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NASA NTRS1985

Implementation of design sensitivity analysis with existing finite element codes

A numerical method is presented to implement structural design sensitivity analysis theory, using the versatility and convenience of existing finite element structural analysis programs. Design variables such as thickness and cross-sectional areas of components of individual members and built-up structures are considered. Structural performance functionals considered include displacement and stress. The method is also applicable for eigenvalue problem design sensitivity analysis. It is shown that calculations can be carried out outside existing finite element codes, using postprocessing data only. Thus, design sensitivity analysis software does not have to be embedded in an existing finite element code. Feasability of the method is shown through analysis of several problems, including a built-up structure. Accurate design sensitivity results are obtained without the uncertainty of numerical accuracy associated with selection of finite difference perturbations.

Choi, Kyung K.↗
NASA NTRS1989

Parallel iterative methods for sparse linear and nonlinear equations

As three-dimensional models are gaining importance, iterative methods will become almost mandatory. Among these, preconditioned Krylov subspace methods have been viewed as the most efficient and reliable, when solving linear as well as nonlinear systems of equations. There has been several different approaches taken to adapt iterative methods for supercomputers. Some of these approaches are discussed and the methods that deal more specifically with general unstructured sparse matrices, such as those arising from finite element methods, are emphasized.

Saad, Youcef↗
NASA NTRS1991

Navier-Stokes computation of wing/rotor interaction for a tilt rotor in hover

A method has been developed to analyze the wing/rotor interaction of tilt rotor aircraft in hover. The unsteady, thin-layer compressible Navier-Stokes equations are solved using an implicit, finite difference scheme that employs LU-ADI factorization. The rotor is modeled as an actuator disk which imparts a radial and azimuthal distribution of pressure rise and swirl to the flowfield. The 'chimera' approach of grid point blanking is used to update the rotor boundary conditions. Results are presented for both a rotor alone and for wing/rotor interaction where the thrust coefficient is 0.0164 and wing flap deflection is 67 degrees. Many of the complex flow features are captured including the fountain effect, leading and trailing edge separation, and the unsteady wake beneath the wing. Wing surface pressures compare fairly well with experimental data although the time-averaged download is about twenty percent higher than the measured value. This discrepancy is due to a combination of factors that are discussed.

Fejtek, Ian↗
NASA NTRS1988

Dynamic structural correlation via nonlinear programming techniques

A solution to the correlation between structural dynamic test results and finite element analyses of the same components is presented in this paper. Basically, the method can be categorized as a Levenberg-Marquardt type Gauss-Newton method which requires only the differences between FE modal analyses and test results and their first derivatives with respect to preassigned design variables. With proper variable normalization and equation scaling, the method has been made numerically better-conditioned and the inclusion of the Levenberg-Marquardt technique overcomes any remaining difficulty encountered in inverting singular or near-singular matrices. An important feature is that each iteration requires only one function evaluation along with the associated design sensitivity analysis and so the procedure is computationally efficient.

Ting, T.↗
NASA NTRS1995

The multidimensional Self-Adaptive Grid code, SAGE, version 2

This new report on Version 2 of the SAGE code includes all the information in the original publication plus all upgrades and changes to the SAGE code since that time. The two most significant upgrades are the inclusion of a finite-volume option and the ability to adapt and manipulate zonal-matching multiple-grid files. In addition, the original SAGE code has been upgraded to Version 1.1 and includes all options mentioned in this report, with the exception of the multiple grid option and its associated features. Since Version 2 is a larger and more complex code, it is suggested (but not required) that Version 1.1 be used for single-grid applications. This document contains all the information required to run both versions of SAGE. The formulation of the adaption method is described in the first section of this document. The second section is presented in the form of a user guide that explains the input and execution of the code. The third section provides many examples. Successful application of the SAGE code in both two and three dimensions for the solution of various flow problems has proven the code to be robust, portable, and simple to use. Although the basic formulation follows the method of Nakahashi and Deiwert, many modifications have been made to facilitate the use of the self-adaptive grid method for complex grid structures. Modifications to the method and the simple but extensive input options make this a flexible and user-friendly code. The SAGE code can accommodate two-dimensional and three-dimensional, finite-difference and finite-volume, single grid, and zonal-matching multiple grid flow problems.

Davies, Carol B.↗
NASA NTRS1977

Numerical methods for inviscid/viscous flows

A study of numerical schemes for solving viscous fluid flow problems with sizable regions of predominantly inviscid flow is presented. Difficulties associated with the familiar central difference approach for such problems were analyzed and alternative finite difference approaches employing windward concepts are presented. In addition, difference relations based on exponential operators were developed. All such schemes were demonstrated and evaluated through application to the case of Falkner Skan flow with blowing - a problem in which a sizable region of predominantly inviscid flow develops near the injection surface that traditionally causes numerical difficulty.

El-Mistikawy, T. M.↗
NASA NTRS1999

Practical Aerodynamic Design Optimization Based on the Navier-Stokes Equations and a Discrete Adjoint Method

The technical details are summarized below: Compressible and incompressible versions of a three-dimensional unstructured mesh Reynolds-averaged Navier-Stokes flow solver have been differentiated and resulting derivatives have been verified by comparisons with finite differences and a complex-variable approach. In this implementation, the turbulence model is fully coupled with the flow equations in order to achieve this consistency. The accuracy demonstrated in the current work represents the first time that such an approach has been successfully implemented. The accuracy of a number of simplifying approximations to the linearizations of the residual have been examined. A first-order approximation to the dependent variables in both the adjoint and design equations has been investigated. The effects of a "frozen" eddy viscosity and the ramifications of neglecting some mesh sensitivity terms were also examined. It has been found that none of the approximations yielded derivatives of acceptable accuracy and were often of incorrect sign. However, numerical experiments indicate that an incomplete convergence of the adjoint system often yield sufficiently accurate derivatives, thereby significantly lowering the time required for computing sensitivity information. The convergence rate of the adjoint solver relative to the flow solver has been examined. Inviscid adjoint solutions typically require one to four times the cost of a flow solution, while for turbulent adjoint computations, this ratio can reach as high as eight to ten. Numerical experiments have shown that the adjoint solver can stall before converging the solution to machine accuracy, particularly for viscous cases. A possible remedy for this phenomenon would be to include the complete higher-order linearization in the preconditioning step, or to employ a simple form of mesh sequencing to obtain better approximations to the solution through the use of coarser meshes. . An efficient surface parameterization based on a free-form deformation technique has been utilized and the resulting codes have been integrated with an optimization package. Lastly, sample optimizations have been shown for inviscid and turbulent flow over an ONERA M6 wing. Drag reductions have been demonstrated by reducing shock strengths across the span of the wing.

Grossman, Bernard↗
NASA NTRS1984

Verification and application of the Iosipescu shear test method

Finite element models were used to study the effects of notch angle variations on the stress state within an Iosipescu shear test speciment. These analytical results were also studied to determine the feasibility of using strain gage rosettes and a modified extensometer to measure shear strains in this test specimen. Analytical results indicate that notch angle variations produced only small differences in simulated shear properties. Both strain gage rosettes and the modified extensometer were shown to be feasible shear strain transducers for the test method. The Iosipoescu shear test fixture was redesigned to incorporate several improvements. These improvements include accommodation of a 50 percent larger specimen for easier measurement of shear train, a clamping mechanism to relax strict tolerances on specimen width, and a self contained alignment tool for use during specimen installation. A set of in-plane and interlaminar shear properties were measured for three graphite fabric/epoxy composites of T300/934 composite material. The three weave patterns were Oxford, 5-harness satin, and 8-harness satin.

Walrath, D. E.↗
NASA NTRS1975

Finite difference identification of noisy distributed systems using scanning measurements

Most of the present-day literature concerned with identification theory and techniques is directed toward lumped parameter systems, and many comprehensive surveys of the field are available. Relatively little has appeared in the literature concerning distributed identification, and even more noticeable is the scarcity of papers dealing with systems described by the one-dimensional wave equation. Perdeauville and Goodson were perhaps the first researchers with a workable but time consuming method for the identification of coefficients of the wave equation. Fairman and Shen, also considering the wave equation, used the technique of finite differencing to approximate spatial derivatives, and Poisson filter chains to approximate temporal derivatives.

Hughes, R. O.↗
NASA NTRS1981

Influence of exit impedance on finite difference solutions of transient acoustic mode propagation in ducts

The time-dependent governing acoustic-difference equations and boundary conditions are developed and solved for sound propagation in an axisymmetric (cylindrical) hard-wall duct without flow and with spinning acoustic modes. The analysis begins with a harmonic sound source radiating into a quiescent duct. This explicit iteration method then calculates stepwise in real time to obtain the steady solutions of the acoustic field. The transient method did not converge to the steady-state solution for cutoff acoustic duct modes. This has implications as to its use in a variable-area duct, where modes may become cutoff in the smal-area portion of the duct. For single cutoff mode propagation the steady-state impedance boundary condition produced acoustic reflections during the initial transient that caused finite instabilities in the numerical calculations. The stability problem is resolved by reformulating the exit boundary condition. Example calculations show good agreement with exact analytical and numerical results for forcing frequencies above, below, and nearly at the cutoff frequency.

Baumeister, K. J.↗
NASA NTRS1990

Matrix-based, finite-difference algorithms for computational acoustics

A compact numerical algorithm is introduced for simulating multidimensional acoustic waves. The algorithm is expressed in terms of a set of matrix coefficients on a three-point spatial grid that approximates the acoustic wave equation with a discretization error of O(h exp 5). The method is based on tracking a local phase variable and its implementation suggests a convenient coordinate splitting along with natural intermediate boundary conditions. Results are presented for oblique plane waves and compared with other procedures. Preliminary computations of acoustic diffraction are also considered.

Davis, Sanford↗
NASA NTRS1995

Development of Fast Algorithms Using Recursion, Nesting and Iterations for Computational Electromagnetics

In the first phase of our work, we have concentrated on laying the foundation to develop fast algorithms, including the use of recursive structure like the recursive aggregate interaction matrix algorithm (RAIMA), the nested equivalence principle algorithm (NEPAL), the ray-propagation fast multipole algorithm (RPFMA), and the multi-level fast multipole algorithm (MLFMA). We have also investigated the use of curvilinear patches to build a basic method of moments code where these acceleration techniques can be used later. In the second phase, which is mainly reported on here, we have concentrated on implementing three-dimensional NEPAL on a massively parallel machine, the Connection Machine CM-5, and have been able to obtain some 3D scattering results. In order to understand the parallelization of codes on the Connection Machine, we have also studied the parallelization of 3D finite-difference time-domain (FDTD) code with PML material absorbing boundary condition (ABC). We found that simple algorithms like the FDTD with material ABC can be parallelized very well allowing us to solve within a minute a problem of over a million nodes. In addition, we have studied the use of the fast multipole method and the ray-propagation fast multipole algorithm to expedite matrix-vector multiplication in a conjugate-gradient solution to integral equations of scattering. We find that these methods are faster than LU decomposition for one incident angle, but are slower than LU decomposition when many incident angles are needed as in the monostatic RCS calculations.

Chew, W. C.↗
NASA NTRS1978

H-H, C-H, and C-C NMR spin-spin coupling constants calculated by the FP-INDO method for aromatic hydrocarbons

The FP-INDO (finite perturbation-intermediate neglect of differential overlap) method is used to calculate the H-H, C-H, and C-C coupling constants in hertz for molecules of six different benzenoid hydrocarbons: benzene, naphthalene, biphenyl, anthracene, phenanthrene, and pyrene. The calculations are based on both the actual and the average molecular geometries. It is found that only the actual molecular geometries can always yield the correct relative order of values for the H-H coupling constants. For the calculated C-C coupling constants, as for the calculated C-H coupling constants, the signs are positive (negative) for an odd (even) number of bonds connecting the two nuclei. Agreements between the calculated and experimental values of the coupling constants for all six molecules are comparable to those reported previously for other molecules.

Long, S. A. T.↗
NASA NTRS1978

Application of the method of integral relations to boundary layer flows over blunt bodies

Calculations of boundary layer flows past blunt bodies at angles of incidence are presented. Using the method of integral relations together with the method of lines, the full three-dimensional boundary layer equations are reduced to a system of first-order ordinary differential equations. The streamwise shear stress function and the cross-flow velocity component are represented as suitable functions of the streamwise velocity component. The role of the zone of dependence is automatically satisfied by the choice of differencing in the method of lines. Solutions correct to the second order are obtained in the positive shear region for flow over an ellipsoid at 30-deg incidence. The results are compared with corresponding finite difference solutions.

Modarress, D.↗
NASA NTRS1980

A study of the sonic-boom characteristics of a blunt body at a Mach number of 6

An experimental and analytical study of the sonic boom static pressure signatures generated by a blunt body at mach 6 has shown that finite difference computer programs can be used to give reasonable estimates of the signatures. The calculated near field static pressure signature was extrapolated to the far field by a program using the method of characteristics. A comparison of this extrapolated signature with the signature predicted by far field sonic boom theory (linearized) shows that peak overpressures are about the same, at least up to mach, but the far field theory overestimates the length of the signature.

Ashby, G. C., Jr.↗
NASA NTRS1981

Test problems for inviscid transonic flow

The paper discusses some results obtained in the solving of test problems for inviscid transonic flow with shock waves using the TRANDES program. The method used employs the full inviscid perturbation-potential flow equation in a Cartesian grid system that is stretched to infinity. The equation is represented by a nonconservative system of finite-difference equations that includes at supersonic points a rotated difference scheme and is solved by column relaxation. Except for the NACA 0012 case with the freestream Mach number equal to 0.95, all test problems were solved straightforwardly and appeared to be converged or close to convergence. The only difficulty was some sensitivity to grid placement, which is typical of the Cartesian formulation

Carlson, L. A.↗
NASA NTRS1988

Dynamic delamination fracture toughness of a graphite/epoxy laminate under impact

Dynamic delamination fracture toughness in a (90/0)5s T300/934 graphite/epoxy laminate was investigated using impact loading. Delamination cracks of three different sizes were embedded at the midplane of the composite specimen. The threshold impact velocity that causes propagation of the delamination crack was used in the dynamic analysis with the finite element method. From the finite element solution, the time-history of the strain energy release rate was calculated. The critical strain energy release rate was taken to equal the maximum value of the response history.

Sun, C. T.↗
NASA NTRS1991

High-order ENO schemes applied to two- and three-dimensional compressible flow

High order essentially non-oscillatory (ENO) finite difference schemes are applied to the 2-D and 3-D compressible Euler and Navier-Stokes equations. Practical issues, such as vectorization, efficiency of coding, cost comparison with other numerical methods, and accuracy degeneracy effects, are discussed. Numerical examples are provided which are representative of computational problems of current interest in transition and turbulence physics. These require both nonoscillatory shock capturing and high resolution for detailed structures in the smooth regions and demonstrate the advantage of ENO schemes.

Shu, Chi-Wang↗