Semi-analytical sensitivity analysis for nonlinear transient problems

Efficient analytical sensitivity computations are essential elements of gradient-based optimization schemes; unfortunately, they can be difficult to implement. This implementation issue is often resolved by adopting the semi-analytical method which exhibits the efficiency of the analytical methods and the ease of implementation of the finite difference method. However, care must be taken as semi-analytical sensitivities may exhibit errors due to truncation and round-off. Additional errors are introduced if the convergence tolerance of the primal analysis is not sufficiently small. This paper gives a general overview and some new developments of the analytical and semi-analytical sensitivity analyses for nonlinear steady-state, transient, and dynamic problems. We discuss the restrictive assumptions, accuracy, and consistency of these methods. Both adjoint and direct differentiation methods are studied. Numerical examples are provided.     

Structural design sensitivity using boundary elements and polynomial response function

This main issue of this paper is a conjunction of the structural design sensitivity analysis using the Boundary Element Method with the polynomial response function determination. The procedure is so general that it enables sensitivity analysis for potential and elasticity problems within both homogeneous and heterogeneous plane and 3D problems. The essential difference with respect to the previous approaches like the Direct Differentiation Method or the Adjoint Variable Method is in discrete evaluation of the structural response using the response polynomials of some state parameters and design variable as the independent parameter. Such a determination is carried out via the several solutions of the given boundary value problem, where design parameter mean value is regularly perturbed in each of the solutions to cover the closest neighborhood of this mean value. Those few solutions make it possible to recover the polynomial response function from node-to node within the boundary elements, so that further symbolic differentiation using MAPLE returns the sensitivity gradients particular values. The entire procedure is tested here twice—first example deals with the homogeneous cantilever beam, where comparison against pure analytical differentiation is done and, separately, for two-component composite cantilever, where such a comparison is made against the central difference method linked with the same BEM solution.     

Second-order shape sensitivity analysis for nonlinear problems

First- and second-order shape sensitivity analyses in a fully nonlinear framework are presented in this paper. Using the fixed domain technique and the adjoint approach, integral expressions over the domain are obtained. The Guillaume-Masmoudi lemma allows these expressions to be rewritten as integrals over the domain boundary. The formalism is then applied to the steady creep of a bar in torsion, as an example of power-law nonlinearities that occur not only in creep problems but also in viscoplastic fluid flow. Finally, a problem with known analytical solution is presented in order to show the equivalence between exact differentiation and the shape sensitivity approach.     

Boundary element analysis of nonlinear transient heat conduction problems

This paper is concerned with a boundary element formulation and its numerical implementation for the nonlinear transient heat conduction problems with temperature-dependent material properties. By using the Kirchhoff transformation for the material properties a set of pseudo-linear integral equations is obtained in space and time for the fully three-dimensional nonlinear problems under consideration. The resulting boundary integral equations are solved by means of the usual boundary element method. Emphasis is placed on the numerical solution procedure employing constant elements with respect to time. It is shown that in this case there is no need to evaluate the domain integrals resulting from the nonlinearity of the problem. Finally, the powerful usefulness of the proposed method is demonstrated through the numerical computation of several sample problems.     

基于逆阵更新方法的局部非线性结构频响分析

针对含有非线性连接的大型局部非线性结构,采用描述函数表示其所连接的非线性内力,将非线性结构频响表示为拟线性动柔度矩阵(Quasilinear Receptance Matrix),提出一种逆阵更新(Inverse Matrix Updating Method,IMU)方法,将求解系统动柔度(频响特性)的高阶矩阵求逆转化为低阶矩阵的求逆,从而获得大型局部非线性结构主频响应的快速计算方法.仿真结果表明,本文的分析方法具有较好的稳定性,并能大幅提高大型局部非线性结构主频响应的计算效率.     

An operator splitting-radial basis function method for the solution of transient nonlinear Poisson problems

This paper presents an operator splitting-radial basis function (OS-RBF) method as a generic solution procedure for transient nonlinear Poisson problems by combining the concepts of operator splitting, radial basis function interpolation, particular solutions, and the method of fundamental solutions. The application of the operator splitting permits the isolation of the nonlinear part of the equation that is solved by explicit Adams-Bashforth time marching for half the time step. This leaves a nonhomogeneous, modified Helmholtz type of differential equation for the elliptic part of the operator to be solved at each time step. The resulting equation is solved by an approximate particular solution and by using the method of fundamental solution for the fitting of the boundary conditions. Radial basis functions are used to construct approximate particular solutions, and a grid-free, dimension-independent method with high computational efficiency is obtained. This method is demonstrated for some prototypical nonlinear Poisson problems in heat and mass transfer and for a problem of transient convection with diffusion. The results obtained by the OS-RBF method compare very well with those obtained by other traditional techniques that are computationally more expensive. The new OS-RBF method is useful for both general (irregular) two- and three-dimensional geometry and provides a mesh-free technique with many mathematical flexibilities, and can be used in a variety of engineering applications.     

Estimation of individual prediction reliability using the local sensitivity analysis

For a given prediction model, some predictions may be reliable while others may be unreliable. The average accuracy of the system cannot provide the reliability estimate for a single particular prediction. The measure of individual prediction reliability can be important information in risk-sensitive applications of machine learning (e.g. medicine, engineering, business). We define empirical measures for estimation of prediction accuracy in regression. Presented measures are based on sensitivity analysis of regression models. They estimate reliability for each individual regression prediction in contrast to the average prediction reliability of the given regression model. We study the empirical sensitivity properties of five regression models (linear regression, locally weighted regression, regression trees, neural networks, and support vector machines) and the relation between reliability measures and distribution of learning examples with prediction errors for all five regression models. We show that the suggested methodology is appropriate only for the three studied models: regression trees, neural networks, and support vector machines, and test the proposed estimates with these three models. The results of our experiments on 48 data sets indicate significant correlations of the proposed measures with the prediction error.     

Structural shape sensitivity analysis for nonlinear material models with strain softening

This paper describes some considerations around the analytical structural shape sensitivity analysis when the structural behaviour is computed using the finite element method with a nonlinear constitutive material model. Traditionally, the structural sensitivity analysis is computed using an incremental approach based on the incremental procedures for the solution of the structural equilibrium problem. In this work, a direct (nonincremental) formulation for computing these structural sensitivities, that is valid for some specific nonlinear material models, is proposed. The material models for which the presented approach is valid are characterized by the fact that the stresses at any timet can be expressed in terms of the strains at the timet and, in some cases, the strains at a specific past timet ^u (t ^u <t). This is the case of elasticity (linear as well as nonlinear), perfect plasticity and damage models. A special strategy is also proposed for material models with strain softening.For the cases where it is applicable, the sensitivity analysis proposed here allows us to compute the structural sensitivities around any structural equilibrium point after finishing the solution process and it is completely independent of the numerical scheme used to solve the structural equilibrium problem. This possibility is particularized for the case of a damage model considering a strain-softening behaviour. Finally, the quality and reliability of the proposed approach is assessed through its application to some examples.     

Design sensitivity analysis for transient response of non-viscously damped dynamic systems

A design sensitivity analysis for the transient response of the non-viscously damped dynamic systems is presented. The non-viscously (viscoelastically) damped system is widely used in structural vibration control. The damping forces in the system depend on the past history of motion via convolution integrals. The non-viscos damping is modeled by the generalized Maxwell model. The transient response is calculated with the implicit Newmark time integration scheme. The design sensitivity analysis method of the history dependent system is developed using the adjoint variable method. The discretize-then-differentiate approach is adopted for deriving discrete adjoint equations. The accuracy and the consistency of the proposed method are demonstrated through a single dof system. The proposed method is also applied to a multi-dof system. The validity and accuracy of the sensitivities from the proposed method are confirmed by finite difference results.     

Reduced basis technique for evaluating the sensitivity of the nonlinear vibrational response of composite plates

A reduced basis technique and a computational procedure are presented for generating the nonlinear vibrational response, and evaluating the first-order sensitivity coefficients of composite plates (derivatives of the nonlinear frequency with respect to material and geometric parameters of the plate). The analytical formulation is based on a form of the geometrically nonlinear shallow shell theory with the effects of transverse shear deformation, rotatory inertia and anisotropic material behavior included. The plate is discretized by using mixed finite element models with the fundamental unknowns consisting of both the nodal displacements and the stress-resultant parameters of the plate. The computational procedure can be conveniently divided into three distinct steps. The first step involves the generation of various-order perturbation vectors, and their derivatives with respect to the material and lamination parameters of the plate, using Linstedt-Poincaré perturbation technique. The second step consists of using the perturbation vectors as basis vectors, computing the amplitudes of these vectors and the nonlinear frequency of vibration, via a direct variational procedure. The third step consists of using the perturbation vectors, and their derivatives, as basis vectors and computing the sensitivity coefficients of the nonlinear frequency via a second application of the direct variational procedure. The effectiveness of the proposed technique is demonstrated by means of numerical examples of composite plates.     

Updating response sensitivity models of nonlinear vibrating structures using particle filters

The problem of updating response gradients with respect to chosen system parameters based on spatially sparse measurements is considered. The measurement noise and imperfections in mathematical modeling are treated as Gaussian white noise processes. The system states are augmented by response gradients with respect to system parameters and an extended set of equations in the state space is formulated. These equations are cast in the form of Ito’s stochastic differential equations and measured data are assimilated into this model using Monte Carlo based Bayesian filtering tools. Illustrative examples include a few low dimensional dynamical systems with cubic and hereditary nonlinearities.     

Design sensitivity analysis of nonlinear dynamic response of structural and mechanical systems

This paper describes a unified variational theory for design sensitivity analysis of nonlinear dynamic response of structural and mechanical systems for shape, nonshape, material and mechanical properties selection, as well as control problems. The concept of an adjoint system, the principle of virtual work and a Lagrangian-Eulerian formulation to describe the deformations and the design variations are used to develop a unified view point. A general formula for design sensitivity analysis is derived and interpreted for usual performance functionals. Analytical examples are utilized to demonstrate the use of the theory and give insights for application to more complex problems that must be treated numerically.Derivatives The comma notation for partial derivatives is used, i.e. G,_u = G/u. An upper dot represents material time derivative, i.e. ü = ²u/t². A prime implies derivative with respect to the time measured in the reference time-domain, i.e. u = du/d.     

Energy conservation in the transient response of nonlinear beam vibration problems subjected to pulse loading—a numerical approach

The nonlinear vibration response of a double cantilevered beam subjected to pulse loading over a central sector is studied. The initial response is generated in detail to ascertain the energetics of the response. The total energy is used as a gauge of the stability and accuracy of the solution. It is shown that to obtain accurate and stable initial solutions an extremely high spatial and time resolution is required. This requirement was only evident through an examination of the energy of the system. It is proposed, therefore, to use the total energy of the system as a necessary stability and accuracy criterion for the nonlinear response of conservative systems. The results also demonstrate that even for moderate nonlinearities, the effects of membrane forces have a significant influence on the system. It is also shown that while the fundamental response is contained in a first mode envelope, the fluctuations caused by the higher order modes must be resolved.     

On the numerical implementation of continuous adjoint sensitivity for transient heat conduction problems using an isogeometric approach

A crucial problem of continuous adjoint shape sensitivity analysis is the numerical implementation of its lengthy formulations. In this paper, the numerical implementation of continuous adjoint shape sensitivity analysis is presented for transient heat conduction problems using isogeometric analysis, which can serve as a tutorial guide for beginners. Using the adjoint boundary and loading conditions derived from the design objective and the primary state variable fields, the numerical analysis procedure of the adjoint problem, which is solved backward in time, is demonstrated. Following that, the numerical integration algorithm of the shape sensitivity using a boundary approach is provided. Adjoint shape sensitivity is studied with detailed explanations for two transient heat conduction problems to illustrate the numerical implementation aspects of the continuous adjoint method. These two problems can be used as benchmark problems for future studies.     

Dynamic response analysis of nonlinear structures using step-by-step integration techniques

The paper describes step-by-step integration techniques to predict the dynamic response of nonlinear tension structures. The analytical procedure and the mathematical formulation of selected potential algorithms used for the solution of the equation of motion are presented. Methods of introducing the wind and earthquake loading into the step-by-step method are described. The response analysis of vibrations of a flat pretensioned cable net excited by simple harmonic loading and by a falling load are also presented, followed by a numerical example of an experimental portal frame solved by the linear acceleration method.     

Dynamic analysis of a nonlinear pressure regulator using bondgraph simulation technique

The dynamics of a direct pressure regulator valve have been studied through bondgraph simulation technique. The governing equations of the system have been derived from the obtained model. While solving the system equations numerically, various pressure-flow characteristics across the regulator ports and the orifices are taken into consideration. The simulation study identifies some critical parameters, which have significant effect on the transient response of the system. The simulation results are determined using the MATLAB-SIMULINK environment. The main novelty of this work is to present an analytic solution in analyzing a nonlinear complex system with interaction of several energy domains. The other conventional attempts employed before this solution resulted to inaccurate simulation results which can not predict the dynamical response of the system. Numerical results implied to the good accuracy of the bondgraph study, while comparing with experimental results.     

Defining the transfer function of a plant using experimental transient response based on the scaling principle

In this paper, we describe an efficient method of defining the transfer function of a plant using experimental data on its transient response. The method involves the scaling principle. An example illustrates the method and its relevant advantages in applications. Among such advantages, simplicity and controlled accuracy of the results should be mentioned. Finally, we present a new computer program, MM-approximation which assists practitioners in fast and accurate definition of the transfer function using the available transient response of a plant.     

Topological sensitivity analysis in large deformation problems

The aim of the present work is to apply the topological sensitivity analysis (TSA) to large-deformation elasticity based on the total Lagrangian formulation. The TSA results in a scalar function, denominated topological derivative, that gives for each point of the domain the sensitivity of a given cost function when a small hole is created. An approximated expression for the topological derivative is obtained by numerical asymptotic analysis. Numerical results of the presented approach are considered for elastic plane problems.