Satchi Venkataraman

Deterministic and Reliability-Based Optimization of Composite Laminates for Cryogenic Environments

Designs of composite laminates are investigated for hydrogen tanks in cryogenic environments. Large residual strains, which can develop due to thermal mismatch between matrix and fibers, result in matrix cracking at cryogenic temperatures and increase hydrogen leakage through the tank wall. To reduce thermal mismatch, ply angles need to be close to each other, but this leads to a substantial weight increase under biaxial loading. First deterministic optimization is used to investigate possible weight reduction measures. Reducing axial loads on walls by auxiliary stiffening mechanisms led to significant weight reduction. Reliability-based optimizations were performed to identify the uncertainties in composite material properties with the largest influences on the optimum design. Then measures for reducing uncertainty in important parameters are examined. The results indicate that the most effective measure for reducing thickness is quality control.

Structural Optimization: What Has Moore's Law Done For Us?

Rapid increases in computer processing power, memory, and storage space have not eliminated computational cost and time constraints faced by engineers who use structural optimization for design. This is due to the constant increase in the required fidelity (and hence complexity) of analysis models. Anecdotal evidence seems to indicate that analysis models of acceptable accuracy have required six to eight hours of computer time (an overnight run) throughout the last thirty years. This poses a severe challenge for global optimization or reliability based design. In this paper, we review how increases in computer power were utilized in structural optimization. We resolve problem complexity to components relating to complexity of analysis model, analysis procedure and optimization methodology. We explore the structural optimization problems that we are capable of solving at present and conclude that we can solve problems that have high complexity index in only one of the three components of model, analysis procedure, or optimization. We use examples of optimum design of composite structures to guide the discussion due to our familiarity with such problems. However, these are supplemented with other structural optimization examples to illustrate the universality of the message.

Challenges in comparing numerical solutions for optimum weights of stiffened shells

Optimizations of stiffened shells with different stiffener shapes performed to rank and identify the optimum designs during the preliminary design trade studies require a large number of analyses and hence rely on the useof efficient but approximate analysis methods. In the design of shells, the treatment of imperfections on buckling loads and stresses is of paramount importance. It is demonstrated how conservativeness of the approximate analyses used in buckling load calculation, the number of variables optimized (design freedom), and nonstructural constraints influence the weight of optimum designs. This demonstration is based on the results of a trade study performed to compare minimum weight designs of stiffened shells optimized under stress and buckling constraints for a reusable launch vehicle tank. PANDA2 was selected for the present study because it uses approximate analysis procedures that permit the many thousands of structural analyses needed for global optimization and it also has sophisticated machinery for generating imperfections and accounting for their effects. Optimum weights were influenced not only by material choice, number of optimization variables, and manufacturing constraints, but also by the analysis model conservativeness. Optimization of shells with effect of initial imperfections exhibited substantial weight differences between different stiffened-shell concepts, partly because of conservativeness in the analysis.

Optimal Functionally Graded Metallic Foam Thermal Insulation

Optimum density profiles that minimize heat transmission through a metal foam thermal insulation under onedimensional steady-state conditions are investigated. The effective thermal conductivity of the foam is derived in terms of cell parameters and the temperature. Maximizing the temperature at the outside wall of the insulation minimizes the heat conduction through the insulation because this maximizes the radiated heat. An optimality condition is derived, and the optimization problem is reduced to that of an ordinary, but a nonlinear differential equation, which is solved numerically. The optimum density variation through the thickness of the insulation for a given incident heat flux and the transmitted heat are presented for graded and uniform foams with open and closed cells. For open-cell foams, functional grading of the foam density can reduce the heat transfer through the foam for given thickness. Conversely, for a specified amount of heat transmission through the foam, the functionally graded foam insulation can be made thinner than uniform density foam insulation.

Minimum mass design of insulation made of functionally graded material

The problem of steady state heat conduction in a functionally graded open-cell metal foam thermal insulation is studied. The mass is minimized by varying the solidity profile for a given thickness. An optimality condition is derived and the optimization problem is reduced to that of an ordinary, nonlinear differential equation, which is solved numerically. The results include optimum cell size variation through the thickness of the insulation for given aerodynamic heating and the corresponding temperature distribution. It is shown that for a given thickness using a functionally graded insulation is predictably lighter than uniform one INTRODUCTION Metal foams [1] and advanced metallic thermal protection systems [2] are being investigated for use in multifunctional structures for reusable launch vehicles. Such multifunctional structures would insulate the vehicle interior from aerodynamic heating as well as carry primary vehicle loads. Varying the density, geometry, and/or material composition from point to point within the foam can produce functionally graded materials (FGM) that may be superior to uniform materials. To develop and test FGM for thermal protection systems, it is important to develop an understanding of what material property distributions offer significant efficiency gains. Satchi Venkataraman et al. [3] developed criterion for minimizing heat conduction through an open-cell titanium foam with variable cell size through its thickness. For a fixed inner wall temperature and foam thickness the outside wall temperature is maximized. Maximizing the outside wall temperature maximizes the heat radiated at the surface and therefore corresponds to minimizing the transmitted heat. The current study seeks to identify density profiles that may yield large improvements in weight efficiency compared to materials with uniform density. These results will be then used to direct research into improved modeling of FGM that will be used to refine the initial optimization. Finally, it is hoped that the results could be used to direct testing of promising configurations. The thermal protection systems (TPS) problem is inherently a transient one. However, we are first solving the simpler steady state problem to gain understanding of the effects of using functionally graded insulations. In this paper an optimality criterion is derived for minimizing the mass of an open-cell titanium foam with variable cell size through its thickness. The effective thermal conductivity of the foam is a function of temperature, pressure, properties of the foam material, and the foam geometry. The objective of optimization problem is to determine the density distribution that minimizes the mass of a titanium foam of given thickness for a fixed inner and outer wall temperatures. The optimality condition is developed and used to obtain the optimum density profile. The minimum mass obtained using a functionally graded foam and uniform density foam are compared to illustrate performance payoffs provided by optimization of graded foam properties. 43rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Con 22-25 April 2002, Denver, Colorado AIAA 2002-1425 Copyright

Optimization of Composite Laminates for Robust and Predictable Progressive Failure Response

Increasing robustness of structures with nonlinear history-dependent behavior requires their response to be very predictable. Predictability is made difficult by lack -of good physical models (e.g., material failure), inaccurate analysis models, and insufficient resolution used in the discretized solutions. In complex structures, the problem is compounded by the complex interactions between the various individual failure events that happen at different scales. The systems that are highly nonlinear and exhibit competing failure paths are sensitive to small design variations and exhibit poor failure predictability. Small design variations significantly alter the failure paths in designs having competing failure modes, reducing predictability. Progressive failure of a composite laminate resembles complex systems with many failing components (plies) and multiple failure modes (plies can fail by shear, matrix, and/or fiber failure) that exhibit problems in failure predictability. In this paper, we investigate the robustness of energy absorption and predictability of the failure sequence of composite laminates in progressive failure response. This investigation demonstrates that deterministic optimization makes predictability poor due to the coalescence in failure modes. A traditional reliability optimization was performed to improve failure predictability. Analyzing designs obtained revealed that robust and predictable progressive failure requires the elimination of competing modes. Strain separation between successive failure modes was identified as a surrogate deterministic measure to eliminate competing failures. A deterministic optimization for maximizing energy absorption with a constraint for strain separation between successive failures in different modes was performed. The deterministic design obtained with the surrogate measure for predictability was comparable to the nondeterministic design in performance and predictability, indicating that this is a sufficient condition for improving predictability. The paper demonstrates an approach for investigating the mechanics that affect progressive failure predictability and developing simple and efficient surrogate measures to use in deterministic optimization to maximize performance and predictability.

Quantifying uncertainty in statistical distribution of small sample data using Bayesian inference of unbounded Johnson distribution

Probabilistic analysis of physical systems requires information on the distributions of random variables. Distributions are typically obtained from testing or field data. In engineering design where tests are expensive, the sample size of such data is small O(10). Identifying correct distributions with small number of samples is difficult. Furthermore, parameters of assumed distributions obtained from small sample data themselves contain some uncertainty. In this study a Johnson SU family distribution function is used to identify shape, location and scale parameters of distribution that can best fit small sample data. A Bayesian inference procedure is used to determine distributions of the parameters. We show that the procedure correctly bounds the tail regions of the distributions and is less conservative than bounds obtained using bootstrap methods.

Calculating Confidence Bounds for Reliability Index to Quantify Effect of Distribution Parameter Uncertainty

Probabilistic methods for risk and reliability assessment require knowledge of statistical variation of design parameters. Often the parameters themselves are uncertain. In such cases it is important to quantify the effect of parameter uncertainty on the reliability calculations. Since parameter uncertainty is a reducible uncertainty, the use of confidence interval bounds to quantify the uncertainty in the reliability predictions is preferred. Obtaining confidence bounds for reliability requires characterizing the uncertainty of distribution parameters of the uncertain variables, and calculating confidence intervals for reliability index based on parameter uncertainty. In this paper, we present different options for calculating confidence intervals for reliability when the distributions of the distribution parameters of the uncertain variables are specified. This accuracy, computational cost and limitations of the different methods presented to calculate lower confidence bound of reliability index ( β ) are discussed. The methods are applied to calculating lower confidence bound of reliability index for a simple beam example.

Reliability optimization using probabilistic sufficiency factor and correction response surface

Reliability-based design optimization for low failure probability often requires millions of function analyses. Response surface approximation of the response functions (analysis response surface(ARS)) is often used to reduce the cost of failure probability calculations. Failure probabilities obtained from numerical sampling schemes are noisy and unsuitable for gradient-based optimization. To overcome this, response surfaces have been fitted to the failure probability of the designs (design response surface (DRS)) as a function of the design variables and used in optimization. Two shortcomings of the approach are that (i) the ARS fitting is extremely expensive for a large number of variables, especially for the high accuracy required to obtain very accurate reliability estimates, and (ii) DRS introduces fitting errors which affect the tails of the distributions which are significant for low failure probabilities. An approach to obtaining high-accuracy reliability estimates using the probabilistic sufficiency factor and correction response surface is investigated in this article. The method is demonstrated using a thin-walled box beam structure designed for minimum weight with failure probability constraints. The design is subjected to buckling, strength, and displacement constraints. Two methods of correcting low-fidelity analyses are compared for accuracy and efficiency. It is shown that correction to the response function is more accurate than the correction fitted to the probabilistic sufficiency factor.

Optimization of Performance and Failure Predictability in Composite Laminates Undergoing Progressive Failure

Increasing robustness of structures with non-linear history dependent behavior requires that their response to be very predictable. Predictability is made difficult by lack of good physical models (e.g. material failure), inaccurate analysis models, and insufficient resolution used in the discretized solutions. In complex structures the problem is compounded because numerous bifurcation events in non-linear response lead to competing failure paths. Introducing small changes to designs parameters or loading in designs having competing failure modes significantly alters the failure paths, reducing predictability. Progressive failure of a composite laminate is a system with many failing components (plies) and multiple failure modes (plies can fail by shear, matrix and/or fiber failure) that exhibits this behavior. This paper demonstrates that deterministic optimization makes predictability poor due the coalescence in failure modes. A deterministic approach is developed to overcome this problem and simultaneously optimize the laminate to maximize energy absorption (performance) and improve failure predictability. The approach is contrasted with traditional reliability-based optimization. The approach developed for eliminating competing failure modes is effective in increasing predictability and robustness, requiring very small computational effort compared to traditional non-deterministic methods used for such problems.