Subdivision schemes for fluid flow

IEEE Transactions on Visualization and Computer Graphics, 2000

We present, extend, and apply a method to extract the contribution of a subregion of a data set to the global flow. To isolate this contribution, we decompose the flow in the subregion into a potential flow that is induced by the original flow on the boundary and a localized flow. The localized flow is obtained by subtracting the potential flow from the original flow. Since the potential flow is free of both divergence and rotation, the localized flow retains the original features and captures the region-specific flow that contains the local contribution of the considered subdomain to the global flow. In the remainder of the paper, we describe an implementation on unstructured grids in both two and three dimensions for steady and unsteady flow fields. We discuss the application of some widely used feature extraction methods on the localized flow and describe applications like reverse-flow detection using the potential flow. Finally, we show that our algorithm is robust and scalable by applying it to various flow data sets and giving performance figures.

Computer Aided Geometric Design, 2005

Curve subdivision schemes on manifolds and in Lie groups are constructed from linear subdivision schemes by first representing the rules of affinely invariant linear schemes in terms of repeated affine averages, and then replacing the operation of affine average either by a geodesic average (in the Riemannian sense or in a certain Lie group sense), or by projection of the affine averages onto a surface. The analysis of these schemes is based on their proximity to the linear schemes which they are derived from. We verify that a linear scheme S and its analogous nonlinear scheme T satisfy a proximity condition. We further show that the proximity condition implies the convergence of T and continuity of its limit curves, if S has the same property, and if the distances of consecutive points of the initial control polygon are small enough. Moreover, if S satisfies a smoothness condition which is sufficient for its limit curves to be C 1 , and if T is convergent, then the curves generated by T are also C 1. Similar analysis of C 2 smoothness is postponed to a forthcoming paper.

Computer Graphics Forum, 2012

We present the first general scheme to describe all four types of characteristic curves of flow fields-stream, path, streak, and time lines-as tangent curves of a derived vector field. Thus, all these lines can be obtained by a simple integration of an autonomous ODE system. Our approach draws on the principal ideas of the recently introduced tangent curve description of streak lines. We provide the first description of time lines as tangent curves of a derived vector field, which could previously only be constructed in a geometric manner. Furthermore, our scheme gives rise to new types of curves. In particular, we introduce advected stream lines as a parameterfree variant of the time line metaphor. With our novel mathematical description of characteristic curves, a large number of feature extraction and analysis tools becomes available for all types of characteristic curves, which were previously only available for stream and path lines. We will highlight some of these possible applications including the computation of time line curvature fields and the extraction of cores of swirling advected stream lines.

Journal of Computational Physics, 2001

A Cartesian grid method for computing flows with complex immersed, moving boundaries is presented. The flow is computed on a fixed Cartesian mesh and the solid boundaries are allowed to move freely through the mesh. A mixed Eulerian-Lagrangian framework is employed, which allows us to treat the immersed moving boundary as a sharp interface. The incompressible Navier-Stokes equations are discretized using a second-order-accurate finite-volume technique, and a second-orderaccurate fractional-step scheme is employed for time advancement. The fractionalstep method and associated boundary conditions are formulated in a manner that properly accounts for the boundary motion. A unique problem with sharp interface methods is the temporal discretization of what are termed "freshly cleared" cells, i.e., cells that are inside the solid at one time step and emerge into the fluid at the next time step. A simple and consistent remedy for this problem is also presented. The solution of the pressure Poisson equation is usually the most time-consuming step in a fractional step scheme and this is even more so for moving boundary problems where the flow domain changes constantly. A multigrid method is presented and is shown to accelerate the convergence significantly even in the presence of complex immersed boundaries. The methodology is validated by comparing it with experimental data on two cases: (1) the flow in a channel with a moving indentation on one wall and (2) vortex shedding from a cylinder oscillating in a uniform free-stream. Finally, the application of the current method to a more complicated moving boundary situation is also demonstrated by computing the flow inside a diaphragm-driven micropump with moving valves.

: This work presents a geometrical domain decomposition (DD) strategy for the finite element solution of incompressible fluid problems involving moving subregions. The DD algorithm is based on the iterative update of the boundary conditions on the interfaces between the subregions, known as iteration-by-subdomain method. Some subregions may be in relative motion. Therefore, in order to treat the constant change of the interfaces configuration, an embedding technique is used together with a Dirichlet/Neumann method. This method enables to couple di#erent flow regions, each one running on separate processes, without major alteration of the original fluid code. The coupling is performed by a master program which coordinates the processes and passes the data back and forth to the slaves. In the first section, the governing equations of the fluid solver are introduced. The following section presents theoretical as well as implementation aspects of the DD algorithm. Finally, the last sect...

We link regularity and smoothness analysis of multivariate vector subdivision schemes with network flow theory and with special linear optimization problems. This connection allows us to prove the existence of what we call optimal difference masks that posses crucial properties unifying the regularity analysis of univariate and multivariate subdivision schemes. We also provide efficient optimization algorithms for construction of such optimal masks. Integrality of the corresponding optimal values leads to purely analytic proofs of $C^k-$regularity of subdivision.

IEEE Transactions on Visualization and Computer Graphics, 2007

We present a new fluid simulation technique that significantly reduces the non-physical dissipation of velocity. The proposed method is based on an apt use of particles and derivative information. We note that a major source of numerical dissipation in the conventional Navier-Stokes equations solver lies in the advection step. Hence, starting with the conventional grid-based simulator, when the details of fluid movements need to be simulated, we replace the advection part with a particle simulator. When swapping between the grid-based and particle-based simulators, the physical quantities such as the level set and velocity must be converted. For this purpose, we develop a novel dissipation-suppressing conversion procedure that utilizes the derivative information stored in the particles as well as in the grid points. For the fluid regions where such details are not needed, the advection is simulated using an octree-based constrained interpolation profile (CIP) solver, which we develop in this work. Through several experiments, we show that the proposed technique can reproduce the detailed movements of high-Reynoldsnumber fluids, such as droplets/bubbles, thin water sheets, and whirlpools. The increased accuracy in the advection, which forms the basis of the proposed technique, can also be used to produce better results in larger scale fluid simulations.

Mathematical and Computer Modelling

A numerical model for the simulation of three-dimensional liquid–gas flow with free surfaces is presented. The incompressible Navier–Stokes equations are assumed to hold in the liquid domain, while the surrounding gas is assumed to be compressible, with constant pressure in each bubble of gas. An implicit time splitting scheme couples a method of characteristics for the solution of advection problems, the continuum surface force model for the computation of surface tension effects, and an implicit scheme for the solution of a time dependent Stokes problem. A two-grid method that couples a structured grid of small cells and a finite element mesh of tetrahedrons is used. A novel interface tracking technique involving local adaptive mesh refinement around the interface is detailed to obtain a more accurate approximation of the free surfaces and the surface forces. Numerical experiments, including sloshing and oscillations problems, illustrate the accuracy improvement when using the adaptive Eulerian grid subdivision.

International Journal for Numerical Methods in Fluids, 2008

This article introduces a new semi-implicit, staggered finite volume scheme on unstructured meshes for the modelling of rapidly varied shallow water flows. Rapidly varied flows occur in the inundation of dry land during flooding situations. They typically involve bores and hydraulic jumps after obstacles such as road banks. Near such sudden flow transitions, the grid resolution is often low compared with the gradients of the bathymetry. Locally the hydrostatic pressure assumption may become invalid. In these situations, it is crucial to apply the correct conservation properties to obtain accurate results. An important feature of this scheme is therefore its ability to conserve momentum locally or, by choice, preserve constant energy head along a streamline. This is achieved using a special interpolation method and control volumes for momentum.

Computing and Visualization in Science, 2000

Vector field visualization is an important topic in scientific visualization. The aim is to graphically represent field data in an intuitively understandable and precise way. Two novel methods are described which enable an easy perception of flow data. The texture transport method especially applies to timedependent velocity fields. Lagrangian coordinates are computed solving the corresponding linear transport equations numerically. Choosing an appropriate texture on the reference frame the coordinate mapping can be applied as a suitable texture mapping. Alternatively, the aligned diffusion methods serves as an appropriate scale space method for the visualization of complicated flow patterns. It is closely related to nonlinear diffusion methods in image analysis where images are smoothed while still retaining and enhancing edges. Here an initial noisy image is smoothed along streamlines, whereas the image is sharpened in the orthogonal direction. The two methods have in common that they are based on a continuous model and discretized only in the final implementational step. Therefore, many important properties are naturally established already in the continuous model.

Journal of computational and applied …, 2006

2007

In this work we present a technique of fast numerical computation for solutions of Navier-Stokes equations in the case of flows of industrial interest. At first the partial differential equations are translated into a set of nonlinear ordinary differential equations using the geometrical shape of the domain where the flow is developing, then these ODEs are numerically resolved using a set of computations distributed among the available processors. We present some results from simulations on a parallel hardware architecture using native multithreads software and simulating a shared-memory or a distributed-memory environment.

We introduce a pure-streamfunction formulation for the incompressible Navier-Stokes equations. The idea is to replace the vorticity in the vorticity-streamfunction evolution equation by the Laplacianof the streamfunction. The resulting formulation includes the streamfunction only, thus no inter-function relations need to invoked. A compact numerical scheme, which interpolates streamfunction values as well as its first order derivatives, is presented and analyzed.

Lecture Notes in Computational Science and Engineering, 2010

A discrete version of the pure streamfunction formulation of the Navier-Stokes equation is presented. The proposed scheme is fourth order in both two and three spatial dimensions. 1 Fourth order scheme for the Navier-Stokes equations in two dimensions We consider the Navier-Stokes equations in pure streamfunction form, which in the two-dimensional case leads to the scalar equation ∂ t ∆ ψ + ∇ ⊥ ψ • ∇∆ ψ − ν∆ 2 ψ = f (x, y,t), ψ(x, y,t) = ψ 0 (x, y). (1) Recall that ∇ ⊥ ψ = (−∂ y ψ, ∂ x ψ) is the velocity vector. The no-slip boundary condition associated with this formulation is ψ = ∂ ψ ∂ n = 0 , (x, y) ∈ ∂ Ω , t > 0 (2)

1999