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NASA NTRSDate not supplied

Comparison of Two Approaches to Constructing Second- and Third-Order Nodal-Gradient Cell-Centered Finite-Volume Methods for Mixed-Element Grids

In this paper, we introduce a third-order nodal-gradient cell-centered ﬁnite volume scheme applicable to mixed-element grids based on a single numerical ﬂux per face. Previously, we considered a split-quadrilateral-face approach, where we split each quadrilateral face of a non-tetrahedral cell into two triangles and then apply second- and third-order schemes designed for tetrahedral grids. This approach is relatively simple to implement but expensive because it requires two numerical ﬂux evaluations per quadrilateral face. In this study, we consider another approach, where the surface ﬂux quadrature is performed with a single numerical ﬂux per face with ﬂux corrections. It requires a 3×3 correction matrix to be stored at each face but is more economical than the split-quadrilateral-face approach. The two approaches are compared for three-dimensional ﬂows on irregular hexahedral grids, and their relative merits are discussed.

Third-order scheme↗

NASA NTRSDate not supplied

Microscale Constitutive Model Sensitivity on Multiscale Modeling of Fiber Reinforced Composites

Fiber reinforced composites are desirable in applications where low weight and high strength are needed, but are susceptible to microscale variability during manufacturing, making failure predictions difficult. The impact microscale variability has on macroscale mechanical response is difficult to predict due to the computational efficiency needed to simulate many, large, high fidelity, microscale models. In this study, a multiscale approach was taken to model 3-point bend, 4-point bend, and tensile experiments of a unidirectional composite from only having microstructure scans of these samples and constituent properties from literature. These scans were sampled with different sized windows, and statistically equivalent microstructures were generated, then simulated for stiffness, strength, and fracture toughness using an efficient micromechanical. Mesoscale models were created where element sizes equaled microstructure size, and properties were assigned through sampling of microscale simulation results. First, this study showed the effect of using Weibull scaling on constituent matrix strength on macroscale response. Then, a comparison was made between different element sizes and experiments. Finally, model dimensions were fixed, and the effect of randomly distributed local properties alone was examined. Results showed that the scatter of strength and stiffness in the experiments could be predicted well using images of the microscale fiber morphologies and that using stochastic properties produced a 3% coefficient of variation of strength for all experiments.

statistical microstructure↗

NASA NTRS1985

Vibrations of twisted cantilever plates - A comparison of theoretical results

As a result of significant differences in the published results for various methods of analysis involving the use of finite element techniques, there are now some questions regarding the adequacy of these methods to predict accurately the vibratory characteristics of highly twisted cantilever plates. In an attempt to help in a resolution of the arising problems, a joint government/industry/university research effort was initiated. The primary objective of the present paper is to summarize the theoretical methods used in the study and show samples of the obtained results. The study provided 19 sets of theoretical results which are derived from beam theory, shell theory, and finite element methods.

Kielb, R. E.↗

NASA NTRS1991

How to determine spiral bevel gear tooth geometry for finite element analysis

An analytical method was developed to determine gear tooth surface coordinates of face milled spiral bevel gears. The method combines the basic gear design parameters with the kinematical aspects for spiral bevel gear manufacturing. A computer program was developed to calculate the surface coordinates. From this data a 3-D model for finite element analysis can be determined. Development of the modeling method and an example case are presented.

Handschuh, Robert F.↗

NASA NTRS1998

Finite Element and Plate Theory Modeling of Acoustic Emission Waveforms

A comparison was made between two approaches to predict acoustic emission waveforms in thin plates. A normal mode solution method for Mindlin plate theory was used to predict the response of the flexural plate mode to a point source, step-function load, applied on the plate surface. The second approach used a dynamic finite element method to model the problem using equations of motion based on exact linear elasticity. Calculations were made using properties for both isotropic (aluminum) and anisotropic (unidirectional graphite/epoxy composite) materials. For simulations of anisotropic plates, propagation along multiple directions was evaluated. In general, agreement between the two theoretical approaches was good. Discrepancies in the waveforms at longer times were caused by differences in reflections from the lateral plate boundaries. These differences resulted from the fact that the two methods used different boundary conditions. At shorter times in the signals, before reflections, the slight discrepancies in the waveforms were attributed to limitations of Mindlin plate theory, which is an approximate plate theory. The advantages of the finite element method are that it used the exact linear elasticity solutions, and that it can be used to model real source conditions and complicated, finite specimen geometries as well as thick plates. These advantages come at a cost of increased computational difficulty, requiring lengthy calculations on workstations or supercomputers. The Mindlin plate theory solutions, meanwhile, can be quickly generated on personal computers. Specimens with finite geometry can also be modeled. However, only limited simple geometries such as circular or rectangular plates can easily be accommodated with the normal mode solution technique. Likewise, very limited source configurations can be modeled and plate theory is applicable only to thin plates.

Prosser, W. H.↗

NASA NTRS1993

Computational mechanics needs study

In order to assess the needs in computational mechanics over the next decade, we formulated a questionnaire and contacted computational mechanics researchers and users in industry, government, and academia. As expected, we found a wide variety of computational mechanics usage and research. This report outlines the activity discussed with those contacts, as well as that in our own organizations. It should be noted that most of the contacts were made before the recent decline of the defense industry. Therefore, areas which are strongly defense-oriented may decrease in relative importance. In order to facilitate updating of this study, names of a few key researchers in each area are included as starting points for future literature surveys. These lists of names are not intended to represent those persons doing the best research in that area, nor are they intended to be comprehensive. They are, as previously stated, offered as starting points for future literature searches. Overall, there is currently a broad activity in computational mechanics in this country, with the breadth and depth increasing as more sophisticated software and faster computers become more available. The needs and desires of the workers in this field are as diverse as their background and organizational products. There seems to be some degree of software development in any organization (although the level of activity is highly variable from one organization to another) which has any research component in its mission. It seems, however, that there is considerable use of commercial software in almost all organizations. In most industrial research organizations, it appears that very little actual software development is contracted out, but that most is done in-house, using a mixture of funding sources. Government agencies vary widely in the ratio of in-house to out-house ratio. There is a considerable amount of experimental verification in most, but not all, organizations. Generally, the amount of experimental verification is more than we expected. Of all the survey contacts, one or two believe that the resources they are allocated are sufficient, but most do not. Some believe they have only half the resources they need. Some see their resource deficits as short-term, while others see it as a trend which will continue or perhaps worsen. The pessimism is stronger in the defense and aerospace industry. When considering only the nonlinear development efforts, there appears to be an even mix of geometric and material nonlinearity. There is not much particular emphasis in linear analysis unless it is in extension of current analysis capabilities to larger problems. The primary exception is concern about modeling of composites, where proven methodologies have trailed element and computer hardware development. Most of the people we spoke to use finite element techniques, but there is some finite difference and boundary element work ongoing. There is also some interest in multiple methods. Coupling of finite elements and boundary elements appears to be of high interest, since the two analysis types are complementary.

Griffin, O. Hayden, Jr.↗

NASA NTRS1996

FEM R-Matrix Calculations of Electron-H2 Collisions

Electron-H2 Collisions have been studied using the finite-element R-matrix method. Our approach uses a mixed finite-element and Gaussian basis to describe the continuum electron. In the inner region, the multi-centered nature of the Gaussian basis provides an efficient representation in the regions of space near the nuclei, whereas the piecemeal and energy-independent nature of the FEM basis is particular well suited for R-matrix calculations. Fixed nuclei calculations have been carried out at 1.4 a(sub 0), the equilibrium internuclear distance of the ground state. Up to six target states are included: the X(sup 1)E+(sub g), b(sup 3)E+(sub u), a(sup 3)E+(sub g), B(sup 1)E+(sub u), c(sup 3)II(sub u), and C(sup 1)II(sub u) states, and configuration-interaction functions are used to describe the target states. The results will be compared with available theoretical and experimental data.

Huo, Winifred M.↗

NASA NTRS1990

Evaluation of finite-element models and stress-intensity factors for surface cracks emanating from stress concentrations

Tan, P. W.↗

NASA NTRS1988

A re-evaluation of finite-element models and stress-intensity factors for surface cracks emanating from stress concentrations

A re-evaluation of the 3-D finite-element models and methods used to analyze surface crack at stress concentrations is presented. Previous finite-element models used by Raju and Newman for surface and corner cracks at holes were shown to have ill-shaped elements at the intersection of the hole and crack boundaries. These ill-shaped elements tended to make the model too stiff and, hence, gave lower stress-intensity factors near the hole-crack intersection than models without these elements. Improved models, without these ill-shaped elements, were developed for a surface crack at a circular hole and at a semi-circular edge notch. Stress-intensity factors were calculated by both the nodal-force and virtual-crack-closure methods. Both methods and different models gave essentially the same results. Comparisons made between the previously developed stress-intensity factor equations and the results from the improved models agreed well except for configurations with large notch-radii-to-plate-thickness ratios. Stress-intensity factors for a semi-elliptical surface crack located at the center of a semi-circular edge notch in a plate subjected to remote tensile loadings were calculated using the improved models. The ratio of crack depth to crack length ranged form 0.4 to 2; the ratio of crack depth to plate thickness ranged from 0.2 to 0.8; and the ratio of notch radius to the plate thickness ranged from 1 to 3. The models had about 15,000 degrees-of-freedom. Stress-intensity factors were calculated by using the nodal-force method.

Tan, P. W.↗

NASA NTRS2004

Three-Dimensional High-Order Spectral Volume Method for Solving Maxwell's Equations on Unstructured Grids

A three-dimensional, high-order, conservative, and efficient discontinuous spectral volume (SV) method for the solutions of Maxwell's equations on unstructured grids is presented. The concept of discontinuous 2nd high-order loca1 representations to achieve conservation and high accuracy is utilized in a manner similar to the Discontinuous Galerkin (DG) method, but instead of using a Galerkin finite-element formulation, the SV method is based on a finite-volume approach to attain a simpler formulation. Conventional unstructured finite-volume methods require data reconstruction based on the least-squares formulation using neighboring cell data. Since each unknown employs a different stencil, one must repeat the least-squares inversion for every cell at each time step, or to store the inversion coefficients. In a high-order, three-dimensional computation, the former would involve impractically large CPU time, while for the latter the memory requirement becomes prohibitive. In the SV method, one starts with a relatively coarse grid of triangles or tetrahedra, called spectral volumes (SVs), and partition each SV into a number of structured subcells, called control volumes (CVs), that support a polynomial expansion of a desired degree of precision. The unknowns are cell averages over CVs. If all the SVs are partitioned in a geometrically similar manner, the reconstruction becomes universal as a weighted sum of unknowns, and only a few universal coefficients need to be stored for the surface integrals over CV faces. Since the solution is discontinuous across the SV boundaries, a Riemann solver is thus necessary to maintain conservation. In the paper, multi-parameter and symmetric SV partitions, up to quartic for triangle and cubic for tetrahedron, are first presented. The corresponding weight coefficients for CV face integrals in terms of CV cell averages for each partition are analytically determined. These discretization formulas are then applied to the integral form of the Maxwell equations. All numerical procedures for outer boundary, material interface, zonal interface, and interior SV face are unified with a single characteristic formulation. The load balancing in a massive parallel computing environment is therefore easier to achieve. A parameter is introduced in the Riemann solver to control the strength of the smoothing term. Important aspects of the data structure and its effects to communication and the optimum use of cache memory are discussed. Results will be presented for plane TE and TM waves incident on a perfectly conducting cylinder for up to fifth order of accuracy, and a plane wave incident on a perfectly conducting sphere for up to fourth order of accuracy. Comparisons are made with exact solutions for these cases.

Liu, Yen↗

NASA NTRS2004

Error Analysis for Discontinuous Galerkin Method for Parabolic Problems

In the proposal, the following three objectives are stated: (1) A p-version of the discontinuous Galerkin method for a one dimensional parabolic problem will be established. It should be recalled that the h-version in space was used for the discontinuous Galerkin method. An a priori error estimate as well as a posteriori estimate of this p-finite element discontinuous Galerkin method will be given. (2) The parameter alpha that describes the behavior double vertical line u(sub t)(t) double vertical line 2 was computed exactly. This was made feasible because of the explicitly specified initial condition. For practical heat transfer problems, the initial condition may have to be approximated. Also, if the parabolic problem is proposed on a multi-dimensional region, the parameter alpha, for most cases, would be difficult to compute exactly even in the case that the initial condition is known exactly. The second objective of this proposed research is to establish a method to estimate this parameter. This will be done by computing two discontinuous Galerkin approximate solutions at two different time steps starting from the initial time and use them to derive alpha. (3) The third objective is to consider the heat transfer problem over a two dimensional thin plate. The technique developed by Vogelius and Babuska will be used to establish a discontinuous Galerkin method in which the p-element will be used for through thickness approximation. This h-p finite element approach, that results in a dimensional reduction method, was used for elliptic problems, but the application appears new for the parabolic problem. The dimension reduction method will be discussed together with the time discretization method.

Kaneko, Hideaki↗

NASA NTRS2000

Real-time simulation of dynamically deformable finite element models using modal analysis and spectral Lanczos decomposition methods

We present two efficient methods for simulating real-time behavior of a dynamically deformable 3D object modeled by finite element equations. The first method is based on modal analysis, which uses the most significant vibration modes of the object to compute the deformations in real-time for applied forces. The second method uses the spectral Lanczos decomposition to obtain the explicit solutions of the finite element equations.

medical↗

NASA NTRS2001

Thermal Analysis of a High-Speed Aircraft Wing Using p-Version Finite Elements

This paper presents the results of conceptual level thermal analyses of a High Speed Civil Transport (HSCT) wing using p-version finite elements. The work was motivated by a thermal analysis of a HSCT wing structure which showed the importance of radiation heat transfer throughout the structure. The analysis also showed that refining a traditional finite element mesh to accurately capture the temperature distribution on the internal structure led to very large meshes with unacceptably long execution times. Further study indicated using p-version finite elements might improve computation performance for this class of problem. Methods for determining internal radiation heat transfer were then developed and demonstrated on test problems representative of the geometry found in an aircraft wing structure. This paper presents the results of the application of these new methods to the analysis of a high speed aircraft wing. Results for both a wing box model as well as a full wing model are presented. 'Me reduced wing box model allows for a comparison of the traditional finite element method with mesh refinement (h-refinement) to the new p-version finite elements while the full wing model demonstrates the applicability and efficiency of p-version finite elements for large models.

Gould, Dana C.↗

NASA NTRS1992

Minimization of the vibration energy of thin-plate structure

An optimization method is proposed to reduce the vibration of thin plate structures. The method is based on a finite element shell analysis, a modal analysis, and a structural optimization method. In the finite element analysis, a triangular shell element with 18 dof is used. In the optimization, the overall vibration energy of the structure is adopted as the objective function, and it is minimized at the given exciting frequency by varying the thickness of the elements. The technique of modal analysis is used to derive the sensitivity of the vibration energy with respect to the design variables. The sensitivity is represented by the sensitivities of both eigenvalues and eigenvectors. The optimum value is computed by the gradient projection method and a unidimensional search procedure under the constraint condition of constant weight. A computer code, based on the proposed method, is developed and is applied to design problems using a beam and a plate as test cases. It is confirmed that the vibration energy is reduced at the given exciting frequency. For the beam excited by a frequency slightly less than the fundamental natural frequency, the optimized shape is close to the beam of uniform strength.

Inoue, Katsumi↗

NASA NTRS2002

The Fractional Step Method Applied to Simulations of Natural Convective Flows

This paper describes research done to apply the Fractional Step Method to finite-element simulations of natural convective flows in pure liquids, permeable media, and in a directionally solidified metal alloy casting. The Fractional Step Method has been applied commonly to high Reynold's number flow simulations, but is less common for low Reynold's number flows, such as natural convection in liquids and in permeable media. The Fractional Step Method offers increased speed and reduced memory requirements by allowing non-coupled solution of the pressure and the velocity components. The Fractional Step Method has particular benefits for predicting flows in a directionally solidified alloy, since other methods presently employed are not very efficient. Previously, the most suitable method for predicting flows in a directionally solidified binary alloy was the penalty method. The penalty method requires direct matrix solvers, due to the penalty term. The Fractional Step Method allows iterative solution of the finite element stiffness matrices, thereby allowing more efficient solution of the matrices. The Fractional Step Method also lends itself to parallel processing, since the velocity component stiffness matrices can be built and solved independently of each other. The finite-element simulations of a directionally solidified casting are used to predict macrosegregation in directionally solidified castings. In particular, the finite-element simulations predict the existence of 'channels' within the processing mushy zone and subsequently 'freckles' within the fully processed solid, which are known to result from macrosegregation, or what is often referred to as thermo-solutal convection. These freckles cause material property non-uniformities in directionally solidified castings; therefore many of these castings are scrapped. The phenomenon of natural convection in an alloy under-going directional solidification, or thermo-solutal convection, will be explained. The development of the momentum and continuity equations for natural convection in a fluid, a permeable medium, and in a binary alloy undergoing directional solidification will be presented. Finally, results for natural convection in a pure liquid, natural convection in a medium with a constant permeability, and for directional solidification will be presented.

Westra, Douglas G.↗

NASA NTRS1989

Parallel processing techniques for finite element analysis of nonlinear large truss structures

Methods were developed for parallel processing of finite element solutions of large truss structures. The parallel processing techniques were implemented in two stages, i.e., the repeated forming of the nonlinear global stiffness matrix and the solving of the global system of equations. The Sequent Balance 21000 parallel computer was employed to demonstrate the procedures and the speed-up.

Chien, L. S.↗

NASA NTRS1982

Computation of consistent boundary quantities in finite element thermal-fluid solutions

The consistent boundary quantity method for computing derived quantities from finite element nodal variable solutions is investigated. The method calculates consistent, continuous boundary surface quantities such as heat fluxes, flow velocities, and surface tractions from nodal variables such as temperatures, velocity potentials, and displacements. Consistent and lumped coefficient matrix solutions for such problems are compared. The consistent approach may produce more accurate boundary quantities, but spurious oscillations may be produced in the vicinity of discontinuities. The uncoupled computations of the lumped approach provide greater flexibility in dealing with discontinuities and provide increased computational efficiency. The consistent boundary quantity approach can be applied to solution boundaries other than those with Dirichlet boundary conditions, and provides more accurate results than the customary method of differentiation of interpolation polynomials.

Thornton, E. A.↗

NASA NTRS1976

Approximation concepts for efficient structural synthesis

It is shown that efficient structural synthesis capabilities can be created by using approximation concepts to mesh finite element structural analysis methods with nonlinear mathematical programming techniques. The history of the application of mathematical programming techniques to structural design optimization problems is reviewed. Several rather general approximation concepts are described along with the technical foundations of the ACCESS 1 computer program, which implements several approximation concepts. A substantial collection of structural design problems involving truss and idealized wing structures is presented. It is concluded that since the basic ideas employed in creating the ACCESS 1 program are rather general, its successful development supports the contention that the introduction of approximation concepts will lead to the emergence of a new generation of practical and efficient, large scale, structural synthesis capabilities in which finite element analysis methods and mathematical programming algorithms will play a central role.

Schmit, L. A., Jr.↗