Regular Rectangular Grid Research Articles

Numerical modelling of convection interacting with a melting and solidification front: Application to the thermal evolution of the basal magma ocean

Melting and solidification are fundamental to geodynamical processes like inner core growth, magma chamber dynamics, and ice and lava lake evolution. Very often, the thermal history of these systems is controlled by convective motions in the melt. Computing the evolution of convection with a solid–liquid phase change requires specific numerical methods to track the phase boundary and resolve the heat transfer within and between the two separate phases. Here we present two classes of method to model the phase transition coupled with convection. The first, referred to as the moving boundary method, uses the finite element method and treats the liquid and the solid as two distinct grid domains. In the second approach, based on the enthalpy method, the governing equations are solved on a regular rectangular grid with the finite volume method. In this case, the solid and the liquid are regarded as one domain in which the phase change is incorporated implicitly by imposing the liquid fraction fL as a function of temperature and a viscosity that varies strongly with fL. We subject the two modelling frameworks to thorough evaluation by performing benchmarks, in order to ascertain their range of applicability. With these tools we perform a systematic study to infer heat transfer characteristics of a solidifying convecting layer. Parametrized relations are then used to estimate the super-isentropic temperature difference maintained across a basal magma ocean (BMO) (Labrosse et al., 2007), which happens to be minute (<0.1K), implying that the Earth’s core must cool at the same pace as the BMO.

A Weak Formulation of the Immersed Boundary Method

A new method of spatial discretization for immersed boundary computations is introduced. Fluid velocity and pressure are obtained as weak solutions of the discretized fluid equations with respect to a wavelet basis of functions. The scaling function of the fluid velocity basis may be chosen to be identical to Peskin's discrete approximation to the Dirac delta function. On a regular rectangular grid the discretized equations are solved using the fast Fourier transform, retaining the efficiency of the immersed boundary method. We show experimental numerical evidence that the rate of volume loss of our method is better than that of the finite difference immersed boundary method. Our formulation offers new insights into the immersed boundary method and leads to new extensions and applications.

Increasing energy efficiency in sensor networks: blue noise sampling and non-convex matrix completion

The energy cost of a sensor network is dominated by the data acquisition and communication cost of individual sensors. At each sampling instant it is unnecessary to sample and communicate the data at all sensors since the data is highly redundant. We find that, if only (random) subset of the sensors acquires and transmits the sample values, it is possible to estimate the sample values at all the sensors under certain realistic assumptions. Since only a subset of all the sensors is active at each sampling instant, the energy cost of the network is reduced over time. When the sensor nodes are assumed to lie on a regular rectangular grid, the problem can be recast as a low-rank matrix completion problem. Current theoretical work on matrix completion relies on purely random sampling strategies and convex estimation algorithms. In this work, we will empirically show that better reconstruction results are obtained when more sophisticated sampling schemes are used followed by non-convex matrix completion algorithms. We find that the proposed approach gives surprisingly good results.

Mathematical modeling and simulation of pigging operation in gas and liquid pipelines

The pigging operation is a common practice in the petroleum and gas industry. Pigging flow lines is employed for many reasons including cleaning deposits such as wax layers, the removal of liquids and condensate, the separation of products pumped one after the other in the same pipeline, measurement, and control and flow line inspection. In this paper, the mathematical modeling of the transient motion of a pig through liquid and gas pipelines is presented. For this purpose, the fluid flow equations were combined with a linear momentum equation for the pig. The nonlinear equations are solved under an unsteady state condition by the method of characteristics (MOC) with a regular rectangular grid through the pipeline under appropriate initial and boundary conditions. From this simulation, the pig position, optimum flow rate in upstream flow and the time that the pig reaches the end of the pipeline are obtained. Comparison of the simulation results with the field data of liquid flow through the pipeline from KG to AG located in Iran show that the derived mathematical models are effective for the prediction of position and pig velocity under the given operational conditions of a pipeline. Similar results are also obtained for gas flow through the pipeline from Nar-1 to Nar-2 located in Iran, in comparison with the field data.

BiGlobal stability analysis in curvilinear coordinates of massively separated lifting bodies

A methodology based on spectral collocation numerical methods for global flow stability analysis of incompressible external flows is presented. A potential shortcoming of spectral methods, namely the handling of the complex geometries encountered in global stability analysis, has been dealt with successfully in past works by the development of spectral-element methods on unstructured meshes. The present contribution shows that a certain degree of regularity of the geometry may be exploited in order to build a global stability analysis approach based on a regular spectral rectangular grid in curvilinear coordinates and conformal mappings. The derivation of the stability linear operator in curvilinear coordinates is presented along with the discretisation method. Unlike common practice to the solution of the same problem, the matrix discretising the eigenvalue problem is formed and stored. Subspace iteration and massive parallelisation are used in order to recover a wide window of its leading Ritz system. The method is applied to two external flows, both of which are lifting bodies with separation occurring just downstream of the leading edge. Specifically the flow configurations are a NACA 0015 airfoil, and an ellipse of aspect ratio 8 chosen to closely approximate the geometry of the airfoil. Both flow configurations are at an angle of attack of 18° with a Reynolds number based on the chord length of 200. The results of the stability analysis for both geometries are presented and illustrate analogous features.

From Maxwell's Equations to Polarimetric SAR Images: A Simulation Approach.

A new electromagnetic approach for the simulation of polarimetric SAR images is proposed. It starts from Maxwell's equations, employs the spectral domain full-wave technique, the moment method, and the stationary phase method to compute the far electromagnetic fields scattered by multilayer structures. A multilayer structure is located at each selected position of a regular rectangular grid of coordinates, which defines the scene area under imaging. The grid is determined taking into account the elementary scatter size and SAR operational parameters, such as spatial resolution, pixel spacing, look angle and platform altitude. A two-dimensional separable “sinc” function to represent the SAR spread point function is also considered. Multifrequency sets of single-look polarimetric SAR images are generated, in L-, C- and X-bands and the images are evaluated using several measurements commonly employed in SAR data analysis. The evaluation shows that the proposed simulation process is working properly, since the obtained results are in accordance with those presented in the literature. Therefore, this new approach becomes suitable for carrying out theoretical and practical studies using polarimetric SAR images.

Rectangular and hexagonal grids used for observation, experiment and simulation in ecology

Regular grids or lattices are frequently used to study ecosystems, for observations, experiments and simulations. The regular rectangular or square grid is used more often than the hexagonal grid, but their relative merits have been little discussed. Here we compare rectangular and hexagonal grids for ecological applications. We focus on the reasons some researchers have preferred hexagonal grids and methods to facilitate the use of hexagonal grids. We consider modelling and other applications, including the role of nearest neighbourhood in experimental design, the representation of connectivity in maps, and a new method for performing field surveys using hexagonal grids, which was demonstrated on montane heath vegetation. The rectangular grid is generally preferred because of its symmetrical, orthogonal co-ordinate system and the frequent use of rasters from Geographic Information Systems. Cells in a rectangular grid can also easily be combined to produce new grids with lower resolutions. However, efficient co-ordinate systems and multi-resolution partitions using the hexagonal grid are available. The nearest neighbourhood in a hexagonal grid is simpler and less ambiguous than in a rectangular grid. When nearest neighbourhood, movement paths or connectivity are important, the rectangular grid may not be suitable. We also investigate important differences between visualizations using hexagonal and rectangular grids. A survey of recent uses of grids in Ecological Modelling suggested that hexagonal grids are rarely used, even in applications for which they are more suitable than rectangular grids, e.g. connectivity and movement paths. Researchers should consider their choice of grid at an early stage in project development, and authors should explain the reasons for their choices.

Cover Picture: Two‐Dimensional Rectangular and Three‐Dimensional Rhombic Grids Created by Self‐Organization of Random Nanoextrusions (Adv. Eng. Mater. 10/2006)

AbstractNanoextrusions on single crystalline Al and Al‐Mg plates under constrained cyclic tension evolve to regular 2D surface rectangular grids and 3D porous rhombic grids. The cover shows AFM images of the plates along with their 3D reconstructions and associated digital Fourier transforms. More about this type of self‐organization can be found in the paper by Y. G. Gordienko et al. on page 957.

Comparison of methods to model the gravitational gradients from topographic data bases

A number of methods have been developed over the last few decades to model the gravitational gradients using digital elevation data. All methods are based on second-order derivatives of the Newtonian mass integral for the gravitational potential. Foremost are algorithms that divide the topographic masses into prisms or more general polyhedra and sum the corresponding gradient contributions. Other methods are designed for computational speed and make use of the fast Fourier transform (FFT), require a regular rectangular grid of data, and yield gradients on the entire grid, but only at constant altitude. We add to these the ordinary numerical integration (in horizontal coordinates) of the gradient integrals. In total we compare two prism, two FFT and two ordinary numerical integration methods using 1″ elevation data in two topographic regimes (rough and moderate terrain). Prism methods depend on the type of finite elements that are generated with the elevation data; in particular, alternative triangulations can yield significant differences in the gradients (up to tens of Eötvös). The FFT methods depend on a series development of the topographic heights, requiring terms up to 14th order in rough terrain; and, one popular method has significant bias errors (e.g. 13 Eötvös in the vertical–vertical gradient) embedded in its practical realization. The straightforward numerical integrations, whether on a rectangular or triangulated grid, yield sub-Eötvös differences in the gradients when compared to the other methods (except near the edges of the integration area) and they are as efficient computationally as the finite element methods.

Microarray image gridding with stochastic search based approaches

The paper reports a novel approach for the problem of automatic gridding in Microarray images. Such problem often requires human intervention; therefore, the development of automated procedures is a fundamental issue for large-scale functional genomic experiments involving many microarray images. Our method uses a two-step process. First a regular rectangular grid is superimposed on the image by interpolating a set of guide spots, this is done by solving a non-linear optimization process with a stochastic search producing the best interpolating grid parameterized by a six values vector. Second, the interpolating grid is adapted, with a Markov Chain Monte Carlo method, to local deformations. This is done by modeling the solution a Markov random field with a Gibbs prior possibly containing first order cliques (1-clique). The algorithm is completely automatic and no human intervention is required, it efficiently accounts arbitrary grid rotations, irregularities and various spot sizes.

Seismic tomography imaging of an unstable embankment

Seismic tomography imaging was employed to make a diagnosis and choose a remedy for an embankment supported by a retaining wall showing clear evidence of structural instability. The geometry and structural characteristics of the wall, the inside geometry of cracks, and the physical parameters of the underground materials were the primary objectives of the geophysical survey. Seismic data were acquired along two vertical sections each one delimited by the lines of the sources within an inclined borehole and of the receivers lying on the ground. For each section, a total of 744 travel times were inverted to obtain compressional-wave velocities on a regular rectangular grid of squared cells (1 m × 1 m) using an inversion algorithm which is based on the simultaneous iterative reconstruction technique (SIRT). Ultrasonic tests carried out in laboratory on intact specimens, together with other supplemental site information, improved tomography resolution, allowing global and node constraints that forced the solution of the inverse problem to match known boundary values. Tomography imaged two high-velocity zones separated by a curved-shape low-velocity zone. The first ones were found to be compatible with the concrete retaining wall and with the schist materials, ranging from highly weathered to intact schist, while the low-velocity zone was interpreted as filling materials and/or completely decomposed rock schist. The combination of seismic tomography and laboratory measurements allowed extrapolation of important parameters over a large volume of rock mass, otherwise only representative of small rock samples near boreholes. It also enabled engineering characterization of subsurface rock mass, providing useful and accurate information to design a remedy for the embankment.

A feasibility study for discontinuous Galerkin discretization with embedded Dirichlet boundary condition

In this paper we introduce a discretization of Discontinuous Galerkin (DG) type for solving 2-D second-order elliptic PDEs on a regular rectangular grid, while the boundary value problem has a curved Dirichlet boundary. According to the same principles that underlie DG-methods, we adapt the discretization in the cell in which the (embedded) Dirichlet boundary cannot follow the gridlines of the orthogonal grid. The DG-discretization aims at a high order of accuracy. We discretize with tensor products of cubic polynomials and we parameterize the embedded boundary by cubic polynomials. Then, by construction, such a DG discretization is fourth-order consistent, both in the interior and at the boundaries. The proposed discretization technique is motivated by the fact that it results in only a slight modification of DG discretization at embedded boundaries and requires a much simpler mesh generation than needed for boundary conforming meshes. In particular the method can show its use in cases where regular rectangular grids are preferred. To illustrate the possibilities of our DG-discretization, we solve a convection dominated boundary value problem on a regular rectangular grid with a circular embedded boundary condition [J. Comput. Appl. Math. 76 (1997) 277]. We show how accurately the boundary and shear layer, emanating from the curved embedded boundary, can be captured by means of the tensor product polynomials on the structured orthogonal mesh.

Seismic propagation simulation in complex media with non-rectangular irregular-grid finite-difference

This paper presents a finite-difference (FD) method with spatially non-rectangular irregular grids to simulate the elastic wave propagation. Staggered irregular grid finite difference operators with a second-order time and spatial accuracy are used to approximate the velocity-stress elastic wave equations. This method is very simple and the cost of computing time is not much. Complicated geometries like curved thin layers, cased borehole and nonplanar interfaces may be treated with non-rectangular irregular grids in a more flexible way. Unlike the multi-grid scheme, this method requires no interpolation between the fine and coarse grids and all grids are computed at the same spatial iteration. Compared with the rectangular irregular grid FD, the spurious diffractions from “staircase” interfaces can easily be eliminated without using finer grids. Dispersion and stability conditions of the proposed method can be established in a similar form as for the rectangular irregular grid scheme. The Higdon's absorbing boundary condition is adopted to eliminate boundary reflections. Numerical simulations show that this method has satisfactory stability and accuracy in simulating wave propagation near rough solid-fluid interfaces. The computation costs are less than those using a regular grid and rectangular grid FD method.

A new modification of the immersed boundaries method for fluid–solid flows: moderate Reynolds numbers

In this study we consider a combination of an interpolation technique with the D’Alambert principle that allows the direct numerical simulation of fluid-solid flows on a regular rectangular grid. The method models a no-slip boundary with the second-order accuracy interpolation scheme without explicit calculation of the forces that the particles exert at the fluid. The method is applicable for a wide range of Reynolds numbers from creeping flows up to Re≈100.

Chaotic advection of finite-size bodies in a cavity flow

We considered advection of neutrally buoyant discs in two-dimensional chaotic Stokes flow. The goal of the study is to explore a possibility to enhance laminar mixing in batch-flow mixers. Addition of freely moving bodies to periodically driven chaotic flow renders the flowfield nonperiodic [D. F. Zhang and D. A. Zumbrunnen, AIChE J. 42, 3301 (1996)], i.e., the Lagrangian chaos of the bodies motion induces Eulerian chaos of the flow that makes mixing more intensive. The presence of three bodies creates new topological features that do not exist in “pure” fluid. The trajectories of the discs in the augmented phase space tangle and form a braid that leads to so-called topological chaos [P. L. Boyland, H. Aref, and M. A. Stremler, J. Fluid Mech. 403, 277 (2000)]. Simulations were performed using a new variant of the immersed boundaries method that allows the direct numerical simulation of fluid–solid flows on a regular rectangular grid without explicit calculation of the forces that the particles exert on the fluid.

Voronoi cell finite difference method for the diffusion operator on arbitrary unstructured grids

AbstractVoronoi cells and the notion of natural neighbours are used to develop a finite difference method for the diffusion operator on arbitrary unstructured grids. Natural neighbours are based on the Voronoi diagram, which partitions space into closest‐point regions. The Sibson and the Laplace (non‐Sibsonian) interpolants which are based on natural neighbours have shown promise within a Galerkin framework for the solution of partial differential equations. In this paper, we focus on the Laplace interpolant with a two‐fold objective: first, to unify the previous developments related to the Laplace interpolant and to indicate its ties to some well‐known numerical methods; and secondly to propose a Voronoi cell finite difference scheme for the diffusion operator on arbitrary unstructured grids. A conservation law in integral form is discretized on Voronoi cells to derive a finite difference scheme for the diffusion operator on irregular grids. The proposed scheme can also be viewed as a point collocation technique. A detailed study on consistency is conducted, and the satisfaction of the discrete maximum principle (stability) is established. Owing to symmetry of the Laplace weight, a symmetric positive‐definite stiffness matrix is realized which permits the use of efficient linear solvers. On a regular (rectangular or hexagonal) grid, the difference scheme reduces to the classical finite difference method. Numerical examples for the Poisson equation with Dirichlet boundary conditions are presented to demonstrate the accuracy and convergence of the finite difference scheme. Copyright © 2003 John Wiley &amp; Sons, Ltd.

A universal cellular automaton in the hyperbolic plane

The paper gives the construction of a universal CA with 22 states in the regular rectangular pentagonal grid of the hyperbolic plane. The CA implements a railway circuit which simulates a register machine and which improves a bit already known railway simulations of a Turing machine.

Discontinuous Galerkin Discretization with Embedded Boundary Conditions

AbstractThe purpose of this paper is to introduce discretization methods of discontinuous Galerkin type for solving second-order elliptic PDEs on a structured, regular rectangular grid, while the problem is defined on a curved boundary. The methods aim at high-order accuracy and the difficulty arises since the regular grid cannot follow the curved boundary. Starting with the Lagrange multiplier formulation for the boundary conditions, we derive variational forms for the discretization of 2-D elliptic problems with embedded Dirichlet boundary conditions. Within the framework of structured, regular rectangular grids, we treat curved boundaries according to the principles that underlie the discontinuous Galerkin method. Thus, the high-order DGdiscretization is adapted in cells with embedded boundaries. We give examples of approximation with tensor products of cubic polynomials. As an illustration, we solve a convection-dominated boundary-value problem on a complex domain. Although, of course, it is impossible to accurately represent a boundary layer with a complex structure by means of cubic polynomials, the boundary condition treatment appears quite effective in handling such complex situations.

Sampling Strategies for Mapping ‘Within-field’ Variability in the Dry Matter Yield and Mineral Nutrient Status of Forage Grass Crops in Cool Temperate Climes

In the absence of suitable technology to measure and map the dry matter (DM) yield distributions of forage grass crops within individual fields, a ‘manual’ procedure of yield mapping has been developed. Samples of herbage are collected just prior to each silage harvest from known grid points within a field, and sward DM yields at each point are predicted from the mineral composition of the herbage, using empirical mathematical models. Yield maps (and maps of sward nutrient status) are then produced by kriging interpolation between the point data. To make the most efficient use of time and resources, however, sampling intensity needs to be kept to the absolute minimum necessary for interpolation purposes. The aim of the present study was to examine the spatial variability in sward DM yield and mineral nutrient status in a large grass silage field under a three-cut system, and devise ‘optimal’ sampling strategies for mapping the distributions of these parameters at each cut. Herbage samples were collected from the field, prior to each harvest, at 25 m intervals in a regular rectangular grid to provide databases of herbage nutrient contents and DM yields. Different data combinations were abstracted from these databases for comparison purposes, and ordinary kriging used to produce interpolated maps of DM yield and sward N, P, K and S statuses. The results suggested that a sampling density of just seven samples per hectare was adequate for estimating the ‘true’ population means of sward DM yield and sward N, P, K, and S statuses. For mapping purposes, it was found that the best compromise between interpolation accuracy and sampling efficiency was to collect herbage samples in a 35.4 m×35.4 m equilateral triangular sampling pattern.

Automatic mesh generation on a regular background grid

This paper presents an automatic mesh generation procedure on a 2D domain based on a regular background grid. The idea is to devise a robust mesh generation scheme with equal emphasis on quality and efficiency. Instead of using a traditional regular rectangular grid, a mesh of equilateral triangles is employed to ensure triangular element of the best quality will be preserved in the interior of the domain.As for the boundary, it is to be generated by a node/segment insertion process. Nodes are inserted into the background mesh one by one following the sequence of the domain boundary. The local structure of the mesh is modified based on the Delaunay criterion with the introduction of each node. Those boundary segments, which are not produced in the phase of node insertion, will be recovered through a systematic element swap process. Two theorems will be presented and proved to set up the theoretical basic of the boundary recovery part. Examples will be presented to demonstrate the robustness and the quality of the mesh generated by the proposed technique.