Performance Analysis of FEM Solvers on Practical Electromagnetic Problems

Adaptive characteristic spatial quadrature schemes for use in discrete ordinates neutral particle transport computer codes in slab geometry have been introduced recently; the extension to two-dimensional Cartesian coordinates with rectangular cells is presented here. These quadratures differ from previous schemes in using strictly non-negative finite-element-like representations of the entering fluxes and scattering source, which adapt to match both the zeroth and first spatial moments. (These input moments are obtained from the boundary conditions or as output moments of the upstream adjacent cells and from the angular quadrature of the spatial flux moments in the outer iteration, respectively.) The flux throughout the cell is found by integration of the transport equation along streaming (characteristic) lines. The moments of the flux through the cell and along the exiting boundaries are then obtained. The resulting quadratures are nonlinear and strictly non-negative, while satisfying zeroth-spatial-moment and first-spatial-moment balances within each cell and globally. For a square test problem 32 mean free paths thick, the variation in results over a range of meshes from 1 by 1 to 8 by 8 was comparable to (usually less than) the corresponding variation due to angular quadrature (S4, S8, S16). Linear adaptive quadrature performed better than step adaptive, with few exceptions. Using a mesh consisting of a single cell, both adaptive methods performed as well as or better than linear nodal and linear characteristic methods did with an 8 by 8 mesh of cells. Although they are substantially more complicated to derive and implement, the adaptive methods cost only three to nine times more per cell to compute, and require no additional storage, as compared to linear nodal and linear characteristic methods. Thus, by using these coarser meshes, step and linear adaptive were more cost-effective than linear nodal or linear characteristic (with or without negative fixups). Linear adaptive is similar to linear characteristic (but with a bilinear term) wherever linear characteristic requires no negative flux or source fixups. Thus, it should share the excellent convergence of linear characteristic. For an optically thin test problem, with a fine mesh (up to 32 by 32 cell mesh on a square 2 mfp thick), we found that linear adaptive was slightly more accurate than linear characteristic without fixup, which was substantially different than linear characteristic with fixup. This test showed that, even with a fine mesh, fixups can be required by conventional methods (linear nodal and linear characteristic), which either have negative fluxes or violate first-spatial-moment balance. Linear adaptive has no negative fluxes and conserves first spatial moments; its results were very close to those of linear characteristic (without fixup), for optically thin cells. Step adaptive converges somewhat more slowly, however; so it is less efficient than the other methods when both are used on a fine enough mesh. We conclude that linear adaptive quadrature can be an accurate and cost-effective method for problems requiring the use of optically thick cells.

Physical interactions between turbulence and shock waves are very complex phenomena. If these interactions take place in chemically reacting flows, the degree of complexity increases dramatically. Examples of applications may be cited in the area of supersonic combustion, in which the controlled generation of turbulence and/or large scale vortices in the mixing and flame-holding zones is crucial for efficient combustion. Equally important, shock waves interacting with turbulence and chemical reactions affect the combustor flowfield resulting in enhanced relaxation and chemical reaction rates. Chemical reactions in turn contribute to dispersion of shock waves and reduction of turbulent kinetic energies. Computational schemes to address these physical phenomena must be capable of resolving various length and time scales. These scales are widely disparate and the most optimum approach is found in explicit/implicit adjustable schemes for the Navier-Stokes solver. This is accomplished by means of the generalized Taylor-Galerkin (gtg) finite element formulations. Adaptive meshes are used in order to assure efficiency and accuracy of solutions. Various benchmark problems are presented for illustration of the theory and applications. Geometries of ducted rockets, supersonic diffusers, flame holders, and hypersonic inlets are included. Merits of proposed schemes are demonstrated through these example problems.

A dimensionally adaptive technique for computationally efficient cylindrical discrete ordinates radiation calculations

ABSTRACTA framework is proposed to bridge coarse and fine meshes in a single simulation within the phase field method for fracture. Fine meshes are used in the vicinity of localized defects to accurately capture crack initiation, while coarse meshes are used away from initial defects and include only crack propagation paths. This reduces the prohibitive computational times associated with uniformly fine meshes over the entire domain, or with the use of complex adaptive meshes in the phase field method. The coupling between the two overlapping meshes is achieved using a variational formulation in which the energies of the models associated with the fine and coarse meshes are weighted in the superposition zone. Two situations are considered. The first includes the resolution of the phase field problem only in the fine mesh, while the coarse mesh is limited to the undamaged elastic problem. In the second situation, cracks can propagate in both the fine and coarse mesh. Variational formulations and associated finite element implementations are detailed. Numerical examples are presented, showing the potential of this approach to significantly reduce computational costs in the phase field method for cracking without affecting the accuracy.

An adaptive mesh strategy for singularly perturbed convection diffusion problems

The present work is devoted to the damped Newton method applied for solving a class of non-linear elasticity problems. Following the approach suggested in earlier related publications, we consider a two-level procedure which involves (i) solving the non-linear problem on a coarse mesh, (ii) interpolating the coarse-mesh solution to the fine mesh, (iii) performing non-linear iterations on the fine mesh. Numerical experiments suggest that in the case when one is interested in the minimization of the L2-norm of the error rather than in the minimization of the residual norm the coarse-mesh solution gives sufficiently accurate approximation to the displacement field on the fine mesh, and only a few (or even just one) of the costly non-linear iterations on the fine mesh are needed to achieve an acceptable accuracy of the solution (the accuracy which is of the same order as the accuracy of the Galerkin solution on the fine mesh). Copyright © 2000 John Wiley & Sons, Ltd.

Some aspects of adaptive grid technology related to boundary and interior layers

An adaptive accuracy-based a posteriori error estimator

Robust adaptive mesh refinement for transonic conditions is one of the most difficult challenges to advancing CFD usage for certification by analysis. Transonic fluid-structure interaction cases represent an even more difficult challenge for adaptive mesh refinement. Production aircraft furthermore have structural models that have gaps and discontinuities that make the interdomain transfer of loads and deflections difficult. This adds to the challenge of adaptive mesh refinement. An initial attempt to create an adaptive mesh refinement process that refines regions of high flow gradient based around the HeldenMesh software is presented. The method is applied to the Integrated Adaptive Wing Technology Maturation (IAWTM) configuration. That has developed an aspect ratio 13.5 wing aeroelastic wind-tunnel model for testing in the Transonic Dynamics Tunnel (TDT). The high levels of deformation of this wing make the adaptation of meshes for the static shape difficult and has necessitated some creativity in how to efficiently adapt meshes. A time-domain identification of the unsteady system based on flexible modes using the adapted meshes is performed.

The BEM-based Finite Element Method is one of the promising strategies which are applicable for the approximation of boundary value problems on general polygonal and polyhedral meshes. The flexibility with respect to meshes arises from the implicit definition of trial functions in a Trefftz-like manner. These functions are treated locally by means of Boundary Element Methods (BEM). The following presentation deals with the formulation of higher order trial functions and their application in uniform and adaptive mesh refinement strategies. For the adaptive refinement, a residual based error estimate is used on general polygonal meshes for the higher order, conforming trial functions. The first numerical results, in the context of adaptive refinement, show optimal rates of convergence with respect to the number of degrees of freedom.

Opposition-based Magnetic Optimization Algorithm with parameter adaptation strategy

This chapter focuses on numerical treatment of a special type of boundary value problems referred to as singularly perturbed differential equations. The solutions of this type of differential equations exhibit sharp layers and pose challenges for conventional numerical methods. In this chapter, the construction of layer-adapted meshes—(a) a priori refined meshes, and (b) a posteriori refined meshes have been discussed. We highlight the difference between the construction of these layer-adapted meshes. The main constituent for constructing a posteriori refined mesh is to choose an appropriate monitor function. We discuss how to extract probable monitor functions from given error estimates for mesh adaptation. Furthermore, in conjunction with an algorithm based on equidistribution principle, we show different stages of mesh adaptation for various test problems. The extension of these layer-adapted meshes has been discussed, and the role of a monitor metric for two-dimensional mesh adaptation is demonstrated by employing BAMG (Bidimensional Anisotropic Mesh Generator), a widely used code for mesh adaptation for two-dimensional mesh adaptation. The codes are provided for further investigations and research.

Mesh refinement for anisotropic diffusion in magnetized plasmas

ABSTRACTMesh adaption is a powerful tool for efficient unstructured-grid computations but causes load imbalance among proces-sors on a parallel machine. We present a novel method todynamically balance the processor workloads with a globalview. This paper presents, for the first time, the imple-mentation and integration of all major components withinour dynamic load balancing strategy for adaptive grid cal-culations. Mesh adaption, repartitioning, processor assign-ment, and remapping are critical components of the frame-work that must be accomplished rapidly and efficiently soas not to cause a significant overhead to the numerical sim-ulation. Previous results indicated that mesh repartitioningand data remapping are potential bottlenecks for perform-ing large-scale scientific calculations. We resolve these issuesand demonstrate that our framework remains viable on alarge number of processors.1 INTRODUCTIONDynamic mesh adaption on unstructured grids is a power-ful tool for computing unsteady three-dimensional problemsthat require grid modifications to efficiently resolve solu-tion features. By locally refining and coarsening the meshto capture flowfield phenomena of interest, such proceduresmake standard computational methods more cost effective.Highly refined meshes are required to accurately captureshock waves, contact discontinuities, vortices, and shear lay-ers. Local mesh adaption provides the opportunity to obtainsolutions that are comparable to those obtained on globally-refined grids but at a much lower cost.Unfortunately, the adaptive solution of unsteady prob-lems causes load imbalance among processors on a paral-lel machine. This is because the computational intensity isboth space and time dependent. An efficient parallel imple-mentation of such methods is extremely difficult to achieve,primarily because of the dynamically-changing nonuniformgrid. Various methods on dynamic load balancing have beenreported to date [5-9,11-14,16-18,24-26]; however, most ofthem either lack a global view of loads across processors orProceedings of the 9th ACM Symposium on Parallel Al-gorithms and Architectures, Newport, Rhode Island, June22-25, 1997.do not apply their techniques to realistic large-scale appli-cations.Figure 1 depicts our framework for parallel adaptive flowcomputation. It consists of a flow solver and a mesh adap-tor, with a partitioner and a remapper that load balancesand redistributes the computational mesh when necessary.Our goal is to build a portable system for efficiently per-forming adaptive large-scale flow calculations in a parallelmessage-passing environment. The mesh is first partitionedand mapped among the available processors. The flow solverthen runs for several iterations, updating solution variables.Once an acceptable solution is obtained, the mesh adaptionprocedure is invoked. It first targets edges for coarseningand refinement based on an error indicator computed fromthe flow solution. The old mesh is then coarsened, resultingin a smaller grid. Since edges have already been marked forrefinement, it is possible to exactly predict the new meshbefore actually performing the refinement step. Programcontrol is thus passed to the load balancer at this time. Aquick evaluation step determines if the new mesh will be sounbalanced as to warrant a repartitioning. If the currentpartitions will remain adequately load balanced, control mpassed back to the subdivision phase of the mesh adaptor.Otherwise, a repartitioning procedure is used to divide thenew mesh into subgrids. The new partitions are then reas-signed to the processors in a way that minimizes the costof data movement. If the remapping cost is less than thecomputational gain that would be achieved with balancedpartitions, all necessary data is appropriately redistributed.Otherwise, the new partitioning is discarded. The computa-tional mesh is then actually refined and the flow calculationis restarted.Notice from the framework in Fig. 1 that splitting themesh refinement step into two distinct phases of edge mark-ing and mesh subdivision allows the subdivision phase tooperate in a more load balanced fashion. In addition, sincedata remapping is performed before the mesh grows in sizedue to refinement, a smaller volume of data is moved. Thiscan lead to a potentially significant savings in the redistribu-tion cost. The load balancer also balances the computationalload for the flow solver while reducing the runtime commu-nication. This is important because flow solvers are usuallyseveral times more expensive than mesh adaptors. In anycase, it is obvious that mesh adaption, repartitioning, pro-cessor assignment, and remapping are critical componentsof the framework and must be accomplished rapidly and ef-ficiently so as not to cause a significant overhead to the flowcomputation.

On the determination of the mesh size for numerical simulations of shock wave propagation in near field underwater explosion