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NASA NTRS1989

A least-squares finite element method for incompressible Navier-Stokes problem

Most finite element schemes for solving the Navier-Stokes equations can be categorized into the Galerkin mixed method and the penalty method. The mixed method leads to a saddle-point problem. In order to guarantee the existence of a solution, the combination of velocity and pressure interpolations requires satisfaction of the Ladyzhenskaya Babuska Brezzi (LBB) consistency condition which precludes the use of equal order interpolations and many seemingly natural pairs of velocity and pressure elements. In a previous paper a least-squares finite element method based on the first order velocity-pressure-vorticity formulation for the Stokes problem was proposed. This method leads to a minimization problem. The choice of combination of elements is thus not subject to the LBB condition. The numerical experiments exhibit the optimal rate of convergence for all variables with equal order interpolations. A theoretical error analysis supports the numerical results. In this paper the least-squares finite element method is extended to solving the incompressible Navier-Stokes problem.

Jiang, Bonan↗
NASA NTRS1992

Direct numerical simulation of instabilities in parallel flow with spherical roughness elements

Results from a direct numerical simulation of laminar flow over a flat surface with spherical roughness elements using a spectral-element method are given. The numerical simulation approximates roughness as a cellular pattern of identical spheres protruding from a smooth wall. Periodic boundary conditions on the domain's horizontal faces simulate an infinite array of roughness elements extending in the streamwise and spanwise directions, which implies the parallel-flow assumption, and results in a closed domain. A body force, designed to yield the horizontal Blasius velocity in the absence of roughness, sustains the flow. Instabilities above a critical Reynolds number reveal negligible oscillations in the recirculation regions behind each sphere and in the free stream, high-amplitude oscillations in the layer directly above the spheres, and a mean profile with an inflection point near the sphere's crest. The inflection point yields an unstable layer above the roughness (where U''(y) is less than 0) and a stable region within the roughness (where U''(y) is greater than 0). Evidently, the instability begins when the low-momentum or wake region behind an element, being the region most affected by disturbances (purely numerical in this case), goes unstable and moves. In compressible flow with periodic boundaries, this motion sends disturbances to all regions of the domain. In the unstable layer just above the inflection point, the disturbances grow while being carried downstream with a propagation speed equal to the local mean velocity; they do not grow amid the low energy region near the roughness patch. The most amplified disturbance eventually arrives at the next roughness element downstream, perturbing its wake and inducing a global response at a frequency governed by the streamwise spacing between spheres and the mean velocity of the most amplified layer.

Deanna, R. G.↗
NASA NTRS1990

Using a multifrontal sparse solver in a high performance, finite element code

We consider the performance of the finite element method on a vector supercomputer. The computationally intensive parts of the finite element method are typically the individual element forms and the solution of the global stiffness matrix both of which are vectorized in high performance codes. To further increase throughput, new algorithms are needed. We compare a multifrontal sparse solver to a traditional skyline solver in a finite element code on a vector supercomputer. The multifrontal solver uses the Multiple-Minimum Degree reordering heuristic to reduce the number of operations required to factor a sparse matrix and full matrix computational kernels (e.g., BLAS3) to enhance vector performance. The net result in an order-of-magnitude reduction in run time for a finite element application on one processor of a Cray X-MP.

King, Scott D.↗
NASA NTRS1993

Composite micromechanical modeling using the boundary element method

The use of the boundary element method for analyzing composite micromechanical behavior is demonstrated. Stress-strain, heat conduction, and thermal expansion analyses are conducted using the boundary element computer code BEST-CMS, and the results obtained are compared to experimental observations, analytical calculations, and finite element analyses. For each of the analysis types, the boundary element results agree reasonably well with the results from the other methodologies, with explainable discrepancies. Overall, the boundary element method shows promise in providing an alternative method to analyze composite micromechanical behavior.

Goldberg, Robert K.↗
NASA NTRS1992

Membrane triangles with corner drilling freedoms. II - The ANDES element

This is the second article in a three-part series on the construction of 3-node, 9-dof membrane elements with normal-to-its-plane rotational freedoms (the so-called drilling freedoms) using parametrized variational principles. In this part, one such element is derived within the context of the assumed natural deviatoric strain (ANDES) formulation. The higher-order strains are obtained by constructing three parallel-to-sides pure-bending modes from which natural strains are obtained at the corner points and interpolated over the element. To attain rank sufficiency, an additional higher-order 'torsional' mode, corresponding to equal hierarchical rotations at each corner with all other motions precluded, is incorporated. The resulting formulation has five free parameters. When these parameters are optimized against pure bending by energy balance methods, the resulting element is found to coalesce with the optimal EFF element derived in Part I. Numerical integration as a strain filtering device is found to play a key role in this achievement.

Felippa, Carlos A.↗
NASA NTRS1991

An accurate nonlinear finite element analysis and test correlation of a stiffened composite wing panel

State-of-the-art nonlinear finite element analysis techniques are evaluated by applying them to a realistic aircraft structural component. A wing panel from the V-22 tiltrotor aircraft is chosen because it is a typical modern aircraft structural component for which there is experimental data for comparison of results. From blueprints and drawings, a very detailed finite element model containing 2284 9-node Assumed Natural-Coordinate Strain elements was generated. A novel solution strategy which accounts for geometric nonlinearity through the use of corotating element reference frames and nonlinear strain-displacement relations is used to analyze this detailed model. Results from linear analyses using the same finite element model are presented in order to illustrate the advantages and costs of the nonlinear analysis as compared with the more traditional linear analysis.

Davis, D. D., Jr.↗
NASA NTRS1993

Probabilistic nonlinear finite element analysis of composite structures

A probabilistic finite element analysis procedure for laminated composite shells is developed. A total Lagrangian finite element formulation, employing a degenerated three-dimensional laminated composite shell element with the full Green-Lagrange strains and first-order shear deformable kinematics, is used. The first-order second-moment technique for probabilistic finite element analysis of random fields is employed, and results are presented in the form of mean and variance of the structural response. Reliability calculations are made by using the first-order reliability method combined with sensitivity derivatives from the finite element analysis. Both ply-level and micromechanics-level random variables are incorporated, the latter by means of the Aboudi micromechanics model. Two sample problems are solved to verify the accuracy of the procedures developed and to quantify the variability of certain material type/structure combinations. In general, the procedure is quite effective in determining the response statistics and reliability for linear and geometric nonlinear behavior of laminated composite shells.

Engelstad, S. P.↗
NASA NTRS1993

Metal-silicate thermochemistry at high temperature - Magma oceans and the 'excess siderophile element' problem of the earth's upper mantle

Low-temperature metal-silicate partition coefficients are extrapolated to magma ocean temperatures. If the low-temperature chemistry data is found to be applicable at high temperatures, an important assumption, then the results indicate that high temperature alone cannot account for the excess siderophile element problem of the upper mantle. For most elements, a rise in temperature will result in a modest increase in siderophile behavior if an iron-wuestite redox buffer is paralleled. However, long-range extrapolation of experimental data is hazardous when the data contains even modest experimental errors. For a given element, extrapolated high-temperature partition coefficients can differ by orders of magnitude, even when data from independent studies is consistent within quoted errors. In order to accurately assess siderophile element behavior in a magma ocean, it will be necessary to obtain direct experimental measurements for at least some of the siderophile elements.

Capobianco, Christopher J.↗
NASA NTRS1993

Recent advances and progress towards an integrated interdisciplinary thermal-structural finite element technology

An integrated finite element approach is presented for interdisciplinary thermal-structural problems. Of the various numerical approaches, finite element methods with direct time integration procedures are most widely used for these nonlinear problems. Traditionally, combined thermal-structural analysis is performed sequentially by transferring data between thermal and structural analysis. This approach is generally effective and routinely used. However, to solve the combined thermal-structural problems, this approach results in cumbersome data transfer, incompatible algorithmic representations, and different discretized element formulations. The integrated approach discussed in this paper effectively combines thermal and structural fields, thus overcoming the above major shortcomings. The approach follows Lax-Wendroff type finite element formulations with flux and stress based representations. As a consequence, this integrated approach uses common algorithmic representations and element formulations. Illustrative test examples show that the approach is effective for integrated thermal-structural problems.

Namburu, Raju R.↗
NASA NTRS1991

The L sub 1 finite element method for pure convection problems

The least squares (L sub 2) finite element method is introduced for 2-D steady state pure convection problems with smooth solutions. It is proven that the L sub 2 method has the same stability estimate as the original equation, i.e., the L sub 2 method has better control of the streamline derivative. Numerical convergence rates are given to show that the L sub 2 method is almost optimal. This L sub 2 method was then used as a framework to develop an iteratively reweighted L sub 2 finite element method to obtain a least absolute residual (L sub 1) solution for problems with discontinuous solutions. This L sub 1 finite element method produces a nonoscillatory, nondiffusive and highly accurate numerical solution that has a sharp discontinuity in one element on both coarse and fine meshes. A robust reweighting strategy was also devised to obtain the L sub 1 solution in a few iterations. A number of examples solved by using triangle and bilinear elements are presented.

Jiang, Bo-Nan↗
NASA NTRS1992

Coupling finite element and integral equation solutions using decoupled boundary meshes

A method is outlined for calculating scattered fields from inhomogeneous penetrable objects using a coupled finite element-integral equation solution. The finite element equation can efficiently model fields in penetrable and inhomogeneous regions, while the integral equation exactly models fields on the finite element mesh boundary and in the exterior region. By decoupling the interior finite element and exterior integral equation meshes, considerable flexibility is found in both the number of field expansion points as well as their density. Only the nonmetal portions of the object need be modeled using a finite element expansion; exterior perfect conducting surfaces are modeled using an integral equation with a single unknown field since E(tan) is identically zero on these surfaces. Numerical convergence, accuracy, and stability at interior resonant frequencies are studied in detail.

Cwik, Tom↗
NASA NTRS1993

A combined experimental and finite element study to predict the failure mechanisms in SiC coated carbon/carbon composites at room and elevated temperatures under flexural loading

Response of quasi-isotropic laminates of SiC coated Carbon/Carbon (C/C) composites have been investigated under flexural loading at various temperatures. Variation of load-deflection behavior with temperatures are studied. Increase in flexural strength and stiffness are observed with the rise in temperature. Extensive analyses through Optical Microscope (OM) and Non-Destructive Evaluation (NDE) have been performed to understand the failure mechanisms. Damage zone is found only within the neighborhood of the loading plane. Isoparametric layered shell elements developed on the basis of the first order shear deformation theory have been used to model the thin laminates of C/C under flexural loading. Large deformation behavior has been considered in the finite element analysis to account for the non-linearities encountered during the actual test. Data generated using finite element analysis are presented to corroborate the experimental findings, and a comparison in respect of displacement and stress-strain behavior are given to check the accuracy of the finite element analysis. Reasonable correlation between the experimental and finite element results have been established.

Mahfuz, Hassan↗
NASA NTRS1993

The interstellar abundances of tin and four other heavy elements

Spectra recorded at 1150-1600 A with an instrumental resolution near 16 km/s were obtained with the Goddard High-Resolution Spectrograph on board the HST. The gaseous interstellar abundances of five heavy elements along the light paths to 23 Ori, 15 Mon, 1 Sco, Pi Sco, and Pi Aqr were determined from the observations. The 1400.450 A line of Sn II was detected and identified toward three stars; at Z = 50, tin is the first element from the fifth row of the periodic table to be identified in the interstellar medium. One spectral line of each of Cu II (Z = 29) and Ga II (Z = 31), three lines of Ge II (Z = 32), and two lines of Kr I (Z = 36) were also detected toward some or all of the five stars. The depletions of these five heavy elements generally decrease monotonically with increasing atomic number toward each of the six stars, and tin is generally undepleted within the observational errors. The depletions of 26 elements from the interstellar gas in an average dense interstellar cloud appear to correlate with the elemental 'nebular' condensation temperatures more closely than with the first ionization potentials.

Hobbs, L. M.↗
NASA NTRS1994

Modal element method for potential flow in non-uniform ducts: Combining closed form analysis with CFD

An analytical procedure is presented, called the modal element method, that combines numerical grid based algorithms with eigenfunction expansions developed by separation of variables. A modal element method is presented for solving potential flow in a channel with two-dimensional cylindrical like obstacles. The infinite computational region is divided into three subdomains; the bounded finite element domain, which is characterized by the cylindrical obstacle and the surrounding unbounded uniform channel entrance and exit domains. The velocity potential is represented approximately in the grid based domain by a finite element solution and is represented analytically by an eigenfunction expansion in the uniform semi-infinite entrance and exit domains. The calculated flow fields are in excellent agreement with exact analytical solutions. By eliminating the grid surrounding the obstacle, the modal element method reduces the numerical grid size, employs a more precise far field boundary condition, as well as giving theoretical insight to the interaction of the obstacle with the mean flow. Although the analysis focuses on a specific geometry, the formulation is general and can be applied to a variety of problems as seen by a comparison to companion theories in aeroacoustics and electromagnetics.

Baumeister, Kenneth J.↗
NASA NTRS1993

Massively parallel computation of RCS with finite elements

One of the promising combinations of finite element approaches for scattering problems uses Whitney edge elements, spherical vector wave-absorbing boundary conditions, and bi-conjugate gradient solution for the frequency-domain near field. Each of these approaches may be criticized. Low-order elements require high mesh density, but also result in fast, reliable iterative convergence. Spherical wave-absorbing boundary conditions require additional space to be meshed beyond the most minimal near-space region, but result in fully sparse, symmetric matrices which keep storage and solution times low. Iterative solution is somewhat unpredictable and unfriendly to multiple right-hand sides, yet we find it to be uniformly fast on large problems to date, given the other two approaches. Implementation of these approaches on a distributed memory, message passing machine yields huge dividends, as full scalability to the largest machines appears assured and iterative solution times are well-behaved for large problems. We present times and solutions for computed RCS for a conducting cube and composite permeability/conducting sphere on the Intel ipsc860 with up to 16 processors solving over 200,000 unknowns. We estimate problems of approximately 10 million unknowns, encompassing 1000 cubic wavelengths, may be attempted on a currently available 512 processor machine, but would be exceedingly tedious to prepare. The most severe bottlenecks are due to the slow rate of mesh generation on non-parallel machines and the large transfer time from such a machine to the parallel processor. One solution, in progress, is to create and then distribute a coarse mesh among the processors, followed by systematic refinement within each processor. Elimination of redundant node definitions at the mesh-partition surfaces, snap-to-surface post processing of the resulting mesh for good modelling of curved surfaces, and load-balancing redistribution of new elements after the refinement are auxiliary steps expected to result in a robust low i/o system for very large finite element problems.

Parker, Jay↗
NASA NTRS1993

Multi-band reflector antenna with double-ring element frequency selective subreflector

Frequency selective subreflectors (FSS) are often employed in the reflector antenna system of a communication satellite or a deep space exploration vehicle for multi-frequency operations. In the past, FSS's have been designed for diplexing two frequency bands. For example, the Voyager FSS was designed to diplex S and X bands and the TDRSS FSS was designed to diplex S and Ku bands. Recently, NASA's CASSINI project requires an FSS to multiplex four frequency (S/X/Ku/Ka) bands. Theoretical analysis and experimental verifications are presented for a multi-band flat pannel FSS with double-ring elements. Both the exact formulation and the thin-ring approximation are described for analyzing and designing this multi-ring patch element FSS. It is found that the thin-ring approximation fails to predict the electrically wide ring element FSS's performance. A single screen double-ring element FSS is demonstrated for the tri-band system that reflects the X-band signal while transmitting through the S- and Ku-band signals. In addition, a double screen FSS with non-similar double-ring elements is presented for the Cassini's four-band system which reflects the X- and Ka-band signals while passing the S- and Ku-band signals. To accurately predict the FSS effects on a dual reflector antenna's radiation pattern, the FSS subreflector's transmitted/reflected field variation as functions of the polarization and incident angles with respect to the local coordinates was taken into account. An FSS transmission/reflection coefficient table is computed for TE and TM polarizations at various incident angles based on the planar FSS model. Next, the hybrid Geometric Optics (GO) and Physical Optics (PO) technique is implemented with linearly interpolating the FSS table to efficiently determine the FSS effects in a dual reflector antenna.

Wu, Te-Kao↗
NASA NTRS1993

Domain decomposition methods for nonconforming finite element spaces of Lagrange-type

In this article, we consider the application of three popular domain decomposition methods to Lagrange-type nonconforming finite element discretizations of scalar, self-adjoint, second order elliptic equations. The additive Schwarz method of Dryja and Widlund, the vertex space method of Smith, and the balancing method of Mandel applied to nonconforming elements are shown to converge at a rate no worse than their applications to the standard conforming piecewise linear Galerkin discretization. Essentially, the theory for the nonconforming elements is inherited from the existing theory for the conforming elements with only modest modification by constructing an isomorphism between the nonconforming finite element space and a space of continuous piecewise linear functions.

Cowsar, Lawrence C.↗
NASA NTRS1993

Numerical investigation of multi-element airfoils

The flow over multi-element airfoils with flat-plate lift-enhancing tabs was numerically investigated. Tabs ranging in height from 0.25 percent to 1.25 percent of the reference airfoil chord were studied near the trailing edge of the main-element. This two-dimensional numerical simulation employed an incompressible Navier-Stokes solver on a structured, embedded grid topology. New grid refinements were used to improve the accuracy of the solution near the overlapping grid boundaries. The effects of various tabs were studied at a constant Reynolds number on a two-element airfoil with a slotted flap. Both computed and measured results indicated that a tab in the main-element cove improved the maximum lift and lift-to-drag ratio relative to the baseline airfoil without a tab. Computed streamlines revealed that the additional turning caused by the tab may reduce the amount of separated flow on the flap. A three-element airfoil was also studied over a range of Reynolds numbers. For the optimized flap rigging, the computed and measured Reynolds number effects were similar. When the flap was moved from the optimum position, numerical results indicated that a tab may help to reoptimize the airfoil to within 1 percent of the optimum flap case.

Cummings, Russell M.↗