Connection between Volterra series and perturbation method in nonlinear systems analyses

This paper is concerned with the connection between the Volterra series and the regular perturbation method in nonlinear systems analyses. It is revealed for the first time that, for a forced polynomial nonlinear system, if its derived linear system is a damped dissipative system, the steady response obtained through the regular perturbation method is exactly identical to the response given by the Volterra series. On the other hand, if the derived linear system is an undamped conservative system, then the Volterra series is incapable of modeling the forced polynomial nonlinear system. Numerical examples are further presented to illustrate these points. The results provide a new criterion for quickly judging whether the Volterra series is applicable for modeling a given polynomial nonlinear system.     

Polytope bounds on multivariate value sets

We improve the upper bounds for the cardinality of the value set of a multivariable polynomial map over a finite field using the polytope of the polynomial. This generalizes earlier bounds only dependent on the degree of a polynomial.     

Optimization over structured subsets of positive semidefinite matrices via column generation

We develop algorithms to construct inner approximations of the cone of positive semidefinite matrices via linear programming and second order cone programming. Starting with an initial linear algebraic approximation suggested recently by Ahmadi and Majumdar, we describe an iterative process through which our approximation is improved at every step. This is done using ideas from column generation in large-scale linear programming. We then apply these techniques to approximate the sum of squares cone in a nonconvex polynomial optimization setting, and the copositive cone for a discrete optimization problem.     

On the impact of cargo weight, vehicle parameters, and terrain characteristics on the prediction of traction for off-road vehicles

A realistic prediction of the traction capacity of vehicles operating in off-road conditions must account for stochastic variations in the system itself, as well as in the operational environment. Moreover, for mobility studies of wheeled vehicles on deformable soil, the selection of the tire model used in the simulation influences the degree of confidence in the output. Since the same vehicle may carry various loads at different times, it is also of interest to analyze the impact of cargo weight on the vehicle’s traction.This study focuses on the development of an algorithm to calculate the tractive capacity of an off-road vehicle with stochastic vehicle parameters (such as suspension stiffness, suspension damping coefficient, tire stiffness, and tire inflation pressure), operating on soft soil with an uncertain level of moisture, and on a terrain topology that induces rapidly changing external excitations on the vehicle. The analysis of the vehicle–soil dynamics is performed for light cargo and heavy cargo scenarios. The algorithm relies on the comparison of the ground pressure and the calculated critical pressure to decide if the tire can be approximated as a rigid wheel or if it should be modeled as a flexible wheel. It also involves using previously-developed vehicle and stochastic terrain models, and computing the vehicle sinkage, resistance force, tractive force, drawbar pull, and tractive torque.The vehicle model used as a case study has seven degrees of freedom. Each of the four suspension systems is comprised of a nonlinear spring and a viscous (linear or magneto-rheological) damper. An off-road terrain profile is simulated as a 2-D random process using a polynomial chaos approach [Sandu C, Sandu A, Li L. Stochastic modeling of terrain profiles and soil parameters. SAE 2005 transactions. J Commer Vehicles 2005-01-3559]. The soil modeling is concerned with the efficient treatment of the impact of the moisture content on relationships critical in defining the mobility of an off-road vehicle (such as the pressure–sinkage [Sandu C et al., 2005-01-3559] and the shear stress–shear displacement relations). The uncertainties in vehicle parameters and in the terrain profile are propagated through the vehicle model, and the uncertainty in the output of the vehicle model is analyzed [Sandu A, Sandu C, Ahmadian M. Modeling multibody dynamic systems with uncertainties. Part I: theoretical and computational aspects, Multibody system dynamics. Publisher: Springer Netherlands; June 29, 2006. p. 1–23 (23), ISSN: 1384-5640 (Paper) 1573-272X (Online). doi:10.1007/s11044-006-9007-5; Sandu C, Sandu A, Ahmadian M. Modeling multibody dynamic systems with uncertainties. Part II: numerical applications. Multibody system dynamics, vol. 15, No. 3. Publisher: Springer Netherlands; 2006. p. 241–62 (22). ISSN: 1384-5640 (Paper) 1573-272X (Online). doi:10.1007/s11044-006-9008-4]. Such simulations can provide the basis for the study of ride performance, handling, and mobility of the vehicle in rough off-road conditions.     

Reliability‐targeted HPLC–UV method validation—A protocol enrichment perspective

Method validation is important in analytical chemistry to obtain the reliability of an analytical method. Guidelines provided by the regulatory bodies can be used as a general framework to assess the validity of a method. Since these guidelines do not focus on the reliability of analytical results exclusively, this study was aimed to combine a few recently evolved strategies that may render analytical method validation more reliable and trustworthy. In this research, the analytical error function was determined by appropriate polynomial regression statistics that determine the range of analyte concentration that may lead to more accurate measurements by producing the least possible total error in the assay and can be regarded as a reliable weighting method. The reliability of the analytical results over a particular concentration range has been proposed by a Bayesian probability study. In order to ensure the applicability of this approach, it was applied for the validation of an HPLC–UV assay method dedicated to the quantification of cefepime and tazobactam in human plasma. A comparison between the newer approach and the usual method validation revealed that the application of analytical error function and Bayesian analysis at the end of the validation process can produce significant improvements in the analytical results.     

Développement en polynômes de chaos d'un modèle lagrangien d'écoulement autour d'un profilPolynomial chaos expansion of a Lagrangian model for the flow around an airfoil

A polynomial chaos (PC) expansion a the Lagrangian model for the stochastic incompressible inviscid flow around an airfoil is presented. The flow field is modeled using a distribution of lumped vortices on the airfoil surface while the wake is modeled with Lagrangian point vortices. An original technique is proposed for the computation of the PC coefficients of the velocities induced by the vortices. Two computational examples for random airfoil motions are provided to illustrate the capability of the method to deal with complex situations. To cite this article: O. Le Maître, C. R. Mecanique 334 (2006).