A penalty approach for nonlinear optimization with discrete design variables

Introduced here is a simple approach to minimization problems with discrete design variables by modifying the penaly function approach of converting the constrained problems into sequential unconstrained minimization technique (SUMT) problems. It was discovered, during the course of the present work, that a similar idea was suggested by Marcal and Gellatly. However, no further work has been encountered. A brief description of the SUMT is presented. The form of the penalty function for the discrete-valued design variables and strategy used for the implementation of the procedure is discussed next. Finally, several design examples are used to demonstrate the procedure, and results are compared with the ones available in the literature

Evaluation of a Highly Anticlastic Panel with Tow Overlaps

A rectangular, variable-stiffness panel with tow overlaps was manufactured using an advanced tow placement machine. The cured panel had large anticlastic imperfections, with measured amplitudes of over two times the average panel thickness. These imperfections were not due to the overall steered-fiber layup or the tow overlaps, but instead resulted from local asymmetries in the laminate that were caused by a manufacturing oversight. In the nominal panel layup, fiber angles vary linearly from 60 degrees on the panel axial centerline to 30 degrees on the parallel edges. A geometrically nonlinear analysis was performed with a -280 degree Fahrenheit thermal load to simulate the postcure cooldown to room temperature. The predicted geometric imperfections correlated well with the measured panel shape. Unique structural test fixtures were then developed which greatly reduced these imperfections, but they also caused prestresses in the panel. Surface imperfections measured after the panel was installed in the test fixtures were used with nonlinear finite element analyses to predict these fixturing-induced prestresses. These prestresses were also included in structural analyses of panel end compression to failure, and the analytical results compared well with test data when both geometric and material nonlinearities were included

Structural efficiency study of composite wing rib structures

A series of short stiffened panel designs which may be applied to a preliminary design assessment of an aircraft wing rib is presented. The computer program PASCO is used as the primary design and analysis tool to assess the structural efficiency and geometry of a tailored corrugated panel, a corrugated panel with a continuous laminate, a hat stiffened panel, a blade stiffened panel, and an unstiffened flat plate. To correct some of the shortcomings in the PASCO analysis when shear is present, a two step iterative process using the computer program VICON is used. The loadings considered include combinations of axial compression, shear, and lateral pressure. The loading ranges considered are broad enough such that the designs presented may be applied to other stiffened panel applications. An assessment is made of laminate variations, increased spacing, and nonoptimum geometric variations, including a beaded panel, on the design of the panels

Automated Fiber Placement Defect Identity Cards: Cause, Anticipation, Existence, Significance, and Progression

Automated Fiber Placement (AFP), a major composite manufacturing process, can result in many defects during the layup process that often require manual corrective action to produce a part with acceptable quality. These defects are the main limitation of the technology and can be hard to categorize or define in many situations. This paper provides a thorough definition and classification of all AFP defects. This effort constitutes a comprehensive and extensive library relevant to AFP defects. The defects selected and defined in this work are based on understanding and experience from the manufacture and research of advanced composite structure. Proper classification of these defects required an in-depth literature review and consideration of various viewpoints ranging from designers, manufacturers, analysts, and inspection professionals. Collectively, these sources were utilized to develop the most accurate view of each of the individual defect types. The results are presented as identity cards for each defect type, intended to provide researchers and the manufacturing industry a clear understanding of the (1) cause, (2) anticipation, (3) existence, (4) significance, and (5) progression of the defined AFP defects. The link between AFP defects and process planning, layup strategies, and machining was also investigated. Categorization of all important automated fiber placement defects is presented

Enhancements of Tow-Steering Design Techniques: Design of Rectangular Panel Under Combined Loads

An extension to existing design tools that utilize tow-steering is presented which is used to investigate the use of elastic tailoring for a flat panel with a central hole under combined loads of compression and shear. The elastic tailoring is characterized by tow-steering within individual lamina as well as a novel approach based on selective reinforcement, which attempts to minimize compliance through the use of Cellular Automata design concepts. The selective reinforcement designs lack any consideration of manufacturing constraints, so a new tow-steered path definition was developed to translate the prototype selective reinforcement designs into manufacturable plies. The minimum weight design of a flat panel under combined loading was based on a model provided by NASA-Langley personnel and analyzed by STAGS within the OLGA design environment. Baseline designs using traditional straight fiber plies were generated, as well as tow-steered designs which incorporated parallel, tow-drop, and overlap plies within the laminate. These results indicated that the overlap method provided the best improvement with regards to weight and performance as compared to traditional constant stiffness monocoque panels, though the laminates did not measure up to similar designs from the literature using sandwich and isogrid constructions. Further design studies were conducted using various numbers of the selective reinforcement plies at the core and outer surface of the laminate. None of these configurations exhibited notable advantages with regard to weight or buckling performance. This was due to the fact that the minimization of the compliance tended to direct the major stresses toward the center of the panel, which decreased the ability of the structure to withstand loads leading to instability

Finite Element Analysis of Geodesically Stiffened Cylindrical Composite Shells Using a Layerwise Theory

Layerwise finite element analyses of geodesically stiffened cylindrical shells are presented. The layerwise laminate theory of Reddy (LWTR) is developed and adapted to circular cylindrical shells. The Ritz variational method is used to develop an analytical approach for studying the buckling of simply supported geodesically stiffened shells with discrete stiffeners. This method utilizes a Lagrange multiplier technique to attach the stiffeners to the shell. The development of the layerwise shells couples a one-dimensional finite element through the thickness with a Navier solution that satisfies the boundary conditions. The buckling results from the Ritz discrete analytical method are compared with smeared buckling results and with NASA Testbed finite element results. The development of layerwise shell and beam finite elements is presented and these elements are used to perform the displacement field, stress, and first-ply failure analyses. The layerwise shell elements are used to model the shell skin and the layerwise beam elements are used to model the stiffeners. This arrangement allows the beam stiffeners to be assembled directly into the global stiffness matrix. A series of analytical studies are made to compare the response of geodesically stiffened shells as a function of loading, shell geometry, shell radii, shell laminate thickness, stiffener height, and geometric nonlinearity. Comparisons of the structural response of geodesically stiffened shells, axial and ring stiffened shells, and unstiffened shells are provided. In addition, interlaminar stress results near the stiffener intersection are presented. First-ply failure analyses for geodesically stiffened shells utilizing the Tsai-Wu failure criterion are presented for a few selected cases

Global/Local Iteration for Blended Composite Laminate Panel Structure Optimization Subproblems

Composite panel structure optimization is commonly decomposed into panel optimization subproblems. Previous work applied a guide based design approach to the problem for a structure where the local loads were assumed to be fixed for each panel throughout the design process. This paper examines the application of guide based design to a more realistic representation of the structure where the local loads for each panel are determined through a global level analysis that is coupled with the stacking sequence for every design panel. A small problem is selected for which an exhaustive search of the subproblem design space verifies the optimality of the solution found through the global/local iteration process introduced in this work. The efficient discovery of solutions to these guide based design subproblems creates an opportunity to incorporate the solutions into a global level optimization process. A parallel genetic algorithm is proposed to control global optimization in which evaluating the fitness of each member of the population requires the solution of a guide based design subproblem where parallelism is solely within fitness evaluations. Results are presented for a wingbox design problem and compared with known solutions for the same problem to demonstrate weight reductions in a problem thought to already be near optimally solved

Variable Stiffness Panel Structural Analyses With Material Nonlinearity and Correlation With Tests

Results from structural analyses of three tow-placed AS4/977-3 composite panels with both geometric and material nonlinearities are presented. Two of the panels have variable stiffness layups where the fiber orientation angle varies as a continuous function of location on the panel planform. One variable stiffness panel has overlapping tow bands of varying thickness, while the other has a theoretically uniform thickness. The third panel has a conventional uniform-thickness [plus or minus 45](sub 5s) layup with straight fibers, providing a baseline for comparing the performance of the variable stiffness panels. Parametric finite element analyses including nonlinear material shear are first compared with material characterization test results for two orthotropic layups. This nonlinear material model is incorporated into structural analysis models of the variable stiffness and baseline panels with applied end shortenings. Measured geometric imperfections and mechanical prestresses, generated by forcing the variable stiffness panels from their cured anticlastic shapes into their flatter test configurations, are also modeled. Results of these structural analyses are then compared to the measured panel structural response. Good correlation is observed between the analysis results and displacement test data throughout deep postbuckling up to global failure, suggesting that nonlinear material behavior is an important component of the actual panel structural response

A genetic algorithm with memory for mixed discrete-continuous design optimization

This paper describes a new approach for reducing the number of the fitness function evaluations required by a genetic algorithm (GA) for optimization problems with mixed continuous and discrete design variables. The proposed additions to the GA make the search more effective and rapidly improve the fitness value from generation to generation. The additions involve memory as a function of both discrete and continuous design variables, multivariate approximation of the fitness function in terms of several continuous design variables, and localized search based on the multivariate approximation. The approximation is demonstrated for the minimum weight design of a composite cylindrical shell with grid stiffeners

A Genetic Algorithm for Mixed Integer Nonlinear Programming Problems Using Separate Constraint Approximations

This paper describes a new approach for reducing the number of the fitness and constraint function evaluations required by a genetic algorithm (GA) for optimization problems with mixed continuous and discrete design variables. The proposed additions to the GA make the search more effective and rapidly improve the fitness value from generation to generation.The additions involve memory as a function of both discrete and continuous design variables, and multivariate approximation of the individual functions' responses in terms of several continuous design variables. The approximation is demonstrated for the minimum weight design of a composite cylindrical shell with grid stiffeners