Sensitivity of a Subject-specific Ankle Sprain Simulation to Extrinsic Versus Intrinsic Biomechanical Factors

Ankle sprains are the most common musculoskeletal injury in sport and military activity, despite existing prophylactic strategies. The purpose of this report was to develop a probabilistic simulation of lateral ankle sprains during single-limb drop landing, towards accelerating innovation in ankle sprain prevention. A deterministic, subject-specific musculoskeletal model was extended with automation and probabilistic distributions on sprain-related biomechanical factors. Probabilistic simulations were generated using traditional Monte Carlo techniques and the advanced mean value method, a more computationally-efficient approach. Predicted distributions of peak ankle joint rotations, velocities, and moments borne by supporting passive structures agreed favorably with the deterministic model and with reports of real sprain biomechanics. Parameter sensitivities identified that predictions were most strongly influenced by drop height, subtalar joint posture at contact, invertor/evertor co-activation, and passive ankle stiffness. The advanced mean value method predicted confidence bounds comparable to a 1000-trial Monte Carlo simulation, and required only 14 model evaluations and 4-min processing time. The extended probabilistic simulation may be useful to virtually test new prophylactic strategies for ankle sprains, and is made available for open-source use (https://simtk.org/projects/sprain-sim).

Reliability Assessment of a Turbogenerator Coil Retaining Ring Based on Low Cycle Fatigue Data

Turbogenerator coil retaining rings are shrunk-fitted onto the rotor over the coils, in order to restrain them against the centrifugal force. They are typically subjected to low cycle fatigue, with a cycle being completed at every machine switch-on and switch-off. The subject of this paper consists in the determination of the failure probability of a coil retaining ring. The failure mode of the ring cracking, when it swells in tension, due to the centrifugal force is here considered. The reliability assessment is preceded by the study of the input variables affecting the low-cycle fatigue load and of their stochastic distributions. This question is tackled by the experimental determination of the static, cyclic and fatigue curves of the involved material and by the application of a statistical model to compute related parameters and their standard deviations. Upon the determination of variable distributions, the probability of failure is estimated in the form of a cumulative distribution function by a computationally efficient methodology, based on the Advanced Mean Value approach. The obtained results account for the material response and the local stressstrain states at the most loaded coil retaining ring region. The determined probability at the end of the machine life, in the order of 10-12, is compatible with reference values for structures under fatigue in the mechanical and aeronautical fields.

A Methodology for Trading-Off Performance and Robustness Under Uncertainty

Mathematical optimization plays an important role in engineering design, leading to greatly improved performance. Deterministic optimization, however, may result in undesired choices because it neglects uncertainty. Reliability-based design optimization (RBDO) and robust design can improve optimization by considering uncertainty. This paper proposes an efficient design optimization method under uncertainty, which simultaneously considers reliability and robustness. A mean performance is traded-off against robustness for a given reliability level of all performance targets. This results in a probabilistic multiobjective optimization problem. Variation is expressed in terms of a percentile difference, which is efficiently computed using the advanced mean value method. A preference aggregation method converts the multiobjective problem to a single-objective problem, which is then solved using an RBDO approach. Indifference points are used to select the best solution without calculating the entire Pareto frontier. Examples illustrate the concepts and demonstrate their applicability.

Design Optimization of Hierarchically Decomposed Multilevel Systems Under Uncertainty

This paper presents a methodology for design optimization of hierarchically decomposed systems under uncertainty. We propose an extended, probabilistic version of the deterministic analytical target cascading (ATC) formulation by treating uncertain quantities as random variables and posing probabilistic design constraints. A bottom-to-top coordination strategy is used for the ATC process. Given that first-order approximations may introduce unacceptably large errors, we use a technique based on the advanced mean value method to estimate uncertainty propagation through the multilevel hierarchy of elements that comprise the decomposed system. A simple yet illustrative hierarchical bilevel engine design problem is used to demonstrate the proposed methodology. The results confirm the applicability of the proposed probabilistic ATC formulation and the accuracy of the uncertainty propagation technique.

Design Optimization of Hierarchically Decomposed Multilevel Systems Under Uncertainty

This paper presents a methodology for design optimization of decomposed systems in the presence of uncertainties. We extend the analytical target cascading (ATC) formulation to probabilistic design by treating stochastic quantities as random variables and parameters and posing reliability-based design constraints. We model the propagation of uncertainty throughout the multilevel hierarchy of elements that comprise the decomposed system by using the advanced mean value (AMV) method to generate the required probability distributions of nonlinear responses. We utilize appropriate metamodeling techniques for simulation-based design problems. A simple yet illustrative hierarchical bi-level engine design problem is used to demonstrate the proposed methodology.

Hybrid Analysis Method for Reliability-Based Design Optimization

Reliability-based design optimization (RBDO) involves evaluation of probabilistic constraints, which can be done in two different ways, the reliability index approach (RIA) and the performance measure approach (PMA). It has been reported in the literature that RIA yields instability for some problems but PMA is robust and efficient in identifying a probabilistic failure mode in the optimization process. However, several examples of numerical tests of PMA have also shown instability and inefficiency in the RBDO process if the advanced mean value (AMV) method, which is a numerical tool for probabilistic constraint evaluation in PMA, is used, since it behaves poorly for a concave performance function, even though it is effective for a convex performance function. To overcome difficulties of the AMV method, the conjugate mean value (CMV) method is proposed in this paper for the concave performance function in PMA. However, since the CMV method exhibits the slow rate of convergence for the convex function, it is selectively used for concave-type constraints. That is, once the type of the performance function is identified, either the AMV method or the CMV method can be adaptively used for PMA during the RBDO iteration to evaluate probabilistic constraints effectively. This is referred to as the hybrid mean value (HMV) method. The enhanced PMA with the HMV method is compared to RIA for effective evaluation of probabilistic constraints in the RBDO process. It is shown that PMA with a spherical equality constraint is easier to solve than RIA with a complicated equality constraint in estimating the probabilistic constraint in the RBDO process.

Structural/Acoustic Sensitivity Analysis of a Panel With Manufacturing Variability

Sensitivity analysis is an essential component in the design of structural/acoustic systems. Current structural/acoustic sensitivity algorithms assume deterministic system design parameters and therefore predict changes in the deterministic response of the system. Due to variability associated with manufacturing tolerances most structural/acoustic systems consist of random and deterministic structural design parameters and therefore produce a probabilistic acoustic response. In this work, a structural/acoustic sensitivity algorithm is presented that considers systems comprised of random and deterministic design parameters. The new sensitivity algorithm calculates the change in the probabilistic vibro-acoustic response of a system due to a change in a deterministic design parameter. The new algorithm uses boundary element analysis, finite element analysis, and an advanced mean value method. An elastically supported panel subject to a non-uniform load is used to illustrate the algorithm. Variability is considered in the elastic support while structural sizing parameters are taken to be deterministic. The response of the panel is defined in terms of probabilistic radiated sound power. Probabilistic sound power sensitivity results are calculated and used to predict new sound power values via finite difference analysis. These results are compared successfully to results obtained through re-analysis.

Numerical Probabilistic Analysis of Structural/Acoustic Systems

A formulation that accounts for manufacturing variability in the analysis of structural/acoustic systems is presented. The methodology incorporates the concept of fast probability integration with finite element (FEA) and boundary element analysis (BEA) for producing the probabilistic acoustic response of a structural/acoustic system. The advanced mean value method is used for integrating the system probability density function. FEA and BEA are combined for producing the acoustic response that constitutes the performance function. The probabilistic acoustic response is calculated in terms of a cumulative distribution function. The new methodology is used to illustrate the difference between the results from a probabilistic analysis that accounts for manufacturing uncertainty, and an equivalent deterministic simulation through applications. The probabilistic computations are validated by comparison to Monte Carlo simulations. Based on its computational efficiency and its accuracy the new methodology is concluded to be a viable method of calculating numerically the probabilistic response of structural/acoustic systems due to manufacturing variability.

Hybrid Analysis Method for Reliability-Based Design Optimization

Reliability-Based Design Optimization (RBDO) involves evaluation of probabilistic constraints, which can be done in two different ways, the Reliability Index Approach (RIA) and the Performance Measure Approach (PMA). It has been reported in the literature that RIA yields instability for some problems but PMA is robust and efficient in identifying a probabilistic failure mode in the RBDO process. However, several examples of numerical tests of PMA have also shown instability and inefficiency in the RBDO process if the Advanced Mean Value (AMV) method, which is a numerical tool for probabilistic constraint evaluation in PMA, is used, since it behaves poorly for a concave performance function, even though it is effective for a convex performance function. To overcome difficulties of the AMV method, the Conjugate Mean Value (CMV) method is proposed in this paper for the concave performance function in PMA. However, since the CMV method exhibits the slow rate of convergence for the convex function, it is selectively used for concave-type constraints. That is, once the type of the performance function is identified, either the AMV method or the CMV method can be adaptively used for PMA during the RBDO iteration to evaluate probabilistic constraints effectively. This is referred to as the Hybrid Mean Value (HMV) method. The enhanced PMA with the HMV method is compared to RIA for effective evaluation of probabilistic constraints in the RBDO process. It is shown that PMA with a spherical equality constraint is easier to solve than RIA with a complicated equality constraint in estimating the probabilistic constraint in the RBDO process.