High Performance Scientific and Engineering Computing

One of the main principles in the new FEM package FEAST [3] is a recursive Divide and Conquer strategy which in combination with the generalized solver approach ScaRC [2] allows a successive decomposition of ‘global’ problems into smaller ‘local’ subproblems. On the lowest-level ‘local’ patches, highly regular data structures are preferred which allow the use of high-performance Numerical Linear Algebra tools as ingredients in multigrid solvers. Realistic MFLOP rates are demonstrated for typical components: matrix-vector multiplication, preconditioned, vector operations such as DAXPY, etc. Finally, we briefly introduce our new matrixvector techniques which are able to exploit a high percentage of the available peak performance on modern hardware platforms.

We consider the inviscid Burgers’ equation augmented by a control term and its adjoint equation. For several discretization schemes of the inviscid Burgers’ equation the adjoint finite difference methods are derived. Applying these discretization methods and the checkpointing routine treeverse, approximations of the solution of both differential equations are calculated and compared.

For the compressible Navier-Stokes equations, different methods for the discretization of the convective and viscous terms on different triangular and quadrilateral grids are compared for the flow over a flat plate boundary layer. It turns out that we obtained the best result for the improved advection upstream splitting method (AUSMDV) for the convective terms on unstructured quadrilateral grids. For the viscous terms discretizations which satisfy a discrete maximum principle are the most successful. We discuss the parallelization of such schemes and a posteriori error estimates.

In this paper we perform a numerical bifurcation analysis of a one-dimensional problem arising in premixed combustion in porous inert media. The analysis shows that multiple steady solutions may occur, that a minimal mass flow rate is required for steady combustion, and that solutions with complete combustion exist for a large range of mass flow rates. Moreover, we show that a jump discontinuity in porosity and cooling contributes to the stability of the combustion zone. The one-dimensional results are compared with computations for a corresponding two-dimensional problem. The qualitative agreement between the solutions is good along the cooling boundary, and satisfactory along the axis of symmetry. The one-dimensional problem can therefore be used as preparation for a more complex two-dimensional simulation.

The application of linear multigrid methods in conjunction with Krylov subspace methods on unstructured grids for two- and three-dimensional problems is shown. Also the use of such methods on hybrid grids for two-dimensional incompressible laminar and turbulent flows is presented. The necessary hierarchy of uniformly or locally refined grids is generated by a grid generation procedure especially useful for boundary layer flows. In the second part, emphasis is placed on the effect of parallelism on solvers and problems. Corresponding examples are a 2D driven cavity flow at Re=7500 and a 3D flow around a cylinder at Re=20.

In this paper Large-eddy simulations (LES) of several flow problems are presented. First, the performance of two different solution schemes, both formulated for compressible flows, is compared for the case of a spatially developing plane turbulent jet. A second-order scheme based on an AUSM method with a central difference of the pressure derivatives and a compact finite difference scheme of sixth order are used with a dynamic model and also without any subgrid scale model. The boundary conditions correspond to a jet evolving into a fluid at rest; at the outflow plane non-reflecting conditions following Poinsot and Lele were used. Simulations were carried out for a Mach number of 0.1 and Reynolds number of 7600 and 22.000. The analysis of the flow field shows that both schemes produce results of comparable accuracy. The main reason why the higher-order scheme does not provide more accurate results than the second-order method, is probably the application of an explicit filter, which had to be used to remove high-frequency oscillations. The other flow problems presented are therefore simulated with the computationally less expensive second-order scheme. Results are presented for internal flows in straight and curved pipes, as well as a flow around a circular cylinder at a Reynolds number of 3900. Good agreement with reference data was found in all cases.

The accurate numerical simulation of aircraft components especially at off-design conditions requires an appropriate computational modeling of the entire flow problem. In such circumstances, the predictive quality of the flow’s gross characteristics is governed by seemingly subtle details pertaining to the representation of turbulence in apparently innocuous portions of the flow. In this study, the performance of a family of advanced turbulence-closure models in flows of aerodynamic relevance is analyzed with respect to the accuracy of flow physics representation and the applicability in industrial parallel computing. Several exemplary threedimensional test cases employing up to 6.8 million grid nodes using 128 PEs on a CRAY T3E were investigated to assess these issues. First results demonstrate the superior predicitive accuracy of advanced models in complex flow situations as well as the feasibility of such computations employing MPP systems even for large-size application-oriented aerodynamic configurations.

The solution of coupled problems is one of the main challenges in scientific computing. Multigrid methods are known to be highly efficient. The use of parallel computers is necessary. Some benefits of the application of multigrid to coupled problems are described.

The fluid damped oscillations of a torsion spring pendulum were investigated applying a fully implicit, partitioned scheme for the coupled numerical computation of the structure and the fluid domain. The two-dimensional Navier-Stokes equations were solved by a Lagrangian-Eulerian finite volume approach on arbitrarily moving and deforming grids to obtain the moment required for the prediction of the single-degree-of-freedom structure oscillation. A new time discretisation scheme provided the conservative formulation of the flow solver. At high initial amplitudes strong structure-vortex and vortex-vortex interaction phenomena were observed in water yielding a nonlinear damping behaviour of the lamina structure.

The efficient representation and handling of complicated geometries is one of the main challenges of today’s numerical simulation. In the area of computational fluid dynamics, e.g., geometry plays a more and more predominant part. Especially for problems with moving interfaces, when we think of the cost of a successive remeshing in a finite element context, e. g., those geometric aspects seem to be at least as important as the construction of appropriate discretization schemes and solvers.

The direct transcription or collocation method has demonstrated notable success in the solution of trajectory optimization and optimal control problems. This approach combines a sparse nonlinear programming algorithm with a discretization of the trajectory dynamics. A challenging class of optimization problems occurs when the spacecraft trajectories are characterized by thrust levels that are very low relative to the vehicle weight. Low thrust trajectories are demanding because realistic forces due to oblateness, aerodynamic drag, and third-body perturbions often dominate the thrust. Furthermore because the thrust is so low, significant changes to the orbits require very long duration trajectories. When a collocation method is applied to a problem of this type, the resulting nonlinear program is very large because the trajectories are long, and very nonlinear because of the perturbing forces.

The multibody system approach provides enhanced models of vehicles, robots, and air- and spacecrafts. Mixed systems consisting of both rigid and deformable bodies are aimed at growing demands for refined simulation. A basic modeling framework for this class of mechanical systems is presented which covers also inelastic material behavior. Moreover, the Differential-Algebraic Equations (DAEs) obtained from semidiscretization in space are classified and the application of DAE solvers is discussed. Two examples illustrate the simulation tasks and show the state-of-the-art in this field of scientific computing.

Virtual prototyping plays a key role in modern car engineering. For virtual test drives of entire cars in the computer, mathematical and computational models of the vehicle, the road, and the driver are presented. The numerical simulation must be performed in real time for application in Hardware-in-the-Loop experiments. Numerical results are presented for the ISO slalom test.

The construction and design of the axles of a new vehicle is one of the most important parts in developing a car. Lightweight structures are used in order to reduce the weight of cars. This can be reached using new materials that are often more flexible, e.g., using aluminium instead of steel. So the conventional rigid multibody models seem to be no more sufficient for the accurate modeling of all physical properties, in particular the vibrations. Therefore, we investigate a multibody model of a rear axle, which contains an elastic subframe. In order to reduce the degrees of freedom of the finite element model, the dynamic behavior is approximated by the modal Ritz ansatz. This approach is a good approximation of the deformation of the subframe, because the deformations are small. In the software package ADAMS that can be used for solving multibody systems the Craig-Bampton method is used for this approach. We focus our investigations on the lateral vibrations of the rear axle. Numerical results for a rigid and an elastic multibody formulation are presented and discussed.

Computer generated synthetic vision basically provides a realistic three dimensional image of the outside world and integrated guidance symbologies including innovative elements like a tunnel display. An efficient computer software is necessary for generating the pictorial elements in real-time. Data fusion is a means for providing the pilot with additional visual information. Such systems are aiming for an all-weather guidance capability.

A new Predictor-Corrector homotopy method is proposed based on a variable order predictor and a stability oriented steplength control. Predictors of different orders are combined by order controls, is developped that gives way to a significant speedup in the solution of optimal control problems. The proposed method is applied to the problem of escaping from a microburst encountered during landing approach. It can be shown that the optimized three-dimensional escape trajectory is very efficient and superior to the two-dimensional trajectory restricted to the vertical plane. Variations of aircraft and problem parameters show that superiority of lateral escape holds for many cases.

The paper starts explaining the motivation for realizing a new user environment for numerical solution of optimal control problems. After specification of the problem class addressed, the traditional process of solving these problems is shortly described. A modification of this process is derived, where the documentation of the optimization problem is used as an entry point. The modules of the integrated user environment and their tasks are denoted. An essential part of the user environment is the automatic transformation from documentation in LATEX to FORTRAN subroutines. An overview of the underlying concept is given and some results on the efficiency are provided. The appendix shows an example for application of the transparent programming concept.

This paper is concerned with the logistic transport of open, fluid filled containers within a warehouse environment. In particular an optimal trajectory path planning in two dimensional space for the motion of the cart, transporting the container will be proposed. The fluid and the mechanical facility that moves the container are subject to several constraints. The objective of the optimization is a minimization of time to transport the container from an initial position to its final destination within the warehouse. Optimization criteria are investigated to control the movement of the fluid within the container. The system of ordinary differential equations and of partial differential equations, representing the dynamics of the models is solved numerically using a direct shooting method. The optimization problem is solved using sequential quadratic programming (SQP).

Numerical simulation has already become an indispensable tool in the chemical engineering industry. In this paper, the extension of an already existing simulation package to the efficient and reliable solution of optimal control problems for optimal plant operation is discussed.

We present new algorithmic methods, highly efficient sequential and scalable parallel implementations of extrapolation algorithms and demonstrate their value for challenging problems of chemical engineering.

Spreading viscous films are of considerable interest in industrial applications, especially in coating processes, where, in general, a uniform quality of the final coating is preferred. Experiments show that the contact-line of, in this case, liquid films driven down an inclined plane through gravity, rapidly destabilizes, forming either sawtooth or finger patterns. For the latter situation, portions of the contact-line may stop to move at all, leaving uncovered regions on the plane. Theoretical investigations using linear stability analysis for the lubrication approximation are in good agreement with experimental results for large to moderate inclination angles [2,4,8,12]. However, for small inclination angles, they predict stability in contrast to experimental observations.

The program package CrysVUN++ is introduced. It is designed for the needs of crystal growers to have an efficient computer code for the simulation of the growth processes and equipment. Heat transfer by nonlinear anisotropic conduction and radiation as well as the analysis of thermoelastic stress are calculated for axisymmetric geometries. The problem of heater control is treated in an integrated software module. The use of CrysVUN++ is supported by an easy to handle graphical interface.

The present paper deals with the numerical simulation of the growth of GaAs epitaxial layers in the Planetary Reactor™ using 3D adaptive unstructured grids.

Direct numerical simulations of turbulent convection in an idealized Czochralski crystal growth configuration were performed with a three-dimensional, time-dependent Navier-Stokes solver. The analysis of the flow data focusses on the influence of crystal and crucible rotation on the flow structures and the development of temperature fluctuations. Therefore, simulations were performed for different melt heights and various crystal and crucible rotation rates. It was found that the counter-rotation of the crystal and crucible leads to a complex flow with three major recirculation zones, when crucible rotation dominates the flow. The dynamics of the flow are controlled by centrifugal forces counteracting buoyancy and surface tension effects. High temperature fluctuations are created within or close to the cristalization zone. Neither variation of the melt height nor reducing the crystal rotation rate has a major effect on the bulk flow structure and overall heat transfer, but temperature fluctuations are reduced due to the decreased rotation of the flow. Increasing rotation of the crystal changes the bulk flow structure tremendously. Additionally the value of maximum rms temperature fluctuations is increased and its position is shifted towards the crucible bottom.

This contribution deals with the Austrian research project AURORA and the application of high performance computing (HPC) in the field of microelectronics. In the first part the ‘Spezialforschungsbereich’ AURORA - Advanced Models, Applications, and Software Systems for High Performance Computing - is presented which is funded by the Austrian ‘Fonds zur Forderung der wissenschaftlichen Forschung’. Seven research groups belonging to different institutes of the ‘Universitat Wien’ and the ‘Technische Universitat Wien’ are participating in AURORA, thus covering the fields computer science, statistics and operations research, numerical mathematics, electrochemistry, and microelectronics. The second part deals with the activities concerning the application of HPC to the simulation of the behaviour of microelectronic devices and their technological process steps in order to intensify the research capabilities of the simulation tools.

The numerical simulation of microstructured semiconductor devices, transducers, and systems aiming at an optimal layout and design often has to take into account that the operating behaviour is based on the interaction of various physical phenomena. This requires on one hand the simulation to be based on a consistent, tailored modeling of the underlying physical processes and on the other hand the use of modern methods in the numerical solution of PDEs and systems thereof such as efficient iterative solvers and adaptive grid refinement and coarsening. In this contribution, the development and implementation of such techniques will be outlined for three industrially relevant case studies. The first one is concerned with the minimization of parasitic effects in converter modules used in high power electronics which amounts to the solution of a shape and topology optimization problem. Here, we consider the efficient computation of electromagnetic potentials related to Maxwell’s equations based on a discretization in terms of curl-conforming edge elements. The second problem deals with electrostatically driven micromembrane pumps that are intended to be used in medical sciences to control metabolism or in the chemical analysis of freshwater bodies. In particular, we will address the simulation of the electromechanical coupling that characterizes the operating behaviour of the electrostatic drive and the fluid-structure interaction between the fluid flow and the deformation of the passive valves. Finally, we consider the computation of the temperature and heat flow distribution in micromachined deformable mirrors that can be used for the positioning of laser beams in optical eye surgery. Emphasis will be laid on a combined time-step selection and adaptivity in space for a primal mixed discretization of the underlying heat equation.

A Schur complement based algebraic multigrid method is discussed in view of the special conditions of the continuity equations for electrons and holes in semiconductors. The so called Scharfetter-Gummel discretization, the practically arbitrary large jumps in the solutions and the structure of the preconditioners impose special problems compared with standard finite element discretizations. The final goal is the application of the method to large three-dimensional problems - in other words: parallelization properties are an issue. The present techniques used in device simulation can not guarantee a close relation of the grids used and the coefficients of the equations. Hence a correction technique for the basis transformation is introduced. The algorithm is implemented on parallel Cray vector machines and on RISC SMPs. The performance for two- and three-dimensional examples is given.

Partitioning strategies are commonly used in network analysis packages for simulating highly integrated circuits such as dynamic memories. These methods allow chip designers to simulate circuits consisting of millions of transistors in reasonable computation time. Using a standard benchmark example, the inverter chain, we consider different approaches used in network analysis to split the system on circuit level. We will show the connection with split numerical methods for ordinary differential equations with partitioned right-hand sides.

A new numerical integration scheme for the simulation of stochastic differential equations is presented. In the context of the computer-aided design of electronic circuits, the modeling of circuits under the influence of electronic noise leads to stochastic differential equations. The available schemes can not solve these equations efficiently. The new discretiziation scheme is constructed similar to multistep schemes for ordinary differential equations. A Fortran77 implementation of the presented integration scheme reduces the simulation time for a ring oscillator to about 17.5% compared with common methods. Therefore, it is an efficient tool for the design of noisy circuits.

We report on a comparison of the implicitly restarted Lanczos algorithm as implemented in ARPACK and the Jacobi-Davidson algorithm for solving large sparse generalized symmetric matrix eigenvalue problems. These problems occur in the computation of a few of the lowest frequencies of standing electro-magnetic waves in cavity resonators. The computational domain is discretized by a finite element method based on edge elements to avoid spurious modes.

Gajewski and Gröger used a convex thermodynamic potential, the free energy, in the analysis of the drift-diffusion model. We consider an energy model as an example of a system in which the temperature is a dynamic variable. In such systems the free energy is no convex functional. We introduce an approach to the transient problem for the model which is based on convex functionals which are related to the entropy and its conjugate potential and which allow the application of the tools of convex analysis. A semi-implicit method for the initial-boundary value problem arises on a rather natural way and basic estimates are proved for the solutions.

We describe and demonstrate the capabilities of a new coupled-field simulator which is based on the industrial CAD platform TP2000 and especially dedicated to the multidimensional numerical analysis of the operation of electromechanical microdevices. The coupling of the finite element and the boundary element methods makes the simulation tool particularly suited for microdevices where movable parts are deflected, displaced, or rotated by electrostatic forces. As an illustrative example, we study the fully coupled electro-mechanical behaviour of a deflectable micromirror.

In this paper, a practical methodology for extracting distributed electromagnetic parasitics and other related quantities for the analysis of bus bar structures in high power modules is presented. It is shown that in consequence of the use of fast switching semiconductor devices in high power applications only a full three-dimensional transient simulation of the entire module under realistic switching conditions can give the necessary insight in the time-dependent behaviour of the electric and magnetic fields in- and outside the interconnects. The usefullness of this methodology is demonstrated by an illustrative example of a bus bar encountered in an industrial application.

A one-dimensional, self-consistent hybrid plasma model is presented which describes both electro-positive and electro-negative radio frequency (RF) discharges within a unified formalism. The model allows for the simultaneous application of inductive and capacitive heating and is capable of representing RF-driven low pressure discharges as used in the processing of VLSI semiconductors. Our hybrid approach combines a streamlined fluid-dynamic plasma description with a microscopic Monte-Carlo scheme, and provides a more efficient method of plasma modeling than other methods which resolve the electron energy distribution. As an application, Id simulation results for an electro-negative gas (CI) and an electro-positive gas (Ar) are presented. A comparision with experimental Langmuir probe data shows a qualitative agreement.

A brief description of the Smoothed Particle Hydrodynamics method is presented together with some test calculations and astrophysical examples of accretion phenomena.

This paper investigates the numerical solution of linear Boltzmann equations. Finite Element Methods are applied to the advection and the scattering operator. An efficient error estimation technique is presented together with a consistent mesh refinement criterion. The solution process on a parallel computer is analyzed and the applicability of the theory is validated by numerical results for problems derived from astrophysical settings.

A density functional method for electronic structure calculations of atoms, molecules and clusters has been parallelized and newly implemented in the program ParaGauss. Parallelization strategies and performance aspects are discussed. The capabilities of this new quantum chemical code, which includes an option for scalar-relativist ic calculations, are demonstrated by all-electron results for large transition metal clusters (Pd₃₀₉, Au₃₈(SH)₂₄).

Semiconductor technology is advancing at an amazing rate. The changes predicted for the coming 15 years will also have a profound impact on scientific computing. Two of the most important trends are a dramatically increased potential for parallel execution within a single chip, and a serious gap between the speed of memory and the processor. Future computer architectures will force the developers of scientific computing applications to address problems originating from fine grain parallelism and problems caused by the memory bottleneck.