid arteries. These capabilities may eventually allow physicians to enhance current image-based diagnosis, and to predict and evaluate the outcome of interventional procedures non- invasively.

A multigrid algorithm for implementation on unstructured meshes is proposed. The algorithm uses a sequence of unnested grids and requires the development of efficient inter-grid interpolation procedures. It is demonstrated how elliptic problems can be solved in this fashion by using Jacobi smoothers

A new heat transfer simulation capability is described. Non-traditional features of this capability include: a seamless link to CAD-CAM for rapid problem specification/description, integrated automatic grid generator tools for rapid mesh generation, nonlinear and/or varying material properties, source-terms and boundary conditions, a one-element type approach for simplicity and efficiency, automatic self-adaptive mesh refinement and coarsening with accurate error estimation, heavy reliance on iterative solvers, and on-line display on workstations for immediate visualization and user feedback. These innovations are documented on several examples that demonstrate the usefulness of the developed capability.

A transient two-dimensional, finite elemnt shock capturing scheme on unstructured grids was applied to the study of a shock interacting with a box suspended above a rigid elevated surface. The area between the box and the surface was partially blocked by the box support beams, resulting in complex shock diffraction processes. The results demonstrate the capability of the developed adaptive refinement/corsening algorithm to properly adapt to weak shocks, expansions, and contact discontinuities, and highlight the resulting excellent resolution of the captured flow features. In addition to interesting shock diffraction and propagation phenomena, the results demonstrate the capability of the new code to capture, and define in great detail, vortex sheets shed from sharp corners. We show that the baroclinic effect, an inviscid process, controls the shedding phenomenon during the diffraction phase. Hence, the Eulerian model is able to correctly predict the process.

This paper describes a Finite Element Method - Flux Corrected Transport (FEM-FCT) approach with an unstructured adaptive grid scheme to the simulation of two-phase flow. The gas equations are computed using an Eulerian frame of reference, while the particle transport is computed using both Eulerian and Lagrangian frames of reference. One dimensional shock wave attenuation was investigated to evaluate the performance of these methodologies and the gas-solid interphase transport models

In previous papers [1,2] we have described an explicit finite element solution procedure for the compressible Euler and Navier-Stokes equations. The approach was a finite element equivalent of a two-step Lax-Wendroff scheme and was implemented on unstructured triangular or tetrahedral grids. An important feature of the work was the use of adaptive mesh refinement methods for the solution of steady state problems in two dimensions, using error indicators based upon interpolation theory. &nbsp

Several data structures for the generation of unstructured grids are described. Their usefulness stems from the fact that they enable the necessary search operations to be performed in an optimal way. In particular, we describe heap lists, quad- and octrees, and linked lists. Combining these data structures, the important problem of interpolating information between unstructured grids is also solved.

This paper describes an extensive comparison study of two high-order-accuracy finite-element schemes: one of upwind type, the other of flux-corrected transport type, for the simulation of shock wave propagation on unstructured meshes. The performance of these two schemes for the solution of the Euler equations on unstructured meshes is compared on the basis of solution accuracy and computational efficiency for some well-known test cases. Advantages and disadvantages of the two schemes are discussed.

A three-dimensional space-marching algorithm using an unstructured discretization is proposed. The method utilizes a two-dimensional unstructured grid generator to construct grids in a crossflow plane while maintaining structure in the marching direction. The spatial discretization is obtained by applying a characteristic-based, upwind, finite volume scheme for the solution of the Euler equations. Solutions are presented for several different geometries comparing the results with existing numerical techniques and experiment.

The current trends for the simulation of large-scale incompressible flow fields using finite elements are discussed. The main items are: a) the use of simple elements through stabilization and analogy with LBB-satisfying elements, and b) the development of fast solvers for general grids.

A recently developed transient, two-dimensional, finite-element shock capturing scheme is applied to the study of shock interaction with a box suspended above a rigid surface. The results demonstrate the excellent resolution of the captured shocks and the application of a newly developed adaptive refinement/coarsening algorithm. In addition to interesting shock propagation and interaction processes, the results demonstrate the capability of the new code to capture, and define in great detail, vortices shed from the downstream corners of the box. The ability of the new scheme to model such fluid dynamic phenomena as shock-shock, shock-surface, and shock-vortex interactions, as well as vortex dynamics, are demonstrated.

A vectorized particle tracer for unstructured grids is described. The basic approach is to use elementary properties of the linear basis functions to search for particles on the grid using the element last occupied as an initial guess. To permit vectorization, a simple binary sort of the particles is performed every timestep such that all particles that have as yet not found their host element remain at the top of the list. In this way, vector-loops can be easily formed. Timings taken from a numerical example indicate that speed-ups of the order of 1:14 can be obtained on vector-machines when using this algorithm.

The current status of CFD with regard to unstructured grids employing finite element methods and Eulerian frames is reviewed. Algorithms suitable for the computation of large three-dimensional problems involving flow past arbitrary geometries are developed. Adaptive mesh refinement strategy is reviewed, and domain splitting or local time-stepping are briefly addressed. The development of search algorithms of optimal order, variable time-stepping Jacobi smoothers for elliptic problems, and transport concepts for hyperbolics to help achieve good performance for unstructured multigrid processes is discussed. As examples, transient supersonic flow in a channel, regular shock reflection of a wall, viscous flow past a protruberance, potential flow past a cylinder, and Burgers equation are considered.

The difficulties experienced in the treatment of hyperbolic systems of equations by the finite element method (or other) spatial discretization procedures are well known. In this paper a temporal discretization precedes the spatial one which in principle is considered along the characteristics to achieve a self adjoint form. By a suitable expansion, the original co‐ordinates are preserved and combined with the use of a standard Galerkin process to achieve an accurate discretization. It is shown that the process is equivalent to the Taylor‐Galerkin methods of Donea.17 Several examples illustrate the accuracy and efficiency attainable in such problems as transport, shallow water equations, transonic flow etc.

Traditional techniques for computing electromagnetic solutions in the time domain rely on finite differences. These so-called finite-difference time-domain (FDTD) methods are usually defined only on regular lattices of points and can be too restrictive for geometrically demanding problems. Great geometric flexibility can be achieved by abandoning the regular latticework of sample points and adopting an unstructured grid. An unstructured grid allows one to place the grid points anywhere one chooses, so that curved boundaries can be fit with ease and local regions in which the field gradients are steep can be selectively resolved with a fine mesh. In this paper we present a technique for solving Maxwell's equations on an unstructured grid based on the Taylor-Galerkin finite-element method. We present several numerical examples which reveal the fundamental accuracy and adaptability of the method. Although our examples are in two dimensions, the techniques and results generalize readily to 3D.

This paper describes the development, validation, and application of a new finite element scheme for the solution of the compressible Euler equations on unstructured grids. The implementation of the numerical scheme is based on an edge-based data structure, as opposed to a more element-based data structure. The use of this edge-based data structure not only improves the efficiency of the algorithm but also enables a straightforward implementation of the upwind schemes in the context of finite element methods. The algorithm has been tested and validated on some well documented configurations. A flow solution about a complete F-18 fighter is shown to demonstrate the accuracy and robustness of the proposed algorithm.

An algorithm for the parallel generation of 3-D unstructured grids is presented. The technique is an extension of the algorithm presented in Reference 21 for the 2-D case. The method uses a background grid as the means to separate spatially different regions, enabling the concurrent, parallel generation of elements in different domains and interdomain regions. The parallel 3-D grid generator was implemented and tested on the INTEL hypercube and Touchstone Delta parallel computers. The results obtained demonstrate the effectiveness of the algorithm developed. The methodology is applicable to the parallel implementation of a wide range of problems that are, in principle, scalar by nature, and do not lend themselves to SIMD parallelization.

A new ray‐tracing technique is presented which does not work with ray–object intersectionsper se, but is based on the traversal of an unstructured tetrahedral mesh providing a convex enclosure of a scene of polyhedral objects. The tetrahedral mesh provides tight bounding and an adaptive subdivision of space. This non‐hierarchical data structure is traversed adaptively until one is led directly and unconditionally to the first object intersected. Rendering times are directly related to the average thickness of the enclosing mesh since all tetrahedra are traversed in constant time. Since the proposed algorithm operates directly with volume elements, it allows for volumetric rendering effects. Volume rendering or anisotropic media can be implemented without any further effort. This is an important advantage as compared to usual techniques, which only operate on surface data. Timings for several examples show that the use of this type of ray‐tracing technique, which is more suitable for general purpose visualization codes than traditional techniques, results in CPU times that are comparable with the best ray‐tracing techniques presently used. This is an unexpected and important result, as the vectorization and parallclization of the proposed technique are straightforward, in contrast with traditional ray‐tracing techniques

Numerical methods for the solution of field problems using unstructured grids have reached a high degree of maturity. Although computational fluid dynamics (CFD) have been dominated by structured grids, with the increase in computational ability, unstructured grids have started to have an impact on complex geometrical problems in CFD. The following paper describes grid generation, adaptive refinement schemes, visualization and parallelization issues. It also includes a short chapter on CFD among related disciplines that will help the newcomer with flow simulation codes

There is an increasing demand to use kinetic plasma simulations to model real plasma devices such as e-beam diodes, plasma torches, opening switches, and so on. With this demands comes the realization that standard simulation tools based on rectangular structured meshes are often too inflexible to accommodate real device geometries. To gain geometric flexibility, one typically must abandon structured rectangular meshes. Some have had considerable success by distorting the grid to conform to boundary-fitted curvilinear coordinates. This gives substantial flexibility while retaining a logically rectangular data structure. However, to achieve the greatest flexibility, an unstructured mesh is necessary. For the last several years, we have been engaged in designing electromagnetic particle-in-cell (PIC) simulations that can be performed on unstructured grids of triangles. We feel this is a powerful strategy for matching kinetic plasma simulations to real problem geometries. Unstructured meshes not only accommodate complicated boundary shapes with ease, but also allow extreme local refinement without affecting resolution elsewhere. The basic challenges to overcome in formulating EM-PIC on these new meshes are: (1) grid generation; (2) particle interpolation and tracking, and (3) field solution. In this paper we briefly describe our techniques and give a simple example.

An adaptive finite element scheme for transient problems is presented. The classic h‐enrichment/coarsening is employed in conjunction with a tetrahedral finite element discretization in three dimensions. A mesh change is performed every n time steps, depending on the Courant number employed and the number of ‘protective layers’ added ahead of the refined region. In order to simplify the refinement/coarsening logic and to be as fast as possible, only one level of refinement/coarsening is allowed per mesh change. A high degree of vectorizability has been achieved by pre‐sorting the elements and then performing the refinement/coarsening groupwise according to the case at hand. Further reductions in CPU requirements arc realized by optimizing the identification and sorting of elements for refinement and deletion. The developed technology has been used extensively for shock‐shock and shock‐object interaction runs in a production mode. A typical example of this class of problems is given.

The passage of planar shocks in a dusty gas was investigated to note effects due to particle loading and initial shock Mach number. Two-phase flow equations have been added to a conservative, monotonic flow solver to allow study of compressible particle and droplet flows, which are of importance for shock propagation in two-phase flows and spray propulsion systems. The formulation developed herein employed a conservative Eulerian treatment for the gas and particle phases. The computations were performed using the finite element method-flux corrected transport (FEM-FCT) scheme, which has shown excellent predictive capability of various compressible flows which include both strong and weak shocks. The flux limiting technique was modified to provide monotonic particle velocity fields to increase the scheme's computational stability. Adaptive unstructured methodology based on adapting to high gradients of both the fluid and particle densities was used in conjunction with the conservative shock-capturing scheme to adequately resolve strong flowfield gradients. The shock attenuation of this scheme was then compared with previous experimental and numerical results and was found to yield robust predictions. Various interphase coupling terms were also considered to note their effect on the shock attenuation.

Only recently has a capability for computing three-dimensional, unsteady incompressible flow over a changing geometry become available. In that work, the flow field generated during a torpedo launch from a submarine was computed. This paper describes the extension of that work to carry out a direct computation, making no geometric simplifications, of the unsteady flow past a tuna, including the oscillatory caudal fin motion. In particular, described is the computation of the time variation of the pressure distribution over the entire body and the integration over all surfaces to obtain the unsteady thrust. &nbsp

An incompressible flow solver based on unstructured grids is implemented on parallel distributed-memory computer architecture. An important phase in the flow solver is the solution of the elliptic equations for the velocities and pressures. This elliptic solver is parallelized and incorporated into both the explicit and implicit versions of the incompressible flow solver. Performance and scalability studies are carried out on both Intel iPSC 860 and the Intel Delta prototype and show that the code is scalable. A parallelizable load balancing algorithm is developed to be used in conjunction with the incompressible flow solver. Steady and unsteady flows over a tri-element airfoil and NACA0012airfoil are computed using the parallel incompressible flow solver. The parallel flow solver is extended to 3D for simulation of flow past a fully appended submarine. The chapter concludes that the future developments will be focused on the incorporation of adaptive regridding in 2D and the parallel implementation of 3-D incompressible flow solver.

A finite element MHD algorithm is used to simulate a two-dimensional, viscous and resistive turbulent model, namely the Orszag-Tang vortex. The results are compared to a pseudo-spectral simulation of the same system reported by Dahlburg and Picone (Phys. Fluids B 1 (1989) 2153). The agreement of results from both methods supports the contention that the finite element method can appropriately simulate systems exhibiting turbulence, thus enabling the use of realistic geometries and boundary conditions, as well as adaptive refinement on simulations of turbulent systems. A short discussion on the behavior of ▿·Bis presented. An inverse correlation between spatial resolution and the magnitude of ▿·Bwas found. The relevance of our findings to Adaptive Mesh Refinement is briefly discussed.

This paper describes the application of FEFL096, a three-dimensional, adaptive, finite element, edge-based, ALE shock capturing methodology on unstructured tetrahedral grids, to large-scale simulations of blast wave interaction with structures. The first simulation applied the CFD methodology to the numerical simulation of blast wave diffraction within the B-2 level of the World Trade Center garage. This simulation modeled blast wave diffraction about hundreds of rigidly-modeled structures, spread over a large area. The second simulation applied a new loose-coupling algorithm that combined FEFL096 and DYNA3D, a state-of-the-art Computational Structural Dynamics (CSD) methodology, to the simulation of shock interaction with a structurally-responding truck. &nbsp

Problems related to automatic generation of highly stretched unstructured grids suitable for 3‐D Reynolds‐averaged Navier–Stokes computations are addressed. Special attention is given to treatment of such geometrical irregularities as convex and concave ridges as well as corners where the ridges meet. The existing unstructured grid generation approaches may fail or produce poor quality meshes in such geometrical regions. The proposed solution is based on special meshing of non‐slip body surfaces resulting in smooth and robust volume meshing and high overall quality of generated grids. Several examples demonstrate the efficiency of the method for complex 3‐D geometries.

A simple CFD tool, coupled to a discrete surface representation and a gradient-based optimization procedure, is applied to the design of optimal hull forms and optimal arrangement of hulls for a wave cancellation multihull ship. The CFD tool, which is used to estimate the wave drag, is based on the zeroth-order slender ship approximation. The hull surface is represented by a triangulation, and almost every grid point on the surface can be used as a design variable. A smooth surface is obtained via a simplified pseudo-shell problem. The optimal design process consists of two steps. The optimal center and outer hull forms are determined independently in the first step, where each hull keeps the same displacement as the original design while the wave drag is minimized. The optimal outer-hull arrangement is determined in the second step for the optimal center and outer hull forms obtained in the first step. Results indicate that the new design can achieve a large wave drag reduction in comparison to the original design configuration.

Surface mesh generation over intersecting triangulations is a problem common to many branches of biomechanics. A new strategy for merging intersecting triangulations is described. The basis of the method is that object surfaces are represented as the zero-level iso-surface of the distance-to-surface function defined on a background grid. Thus, the triangulation of intersecting objects reduces to the extraction of an iso-surface from an unstructured grid. In a first step, a regular background mesh is constructed. For each point of the background grid, the closest distance to the surface of each object is computed. Background points are then classified as external or internal by checking the direction of the surface normal at the closest location and assigned a positive or negative distance, respectively. Finally, the zero-level iso-surface is constructed. This is the final triangulation of the intersecting objects. The overall accuracy is enhanced by adaptive refinement of the background grid elements. The resulting surface models are used as support surfaces to generate three-dimensional grids for finite element analysis. The algorithms are demonstrated by merging arterial branches independently reconstructed from contrast-enhanced magnetic resonance images and by adding extra features such as vascular stents. Although the methodology is presented in the context of finite element analysis of blood flow, the algorithms are general and can be applied in other areas as well.

A design methodology based on the adjoint approach for flow problems governed by the incompressible Euler equations is presented. The main feature of the algorithm is that it avoids solving the adjoint equations, which saves an important amount of CPU time. Furthermore, the methodology is general in the sense it does not depend on the geometry representation. All the grid points on the surface to be optimized can be chosen as design parameters. In addition, the methodology can be applied to any type of mesh. The partial derivatives of the Row equations with respect to the design parameters are computed by finite differences. In this way, this computation is independent of the numerical scheme employed to obtain the Row solution. Once the design parameters have been updated, the new solid surface is obtained with a pseudo-shell approach in such a way that local singularities, which can degrade or inhibit the convergence to the optimal solution, are avoided. Some 2D and 3D numerical examples are shown to demonstrate the proposed methodology.&nbsp

An unstructured grid matrix-free GMRES+LU-SGS scheme is used to simulate unsteady CFD problems with moving bodies. The method is applied on sharedmemory, cache-based parallel machines. A special grid renumbering technique is used for the parallelization rather than the traditional method of partitioning the computational domain. Moving mesh with recurrent remeshing models the body motion.

Several classes of important engineering problems require the concurrent application of CFD and CSD techniques. Currently, attempts to model these problems are solved either iteratively, requiring several cycles of CFD run followed by CSD run, or by assuming that the CFD and CSD solutions can be decoupled. The various efforts to develop a fluid/structure coupling can be classified according to the complexity level of the approximations used for each of the domains. These range from simple 6 DOF integration to finite elements with complex models for elasto-plastic materials with rupture laws and contact. Similarly, the fluid dynamics approximations range from the potential flow (irotational, inviscid, isentropic flows) to the full Navier-Stokes set of equations. The present research interests focus on non-linear applications, in particular, structures that experience severe deformations due to blast loads. Hence, the fluid applies either the Euler or Reynolds-Averaged Navier-Stokes equations, while elasto-plastic materials with rupture criteria are used for the structural modeling.

Against the general feeling that ‘CFD is solved’, I offer a few remarks from the trenches. When performing large-scale simulations of geometrically and physically complex flows, the notion of reliable, cost-effective, widely applicable and easy-to-use CFD still appears more as fiction than reality. The present monograph addresses some major areas where progress is urgently required. These areas are listed in the order a typical CFD run proceeds: model import/creation, grid generation, flow solvers, mesh adaptation and quality control. Finally, with a view towards the coming decade, code, data and project management are discussed.

A point renumbering scheme that halves the number of indirect addressing operations required for edge‐based field solvers has been developed. At the same time, an alternative (and entirely serendipitous) assembly of right‐hand side vectors was found to enhance performance considerably. The new loop structure is especially attractive for vector‐parallel machines with direct memory access. Timings on the CRAY‐SV1 and NEC‐SX5 show that for Laplacian loops, which appear in many CFD and CEM codes, CPU can be reduced considerably. The new renumbering scheme does not require any new data structures or arrays, allowing for progressive porting of loops in large‐scale production codes.

A procedure for the generation of highly stretched grids suitable for Reynolds‐averaged Navier–Stokes (RANS) calculations is presented. In a first stage, an isotropic (Euler) mesh is generated. In a second stage, this grid is successively enriched with points in order to achieve highly stretched elements. The element reconnection is carried out using a constrained Delaunay approach. Points are introduced from the regions of lowest stretching towards the regions of highest stretching. The procedure has the advantages of not requiring any type of surface recovery, not requiring extra passes or work to mesh concave ridges/corners, and guarantees a final mesh, an essential requirement for industrial environments. Given that point placement and element quality are highly dependent for the Delaunay procedure, special procedures were developed in order to obtain optimal point placement.