Surya N. Patnaik

Optimum Design of Aerospace Structural Components Using Neural Networks

The application of artificial neural networks to capture structural design expertise is demonstrated. The principal advantage of a trained neural network is that it requires a trivial computational effort to produce an acceptable new design. For the class of problems addressed, the development of a conventional expert system would be extremely difficult. In the present effort, a structural optimization code with multiple nonlinear programming algorithms and an artificial neural network code NETS were used. A set of optimum designs for a ring and two aircraft wings for static and dynamic constraints were generated using the optimization codes. The optimum design data were processed to obtain input and output pairs, which were used to develop a trained artificial neural network using the code NETS. Optimum designs for new design conditions were predicted using the trained network. Neural net prediction of optimum designs was found to be satisfactory for the majority of the output design parameters. However, results from the present study indicate that caution must be exercised to ensure that all design variables are within selected error bounds.

The Variational Energy Formulation for the Integrated Force Method

The integrated force method (IFM) is one of the five formulations of mechanics, the others being the flexibility, stiffness, mixed, and total methods. To date, all but the IFM have been associated with variational functionals. A variational functional (VF) has been developed for the IFM. The stationary condition of the VF for the IFM yields the equilibrium and compatibility equations as well as the force and displacement boundary conditions. The stationary condition also yields a new set of boundary equations identified as the boundary compatibility conditions. This paper presents the theory of the variational functional for the IFM. It is illustrated by examples from discrete structures, the plane stress problem of elasticity, and Kirchhoff s plate bending problem. The properties of the VF and its relationship to the potential and complementary energy functionals are shown for discrete analysis.

Integrated force method versus displacement method for finite element analysis

Abstract A novel formulation termed the ‘integrated force method’ (IFM) has been developed in recent years for analyzing structures. In this method all the internal forces are taken as independent variables, and the system equilibrium equations (EEs) are integrated with the global compatability conditions (CCs) to form the governing set of equations. In IFM the CCs are obtained from the strain formulation of St. Venant, and no choices of redundant load systems have to be made, in contrast to the standard force method (SFM). This property of IFM allows the generation of the governing equation to be automated straightforwardly, as it is in the popular stiffness method (SM). In this paper IFM and SM are compared relative to the structure of their respective equations, their conditioning, required solution methods, overall computational requirements, and convergence properties as these factors influence the accuracy of the results. Overall, this new version of the force method produces more accurate results than the stiffness method for comparable computational cost.

COMPARATIVE EVALUATION OF DIFFERENT OPTIMIZATION ALGORITHMS FOR STRUCTURAL DESIGN APPLICATIONS

Non-linear programming algorithms play an important role in structural design optimization. Fortunately, several algorithms with computer codes are available. At NASA Lewis Research Centre, a project was initiated to assess the performance of eight different optimizers through the development of a computer code CometBoards. This paper summarizes the conclusions of that research. CometBoards was employed to solve sets of small, medium and large structural problems, using the eight different optimizers on a Cray-YMP8E/8128 computer. The reliability and efficiency of the optimizers were determined from the performance of these problems. For small problems, the performance of most of the optimizers could be considered adequate. For large problems, however, three optimizers (two sequential quadratic programming routines, DNCONG of IMSL and SQP of IDESIGN, along with Sequential Unconstrained Minimizations Technique SUMT) outperformed others. At optimum, most optimizers captured an identical number of active displacement and frequency constraints but the number of active stress constraints differed among the optimizers. This discrepancy can be attributed to singularity conditions in the optimization and the alleviation of this discrepancy can improve the efficiency of optimizers.

Modified Fully Utilized Design (MFUD) Method for Stress and Displacement Constraints

The traditional fully stressed method performs satisfactorily for stress-limited structural design. When this method is extended to include displacement limitations in addition to stress constraints, it is known as the Fully Utilized Design (FUD). Typically, the FUD produces an overdesign, which is the primary limitation of this otherwise elegant method. We have modified FUD in an attempt to alleviate the limitation. This new method, called the Modified Fully Utilized Design (MFUD) method, has been tested successfully on a number of problems that were subjected to multiple loads and had both stress and displacement constraints. The solutions obtained with MFUD compare favourably with the optimum results that can be generated by using non-linear mathematical programming techniques. The MFUD method appears to have alleviated the overdesign condition and offers the simplicity of a direct, fully stressed type of design method that is distinctly different from optimization and optimality criteria formulations. The MFUD method is being developed for practicing engineers who favour traditional design methods rather than methods based on advanced calculus and non-linear mathematical programming techniques. The Integrated Force Method (IFM) was found to be the appropriate analysis tool in the development of the MFUD method. In this paper, the MFUD method and its optimality are examined along with a number of illustrative examples.

Improved accuracy for finite element structural analysis via an integrated force method

Abstract Finite element structural analysis based on the original displacement (stiffness) method has been researched and developed for over three decades. Although today it dominates the scene in terms of routine engineering use, the stiffness method does suffer from certain deficiencies. Various alternate analysis methods, commonly referred to as the mixed and hybrid methods, have been promoted in an attempt to compensate for some of these limitations. In recent years two methods for finite element analyses of structures, within the framework of the original force method concept, have been introduced. These are termed the ‘integrated force method’ and the ‘dual integrated force method’. A comparative study was carried out to determine the accuracy of finite element analyses based on the stiffness method, a mixed method, and the new integrated force and dual integrated force methods. The numerical results were obtained with the following software: MSC/NASTRAN and ASKA for the stiffness method; an MHOST implementation for a mixed method; a GIFT for the integrated force methods. For the cases considered, the results indicate that, on an overall basis, the stiffness and mixed methods present some limitations. The stiffness method typically requires a large number of elements in the model to achieve acceptable accuracy. The MHOST mixed method tends to achieve a higher level of accuracy for coarse models than does the stiffness method as implemented by MSC/NASTRAN and ASKA. The two integrated force methods, which bestow simultaneous emphasis on stress equilibrium and strain compatibility, yield accurate solutions with fewer elements in a model. The full potential of these new integrated force methods remains largely unexploited, and they hold the promise of spawning new finite element structural analysis tools.

Optimality of a fully stressed design

For a truss a fully stressed state is reached when all its members are utilized to their full strength capacity. Historically, engineers considered such a design optimum. But recently this optimality has been questioned, especially since the weight of the structure is not explicitly used in fully stressed design calculations. This paper examines optimality of the fully stressed design (FSD) with analytical and graphical illustrations. Solutions for a set of examples obtained by using the FSD method and optimization methods numerically confirm the optimality of the FSD. The FSD, which can be obtained with a small amount of calculation, can be extended to displacement constraints and to nontruss-type structures.

Development of Finite Elements for Two-Dimensional Structural Analysis Using the Integrated Force Method

The integrated force method has been developed in recent years for the analysis of structural mechanics problems. In the intgrated force method all independent forces are treated as unknown variables, which are calculated by simultaneously imposing equations of equilibrium and compatibility conditions. The development of a finite element library for the analysis of two-dimensional problems using the integrated force method is presented in this paper. Elements of triangular and quadrilateral shapes, capable of modeling arbitrary domain configurations are developed. The element equilibrium and flexibility matrices are derived by discretizing expressions for corresponding potential and complementary energies, respectively. Independent approximations of displacement and stress fields within finite elements are performed. Interpolation of the displacement field is done similarly as in the standard displacement method. The stress field is approximated using full polynomials of correct orders. A procedure for deriving the stress interpolation polynomials that utilizes the definitions of stress components in terms of Airys stress function is developed. Such derived stress fields identically satisfy equations of equilibrium, and the resulting element matrices are insensitive to the orientation of local coordinate systems. A method to calculate the number of rigid body modes is devised, and it is shown that the present elements do not possess spurious zero energy modes. A number of example problems are solved using the present library and the results are compared with corresponding analytical solutions and those obtained from the standard displacement finite element method. A good agreement of the results, and better performance of the integrated force method, compared to the displacement method, in stress calculations, is observed.

Structural optimization with approximate sensitivities

Computational efficiency in structural optimization can be enhanced if the intensive computations associated with the calculation of the sensitivities, that is, gradients of the behavior constraints, are reduced. Approximation to gradients of the behavior constraints that can be generated with a small amount of numerical calculations is proposed. Structural optimization with these approximate sensitivities produced correct optimum solution. Approximate gradients performed well for different nonlinear programming methods, such as the sequence of unconstrained minimization technique, method of feasible directions, sequence of quadratic programming and sequence of linear programming. Structural optimization with approximate gradients can reduce by one third the CPU time that would otherwise be required to solve the problem with explicit closed-form gradients. The proposed gradient approximation shows potential to reduce intensive computation that has been associated with traditional structural optimization.

Compatibility conditions of structural mechanics for finite element analysis

The equilibrium equations and the compatibility conditions are fundamental to the analyses of structures. However, anyone who undertakes even a cursory generic study of the compatibility conditions can discover, with little effort, that historically this facet of structural mechanics had not been adequately researched by the profession. Now the compatibility conditions (CCs) have been researched and are understood to a great extent. For finite element discretizations, the CCs are banded and can be divided into three distinct categories: (1) the interface CCs; (2) the cluster or field CCs; and (3) the external CCs. The generation of CCs requires the separating of a local region, then writing the deformation displacement relation (ddr) for the region, and finally, the eliminating of the displacements from the ddr. The procedure to generate all three types of CCs is presented and illustrated through examples of finite element models. The uniqueness of the CCs thus generated is shown.