Aprocedure has been developed to couple a hypersonic reacting flowmodel, a radiative heat transfermodel, and a surface ablation model to study the surface heat transfer and surface ablation rate of atmospheric reentry vehicles. The two-way loose-coupling algorithm is described for each of the models, as is the solution procedure to achieve convergence. Observations on the challenges of the loose-coupling strategy are given. Representative results are presented for two-dimensional benchmark examples and for three-dimensional flow at an angle of attack past a symmetric capsule based on the Crew Exploration Vehicle reentry vehicle. Effects due to the interaction with radiation and ablation are shown for two quantities of interest: the predicted peak surface heat flux and the ablation rate on the vehicle heat shield. Uncertain parameters are identified in each of the submodels, and a preliminary parameter sensitivity study is carried out by varying these values to examine their effects on the heat transfer and ablation rates in the coupled problem.

This study considers local post-processing strategies on contiguous patches of cells (or elements) and the properties and applications of the resulting approximations. For instance, one is interested in accuracy of recovered gradients and in their use in constructing error estimators or error indicators for adaptivity. Hence, there are two distinct concepts: (1) The improved quality of recovered gradients on a local patch; and/or (2) the use of local recovered quantities to construct error indicators to guide adaptive refinement schemes. We present a framework generalizing existing strategies for averaging operators and discuss the advantages of this framework in a parallel computing environment. Copyright (C) 2011 John Wiley & Sons, Ltd.

The behavior and properties of vortex pattern solutions to a benchmark Ginzburg-Landau model are investigated using hybrid continuation algorithms in conjunction with parallel adaptive mesh refinement (AMR) schemes to resolve the local vortices. The model is related to phase transition models arising in superconductivity and superfluids. The approach is based on a coupled variational formulation and finite element approximation scheme for the complex-valued solution. The associated algorithms implement continuation treatments based on the vortex scale coherence parameter and the winding number parameter in this model. Simulation results demonstrate the behavior of non-unique solutions, as characterized by different vortex configurations, and energy plots are used to display hysteresis effects. The complex-valued nature of the solution also serves to illustrate some interesting open questions related to AMR strategies and error indicators for complex-valued solution fields as well as other implications for such coupled systems. Copyright (C) 2009 John Wiley & Sons, Ltd.

case of a hollow-cylinder flare. Local predicted values of surface pressure and heat transfer are compared with available experimentally measured values for both geometries. These results are interpreted in the context of validation. Predicted results for complex flowfield/shock interactions and the viscous slip surface are illustrated through a computed schlieren image for the double cone. The present work presents the first known comparison of predictions from the finite element method to this set of data, and the validation study provides the comparative details to motivate future computational investigations and experimental studies.

We analyze the effect of mesh distortion on the condition number of the representative mass matrix M and stiffness matrix K arising in a typical finite element scheme. Bounds are stated for the respective condition numbers in terms of the Jacobian of the map from a reference element. These results are then used to construct the related bounds in terms of representative metrics for mesh distortion. These bounds are easily pre‐computable and provide a new explicit mathematical relation between matrix conditioning and mesh quality metrics. Numerical studies for a 2D test problem using a representative cell quality metric demonstrate the upper bound property and the dependence on cell quality for a quadrilateral cell. Analogous results for a 3D test problem under progressive symmetric mesh distortion of an interior hexahedral cell are also provided, as well as a study on a complex 3D geometry. We conclude by presenting practical adaptive mesh grading applications employing aforementioned mesh quality metrics. Copyright © 2010 John Wiley & Sons, Ltd.

Summary In this work we introduce a geometrical diagram to study the geometric quality of triangles generated by iterative application of the four Triangles Longest Edge (4TLE) partition. The diagram provides a convenient graphic tool to visualize the evolution and migration of element shapes leading to a better understanding of the improvement process and the efiect of recursive subdivision schemes. A complex variable mapping analysis supports the diagram and similarity class speciflcations. In addition, it is introduced a mesh subdivision method (hybrid 4TLE-SS) that combines the four Triangles Longest Edge (4TLE) subdivision pattern and the self-similar 4TSS. It is showed that the number of triangles of superior quality is greater than in the 4TLE method. The presented work is of interest in mesh generation and reflnement for triangle meshes.

A proportional‐integral‐derivative (PID) control approach is developed, implemented and investigated numerically in conjunction with continuation techniques for nonlinear problems. The associated algorithm uses PID control to adapt parameter stepsize for branch—following strategies such as those applicable to turning point and bifurcation problems. As representative continuation strategies, incremental Newton, Euler–Newton and pseudo‐arclength continuation techniques are considered. Supporting numerical experiments are conducted for finite element simulation of the ‘driven cavity’ Navier–Stokes benchmark over a range in Reynolds number, the classical Bratu turning point problem over a reaction parameter range, and for coupled fluid flow and heat transfer over a range in Rayleigh number. Computational performance using PID stepsize control in conjunction with inexact Newton–Krylov solution for coupled flow and heat transfer is also examined for a 3D test case. Copyright © 2009 John Wiley & Sons, Ltd.

A patch iteration approach and class of algorithms are described for inexpensively enhancing error indicators based on residuals and local boundary-value problems. The scheme is applicable to finite element, finite volume, finite difference and similar discretization schemes. It exploits ideas from multicolor/multilevel iterative relaxation at the patch level and more generally overlapping Schwarz subdomain iteration concepts. The scheme can be 'retrofitted' to existing serial or parallel codes to enhance local indicators, reduce pollution effects, yield improved stopping criteria for refinement or even simply to assess the reliability of a Computation on a given grid. Extrapolation techniques for further enhancing the patch iteration indicator and other extensions such as application to the adjoint solve step for dual formulations are also described. Copyright (C) 2007 John Wiley & Sons, Ltd.

This paper considers the numerical approximation of complex spatial patterns and rapidly evolving transients in chemotactic biological systems using parallel adaptive multiscale schemes and algorithms Transport processes in such biological systems are typically modeled by Coupled systems of nonlinear reaction-diffusion equations. For example, a model of this form has been proposed for Studying chemotaxis in bacteria colonies. In the present study, we develop a variational formulation for this model leading to an approximate finite element scheme with adaptive time stepping and local adaptive mesh refinement/coarsening algorithms. The parallel adaptive Solution algorithm is presented in detail and applied to investigate the effect of chemotaxis in spot formation behind concentric advancing concentrations fronts. Numerical results Concerning the accuracy, efficiency, and performance of the algorithm are also presented. Copyright (C) 2008 John Wiley & Sons, Ltd.

We examine Proportional-Integral- Derivative (PID) feedback control to enhance algorithm performance via parameter adap- tion. Of particular interest is use in branch- following algorithms applicable to turning point and bifurcation problems. Results of numeri- cal experiments are described for finite element approximation of a benchmark Navier-Stokes problem and for the classical Bratu reaction- diffusion problem. Continuation techniques provide a means of extending the range of flows for which a so- lution can be computed using a given nonli-

The Elder‐Voss‐Souza (EVS) problem is a standard benchmark for the numerical simulation of variable‐density flow and solute transport in a porous medium. However, a review of prior work shows that there is a wide variation in published results on plume structure that cannot entirely be explained by different choices of governing and constitutive equation or by discretization error and numerical error. This work presents results using a new numerical model and algorithms for a series of well‐calibrated comparison tests that are designed to elucidate the issue. In addition to the factors mentioned above, we consider the effects of initial and boundary conditions, domain geometry, and problem parameters such as the length of the source boundary segment and the buoyancy parameter in the associated Rayleigh number. It is found that these are key parameters determining the upwelling/downwelling plume behavior. More specifically, the values chosen for the EVS problem lie close to critical values that separate odd and even plume numbers. Consequently, perturbations to the model, mesh, and simulation variables may lead to different plume behavior.

An efficient numerical solution scheme based on a new generalized finite difference discretization and iterative strategies is developed for submicron semiconductor devices. As a representative model we consider a non-parabolic hydrodynamic system. The discretization is formulated in a mapped reference domain, and incorporates a transformed Scharfetter-Gummel treatment for the current density and energy flux. This permits the use of graded, nonuniform curvilinear grids in the physical domain of interest, which has advantages when gridding irregular domain shapes or solution profiles. The solution of the discrete system is carried out in a fully-coupled, implicit form, and non-symmetric gradient-type iterative strategies are investigated. Numerical results demonstrating the performance and reliability of the scheme are presented for ID and 2D test problems.

This paper evaluates the effects of reordering the unknowns on the convergence of preconditioned Krylov subspace methods for the solution of nonsymmetric linear sys- tems that arise from the finite element discretization of flow and transport. Of particular interest is the iterative solver behavior when adaptive mesh refinement (AMR) is utilized. Numerical studies are conducted using the object oriented AMR software system LibMesh with the PETSc Library. Using incomplete factorization preconditioners with several lev- els of fill-in, we investigate the effects of the Reverse Cuthill-McKee algorithm on GMRES, LCD and BICGSTAB methods. It is shown that the reordering applied in this finite ele- ment implementation with adaptive mesh refinement can reduce the number of iterations and, consequently, improve CPU time for some incomplete factorization preconditioners.

An adaptive mesh refmement (AMR) version of CTH is currently under development. This project is being conducted jointly by researchers at the University of Texas and at Sandia National Laboratories. The AMR version of CTH represents a significant milestone in the ten-year development of this legacy code. CTH is a multi-material wave propagation code used by many analysts in the DoD user community to simulate large deformations, large strain rates, and strong shocks in solid, liquids and gases. The numerical procedure is based on a finite volume formulation of very general forms of the continuum equations. As such, it can be applied to a wide variety of problems. The computational mesh used in CTH is Eulerian; materials and material interfaces are permitted to flow through the mesh as the calculation proceeds. The incorporation of new algorithms into the AMR version of CTH is described. A block refinement algorithm that preserves the location of material interfaces has been implemented into a working version of the code. This algorithm uses advanced interface tracking to map materials; this minimizes the dispersion normally associated with the process. A multimaterial advection algorithm, which is basically a generalization of Youngs' interface reconstruction method to cells with mismatched faces, is described. Results from three-dimensional examples problems are shown that effectively illustrates the improvements afforded by these new algorithms. ADAPTIVE STRATEGY When implementing adaptive refinement into an existing code, it is very important to consider the organization and data structure of the target application code. Here, the application code is CTH (McGlaun et al. 1990), a threedimensional multi-material Eulerian wave propagation code designed for modeling very large deformations and strong shocks. The data in CTH is organized in (l,J,K) logical blocks that correspond to the mesh used in the problem. Within a block, the mesh contours are constrained to remain parallel with the coordinate axes, and the introduction of hanging, or constrained, nodes is not permitted. However, adjacent blocks are permitted to have different values of l, J and K. Thus, a reasonable approach for the implementation of adaptivity, which preserves the original data structure used in CTH, is to limit refinement/unrefinement to the block level. Furthermore, in order to simplify the algorithms for communication between blocks, the refinement/unrefmement was limited to isotropic 2: I ratios between adjacent blocks. This process is illustrated in Fig. I, where a set of communicating blocks is shown. The contents of the ghost cells along the periphery of the blocks are provided by information coming from adjacent blocks. Since these adjacent blocks may be at a different resolution, calculating the contents of the ghost cells may involve a cell split/combine process. r-r-,..··T ·r0.: 'TT.'·''''.lTTT ..'::I i ''''"";",,),, " ....... ! ,~It "1 i·i······ ;-"i I"" ,J , ' 1.1 i ~1 '-i ~ :! tl.······.. ,.· ... · ·,······!··· ;I~VC! ,····'·i::

Computational results are compared to experimental benchmark results for natural convection of a Newtonian fluid in a cubical cavity. These results are then extended to Powell-Eyring and extended Williamson fluids. Good agreement is seen between the experimental and computational results for most of the Newtonian cases. Results from the non-Newtonian cases are proposed for comparison to future experiments. An issue raised by the Newtonian experimental study is identified and a probable resolution is described. Comparison of Newtonian and non-Newtonian results shows increased heat flux with increasing nonlinearity of the models. The non-Newtonian cases for the diagonally inclined orientation also show periodic behavior not seen in the Newtonian cases.

We examine the propagation of local adaptive mesh refinement (AMR) under a longest edge conformity scheme. Supporting numerical studies are included and discussed. Of specific interest is the statistical behaviour of the propagation zone in AMR of simplicial meshes. To this end three propagation metrics are used: the total number of original triangles in the propagation paths emanating from any target element, the longest individual edge path, and the extent of secondary refinement due to the conformity. Copyright (c) 2006 John Wiley & Sons, Ltd.

In this part we consider the dilute surfactant model developed in Part I and construct a variational formulation and mixed finite element scheme to obtain approximate solutions. In particular, we consider the stability regimes identified in the linear stability analysis of Part I and conduct numerical experiments to explore the nature of stability for the approximate solutions in these regimes. Both 1D and 2D simulation results are provided to illustrate the behaviour. Copyright (c) 2004 John Wiley & Sons, Ltd.

The purpose of this work is to analyse the parameter sensitivity problem for a class of nonlinear elliptic partial differential equations, and to show how numerical simulations can help to optimize experiments for the estimation of parameters in such equations. As a representative example we consider the Laplace–Young problem describing the free surface between two fluids in contact with the walls of a bounded domain, with the parameters being those associated with surface tension and contact. We investigate the sensitivity of the solution and associated functionals to the parameters, examining in particular under what conditions the solution is sensitive to parameter choice. From this, the important practical question of how to optimally design experiments is discussed; i.e. how to choose the shape of the domain and the type of measurements to be performed, such that a subsequent inversion of the measured data for the model parameters yields maximal accuracy in the parameters. We investigate this through numerical studies of the behaviour of the eigenvalues of the sensitivity matrix and their relation to experimental design. These studies show that the accuracy with which parameters can be identified from given measurements can be improved significantly by numerical experiments. Copyright © 2005 John Wiley & Sons, Ltd.

We develop a model for surface tension driven flow induced by an insoluble surfactant monolayer on a heated thin fluid layer. The mathematical model is based on a perturbation analysis for a thin fluid layer. The resulting model involves coupling of flow and heat transfer to an additional transport equation for surfactant concentration on the surface. We develop the stability analysis of this coupled system. We characterize the stability behaviour and induced wave motion into four parametric regions based on linear stability analysis. A finite element formulation and numerical studies of the behaviour in the various stability regimes are given in Part II. Copyright (c) 2004 John Wiley & Sons, Ltd.