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NASA NTRS1990

Finite element mesh refinement criteria for stress analysis

This paper discusses procedures for finite-element mesh selection and refinement. The objective is to improve accuracy. The procedures are based on (1) the minimization of the stiffness matrix race (optimizing node location); (2) the use of h-version refinement (rezoning, element size reduction, and increasing the number of elements); and (3) the use of p-version refinement (increasing the order of polynomial approximation of the elements). A step-by-step procedure of mesh selection, improvement, and refinement is presented. The criteria for 'goodness' of a mesh are based on strain energy, displacement, and stress values at selected critical points of a structure. An analysis of an aircraft lug problem is presented as an example.

Kittur, Madan G.↗

NASA NTRS1990

Behaviour of plate elements based on the first-order shear deformation theory

A new analytical technique to assess the performance of shear deformable elements is presented, which makes it possible to determine a priori whether a given element will lock when used to model thin structures. The role that shear constraints play in determining the behavior of thin elements was established by comparing the results of key numerical tests with the predictions of element behavior made by studying the form of the shear constraints. Conclusions regarding locking behavior and the effects of reduced integration in thin shear deformable elements are presented, including the findings (1) that singularity of the shear stiffness matrix is not sufficient to avoid locking; (2) that the effect of mesh refinement on an element that contains spurious constraints is two-fold; and (3) that reduced integration does not remove spurious constraints but rather relaxes them. The results of the study are in agreement with previous studies of Mindlin plate elements in regarding Lagrangian elements as superior to serendipity elements when either full or reduced integration is employed.

Averill, R. C.↗

NASA NTRS1991

Triangular Element For Analyzing Elasticity Of Laminates

Flat triangular element developed for use in finite-element analyses of stress and strain in laminated plates made of such materials as plywood or advanced fiber/epoxy composite materials. Has multiple layers, each of which can have different isotropic or orthotropic elastic properties. Many such elements used in finite-element mesh to calculate stiffness of plate. Formulation of element straight-forward, and calculation of its stiffness matrix simple and fast.

Martin, C. Wayne↗

NASA NTRS1990

Parallel-vector computation for CSI-design code

Computational aspects of Control-Structure Interaction (CSI) DESIGN code is reviewed. Numerical intensive computation portions of CSI-DESIGN code were identified. Improvements in computational speed for the CSI-DESIGN code can be achieved by exploiting parallel and vector capabilities offered by modern computers, such as the Alliant, Convex, Cray-2, and Cray-YMP. Four options to generate the coefficient stiffness matrix and to solve the system of linear, simultaneous equations are currently available in the CSI-DESIGN code. A preprocessor to use RCM (Reverse Cuthill-Mackee) algorithm for bandwidth minimization was also developed for the CSI-DESIGN code. Preliminary results obtained by solving a small-scale, 97 node CSI finite element model (for eigensolution) have indicated that this new CSI-DESIGN code is 5 to 6 times faster (using 1 Alliant processor) than the old version of CSI-DESIGN code. This speed-up was achieved due to the RCM algorithm and the use of a new skyline solver. Efforts are underway to further improve the vector speed for CSI-DESIGN code, to evaluate its performance on a larger scale CSI model (such as phase zero CSI model) to make the code run efficiently on multiprocessor, parallel computer environment, and to make the code portable among different parallel computers available at NASA LaRC, such as Alliant, Convex, and Cray computers.

Nguyen, Duc T.↗

NASA NTRS1982

Equations of motion for a flexible spacecraft-lumped parameter idealization

The equations of motion for a flexible vehicle capable of arbitrary translational and rotational motions in inertial space accompanied by small elastic deformations are derived in an unabridged form. The vehicle is idealized as consisting of a single rigid body with an ensemble of mass particles interconnected by massless elastic structure. The internal elastic restoring forces are quantified in terms of a stiffness matrix. A transformation and truncation of elastic degrees of freedom is made in the interest of numerical integration efficiency. Deformation dependent terms are partitioned into a hierarchy of significance. The final set of motion equations are brought to a fully assembled first order form suitable for direct digital implementation. A FORTRAN program implementing the equations is given and its salient features described.

Storch, Joel↗

NASA NTRS1989

Structure-borne noise transmission through rolling element bearings

Numerical modeling techniques for the transmission of vibration through ball or roller bearings are developed analytically and applied to several test problems. The method is based on a novel formulation of the bearing stiffness matrix, which is outlined. Numerical results for sample problems are presented graphically, and good agreement with experimental data is demonstrated.

Lim, Teik Chin↗

NASA NTRS1991

Sensitivity analysis of a wing aeroelastic response

A variation of Sobieski's Global Sensitivity Equations (GSE) approach is implemented to obtain the sensitivity of the static aeroelastic response of a three-dimensional wing model. The formulation is quite general and accepts any aerodynamics and structural analysis capability. An interface code is written to convert one analysis's output to the other's input, and visa versa. Local sensitivity derivatives are calculated by either analytic methods or finite difference techniques. A program to combine the local sensitivities, such as the sensitivity of the stiffness matrix or the aerodynamic kernel matrix, into global sensitivity derivatives is developed. The aerodynamic analysis package FAST, using a lifting surface theory, and a structural package, ELAPS, implementing Giles' equivalent plate model are used.

Kapania, Rakesh K.↗

NASA NTRS1991

New displacement-based methods for optimal truss topology design

Two alternate methods for maximum stiffness truss topology design are presented. The ground structure approach is used, and the problem is formulated in terms of displacements and bar areas. This large, nonconvex optimization problem can be solved by a simultaneous analysis and design approach. Alternatively, an equivalent, unconstrained, and convex problem in the displacements only can be formulated, and this problem can be solved by a nonsmooth, steepest descent algorithm. In both methods, the explicit solving of the equilibrium equations and the assembly of the global stiffness matrix are circumvented. A large number of examples have been studied, showing the attractive features of topology design as well as exposing interesting features of optimal topologies.

Bendsoe, Martin P.↗

NASA NTRS1990

Force and moment transmissibilities through rolling element bearings in a single-stage rotor system

A lumped parameter model of a single-stage rotor system is developed which includes a new mathematical model for precision rolling element bearings to clarify this issue quantitatively and qualitatively. A comprehensive bearing stiffness matrix of dimension six is proposed which clearly demonstrates the coupling between the rotating shaft and casing motion. An eigensolution and the forced harmonic solution are obtained numerically. The natural frequencies predicted are found to be in excellent agreement with a finite element model.

Lim, Teik Chin↗

NASA NTRS1991

Space Shuttle Redesigned Solid Rocket Motor nozzle natural frequency variations with burn time

The effects of erosion and thermal degradation on the Space Shuttle Redesigned Solid Rocket Motor (RSRM) nozzle's structural dynamic characteristics were analytically evaluated. Also considered was stiffening of the structure due to internal pressurization. A detailed NASTRAN finite element model of the nozzle was developed and used to evaluate the influence of these effects at several discrete times during motor burn. Methods were developed for treating erosion and thermal degradation, and a procedure was developed to account for internal pressure stiffening using differential stiffness matrix techniques. Results were verified using static firing test accelerometer data. Fast Fourier Transform and Maximum Entropy Method techniques were applied to the data to generate waterfall plots which track modal frequencies with burn time. Results indicate that the lower frequency nozzle 'vectoring' modes are only slightly affected by erosion, thermal effects and internal pressurization. The higher frequency shell modes of the nozzle are, however, significantly reduced.

Lui, C. Y.↗

NASA NTRS1991

On the thermodynamic framework of generalized coupled thermoelastic-viscoplastic-damage modeling

A complete potential based framework using internal state variables is put forth for the derivation of reversible and irreversible constitutive equations. In this framework, the existence of the total (integrated) form of either the (Helmholtz) free energy or the (Gibbs) complementary free energy are assumed a priori. Two options for describing the flow and evolutionary equations are described, wherein option one (the fully coupled form) is shown to be over restrictive while the second option (the decoupled form) provides significant flexibility. As a consequence of the decoupled form, a new operator, i.e., the Compliance operator, is defined which provides a link between the assumed Gibb's and complementary dissipation potential and ensures a number of desirable numerical features, for example the symmetry of the resulting consistent tangent stiffness matrix. An important conclusion reached, is that although many theories in the literature do not conform to the general potential framework outlined, it is still possible in some cases, by slight modifications of the used forms, to restore the complete potential structure.

Arnold, S. M.↗

NASA NTRS1992

Improvement of structural models using covariance analysis and nonlinear generalized least squares

The next generation of large, flexible space structures will be too light to support their own weight, requiring a system of structural supports for ground testing. The authors have proposed multiple boundary-condition testing (MBCT), using more than one support condition to reduce uncertainties associated with the supports. MBCT would revise the mass and stiffness matrix, analytically qualifying the structure for operation in space. The same procedure is applicable to other common test conditions, such as empty/loaded tanks and subsystem/system level tests. This paper examines three techniques for constructing the covariance matrix required by nonlinear generalized least squares (NGLS) to update structural models based on modal test data. The methods range from a complicated approach used to generate the simulation data (i.e., the correct answer) to a diagonal matrix based on only two constants. The results show that NGLS is very insensitive to assumptions about the covariance matrix, suggesting that a workable NGLS procedure is possible. The examples also indicate that the multiple boundary condition procedure more accurately reduces errors than individual boundary condition tests alone.

Glaser, R. J.↗

NASA NTRS1992

Vibration transmission through rolling element bearings. IV - Statistical energy analysis

A theoretical broadband coupling-loss factor is developed analytically for use in the statistical energy analysis (SEA) of a shaft-bearing-plate system. The procedure is based on the solution of the boundary-value problem at the plate-bearing interface and incorporates a bearing-stiffness matrix developed by the authors. Three examples are utilized to illustrate the SEA incorporating the coupling-loss factor including: (1) a shaft-bearing-plate system; (2) a plate-cantilevered beam; and (3) a circular-shaft-bearing plate. The coupling-loss factor in the case of the thin plate-cantilevered beam is found to be more accurate than that developed by Lyon and Eichler (1964). The coupling-loss factor is described for the bearing system and extended to describe the mean-square vibratory response of a rectangular plate. The proposed techniques are of interest to the study of vibration and noise in rotating machinery such as gearboxes.

Lim, T. C.↗

NASA NTRS1992

An independent refinement and integration procedure in multiregion finite element analysis

An independent refinement and integration procedure is developed to couple together independently modeled (global and local) regions in a single analysis. The models can have different levels of refinement and along the interface between them the finite element nodes need not coincide with one another. In the local model all the nodes except the nodes at the interface are statically condensed and the reduced stiffness matrix is obtained. For this static condensation a modified frontal solution technique is employed. A spline interpolation function that satisfies the linear isotropic plate bending differential equation is used to relate the local model interface nodal displacements to the global model interface displacements. The proposed independent refinement and integration procedure is evaluated by applying it to two- and three-dimensional cases involving inplane and out-of-plane deformation. The procedure yielded very accurate results for all the examples studied.

Krishnamurthy, T.↗

NASA NTRS1993

Hierarchical Poly Tree Configurations for the Solution of Dynamically Refined Finte Element Models

This paper demonstrates how a multilevel substructuring technique, called the Hierarchical Poly Tree (HPT), can be used to integrate a localized mesh refinement into the original finite element model more efficiently. The optimal HPT configurations for solving isoparametrically square h-, p-, and hp-extensions on single and multiprocessor computers is derived. In addition, the reduced number of stiffness matrix elements that must be stored when employing this type of solution strategy is quantified. Moreover, the HPT inherently provides localize 'error-trapping' and a logical, efficient means with which to isolate physically anomalous and analytically singular behavior.

Gute, G. D.↗

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 NTRS1992

A robust direct-integration method for rotorcraft maneuver and periodic response

The Newmark-Beta method and the Newton-Raphson iteration scheme are combined to develop a direct-integration method for evaluating the maneuver and periodic-response expressions for rotorcraft. The method requires the generation of Jacobians and includes higher derivatives in the formulation of the geometric stiffness matrix to enhance the convergence of the system. The method leads to effective convergence with nonlinear structural dynamics and aerodynamic terms. Singularities in the matrices can be addressed with the method as they arise from a Lagrange multiplier approach for coupling equations with nonlinear constraints. The method is also shown to be general enough to handle singularities from quasisteady control-system models. The method is shown to be more general and robust than the similar 2GCHAS method for analyzing rotorcraft dynamics.

Panda, Brahmananda↗

NASA NTRS1993

Modelling of thick composites using a layerwise laminate theory

The layerwise laminate theory of Reddy (1987) is used to develop a layerwise, two-dimensional, displacement-based, finite element model of laminated composite plates that assumes a piecewise continuous distribution of the tranverse strains through the laminate thickness. The resulting layerwise finite element model is capable of computing interlaminar stresses and other localized effects with the same level of accuracy as a conventional 3D finite element model. Although the total number of degrees of freedom are comparable in both models, the layerwise model maintains a 2D-type data structure that provides several advantages over a conventional 3D finite element model, e.g. simplified input data, ease of mesh alteration, and faster element stiffness matrix formulation. Two sample problems are provided to illustrate the accuracy of the present model in computing interlaminar stresses for laminates in bending and extension.

Robbins, D. H., Jr.↗