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All sourcesNASA NTRSDOE OSTI
NASA NTRS1976

The Relation of Finite Element and Finite Difference Methods

Finite element and finite difference methods are examined in order to bring out their relationship. It is shown that both methods use two types of discrete representations of continuous functions. They differ in that finite difference methods emphasize the discretization of independent variable, while finite element methods emphasize the discretization of dependent variable (referred to as functional approximations). An important point is that finite element methods use global piecewise functional approximations, while finite difference methods normally use local functional approximations. A general conclusion is that finite element methods are best designed to handle complex boundaries, while finite difference methods are superior for complex equations. It is also shown that finite volume difference methods possess many of the advantages attributed to finite element methods.

Vinokur, M.↗
NASA NTRS1978

Finite elements and finite differences for transonic flow calculations

The paper reviews the chief finite difference and finite element techniques used for numerical solution of nonlinear mixed elliptic-hyperbolic equations governing transonic flow. The forms of the governing equations for unsteady two-dimensional transonic flow considered are the Euler equation, the full potential equation in both conservative and nonconservative form, the transonic small-disturbance equation in both conservative and nonconservative form, and the hodograph equations for the small-disturbance case and the full-potential case. Finite difference methods considered include time-dependent methods, relaxation methods, semidirect methods, and hybrid methods. Finite element methods include finite element Lax-Wendroff schemes, implicit Galerkin method, mixed variational principles, dual iterative procedures, optimal control methods and least squares.

Hafez, M. M.↗
NASA NTRS1972

Evaluation of the use of a singularity element in finite element analysis of center-cracked plates

Two different methods are applied to the analyses of finite width linear elastic plates with central cracks. Both methods give displacements as a primary part of the solution. One method makes use of Fourier transforms. The second method employs a coarse mesh of triangular second-order finite elements in conjunction with a single singularity element subjected to appropriate additional constraints. The displacements obtained by these two methods are in very good agreement. The results suggest considerable potential for the use of a cracked element for related crack problems, particularly in connection with the extension to nonlinear material behavior.

Mendelson, A.↗
NASA NTRS1978

Numerical computation of transonic flows by finite-element and finite-difference methods

Studies on applications of the finite element approach to transonic flow calculations are reported. Different discretization techniques of the differential equations and boundary conditions are compared. Finite element analogs of Murman's mixed type finite difference operators for small disturbance formulations were constructed and the time dependent approach (using finite differences in time and finite elements in space) was examined.

Hafez, M. M.↗
NASA NTRS1981

Application of steady state finite element and transient finite difference theory to sound propagation in a variable area duct: A comparison with experiment

Sound propagation without flow in a rectangular duct with a converging-diverging area variation was studied experimentally and theoretically. The area variation was of sufficient magnitude to produce large reflections and induce modal scattering. The rms (root-mean-squared) pressure and phase angle on both the flat and curved surface were measured and tabulated. The steady state finite element theory and the transient finite difference theory are in good agreement with the data. It is concluded that numerical finite difference and finite element theories appear ideally suited for handling duct propagation problems which encounter large area variations.

Baumeister, K. J.↗
NASA NTRS1981

Application of 'steady' state finite element and transient finite difference theory to sound propagation in a variable duct - A comparison with experiment

Experimental data are presented for sound propagation in a simulated infinite hard wall duct with a large change in duct cross sectional area. The data are conveniently tabulated for further use. The 'steady' state finite element theory of Astley and Eversman (1981) and the transient finite difference theory of White (1981) are in good agreement with the data for both the axial and transverse pressure profiles and the axial phase angle. Therefore, numerical finite difference and finite element theories appear to be ideally suited for handling duct propagation problems which encounter large axial gradients in acoustic parameters. The measured energy reflection coefficient agrees with the values from the Astley-Eversman modal coupling model.

Baumeister, K. J.↗
NASA NTRS1972

Finite element analysis of elasto-plastic soils. Report no. 4: Finite element analysis of elasto-plastic frictional materials for application to lunar earth sciences

The behavior of finite element models employing different constitutive relations to describe the stress-strain behavior of soils is investigated. Three models, which assume small strain theory is applicable, include a nondilatant, a dilatant and a strain hardening constitutive relation. Two models are formulated using large strain theory and include a hyperbolic and a Tresca elastic perfectly plastic constitutive relation. These finite element models are used to analyze retaining walls and footings. Methods of improving the finite element solutions are investigated. For nonlinear problems better solutions can be obtained by using smaller load increment sizes and more iterations per load increment than by increasing the number of elements. Suitable methods of treating tension stresses and stresses which exceed the yield criteria are discussed.

Marr, W. A., Jr.↗
NASA NTRS1976

Survey and development of finite elements for nonlinear structural analysis. Volume 2: Nonlinear shell finite elements

The development of two new shell finite elements for applications to large deflection problems is considered. The elements in question are doubly curved and of triangular and quadrilateral planform. They are restricted to small strains of elastic materials, and can accommodate large rotations. The elements described, which are based on relatively simple linear elements, make use of a new displacement function approach specifically designed for strongly nonlinear problems. The displacement function development for nonlinear applications is based on certain beam element formulations, and the strain-displacement equations are of a shallow shell type. Additional terms were included in these equations in an attempt to avoid the large errors characteristic of shallow shell elements in certain types of problems. An incremental nonlinear solution procedure specifically adopted to the element formulation was developed. The solution procedure is of combined incremental and total Lagrangian type, and uses a new updating scheme. A computer program was written to evaluate the developed formulations. This program can accommodate small element groups in arbitrary arrangements. Two simple programs were successfully solved. The results indicate that this new type of element has definite promise and should be a fruitful area for further research.

Source record↗
DOE OSTI2025

Parametric Finite Element Analysis of Naturally Corroded Steel Specimens Using 3D Surface Laser Scans

Corrosion is considered a uniform thickness reduction design guideline of the maritime industry. However, additionally, the corroded and irregular morphology of the surface affects the steel's load-bearing capacity and its impact on the strength and elongation behaviour of the steel is not yet fully understood. These effects on the local behaviour of steel structures under tensile loading were investigated with tensile tests on naturally corroded steel specimens and nonlinear finite element simulations including the corroded surface morphology with a uniform surface idealation. The models also include the deformed specimen shape. The developed approach led to highly accurate parametric finite element models predicting the ultimate tensile strength and longitudinal position of fracture. The results show that all included aspects are essential for accurate simulations, while solely the maximum available surface resolution was not as decisive.

corrosion↗
NASA NTRS1976

Survey and development of finite elements for nonlinear structural analysis. Volume 1: Handbook for nonlinear finite elements

A survey of research efforts in the area of geometrically nonlinear finite elements is presented. The survey is intended to serve as a guide in the choice of nonlinear elements for specific problems, and as background to provide directions for new element developments. The elements are presented in a handbook format and are separated by type as beams, plates (or shallow shells), shells, and other elements. Within a given type, the elements are identified by the assumed displacement shapes and the forms of the nonlinear strain equations. Solution procedures are not discussed except when a particular element formulation poses special problems or capabilities in this regard. The main goal of the format is to provide quick access to a wide variety of element types, in a consistent presentation format, and to facilitate comparison and evaluation of different elements with regard to features, probable accuracy, and complexity.

Source record↗
NASA NTRS1971

Analysis of finite deformations of elastic solids by the finite element method.

Finite element applications, particularly to analyses of finite deformations in elastic solids, are reviewed, along with the difficulties encountered in the formulation of certain problems and in their numerical solution. Various approaches are discussed for overcoming these and other difficulties. A computer program designed for finite elasticity problems is described, and several numerical examples are presented.

Oden, J. T.↗
NASA NTRS1975

On a numerical sufficiency test for monotonic convergence of finite element models

Finite element analyses characterized by monotonic convergence include the discipline for meaningful measurements of convergence rate and consequently economical extrapolation. Few proposers of element models guarantee monotonic convergence for their elements. Thus, a need exists for an automatic test to classify available element models. This paper describes such a test - a test can be performed using a digital computer to guarantee that a particular element model imbues monotonicity. It describes the test and its basis. It examines seven element models for a rectangular membrane to illustrate the value of the tests. Besides confirming results already known, the application yields new data. It 'proves' monotonicity for two improved models, defines the range of element proportions for which another element can be guaranteed to exhibit monotonicity, and suggests that another element is deficient. In the special case of absolutely convergent membrane displacement models, proof of monotonicity is a necessary and sufficient condition to insure that upper bound estimates of strain energy are developed. Accordingly, the test furnishes a proof of bound solutions independently of requirements on displacement continuity the element basis may or may not satisfy.

Melosh, R. J.↗
NASA NTRS1980

Surface temperatures in sliding systems - A finite element analysis

Finite element equations are developed for studying surface temperatures resulting from frictional heating in sliding systems. The equations include the effect of velocity of moving components, an effect which is found to be quite significant, even at low sliding velocities. A program was written using the equations and it was applied to the study of surface temperatures in two different sliding systems: dry or boundary lubricated sleeve bearings and a labyrinth gas path seal configuration. Very good agreement was achieved between analytical predictions using the program and experimental temperature measurements. The program was used to study the influence of various material parameters on surface temperatures in the two sliding systems.

Kennedy, F. E., Jr.↗