On third-order radiation theory for axisymmetric bodies undergoing heave motion of multiple frequencies

Summary As a first step in the development of a nonlinear theory for calculating the response of an axisymmetric system to an irregular sea, the radiation problem for axisymmetric floating or immersed bodies in a periodic heave motion, composed of a number of harmonic components, is considered by means of a third-order potential theory.It is shown that the knowledge of only first- and second-order potential functions is required for the calculation of all forces up to the third order. A boundary integral equation method is proposed for the determination of these potential functions.     

Second-order design sensitivity analysis of mechanical system dynamics

Dependence of dynamic response of nonlinear mechanical systems on design variables is analysed. An adjoint variable method is used to derive first- and second-order derivatives of measures of dynamic response with respect to design variables. A computational algorithm is presented for numerical calculation of first and second design derivatives. A simple oscillator example is solved analytically and by the adjoint variable method, with identical results. A burst fire automatic weapon mechanism with linear and nonlinear damping is treated numerically. It is shown that quadratic appriximations of dynamic response, using results of second-order design sensitivity analysis, can be substantially better than conventional linear approximations.     

Subcycling first- and second-order generalizations of the trapezoidal rule

Subcycling algorithms which employ multiple timesteps have been previously proposed for explicit direct integration of first- and second-order systems of equations arising in finite element analysis, as well as for integration using explicit/implicit partitions of a model. The author has recently extended this work to implicit/implicit multi-timestep partitions of both first- and second-order systems. In this paper, improved algorithms for multi-timestep implicit integration are introduced, that overcome some weaknesses of those proposed previously. In particular, in the second-order case, improved stability is obtained. Some of the energy conservation properties of the Newmark family of algorithms are shown to be preserved in the new multi-timestep extensions of the Newmark method. In the first-order case, the generalized trapezoidal rule is extended to multiple timesteps, in a simple way that permits an implicit/implicit partition. Explicit special cases of the present algorithms exist. These are compared to algorithms proposed previously. © 1998 John Wiley & Sons, Ltd.     

A dimensional decomposition method for stochastic fracture mechanics

This paper presents a new dimensional decomposition method for obtaining probabilistic characteristics of crack-driving forces and reliability analysis of general cracked structures subject to random loads, material properties, and crack geometry. The method involves a novel function decomposition permitting lower-variate approximations of a crack-driving force or a performance function, Lagrange interpolations for representing lower-variate component functions, and Monte Carlo simulation. The effort required by the proposed method can be viewed as performing deterministic fracture analyses at selected input defined by sample points. Compared with commonly-used first- and second-order reliability methods, no derivatives of fracture response are required by the new method developed. Results of three numerical examples involving both linear-elastic and nonlinear fracture mechanics of cracked structures indicate that the decomposition method provides accurate and computationally efficient estimates of probability density of the J-integral and probability of fracture initiation for various cases including material gradation characteristics and magnitudes of applied stresses and loads.     

Recursive approach for random response analysis using non-orthogonal polynomial expansion

Using non-orthogonal polynomial expansions, a recursive approach is proposed for the random response analysis of structures under static loads involving random properties of materials, external loads, and structural geometries. In the present formulation, non-orthogonal polynomial expansions are utilized to express the unknown responses of random structural systems. Combining the high-order perturbation techniques and finite element method, a series of deterministic recursive equations is set up. The solutions of the recursive equations can be explicitly expressed through the adoption of special mathematical operators. Furthermore, the Galerkin method is utilized to modify the obtained coefficients for enhancing the convergence rate of computational outputs. In the post-processing of results, the first- and second-order statistical moments can be quickly obtained using the relationship matrix between the orthogonal and the non-orthogonal polynomials. Two linear static problems and a geometrical nonlinear problem are investigated as numerical examples in order to illustrate the performance of the proposed method. Computational results show that the proposed method speeds up the convergence rate and has the same accuracy as the spectral finite element method at a much lower computational cost, also, a comparison with the stochastic reduced basis method shows that the new method is effective for dealing with complex random problems.     

Laplace变换在非线性系统中的应用

本文通过提出计算函数乘积的拉氏变换的新算法─—Ｓ积＊运算，使一维拉氏变换解决非线性系统问题成为可能。通过此方法和Ｖｏｌｔｅｒｒａ方法求得的一阶、二阶非线性系统的传递函数和脉冲响应解析式是相同的。但计算更为方便。     

A univariate dimension-reduction method for multi-dimensional integration in stochastic mechanics

This paper presents a new, univariate dimension-reduction method for calculating statistical moments of response of mechanical systems subject to uncertainties in loads, material properties, and geometry. The method involves an additive decomposition of a multi-dimensional response function into multiple one-dimensional functions, an approximation of response moments by moments of single random variables, and a moment-based quadrature rule for numerical integration. The resultant moment equations entail evaluating N number of one-dimensional integrals, which is substantially simpler and more efficient than performing one N-dimensional integration. The proposed method neither requires the calculation of partial derivatives of response, nor the inversion of random matrices, as compared with commonly used Taylor expansion/perturbation methods and Neumann expansion methods, respectively. Nine numerical examples involving elementary mathematical functions and solid-mechanics problems illustrate the proposed method. Results indicate that the univariate dimension-reduction method provides more accurate estimates of statistical moments or multidimensional integration than first- and second-order Taylor expansion methods, the second-order polynomial chaos expansion method, the second-order Neumann expansion method, statistically equivalent solutions, the quasi-Monte Carlo simulation, and the point estimate method. While the accuracy of the univariate dimension-reduction method is comparable to that of the fourth-order Neumann expansion, a comparison of CPU time suggests that the former is computationally far more efficient than the latter.     

AC network state estimation using linear measurement functions

The real/reactive power and current magnitude measurements can be accounted for in an AC network state estimator using linear measurement functions. The nonlinearity in the conventional AC state estimator equations is transferred from the measurement functions into a system of nonlinear equality constraints which is independent of the measurement set. The new format of equations entails two advantages. First, it can be easily integrated in optimisation routines which employ first- and second-order derivative functions. Second, the linear measurement functions can benefit from scaling techniques which are well documented in the linear programming literature. This research details the implementation of a least absolute value state estimator employing the new format of equations. The optimisation method is based on a primal-dual interior-point method that can accurately account for zero injection measurements and power directions. Numerical testing is used to validate the approach for networks with measurement sets that are (i) conventional and (ii) have a high proportion of current magnitude measurements and power signs.     

时域稳定的基础频响连续有理近似参数识别方法EI北大核心CSCD

采用连续有理近似函数描述基础时域动力模型是土-结动力相互作用时程分析的重要方法。该近似函数的稳定性和精度决定了土-结相互作用系统时域分析的稳定性和精度。目前,连续时间有理近似函数的参数识别方法无法同时确保精度、稳定性和计算效率。该文基于线性系统控制理论,将有理近似函数分解为一系列一阶和二阶子系统的组合,并通过各子系统的稳定条件提出了被识别参数的理论稳定边界。基于此,利用遗传算法和序列二次规划算法建立了时域稳定的连续有理函数参数识别方法。不同基础频响函数数值仿真结果表明:对于简单函数,该文方法与既有方法均可保持高精度,但该文方法在确保稳定性的同时可以提高近35%的计算效率。对于复杂函数,采用高阶有理函数时,该文方法用时仅为既有方法的25%,同时精度大于95%。证明了该方法可以在保证函数稳定性与拟合精度的同时提高了计算效率,从而使得连续有理近似函数适用性更好。     

A comparison of approximate response functions in structural reliability analysis

In order to reduce computational costs in structural reliability analysis, it has been suggested to utilize approximate response functions for reliability assessment. One well-established class of methods to deal with suitable approximations is the Response Surface Method. The basic idea in utilizing the response surface method is to replace the true limit state function by an approximation, the so-called response surface, whose function values can be computed more easily. The functions are typically chosen to be first- or second-order polynomials. Higher-order polynomials on the one hand tend to show severe oscillations, and on the other hand they require too many support points. This may be overcome by applying smoothing techniques such as the moving least-squares method. An alternative approach is given by Artificial Neural Networks. In this approach, the input and output parameters are related by means of relatively simple yet flexible functions, such as linear, step, or sigmoid functions which are combined by adjustable weights. The main feature of this approach lies in the possibility of adapting the input–output relations very efficiently. A further possibility lies in the utilization of radial basis functions. This method also allows for a flexible adjustment of the interpolation scheme. In all approaches as presented it is essential to achieve high quality of approximation primarily in the region of the random variable space which contributes most significantly to the probability of failure. The paper presents an overview of these approximation methods and demonstrates their potential by application to several examples of nonlinear structural analysis.     

Improved dynamic analysis of structures with mechanical uncertainties under deterministic input

This paper addresses the dynamic analysis of linear systems with uncertain parameters subjected to deterministic excitation. The conventional methods dealing with stochastic structures are based on series expansion of stochastic quantities with respect to uncertain parameters, by means of either Taylor expansion, perturbation technique or Neumann expansion and evaluate the first- and second-order moments of the response by solving deterministic equations. Unfortunately, these methods lead to significant error when the coefficients of variation of uncertainties are relatively large. Herein, an improved first-order perturbation approach is proposed, which considers the stochastic quantities as the sum of their mean and deviation. Comparisons with conventional second-order perturbation approach and Monte Carlo simulations illustrate the efficiency of the proposed method. Applications are discussed in order to investigate the influence of mass, damping and stiffness uncertainty on the dynamic response of the system.     

Modeling of finite-amplitude sound beams: second order fields generated by a parametric loudspeaker

The nonlinear interaction of sound waves in air has been applied to sound reproduction for audio applications. A directional audible sound can be generated by amplitude-modulating the ultrasound carrier with an audio signal, then transmitting it from a parametric loudspeaker. This brings the need of a computationally efficient model to describe the propagation of finite-amplitude sound beams for the system design and optimization. A quasilinear analytical solution capable of fast numerical evaluation is presented for the second-order fields of the sum-, difference-frequency and second harmonic components. It is based on a virtual-complex-source approach, wherein the source field is treated as an aggregation of a set of complex virtual sources located in complex distance, then the corresponding fundamental sound field is reduced to the computation of sums of simple functions by exploiting the integrability of Gaussian functions. By this result, the five-dimensional integral expressions for the second-order sound fields are simplified to one-dimensional integrals. Furthermore, a substantial analytical reduction to sums of single integrals also is derived for an arbitrary source distribution when the basis functions are expressible as a sum of products of trigonometric functions. The validity of the proposed method is confirmed by a comparison of numerical results with experimental data previously published for the rectangular ultrasonic transducer.     

Interaction of first- and second-order direction in motion-defined motion

Motion-defined motion can play a special role in the discussion of whether one or two separate systems are required to process first- and second-order information because, in contrast to other second-order stimuli, such as contrast-modulated contours, motion detection cannot be explained by a simple input nonlinearity but requires preprocessing by motion detectors. Furthermore, the perceptual quality that defines an object (motion on the object surface) is identical to that which is attributed to the object as an emergent feature (motion of the object), raising the question of how these two object properties are linked. The interaction of first- and second-order information in such stimuli has been analyzed previously in a direction-discrimination task, revealing some cooperativity. Because any comprehensive integration of these two types of motion information should be reflected in the most fundamental property of a moving object, i.e., the direction in which it moves, we now investigate how motion direction is estimated in motion-defined objects. Observers had to report the direction of moving objects that were defined by luminance contrast or in random-dot kinematograms by differences in the spatiotemporal properties between the object region and the random-noise background. When the dots were moving coherently with the object (Fourier motion), direction sensitivity resembled that for luminance-defined objects, but performance deteriorated when the dots in the object region were static (drift-balanced motion). When the dots on the object surface were moving diagonally relative to the object direction (theta motion), the general level of accuracy declined further, and the perceived direction was intermediate between the veridical object motion direction and the direction of dot motion, indicating that the first- and second-order velocity vectors are somehow pooled. The inability to separate first- and second-order directional information suggests that the two corresponding subsystems of motion processing are not producing independent percepts and provides clues for possible implementations of the two-layer motion-processing network.     

An effective evidence theory-based reliability analysis algorithm for structures with epistemic uncertainty

The purpose of this article is to develop an effective method to evaluate the reliability of structures with epistemic uncertainty so as to improve the applicability of evidence theory in practical engineering problems. The main contribution of this article is to establish an approximate semianalytic algorithm, which replaces the process of solving the extreme value of performance function and greatly improve the efficiency of solving the belief measure and the plausibility measure. First, the performance function is decomposed as a combination of a series of univariate functions. Second, each univariate function is approximated as a unary quadratic function by the second-order Taylor expansion. Finally, based on the property of the unary quadratic function, the maximum and minimum values of each univariate function are solved, and then the maximum and minimum values of performance function are obtained according to the monotonic relationship between each univariate function and their combination. As long as the first- and second-order partial derivatives of the performance function with respect to each input variable are obtained, the belief measure and plausibility measure of the structure can be estimated effectively without any additional computational cost. Two numerical examples and one engineering application are investigated to demonstrate the accuracy and efficiency of the proposed method.     

Nonlinear predictors and hybrid corrector for fast continuation power flow

Continuation power flow is a powerful tool to simulate power system steady-state stationary behaviours with respect to a given power injection variation scenario. Although continuation power flow methods have been implemented in several commercial packages, they may be still too slow for online applications. The authors aim to improve the continuation power flow methods, mainly their speed and, to a less extent, their reliability. Nonlinear predictors are developed based on the polynomial interpolations. The authors' numerical studies show that continuation power flow with the proposed nonlinear predictors can be much faster than that with traditional linear predictors such as tangent or secant predictors. Of the nonlinear predictors, second-order polynomial approximation-based and third-order-based nonlinear predictors show their superior performance in speed. Continuation power flow with second-order nonlinear predictors is generally slightly faster than that with third-order nonlinear predictors. In addition, a hybrid corrector is developed and incorporated into continuation power flow. It is numerically shown on several test systems ranging from 118-bus to 1648-bus that continuation power flow with the proposed hybrid corrector can be much faster than that with traditional correctors such as the Newton method and the fast decoupled method. Finally, an improved continuation power flow with the developed nonlinear predictor and hybrid corrector is presented and evaluated.     

Second-order sensitivity analysis in vibration and buckling problems

A perturbation approach for the first- and second-order sensitivity analysis of eigenvalue problems, as they arise in buckling and vibration of structural systems, is presented. A particular feature introduced in the formulation is the dependence of all matrices that model the problem not only on the design parameter, but also on a pre-critical state. Thus, the sensitivity of the response of the fundamental path in buckling problems, or a pre-stressed state, in vibration problems, are introduced in the analysis. Two techniques are considered in this work: the direct and the adjoint methods. First- and second-order sensitivity equations for eigenvalues and eigenvectors are obtained, and it is shown that second-order sensitivity of an eigenvalue can be accomplished by using only one adjoint problem.     

Assessment of the efficiency of Kriging surrogate models for structural reliability analysis

This paper presents an assessment of the efficiency of the Kriging interpolation models as surrogate models for structural reliability problems involving time-consuming numerical models such as nonlinear finite element analysis structural models. The efficiency assessment is performed through a systematic comparison of the accuracy of the failure probability predictions based on the first-order reliability method using the most common first- and second-order polynomial regression models and the Kriging interpolation models as surrogates for the true limit state function. An application problem of practical importance in the field of marine structures that requires the evaluation of a nonlinear finite element structural model is adopted as numerical example. The accuracy of the failure probability predictions is characterised as a function of the number of support points, dispersion of the support points in relation to the so-called design point and order of the Kriging basis functions. It is shown with the application problem considered that the Kriging interpolation models are efficient surrogate models for structural reliability problems and can provide significantly more accurate failure probability predictions as compared with the most common polynomial regression models.     

Three-systems theory of human visual motion perception: review and update

Lu and Sperling [Vision Res. 35, 2697 (1995)] proposed that human visual motion perception is served by three separate motion systems: a first-order system that responds to moving luminance patterns, a second-order system that responds to moving modulations of feature types-stimuli in which the expected luminance is the same everywhere but an area of higher contrast or of flicker moves, and a third-order system that computes the motion of marked locations in a "salience map," that is, a neural representation of visual space in which the locations of important visual features ("figure") are marked and "ground" is unmarked. Subsequently, there have been some strongly confirmatory reports: different gain-control mechanisms for first- and second-order motion, selective impairment of first- versus second- and/or third-order motion by different brain injuries, and the classification of new third-order motions, e.g., isoluminant chromatic motion. Various procedures have successfully discriminated between second- and third-order motion (when first-order motion is excluded): dual tasks, second-order reversed phi, motion competition, and selective adaptation. Meanwhile, eight apparent contradictions to the three-systems theory have been proposed. A review and reanalysis here of the new evidence, pro and con, resolves the challenges and yields a more clearly defined and significantly strengthened theory.     

Optimization of large structures subjected to dynamic loads with the multiplier method

The multiplier method for optimization of large-scale mechanical and structural systems subjected to dynamic loads is investigated. A large-scale dynamic response optimization problem is formulated and solved in several alternate ways using the first- and second-order forms of the equations of motion. Results are compared with those obtained with the sequential quadratic programming algorithm—a primal method. In all the cases investigated, the multiplier algorithm is more efficient than the primal method. Therefore, it is concluded that the multiplier method is more appropriate for dynamic response optimization of large-scale problems.     

Comparison of two iteration procedures for a class of nonlinear jerk equations

A modified Michens iteration procedure and a direct Michens iteration procedure are applied to determine approximate periods and analytical approximate periodic solutions of a class of nonlinear jerk equations. When we use the modified Michens iteration procedure to deal with the nonlinear jerk equations, we need to solve the nonlinear algebra equation for determining the approximate angular frequency. The higher the number of iterations, the more difficult it is to deal with the nonlinear algebra equation. This is because the kth-order approximate solution obtained by the modified Michens iteration procedure is a function of the angular frequency. However, since the kth-order approximate solution obtained by the direct Michens iteration procedure is independent on the kth-order approximate angular frequency, the form of nonlinear algebra equation that determines the kth-order approximate angular frequency of nonlinear jerk equation is similar. The second-order approximate period obtained by the direct Michens iteration procedure provides very accurate results for the large initial velocity amplitudes. But the second-order approximate period obtained by the modified Michens iteration procedure is invalid for all amplitudes B of the initial velocity. A comparison of the first- and second-order approximate periodic solutions obtained by the direct Michens iteration procedure with the numerically exact solutions shows that the second-order approximate periodic solution is much more accurate than the first one. Thus, the direct Michens iteration procedure is very effective for the class of nonlinear jerk equations.