Journal articles on the topic 'Grid solving'

An enhanced multi-grid method eliminating the error correction process of the conventional multi-grid method is presented for solving Poissonian problems and tested on two simple two-dimensional magnetostatic field problems. The finite element method (FEM) was used to solve for the vector potential in a sequence of grids. The gains in computation time are shown to be immense compared to the standard multi-grid methods, especially as the matrix system grows in size. These gains are very useful in solving electromagnetic problems using the finite element method.

Results of theoretical analysis of the geometric multigrid algorithms convergence are presented for solving the linear boundary value problems on a two-block grid. In this case, initial domain could be represented as a union of intersecting subdomains, in each of them a structured grid could be constructed generating a hierarchy of coarse grids. Multigrid iteration matrix is obtained using the damped nonsymmetric iterative method as a smoother. The multigrid algorithm contains a new problem-dependent component --- correction interpolation between grid blocks. Smoothing property for the damped nonsymmetric iterative method and convergence of the robust multigrid technique are proved. Estimation of the multigrid iteration matrix norm is obtained (sufficient convergence condition). It is shown that the number of multigrid iterations does not depend on either the step or the number of grid blocks, if interpolation of the correction between grid blocks is sufficiently accurate. Results of computational experiments are presented on solving the three-dimensional Dirichlet boundary value problem for the Poisson equation illustrating the theoretical analysis. Results obtained could be easily generalized to multiblock grids. The work is of interest for developers of highly efficient algorithms for solving the (initial-) boundary value problems describing physical and chemical processes in complex geometry domains

Grid generation techniques have contributed significantly toward the application of mathematical modeling in large-scale engineering problems. The structured grids have the advantage that very robust and parallel computational algorithms have been proposed for solving (initial-)boundary value problems. Orthogonal grids make it possible to simplify an approximation of the differential equations and to increase computation accuracy. Opportunity of the orthogonal structured grid generation for solving two- and three-dimensional (initial-)boundary value problems is analyzed in the article in assumption that isolines or isosurfaces of d (=2,3) functions form this grid. Condition of the isolines/isosurfaces orthogonality is used for formulation of the boundary value problems, the solutions of which will be form the orthogonal grid. A differential substitution is proposed to formulate the boundary value problems directly from the orthogonality condition of the grid. The substitution leads to the general partial differrential equations with undetermined coefficients. In the two-dimensional case, it is shown that the orthogonal grid generation is equivalent to the solution of partial differential equations of either elliptic or hyperbolic type. In three-dimensional domains, an orthogonal grid can be generated only in special cases. The obtained results are useful for mathematical modeling of the complex physicochemical processes in the technical devices

Summary The two-point flux finite-volume method (2P-FVM) is the most widely used method for solving the flow equation in reservoir simulations. For 2P-FVM to be consistent, the simulation grid needs to be orthogonal (or k-orthogonal if the permeability field is anisotropic). It is well known that corner-point grids can introduce large errors in the flow solutions because of the lack of orthogonality in general. Multipoint flux formulations that do not rely on grid orthogonality have been proposed, but these methods add significant computational cost to solving the flow equation. Recently, 2.5D unstructured grids that combine 2D Voronoi areal grids with vertical projections along deviated coordinate lines have become an attractive alternative to corner-point gridding. The Voronoi grid helps maintain orthogonality areally and can mitigate grid orientation effects. However, experience with these grids is limited. In this paper, we present an analytical and numerical study of these 2.5D unstructured grids. We focus on the effect of grid deviation on flow solutions in homogeneous, but anisotropic, permeability fields. In particular, we consider the grid deviation that results from gridding to sloping faults. We show that 2P-FVM does not converge to the correct solution as the grid refines. We further quantify the errors for some simple flow scenarios using a technique that combines numerical analysis and asymptotic expansions. Analytical error estimates are obtained. We find that the errors are highly flow dependent and that they can be global with no strong correlation with local nonorthogonality measures. Numerical tests are presented to confirm the analytical findings and to show the applicability of our conclusions to more-general flow scenarios.

Downbursts, which resulted from the flow downdraft in thunderstorms, have become one of the most destructive disasters to buildings including transmission towers, etc. This disaster has drawn researchers’ interests and progresses have been continuously made by employing test and numerical tools. Accounting for the grid validations in the numerical simulation of downbursts, eight grids with different grid point distributions are generated, and then their corresponding flow fields are calculated by solving Navier-Stokes equations. The numerical results are compared with test results to investigate the influence of grid distributions onto numerical results. The results indicate that, numerical fidelity could be improved by refining grids in the zone with strong horizontal wind; while local grid refinement at inlet boundary could deteriorate numerical accuracy when the grid point number is kept constant, hence uniform grid distribution is recommended at inlet boundary without any grid refinement.

A multicomponent iterative method of domain decomposition on adaptive grids for solution of two‐dimensional heat transfer equation is proposed. The adaptive grid is constructed in curvilinear space where Cartesian grid is non‐stationary and depends on the solution behavior. In curvilinear space the initial two‐dimensional heat transfer equation is converted to the system of nonlinear parabolic equations with mixed derivatives, a source and convective transfer.

To avoid grid degradation, the numerical analysis of the j-solution of the Navier–Stokes equation has been studied. The Navier–Stokes equations describe the motion of viscous fluid substances. On the basis of the advantages and disadvantages of the Navier–Stokes equations, the incompressible terms and the nonlinear terms are separated, and the original boundary conditions satisfying the j-solution of the Navier–Stokes equation are analyzed. Secondly, the development of a computational grid has been introduced; the turbulence model has also been described. The fluid form and the initial value of the j-solution of the Navier–Stokes equation are combined. The original boundary conditions are solved by a computer, and the nonlinear turbulence equations are derived, which control the fluid flow. The simulation of the fine grid is comprehended to analyze the research outcome. Simulation analysis is carried out to generate multiblock-structured grids with high quality. The j-solution on the grid points is the j-solution that can be used with a fewer number of meshes under the same conditions. The proposed work is easy to implement, and it consumes lesser memory. The results obtained are able to avoid mesh degradation skillfully, and the generated mesh exhibits the characteristics of smoothness, orthogonality, and controllability, which eventually improves the calculation accuracy.

The higher order Haar wavelet method (HOHWM) is used with a nonuniform grid to solve nonlinear partial differential equations numerically. The Burgers’ equation, the Korteweg–de Vries equation, the modified Korteweg–de Vries equation and the sine–Gordon equation are used as model equations. Adaptive as well as nonadaptive nonuniform grids are developed and used to solve the model equations numerically. The numerical results are compared to the known analytical solutions as well as to the numerical solutions obtained by application of the HOHWM on a uniform grid. The proposed methods of using nonuniform grid are shown to significantly increase the accuracy of the HOHWM at the same number of grid points.

In this paper we propose a technique accelerating the convergence rate of the known iterative schemes for solving grid equations such as the alternately triangular method and the alternating direction method. Our idea is by the parametric extrapolation of the solutions of equations, which can be solved faster than the original ones. The effeciancy of the accelerated methods is shown on example.

In this paper, a method for solving grid mismatch or off-grid target is presented for direction of arrival (DOA) estimation problem using compressive sensing (CS) technique. Location of the sources are at few angles as compare to the entire angle domain, i.e., spatially sparse sources, and their location can be estimated using CS methods with ability of achieving super resolution and estimation with a smaller number of samples. Due to grid mismatch in CS techniques, the source energy is distributed among the adjacent grids. Therefore, a fitness function is introduced which is based on the difference of the source energy among the adjacent grids. This function provides the best discretization value for the grid through iterative grid refinement. The effectiveness of the proposed scheme is verified through extensive simulations for different number of sources.

AbstractA single software framework is introduced to evaluate numerical accuracy of the A-grid (NICAM) versus C-grid (MPAS) shallow-water model solvers on icosahedral grids. The C-grid staggering scheme excels in numerical noise control and total energy conservation, which results in exceptional stability for long time integration. Its weakness lies in the lack of model error reduction with increasing resolution in specific test cases (especially the root-mean-square error). The A-grid method conserves well potential enstrophy and shows a linear reduction of error with increasing resolution. The gridpoint noise manifests itself clearly on A-grid, but much less on C-grid. We show that the Coriolis force term on C-grid has a larger error than on A-grid. To treat the Coriolis term and kinetic energy gradient on an equal footing on C-grid, we propose combining these two quantities into a single tendency term and computing its value by a linear combination operation. This modification alone reduces numerical errors but still fails to converge the maximum error with resolution. The method of Peixoto can solve the maximum-error nonconvergence problem on C-grid but degrades the numerical stability. For the steady-state thin-layer test (0.01 m in depth), the A-grid method is less susceptible than C-grid methods, which are presumably disrupted by the Hollingsworth instability. The effect of horizontal diffusion on model accuracy and energy conservation is shown in detail. Programming experience shows that software implementation and optimization can strongly influence computational performance for models, although memory requirement and computational load of the two schemes are comparable.

Abstract Sudoku grids are often cited as being useful in cryptography as a key for some encryption process. Historically transporting keys over an alternate channel has been very difficult. This article describes how a Sudoku grid key can be secretly transported using quantum key distribution methods whereby partial grid (or puzzle) can be received and the full key can be recreated by solving the puzzle.

AbstractTo obtain an approximate solution of the steady-state convectiondiffusion problem, it is necessary to solve the corresponding system of linear algebraic equations. The basic peculiarity of these LA systems is connected with the fact that they have non-symmetric matrices. We discuss the questions of approximate solution of 2D convection-diffusion problems on the basis of two- and three-level iterative methods. The general theory of iterative methods of solving grid equations is used to present the material of the paper. The basic problems of constructing grid approximations for steady-state convection-diffusion problems are considered. We start with the consideration of the Dirichlet problem for the differential equation with a convective term in the divergent, nondivergent, and skew-symmetric forms. Next, the corresponding grid problems are constructed. And, finally, iterative methods are used to solve approximately the above grid problems. Primary consideration is given to the study of the dependence of the number of iteration on the Peclet number, which is the ratio of the convective transport to the diffusive one.

This paper presents the application of a class of multi-grid methods to the solution of the Navier-Stokes equations for two-dimensional laminar flow problems. The methods consist of combining the full approximation scheme-full multi-grid technique (FAS-FMG) with point-, line- or plane-relaxation routines for solving the Navier-Stokes equations in primitive variables. The performance of the multi-grid methods is compared to those of several single-grid methods. The results show that much faster convergence can be procured through the use of the multi-grid approach than through the various suggestions for improving single-grid methods. The importance of the choice of relaxation scheme for the multi-grid method is illustrated.

Abstract This paper presents a new robust multigrid technique for solving boundary value problems in a black box manner. To overcome the problem of robustness, the technique is based on the incorporation of adaptation of boundary value problems to numerical methods, control volume discretization and a new multigrid solver into a united computational algorithm. The special multiple coarse grid correction strategy makes it possible to obtain problem-independent transfer operators. As a result, most modes are approximated on coarse grids to make the task of the smoother on the finest grid the least demanding. A detailed description of the robust multigrid technique and examples of its application for solving benchmark problems are given in the paper.

A series of compact implicit schemes of fourth and sixth orders are developed for solving differential equations involved in geodynamics simulations. Three illustrative examples are described to demonstrate that high-order convergence rates are achieved while good efficiency in terms of fewer grid points is maintained. This study shows that high-order compact implicit difference methods provide high flexibility and good convergence in solving some special differential equations on nonuniform grids.

This paper presents the developed algorithm for numerical grid deformation for solving the problems of modeling the flow around the helicopter main rotor in the horizontal flight mode with allowance for flapping movements and cyclic changes in the angle of the blade installation. In general, this algorithm can be applied to simulate the aerodynamics of solid bodies deviating from its initial position at angles up to 90 degrees in the vertical and horizontal planes relative to the origin point, and also performing a rotational motion at an angle up to 90 degrees around the axis through the center of coordinates and the body mass center. The first part provides a brief overview of the existing methods of the computational grid deformation for solving various problems of numerical simulation. These include methods for rebuilding the grid, moving grids and "Chimera" grids. The second part describes the algorithms for allocating of grid deformation and for finding the final coordinate of the computational grid nodes in the presence of a predetermined blade control law. The equations of the deformation zones shape in numerical grid are given. The influence of variables on zones sizes is shown. The third part presents the results of methodological calculations confirming the performance and limitations when choosing mesh deformation zones. The influence of the size and shape of the deformation zones of the numerical grid on the quality of the mesh elements is also shown. This work is methodical in nature and is a preliminary stage in the numerical modeling of the flow around the helicopter main rotor taking into account the automatic main rotor balancing and blades flapping.

With no less than half a billion people in the world without electricity supply, and electricity being the back bone for technological development, it makes sense that electricity is the center of discussion. While innovation and technology have radically transformed other industrial sectors, the electric system, has continued to operate the same way for decades. The real challenge today is not to meet the minimum functionality but to be prepared for future demands. These demands make it necessary for the transformation from regular grid to a Smart Grid. In this paper, the Smart Grid was evaluated for its impacts on the environment, industry, and the global population. Additionally, the role of ICTs in solving the hurdles of Smart Grid has been examined.

AbstractIn this paper, we propose a new method for solving two-dimensional elliptic interface problems with nonhomogeneous flux jump condition and nonlinear jump condition. The method we used is traditional finite element method coupled with Newton’s method, it is very simple and easy to implement. The grid we used here is body-fitting grids based on the idea of semi-Cartesian grid. Numerical experiments show that this method is about second order accurate in the $L^\infty$ norm.

Bioinformatics can be considered as a bridge between life science and computer science, where high performance computational platforms and software are required to manage complex biological data. In this paper we present PROTEUS, a Grid-based Problem Solving Environment that integrates ontology and workflow approaches to enhance composition and execution of bioinformatics application on the Grid. Architecture and preliminary experimental results are reported.

The study of non-stationary rarefied gas flows is, currently, attracting a great deal of attention. Such an interest arises from creating the pulsed jets used for deposition of thin films and special coatings on the solid surfaces. However, the problems of non-stationary rarefied gas flows are still understudied because of their large computational complexity. The paper considers the computational aspects of investigating non-stationary movement of gas reflected from a wall and flowing through a suddenly formed gap. The study objective is to analyse the possible numerical kinetic approaches to solve such problems and identify the difficulties in their solving. When modeling the gas flows in strong rarefaction one should consider the Boltzmann kinetic equation, but its numerical implementation is rather time-consuming. In order to use more simple approaches based, for example, on approximation kinetic equations (Ellipsoidal-Statistical model, Shakhov model), it is important to estimate the difference between the solutions of the model equations and of the Boltzmann equation. For this purpose, two auxiliary problems are considered, namely reflection of the gas flow from the wall and outflow of the free jet into the rarefied background gas.A numerical solution of these problems shows a weak dependence of the solution on the type of the collision operator in the rarefied region, but at the same time a strong dependence of a behavior of the macro-parameters on the velocity grid step. The detailed velocity grid is necessary to avoid a non-monotonous behavior of the macro-parameters caused by so-called ray effect. To reduce computational costs of the detailed velocity grid solution, a hybrid method based on the synthesis of model equations and the Boltzmann equation is proposed. Such an approach can be promising since it reduces the domain in which the Boltzmann collision integral should be used.The article presents the results obtained using two different software packages, namely a Unified Flow Solver (UFS) [13] and a Nesvetay 3D software complex [14-15]. Note that the UFS uses the discrete ordinate method for velocity space on a uniform grid and a hierarchical adaptive mesh refinement in physical space. The possibility to calculate both the Boltzmann equation and the model equations is realized. The Nesvetay 3D software complex was created to solve the Shakhov model equation (S-model) for calculations based on non-structured non-uniform grids, both in velocity space and in physical one.

The study of nonstationary rarefied gas flows is currently paid much attention. Such interest to these problems is caused by the creation of pulsed jets used for the deposition of thin films and special coatings on solid surfaces. However the problems of nonstationary rarefied gas flows have not been studied sufficiently fully because of their large computational complexity. In this paper the computational aspects of investigating the nonstationary flows of a reflected gas from a wall and flowing through a suddenly formed gap is considering. The objective of this study is to analyze the possible numerical kinetic approaches for solving such nonstationary problems and to identify the difficulties encountered in solving.When studying the gas flows in strong rarefaction regimes one should consider the Boltzmann kinetic equation, but its numerical implementation is rather laborious. In order to use more simple approaches based for example on approximation kinetic equations (Ellipsoidal-Statistical model, Shakhov model), it is important to estimate the difference of the solutions of the model equations and the Boltzmann equation. For this purpose two auxiliary problems are considered: reflection of the gas flow from the wall and outflow of the free jet into the rarefied background gas. Numerical solution of these problems shows a weak dependence of the solution on the type of the collision operator in the rarefied region, but a strong dependence on the velocity grid step . The detailed velocity grid is necessary to avoid non-monotonous behavior of macroparameters caused by the “ray effect”. To reduce numerical costs on detailed grid a hybrid method based on the synthesis of model equation and the Boltzmann equation is proposed. Such approach can be promising since it reduces the domain in which the Boltzmann collision integral should be used.The results presented in this paper were obtained using two different software packages Unified Flow Solver (UFS) [13] and Nesvetay 3D [14-15]. Note that UFS uses the discrete ordinate method for velocity space on a uniform grid and a hierarchical adaptive mesh refinement in physical space. The possibility of calculating both the Boltzmann equation and model equations is realized. The Nesvetay 3D complex was created to solve the Shakhov model equation, (S-model) and makes it possible to calculate on non-structured non uniform grids in velocity and physical spaces.Translated from Russian. Original text: Mathematics and Mathematical Modeling. 2018. no. 4. Pp. 27-44.

Robotic vehicles working in unknown environments require the ability to determine their location while learning about obstacles located around them. In this paper a method of solving the SLAM problem that makes use of compressed occupancy grids is presented. The presented approach is an extension of the FastSLAM algorithm which stores a compressed form of the occupancy grid to reduce the amount of memory required to store the set of occupancy grids maintained by the particle filter. The performance of the algorithm is presented using experimental results obtained using a small inexpensive ground vehicle equipped with LiDAR, compass, and downward facing camera that provides the vehicle with visual odometry measurements. The presented results demonstrate that although with our approach the occupancy grid maintained by each particle uses only40%of the data needed to store the uncompressed occupancy grid, we can still achieve almost identical results to the approach where each particle filter stores the full occupancy grid.

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Locked="false" Priority="19" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Subtle Emphasis" /> <w:LsdException Locked="false" Priority="21" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Intense Emphasis" /> <w:LsdException Locked="false" Priority="31" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Subtle Reference" /> <w:LsdException Locked="false" Priority="32" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Intense Reference" /> <w:LsdException Locked="false" Priority="33" SemiHidden="false" UnhideWhenUsed="false" QFormat="true" Name="Book Title" /> <w:LsdException Locked="false" Priority="37" Name="Bibliography" /> <w:LsdException Locked="false" Priority="39" QFormat="true" Name="TOC Heading" /> </w:LatentStyles> </xml><![endif]--><!--[if gte mso 10]> <style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-priority:99; mso-style-qformat:yes; mso-style-parent:""; mso-padding-alt:0cm 5.4pt 0cm 5.4pt; mso-para-margin:0cm; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:11.0pt; font-family:"Calibri","sans-serif"; mso-ascii-font-family:Calibri; mso-ascii-theme-font:minor-latin; mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast; mso-hansi-font-family:Calibri; mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:minor-bidi;} </style> <![endif]--><span style="font-size: 10pt; font-family: &quot;Times New Roman&quot;,&quot;serif&quot;;" lang="EN-US">Recent studies have argued the increasing complexity of engineering practice brings challenges to engineers. Creativity has been thought as one necessary element in engineering education. And to let students work with project work has been regarded as a good strategy of developing creativity. However, the literature shows studies on engineering education, most previous efforts take a departure of psychological perspective to discuss developing creative skills through project work; there is a lack of discussion on how project work supports achieving new knowledge and creativity in a social theory framework. Therefore, this paper aims to answer this lack and provide a literature review by focusing on three questions: 1) Which kinds of knowledge are needed for engineering students? 2) What are relationships between knowledge, learning and creativity? And 3) why does a solving project context support creativity and learning? Based on answering the three questions, the literature review underpins the needs of developing creativity in engineering education and strengths of solving project context in stimulating creativity and learning. So this paper contributes to future studies and practical strategies of fostering creative engineers.<span>&nbsp; </span><span>&nbsp;</span><span> <br /></span></span>

In this paper we construct an iteractive method for solving degenerate system of grid equations and after that we use parametric extrapolation technique for accerlerating its convergence rate. The efficiency of the method is shown on examples.

Problem Statement. The advances in human civilization lead to more complications in problem solving. Grid computing serves as an efficient technology in solving those complicated problems. In computational grids, the grid scheduler schedules the task and finds the appropriate resource for each task. The scheduler must consider several factors such as user demand, communication time, failure handling mechanisms, and reduced makespan. Most of the existing algorithms do not consider user satisfaction. Thus a scheduling algorithm that handles failure of resources and achieves user satisfaction gains more importance.Approach. A new bicriteria scheduling algorithm (BSA) that considers user satisfaction along with fault tolerance has been introduced. The main contribution of this paper includes achieving user satisfaction along with fault tolerance and minimizing the makespan of jobs.Results. The performance of this proposed algorithm is evaluated using GridSim based on makespan and number of jobs completed successfully within user deadline.Conclusions/Recommendations. The proposed BSA algorithm achieves reduced makespan and better hit rate with higher user satisfaction and fault tolerance.

AbstractBased on two-grid discretization, a simplified parallel iterative finite element method for the simulation of incompressible Navier-Stokes equations is developed and analyzed. The method is based on a fixed point iteration for the equations on a coarse grid, where a Stokes problem is solved at each iteration. Then, on overlapped local fine grids, corrections are calculated in parallel by solving an Oseen problem in which the fixed convection is given by the coarse grid solution. Error bounds of the approximate solution are derived. Numerical results on examples of known analytical solutions, lid-driven cavity flow and backward-facing step flow are also given to demonstrate the effectiveness of the method.

In order to build a truly open OpenDSS and enhance the complexity of decision-making DSS for solving the problem of Fire spreading. a new model was given based on mobile Agent Open Decision Support System in grid environment. Making use of the intelligence of MAS and adaptive capacity, based on the reasoning mechanisms for task decomposition and resource decision-making grid matching, described in detail layer program CBR-based reasoning mechanism of the Agent and its operation flow and interactive mechanism, put the complexity of distribution of decision-making problem solving to the grid nodes on the environment, implementation of parallel asynchronous decision-making problem solving. Optimize distribution through the logistics of the problem MABODSS design, it proved to improve the system's intelligence and operational efficiency.

AbstractWe analyze here, a two-grid finite element method for the two dimensional time-dependent incompressible Navier-Stokes equations with non-smooth initial data. It involves solving the non-linear Navier-Stokes problem on a coarse grid of size H and solving a Stokes problem on a fine grid of size h, h « H. This method gives optimal convergence for velocity in H1-norm and for pressure in L2-norm. The analysis mainly focuses on the loss of regularity of the solution at t = 0 of the Navier-Stokes equations.

In this paper, application of multi-conductor transmission line model (MTL) in transient analysis of grounding grids buried in soils with frequency-dependent electrical parameters (dispersive soil) is investigated. In this modeling approach, each set of parallel conductors in the grounding grid is considered as a multi-conductor transmission line (MTL). Then, a two-port network for each set of parallel conductors in the grid is then defined. Finally, the two-port networks are interconnected depending upon the pattern of connections in the grid and its representative equations are then reduced. Via solving these simplified equations, the transient analyses of grounding grids is efficiently carried out. With the aim of validity, a number of examples previously published in literature are selected. The comparison of simulation results based on the MTL shows good agreement with numerical and experimental results. Moreover, in despite of numerical methods computational efficiency is considerably increased.

Vectors are a key type of geospatial data, and their discretization, which involves solving the problem of generating a discrete line, is particularly important. In this study, we propose a method for constructing a discrete line mathematical model for a triangular grid based on a “weak duality” hexagonal grid, to overcome the drawbacks of existing discrete line generation algorithms for a triangular grid. First, a weak duality relationship between triangular and hexagonal grids is explored. Second, an equivalent triangular grid model is established based on the hexagonal grid, using this weak duality relationship. Third, the two-dimensional discrete line model is solved by transforming it into a one-dimensional optimal wandering path model. Finally, we design and implement the dimensionality reduction generation algorithm for a discrete line in a triangular grid. The results of our comparative experiment indicate that the proposed algorithm has a computation speed that is approximately 10 times that of similar existing algorithms; in addition, it has better fitting effectiveness. Our proposed algorithm has broad applications, and it can be used for real-time grid transformation of vector data, discrete global grid system (DGGS), and other similar applications.

For solving the eikonal equation in the regions near the curved earth’s surface and the curved interface, we find a second order upwind finite difference scheme that uses nonuniform grid spacing in the regions near the earth’s surface and the interface, respectively. Specifically, in the direct neighborhood of the earth’s surface and of the considered interface, we replace the regular grid spacing in the vertical direction by the vertical distance between the surface (interface) point and the grid point under consideration. For the horizontal direction, however, only the regular grid points are used. As a result, the conventional upwind finite difference formulas are changed into the ones with nonuniform grid spacing. Furthermore, for capturing and propagating the local wavefront near the curved earth’s surface (interface), we adapt the fast marching method by introducing new point types, namely the surface point, the point above the surface, the interface point, and the point under the interface. If we use the scheme in a multistage fashion, we can compute not only the traveltimes of the first arrivals but also the traveltimes of the reflected and transmitted events. In comparison to the published schemes, our scheme has the following two advantages: (1) there is no need to construct a local unstructured grid for suturing the surface or the interface points to the neighboring regular grid points; (2) there is no need to make a local coordinate transform for capturing the local wavefront. Numerical results show that our scheme can treat the irregular region problem caused by the curved earth’s surface and by the curved interface with satisfactory effectiveness and flexibility.

Purpose The purpose of this paper is to develop a new finite difference method for solving the seismic wave propagation in fluid-solid media, which can be described by the acoustic and viscoelastic wave equations for the fluid and solid parts, respectively. Design/methodology/approach In this paper, the authors introduced a coordinate transformation method for seismic wave simulation method. In the new method, the irregular fluid–solid interface is transformed into a horizontal interface. Then, a multi-block coordinate transformation method is proposed to mesh every layer to curved grids and transforms every interface to horizontal interface. Meanwhile, a variable grid size is used in different regions according to the shape and the velocity within each region. Finally, a Lebedev-standard staggered coupled grid scheme for curved grids is applied in the multi-block coordinate transformation method to reduce the computational cost. Findings The instability in the auxiliary coordinate system caused by the standard staggered grid scheme is resolved using a curved grid viscoelastic wave field separation strategy. Several numerical examples are solved using this new method. It has been shown that the new method is stable, efficient and highly accurate in solving the seismic wave equation defined on domain with irregular fluid–solid interface. Originality/value First, the irregular fluid–solid interface is transformed into a horizontal interface by using the coordinate transformation method. The conversion between pressures and stresses is easy to implement and adaptive to different irregular fluid–solid interface models, because the normal stress and shear stress vanish when the normal angle is 90° in the interface. Moreover, in the new method, the strong false artificial boundary reflection and instability caused by ladder-shaped grid discretion are resolved as well.

AbstractIn this paper, we propose a multigrid algorithm based on the full approximate scheme for solving the membrane constrained obstacle problems and the minimal surface obstacle problems in the formulations of HJB equations. A Newton-Gauss-Seidel (NGS) method is used as smoother. A Galerkin coarse grid operator is proposed for the membrane constrained obstacle problem. Comparing with standard FAS with the direct discretization coarse grid operator, the FAS with the proposed operator converges faster. A special prolongation operator is used to interpolate functions accurately from the coarse grid to the fine grid at the boundary between the active and inactive sets. We will demonstrate the fast convergence of the proposed multigrid method for solving two model obstacle problems and compare the results with other multigrid methods.

This paper presents the thoughts about application of composite grid method to solve for current distribution of the dipole.It discusses the basic principle of the method and calculation steps, compares the calculated results with the normal method (FEM) of single set of grid computing , and summarizes the application process and development direction in the future.

Grid, as a new tool for solving problems, is applied to science, engineering, industries and commerce. More and more application programs are making use of grid infrastructure to meet their needs of computing, storage and other aspects. The effectiveness of a grid environment depends on the effectiveness and efficiency of its scheduler. Grid scheduling is the process that maps grid jobs to the resource in multiple administrative domains. The article does research on the principles and methods of grid scheduling and gives a description about grid resource section by way of quantitative evaluation.

The numerical solution of ElastoHydrodynamic Lubrication (EHL) point contact problems requires the solution of highly nonlinear systems of equations which pose a formidable computational challenge. Multigrid methods provide one efficient approach. EHL problems solved using a single grid and multigrid will be compared and contrasted with a homotopy method which works on the concept of deforming one problem into another by the continuous variation of a single parameter. Both the multigrid and the single grid method employ a new relaxation scheme. Numerical results on demanding test problems will be used to compare these methods and suggestions for future developments to produce robust solvers will be made.