Variational method for adaptive mesh generation

A variational method is suggested for generating adaptive grids composed of hexahedral cells. The method is based on the minimization of a functional written on a manifold in a space whose variables are usual spatial coordinates in a physical domain and the components of a monitor vector function. A grid is constructed in the manifold, and its projection onto the physical domain yields an adaptive grid. Examples of adaptive grid generation are given.     

On 2D structured mesh generation by using mappings

A grid generation problem in two‐dimensional domains is considered by using a quasi‐conformal mapping of the parametric domain with a given square mesh onto the physical domain where the grid is required. To this end, a harmonic mapping is first applied, which, by the Radó theorem, is a diffeomorphism subject to some known conditions. However, the discrete harmonic mapping may produce folded meshes on a nonconvex domain with a strongly bent boundary. We demonstrate that it is caused by the truncation error. With the aim of controlling grid node location, an additional mapping is used. The Dirichlet problem for the universal elliptic partial differential equations is solved to construct the mapping. This allows to produce unfolded grids with a prescribed cell shape. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1072–1091, 2011     

Grid approximation of a parabolic convection-diffusion equation on a priori adapted grids: ε-uniformly convergent schemes

The boundary value problem for a singularly perturbed parabolic convection-diffusion equation is considered. A finite difference scheme on a priori (sequentially) adapted grids is constructed and its convergence is examined. The construction of the scheme on a priori adapted grids is based on a majorant of the singular component of the grid solution that makes it possible to a priori find a subdomain in which the grid solution should be further refined given the perturbation parameter ε, the size of the uniform mesh in x, the desired accuracy of the grid solution, and the prescribed number of iterations K used to refine the solution. In the subdomains where the solution is refined, the grid problems are solved on uniform grids. The error of the solution thus constructed weakly depends on ε. The scheme converges almost ε-uniformly; namely, it converges under the condition N ^?1 = o(ε^v), where v = v(K) can be chosen arbitrarily small when K is sufficiently large. If a piecewise uniform grid is used instead of a uniform one at the final Kth iteration, the difference scheme converges ε-uniformly. For this piecewise uniform grid, the ratio of the mesh sizes in x on the parts of the mesh with a constant size (outside the boundary layer and inside it) is considerably less than that for the known ε-uniformly convergent schemes on piecewise uniform grids.     

Least‐squares finite element method on adaptive grid for PDEs with shocks

We apply the least‐squares finite element method with adaptive grid to nonlinear time‐dependent PDEs with shocks. The least‐squares finite element method is also used in applying the deformation method to generate the adaptive moving grids. The effectiveness of this method is demonstrated by solving a Burgers' equation with shocks. Computational results on uniform grids and adaptive grids are compared for the purpose of evaluation. The results show that the adaptive grids can capture the shock more sharply with significantly less computational time. For moving shock, the adaptive grid moves correctly with the shock. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

An analytical methodology for assessment of smart monitoring impact on future electric power distribution system reliability

Smart grid is referred to a modernized power grid which can mitigate fault detection and allow self‐healing of the system without the intervention of operators. This article proposes an innovative analytical formulation using Markov method to evaluate electric power distribution system reliability in smart grids, which incorporates the impact of smart monitoring on the overall system reliability. An accurate reliability model of the main network components and the communication infrastructure have been also considered in the methodology. The proposed approach was applied to a well‐known test bed (Roy Billinton Test System) and various reliability case studies with monitoring provision and monitoring deficiency are analyzed. This article involves the developing possibilities of communication technologies and next‐generation control systems of the entire smart network based on the real‐time monitoring and modern control system to achieve a reliable, economical, safe, and high efficiency of electricity. The implementations indicate that using an appropriate set of the smart grid monitoring devices for power system components can virtually influence all the reliability indices although the amount of improvement varies between techniques. The proposed approach determined that smart monitoring for which components of the electric power distribution systems are tailored and deduce to major economical benefits. The described approach also reveals which reliability indices drastically influenced using monitoring. © 2014 Wiley Periodicals, Inc. Complexity 21: 99–113, 2015     

A numerical study of 2D integrated RBFNs incorporating Cartesian grids for solving 2D elliptic differential problems

This article reports a numerical discretization scheme, based on two‐dimensional integrated radial‐basis‐function networks (2D‐IRBFNs) and rectangular grids, for solving second‐order elliptic partial differential equations defined on 2D nonrectangular domains. Unlike finite‐difference and 1D‐IRBFN Cartesian‐grid techniques, the present discretization method is based on an approximation scheme that allows the field variable and its derivatives to be evaluated anywhere within the domain and on the boundaries, regardless of the shape of the problem domain. We discuss the following two particular strengths, which the proposed Cartesian‐grid‐based procedure possesses, namely (i) the implementation of Neumann boundary conditions on irregular boundaries and (ii) the use of high‐order integration schemes to evaluate flux integrals arising from a control‐volume discretization on irregular domains. A new preconditioning scheme is suggested to improve the 2D‐IRBFN matrix condition number. Good accuracy and high‐order convergence solutions are obtained. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

A survey of dynamically-adaptive grids in the numerical solution of partial differential equations

The construction of dynamically-adaptive curvilinear coordinate systems based on numerical grid generation and the use thereof in the numerical solution of partial differential equations is surveyed, and correlations are made among the various approaches. These adaptive grids are coupled with the physical solution being done on the grid so that the grid points continually move in the course of the solution in order to resolve developing gradients, or higher variations, in the solution. Particular attention is given to systems using elliptic grid generation based on variational principles. It is noted that dynamic grid adaption can remove the oscillations common when strong gradients occur on fixed grids, and that it appears that when the grid adapts to the solution most numerical solution algorithms work well. Particular applications in computational fluid dynamics and heat transfer are noted.     

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.     

Two‐grid mixed finite‐element methods for nonlinear Schrödinger equations

Two‐grid mixed finite element schemes are developed for solving both steady state and unsteady state nonlinear Schrödinger equations. The schemes use discretizations based on a mixed finite‐element method. The two‐grid approach yields iterative procedures for solving the nonlinear discrete equations. The idea is to relegate all of the Newton‐like iterations to grids much coarser than the final one, with no loss in order of accuracy. Numerical tests are performed. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 63‐73, 2012     

Block mesh refinement for incompressible flows in curvilinear domains

A method for the solution of the Navier–Stokes equation for the prediction of flows inside domains of arbitrary shaped bounds by the use of Cartesian grids with block-refinement in space is presented. In order to avoid the complexity of the body fitted numerical grid generation procedure, we use a saw tooth method for the curvilinear geometry approximation. By using block-nested refinement, we achieved the desired geometry Cartesian approximation in order to find an accurate solution of the N–S equations. The method is applied to incompressible laminar flows and is based on a cell-centred approximation. We present the numerical simulation of the flow field for two geometries, driven cavity and stenosed tubes. The utility of the algorithm is tested by comparing the convergence characteristics and accuracy to those of the standard single grid algorithm. The Cartesian block refinement algorithm can be used in any complex curvilinear geometry simulation, to accomplish a reduction in memory requirements and the computational time effort.     

Area control in generating smooth and convex grids over general plane regions

A new method (adaptive smoothness functional) to produce convex and smooth grids over general plane regions has been introduced in a recent work by the authors [10]; this method belongs to the variational grid generation approach. Theoretical results showing that a convex grid over a region is obtained when this method is applied, were presented; the basic assumption was that at least one convex grid exists. A procedure to control large cells (bilateral smoothness functional), in addition to smoothness and convexity, was also presented. Experimental results, showing the effectiveness of these methods, were reported; however, no theoretical results were reported assuring that the area control can be always exerted. This article continues the same line of research, introducing a new version of the bilateral smoothness functional that improves the control of large areas. Unlike the former method, theoretical results show the effectiveness of the new bilateral smoothness functional to exert such control. Optimal grids obtained with the new functional are compared with those reached using the older version, demostrating the improvement.     

Some aspects of grid generation

New results concerning the development of a universal method for grid generation based on the numerical solution of the inverted Beltrami and diffusion equations with respect to the monitor metric are obtained. In order to build monitor metrics, layer-type functions are used. Algorithms for generating smoothly matched block grids are proposed. Examples of two-and three-dimensional grids for the tokamak edge region, for calculation of a passive impurity in the atmosphere, and for the numerical solution of two-dimensional singularly perturbed problems are presented.     

A decoupling two‐grid algorithm for the mixed Stokes‐Darcy model with the Beavers‐Joseph interface condition

In this article, a decoupling scheme based on two‐grid finite element for the mixed Stokes‐Darcy problem with the Beavers‐Joseph interface condition is proposed and investigated. With a restriction of a physical parameter α, we derive the numerical stability and error estimates for the scheme. Numerical experiments indicate that such two‐grid based decoupling finite element schemes are feasible and efficient. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1066–1082, 2014     

Automatic parallel generation of tetrahedral grids by using a domain decomposition approach

An algorithm for the automatic parallel generation of three-dimensional unstructured grids based on geometric domain decomposition is proposed. A software package based on this algorithm is described. Examples of generating meshes for some application problems on a multiprocessor computer are presented. It is shown that the parallel algorithm can significantly (by a factor of several tens) reduce the mesh generation time. Moreover, it can easily generate meshes with as many as 5 × 10⁷ elements, which can hardly be generated sequentially. Issues concerning the speedup and the improvement of the efficiency of the computations and of the quality of the resulting meshes are discussed.     

Semi‐Unstructured Grids in the Laser Simulation Program LASCAD

Automatic grid generation is very important in industrial applications. Furthermore, in several applications discretization grids are needed which provide an anisotropic refinement and an accurate approximation of the derivatives of the solution. These requirements are fulfilled by semi‐unstructured grids. In this paper, we report how semi‐unstructured grids are used in the laser simulation program LASCAD [3].     

Efficient modelling of rotating disks and cylinders using a cartesian grid

Calculations are presented of flow characteristics in the vicinity of disks and cylinders rotating at speeds typical of those found in modern mechatronics machinery. The rotational speeds are slow or intermittent, and the generated boundary layers are laminar and transitional. Comparison is made with existing experimental data and exact, though idealised, analytical solutions. A three-dimensional finite volume procedure with time dependence was employed as the solution method, and two grid geometries were used, namely, axisymmetric and cartesian. Use of a cartesian grid is very important, as it is compatible with the design of the interiors of mechatronics machinery, and present practice is to model these interiors with computationally economical cartesian grids. Expanding grids were generated normal to surfaces for each of the grid geometries so as to capture the thin boundary layers. To alleviate numerical difficulties, when using the cartesian geometry, an expanding and contracting grid was generated normal to the axis of the disks and cylinders with the grid spacing based on a shifted Chebyshev polynomial.     

Boundary‐conforming coordinates with grid line control for acoustic scattering from complexly shaped obstacles

The current work sets forth a practical approach to numerically solve two‐dimensional direct acoustic scattering problems from complexly shaped scatterers with severe singularities, such as corners and cusps. First, boundary conforming coordinates are generated. This generation is performed through an elliptic grid generator algorithm, including control of the coordinate lines. The grid line control solely depends on the initial distribution of grid points. Following the grid generation process, the initial boundary value problem, modelling the scattering phenomenon, is formulated in terms of the new curvilinear coordinates, and a finite‐difference time domain method is implemented. The presence of the boundary singularities causes instability of the numerical method. However, by appropriately controlling the distance between grid lines in the vicinity of these singularities, stability and convergence are achieved. A semianalytical formula for the differential scattering cross‐section is obtained from the discrete Fourier transform of the computed scattered pressure field. The method is successfully applied to several interesting scatterers of various shapes. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Application of the spherical metric tensor to grid adaptation and the solution of applied problems

New results concerning the construction and application of adaptive numerical grids for solving applied problems are presented. The grid generation technique is based on the numerical solution of inverted Beltrami and diffusion equations for a monitor metric. The capabilities of the spherical metric tensor as applied to adaptive grid generation are examined in detail. Adaptive hexahedral grids are used to numerically solve a boundary value problem for the three-dimensional heat equation with a moving boundary in a continuous medium with discontinuous thermophysical parameters; this problem models the interaction of a thermal wave with a thermocouple embedded in the solid.     

Two‐Grid method for nonlinear parabolic equations by expanded mixed finite element methods

In this article, we develop a two‐grid algorithm for nonlinear reaction diffusion equation (with nonlinear compressibility coefficient) discretized by expanded mixed finite element method. The key point is to use two‐grid scheme to linearize the nonlinear term in the equations. The main procedure of the algorithm is solving a small‐scaled nonlinear equations on the coarse grid and dealing with a linearized system on the fine space using the Newton iteration with the coarse grid solution. Error estimation to the expanded mixed finite element solution is analyzed in detail. We also show that two‐grid solution achieves the same accuracy as long as the mesh sizes satisfy H = O(h^1/2). Two numerical experiments are given to verify the effectiveness of the algorithm. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013