Shantaram S. Pai

Probabilistic Study of Fluid Structure Interaction

A combustor liner was computationally simulated and probabilistically evaluated in view of the several uncertainties in the aerodynamic, structural, material and thermal variables that govern the combustor liner. The interconnection between the computational fluid dynamics code and the finite element structural analysis codes was necessary to couple the thermal profiles with structural design. The stresses and their variations were evaluated at critical points on the liner. Cumulative distribution functions and sensitivity factors were computed for stress responses due to the aerodynamic, mechanical and thermal random variables. It was observed that the inlet and exit temperatures have a lot of influence on the hoop stress. For prescribed values of inlet and exit temperatures, the Reynolds number of the flow, coefficient of thermal expansion, gas emissivity and absorptivity and thermal conductivity of the material have about the same impact on the hoop stress. These results can be used to quickly identify the most critical design variables in order to optimize the design and make it cost effective.

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.

Compatibility Condition in Theory of Solid Mechanics (Elasticity, Structures, and Design Optimization)

Abstract The strain formulation in elasticity and the compatibility condition in structural mechanics have neither been understood nor have they been utilized. This shortcoming prevented the formulation of a direct method to calculate stress and strain, which are currently obtained indirectly by differentiating the displacement. We have researched and understood the compatibility condition for linear problems in elasticity and in finite element structural analysis. This has lead to the completion of the “method of force” with stress (or stress resultant) as the primary unknown. The method in elasticity is referred to as the completed Beltrami-Michell formulation (CBMF), and it is the integrated force method (IFM) in the finite element analysis. The dual integrated force method (IFMD) with displacement as the primary unknown had been formulated. Both the IFM and IFMD produce identical responses. The IFMD can utilize the equation solver of the traditional stiffness method. The variational derivation of the CBMF produced the existing sets of elasticity equations along with the new boundary compatibility conditions, which were missed since the time of Saint-Venant, who formulated the field equations about 1860. The CBMF, which can be used to solve stress, displacement, and mixed boundary value problems, has eliminated the restriction of the classical method that was applicable only to stress boundary value problem. The IFM in structures produced high-fidelity response even with a modest finite element model. Because structural design is stress driven, the IFM has influenced it considerably. A fully utilized design method for strength and stiffness limitation was developed via the IFM analysis tool. The method has identified the singularity condition in structural optimization and furnished a strategy that alleviated the limitation and reduced substantially the computation time to reach the optimum solution. The CBMF and IFM tensorial approaches are robust formulations because both methods simultaneously emphasize the equilibrium equation and the compatibility condition. The vectorial displacement method emphasized the equilibrium, while the compatibility condition became the basis of the scalar stress-function approach. The tensorial approach can be transformed to obtain the vector and the scalar methods, but the reverse course cannot be followed. The tensorial approach outperformed other methods as expected. This paper introduces the new concepts in elasticity, in finite element analysis, and in design optimization with numerical illustrations.

Reliability-Based Design Optimization of a Composite Airframe Component

*† ‡ A stochastic design optimization methodology (SDO) has been developed to design components of an airframe structure that can be made of metallic and composite materials. The design is obtained as a function of the risk level, or reliability, p. The design method treats uncertainties in load, strength, and material properties as distribution functions, which are defined with mean values and standard deviations. A design constraint or a failure mode is specified as a function of reliability p. Solution to stochastic optimization yields the weight of a structure as a function of reliability p. Optimum weight versus reliability p traced out an inverted-S-shaped graph. The center of the inverted-S graph corresponded to 50 percent (p = 0.5) probability of success. A heavy design with weight approaching infinity could be produced for a near-zero rate of failure that corresponds to unity for reliability p (or p = 1). Weight can be reduced to a small value for the most failureprone design with a reliability that approaches zero (p = 0). Reliability can be changed for different components of an airframe structure. For example, the landing gear can be designed for a very high reliability, whereas it can be reduced to a small extent for a raked wingtip. The SDO capability is obtained by combining three codes: (1) The MSC/Nastran code was the deterministic analysis tool, (2) The fast probabilistic integrator, or the FPI module of the NESSUS software, was the probabilistic calculator, and (3) NASA Glenn Research Center’s optimization testbed CometBoards became the optimizer. The SDO capability requires a finite element structural model, a material model, a load model, and a design model. The stochastic optimization concept is illustrated considering an academic example and a real-life raked wingtip structure of the Boeing 767–400 extended range airliner made of metallic and composite materials. ngineers have recognized the existence of uncertainty in material properties, in load, and in structural analysis as well as in design constraints. Consider for example the yield strength of a steel that is required to design a steel structure. Strength is measured in the laboratory from tests conducted on standard coupons. It is commonly observed that repeated tests yield different values for the strength of steel. The test data can be processed to obtain a nominal or mean value and a dispersion range or a standard deviation. The nominal strength along with a safety factor is used to define allowable strength for traditional deterministic design calculations. Alternatively, strength can be considered as a random variable with a mean value and a standard deviation. The experimental data can be processed to obtain a probability distribution function such as, for example, the commonly used normal distribution function that is defined by a mean value and a standard deviation. This concept for strength can be extended to Young’s modulus, Poisson’s ratio, density, and so forth, and a probabilistic material model can be generated. The procedure can be repeated and a probabilistic load model can be developed for mechanical, thermal and initial deformation loads. Likewise, a probabilistic design model can be developed for sizing variables like depth and thickness of a beam.

Probabilistic Flutter Analysis of a Mistuned Bladed Disks

This paper presents the results of a probabilistic flutter study of a mistuned bladed disk using a high fidelity model including both structural and aerodynamic coupling. The approach used in this paper is relatively fast because it does not require any additional information than that required of a tuned flutter analysis, with the exception of the mistuned blade frequencies. The case study shows that the stability of the fleet can be significantly affected by the standard deviation of blade frequencies and the pattern in which they are arranged in the wheel. A method for understanding and identifying the beneficial patterns is presented.Copyright

Precision of Sensitivity in the Design Optimization of Indeterminate Structures

Design sensitivity is central to most optimization methods. The analytical sensitivity expression for an indeterminate structural design optimization problem can be factored into a simple determinate term and a complicated indeterminate component. Sensitivity can be approximated by retaining only the determinate term and setting the indeterminate factor to zero. The optimum solution is reached with the approximate sensitivity. The central processing unit (CPU) time to solution is substantially reduced. The benefit that accrues from using the approximate sensitivity is quantified by solving a set of problems in a controlled environment. Each problem is solved twice: first using the closed-form sensitivity expression, then using the approximation. The problem solutions use the CometBoards testbed as the optimization tool with the integrated force method as the analyzer. The modification that may be required, to use the stiffener method as the analysis tool in optimization, is discussed. The design optimization problem of an indeterminate structure contains many dependent constraints because of the implicit relationship between stresses, as well as the relationship between the stresses and displacements. The design optimization process can become problematic because the implicit relationship reduces the rank of the sensitivity matrix. The proposed approximation restores the full rank and enhances the robustness of the design optimization method.

Probabilistic Design and Analysis for System-Level Application

This paper focuses on the computation of system reliability and its application to probabilistic design. While success has been reported with methods for individual components and failure modes, integrating them to estimate overall system-level reliability faces several hurdles, such as integration across multiple levels and multiple physics. Computational effort is a major challenge, especially when evaluating reliability over time and considering progressive damage and interactions among multiple failure mechanisms. This paper discusses techniques to overcome some of these challenges, considering trade-offs between accuracy and efficiency.

Unsteady Fluid Structure Interaction in a Turbine Blade

An unsteady, three dimensional Navier-Stokes solution in rotating frame formulation for turbomachinery applications has been described. Casting the governing equations in a rotating frame enables the freezing of grid motion and results in substantial savings in computer time. Heat transfer to a gas turbine blade was computationally simulated by finite element methods and probabilistically evaluated in view of the several uncertainties in the performance parameters. The interconnection between the CFD code and finite element structural analysis code was necessary to couple the thermal profiles with the structural design. The stresses and their variations were evaluated at critical points on the turbine blade. Cumulative distribution functions and sensitivity factors were computed for stresses due to the aerodynamic, geometric, material and thermal random variables. These results can be used to quickly identify the most critical design variables in order to optimize the design and make it cost effective. The analysis leads to the selection of the appropriate materials to be used and to the identification of both the most critical measurements and parameters.© 2005 ASME

Probabilistic structural analysis of ultra efficient engine technology ceramic matrix composite combustor liners

A probabilistic structural analysis of NASAs ultra efficient engine technology (UEET) ceramic matrix composite (CMC) combustor liners has been completed using the NESTEM Code. The purpose was to identify the maximum stress locations and perform a probabilistic structural analysis at these locations on the inner and outer liners for given thermal loadings to determine the probability of failure at these locations. The probabilistic structural analysis included quantifying the influence of uncertainties in material stiffness properties and the coefficient of thermal expansion. Results of the analysis indicate that the circumferential component of stress was the most severe stress component and that the inner liner was more likely to fail than the outer liner. Tests of the combustor liners by general electric aerospace engines (GEAE) qualitatively support the results of this analysis.