Richard H. Gallagher

Sensitivity analysis in shape optimization of continuum structures

Abstract This paper describes an efficient method for sensitivity analysis in shape optimum design. One feature is the use of limited number of master nodes to characterize the surfaces of a set of isoparametric finite elements, and the adoption of their coordinates as design variables of the shape optimization. Another is the derivation of analytical formulations of the gradients of both the stiffness terms and the load vectors with respect to the design variables. A finite element analysis code is adapted to the purposes of the method and numerical examples are performed and comparisons made with sensitivity analysis based on forward finite differences.

An approach to the inclusion of transverse shear deformation in finite element plate bending analysis

Abstract An expression for the strain energy of plates, which includes a term for transverse shear deformation and which is expressed in terms of the total transverse displacement and the displacement due to bending alone, is presented. Use of this expression in the construction of finite element representations requires independent descriptions of both these displacements. A triangular plate element is formulated in this manner, using the shape functions of the “discrete-Kirchoff” model. Numerical results for a range of finite element meshes and thickness-to-plate length ratios, for rectangular plates with different support and loading conditions, demonstrate the effectiveness of the proposed approach.

The finite element method in shell stability analysis

Abstract A developmentof the finite element method for thin shell instability analysis is presented, covering three principal aspects: 1. (1) representation of shell geometry 2. (2) representation of element behavior 3. (3) algorithmic tools for solution of the large-order systems of nonlinear algebraic equations which characterize various phases of shell instability. Two shell elements are described, an arbitrary quadrilateral and a triangle, and numerical results are presented for two widely-employed comparison problems for linear (stable) analysis. Two shell problems which include instability effects are also solved.

Higher order finite element analysis of lake circulation

Abstract The finite element method is applied to the analysis of the wind-driven circulation of variable-depth, shallow, homogenous lakes. Attention is concentrated upon higher-order description of the flow phenomena within the individual elements and upon the use of these higher order functions in the definition of curved element boundaries (isoparametric elements). Numerical results are presented for a rectangular basin, for which alternative results are available from both first-order finite element representations and finite difference analyses, and also for Lake Ontario, for which only the first-order finite element solution is available for comparison. These comparisons confirm the accuracy and efficiency of the finite element method in this field of application.

A generalized graphic preprocessor for two-dimensional finite element analysis

Input preprocessors have come to be recognized as important components of modern finite element programs. A method is described which utilizes interactive computer graphics digitizing techniques to create a powerful input preprocessor for finite element analysis. A limited number of general mesh generators based on linear blending functions permit the program to handle virturally all two-dimensional topologies. The processes of geometric input and specification of problem-specific “attributes” have been kept separate so that the mesh generation routines may be used with a variety of analysis programs. Graphical editors have been implemented to specify attributes for structural mechanics problems. Although this type of graphical preprocessor shows considerable promise for applications in three dimensions, there are still unresolved problems in the areas of geometrical description, perception, and interactive hardware.

Deformation und Stabilität windbeanspruchter Kühlturmschalen

ÜbersichtNach der Formulierung des Deformations- und des Stabilitätsproblems auf der Grundlage ursprünglich nicht-konformer, dreiecksförmiger, gekrümmter finiter Elemente wird zunächst das Deformationsproblem für eine ausgeführte Kühlturmschale numerisch gelöst. Die nachfolgende Stabilitätsanalyse dieser Kühlturmschale zeigt gute Übereinstimmung der mittels der Methode der finiten Elemente errechneten Beulsicherheit mit dem aufgrund des Munganschen Beulkriteriums erhaltenen Kleinstwert der Beulsicherheit.SummaryFollowing the formulation of the deformation and stability problem on the basis of originally non-conforming triangular curved finite elements with the help of a variational principle with subsidiary conditions, the deformation problem is solved numerically for an existing cooling tower shell. The subsequent stability analysis of this cooling tower shell shows relatively good agreement of the buckling safety computed by means of the finite element method and the minimum of buckling safety obtained with the help of Mungans stability criterion.

Instability analysis of torispherical pressure vessel heads with triangular thin-shell finite elements

The elastic instability of an internally-pressurized cylindrical tank with a torispherical head is investigated using a triangular, doubly curved, thin-shell finite element. The formulation of the finite element, which is based upon cubic displacement functions and a modified principle of potential energy, is first described. Then, the element is verified by comparing numerical results for the linear, stable analysis to alternative solutions for the same problem. The subsequent instability investigation includes the solution of the linearized problem of equilibrium bifurcation, that is, of the classical eigenvalue problem, and a general nonlinear analysis, based on tracing the nonlinear load-displacement path. The critical pressure, obtained with use of the general nonlinear analysis, agrees closely with an experimental result as well as with a numerical solution stemming from an axisymmetric formulation.

Finite element analysis in brittle material design

Abstract Analysis requirements for the design of structures composed of brittle materials are briefly reviewed and related to the capabilities found in the finite element method. The abilities of the method in dealing with the thermal and elastic analysis aspects of the overall brittle material design problem, as well as the amenability of results obtained with use of the method to interpretation in terms of pertinent failure theories, are emphasized. The scope and limitations of the method with respect to each of these aspects of the brittle material design problem are examined. Based on these examinations, suggestions are presented for future research and development activities.

On the Nonuniqueness of Solutions Obtained With Simplified Variational Principles

The attempt to satisfy subsidiary conditions in boundary value problems without additional independent unknowns in the form of Lagrange multipliers has led to the development of so-called simplified variational principles. They are based on using the Euler-Lagrange equations for the Lagrange multipliers to express the multipliers in terms of the original variables. It is shown that the conversion of a variational principle with subsidiary conditions to such a simplified variational principle may lead to the loss of uniqueness of the solution of a boundary value problem. A particularly simple form of the geometrically nonlinear theory of bending of beams is used as the vehicle for this proof The development given in this paper is entirely analytical. Hence, the deficiencies of the investigated simplified variational principle are fundamental.

Trends and Directions in the Applications of Numerical Analysis

Publisher Summary This chapter examines the utilization of mixed and complementary energy-element formulations in existing large-scale computer programs for finite-element analysis and also reviews the expansion of finite-element analysis concerning structural mechanics along with a wide range of physical problems. The simplification of actual interdisciplinary behavior to independent representation has been a limit imposed historically by analysis capabilities. The removal of these limits is a major contribution of enlarged numerical analysis capabilities, matching that of the inroads made into nonlinear problems. The most desirable situation occurs when the analytical models for transient thermal analysis and structural analysis are in complete correspondence. Any disparity between these representations is a source of error that is amplified in the integration of the transients in time. Furthermore, uncoordinated thermal and elastic analyses are highly inefficient, requiring large costs in the transference of data from the thermal to the thermal stress analysis. Motivation for the extension of the finite-element method to a broader range of physical problems might come from increased computational efficiency, ability to cope with more complex forms of a given class of problem, and savings through a unified general-purpose computer program.