Ravi C. Penmetsa

Adaption of fast Fourier transformations to estimate structural failure probability

Probabilistic analysis with multiple non-normal random variables requires multi-fold integration, for which the closed-form solutions do not exist. Moreover, it is almost impossible to estimate failure probability accurately without the use of numerical integration. A central problem in probabilistic analysis is the computation of the cumulative distribution function of the limit-state. Since the limit-state surface is not available in a closed-form, the convolution theorem is usually thought to be not applicable to the problem. In this paper, a methodology based on function approximations and the convolution theorem is presented to estimate the structural failure probability. The convolution integral is solved efficiently using the fast Fourier transform technique, and the limit-state is approximated using a two-point adaptive nonlinear approximation at the most probable failure point. The proposed technique estimates the failure probability accurately with significantly less computational effort compared to the Monte Carlo simulation. The methodology developed is applicable to structural reliability problems with any number of random variables and any kind of random variable distribution, including normal, log-normal, extreme value, Weibull, etc. The accuracy and robustness of the proposed algorithm is demonstrated by several examples having highly complex, explicit/implicit performance functions.

Computed-tomography-based finite-element models of long bones can accurately capture strain response to bending and torsion

Finite element (FE) models of long bones constructed from computed-tomography (CT) data are emerging as an invaluable tool in the field of bone biomechanics. However, the performance of such FE models is highly dependent on the accurate capture of geometry and appropriate assignment of material properties. In this study, a combined numerical-experimental study is performed comparing FE-predicted surface strains with strain-gauge measurements. Thirty-six major, cadaveric, long bones (humerus, radius, femur and tibia), which cover a wide range of bone sizes, were tested under three-point bending and torsion. The FE models were constructed from trans-axial volumetric CT scans, and the segmented bone images were corrected for partial-volume effects. The material properties (Youngs modulus for cortex, density-modulus relationship for trabecular bone and Poissons ratio) were calibrated by minimizing the error between experiments and simulations among all bones. The R(2) values of the measured strains versus load under three-point bending and torsion were 0.96-0.99 and 0.61-0.99, respectively, for all bones in our dataset. The errors of the calculated FE strains in comparison to those measured using strain gauges in the mechanical tests ranged from -6% to 7% under bending and from -37% to 19% under torsion. The observation of comparatively low errors and high correlations between the FE-predicted strains and the experimental strains, across the various types of bones and loading conditions (bending and torsion), validates our approach to bone segmentation and our choice of material properties.

Uncertainty Propagation Using Possibility Theory and Function Approximations

Abstract Based on the nature and extent of uncertainty existing in an engineering system, different approaches can be used for uncertainty propagation. If the uncertainty of the system is due to imprecise information, and lack of statistical data, the Possibilistic theory can be used. During preliminary design, uncertainties need to be accounted for and due to lack of sufficient information, assigning a probability distribution may not be possible. Moreover, the flight conditions (loads, control surface settings, etc.) during a mission could take values within certain bounds, which do not follow any pattern. The uncertain information in these cases is available as intervals with lower and upper limits. In this case, the fuzzy-arithmetic-based method is suitable to estimate the possibility of failure. The use of surrogate models to improve the efficiency of prediction is presented in this article. Various numerical examples are presented to demonstrate the applicability of the method to practical problems.

Structural System Reliability Quantification Using Multipoint Function Approximations

In structural problems, when dealing with uncertainties, the failure probability of the structure is estimated subject to a particular performance criterion. However, when the failure of a structural system is governed by multiple failure criteria, all of the measures have to be considered in the failure probability estimation. These failure criteria are usually correlated, and the accuracy of the estimated structural failure probability highly depends on the ability to model the joint failure surface. For example, in an aircraft structure, the stresses in each of the members of a wing can be posed as limit-state functions, along with the displacements and the natural frequencies of the wing. There are no criteria to disregard one limit state over the other, or to convert the system reliability problem into component reliability (dealing with displacement, stress, and frequency individually). Each failure criterion is modeled as a limit-state function for the reliability analysis, which is an implicit function of the random variables. The evaluation of this limit state often requires an expensive finite element simulation or a computational fluid dynamics simulation. Therefore, to predict the failure probability of a structural system efficiently, function approximations for the limit states are considered. An accurate way of defining highly nonlinear functions is presented using a new class of approximations. These approximations are used in conjunction with the Monte Carlo simulation to estimate the structural failure probability. Numerical examples are presented to show the applicability of the proposed method.

Topology Optimization for an Evolutionary Design of a Thermal Protection System

An optimal thermal protection system design for a spacecraft operating in extreme environments of thermal and acoustic loading is of significant importance for todays military space missions. These military space missions tend to push the envelope of the spacecrafts capabilities to the extreme, requiring robust performance of all key systems to assure a successful mission. Because thermal protection system protects the entire spacecraft, its survival from these extreme conditions is critical to the safety of the mission. The design criteria for the thermal and acoustic loading conditions tend to be conflicting. To reduce thermal stress, free boundaries that allow for expansion of the thermal protection system are desirable. However, the random sound pressure level fluctuations associated with, for example, engine noise during ascent, subject the thermal protection system to a wide-band random excitation. Therefore, if the thermal protection system possesses low frequency modes, damaging strains could result. To limit the magnitude of the strains, higher frequency (stiffer) designs with fixity at the boundary are desirable. Therefore, when designing the thermal protection system within these two simultaneous operating environments, the safe operating region is often severely restricted. This research aims at evolving a design that satisfies these two design requirements.

Variable Shape Cavitator Design for a Supercavitating Torpedo

A variable shape cavitator is designed for generating the cavity that surrounds a supercavitating torpedo. The design takes into consideration the various speeds that a torpedo travels through during acceleration from rest. This is accomplished by using shape optimization techniques integrated with potential flow theory and boundary element modeling. The intent is to determine the optimum cavitator shape for different stages of cavity growth as the torpedo accelerates to full speed. The cavity is modeled as axisymmetric with a reentrant jet. The shape of the cavity and the corresponding properties are determined by applying potential flow theory. The potential flow theory is applied using boundary elements along the cavitator and cavity shapes. This analysis yields the physical properties of the cavity that correspond to a given cavitator shape. Shape optimization is then performed on the cavitator to determine the shape yielding a minimum drag coefficient for a fixed cavity length and cavitation number. This optimization process is carried out for many cavity lengths to determine the optimum cavitator shape for each length representing cavity growth.

Estimation of Structural System Reliability Using Fast Fourier Transforms

In probabilistic analysis, the failure probability of a structural component is estimated based on a particular performance criterion. However, the failure of a structural system is governed by multiple failure criteria, all of which are to be taken into consideration for the reliability estimation. In a multidisciplinary environment, where all the failure criteria are equally important, there is no methodology to convert the system reliability problem into a component reliability problem. These failure criteria are often correlated and the accuracy of the estimated structural failure probability is highly dependent on the ability to model the joint failure surface. The evaluation of limit states often requires expensive Finite Element Analysis (FEA) or Computational Fluid Dynamics (CFD) simulation. Therefore, to efficiently predict the failure probability of the structural system, the use of high quality function approximations for each of the limit states and the joint failure surface are considered in this paper. Once the joint failure surface is approximated as a closed-form expression, the convolution integral can be solved efficiently using a Fast Fourier Transform (FFT) technique to estimate the structural failure probability. Due to the high non-linearity of the joint failure region, a methodology is presented to solve this convolution integral based on multiple function approximations over several disjointed regions over the design space. Numerical examples are presented to show the applicability, efficiency, and accuracy of the proposed method.

Engineering Design Exploration Using Locally Optimized Covariance Kriging

Surrogate models are used in many engineering applications where actual function evaluations are computationally expensive. Kriging is a flexible surrogate model best suited for interpolating nonlinear system responses with a limited number of training points. It is commonly used to alleviate the high computational cost associated with design exploration techniques: for example, uncertainty quantification and multidisciplinary design optimization. However, when the underlying function shows varying degrees of nonlinear behavior within a design domain of interest, kriging, with a stationary covariance structure, can result in low-quality predictions and an overly conservative expected mean squared error. This effect is often amplified by data collected adaptively and unevenly during iterative design explorations. In this paper, the locally optimized covariance kriging method is proposed to capture the nonstationarity of the underlying function behavior. In locally optimized covariance kriging, the nonstati...

Fast Fourier transform based system reliability analysis

In probabilistic analysis, the failure of a structural system is governed by multiple failure criteria, all of which are to be taken into consideration for the reliability estimation. The accuracy of the estimated structural failure probability highly depends on the joint failure surface and its representation. Moreover, the evaluation of limit-states often requires computationally expensive simulations. To improve the efficiency of reliability analysis methods high quality function approximations for each of the limit-states and the joint failure surface are considered in this paper. Once the joint failure surface is represented using a surrogate model, the convolution integral can be solved efficiently using a Fast Fourier Transform (FFT) technique. Due to the high non-linearity of the joint failure region, a methodology is developed to evaluate the convolution integral based on multiple approximations over several disjoint regions spanning the entire design space. Numerical examples are presented to show the applicability, efficiency and accuracy of the proposed method.

Locally-Optimized Covariance Kriging for Engineering Design Exploration

To alleviate computational challenges in uncertainty quantification and multidisciplinary design optimization, Kriging has gained its popularity due to its high accuracy and flexibility interpolating non-linear system responses with collected data. One of the benefits of using Kriging is the availability of expected mean square error along with a response prediction at any location of interest. However, a stationary covariance structure, as is the case with the typical Kriging methodology, used with data collected adaptively from an optimal data acquisition strategy will result in lower quality predictions across the entire sample space. In this paper, a Locally Optimized Covariance Kriging (LOC-Kriging) method is proposed to address the difficulties of building a Kriging model with unevenly distributed adaptive samples. In the proposed method, the global non-stationary covariance is approximated by constructing and aggregating multiple local stationary covariance structures. An optimization problem is formulated to find a minimum number of LOC windows and a membership weighting function is used to combine the LOC-Krigings across the entire domain. This paper will demonstrate that LOC-Kriging improves efficiency and provides more reliable predictions and estimated error bounds than a stationary covariance Kriging, especially with adaptively collected data.