Grid Vertices Research Articles

A two-step finite volume method for the simulation of multiphase fluid flow in heterogeneous and anisotropic reservoirs

Efficient and accurate simulation of multiphase flow and transport in porous media plays a key role in optimizing recovery of hydrocarbon resources. Two-Point Flux Approximation (TPFA) is widely used in the industry for discretizing the pressure equation because of its robustness and efficiency despite the fact that it introduces O(1) flux errors on non-K-orthogonal grids. Multi-Point Flux Approximation (MPFA) provides a consistent discretization of the flow equations at the cost of larger flux stencils which leads to a denser, asymmetric system of linear equations requiring more computer memory and work for linear solvers. This drawback of MPFA becomes more pronounced for generally unstructured triangular grids and is even worse in 3D applications. In this work, we develop a two-step finite volume method (TSFVM) that is consistent for the anisotropic pressure equation on triangular and tetrahedral grids with improved efficiency as compared to MPFA methods. During each time step, the pressure equation is solved implicitly and saturation is then updated explicitly (IMPES). To discretize the pressure equation with full permeability tensors, Galerkin finite element method (FEM) is employed in the first step to compute pressure at grid vertices. Pressures at grid vertices are then utilized in the second step to derive a two-point flux stencil with a constant for each grid face. Pressure at cell centers and flux across grid faces can then be solved using a TPFA approach. Finally, the flux field at the new time level is used to update saturations explicitly. Although during each time step our TSFVM needs to solve two systems of linear equations, results of the numerical experiments clearly demonstrate that it is still less expensive than the classical MPFA-O method while providing high-quality numerical solutions matching that of MPFA-O. The gained efficiency is especially significant for 3D tetrahedral grid. As the scale of the problem increases, MPFA becomes less appealing because of its large flux stencils that result in much denser matrices while our TSFVM still remains computationally tractable thanks to the flexibility of FEM in the first step and the two-point flux stencil in the second step. On the other hand, the first step of the TSFVM is reminiscent of the Control Volume Finite Element Method (CVFEM) but with important differences. Unlike CVFEM, our TSFVM does not need to construct a dual control volume mesh around grid vertices on the basis of the primary mesh and the saturation unknowns are not associated with node-centered control volumes but are piece-wise constant on grid cells in which reservoir properties are distributed. This ensures that a saturation value does not straddle across material discontinuities which CVFEM suffer from.

An optimized real time algorithm for window frost formation suited to mobile devices

We propose a real time simulation for window frost formation on mobile devices that uses both particles and grids. Previous ice formation methods made heavy demands on both memory and computational capacity because they were designed for a desktop environment. In this paper, a frost skeleton grows around a location touched by the user using particles, and the ice surfaces are constructed using a grid. Using a nonlattice random-walk technique, the frost skeleton grows freely and naturally. A hash grid technique is used to search efficiently for neighbor particles during the crystallization process. Finally, some 2.5D details are added to the ice skeleton by adjusting the height of the grid vertices around the skeleton. Experiments show that our method creates realistic frost in real time. Our method can be used to express ice formation effects in touch-based mobile device applications such as weather forecasts or games.

A GPU-Adapted Structure for Unstructured Grids

A key advantage of working with structured grids e.g., images is the ability to directly tap into the powerful machinery of linear algebra. This is not much so for unstructured grids where intermediate bookkeeping data structures stand in the way. On modern high performance computing hardware, the conventional wisdom behind these intermediate structures is further challenged by costly memory access, and more importantly by prohibitive memory resources on environments such as graphics hardware. In this paper, we bypass this problem by introducing a sparse matrix representation for unstructured grids which not only reduces the memory storage requirements but also cuts down on the bulk of data movement from global storage to the compute units. In order to take full advantage of the proposed representation, we augment ordinary matrix multiplication by means of action maps, local maps which encode the desired interaction between grid vertices. In this way, geometric computations and topological modifications translate into concise linear algebra operations. In our algorithmic formulation, we capitalize on the nature of sparse matrix-vector multiplication which allows avoiding explicit transpose computation and storage. Furthermore, we develop an efficient vectorization to the demanding assembly process of standard graph and finite element matrices.

Low- and high-order accurate boundary conditions: From Stokes to Darcy porous flow modeled with standard and improved Brinkman lattice Boltzmann schemes

The present contribution focuses on the accuracy of reflection-type boundary conditions in the Stokes–Brinkman–Darcy modeling of porous flows solved with the lattice Boltzmann method (LBM), which we operate with the two-relaxation-time (TRT) collision and the Brinkman-force based scheme (BF), called BF-TRT scheme. In parallel, we compare it with the Stokes–Brinkman–Darcy linear finite element method (FEM) where the Dirichlet boundary conditions are enforced on grid vertices. In bulk, both BF-TRT and FEM share the same defect: in their discretization a correction to the modeled Brinkman equation appears, given by the discrete Laplacian of the velocity-proportional resistance force. This correction modifies the effective Brinkman viscosity, playing a crucial role in the triggering of spurious oscillations in the bulk solution. While the exact form of this defect is available in lattice-aligned, straight or diagonal, flows; in arbitrary flow/lattice orientations its approximation is constructed. At boundaries, we verify that such a Brinkman viscosity correction has an even more harmful impact. Already at the first order, it shifts the location of the no-slip wall condition supported by traditional LBM boundary schemes, such as the bounce-back rule.For that reason, this work develops a new class of boundary schemes to prescribe the Dirichlet velocity condition at an arbitrary wall/boundary-node distance and that supports a higher order accuracy in the accommodation of the TRT-Brinkman solutions. For their modeling, we consider the standard BF scheme and its improved version, called IBF; this latter is generalized in this work to suppress or to reduce the viscosity correction in arbitrarily oriented flows. Our framework extends the one- and two-point families of linear and parabolic link-wise boundary schemes, respectively called B-LI and B-MLI, which avoid the interference of the Brinkman viscosity correction in their closure relations.The performance of LBM and FEM is thoroughly evaluated in three benchmark tests, which are run throughout three distinctive permeability regimes. The first configuration is a horizontal porous channel, studied with a symbolic approach, where we construct the exact solutions of FEM and BF/IBF with different boundary schemes. The second problem refers to an inclined porous channel flow, which brings in as new challenge the formation of spurious boundary layers in LBM; that is, numerical artefacts that arise due to a deficient accommodation of the bulk solution by the low-accurate boundary scheme. The third problem considers a porous flow past a periodic square array of solid cylinders, which intensifies the previous two tests with the simulation of a more complex flow pattern. The ensemble of numerical tests provides guidelines on the effect of grid resolution and the TRT free collision parameter over the accuracy and the quality of the velocity field, spanning from Stokes to Darcy permeability regimes. It is shown that, with the use of the high-order accurate boundary schemes, the simple, uniform-mesh-based TRT-LBM formulation can even surpass the accuracy of FEM employing hardworking body-fitted meshes.

Two particle-in-grid realisations on spacetrees

The present paper studies two particle management strategies for dynamically adaptive Cartesian grids at hands of a particle-in-cell code. One holds the particles within the grid cells, the other within the grid vertices. The fundamental challenge for the algorithmic strategies results from the fact that particles may run through the grid without velocity constraints. To facilitate this, we rely on multiscale grid representations. They allow us to lift and drop particles between different spatial resolutions. We call this cell-based strategy particle in tree (PIT). Our second approach assigns particles to vertices describing a dual grid (PIDT) and augments the lifts and drops with multiscale linked cells. Our experiments validate the two schemes at hands of an electrostatic particle-in-cell code by retrieving the dispersion relation of Langmuir waves in a thermal plasma. They reveal that different particle and grid characteristics favour different realisations. The possibility that particles can tunnel through an arbitrary number of grid cells implies that most data is exchanged between neighbouring ranks, while very few data is transferred non-locally. This constraints the scalability as the code potentially has to realise global communication. We show that the merger of an analysed tree grammar with PIDT allows us to predict particle movements among several levels and to skip parts of this global communication a priori. It is capable to outperform several established implementations based upon trees and/or space-filling curves.

Multidimensional Riemann problem with self-similar internal structure. Part II – Application to hyperbolic conservation laws on unstructured meshes

Multidimensional Riemann solvers that have internal sub-structure in the strongly-interacting state have been formulated recently (D.S. Balsara (2012, 2014) [5,16]). Any multidimensional Riemann solver operates at the grid vertices and takes as its input all the states from its surrounding elements. It yields as its output an approximation of the strongly interacting state, as well as the numerical fluxes. The multidimensional Riemann problem produces a self-similar strongly-interacting state which is the result of several one-dimensional Riemann problems interacting with each other. To compute this strongly interacting state and its higher order moments we propose the use of a Galerkin-type formulation to compute the strongly interacting state and its higher order moments in terms of similarity variables. The use of substructure in the Riemann problem reduces numerical dissipation and, therefore, allows a better preservation of flow structures, like contact and shear waves. In this second part of a series of papers we describe how this technique is extended to unstructured triangular meshes. All necessary details for a practical computer code implementation are discussed. In particular, we explicitly present all the issues related to computational geometry. Because these Riemann solvers are Multidimensional and have Self-similar strongly-Interacting states that are obtained by Consistency with the conservation law, we call them MuSIC Riemann solvers. (A video introduction to multidimensional Riemann solvers is available on http://www.nd.edu/~dbalsara/Numerical-PDE-Course.)The MuSIC framework is sufficiently general to handle general nonlinear systems of hyperbolic conservation laws in multiple space dimensions. It can also accommodate all self-similar one-dimensional Riemann solvers and subsequently produces a multidimensional version of the same. In this paper we focus on unstructured triangular meshes. As examples of different systems of conservation laws we consider the Euler equations of compressible gas dynamics as well as the equations of ideal magnetohydrodynamics (MHD). Several stringent test problems are solved for both PDE systems.

Four-dimensional radiotherapeutic dose calculation using biomechanical respiratory motion description

Organ motion due to patient breathing introduces a technical challenge for dosimetry and lung tumor treatment by hadron therapy. Accurate dose distribution estimation requires patient-specific information on tumor position, size, and shape as well as information regarding the material density and stopping power of the media along the beam path. A new 4D dosimetry method was developed, which can be coupled to any motion estimation method. As an illustration, the new method was implemented and tested with a biomechanical model and clinical data. First, an anatomical model of the lung and tumor was synthesized with deformable tetrahedral grids using computed tomography (CT) images. The CT attenuation values were estimated at the grid vertices. Respiratory motion was simulated biomechanically based on nonlinear finite element analysis. Contrary to classical image-based methods where motion is described using deformable image registration algorithms, the dose distribution was accumulated over tetrahedral meshes that are deformed using biomechanical modeling based on finite element analysis. The new method preserves the mass of the objects during simulation with an error between 1.6 and 3.6%. The new method was compared to an existing dose calculation method demonstrating significant differences between the two approaches and overall superior performance using the new method. A unified model of 4D radiotherapy respiratory effects was developed where biomechanical simulations are coupled with dose calculations. Promising results demonstrate that this approach has significant potential for the treatment for moving tumors.

Constructing Isosurfaces with Sharp Edges and Corners using Cube Merging

AbstractA number of papers present algorithms to construct isosurfaces with sharp edges and corners from hermite data, i.e. the exact surface normals at the exact intersection of the surface and grid edges. We discuss some fundamental problems with the previous algorithms and describe a new approach, based on merging grid cubes near sharp edges, that produces significantly better results. Our algorithm requires only gradients at the grid vertices, not at each surface‐edge intersection point. We also give a method for measuring the correctness of the resulting sharp edges and corners in the isosurface.

Coding, stability, and non-spatial inputs in a modular grid-to-place cell model

Grid cells in the medial entorhinal cortex (mEC) and place cells in the hippocampus are paradigms for population coding of spatial information [1]. Both the spatiallyperiodic firing fields of grid cells and the spatially localized firing fields of place cells show systematic increases in spatial scale along the dorso-ventral axes of the mEC and hippocampus, respectively [2,3], which has led to speculation that place field size is determined simply by the spatial scale of a place cell’s grid cell inputs. However, in addition to receiving spatially-modulated entorhinal inputs, place cells receive contextual, non-spatial inputs from sources such as the amygdala and hypothalamus [4], which may be important in determining place cell firing properties. These non-spatial inputs are particularly prominent toward the ventral pole of the hippocampus [4], and thus could also play a role in producing dorso-ventral place cell differences. In order to understand the relative contributions of grid cells and non-spatial inputs in determining place field size and stability, we propose a computational model of the hippocampal-entorhinal network that includes a modular organization of grid cell inputs arranged in order of increasing spatial scale, as is seen experimentally in the mEC. Our underlying place cell model is a firing-rate based model inspired by previous work [5], in which place fields are formed via competition between place cells. We also introduce a dorsoventral gradient in the amount of non-spatial input to place cells, with ventral cells receiving more input from non-spatial sources. Finally, we introduce heterogeneity into the firing rates of grid vertices within individual grid fields. This heterogeneity is observed in experimental recordings [6] but has received relatively little attention in experimental or theoretical work, despite its potential impact on place field stability. Our main findings suggest that: 1.) For a wide range of parameters, the relative amounts of spatial and non-spatial inputs to place cells plays a more important role in determining place field size and stability than the spatial scale of grid cell inputs. This implies that the dorso-ventral gradient in place field size may reflect a dorso-ventral gradient in non-spatial inputs, rather than grid scale, and is agreement with prior suggestions of a functional distinction between the dorsal and ventral regions of the hippocampus [7]. 2.) In our model, place fields are sensitive to changes in the firing rates of the grid vertices of individual grid cells, emphasizing the potential implications of this grid field heterogeneity for place field formation and stability.

Towards augmenting federated wireless sensor networks in forestry applications

Environmental Monitoring (EM) has witnessed significant improvements in recent years due to the great utility of wireless sensor networks (WSNs). Nevertheless, due to harsh operational conditions in such applications, WSNs often suffer large-scale damage in which nodes fail concurrently and the network gets partitioned into disjoint sectors. Thus, reestablishing connectivity between the sectors, via their remaining functional nodes, is of utmost importance in EM, especially in forestry. In this regard, considerable work has been proposed in the literature tackling this problem by deploying Relay Nodes (RNs) aimed at reestablishing connectivity. Although finding the minimum relay count and positions is NP-Hard, efficient heuristic approaches have been anticipated. However, the majority of these approaches ignore the surrounding environment characteristics and the infinite 3-dimensional (3-D) search space that significantly degrades network performance in practice. Therefore, we propose a 3-D grid-based deployment for RNs in which the relays are efficiently placed on grid vertices. We present a novel approach, named fixing augmented network damage intelligently, based on a minimum spanning tree construction to re-connect the disjointed WSN sectors. The performance of the proposed approach is validated and assessed through extensive simulations, and comparisons with two main stream approaches are presented. Our protocol outperforms the related work in terms of the average relay node count and distribution, the scalability of the federated WSNs in large-scale applications, and the robustness of the topologies formed.

Cell-based volume integration for boundary integral analysis

SUMMARY The evaluation of volume integrals that arise in boundary integral formulations for non-homogeneous problems was considered. Using the “Galerkin vector” to represent the Green's function, the volume integral was decomposed into a boundary integral, together with a volume integral wherein the source function was everywhere zero on the boundary. This new volume integral can be evaluated using a regular grid of cells covering the domain, with all cell integrals, including partial cells at the boundary, evaluated by simple linear interpolation of vertex values. For grid vertices that lie close to the boundary, the near-singular integrals were handled by partial analytic integration. The method employed a Galerkin approximation and was presented in terms of the three-dimensional Poisson problem. An axisymmetric formulation was also presented, and in this setting, the solution of a nonlinear problem was considered. Copyright © 2012 John Wiley & Sons, Ltd.

Towards Augmenting Federated Wireless Sensor Networks

Environmental Monitoring (EM) has witnessed significant improvements in recent years due to the great utility of Wireless Sensor Networks (WSNs). Nevertheless, due to harsh operational conditions in such applications, WSNs often suffer large scale damage in which nodes fail concurrently and the network gets partitioned into disjoint sectors. Thus, reestablishing connectivity between the sectors, via their remaining functional nodes, is of utmost importance in EM; especially in forestry. In this regard, considerable work has been proposed in the literature tackling this problem by deploying Relay Nodes (RNs) aimed at re-establishing connectivity. Although finding the minimum relay count and positions is NP-Hard, efficient heuristic approaches have been anticipated. However, the majority of these approaches ignore the surrounding environment characteristics and the infinite 3-Dimensional (3-D) search space which significantly degrades network performance in practice. Therefore, we propose a 3-D grid-based deployment for relay nodes in which the relays are efficiently placed on grid vertices. We present a novel approach, named FADI, based on a minimum spanning tree construction to re-connect the disjointed WSN sectors. The performance of the proposed approach is validated and assessed through extensive simulations, and comparisons with two main stream approaches are presented. Our protocol outperforms the related work in terms of the average relay node count and distribution, the scalability of the federated WSNs in large scale applications, and the robustness of the topologies formed.

A Convergent Finite Volume Scheme for Diffusion on Evolving Surfaces

A finite volume scheme for transport and diffusion problems on evolving hypersurfaces is discussed. The underlying motion is assumed to be described by a fixed, not necessarily normal, velocity field. The ingredients of the numerical method are an approximation of the family of surfaces by a family of interpolating simplicial meshes, where grid vertices move on motion trajectories, a consistent finite volume discretization of the induced transport on the simplices, and a proper incorporation of a diffusive flux balance at simplicial faces. The semi-implicit scheme is derived via a discretization of the underlying conservation law, and discrete counterparts of continuous a priori estimates in this geometric setup are proved. The continuous solution on the continuous family of evolving surfaces is compared to the finite volume solution on the discrete sequence of simplicial surfaces, and convergence of the family of discrete solutions on successively refined meshes is proved under suitable assumptions on the geometry and the discrete meshes. Furthermore, numerical results show remarkable aspects of transport and diffusion phenomena on evolving surfaces and experimentally reflect the established convergence results. Finally, we discuss how to combine the presented scheme with a corresponding finite volume scheme for advective transport on the surface via an operator splitting and present some applications.

How Many Lions Are Needed to Clear a Grid?

We consider a pursuit-evasion problem where some lions have the task to clear a grid graph whose nodes are initially contaminated. The contamination spreads one step per time unit in each direction not blocked by a lion. A vertex is cleared from its contamination whenever a lion moves to it. Brass et al. [5] showed that n/2 lions are not enough to clear the n x n-grid. In this paper, we consider the same problem in dimension d > 2 and prove that Θ(nd-1/√d) lions are necessary and sufficient to clear the nd-grid. Furthermore, we analyze a problem variant where the lions are also allowed to jump from grid vertices to non-adjacent grid vertices.

The Role of Thrombomodulin in Angiogenesis

We investigated the role of the endothelial cell (EC) receptor, thrombomodulin (TM), in angiogenesis by determining its contribution to EC microtubule formation, invasion and cell-cell adhesion. We first established a method to quantitate microtubule formation in vitro using stereology. Digital images of HUVECs grown in Matrigel™ were overlaid with a 4800-pixel grid. The mean number of points where one of the grid's vertices crossed a microtubule was determined and divided by the number of vertices per well (surface area covered by microtubules). Using this quantitative method, we determined that monoclonal antibody (MAb) binding to epidermal growth factor-like domains 4–6 of TM increased microtubule formation. Addition of thrombin, which binds to these EGF-like domains, decreased the augmented microtubule formation. We also investigated TM's involvement in EC invasion in vitro. Binding to the lectin-like domain of TM by MAb decreased invasion of HUVECs. Finally, we determined that binding to TM's extracellular EGF-like domains affected the ability of HUVECs to adhere to each other. We conclude that TM, specifically the EGF- and lectin-like domains of TM, is involved in angiogenesis by regulating the ability of endothelial cells to proliferate, invade, and adhere to each other. Funding: NIH Grant # P20RR-16461, McKay Urology Endowment Fund, and the Winthrop University Research Council.

A practical global distortion correction method for an image intensifier based x‐ray fluoroscopy system

X-ray images acquired on systems with image intensifiers (II) exhibit characteristic distortion which is due to both external and internal factors. The distortion is dependent on the orientation of the II, a fact particularly relevant to II's mounted on C arms which have several degrees of freedom of motion. Previous descriptions of distortion correction strategies have relied on a dense sampling of the C-arm orientation space, and as such have been limited mostly to a single arc of the primary angle, alpha. We present a new method which smooths the trajectories of the segmented vertices of the grid phantom as a function of a prior to solving the two-dimensional warping problem. It also shows that the same residual errors of distortion correction could be achieved without fitting the trajectories of the grid vertices, but instead applying the previously described global method of distortion correction, followed by directly smoothing the values of the polynomial coefficients as functions of the C-arm orientation parameters. When this technique was applied to a series of test images at arbitrary alpha, the root-mean-square (RMS) residual error was 0.22 pixels. The new method was extended to three degrees of freedom of the C-arm motion: the primary angle, alpha; the secondary angle, beta; and the source-to-intensifier distance, lambda. Only 75 images were used to characterize the distortion for the following ranges: alpha, +/- 45 degrees (Deltaalpha = 22.5 degrees); beta, +/- 36 degrees (Deltabeta = 18 degrees); lambda, 98-118 cm (Deltalambda = 10 cm). When evaluated on a series of test images acquired at arbitrary (alpha, beta, lambda), the RMS residual error was 0.33 pixels. This method is targeted at applications such as guidance of catheter-based interventions and treatment planning for brachytherapy, which require distortion-corrected images over a large range of C-arm orientations.

Harmonic averaging of smooth permittivity functions in finite-difference Poisson–Boltzmann Electrostatics

Finite-difference Poisson–Boltzmann calculations offer an efficient and accurate means for electrostatic characterization of solvated molecules. However, discretization of charge and permittivity results in sensitive dependence on molecular position and orientation relative to the finite-difference grid. In this article, an improved method for limiting the error associated with discretization of the molecular volume, combining harmonic averaging between grid vertices and Gaussian-based smooth permittivity functions, is presented. While both these methods have the broader result of a smoothly varying permittivity, the Gaussian model represents a fundamental description of the dielectric boundary while harmonic averaging serves to provide information about the permittivity between grid points. Grid positional error is reduced by an order of magnitude in calculations of Born ion solvation energies, small molecule and protein solvation energies, and the solvation energy contribution to a protein-inhibitor complex.

Hierarchic higher-order hermite elements on hybrid triangular/quadrilateral meshes

In this paper we propose a new family of hierarchic higher-order Hermite elements on hybrid triangular/quadrilateral meshes. Optimal higher-order interior modes are calculated using an appropriate generalized eigenvalue problem. New nonaffine reference mappings preserving the continuity of derivatives at grid vertices are developed.

The heat equation to generate planar grids

An algorithm for the generation of quadrilateral grids on planar domains is presented. This algorithm is given by an iterative procedure, where, starting from an initial grid on the domain under consideration, the coordinates of the grid vertices are iteratively adjusted by using a local discrete variational approach. This procedure resembles the explicit difference scheme for a perturbed heat equation, where the perturbation can be dropped for convex domains. Experimental results on benchmark domains are presented, and show an interesting behavior of the proposed method.

A class of network optimization methods for planar grid generation

The grid generation problem considers the question of the computation of a grid Q over a given domain Ω from the knowledge of Q on the boundary ∂ Ω. It is usually given in terms of the computation of the grid vertices belonging to the interior of Ω, that in the direct optimization formulation are computed as the minimizer of a particular minimization problem. In this paper the grid generation problem is reformulated as a network optimization problem on a particular graph, that is a nonlinear minimum cost flow problem, where in place of the standard Euclidean vector norm is proposed the use of the well-known vector p-norm, where p ⩾ 1 . Some results coming from our numerical experience on the examples proposed in the Rogue's Gallery of Grids are reported.