Sensitivity analysis for multibody systems formulated on a Lie group

The objective of this study is to examine two analytical methods for the design sensitivity analysis of optimal control problems and to develop the analyses necessary for the application of the methods to thermal systems. The two methods are the direct differentiation method and the adjoint variable method. Results for these methods are compared with those obtained using central finite difference approximations for design sensitivity analysis. The adjoint variable method has the potential to reduce significantly computational times for design sensitivity analysis while sacrificing only a small amount of accuracy. As the number of functional requiring design sensitivity analysis increases, however, the adjoint variable method becomes less attractive and the direct differentiation method becomes more attractive.

Three methods for design sensitivity analysis such as finite difference method(FDM), direct differentiation method(DDM) and adjoint variable method(AVM) are well known. FDM and DDM for design sensitivity analysis cost too much when the number of design variables is too large. An AVM is required to compute adjoint variables from the simultaneous linear system equation, the so-called adjoint equation. Because the adjoint equation is independent of the number of design variables, an AVM is efficient for when number of design variables is too large. In this study, AVM has been extended to the eigenproblem of damped systems whose eigenvlaues and eigenvectors are complex numbers. Moreover, this method is implemented into a commercial finite element analysis program by means of the semi-analytical method to show applicability of the developed method into practical structural problems. The proposed method is compared with FDM and verified its accuracy for analytical and practical cases.

Adjoint design sensitivity analysis and optimization of nonlinear structures using geometrical mapping approach

An incremental finite element formulation for structural sensitivity analysis or nonlinear (possibly path dependent) systems with fixed overall shape is proposed. Both the direct differentiation and adjoint variable methods, as well as a mixed differentiation-adjoint technique are employed to evaluate 1st- and 2nd-order sensitivity increments during the load step. Numerical illustration is given.

Design sensitivity analysis of dynamic response of nonviscously damped systems

A continuum-based design sensitivity analysis method for hyperelastic structures is presented. Analysis is performed using a meshless method, called the reproducing kernel particle method. Yeoh’ s energy density function is used to describe thehyperelastic structural behavior. The meshless method eliminates mesh distortion orentanglement encountered in using e nite element analysis for large deformation structural analysis and structural shape design optimization. Both the adjoint variable and direct differentiation methods are developed for material and shape design variables. An ine nitely long rubber tube, a two-dimensional rubber band, and an engine mount are examples used to demonstrate the feasibility and accuracy of the method.

We investigate a natural Lie group structure that correlations possess, interpreted as a special case of the quotient manifold structure for correlation matrices. The one‐dimensional setting of the general theory gives rise to additional structure in the form of a natural associative multiplication of correlations making the space of correlations into a Lie group. We explicitly compute left‐invariant vector fields, exponential and logarithmic mappings, and ultimately, a closed‐form distance formula between correlations. The Lie group formalism is then applied to an application in anomalous correlation detection. The advantage of the Lie group method illustrated with time series data of stock prices.

In this paper, the mean-centered first-order second-moment (FOSM) method is employed to perform probabilistic push-over analysis (POA) of structural and/or soil-structure systems. Approximations of first and second statistical moments (FSSMs) of engineering demand parameters (EDPs) of structural and/or geotechnical systems with random material parameters are computed based on finite-element (FE) response and response sensitivity analysis (RSA) results. The FE RSA is performed accurately and efficiently by using the direct differentiation method (DDM) and is employed to evaluate the relative importance (RI) of the various modeling material parameters in influencing the variability of the EDPs. The proposed approximate methodology is illustrated through probabilistic POA results for nonlinear inelastic FE models of: (1) a three-story reinforced-concrete (RC) frame building and (2) a soil-foundation-structure interaction system consisting of a RC frame structure founded on layered soil. FSSMs of EDPs computed through the FOSM method are compared with the corresponding accurate estimates obtained via Monte Carlo simulation. Results obtained from exact (or local) and averaged (or global) response sensitivities are also compared. The RI of the material parameters describing the systems is studied in both the deterministic and probabilistic sense, and presented in the form of tornado diagrams. Effects of statistical correlation between material parameters are also considered and analyzed by the FOSM method. A simple approximation of the probability density function and cumulative distribution function of EDPs due to a single random parameter at a time (while all the other parameters are fixed to their mean values) is also proposed. Conclusions are drawn on both the appropriateness of using local RSA for simplified probabilistic POA and on the application limits of the FOSM method. It is observed that the FOSM method combined with the DDM provides accurate estimates of FSSMs of EDPs for low-to-moderate level of inelastic structural or system behavior and useful qualitative information on the RI ranking of material parameters on the structural or system response for high level of inelastic behavior.

Direct differentiation method for response sensitivity analysis of a bounding surface plasticity soil model

Design sensitivity analysis of sheet metal forming processes with a direct differentiation method

Sensitivity analysis of the dynamic response of a frame. Part I: Direct differentiation method

In this paper we define ``embedded optimal control problems" which prescribe parametrized families of well defined associated optimal control problems. We show that the extremal generating Hamiltonian equations for an embedded optimal control problem and any associated optimal control problem are simply related by a projection. Furthermore normal extremals project to normal extremals and similarly for abnormal extremals. An interesting class of embedded optimal control problems consists of Clebsch optimal control problems. We provide necessary conditions for a Clebsch optimal control problem to describe a variational problem and thereby a mechanical system. There may be many advantages to analyzing an embedded optimal control problem instead of a particular associated optimal control problem, for example the former being defined on a linear space and the latter on a nonlinear space. The continuous analysis is paralleled by a similar discrete analysis. We define a discrete embedded/Clebsch optimal control problem along with associated discrete optimal control problems and we show results that are analogous to the continuous results. We apply the theory, both in the continuous and the discrete setting, to two example systems: mechanical systems on matrix Lie groups and mechanical systems on $n$-spheres.

Parameters affecting laterally loaded piles in frozen soils by an efficient sensitivity analysis method

SummaryResponse sensitivity is an essential component to understanding the complexity of material and geometric nonlinear finite element formulations of structural response. The direct differentiation method (DDM), a versatile approach to computing response sensitivity, requires differentiation of the equations that govern the state determination of an element and it produces accurate and efficient results. The DDM is applied to a force‐based element formulation that utilizes curvature‐shear‐based displacement interpolation (CSBDI) in its state determination for material and geometric nonlinearity in the basic system of the element. The response sensitivity equations are verified against finite difference computations, and a detailed example shows the effect of parameters that control flexure–shear interaction for a stress resultant plasticity model. The developed equations make the CSBDI force‐based element available for gradient‐based applications such as reliability and optimization where efficient computation of response sensitivities is necessary for convergence of gradient‐based search algorithms. Copyright © 2016 John Wiley & Sons, Ltd.

This paper describes a highly flexible approach to real-time frame-rate tracking in complex camera and structures configurations, including the use of multiple cameras and the tracking of multiple or articulated targets. A powerful and general method is presented for expressing and solving the constraints which exist in these configurations in a principled manner. This method exploits the geometric structure present in the Lie group and Lie algebra formalism to express the constraints that derive from structures such as hinges or a common ground plane. This method makes use of the adjoint representation to simplify the constraints which are then applied by means of Lagrange multipliers.KeywordsVisual ServoingMultiple CameraRigid ComponentEuclidean TransformationVideo FeedThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.