Probability density evolution method for dynamic response analysis of structures with uncertain parameters

Probability density evolution method is proposed for dynamic response analysis of structures with random parameters. In the present paper, a probability density evolution equation (PDEE) is derived according to the principle of preservation of probability. With the state equation expression, the PDEE is further reduced to a one-dimensional partial differential equation. The numerical algorithm is studied through combining the precise time integration method and the finite difference method with TVD schemes. The proposed method can provide the probability density function (PDF) and its evolution, rather than the second-order statistical quantities, of the stochastic responses. Numerical examples, including a SDOF system and an 8-story frame, are investigated. The results demonstrate that the proposed method is of high accuracy and efficiency. Some characteristics of the PDF and its evolution of the stochastic responses are observed. The PDFs evidence heavy variance against time. Usually, they are much irregular and far from well-known regular distribution types. Additionally, the coefficients of variation of the random parameters have significant influence on PDF and second-order statistical quantities of responses of the stochastic structure.The support of the Natural Science Funds for Distinguished Young Scholars of China (Grant No.59825105) and the Natural Science Funds for Innovative Research Groups of China (Grant No.50321803) are gratefully appreciated.     

随机结构反应概率密度演化分析的切球选点法

发展了随机结构反应概率密度演化分析中随机参数空间的切球选点法。密度演化方法是一类直接获取随机结构动力反应概率密度函数及其演化过程的有效方法。在多个随机变量时，随机变量空间中的离散代表点选点规则直接关系到密度演化方法的精度和效率。本文构造了平面内等半径相切圆圆心分布定位的算法，以此为基础，建立了三维空间中等半径相切球球心坐标定位的计算公式。从而给出随机变量空间中的离散代表点及其赋得概率。计算表明，基于空间切球法的选点规则具有良好的精度和效率，在2个和3个随机变量情况下是较为理想的选点方法。     

Strategy for selecting representative points via tangent spheres in the probability density evolution method

A strategy of selecting efficient integration points via tangent spheres in the probability density evolution method (PDEM) for response analysis of non‐linear stochastic structures is studied. The PDEM is capable of capturing instantaneous probability density function of the stochastic dynamic responses. The strategy of selecting representative points is of importance to the accuracy and efficiency of the PDEM. In the present paper, the centers of equivalent non‐overlapping tangent spheres are used as the basis to construct a representative point set. An affine transformation is then conducted and a hypersphere sieving is imposed for spherically symmetric distributions. Construction procedures of centers of the tangent spheres are elaborated. The features of the point sets via tangent spheres, including the discrepancy and projection ratio, are observed and compared with some other typical point sets. The investigations show that the discrepancies of the point sets via tangent spheres are in the same order of magnitude as the point sets by the number theoretical method. In addition, it is observed that rotation transformation could greatly improve the projection ratios. Numerical examples show that the proposed method is accurate and efficient for situations involving up to four random variables. Copyright © 2007 John Wiley & Sons, Ltd.     

Dynamic response and reliability analysis of structures with uncertain parameters

An original approach for dynamic response and reliability analysis of stochastic structures is proposed. The probability density evolution equation is established which implies that incremental rate of the probability density function is related to the structural response velocity. Therefore, the response analysis of stochastic structures becomes an initial‐value partial differential equation problem. For the dynamic reliability problem, the solution can be derived through solving the probability density evolution equation with an initial value condition and an absorbing boundary condition corresponding to specified failure criterion. The numerical algorithm for the proposed method is suggested by combining the precise time integration method and the finite difference method with TVD scheme. To verify and validate the proposed method, a SDOF system and an 8‐storey frame with random parameters are investigated in detail. In the SDOF system, the response obtained by the proposed method is compared with the counterparts by the exact solution. The responses and the reliabilities of a frame with random stiffness, subject to deterministic excitation or random excitation, are evaluated by the proposed method as well. The mean, the standard deviation and the reliabilities are compared, respectively, with the Monte Carlo simulation. The numerical examples verify that the proposed method is of high accuracy and efficiency. Moreover, it is found that the probability transition of structural responses is like water flowing in a river with many whirlpools, showing complexity of probability transition process of the stochastic dynamic responses. Copyright © 2004 John Wiley & Sons, Ltd.     

The dimension-reduction strategy via mapping for probability density evolution analysis of nonlinear stochastic systems

Dynamic response analysis of nonlinear structures involving random parameters has for a long time been an important and challenging problem. In recent years, the probability density evolution method, which is capable of capturing the instantaneous probability density function (PDF) of the dynamic response and its evolution, has been proposed and developed for nonlinear stochastic dynamical systems. In the probability density evolution method, the strategy of selecting representative points is of critical importance to the efficiency especially when the number of random parameters is large. Enlightened by Cantor’s set theory, a strategy of dimension-reduction via mapping is proposed in the present paper. In the strategy, a two-dimensional domain is firstly considered and discretized such that the grid points are assigned with probabilities associated to the joint PDF. These points are then sorted and set on a virtual line according to a certain principle. Partitioning the sorted points on the virtual line into a certain number of intervals and selecting one single point in each interval, the two random variables can be transformed to a single comprehensive random variable. The associated probability of each point is simultaneously transformed accordingly. In the case of multiple random parameters, the above dimension-reduction procedure from two to one could be used recursively such that the random vector is finally transformed to one single comprehensive random variable. Numerical examples are investigated, showing that the proposed method is of high efficiency and fair accuracy.     

Optimum design of stochastically excited non‐linear dynamic systems without geometric constraints

This paper presents an efficient procedure for min–max dynamic response optimization of stochastically excited non‐linear systems with multiple time‐delayed inputs. This procedure employs a stochastic linearization technique to overcome system non‐linearity and an auto‐covariance analysis technique to represent the original stochastic mechanical model in a suitable form for optimization. Special attention is given to the sensitivity analysis, due to the complex nature of the problem. Therefore, exact expressions are obtained in a simple form for the evaluation of the required gradients, which greatly improve the stability and efficiency of the optimization algorithm. The numerical results and performance are presented by means of solving two min–max dynamic response optimization problems. Copyright © 2002 John Wiley & Sons, Ltd.     

Partition of the probability-assigned space in probability density evolution analysis of nonlinear stochastic structures

Based on a partition of probability-assigned space, a strategy for determining the representative point set and the associated weights for use in the probability density evolution method (PDEM) is developed. The PDEM, which is capable of capturing the instantaneous probability density function of responses of linear and nonlinear stochastic systems, was developed in the past few years. The determination of the representative point set and the assigned probabilities is of paramount importance in this approach. In the present paper, a partition of probability-assigned space related to the representative points and the assigned probabilities are first examined. The error in the resulting probability density function of the stochastic responses is then analyzed, leading to two criteria on strategies for determining the representative points and a set of indices in terms of discrepancy of the point sets. A two-step algorithm is proposed, in which an initial uniformly scattered point set is mapped to an optimal set. The implementation of the algorithm is elaborated. Two methods for generating the initial point set are outlined. These are the lattice point sets and the Number-Theoretical nets. A density-related transformation yielding the final point set is then analyzed. Numerical examples are investigated, where the results are compared to those obtained from the standard Monte Carlo simulation and the Latin hyper-cube sampling, demonstrating the accuracy and efficiency of the proposed approach.     

Reliability of deterministic non‐linear systems subjected to stochastic dynamic excitation

Engineering structures react to exceptionally high forces caused by, for example, extreme winds, sea waves, earthquakes, avalanches, etc. in a non‐linear way, before they finally collapse. Mostly these environmental loadings cause dynamic excitations which are adequately modeled by the so‐called stochastic processes. To identify subsets of the excitation, which may trigger failure, methods based on power inputs of the stochastic excitation will be exploited. This procedure is based on the simple consideration that any excitation that maximizes the energy input into the system has the potential to adversely affect the integrity of the structure. This method considers the velocity of the displacement field of the structure and the energy dissipation induced by viscous damping, friction and hysteresis. For an efficient reliability estimation, the n‐dimensional standard normal space S^[0]∈??ⁿ, in which the stochastic excitation is modeled, is split into two disjunct subspaces S^[1]∈??^m and S^[2]∈??^n?m. The subset S^[1]∈??^m represents the space of important directions, which is identified by a procedure based on an approximation of the gradient of the energy input. Directional sampling in the subspace S^[1] and direct Monte Carlo sampling in the subspace S^[2] are combined to established an efficient estimator for the structural reliability. The proposed methodology is generally applicable to finite element models with strong non‐conservative non‐linearities. Copyright © 2010 John Wiley & Sons, Ltd.     

A time‐parallel implicit method for accelerating the solution of non‐linear structural dynamics problems

The parallel implicit time‐integration algorithm (PITA) is among a very limited number of time‐integrators that have been successfully applied to the time‐parallel solution of linear second‐order hyperbolic problems such as those encountered in structural dynamics. Time‐parallelism can be of paramount importance to fast computations, for example, when space‐parallelism is unfeasible as in problems with a relatively small number of degrees of freedom in general, and reduced‐order model applications in particular, or when reaching the fastest possible CPU time is desired and requires the exploitation of both space‐ and time‐parallelisms. This paper extends the previously developed PITA to the non‐linear case. It also demonstrates its application to the reduction of the time‐to‐solution on a Linux cluster of sample non‐linear structural dynamics problems. Copyright © 2008 John Wiley & Sons, Ltd.     

Complementarity framework for non‐linear dynamic analysis of skeletal structures with softening plastic hinges

In this paper, we describe an algorithm for the incremental state update of elasto‐plastic systems with softening. The algorithm uses a complementary pivoting technique and is based on casting the incremental state update as a complementarity problem. In developing the algorithm, we take advantage of the special features of solid and structural mechanics problems to achieve good computational performance, and hence the ability to compute numerical solutions to practical size problems. For example, the notion of a tangent stiffness matrix arises. Numerical examples using models of skeletal structures are presented to demonstrate the practicability of the algorithm. The numerical examples also raise some interesting questions about multiplicity of the solutions. Copyright © 2010 John Wiley & Sons, Ltd.     

A general non‐linear optimization algorithm for lower bound limit analysis

The non‐linear programming problem associated with the discrete lower bound limit analysis problem is treated by means of an algorithm where the need to linearize the yield criteria is avoided. The algorithm is an interior point method and is completely general in the sense that no particular finite element discretization or yield criterion is required. As with interior point methods for linear programming the number of iterations is affected only little by the problem size. Some practical implementation issues are discussed with reference to the special structure of the common lower bound load optimization problem, and finally the efficiency and accuracy of the method is demonstrated by means of examples of plate and slab structures obeying different non‐linear yield criteria. Copyright © 2002 John Wiley & Sons, Ltd.     

An efficient second‐order characteristic finite element method for non‐linear aerosol dynamic equations

In the paper we consider the non‐linear aerosol dynamic equation on time and particle size, which contains the advection process of condensation growth and the process of non‐linear coagulation. We develop an efficient second‐order characteristic finite element method for solving the problem. A high accurate characteristic method is proposed to treat the condensation advection while a second‐order extrapolation along the characteristics is proposed to approximate the non‐linear coagulation. The method has second‐order accuracy in time and the optimal‐order accuracy of finite element spaces in particle size, which improves the first‐order accuracy in time of the classical characteristic method. Numerical experiments show the efficient performance of our method for problems of log‐normal distribution aerosols in both the Euler coordinates and the logarithmic coordinates. Copyright © 2009 John Wiley & Sons, Ltd.     

基于概率密度演化理论的结构抗震可靠性分析

运用随机过程的正交展开方法,将地震动加速度过程表示为由10个左右的独立随机变量所调制的确定性函数的线性组合形式。结合概率密度演化方法和等价极值事件的基本思想,研究了非线性结构的抗震可靠度分析问题。以具有滞回特性的非线性结构为例,对某一多自由度的剪切型框架结构进行了抗震可靠性分析。结果表明：按照复杂失效准则计算的结构抗震可靠度较之结构各层抗震可靠度均低。这一研究为基于概率密度函数的、精细化的抗震可靠度计算提供了新的途径。     

Multi‐time‐step explicit–implicit method for non‐linear structural dynamics

We present a method with domain decomposition to solve time‐dependent non‐linear problems. This method enables arbitrary numeric schemes of the Newmark family to be coupled with different time steps in each subdomain: this coupling is achieved by prescribing continuity of velocities at the interface. We are more specifically interested in the coupling of implicit/explicit numeric schemes taking into account material and geometric non‐linearities. The interfaces are modelled using a dual Schur formulation where the Lagrange multipliers represent the interfacial forces. Unlike the continuous formulation, the discretized formulation of the dynamic problem is unable to verify simultaneously the continuity of displacements, velocities and accelerations at the interfaces. We show that, within the framework of the Newmark family of numeric schemes, continuity of velocities at the interfaces enables the definition of an algorithm which is stable for all cases envisaged. To prove this stability, we use an energy method, i.e. a global method over the whole time interval, in order to verify the algorithms properties. Then, we propose to extend this to non‐linear situations in the following cases: implicit linear/explicit non‐linear, explicit non‐linear/explicit non‐linear and implicit non‐linear/explicit non‐linear. Finally, we present some examples showing the feasibility of the method. Copyright © 2001 John Wiley & Sons, Ltd.     

An explicit discontinuous Galerkin method for non‐linear solid dynamics: Formulation,parallel implementation and scalability properties

An explicit‐dynamics spatially discontinuous Galerkin (DG) formulation for non‐linear solid dynamics is proposed and implemented for parallel computation. DG methods have particular appeal in problems involving complex material response, e.g. non‐local behavior and failure, as, even in the presence of discontinuities, they provide a rigorous means of ensuring both consistency and stability. In the proposed method, these are guaranteed: the former by the use of average numerical fluxes and the latter by the introduction of appropriate quadratic terms in the weak formulation. The semi‐discrete system of ordinary differential equations is integrated in time using a conventional second‐order central‐difference explicit scheme. A stability criterion for the time integration algorithm, accounting for the influence of the DG discretization stability, is derived for the equivalent linearized system. This approach naturally lends itself to efficient parallel implementation. The resulting DG computational framework is implemented in three dimensions via specialized interface elements. The versatility, robustness and scalability of the overall computational approach are all demonstrated in problems involving stress‐wave propagation and large plastic deformations. Copyright © 2007 John Wiley & Sons, Ltd.     

A time domain FEM approach based on implicit Green's functions for non‐linear dynamic analysis

The present paper describes an unconditionally stable algorithm to integrate the equations of motion in time. The standard FEM displacement model is employed to perform space discretization, and the time‐marching process is carried out through an algorithm based on the Green's function of the mechanical system in nodal co‐ordinates. In the present ‘implicit Green's function approach’ (ImGA), mechanical system Green's functions are not explicitly computed; rather they are implicitly considered through an iterative pseudo‐forces process. Under certain simplifying hypothesis, iterations are not necessary and the ImGA becomes cheaper than standard Newmark/Newton–Raphson algorithm. At the end of the paper numerical examples are presented in order to illustrate the accuracy of the present approach. Copyright © 2004 John Wiley & Sons, Ltd.     

Geometrically non‐linear thermoelastic analysis of functionally graded shells using finite element method

A finite element formulation governing the geometrically non‐linear thermoelastic behaviour of plates and shells made of functionally graded materials is derived in this paper using the updated Lagrangian approach. Derivation of the formulation is based on rewriting the Green–Lagrange strain as well as the 2nd Piola–Kirchhoff stress as two second‐order functions in terms of a through‐the‐thickness parameter. Material properties are assumed to vary through the thickness according to the commonly used power law distribution of the volume fraction of the constituents. Within a non‐linear finite element analysis framework, the main focus of the paper is the proposal of a formulation to account for non‐linear stress distribution in FG plates and shells, particularly, near the inner and outer surfaces for small and large values of the grading index parameter. The non‐linear heat transfer equation is also solved for thermal distribution through the thickness by the Rayleigh–Ritz method. Advantages of the proposed approach are assessed and comparisons with available solutions are presented. Copyright © 2007 John Wiley & Sons, Ltd.     

Time discontinuous Galerkin methods with energy decaying correction for non‐linear elastodynamics

In this paper a new time discontinuous Galerkin (TDG) formulation for non‐linear elastodynamics is presented. The new formulation embeds an energy correction which ensures truly energy decaying, thus allowing to achieve unconditional stability that, as shown in the paper, is not guaranteed by the classical TDG formulation. The resulting method is simple and easily implementable into existing finite element codes. Moreover, it inherits the desirable higher‐order accuracy and high‐frequency dissipation properties of the classical formulation. Numerical results illustrate the very good performance of the proposed formulation. Copyright © 2010 John Wiley & Sons, Ltd.     

Fast frequency response analysis of large‐scale structures with non‐proportional damping

Modal frequency response analysis is an economical approach for large and complex structural systems since there is an enormous reduction in dimension from the finite element frequency response problem. However, when non‐proportional damping exists, the modal frequency response problem is expensive to solve at many frequencies because modal damping matrices are fully populated. This paper presents a new algorithm to solve the modal frequency response problem for large and complex structural systems with structural and viscous damping. The newly developed algorithm, fast frequency response analysis (FFRA) algorithm, solves the damped modal frequency response problem with O(m²) operations at each frequency. Then the FFRA algorithm is extended for solving a system of equations in optimization application with the modal correction approach, in which the mass, stiffness and damping matrices of a modified configuration differ from the original configuration. Numerical results show that the FFRA algorithm dramatically improves the performance of the modal frequency response analysis compared to conventional methods in industry while obtaining the same accuracy. Copyright © 2006 John Wiley & Sons, Ltd.