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NASA NTRS1987

Exploiting symmetries in the modeling and analysis of tires

A simple and efficient computational strategy for reducing both the size of a tire model and the cost of the analysis of tires in the presence of symmetry-breaking conditions (unsymmetry in the tire material, geometry, or loading) is presented. The strategy is based on approximating the unsymmetric response of the tire with a linear combination of symmetric and antisymmetric global approximation vectors (or modes). Details are presented for the three main elements of the computational strategy, which include: use of special three-field mixed finite-element models, use of operator splitting, and substantial reduction in the number of degrees of freedom. The proposed computational stategy is applied to three quasi-symmetric problems of tires: linear analysis of anisotropic tires, through use of semianalytic finite elements, nonlinear analysis of anisotropic tires through use of two-dimensional shell finite elements, and nonlinear analysis of orthotropic tires subjected to unsymmetric loading. Three basic types of symmetry (and their combinations) exhibited by the tire response are identified.

Noor, Ahmed K.↗

NASA NTRS1986

Fast algorithm for calculating chemical kinetics in turbulent reacting flow

This paper addresses the need for a fast batch chemistry solver to perform the kinetics part of a split operator formulation of turbulent reacting flows, with special attention focused on the solution of the ordinary differential equations governing a homogeneous gas-phase chemical reaction. For this purpose, a two-part predictor-corrector algorithm which incorporates an exponentially fitted trapezoidal method was developed. The algorithm performs filtering of ill-posed initial conditions, automatic step-size selection, and automatic selection of Jacobi-Newton or Newton-Raphson iteration for convergence to achieve maximum computational efficiency while observing a prescribed error tolerance. The new algorithm, termed CREK1D (combustion reaction kinetics, one-dimensional), compared favorably with the code LSODE when tested on two representative problems drawn from combustion kinetics, and is faster than LSODE.

Radhakrishnan, K.↗

NASA NTRS1987

Nonlinear dynamic analysis of quasi-symmetric anisotropic structures

An efficient computational method for the nonlinear dynamic analysis of quasi-symmetric anisotropic structures is proposed. The application of mixed models simplifies the analytical development and improves the accuracy of the response predictions, and operator splitting allows the reduction of the analysis model of the quasi-symmetric structure to that of the corresponding symmetric structure. The preconditoned conjugate gradient provides a stable and effective technique for generating the unsymmetric response of the structure as the sum of a symmetrized response plus correction modes. The effectiveness of the strategy is demonstrated with the example of a laminated anisotropic shallow shell of quadrilateral planform subjected to uniform normal loading.

Noor, Ahmed K.↗

NASA NTRS1987

Re-analysis procedure based on the mixed formulation

An analysis procedure is presented for large-scale structural systems, with large numbers of degrees of freedom and design variables. The procedure uses a mixed formulation with the fundamental unknowns consisting of both stress and displacement parameters. Other elements of the procedure include: (1) lumping the design variables into a single tracing parameter; (2) operator splitting or additive decomposition of different arrays in the finite element equations into the corresponding arrays of the original structure plus correction terms; and (3) application of a reduction method through the use of the finite element method and the classical Bubnov-Galerkin technique. The re-analysis procedure is applied to the linear static and free vibration problems of plate and shell structures.

Noor, Ahmed K.↗

NASA NTRS1988

A computational strategy for making complicated structural problems simple

An effective computational strategy is presented for the analysis of large and complex structures. The strategy is based on generating the response of the complex structure using large perturbations from that of a lower-order (simpler) model associated with a simpler structure (or a simpler mathematical/discrete model of the original structure). The three key elements of the strategy are: (a) mixed (or primitive variable) formulation with the fundamental unknowns consisting of generalized displacements and stress parameters; (2) operator splitting, or a reduction method to relate the arrays and degrees of freedom of the original complex structure ot those of the simpler system; and (3) efficient iterative process for the generation of the response of the complex structure starting from that of the simpler system. The effectiveness of the proposed strategy is demonstrated by means of two numerical examples.

Noor, Ahmed K.↗

NASA NTRS1987

Parallel processing in finite element structural analysis

A brief review is made of the fundamental concepts and basic issues of parallel processing. Discussion focuses on parallel numerical algorithms, performance evaluation of machines and algorithms, and parallelism in finite element computations. A computational strategy is proposed for maximizing the degree of parallelism at different levels of the finite element analysis process including: 1) formulation level (through the use of mixed finite element models); 2) analysis level (through additive decomposition of the different arrays in the governing equations into the contributions to a symmetrized response plus correction terms); 3) numerical algorithm level (through the use of operator splitting techniques and application of iterative processes); and 4) implementation level (through the effective combination of vectorization, multitasking and microtasking, whenever available).

Noor, Ahmed K.↗

NASA NTRS1988

Reanalysis procedure for large structural systems

Global-basis-vector approximate reanalysis techniques for use in automated structural optimization schemes are developed and demonstrated. The vibrational response of a modified structure (MS) is estimated by lumping the design variables into a single tracing parameter, applying an operator splitting procedure to express the FEM equations of the MS in terms of the original-structure equations plus correction terms, and reducing the MS equations via a classical Bubnov-Galerkin scheme. The sensitivity of the vibrational response to structural modifications is evaluated, and the relationship between this method and the preconditioned conjugate-gradient method (Noor and Peters, 1988) is explored. Numerical results for linear static and free vibration problems involving beamlike lattices, double-layered hexahedral grids, and structural gridworks are presented in tables and graphs and briefly characterized.

Noor, Ahmed K.↗

NASA NTRS1989

Efficient Computation Of Behavior Of Aircraft Tires

NASA technical paper discusses challenging application of computational structural mechanics to numerical simulation of responses of aircraft tires during taxing, takeoff, and landing. Presents details of three main elements of computational strategy: use of special three-field, mixed-finite-element models; use of operator splitting; and application of technique reducing substantially number of degrees of freedom. Proposed computational strategy applied to two quasi-symmetric problems: linear analysis of anisotropic tires through use of two-dimensional-shell finite elements and nonlinear analysis of orthotropic tires subjected to unsymmetric loading. Three basic types of symmetry and combinations exhibited by response of tire identified.

Tanner, John A.↗

NASA NTRS1989

Partitioning strategy for efficient nonlinear finite element dynamic analysis on multiprocessor computers

A computational procedure is presented for the nonlinear dynamic analysis of unsymmetric structures on vector multiprocessor systems. The procedure is based on a novel hierarchical partitioning strategy in which the response of the unsymmetric and antisymmetric response vectors (modes), each obtained by using only a fraction of the degrees of freedom of the original finite element model. The three key elements of the procedure which result in high degree of concurrency throughout the solution process are: (1) mixed (or primitive variable) formulation with independent shape functions for the different fields; (2) operator splitting or restructuring of the discrete equations at each time step to delineate the symmetric and antisymmetric vectors constituting the response; and (3) two level iterative process for generating the response of the structure. An assessment is made of the effectiveness of the procedure on the CRAY X-MP/4 computers.

Noor, Ahmed K.↗

NASA NTRS1989

A partitioning strategy for efficient nonlinear finite element dynamic analysis on multiprocessor computer

Noor, Ahmed K.↗

NASA NTRS1990

Modeling and analysis of the space shuttle nose-gear tire with semianalytic finite elements

A computational procedure is presented for the geometrically nonlinear analysis of aircraft tires. The Space Shuttle Orbiter nose gear tire was modeled by using a two-dimensional laminated anisotropic shell theory with the effects of variation in material and geometric parameters included. The four key elements of the procedure are: (1) semianalytic finite elements in which the shell variables are represented by Fourier series in the circumferential direction and piecewise polynominals in the meridional direction; (2) a mixed formulation with the fundamental unknowns consisting of strain parameters, stress-resultant parameters, and generalized displacements; (3) multilevel operator splitting to effect successive simplifications, and to uncouple the equations associated with different Fourier harmonics; and (4) multilevel iterative procedures and reduction techniques to generate the response of the shell. Numerical results of the Space Shuttle Orbiter nose gear tire model are compared with experimental measurements of the tire subjected to inflation loading.

Kim, Kyun O.↗

NASA NTRS1990

Analysis of aircraft tires via semianalytic finite elements

A computational procedure is presented for the geometrically nonlinear analysis of aircraft tires. The tire was modeled by using a two-dimensional laminated anisotropic shell theory with the effects of variation in material and geometric parameters included. The four key elements of the procedure are: (1) semianalytic finite elements in which the shell variables are represented by Fourier series in the circumferential direction and piecewise polynomials in the meridional direction; (2) a mixed formulation with the fundamental unknowns consisting of strain parameters, stress-resultant parameters, and generalized displacements; (3) multilevel operator splitting to effect successive simplifications, and to uncouple the equations associated with different Fourier harmonics; and (4) multilevel iterative procedures and reduction techniques to generate the response of the shell.

Noor, Ahmed K.↗

NASA NTRS1991

Solution of the Navier-Stokes equations for a driven cavity

The flow field in a lid driven cavity is determined by integration of the incompressible Navier-Stokes equations. The numerical integration is accomplished via an operator splitting method known as the theta-scheme. This splitting separates the problem into the solution of a quasi-stokes problem and a nonlinear convection problem. Some details of solution methods used for the two subproblems and results obtained for the driven cavity are described. The schemes developed for the quasi-Stokes problem are more advanced at this stage than those for the nonlinear problem. However, the approaches used for both parts are outlined. As a model problem, a two dimensional square cavity with sides of unit length and a lid moving with unit velocity from left to right is considered. The Navier-Stokes equations are discretized in space on a uniform staggered or MAC mesh. The time discretization is accomplished via the theta-scheme.

Semeraro, B. D.↗

NASA NTRS1991

A novel gas-droplet numerical method for spray combustion

This paper presents a non-iterative numerical technique for computing time-dependent gas-droplet flows. The method is a fully-interacting combination of Eulerian fluid and Lagrangian particle calculation. The interaction calculations between the two phases are formulated on a pressure-velocity coupling procedure based on the operator-splitting technique. This procedure eliminates the global iterations required in the conventional particle-source-in-cell (PSIC) procedure. Turbulent dispersion calculations are treated by a stochastic procedure. Numerical calculations and comparisons with available experimental data, as well as efficiency assessments are given for some sprays typical of spray combustion applications.

Chen, C. P.↗

NASA NTRS1992

Numerical modeling for dilute and dense sprays

We have successfully implemented a numerical model for spray-combustion calculations. In this model, the governing gas-phase equations in Eulerian coordinate are solved by a time-marching multiple pressure correction procedure based on the operator-splitting technique. The droplet-phase equations in Lagrangian coordinate are solved by a stochastic discrete particle technique. In order to simplify the calculation procedure for the circulating droplets, the effective conductivity model is utilized. The k-epsilon models are utilized to characterize the time and length scales of the gas phase in conjunction with turbulent modulation by droplets and droplet dispersion by turbulence. This method entails random sampling of instantaneous gas flow properties and the stochastic process requires a large number of computational parcels to produce the satisfactory dispersion distributions even for rather dilute sprays. Two major improvements in spray combustion modelings were made. Firstly, we have developed a probability density function approach in multidimensional space to represent a specific computational particle. Secondly, we incorporate the Taylor Analogy Breakup (TAB) model for handling the dense spray effects. This breakup model is based on the reasonable assumption that atomization and drop breakup are indistinguishable processes within a dense spray near the nozzle exit. Accordingly, atomization is prescribed by injecting drops which have a characteristic size equal to the nozzle exit diameter. Example problems include the nearly homogeneous and inhomogeneous turbulent particle dispersion, and the non-evaporating, evaporating, and burning dense sprays. Comparison with experimental data will be discussed in detail.

Chen, C. P.↗

NASA NTRS1992

An efficient pressure-velocity procedure for gas-droplet two-phase flow calculations

This paper presents a noniterative numerical technique for computing time-dependent gas-droplet flows. The method is a fully interacting combination of Eulerian fluid and Lagrangian particle calculations. The interaction calculations between the two phases are formulated on a pressure-velocity-coupling procedure based on the operator-splitting technique. This procedure eliminates the global iterations required in the conventional particle-source-in-cell procedure. Turbulent dispersion calculations are treated by a stochastic procedure. Numerical calculations and comparisons with available experimental data as well as efficiency assessments are given for some sprays typical of spray combustion applications.

Chen, C. P.↗

NASA NTRS1992

Domain decomposition: A bridge between nature and parallel computers

Domain decomposition is an intuitive organizing principle for a partial differential equation (PDE) computation, both physically and architecturally. However, its significance extends beyond the readily apparent issues of geometry and discretization, on one hand, and of modular software and distributed hardware, on the other. Engineering and computer science aspects are bridged by an old but recently enriched mathematical theory that offers the subject not only unity, but also tools for analysis and generalization. Domain decomposition induces function-space and operator decompositions with valuable properties. Function-space bases and operator splittings that are not derived from domain decompositions generally lack one or more of these properties. The evolution of domain decomposition methods for elliptically dominated problems has linked two major algorithmic developments of the last 15 years: multilevel and Krylov methods. Domain decomposition methods may be considered descendants of both classes with an inheritance from each: they are nearly optimal and at the same time efficiently parallelizable. Many computationally driven application areas are ripe for these developments. A progression is made from a mathematically informal motivation for domain decomposition methods to a specific focus on fluid dynamics applications. To be introductory rather than comprehensive, simple examples are provided while convergence proofs and algorithmic details are left to the original references; however, an attempt is made to convey their most salient features, especially where this leads to algorithmic insight.

Keyes, David E.↗

NASA NTRS1992

Characterization and modeling for dilute and dense sprays

The present numerical modeling of dilute and dense spray combustion solves the governing gas-phase equations in Eulerian coordinates by means of a time-marching multiple pressure correction procedure that is based on the operator-splitting technique. A sequence of validating cases involving (1) nonevaporating, (2) evaporating, (3) burning, (4) dilute, and (5) dense spray cases is undertaken; it is found that in the combusting dilute spray case, the present procedure correctly predicts the general features of flows and yields qualitative agreement with experimental data. In the other cases, reasonably good agreement with experimental results is obtained.

Chen, C. P.↗