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The paper pertains to the development of a new relationship expressing the Winkler modulus of subgrade reaction as a function of elastic parameters such as the modulus of elasticity and Poisson’s ratio of the foundation soil, and the relative rigidity of the foundation soil and the beam resting over the same. It ensures that the maximum values of beam deflection and bending moment computed by using both the theory of elastic continuum and lumped parameter modeling are either identical or very close to each other. In this respect the developed expression can be construed to be superior to those proposed by Biot and Vesic, which respectively predict correct values for only one of the quantities (either maximum bending moment or maximum deflection). In addition, the proposed model for subgrade modulus is applicable to multiple load conditions as well unlike the other two approaches as reported.

Landmine detonation continues to be a serious threat to Army military vehicles in theater. One strategy to mitigate its effects is to use engineering design techniques to reshape the underbody of the hull. Previous works have shown that reshaping the hull's underbody can change the vehicle's response to blast loading. In the presented work, a metamodel-based shape optimization methodology is proposed for optimizing shallow-buried blast-loaded underbody structures. The shallow-buried blast load is simulated using an empirical model. Additional studies are conducted on target positions with respect to the blast load and load positioning and the effects of optimization parameters on the optimal solution. It is shown that target vertical position affects impulse response comparison, and for center-buried loads, the v-shaped underbody target produces a lower impulse than that of other shaped targets. Optimal solutions obtained using an adaptive domain reduction strategy produce more accurate solutions than a single iteration strategy. Based on the proposed optimization method, a unique v-like underbody shape is presented as the optimal underbody solution.

Purpose – The purpose of this paper is to present a study of the parallelization of the construction of explicit constraints or limit‐state functions using support vector machines. These explicit boundaries have proven to be beneficial for design optimization and reliability assessment, especially for problems with large computational times, discontinuities, or binary outputs. In addition to the study of the parallelization, the objective of this article is also to provide an approach to select the number of processors.Design/methodology/approach – This article investigates the parallelization in two ways. First, the efficiency of the parallelization is assessed by comparing, over several runs, the number of iterations needed to create an accurate boundary to the number of iterations associated with a theoretical “linear” speedup. Second, by studying these differences, an “appropriate” range of parallel processors can be inferred.Findings – The parallelization of the construction of explicit boundaries ca...

This paper presents a methodology for constrained efficient global optimization (EGO) using support vector machines (SVMs). While the objective function is approximated using Kriging, as in the original EGO formulation, the boundary of the feasible domain is approximated explicitly as a function of the design variables using an SVM. Because SVM is a classification approach and does not involve response approximations, this approach alleviates issues due to discontinuous or binary responses. More importantly, several constraints, even correlated, can be represented using one unique SVM, thus considerably simplifying constrained problems. In order to account for constraints, this paper introduces an SVM-based "probability of feasibility" using a new Probabilistic SVM model. The proposed optimization scheme is constituted of two levels. In a first stage, a global search for the optimal solution is performed based on the "expected improvement" of the objective function and the probability of feasibility. In a second stage, the SVM boundary is locally refined using an adaptive sampling scheme. An unconstrained and a constrained formulation of the optimization problem are presented and compared. Several analytical examples are used to test the formulations. In particular, a problem with 99 constraints and an aeroelasticity problem with binary output are presented. Overall, the results indicate that the constrained formulation is more robust and efficient.

This article presents an improved adaptive sampling scheme for the construction of explicit decision functions (constraints or limit state functions) using Support Vector Machines (SVMs). The proposed work presents substantial modifications to an earlier version of the scheme (Basudhar and Missoum, Comput Struct 86(19–20):1904–1917, 2008). The improvements consist of a different choice of samples, a more rigorous convergence criterion, and a new technique to select the SVM kernel parameters. Of particular interest is the choice of a new sample chosen to remove the “locking” of the SVM, a phenomenon that was not understood in the previous version of the algorithm. The new scheme is demonstrated on analytical problems of up to seven dimensions.

An aneurysm is a gradual and progressive ballooning of a blood vessel due to wall degeneration. Rupture of abdominal aortic aneurysm (AAA) constitutes a significant portion of deaths in the US. In this study, we describe a technique to reconstruct AAA geometry from CT images in an inexpensive and streamlined fashion. A 3D reconstruction technique was implemented with a GUI interface in MATLAB using the active contours technique. The lumen and the thrombus of the AAA were segmented individually in two separate protocols and were then joined together into a hybrid surface. This surface was then used to obtain the aortic wall. This method can deal with very poor contrast images where the aortic wall is indistinguishable from the surrounding features. Data obtained from the segmentation of image sets were smoothed in 3D using a Support Vector Machine technique. The segmentation method presented in this paper is inexpensive and has minimal user-dependency in reconstructing AAA geometry (lumen and wall) from patient image sets. The AAA model generated using this segmentation algorithm can be used to study a variety of biomechanical issues remaining in AAA biomechanics including stress estimation, endovascular stent-graft performance, and local drug delivery studies.

This article presents a methodology to generate explicit decision functions using support vector machines (SVM). A decision function is defined as the boundary between two regions of a design space (e.g., an optimization constraint or a limit-state function in reliability). The SVM-based decision function, which is initially constructed based on a design of experiments, depends on the amount and quality of the training data used. For this reason, an adaptive sampling scheme that updates the decision function is proposed. An accurate approximated explicit decision functions is obtained with a reduced number of function evaluations. Three problems are presented to demonstrate the efficiency of the update scheme to explicitly reconstruct known analytical decision functions. The chosen functions are the boundaries of disjoint regions of the design space. A convergence criterion and error measure are proposed. The scheme is also applied to the definition of an explicit failure region boundary in the case of the buckling of a geometrically nonlinear arch.

This article presents a method for the explicit construction of limit state functions using Support Vector Machines (SVM). Specifically, the approach aims at handling the difficulties associated with the reliability assessment of problems exhibiting discontinuous responses and disjoint failure domains. The SVM-based explicit construction of limit state functions allows for an easy calculation of a probability of failure and enables the association of a specific system behavior with a region of the design space. The explicit limit state function can then be used within a reliability-based design optimization (RBDO) problem. Two problems are presented to demonstrate the successful application of the developed method for explicit construction of limit state function and reliability-based optimum design.

This paper presents an approach to estimate probabilities of failure in cases where the random variables are correlated. An explicit limit state function is constructed in the uncorrelated standard normal space using the Nataf transformation and a support vector machine (SVM). The method of explicitly constructing the limit state function is referred to as explicit design space decomposition (EDSD), which also includes an adaptive sampling strategy to build an accurate SVM approximation. Several analytical examples with various distributions and also multiple failure modes are presented.

This paper presents a methodology for constrained e cient global optimization (EGO) using support vector machines (SVMs). The proposed SVM-based method has several advantages. It is more general because it is applicable to a wider variety of problems compared to current techniques. These include problems with discontinuous and binary (pass/fail) states and multiple constraints. In this paper, the objective function is approximated using Kriging while the constraint boundary is approximated using an SVM classi er. The probability of misclassi cation by the SVM is calculated using a probabilistic support vector machine (PSVM). The existing PSVM models have certain limitations that make them unsuitable for application in the proposed methodology. Therefore, a modi ed PSVM model is also proposed to overcome these limitations. Several constrained EGO formulations are implemented and compared in this paper. The results are also compared to EGO implementations with Kriging-based constraint approximations from the literature.

This paper introduces a new approach for the optimal geometric design and tolerancing of multibody systems. The approach optimizes both the nominal system dimensions and the associated tolerances by solving a reliability-based design optimization (RDBO) problem under the assumption of truncated normal distributions of the geometric properties. The solution is obtained by first constructing the explicit boundaries of the failure regions (limit state function) using a support vector machine, combined with adaptive sampling and uniform design of experiments. The use of explicit boundaries enables the treatment of systems with discontinuous or binary behaviors. The explicit boundaries also allow for an efficient calculation of the probability of failure using importance sampling. The probability of failure is subsequently approximated over the whole design space (the nominal system dimensions and the associated tolerances), thus making the solution of the RBDO problem straightforward. The proposed approach is applied to the optimization of a web cutter mechanism.

This paper presents a new adaptive sampling technique for the construction of locally reflned explicit decision functions. The decision functions can be used for both deterministic and probabilistic optimization, and may represent a constraint or a limit-state function. In particular, the focus of this paper is on reliability-based design optimization (RBDO). Instead of approximating the responses, the method is based on explicit design space decomposition (EDSD), in which an explicit boundary separating distinct regions in the design space is constructed. A statistical learning tool known as support vector machine (SVM) is used to construct the boundaries. A major advantage of using an EDSD-based method lies in its ability to handle discontinuous responses. A separate adaptive sampling scheme for calculating the probability of failure is also developed, which is used within the RBDO process. The update methodology is validated through several test examples with analytical decision functions.

An original technique to incorporate random fields non-intrusively in probabilistic design is presented. The approach is based on the extraction of the main features of a random field using a limited number of experimental observations (snapshots). An approximation of the random field is obtained using proper orthogonal decomposition (POD). For a given failure criterion, an explicit limit state function (LSF) in terms of the coefficients of the POD expansion is obtained using a support vector machine (SVM). An adaptive sampling technique is used to generate samples and update the approximated LSF. The coefficients of the orthogonal decomposition are considered as random variables with distributions determined from the snapshots. Based on these distributions and the explicit LSF, the approach allows for an efficient assessment of the probabilities of failure. In addition, the construction of explicit LSF has the advantage of handling discontinuous responses. Two test-problems are used to demonstrate the proposed methodology used for the calculation of probabilities of failure. The first example involves the linear buckling of an arch structure for which the thickness is a random field. The second problem concerns the impact of a tube on a rigid wall. The planarity of the walls of the tube is considered as a random field.

This article presents a method for the explicit construction of limit state functions using Support Vector Machines (SVM). An algorithm is proposed for updating the SVM decision function by carefully selecting the training samples. This results in the construction of an accurate limit state function with a reduced number of function evaluations. Speciflcally, the SVM-based approach aims at handling the di‐culties associated with the reliability assessment of problems exhibiting discontinuous responses and disjoint failure domains. The explicit construction of limit state functions allows for an easy calculation of a probability of failure and enables the association of a speciflc system behavior with a region of the design space. Three problems are presented to demonstrate the explicit construction of a limit state function. The proposed update scheme is validated by comparing the obtained explicit function to actual analytical limit state functions. Non-linear problems are often characterized by various and sudden behavioral changes which, in structural mechanics, are associated with the presence of critical points. A typical example is a geometrically nonlinear structure which globally buckles for a load larger than the limit load. Because these abrupt changes can be triggered by inflnitesimally small modiflcations of design parameters or loading conditions, the responses of the system are discontinuous in a mathematical sense. In the context of reliability, these slight variations often fall in the range of uncertainties. In simulation-based design, discontinuities present a serious problem for optimization or probabilistic techniques because it is usually assumed that the system’s responses are continuous. In optimization, this limits any traditional gradient-based methods or response surface techniques. When considering reliability, discontinuities might hamper the use of approximation methods such as First Order and Second Order Reliability Methods (FORM and SORM), 1 Advanced Mean Value (AMV), 2 or Monte-Carlo simulations 3 with response surfaces. 4

This paper introduces a new methodology for probabilistic optimal design of multibody systems. Specifically, the effects of dimensional uncertainties on the behavior of a system are considered. The proposed reliability-based optimization method addresses difficulties such as high computational effort and non-smoothness of the system’s responses, for example, as a result of contact events. The approach is based on decomposition of the design space into regions, corresponding to either acceptable or non-acceptable system performance. The boundaries of these regions are defined using Support Vector Machines (SVMs), which are explicit in terms of the design parameters. A SVM can be trained based on a limited number of samples, obtained from a design of experiments, and allows a very efficient estimation of probability of failure, even when Monte Carlo Simulation (MCS) is used. A modularly structured tolerance analysis scheme for automatic estimation of system production cost and probability of system failure is presented. In this scheme, detection of failure is based on multibody system simulation, yielding high computational demand. A SVM-based replication of the failure detection process is derived, which ultimately allows for automatic optimization of tolerance assignments. A simple multibody system, whose performance usually shows high tolerance sensitivity, is chosen as an exemplary system for illustration of the proposed approach. The system is optimally designed for minimum manufacturing cost while satisfying a target performance level with a given probability.Copyright © 2007 by ASME

This paper presents a methodology to calculate probabilities of failure using Probabilistic Support Vector Machines (PSVMs). Support Vector Machines (SVMs) have recently gained attention for reliability assessment because of several inherent advantages. Specifically, SVMs allow one to construct explicitly the boundary of a failure domain. In addition, they provide a technical solution for problems with discontinuities, binary responses, and multiple failure modes. However, the basic SVM boundary might be inaccurate; therefore leading to erroneous probability of failure estimates. This paper proposes to account for the inaccuracies of the SVM boundary in the calculation of the Monte Carlo-based probability of failure. This is achieved using a PSVM which provides the probability of misclassification of Monte Carlo samples. The probability of failure estimate is based on a new sigmoid- based PSVM model along with the identification of a region where the probability of misclassification is large. The PSVM-based probabilities of failure are, by construction, always more conservative than the deterministic SVM-based probability estimates.