A high-order, matrix-free implicit method has been developed for the transient solutions of hyperbolic conservation laws. The discontinuous Galerkin method is applied for temporal discretization. This method has the advantage that its discretization error is O(Delta t(2p+1)) when a polynomial basis of degree p is used for time discretization. The nonlinear system of equations from the implicit time discretization is solved at each time step using a nonlinear Krylov subspace projection method. The system of linear equations is solved by the generalized minimum residual algorithm with a lower-upper symmetric Gauss-Seidel preconditioner. The numerical results from the inviscid Burgers' equation indicate that the implicit method is several times faster in performance relative to explicit integration by the total variation diminishing Runge-Kutta method of order 3. The forward Euler method requires a time step proportional to the square of the spatial step for stability with equations such as Burgers' equation (J. Sci. Comput. 2001; 16:173-261). It would, hence, be much less efficient than other explicit methods, for example, Cockburn and Shu (Math. Comput. 1989; 52:411-435), which would only require a time step proportional to the spatial step. Copyright (C) 2009 John Wiley & Sons, Ltd.

We develop a mathematical model for the oxidation of silicon carbide in a crack or pore. The model consists of a nonlinear partial differential system that is solved by adaptive finite element software that automates many of the computational decisions.

Proceedings of the 18th Annual Conference on Composites and Advanced Ceramic Materials—B: Ceramic Engineering and Science Proceedings, Volume 15 Chapter 39 Adaptive Numerical Techniques for Reactive Vapor Infiltration S. Adjerid, S. Adjerid Scientific Computation Research Center Center for Composite Materials and Structures Rensselaer Polytechnic Institute, Troy, New York 12180, USASearch for more papers by this authorJ. E. Flaherty, J. E. Flaherty Scientific Computation Research Center Center for Composite Materials and Structures Rensselaer Polytechnic Institute, Troy, New York 12180, USASearch for more papers by this authorM. S. Shephard, M. S. Shephard Scientific Computation Research Center Center for Composite Materials and Structures Rensselaer Polytechnic Institute, Troy, New York 12180, USASearch for more papers by this authorY. J. Wang, Y. J. Wang Scientific Computation Research Center Center for Composite Materials and Structures Rensselaer Polytechnic Institute, Troy, New York 12180, USASearch for more papers by this authorW. Hillig, W. Hillig Scientific Computation Research Center Center for Composite Materials and Structures Rensselaer Polytechnic Institute, Troy, New York 12180, USASearch for more papers by this authorJ. Hudson, J. Hudson Scientific Computation Research Center Center for Composite Materials and Structures Rensselaer Polytechnic Institute, Troy, New York 12180, USASearch for more papers by this authorN. Patibandla, N. Patibandla Scientific Computation Research Center Center for Composite Materials and Structures Rensselaer Polytechnic Institute, Troy, New York 12180, USASearch for more papers by this author S. Adjerid, S. Adjerid Scientific Computation Research Center Center for Composite Materials and Structures Rensselaer Polytechnic Institute, Troy, New York 12180, USASearch for more papers by this authorJ. E. Flaherty, J. E. Flaherty Scientific Computation Research Center Center for Composite Materials and Structures Rensselaer Polytechnic Institute, Troy, New York 12180, USASearch for more papers by this authorM. S. Shephard, M. S. Shephard Scientific Computation Research Center Center for Composite Materials and Structures Rensselaer Polytechnic Institute, Troy, New York 12180, USASearch for more papers by this authorY. J. Wang, Y. J. Wang Scientific Computation Research Center Center for Composite Materials and Structures Rensselaer Polytechnic Institute, Troy, New York 12180, USASearch for more papers by this authorW. Hillig, W. Hillig Scientific Computation Research Center Center for Composite Materials and Structures Rensselaer Polytechnic Institute, Troy, New York 12180, USASearch for more papers by this authorJ. Hudson, J. Hudson Scientific Computation Research Center Center for Composite Materials and Structures Rensselaer Polytechnic Institute, Troy, New York 12180, USASearch for more papers by this authorN. Patibandla, N. Patibandla Scientific Computation Research Center Center for Composite Materials and Structures Rensselaer Polytechnic Institute, Troy, New York 12180, USASearch for more papers by this author Book Editor(s):John B. Wachtman Jr., John B. Wachtman Jr.Search for more papers by this author First published: 01 January 1994 https://doi.org/10.1002/9780470314555.ch39Citations: 4Book Series:Ceramic Engineering and Science Proceedings AboutRelatedInformationPDFPDFPDFPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShareShare a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Proceedings of the 18th Annual Conference on Composites and Advanced Ceramic Materials—B: Ceramic Engineering and Science Proceedings, Volume 15 RelatedInformation

Various research applications using the adaptive solution of partial diierential equations and parallel discrete event simulation as well as the investigation of object oriented programming paradigms are described. This work utilizes the IBM SP2 at Rensselaer's Scientiic Computation Research Center (SCOREC). The software is designed using C, C++ and Fortran 90, together with MPI. Tools described include a parallel mesh database, load balancers and run-time code optimizers. Additionally, using plasma codes as an example, we compare compilers and architecture performance of the SP2 to competing distributed memory parallel machines. Applications studying nite element analysis of problems in biomechanics and uid dynamics, the simulation of fusion plasma processes and the spread of Lyme disease are presented. The nite element method (FEM) has become a standard engineering analysis tool for solving partial diierential equations (PDEs). Adaptive FEMs have gained importance because they provide greater accuracy, reliability, robustness, and time and space eeciency. In such an analysis, the computational domain is rst discretized to create a mesh. During the solution process, portions of the discrete domain are spatially reened or coarsened (h-reenement), the method order is varied (p-reenement), and/or the mesh is moved to follow evolving phenomena (r-reenement), each of which concentrates the computational eeort in regions where the solution resolution is inadequate 1]. In order to solve large problems within a reasonable period of time, these methods have been implemented on parallel computers. While parallelism can yield faster solution times or greater accuracy in the same computing time, there are complications such as coordinating interprocessor communication and managing distributed data structures. The standard methodology for optimizing parallel FEM programs relies on the initial partitioning of the meshes involved in the computation. However, in parallel adaptive codes, a good initial partitioning is not suucient to assure high performance throughout the computation. Load imbalance caused by adaptive enrichment necessitates a dynamic redistribution of data. Our focus is to build an eecient load balancer for parallel adaptive methods that can be used to solve complex problems. The tools that have been developed at Rensselaer Polytechnic Institute to facilitate this focus are described in the following sections.

We describe a procedure for the adaptive h-refinement solution of the incompressible MHD equations in stream function form using a stabilized finite element formulation. The mesh is adapted based on a posteriori spatial error estimates of the magnetic field using both recovery and order extrapolation techniques. The step size for time integration is chosen so that temporal discretization errors are small relative to spatial errors. The adaptive procedure is applied to study singular current sheets in the tilt instability problem of ideal magnetohydrodynamics. Numerical results indicate a more accurate resolution of current sheets with higher-order methods than with piecewise-linear approximations.

Data partitioning and load balancing are important components of parallel computations. Many different partitioning strategies have been developed, with great effectiveness in parallel applications. But the load-balancing problem is not yet solved completely; new applications and architectures require new partitioning features. Existing algorithms must be enhanced to support more complex applications. New models are needed for non-square, non-symmetric, and highly connected systems arising from applications in biology, circuits, and materials simulations. Increased use of heterogeneous computing architectures requires partitioners that account for non-uniform computing, network, and memory resources. And, for greatest impact, these new capabilities must be delivered in toolkits that are robust, easy-to-use, and applicable to a wide range of applications. In this paper, we discuss our approaches to addressing these issues within the Zoltan Parallel Data Services toolkit.

We address the problem of partitioning and dynamic load balancing on clusters with heterogeneous hardware resources. We propose DRUM, a model that encapsu- lates hardware resources and their interconnection topology. DRUM provides mon- itoring facilities for dynamic evaluation of communication, memory, and processing capabilities. Heterogeneity is quantifled by merging the information from the monitors to produce a scalar number called \power." This power allows DRUM to be used easily by existing load-balancing procedures such as those in the Zoltan Toolkit while placing minimal burden on application programmers. We demonstrate the use of DRUM to guide load balancing in the adaptive solution of a Laplace equation on a heterogeneous cluster. We observed a signiflcant reduction in execution time compared to traditional methods.

We use an artificial viscosity term to stabilize discontinuous Galerkin solutions of hyperbolic conservation laws in the presence of discontinuities. Viscous coefficients are selected to minimize spurious oscillations when a kinematic wave equation is subjected to piecewise constant initial data. The same strategy is used with a local linearization in more complex situations. Several one and two-dimensional flow problems illustrate performance. A shock detection scheme [L. Krivodonova, J. Xin, J.-F. Remacle, N. Chevaugeon, J.E. Flaherty, Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws, Appl. Numer. Math. 48 (2004) 323-338] further sharpens results near discontinuities.

We present the implementation of well-conditioned hierarchical bases for one-dimensional, triangular and tetrahedral elements in finite element FEMLAB software. Using the domain mesh information provided by FEMLAB, we found an easy way to maintain the continuity of solution across the interelement boundaries. The conditionings of the global stiffness matrices of several standard problems are compared with the Lagrange bases and are smaller for all cases.

We describe procedures to model transient shock interaction problems using discontinuous Galerkin methods to solve the compressible Euler equations. The problems are motivated by blast flows surrounding cannons with perforated muzzle brakes. The goal is to predict shock strengths and blast over pressure. This application illustrates several computational difficulties. The software must handle complex geometries. The problems feature strong interacting shocks, with pressure ratios on the order of 1000 as well as weaker precursor shocks traveling rearward that also must be accurately captured. These aspects are addressed using anisotropic mesh adaptation. A shock detector is used to control the adaptation and limiting. We also describe procedures to track projectile motion in the flow by a level-set procedure.

We consider problems involving high-speed moving rigid objects in a fluid flow. Instead of a more traditional deformable mesh approach, we describe a novel fixed mesh approach which use a level set function to implicitly track the fluid-solid interface, therefore no mesh motion or modification is required. The interface boundary conditions are also captured implicitly by combining a Ghost Fluid technique with the level set approach. This fixed mesh approach is a much more efficient than moving mesh methods since no mesh modification is necessary. It has a wide-spread applicability for problems with complex object motion and/or complex geometry. It is also dimension free and relatively simple to implement relative to moving mesh approaches. The discontinuous Galerkin method (DGM) is used to discretize the Euler equations associated with a compressible inviscid fluid.

In this paper, we consider the partial differential equations approach for valuing European and American style options on multiple assets. We use a method of lines finite element implementation available in the software package Femlab in order to solve the variational inequality that characterizes the American style option, as well as the partial differential equation that defines the European style option, for two and three state variables. A detailed study of the approximation error is provided, including a theoretical estimate, an asymptotic analysis, the space–time distribution, and the dependence on the size of the truncation domain.

Melt flows associated with a Czochralski crystal growth process was investigated to better understand the transition from a steady laminar regime to an unsteady one as the Grashof number increases. The kinetic energy of the flow as a function of time was examined as an indication of stability. Three-dimensional solutions were interpolated onto a two-dimensional unstructured mesh to compute the Reynolds average mean flow and its fluctuations. Our simulations showed that the transition to unsteady three-dimensional flow begins at a Grashof number of approximately 3.0 million. At higher Grashof numbers (e.g., 6.6 million), the melt flow is fully unsteady, three-dimensional turbulent flow. The simulations further indicated that the melt flow at a Grashof number of 6.6 million is statistically stable, which suggested the Reynolds quasi-steady assumption is valid in this case.

We present a high-order formulation for solving hyperbolic conservation laws using the discontinuous Galerkin method (DGM). We introduce an orthogonal basis for the spatial discretization and use explicit Runge--Kutta time discretization. Some results of higher order adaptive refinement calculations are presented for inviscid Rayleigh--Taylor flow instability and shock reflection problems. The adaptive procedure uses an error indicator that concentrates the computational effort near discontinuities.

We study the problem of smoothing finite element meshes of triangles and tetrahedra, where vertices are recursively moved to improve the overall quality of the elements with respect to a given shape quality metric. We propose a geometric approach to solving the local optimization problem. Level sets of the given metric are used to characterize the set of optimal point(s). We also introduce a new mesh quality metric for tetrahedra.

We describe a strategy for detecting discontinuities and for limiting spurious oscillations near such discontinuities when solving hyperbolic systems of conservation laws by high-order discontinuous Galerkin methods. The approach is based on a strong superconvergence at the outflow boundary of each element in smooth regions of the flow. By detecting discontinuities in such variables as density or entropy, limiting may be applied only in these regions; thereby, preserving a high order of accuracy in regions where solutions are smooth. Several one- and two-dimensional flow problems illustrate the performance of these approaches.

A greater understanding of the rate at which emerging disease advances spatially has both ecological and applied significance. Analyzing the spread of vector-borne disease can be relatively complex when the vector's acquisition of a pathogen and subsequent transmission to a host occur in different life stages. A contemporary example is Lyme disease. A long-lived tick vector acquires infection during the larval blood meal and transmits it as a nymph. We present a reaction-diffusion model for the ecological dynamics governing the velocity of the current epidemic's spread. We find that the equilibrium density of infectious tick nymphs (hence the risk of human disease) can depend on density-independent survival interacting with biotic effects on the tick's stage structure. The local risk of infection reaches a maximum at an intermediate level of adult tick mortality and at an intermediate rate of juvenile tick attacks on mammalian hosts. If the juvenile tick attack rate is low, an increase generates both a greater density of infectious nymphs and an increased spatial velocity. However, if the juvenile attack rate is relatively high, nymph density may decline while the epidemic's velocity still increases. Velocities of simulated two-dimensional epidemics correlate with the model pathogen's basic reproductive number (R-0), but calculating R-0 involves parameters of both host infection dynamics and the vector's stage-structured dynamics.

Highly parallelizable domain decomposition Dirichlet-Dirichlet solvers for hp-version finite element methods on angular quasiuniform triangular meshes are studied under different assumptions on a reference element. The edge coordinate functions of a reference element are allowed to be either nodal with special choices of nodes, or hierarchical polynomials of several types. These coordinate functions are defined within elements as being arbitrary or discrete quasi-harmonic coordinate functions. The latter are obtained from explicit and inexpensive prolongation operators. In all situations, we are able to suggest preconditioners which are spectrally equivalent to the global stiffness matrix, which only require element-by-element and edge-by-edge operations, and which reduce computational cost. In this way, elimination is avoided when dealing with the interface problem. The domain decomposition algorithms essentially use prolongation operators from the interface boundary inside the subdomains of the decomposition according to the approach initially used for the hp-version finite element methods with quadrilateral elements [S.A. Ivanov, V.G. Korneev, Izv. Vyssh. Uchebn. Zaved. 395 (1995) 62-81; Technische Universität Chemnitz-Zwickau, Preprint SPC 95-35, 1995, 1-15, and Preprint SPC 95-36, 1995, 1-14; Math. Modeling 8 (1996) 63-73].