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Algebraic multigrid methods applied to problems in computational structural mechanics

The development of algebraic multigrid (AMG) methods and their application to certain problems in structural mechanics are described with emphasis on two- and three-dimensional linear elasticity equations and the 'jacket problems' (three-dimensional beam structures). Various possible extensions of AMG are also described. The basic idea of AMG is to develop the discretization sequence based on the target matrix and not the differential equation. Therefore, the matrix is analyzed for certain dependencies that permit the proper construction of coarser matrices and attendant transfer operators. In this manner, AMG appears to be adaptable to structural analysis applications.

Mccormick, Steve↗

A New Semistructured Algebraic Multigrid Method

Multigrid methods are well suited to large massively parallel computer architectures because they are mathematically optimal and display good parallelization properties. Since current architecture trends are favoring regular compute patterns to achieve high performance, the ability to express structure has become much more important. The hypre software library provides high-performance multigrid preconditioners and solvers through conceptual interfaces, including a semistructured interface that describes matrices primarily in terms of stencils and logically structured grids. This paper presents a new semistructured algebraic multigrid (SSAMG) method built on this interface. The numerical convergence and performance of a CPU implementation of this method are evaluated for a set of semistructured problems. In conclusion, SSAMG achieves significantly better setup times than hypre’s unstructured AMG solvers and comparable convergence. In addition, the new method is capable of solving more complex problems than hypre’s structured solvers.

97 MATHEMATICS AND COMPUTING↗

An algebraic homotopy method for generating quasi-three-dimensional grids for high-speed configurations

A fast and versatile procedure for algebraically generating boundary conforming computational grids for use with finite-volume Euler flow solvers is presented. A semi-analytic homotopic procedure is used to generate the grids. Grids generated in two-dimensional planes are stacked to produce quasi-three-dimensional grid systems. The body surface and outer boundary are described in terms of surface parameters. An interpolation scheme is used to blend between the body surface and the outer boundary in order to determine the field points. The method, albeit developed for analytically generated body geometries is equally applicable to other classes of geometries. The method can be used for both internal and external flow configurations, the only constraint being that the body geometries be specified in two-dimensional cross-sections stationed along the longitudinal axis of the configuration. Techniques for controlling various grid parameters, e.g., clustering and orthogonality are described. Techniques for treating problems arising in algebraic grid generation for geometries with sharp corners are addressed. A set of representative grid systems generated by this method is included. Results of flow computations using these grids are presented for validation of the effectiveness of the method.

Moitra, Anutosh↗

An approach to simultaneous system design. II - Nonswitching gain and dynamic feedback compensation by algebraic geometric methods

This paper studies structured uncertainty problems in feedback system design, considers a compact parameterization of the space of linear dynamical systems and introduces 'base points' and 'critical points' as two algebraic-geometric objects that have significance in sensitivity and robustness studies, respectively. Using the Nevanlinna-Pick interpolation theory, the author obtains a necessary and sufficient condition for simultaneous stabilization of a structured one-parameter family of plants. A recent result due to Kharitonov, on the simultaneous stability of a parameterized family of polynomials, leads to a sufficiency condition for simultaneous stabilization of a structured multiparameter family of plants. Furthermore, the author considers 'simultaneous pole placement' of an r-tuple of plants as a means to arbitrarily tune the natural frequencies of a multimode linear dynamical system. The concept of 'nondegenerate' and 'twisted' r-tuples of plants is introduced as the pole placement problem is studied via Schubert enumerative geometry as an intersection problem on the associated Grassmannian. Various other design problems, viz., the strong stabilization problem and the dead beat control problem, are also considered.

Ghosh, Bijoy K.↗

TAMM: Tensor algebra for many-body methods

Tensor algebra operations such as contractions in computational chemistry consume a significant fraction of the computing time on large-scale computing platforms. The widespread use of tensor contractions between large multi-dimensional tensors in describing electronic structure theory has motivated the development of multiple tensor algebra frameworks targeting heterogeneous computing platforms. In this paper, we present Tensor Algebra for Many-body Methods (TAMM), a framework for productive and performance-portable development of scalable computational chemistry methods. TAMM decouples the specification of the computation from the execution of these operations on available high-performance computing systems. With this design choice, the scientific application developers (domain scientists) can focus on the algorithmic requirements using the tensor algebra interface provided by TAMM, whereas high-performance computing developers can direct their attention to various optimizations on the underlying constructs, such as efficient data distribution, optimized scheduling algorithms, and efficient use of intra-node resources (e.g., graphics processing units). The modular structure of TAMM allows it to support different hardware architectures and incorporate new algorithmic advances. We describe the TAMM framework and our approach to the sustainable development of scalable ground- and excited-state electronic structure methods. We present case studies highlighting the ease of use, including the performance and productivity gains compared to other frameworks.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Two and three dimensional grid generation by an algebraic homotopy procedure

An algebraic method for generating two- and three-dimensional grid systems for aerospace vehicles is presented. The method is based on algebraic procedures derived from homotopic relations for blending between inner and outer boundaries of any given configuration. Stable properties of homotopic maps have been exploited to provide near-orthogonality and specified constant spacing at the inner boundary. The method has been successfully applied to analytically generated blended wing-body configurations as well as discretely defined geometries such as the High-Speed Civil Transport Aircraft. Grid examples representative of the capabilities of the method are presented.

Moitra, Anutosh↗

Flow simulations for nacelle-propeller configurations using Euler equations

Euler codes for both axisymmetric and general three-dimensional nacelle-propeller flow analysis have been developed. Surface-fitted grids are generated either by a finite difference method or by an algebraic method. The propeller is represented by an actuator disk along a computational plane where proper boundary conditions are assigned to simulate the propeller power loading. Computed results for a NASA SR3 propeller as well as a NASA turboprop configuration are compared with test data. Good agreement has been achieved through the present simulation method.

Yu, N. J.↗

Quantum mechanical algebraic variational methods for inelastic and reactive molecular collisions

The quantum mechanical problem of reactive or nonreactive scattering of atoms and molecules is formulated in terms of square-integrable basis sets with variational expressions for the reactance matrix. Several formulations involving expansions of the wave function (the Schwinger variational principle) or amplitude density (a generalization of the Newton variational principle), single-channel or multichannel distortion potentials, and primitive or contracted basis functions are presented and tested. The test results, for inelastic and reactive atom-diatom collisions, suggest that the methods may be useful for a variety of collision calculations and may allow the accurate quantal treatment of systems for which other available methods would be prohibitively expensive.

Schwenke, David W.↗

On the applications of algebraic grid generation methods based on transfinite interpolation

Algebraic grid generation methods based on transfinite interpolation called the two-boundary and four-boundary methods are applied for generating grids with highly complex boundaries. These methods yield grid point distributions that allow for accurate application to regions of sharp gradients in the physical domain or time-dependent problems with small length scale phenomena. Algebraic grids are derived using the two-boundary and four-boundary methods for applications in both two- and three-dimensional domains. Grids are developed for distinctly different geometrical problems and the two-boundary and four-boundary methods are demonstrated to be applicable to a wide class of geometries.

Nguyen, Hung Lee↗

Leapfrog variants of iterative methods for linear algebra equations

Two iterative methods are considered, Richardson's method and a general second order method. For both methods, a variant of the method is derived for which only even numbered iterates are computed. The variant is called a leapfrog method. Comparisons between the conventional form of the methods and the leapfrog form are made under the assumption that the number of unknowns is large. In the case of Richardson's method, it is possible to express the final iterate in terms of only the initial approximation, a variant of the iteration called the grand-leap method. In the case of the grand-leap variant, a set of parameters is required. An algorithm is presented to compute these parameters that is related to algorithms to compute the weights and abscissas for Gaussian quadrature. General algorithms to implement the leapfrog and grand-leap methods are presented. Algorithms for the important special case of the Chebyshev method are also given.

Saylor, Paul E.↗

Probabilistic load model development and validation for composite load spectra for select space propulsion engines

A major task of the program to develop an expert system to predict the loads on selected components of a generic space propulsion engine is the design development and application of a probabilitic loads model. This model is being developed in order to account for the random nature of the loads and assess the variable load ranges' effect on the engine performance. A probabilistic model has been developed. The model is based primarily on simulation methods, but also has a Gaussian algebra method (if all variables are near normal), a fast probability integrator routine (for the calculation of low probability events), and a separate, stand alone program for performing barrier crossing calculations. Each of these probabilistic methods has been verified with theoretical calculations using assumed distributional forms.

Kurth, R.↗

Atomic data for opacity calculations. X - Oscillator strengths and photoionisation cross sections for O III

Energy levels and electric dipole radiative transitions were determined for O III. The wavefunctions for the bound and continuum states were derived by solving coupled integrodifferential equations in the close-coupling approximation with the aid of two methods, the R-matrix and linear algebraic methods. Configuration interaction wavefunctions are presented for eight states of the O IV target with configurations of 2s(x)sp(y) (x + y = 3). Oscillator strengths are calculated for the transitions between, and photoionization cross sections from, bound states of O III with configurations of 2s(x)2p(y)nl (for n of not greater than 10 and l of not greater than 3).

Luo, D.↗

Numerical solution of the Euler equations for high-speed, blended wing-body configurations

Simulation of high-speed three-dimensional flow about blended wing-body combinations is investigated. A finite-volume explicit scheme with Runge-Kutta time integration is used to solve the compressible Euler equations in order to simulate the flow. The method, augmented by carefully chosen dissipative terms and convergence accelerators such as enthalpy damping and maximum local time-stepping, has been found to be very efficient in solving high-speed flows involving strong shocks. An analytic method for generating the body geometry and an algebraic method for quasi-three-dimensional grid generation are described. Results are presented for various blended wing-body configurations at different Mach numbers and angles of attack.

Moitra, A.↗

Grid adaption for bluff bodies

Methods of grid adaptation are reviewed and a method is developed with the capability of adaptation to several flow variables. This method is based on a variational approach and is an algebraic method which does not require the solution of partial differential equations. Also the method was formulated in such a way that there is no need for any matrix inversion. The method is used in conjunction with the calculation of hypersonic flow over a blunt nose. The equations of motion are the compressible Navier-Stokes equations where all viscous terms are retained. They are solved by the MacCormack time-splitting method and a movie was produced which shows simulataneously the transient behavior of the solution and the grid adaptation. The results are compared with the experimental and other numerical results.

Abolhassani, Jamshid S.↗

Grid adaption for hypersonic flow

The methods of grid adaption are reviewed and a method is developed with the capability of adaption to several flow variables. This method is based on a variational approach and is an algebraic method which does not require the solution of partial differential equations. Also the method has been formulated in such a way that there is no need for any matrix inversion. The method is used in conjunction with the calculation of hypersonic flow over a blunt nose body. The equations of motion are the compressible Navier-Stokes equations where all viscous terms are retained. They are solved by the MacCormack time-splitting method. A movie has been produced which shows simultaneously the transient behavior of the solution and the grid adaption.

Abolhassani, Jamshid S.↗

Numerical solutions of three-dimensional Navier-Stokes equations for closed-bluff bodies

Methods of grid adaption are reviewed and a method is developed with the capability of adaption to several flow variables. This method is based on a variational approach and is an algebraic method which does not require the solution of partial differential equations. The method has also been formulated in such a way that there is no need for any matrix inversion. The method is used in conjunction with the calculation of hypersonic flow over a blunt nose. The equations of motion are the compressible Navier-Stokes equations where all viscous terms are retained. They are solved by the MacCormack time-splitting method and a movie has been produced which shows simultaneously the transient behavior of the solution and the grid adaption. The results are compared with the experimental and other numerical results.

Abolhassani, Jamshid S.↗

Topology and grid adaption for high-speed flow computations

This study investigates the effects of grid topology and grid adaptation on numerical solutions of the Navier-Stokes equations. In the first part of this study, a general procedure is presented for computation of high-speed flow over complex three-dimensional configurations. The flow field is simulated on the surface of a Butler wing in a uniform stream. Results are presented for Mach number 3.5 and a Reynolds number of 2,000,000. The O-type and H-type grids have been used for this study, and the results are compared together and with other theoretical and experimental results. The results demonstrate that while the H-type grid is suitable for the leading and trailing edges, a more accurate solution can be obtained for the middle part of the wing with an O-type grid. In the second part of this study, methods of grid adaption are reviewed and a method is developed with the capability of adapting to several variables. This method is based on a variational approach and is an algebraic method. Also, the method has been formulated in such a way that there is no need for any matrix inversion. This method is used in conjunction with the calculation of hypersonic flow over a blunt-nose body. A movie has been produced which shows simultaneously the transient behavior of the solution and the grid adaption.

Abolhassani, Jamshid S.↗