Refinement Criteria Research Articles

A generic software design for Delaunay refinement meshing

This paper describes a generic software designed to implement meshing algorithms based on the Delaunay refinement paradigm. Such a meshing algorithm is generally described through a set of rules guiding the refinement of mesh elements. The central item of the software design is a generic class, called a mesher level, that is able to handle one of the rules guiding the refinement process. Several instantiations of the mesher level class can be stacked and tied together to implement the whole refinement process. As shown in this paper, the design is flexible enough to implement all currently known mesh generation algorithms based on Delaunay refinement. In particular it can be used to generate meshes approximating smooth or piecewise smooth surfaces, as well as to mesh three dimensional domains bounded by such surfaces. It also adapts to algorithms handling small input angles and various refinement criteria. This design highly simplifies the task of implementing Delaunay refinement meshing algorithms. It has been used to implemented several meshing algorithms in the Cgal library.

A p‐hierarchical adaptive procedure for the scaled boundary finite element method

AbstractThis paper describes a p‐hierarchical adaptive procedure based on minimizing the classical energy norm for the scaled boundary finite element method. The reference solution, which is the solution of the fine mesh formed by uniformly refining the current mesh element‐wise one order higher, is used to represent the unknown exact solution. The optimum mesh is assumed to be obtained when each element contributes equally to the global error. The refinement criteria and the energy norm‐based error estimator are described and formulated for the scaled boundary finite element method. The effectivity index is derived and used to examine quality of the proposed error estimator. An algorithm for implementing the proposed p‐hierarchical adaptive procedure is developed. Numerical studies are performed on various bounded domain and unbounded domain problems. The results reflect a number of key points. Higher‐order elements are shown to be highly efficient. The effectivity index indicates that the proposed error estimator based on the classical energy norm works effectively and that the reference solution employed is a high‐quality approximation of the exact solution. The proposed p‐hierarchical adaptive strategy works efficiently. Copyright © 2007 John Wiley & Sons, Ltd.

A Supra-Convergent Finite Difference Scheme for the Poisson and Heat Equations on Irregular Domains and Non-Graded Adaptive Cartesian Grids

We present finite difference schemes for solving the variable coefficient Poisson and heat equations on irregular domains with Dirichlet boundary conditions. The computational domain is discretized with non-graded Cartesian grids, i.e., grids for which the difference in size between two adjacent cells is not constrained. Refinement criteria is based on proximity to the irregular interface such that cells with the finest resolution is placed on the interface. We sample the solution at the cell vertices (nodes) and use quadtree (in 2D) or octree (in 3D) data structures as efficient means to represent the grids. The boundary of the irregular domain is represented by the zero level set of a signed distance function. For cells cut by the interface, the location of the intersection point is found by a quadratic fitting of the signed distance function, and the Dirichlet boundary value is obtained by quadratic interpolation. Instead of using ghost nodes outside the interface, we use directly this intersection point in the discretization of the variable coefficient Laplacian. These methods can be applied in a dimension-by-dimension fashion, producing schemes that are straightforward to implement. Our method combines the ability of adaptivity on quadtrees/octrees with a quadratic treatment of the Dirichlet boundary condition on the interface. Numerical results in two and three spatial dimensions demonstrate second-order accuracy for both the solution and its gradients in the L 1 and L ? norms.

Unstructured, anisotropic mesh generation for the Northwestern European continental shelf, the continental slope and the neighbouring ocean

A new mesh refinement strategy for generating high quality unstructured meshes of the Northwestern European continental shelf, the continental slope and the neighbouring ocean is presented. Our objective is to demonstrate the ability of anisotropic unstructured meshes to adequately address the challenge of simulating the hydrodynamics occurring in these three regions within a unique mesh. The refinement criteria blend several hydrodynamic considerations as the tidal wave propagation on the continental shelf and the hydrostatic consistency condition in steep areas. Several meshes illustrate both the validity and the efficiency of the refinement strategy. The selection of the refinement parameters is discussed. Finally, an attempt is made to take into account tidal ellipses, providing another cause for anisotropy in the mesh.

Controlled cost of adaptive mesh refinement in practical 3D finite element analysis

In this paper, attention is restricted to mesh adaptivity. Traditionally, the most common mesh adaptive strategies for linear problems are used to reach a prescribed accuracy. This goal is best met with an h-adaptive scheme in combination with an error estimator. In an industrial context, the aim of the mechanical simulations in engineering design is not only to obtain greatest quality but more often a compromise between the desired quality and the computation cost (CPU time, storage, software, competence, human cost, computer used). In this paper we propose the use of alternative mesh refinement with an h-adaptive procedure for 3D elastic problems. The alternative mesh refinement criteria allow to obtain the maximum of accuracy for a prescribed cost. These adaptive strategies are based on a technique of error in constitutive relation (the process could be used with other error estimators) and an efficient adaptive technique which automatically takes into account the steep gradient areas. This work proposes a 3D method of adaptivity with the latest version of the INRIA automatic mesh generator GAMHIC3D.

Adaptive finite element buckling analysis of folded plate structures

Publisher Summary This chapter presents the finite element linear buckling analysis of folded plate structures using adaptive h-refinement methods. The variable-node flat shell element used possesses a drilling D.O.F. that, in addition to improvement of the element behavior, permits an easy connection to other elements with six degrees of freedom per node. For the variable-node membrane elements with a drilling D.O.F., the Allman-type interpolations are used for the basic behavior of the element. The variable-node plate bending element is formulated based on Reissner–Mindlin plate theory to account for the shear deformation and has the substitute shear strain fields. The adaptive mesh refinement is normally performed after an initial solution has already been made available, and regions of the solution domain where the accuracy is not satisfactory have been identified according to a prescribed set of criteria. This process can be iterated using the last solution as the new initial solution. Two criteria are essential for adaptive mesh refinement—stopping criterion and refinement criterion. The stopping criterion is used to decide whether the obtained error is within the prescribed maximum error tolerance, and the refinement criterion is used to judge whether an element is to be refined at each iteration step.

Adaptive Mesh Refinement for One-Dimensional Three-Phase Flow with Phase Change in Porous Media

This article presents a detailed analysis of the implementation of the adaptive mesh refinement (AMR) technique to one-dimensional three-phase flows in heterogeneous porous media, including phase change. Since this is a particularly complex, highly nonlinear problem, many difficulties had to be resolved. A specific refining procedure and refinement criterion are proposed to deal with discontinuous saturations across interfaces between different rock types. In addition, we propose an alternative numerical algorithm to avoid difficulties in applying previous multilevel or multigrid algorithms to advance the solution on the AMR grids. Numerical examples, focused on steam injection modeling, show that the AMR technique is fast and has good accuracy.

An adaptive grid refinement strategy for the simulation of negative streamers

The evolution of negative streamers during electric breakdown of a non-attaching gas can be described by a two-fluid model for electrons and positive ions. It consists of continuity equations for the charged particles including drift, diffusion and reaction in the local electric field, coupled to the Poisson equation for the electric potential. The model generates field enhancement and steep propagating ionization fronts at the tip of growing ionized filaments. An adaptive grid refinement method for the simulation of these structures is presented. It uses finite volume spatial discretizations and explicit time stepping, which allows the decoupling of the grids for the continuity equations from those for the Poisson equation. Standard refinement methods in which the refinement criterion is based on local error monitors fail due to the pulled character of the streamer front that propagates into a linearly unstable state. We present a refinement method which deals with all these features. Tests on one-dimensional streamer fronts as well as on three-dimensional streamers with cylindrical symmetry (hence effectively 2D for numerical purposes) are carried out successfully. Results on fine grids are presented, they show that such an adaptive grid method is needed to capture the streamer characteristics well. This refinement strategy enables us to adequately compute negative streamers in pure gases in the parameter regime where a physical instability appears: branching streamers.

A parallel adaptive mesh refinement algorithm for predicting turbulent non-premixed combusting flows

A parallel adaptive mesh refinement (AMR) algorithm is proposed for predicting turbulent non-premixed combusting flows characteristic of gas turbine engine combustors. The Favre-averaged Navier–Stokes equations governing mixture and species transport for a reactive mixture of thermally perfect gases in two dimensions, the two transport equations of the k–ω turbulence model, and the time-averaged species transport equations, are all solved using a fully coupled finite-volume formulation. A flexible block-based hierarchical data structure is used to maintain the connectivity of the solution blocks in the multi-block mesh and facilitate automatic solution-directed mesh adaptation according to physics-based refinement criteria. This AMR approach allows for anisotropic mesh refinement and the block-based data structure readily permits efficient and scalable implementations of the algorithm on multi-processor architectures. Numerical results for turbulent non-premixed diffusion flames for a bluff-body burner are described and compared to available experimental data. The numerical results demonstrate the validity and potential of the parallel AMR approach for predicting complex non-premixed turbulent combusting flows.

Efficient load balancing for parallel adaptive finite-element electromagnetics with vector tetrahedra

The potential benefits of employing optimal discretization-based (ODB) refinement criteria for vector tetrahedra to achieve load balancing in three-dimensional parallel adaptive finite-element electromagnetic analysis are considered. Specifically, the ability of this class of adaption refinement criteria to resolve effective domain decompositions based on initial discretizations with only relatively few tetrahedra is examined for generalized vector Helmholtz systems. The effectiveness of the new load balancing method is demonstrated with adaptively refined finite-element meshes for benchmark systems.

Electromagnetic full particle code with adaptive mesh refinement technique: Application to the current sheet evolution

We have developed a new two and a half dimensional electromagnetic particle code with adaptive mesh refinement (AMR) technique in an effort to give a self-consistent description of the dynamic change of the plasma sheet in association with magnetic reconnection, which includes multi-scale processes from the electron scale to the magnetohydrodynamic scale. The AMR technique subdivides and removes cells dynamically in accordance with a refinement criterion and it is quite effective to achieve high-resolution simulations of phenomena that locally include micro-scale processes. Since the number of particles per cell decreases in the subdivided cells and a numerical noise increases, we subdivide not only cells but also particles therein and control the number of particles per cell. Our code is checked against several well-known processes such as the Landau damping of the Langmuir waves and we show that the AMR technique and particle splitting algorithm are successfully applied to the conventional particle codes. We have also examined the nonlinear evolution of the Harris-type current sheet and realized basically the same properties as those in other full particle simulations. We show that the numbers of cells and particles are greatly reduced so that the time to complete the simulation is significantly shortened in our AMR code, which enables us to conduct large-scale simulations on the current sheet evolution.

Detail control in line drawings of 3D meshes

We address the problem of rendering a 3D mesh in the style of a line drawing, in which little or no shading is used and instead shape cues are provided by silhouettes and suggestive contours. Our specific goal is to depict shape features at a chosen scale. For example, when mesh triangles project into the image plane at subpixel sizes, both suggestive contours and silhouettes may form dense networks that convey shape poorly. The solution we propose is to convert the input mesh to a multiresolution representation (specifically, a progressive mesh), then view-dependently refine or coarsen the mesh to control the size of its triangles in image space. We thereby control the scale of shape features that are depicted via silhouettes and suggestive contours. We propose a novel refinement criterion that achieves this goal and address the problem of maintaining temporal coherence of silhouette and suggestive contours when extracting them from a changing mesh.

Adaptive mesh refinement for the control of cost and quality in finite element analysis

Adaptive strategies are a necessary tool to make finite element analysis applicable to engineering practice. In this paper, attention is restricted to mesh adaptivity. Traditionally, the most common mesh adaptive strategies for linear problems are used to reach a prescribed accuracy. This goal is best met with an h-adaptive scheme in combination with an error estimator. In an industrial context, the aim of the mechanical simulations in engineering design is not only to obtain greatest quality but more often a compromise between the desired quality and the computation cost (CPU time, storage, software, competence, human cost, computer used). In this paper, we propose the use of alternative mesh refinement criteria with an h-adaptive procedure for 3D elastic problems. The alternative mesh refinement criteria (MR) are based on: prescribed number of elements with maximum accuracy, prescribed CPU time with maximum accuracy and prescribed memory size with maximum accuracy. These adaptive strategies are based on a technique of error in constitutive relation (the process could be used with other error estimators) and an efficient adaptive technique which automatically takes into account the steep gradient areas. This work proposes a 3D method of adaptivity with the latest version of the INRIA automatic mesh generator GAMHIC3D.

An adaptive finite element water column model using the Mellor–Yamada level 2.5 turbulence closure scheme

A one-dimensional water column model using the Mellor and Yamada level 2.5 parameterization of vertical turbulent fluxes is presented. The model equations are discretized with a mixed finite element scheme. Details of the finite element discrete equations are given and adaptive mesh refinement strategies are presented. The refinement criterion is an “a posteriori” error estimator based on stratification, shear and distance to surface. The model performances are assessed by studying the stress driven penetration of a turbulent layer into a stratified fluid. This example illustrates the ability of the presented model to follow some internal structures of the flow and paves the way for truly generalized vertical coordinates.

The Use of Solution Adaptive Grid for Modeling Small Scale Turbulent Structures

The use of large eddy simulation (LES) is computationally intensive and various studies demonstrated the considerable range of vortex scales to be resolved in an LES type of simulation. The purpose of this study is to investigate the use of a dynamic grid adaptation (DGA) algorithm. Despite many developments related to adaptive methods and adaptive grid strategies, the use of DGA in the context of turbulence modeling is still not well understood, and various profound problems with DGA in relation to turbulence modeling are still present. The work presented in this paper focuses on the numerical modeling of flow around a circular cylinder in the sub-critical flow regime at a Reynolds number of 3.9∙103. LES simulations with conventional mesh and DGA have been performed with various mesh sizes, refinement criteria and re-meshing frequency, to investigate the effects of re-meshing on the flow field prediction. The results indicate that the turbulent flow field is sensitive to modifications in the mesh and re-meshing frequency, and it is suggested that the re-meshing in the unsteady flow region is affecting the onset of small scale flow motions in the free shear layer.

Full particle simulation of the plasma sheet using adaptive mesh refinement technique

We have developed a new 2-1/2 dimensional electromagnetic particle code with adaptive mesh refinement (AMR) technique in an effort to give a self-consistent description of the dynamic change of the plasma sheet in association with magnetic reconnection, which includes multi-scale processes from the electron scale to the magnetohydrodynamic scale. The AMR technique dynamically subdivides the cells that satisfy a certain refinement criterion and it is quite effective to achieve high-resolution simulations such as those around the X-type neutral line. Our code is checked against several well known processes such as the Landau damping of the Langmuir waves and we show that the AMR technique is successfully applied to the conventional particle code. A preliminary result on the time evolution of the Harris-type current sheet using our code is also presented. The number of cells and particles are dramatically reduced by using the AMR.

Parallel complete remeshing for adaptive schemes

In order to improve the convergence ratio and to automate Finite Element Methods, several strategies have been introduced. One of these is the adaptive scheme. This approximation presents limitations for parallelism since the generation of a conformal, valid and well conditioned finite element mesh is a time consuming task, and now it appears as a main task in each iteration of the adaptation procedure. This work is motivated by the use of the h-adaptive method in its most flexible form, where a complete reconstruction of the whole mesh has to be performed whenever a solution over the current mesh has been obtained and until error criteria are achieved. We focused on the problem of the fast generation of tetrahedral unstructured meshes in a parallel fashion over geometric models with some given refinement criteria. The chosen strategy implies the use of an octal tree, octree, as a key hierarchical data structure to guide the algorithm. The codes have been developed using the MPI library in a SGI Origin 200 multiprocessor.

Refinement of Er3+-doped hole-assisted optical fiber amplifier

This paper deals with design and refinement criteria of erbium doped hole-assisted optical fiber amplifiers for applications in the third band of fiber optical communication. The amplifier performance is simulated via a model which takes into account the ion population rate equations and the optical power propagation. The electromagnetic field profile of the propagating modes is carried out by a finite element method solver. The effects of the number of cladding air holes on the amplifier performance are investigated. To this aim, four different erbium doped hole-assisted lightguide fiber amplifiers having a different number of cladding air holes are designed and compared. The simulated optimal gain, optimal length, and optimal noise fig. are discussed. The numerical results highlight that, by increasing the number of air holes, the gain can be improved, thus obtaining a shorter amplifier length. For the erbium concentration NEr=1.8x1024 ions/m3, the optimal gain G(Lopt) increases up to ~2dB by increasing the number of the air holes from M=4 to M=10.

Computation of Mean Drag for Bluff Body Problems Using Adaptive DNS/LES

We compute the time average of the drag in two benchmark bluff body problems: a surface mounted cube at Reynolds number 40000, and a square cylinder at Reynolds number 22000, using adaptive DNS/LES. In adaptive DNS/LES the Galerkin least-squares finite element method is used, with adaptive mesh refinement until a given stopping criterion is satisfied. Both the mesh refinement criterion and the stopping criterion are based on a posteriori error estimates of a given output of interest, in the form of a space-time integral of a computable residual multiplied by a dual weight, where the dual weight is obtained from solving an associated dual problem computationally, with the data of the dual problem coupling to the output of interest. No filtering is used, and in particular no Reynolds stresses are introduced. We thus circumvent the problem of closure, and instead we estimate the error contribution from subgrid modeling a posteriori, which we find to be small. We are able to predict the mean drag with an estimated tolerance of a few percent using about $10^5$ mesh points in space, with the computational power of a PC.

A parallel solution-adaptive scheme for multi-phase core flows in solid propellant rocket motors

The development of a parallel adaptive mesh refinement (AMR) scheme is described for solving the governing equations for multi-phase (gas–particle) core flows in solid propellant rocket motors (SRMs). An Eulerian formulation is used to describe the coupled motion between the gas and particle phases. A cell-centred upwind finite-volume discretization and the use of limited linear reconstruction, Riemann solver based flux functions for the gas and particle phases, and explicit multi-stage time-stepping allows for high solution accuracy and computational robustness. A Riemann problem is formulated for prescribing boundary data at the burning surface and a mesh adjustment algorithm has been implemented to adjust the multi-block quadrilateral mesh to the combustion interface. A flexible block-based hierarchical data structure is used to facilitate automatic solution-directed mesh adaptation according to physics-based refinement criteria. Efficient and scalable parallel implementations are achieved with domain decomposition on distributed memory multi-processor architectures. Numerical results are described to demonstrate the capabilities of the approach for predicting SRM core flows.