Error evaluations for the computation of failure probability in static structural reliability problems

The determination of mathematical reliability in static structures is still a motivating field of research. On one hand, the failure probability values are of greater importance in engineering activities; on the other hand, the determination of this probability remains a time consuming computational task when real problems are examined. To obtain a significant value for the failure probability while preserving a reasonable computation cost is therefore an objective to be considered. One of the necessary conditions to reach this target is to design a method to determine the computational error. Knowing the error will then give the capacity to limit the computation time, in particular by avoiding a too accurate probability evaluation. This paper presents a method allowing one to deal with these factors. The RGMR method, presented at the ICASP'7 meeting (July 1995, Paris) was designed to make such an error evaluation possible. Examples of numerical error computations with the RGMR method, particularly when low significant variables are eliminated, are given. These new developments are a step towards the goal mentioned above.     

Input-profile-based software failure probability quantification for safety signal generation systems

The approaches for software failure probability estimation are mainly based on the results of testing. Test cases represent the inputs, which are encountered in an actual use. The test inputs for the safety-critical application such as a reactor protection system (RPS) of a nuclear power plant are the inputs which cause the activation of protective action such as a reactor trip. A digital system treats inputs from instrumentation sensors as discrete digital values by using an analog-to-digital converter. Input profile must be determined in consideration of these characteristics for effective software failure probability quantification. Another important characteristic of software testing is that we do not have to repeat the test for the same input value since the software response is deterministic for each specific digital input. With these considerations, we propose an effective software testing method for quantifying the failure probability. As an example application, the input profile of the digital RPS is developed based on the typical plant data. The proposed method in this study is expected to provide a simple but realistic mean to quantify the software failure probability based on input profile and system dynamics.     

The meaning of probability in probabilistic safety analysis

A Probabilistic Safety Analysis expresses uncertainty about the possible future damaging consequences of complex installations, such as chemical or nuclear plants, in terms of probabilities. Often these probabilities are interpreted as measures of physical properties of the installation and how it is operated, and a PSA is seen as a means to carry out these measurements. This view is useful for making comparative statements about the riskiness of installations, particularly when comparing them with standards. It is argued here, however, that this interpretation of probability is inconsistent with all the standard philosophical theories of probability, and so a different interpretation of PSA is necessary. We suggest, alternatively, that by using the standard subjective theory of probability, PSA may be seen as a tool for argument, rather than an objective representation of truth. In this interpretation the problems of expert choice and model validation become less problematic.     

The solution of nonlinear finite element equations

An algorithm is described which appears to give an efficient solution of nonlinear finite element equations. It is a quisi-Nowton method, and we compare it with some of the alternatives. Initial tests of its application to both material and geometric nonlinearities are discussed.     

Integral results for nonlinear diffusion equations

We show how to construct integral results for the multi-dimensional nonlinear diffusion equation c/t=·(D(c)c) and for some generalisations of this. For appropriate boundary conditions these become integral invariants. An application of these results to determining the large-time behaviour of some radially symmetric problems is indicated.     

An analytical-numerical method for fast evaluation of probability densities for transient solutions of nonlinear Itô’s stochastic differential equations

Probability densities for solutions of nonlinear Itô’s stochastic differential equations are described by the corresponding Kolmogorov-forward/Fokker-Planck equations. The densities provide the most complete information on the related probability distributions. This is an advantage crucial in many applications such as modelling floating structures under the stochastic-load due to wind or sea waves. Practical methods for numerical solution of the probability density equations are combined, analytical-numerical techniques. The present work develops a new analytical-numerical approach, the successive-transition (ST) method, which is a version of the path-integration (PI) method. The ST technique is based on an analytical approximation for the transition probability density. It enables the PI approach to explicitly allow for the damping matrix in the approximation. This is achieved by extending another method, introduced earlier for bistable nonlinear reaction-diffusion equations, to the probability density equations. The ST method also includes a control for the size of the time-step. The overall accuracy of the ST method can be tested on various nonlinear examples. One such example is proposed. It is one-dimensional nonlinear Itô’s equation describing the velocity of a ship maneuvering along a straight line under the action of the stochastic drag due to wind or sea waves. Another problem in marine engineering, the rolling of a ship up to its possible capsizing is also discussed in connection with the complicated damping matrix picture. The work suggests a few directions for future research.     

Existence and uniqueness for nonlinear reynolds equations

The pressure distribution in a gas-lubricated bearing is given by the nonlinear Reynolds equation with the boundary value problem ▽ · (H³P▽P) = Λ(HP)_x in Ω, P = G on δω. Various results of existence and uniqueness for this equation are presented. Furthermore the system of nonlinear equations arising when elastic deformation of the bearing surfaces is not neglected is discussed.     

Subspace methods for large scale nonlinear equations and nonlinear least squares

In this paper, we study large scale nonlinear systems of equations and nonlinear least square problems. We present subspace methods for solving these two special optimization problems. The subspace methods have the characteristic to force the next iteration in a low dimensional subspace. The main technique is to construct subproblems in low dimensions so that the computation cost in each iteration can be reduced comparing to standard approaches. The subspace approach offers a possible way to handle large scale optimization problems which are now attracting more and more attention. Actually, quite a few known techniques can be viewed as subspace methods, such as conjugate gradient method, limited memory quasi-Newton method, projected gradient method, and null space method. This work is partially supported by Chinese NSF grants 10231060, 10831006 and by CAS grant kjcx-yw-s7.     

On sampling-based schemes for probability of failure sensitivity analysis

In this paper we discuss the accuracy of probability of failure sensitivity analysis with sampling-based schemes. Three approaches commonly employed in literature are discussed: the Weak sensitivity analysis, the Direct employment of finite difference schemes and the Common Random Variable approach. Theoretical estimates for the bias, the coefficient of variation and the mean square error for each approach are presented. The results hold for a single random variable and the extension to more general situations should be pursued in future works. These results lead to the conclusion that the Common Random Variable approach is superior to the Direct approach from the theoretical point of view. The Weak approach, on the other hand, is equivalent to the Common Random Variable approach with central finite difference formula. The choice between these latter two approaches is then a matter of computational efficiency. The results of this work should contribute to further development of efficient algorithms for the problem under study.     

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.     

Limiting the probability of failure for components containing flaws

A new methodology is proposed for determining the probability of failure of an arbitrarily loaded component with an arbitrary shape, containing internal flaws. An important application area of the proposed equation is developing optimised designs and loading, associated with low probability of failure.Methods have also been developed for specifying the maximum acceptable level of the flaw number density and the maximum size of the stressed volume to guarantee that the probability of failure triggered by flaws remains below a maximum acceptable level. An important parameter called detrimental factor has been introduced to characterise components with internal flaws. Components with identical geometry and material, with the same detrimental factors, are characterised by the same probability of failure. The methods are based on derived equations related to the probability of triggering failure by flaws and the probability of clustering of flaws following a homogeneous Poisson process.Using the developed models a new technique has been created for setting reliability requirements regarding the flaw number density bounds which limit the risk of failure below a maximum acceptable level and minimise the total losses.     

Error computation for capacitive displacement transducers using nonlinear bridge circuits

Translated from Izmeritel'naya Tekhnika, No. 3, pp. 10–11, March, 1989.     

Sample regeneration algorithm for structural failure probability function estimation

An efficient strategy to approximate the failure probability function in structural reliability problems is proposed. The failure probability function (FPF) is defined as the failure probability of the structure expressed as a function of the design parameters, which in this study are considered to be distribution parameters of random variables representing uncertain model quantities. The task of determining the FPF is commonly numerically demanding since repeated reliability analyses are required. The proposed strategy is based on the concept of augmented reliability analysis, which only requires a single run of a simulation-based reliability method. This paper introduces a new sample regeneration algorithm that allows to generate the required failure samples of design parameters without any additional evaluation of the structural response. In this way, efficiency is further improved while ensuring high accuracy in the estimation of the FPF. To illustrate the efficiency and effectiveness of the method, case studies involving a turbine disk and an aircraft inner flap are included in this study.     

Rectangular lattice Boltzmann model for nonlinear convection-diffusion equations

In this paper, a rectangular lattice Boltzmann model is proposed for nonlinear convection-diffusion equations (NCDEs). The model can be used to solve NCDEs with very general form by using a real/complex-valued quadric equilibrium distribution function and relaxation time. Detailed simulations on several examples are performed to validate the model. The numerical results show good agreement with the analytical solutions, and the numerical accuracy is much better than that of the models with a linear equilibrium distribution function.     

A numerical scheme for nonlinear Helmholtz equations with strong nonlinear optical effects

A numerical scheme is presented to solve the nonlinear Helmholtz (NLH) equation modeling second-harmonic generation (SHG) in photonic bandgap material doped with a nonlinear χ((2)) effect and the NLH equation modeling wave propagation in Kerr type gratings with a nonlinear χ((3)) effect in the one-dimensional case. Both of these nonlinear phenomena arise as a result of the combination of high electromagnetic mode density and nonlinear reaction from the medium. When the mode intensity of the incident wave is significantly strong, which makes the nonlinear effect non-negligible, numerical methods based on the linearization of the essentially nonlinear problem will become inadequate. In this work, a robust, stable numerical scheme is designed to simulate the NLH equations with strong nonlinearity.     

Constitutive equations for the nonlinear elastic response of rubbers

Summary A constitutive model is derived for the elastic behavior of rubbers at three-dimensional deformations with finite strains. An elastomer is thought of as an incompressible network of flexible chains bridged by permanent junctions that move affinely with the bulk medium. The constraints imposed by surrounding macromolecules on configurations of an individual chain are introduced by combining the Flory–Erman and Erman–Monnerie approaches. To describe inter-chain interactions in a tractable way, the conventional picture of a tube where a chain is confined is replaced by geometrical restrictions on the positions of its ends and center of mass. The constraints on the chain ends are formulated within the traditional Flory concept, whereas those on the position of center of mass are described following the Ronca–Allegra scenario. Stress–strain relations for a network of constrained chains are derived by using the laws of thermodynamics. The constitutive equations involve four adjustable parameters with transparent physical meaning. The material constants are found by fitting experimental data on elastomers at uniaxial and equi-biaxial tensions and pure shear. It is demonstrated that (i) the model provides an acceptable prediction of stresses in a test with one deformation mode, when its parameters are found by matching observations in an experiment with another mode, and (ii) material constants are affected by chemical composition of elastomers in a physically plausible way.     

Stochastic response of nonlinear system in probability domain

A stochastic averaging procedure for obtaining the probability density function (PDF) of the response for a strongly nonlinear single-degree-of-freedom system, subjected to both multiplicative and additive random excitations is presented. The procedure uses random Van Der Pol transformation, Ito’s equation of limiting diffusion process and stochastic averaging technique as outlined by Zhu and others. However, the equations are rederived in generalized form and arranged in such a way that the procedure lends itself to a numerical computational scheme using FFT. The main objective of the modification is to consider highly irregular nonlinear functions which cannot be integrated in closed form and also to solve problems where analytical expressions for probability density function cannot be obtained. The procedure is applied to obtain the PDF of the response of Duffing oscillator subjected to additive and multiplicative random excitations represented by rational power spectral density functions (PSDFs). The results are verified by digital simulation. It is shown that the procedure provides results which compare very well with those obtained from simulation analysis not only for wide-band excitations but also for very narrow-band excitations, which are weak (when normalized with respect to mass of the system.) This paper is dedicated to Prof R N Iyengar of the Indian Institute of Science on the occasion of his formal retirement.