Almost three decades of advances in theory and software together with the development of large-scale mainframe computers have put the finite element method at the fingertips of the structures design/analysis engineer. Ten of those years have included mechanization and automation of the design process, a trend which continues today. The ability to rapidly design complex structures is the positive result of this trend. However, there is also a negative aspect, viz: structures engineers can lose sight of the facts that the finite element model is only a model and that it may exclude the most significant mechanical characteristics of the real structure. This paper is based on the authors' experience with both theory and applications. Numerous examples are drawn upon to illustrate both the utility of the finite element method and its propensity, when abused, to produce numbers which look plausible but are actually wrong. The paper concludes with some guidelines for finite element modelling of structural systems and details.

The finite element method was used to study the effect of mount-induced aberrations on the optical surface of a lightweight double arch mirror subjected to cryogenic temperatures. The mount design was controlled by the requirements imposed on the optical surface quality and stress levels. The finite element analysis was used to define the feasible range of mount parameters and the selection of a design within the feasible region. The final design consisted of three spring-loaded Invar T-clamps that uniquely define the location of the mirror, three radially compliant parallel spring guides that remove the effect of radial contraction of structure in cryogenic temperatures, and a flexible baseplate that was used to reduce the effect of temperature-induced baseplate tilt errors. The experimental results from the applica-tion of this system to an existing 20-inch fused silica double arch mirror are shown, and possible improvements in system performance are discussed.

To meet the demands of a fraction of a wavelength of light residual error imposed by high resolution optical systems, detailed analytical modeling techniques are required. For large lightweight, optical elements, errors can be significant even in a relatively benign thermal environment with materials exhibiting near zero coefficient of thermal expansion (CTE). This study evaluates a thin, circular, spherical mirror made of Corning Glass Works Ultra Low Expansion (ULE) fused silica glass, subjected to thermal loading. To validate performance characteristics, a detailed finite element mathematical model is made, utilizing the NASTRAN digital routine. Included are typical Corning measured values of CTE, which vary positively and negatively within the mirror about a near zero nominal value. NASTRAN results under thermal soak and linear gradient conditions are com-pared to a theoretical solution of a partial shell under uniform axial thermal gradient.

The thermal structural twist deformation of a graphite epoxy optical alignment tube is minimized under consideration of manufacturing tolerances on layup angles of the laminates. Laminate properties, determined using an advanced laminate computer code, are input to structural analysis finite element codes and thermal analysis codes wherein design parameters are varied under control of an optimization process to minimize twist. Although this is a design peculiar problem, it demonstrates the ability to optimize composite structure using finite element techniques.

The design of the University of California Ten Meter Telescope requires supporting 36 individual hexago-nal mirrors; these 36 mirror segments constitute the primary mirror of the telescope. Each segment is axially supported by a set of whiffletrees, which applies forces at 36 points on the back of the segment. Because these mirror segments are thin (1.8 m diameter, 7.5 em thick), the optimization of the location and magnitude of the supporting forces is a crucial part of the design. We have analyzed this problem by using the finite element method. Careful testing of a finite element model and the associated computer programs is required before the results can be trusted. We began by analyzing the deflection of a thin, flat, circular plate, since this can be calculated independently and then compared with the results of the finite element method. The factors of curvature of the mirror segment, shear, and hexagonal shape were then included one at a time, in order to arrive at the final model. The optimization of the support forces was based on minimizing the r.m.s. surface error, with a design goal of 0.010 Am or less. This is a surface error roughly one hundred times smaller than that resulting from an unoptimized support. The optimization calculations were performed on a large minicomputer. Practical con-siderations required that the physical symmetries of the problem be taken into account, in order to reduce the number of independently optimized variables to a manageable size. The use of the finite element method pro-duced a solution that met the design requirements, and it gave useful information about the sensitivity of mirror deformations to variations in the support structure.

Analysis of convectively cooled optical elements has been traditionally treated by finite element methods (FEM) techniques. NASTRAN, TAP-3, and CINDA are standard programs that have been used for predicting the thermal gradients and resultant distortions of high power laser cooled elements. Due to the high resolution of some optical ray trace codes, thermal/structural analyses using traditional NASTRAN and CINDA FEM are cost-prohibitive. The solution, is therefore, to develop high resolution analytical techniques to reduce cost and still produce the necessary optical distortion required by the optical engineers. Therefore, several thermal/structural codes have been developed to aid in the design and analysis of optical elements. The codes can interface with High Energy Laser (HEL) irradiance maps supplied by optical engineers, solve for the heat exchanger thermal gradients, and establish the necessary boundary conditions for subsequent NASTRAN FEM. The codes have been successfully used with irradiance maps with moderately large array sizes. Their use in predicting wavefront distortion has been verified through thermal testing.

The spectral condition number of the global stiffness and mass matrices set UD with finite elements for the Laplace or Poisson problems is examined in terms of element size and geometry.

The finite element analysis of mirrors is now common practice. The results of such an analysis is the vector displacement of a great number of grid points within the mirror. Evaluation and interpretation of the raw data is difficult, even when represented as contour plots. It is more convenient from an optical engineer's viewpoint to describe the total deformation in terms of its components: tilt, defocus, and common aberrations. A NASTRAN post-processing program is described which performs a least squares fit of Zernike polynomials to a deformed surface.

Actuation system design for adaptive optic devices is based upon the nature of the distortions or aberrations to be controlled, requirinc that the vectors describing these wavefront error sources also aid the designer in selecting the number and locations of actuators. The traditional Zernike polynomials give a good optical description of the aberrations, but tend to obscure actuator requirements. An alternate set of surface vectors, sines and cosines radially and circumferentially, is presented which give the designer a good feel for actuation requirements as well as an accurate description of the wavefront. The constraints on nodal density and arrangement driven by orthogonality and finite element modelling considerations are discussed for both vector sets. In addition, a comparison of wavefront description accuracy is presented.

A method is shown analytically which reduces the effects of epoxy shrinkage for an ultra-high precision X-ray telescope to within the system error budget. The three-dimensional shrinkage effects are discussed with reference to this telescope. The results of the analysis point to the use of an interrupted rather than continuous bond line as the best solution. Discussion of the finite element modelling techniques is included.

In an application of computer graphics, finite element analysis, and servo control theory, this film is part of the development of an engineering tool for studying the active control of the University of Texas 7.6m telescope primary mirror. The mirror is modelled as a ULE meniscus 0.12m thick. A sinusoidal forcing function is applied to a finite element model, and control software by Dr. R. D. Hefner at Aerospace supplies correction forces in real time to the model. The data displayed are (1)a moving three dimensional plot of the finite element mesh, with its collimating points and actuating points delineated, (2)an rms figure error, (3)an intensity plot of the image formed by the mirror surface, and (4)a plot of the encircled energy at various concentration lev-els. Rate damping is used as the inner loop of control, using velocity feedback data from the model, and an output feedback algorithm after a method by Kosut[1] is used for the outer loop. There are 36 actuation points used for the surface error control. The control software uses the first ten natural modes of vibration of the meniscus.

This paper presents a personal view regarding the need for a continued interest and activity in structural methods in general, while viewing finite elements and the computer as simply two specific tools for assisting in this endeavor. An attempt is made to provide some insight as to why finite element methods seem to have "won the war," and to give examples of their more (and less) intelligent use. Items addressed include a highlight of unnecessary limitations of many existing standard finite element codes and where it is felt that further development work is needed.

The problem of supporting an optical element in a l-g force field on a multi-level kinematic mount arises, when a single level 3-point kinematic mount is inadequate for keeping the stress and/or surface deflection within specified design limits. In this paper, a solution of the biharmonic differential equation for the bending of a flat thin circular plate is first derived for m-point (m > 2) supports, equi-spaced on a concentric circle, and then applied to the problem of a two-level kinematic mount, which is also known as a 9-point Hindle mount. From this, normalized design curves are developed for determining nominal locations of the nine support points, associated RMS deflections and support location sensitivity. These design curves provide the practicing engineer with a useful, efficient and accurate means for developing a preliminary Hindle mount design without resorting to FEM analysis. Several cases of the 9-point Hindle mount solution were compared with independent NASTRAN based finite element solutions. Excellent correlation between the two was obtained in all cases. The methodology used in this paper is not limited to flat optical elements. The solution of the problem of curved optical elements on a 9-point Hindle mount in a l-g force field can be similarly obtained by the same approach with E. Reissner's thin shallow spherical shell equations.

The principles of elastic structural behavior and superposition were used in conjunction with finite element analysis to find a suitable test support system for thin meniscus mirrors. The finite element method was used to calculate the deflections of a thin meniscus mirror supported by various mechanisms. The support systems were explicitly represented in the finite element model by including the forces and moments that are induced by the support system. To minimize the number of finite element runs, the basic loading cases were analyzed one at a time, and the deflections in each case were fit to Zernike polynomials. To represent the effect of each support system on the optical surface, the basic loading cases were combined by using the appropriate proportions of the Zernike polynomial coefficients. The analytical results favored airbag support over other possible support systems. If the pressure in the airbag is varied in a controlled manner, optical errors may be separated out from the mechanical errors caused by an imperfect support system even for optical elements of a very high diameter-to-thickness ratio or asymmetric shapes. Using holographic techniques, several sets of experimental results were obtained that strongly confirmed the analytical findings.

An adaptive primary mirror design containing four mirror segments and 56 push-pull actuators is examined for optical correction capability and compatibility with control system dynamics. To perform this analysis and design effort, a finite element model of a primary mirror, its backup structure, a set of tangent bars, and 14 actuators was developed. Influence functions were obtained for motion at the various actuator armature locations. To satisfy dynamic control bandwidth requirements, a mirror modal frequency of 32 Hz (approximately one magnitude higher than the required bandwidth) had to be met for the structural system modelled with unpowered actuators. This paper details the finite element model and model results, as well as addresses the residual error resulting from a best static fit to various aberrations utilizing the influence functions.

Attempts to numerically analyze problems which have been solved by classical methods have been and continue to be the major stimuli for the development and refinement of general finite element analysis strategies. This perhaps is not well appreciated, especially by those new to the field. Demonstrations are given in discussions of three topics of current interest: the performance of distorted isoparametric elements, stress analysis of bonded materials and step size selection in elastic-plastic analysis.

Numerical time integration methods for solving sets of simultaneous dynamic equilibrium equations are reviewed. Both mathematical and physical approximations are used in generating numerical solutions to the governing second-order differential equations. The resulting algorithms are classified into two basic categories: explicit (predictor) and implicit (corrector). Explicit methods are computationally efficient, but all explicit second-order accurate methods are only conditionally stable. On the other hand, many implicit methods are unconditionally stable, but may require iteration for convergence at each time step. For linear problems, implicit methods usually can be reduced to explicit form and iteration can be avoided. Methods which are unconditionally stable for linear problems may become unstable for nonlinear problems depending upon the manner in which nonlinearities are evaluated: exactly, or approximately using a pseudo-force or tangent modulus idealization. Methods which introduce numerical (or algorithmic) damping are used to eliminate "spurious" high frequency noise present in nonlinear, and sometimes even linear, structural dynamics problems. The question still remains whether the manner and extent of filtering at high frequencies is realistic. While attempts to develop new time integration algorithms synthesizing the better features of both explicit and implicit methods continue to be made, a basic controversy over the relative advantages and disadvantages of explicit and implicit schemes still exists.

A hybrid analytical/graphical method is presented to calculate the fundamental natural frequency of rectangular mirrors mounted at 3 points. A NASTRAN assisted parametric approach was used to calculate the characteristic roots of the plate vibration equation for mirrors with aspect ratios ranging from 1.0 x 1.0 to 10.0 x 1.0. Also considered were simply supported boundary conditions at three mirror corner points or at two corner points on one edge and one point along the opposite edge. Experimental varification within 6.0% was achieved for the extreme case tested with approximately a +2.0% average experimental error overall.

Finite element models are often used to predict the optical performance of a structure under a wide variety of conditions. A technique is described in this paper which couples the equations of optics to the finite element model. It assumes only that each optical lens or mirror is a rigid mass so that motion of the vertex defines motion of the element. Examples are drawn from the Space Telescope project.

This paper describes a simplified method to evaluate the optical errors in a high accuracy pointing system. The described method is for a structure excited by random disturbances, such as those created by atmospheric turbulence and cooling systems. The method has been developed specifically for an optical train that includes active control elements to attenuate jitter errors.

This paper presents a case study of the development of a Zinc Sulfide IR dome for a mortar-launched projectile. As a case study, it demonstrates the value of analysis as a means to obtain insight and derive solutions to design problems. It also describes problems solved by a variety of finite element codes: SAAS-3, EPIC-2, and NASTRAN. The paper discusses the determination of inertia loads, joint design, bonding of the dome to the attachment ring, verification testing, mortar test firings, and post test analyses. The paper discusses various FEM analyses performed during the development. Inertia load factors used for design were derived from a transient dynamic response analysis of launch and muzzle exit. Structural integrity of the brittle ceramic dome material was demonstrated by a detailed FEM stress analysis of the joint. This analysis considered modeling techniques for threaded and bonded joints. This FE model was also utilized to define an equivalent proof pressure test to simulate the inertia load. Initial tests resulted in the failure of both the dome and the attachment device of internal ballast. Additional analyses were conducted to evaluate modal response and impact. These analyses established the internal ballast as the cause of failure, and subsequent successful dome test firings validated the analyses.