A data‐driven stochastic collocation approach for uncertainty quantification in MEMS

This work presents a data‐driven stochastic collocation approach to include the effect of uncertain design parameters during complex multi‐physics simulation of Micro‐ElectroMechanical Systems (MEMS). The proposed framework comprises of two key steps: first, probabilistic characterization of the input uncertain parameters based on available experimental information, and second, propagation of these uncertainties through the predictive model to relevant quantities of interest. The uncertain input parameters are modeled as independent random variables, for which the distributions are estimated based on available experimental observations, using a nonparametric diffusion‐mixing‐based estimator, Botev (Nonparametric density estimation via diffusion mixing. Technical Report, 2007). The diffusion‐based estimator derives from the analogy between the kernel density estimation (KDE) procedure and the heat dissipation equation and constructs density estimates that are smooth and asymptotically consistent. The diffusion model allows for the incorporation of the prior density and leads to an improved density estimate, in comparison with the standard KDE approach, as demonstrated through several numerical examples. Following the characterization step, the uncertainties are propagated to the output variables using the stochastic collocation approach, based on sparse grid interpolation, Smolyak (Soviet Math. Dokl. 1963; 4 :240–243). The developed framework is used to study the effect of variations in Young's modulus, induced as a result of variations in manufacturing process parameters or heterogeneous measurements on the performance of a MEMS switch. Copyright © 2010 John Wiley & Sons, Ltd.     

Combining differential quadrature method with simulation technique to solve non‐linear differential equations

Nowadays, most of the ordinary differential equations (ODEs) can be solved by modelica‐based approaches, such as Matlab/Simulink, Dymola and LabView, which use simulation technique (ST). However, these kinds of approaches restrict the users in the enforcement of conditions at any instant of the time domain. This limitation is one of the most important drawbacks of the ST. Another method of solution, differential quadrature method (DQM), leads to very accurate results using only a few grids on the domain. On the other hand, DQM is not flexible for the solution of non‐linear ODEs and it is not so easy to impose multiple conditions on the same location. For these reasons, the author aims to eliminate the mentioned disadvantages of the simulation technique (ST) and DQM using favorable characteristics of each method in the other. This work aims to show how the combining method (CM) works simply by solving some non‐linear problems and how the CM gives more accurate results compared with those of other methods. Copyright © 2008 John Wiley & Sons, Ltd.     

Stochastic uniform observability of general linear differential equations

The aim of this article is to discuss the uniform observability property of general linear differential equations with multiplicative white noise in Hilbert spaces. Based on perturbation theory for evolution operators on Hilbert Schmidt spaces and on the space of nuclear operators, new representations of the covariance operators associated to the mild solutions of the investigated stochastic differential equations are given. Using these results we obtain deterministic characterizations of the stochastic uniform observability property. We also identify an entire class of stochastic differential equations which are never stochastic uniformly observable. Some examples will illustrate the theory.     

Numerical solution of non‐linear Klein–Gordon equations by variational iteration method

In this paper, numerical solution of non‐linear Klein–Gordon equations with power law non‐linearities are obtained by the new application of He's variational iteration method. Numerical illustrations that include non‐linear Klein–Gordon equations and non‐linear partial differential equations are investigated to show the pertinent features of the technique. Copyright © 2006 John Wiley & Sons, Ltd.     

Truncation error and stability analysis of iterative and non‐iterative Thomas–Gladwell methods for first‐order non‐linear differential equations

The consistency and stability of a Thomas–Gladwell family of multistage time‐stepping schemes for the solution of first‐order non‐linear differential equations are examined. It is shown that the consistency and stability conditions are less stringent than those derived for second‐order governing equations. Second‐order accuracy is achieved by approximating the solution and its derivative at the same location within the time step. Useful flexibility is available in the evaluation of the non‐linear coefficients and is exploited to develop a new non‐iterative modification of the Thomas–Gladwell method that is second‐order accurate and unconditionally stable. A case study from applied hydrogeology using the non‐linear Richards equation confirms the analytic convergence assessment and demonstrates the efficiency of the non‐iterative formulation. Copyright © 2004 John Wiley & Sons, Ltd.     

Sparse grids‐based stochastic approximations with applications to aerodynamics sensitivity analysis

This work compares sample‐based polynomial surrogates, well suited for moderately high‐dimensional stochastic problems. In particular, generalized polynomial chaos in its sparse pseudospectral form and stochastic collocation methods based on both isotropic and dimension‐adapted sparse grids are considered. Both classes of approximations are compared, and an improved version of a stochastic collocation with dimension adaptivity driven by global sensitivity analysis is proposed. The stochastic approximations efficiency is assessed on multivariate test function and airfoil aerodynamics simulations. The latter study addresses the probabilistic characterization of global aerodynamic coefficients derived from viscous subsonic steady flow about a NACA0015 airfoil in the presence of geometrical and operational uncertainties with both simplified aerodynamics model and Reynolds‐Averaged Navier‐Stokes (RANS) simulation. Sparse pseudospectral and collocation approximations exhibit similar level of performance for isotropic sparse simulation ensembles. Computational savings and accuracy gain of the proposed adaptive stochastic collocation driven by Sobol' indices are patent but remain problem‐dependent. Copyright © 2015 John Wiley & Sons, Ltd.     

The probability density evolution method for dynamic response analysis of non‐linear stochastic structures

The probability density evolution method (PDEM) for dynamic responses analysis of non‐linear stochastic structures is proposed. In the method, the dynamic response of non‐linear stochastic structures is firstly expressed in a formal solution, which is a function of the random parameters. In this sense, the dynamic responses are mutually uncoupled. A state equation is then constructed in the augmented state space. Based on the principle of preservation of probability, a one‐dimensional partial differential equation in terms of the joint probability density function is set up. The numerical solving algorithm, where the Newmark‐Beta time‐integration algorithm and the finite difference method with Lax–Wendroff difference scheme are brought together, is studied. In the numerical examples, free vibration of a single‐degree‐of‐freedom non‐linear conservative system and dynamic responses of an 8‐storey shear structure with bilinear hysteretic restoring forces, subjected to harmonic excitation and seismic excitation, respectively, are investigated. The investigations indicate that the probability density functions of dynamic responses of non‐linear stochastic structures are usually irregular and far from the well‐known distribution types. They exhibit obvious evolution characteristics. The comparisons with the analytical solution and Monte Carlo simulation method demonstrate that the proposed PDEM is of fair accuracy and efficiency. Copyright © 2005 John Wiley & Sons, Ltd.     

A new hybrid method for solving linear–non‐linear systems of equations

This paper presents a new method for solving any combination of linear–non‐linear equations. The method is based on the separation of linear equations in terms of some selected variables from the non‐linear ones. The linear group is solved by means of any method suitable for the linear system. This operation needs no iteration. The non‐linear group, however, is solved by an iteration technique based on a new formula using the Taylor series expansion. The method has been described and demonstrated in several examples of analytical systems with very good results. The new method needs the initial approximations for non‐linear variables only. This requires far less computation than the Newton–Raphson method. The method also has a very good convergence rate. The proposed method is most beneficial for engineering systems that very often involve a large number of linear equations with limited number of non‐linear equations. Copyright © 2005 John Wiley & Sons, Ltd.     

Galerkin reduced‐order modeling scheme for time‐dependent randomly parametrized linear partial differential equations

In this paper, we consider the problem of constructing reduced‐order models of a class of time‐dependent randomly parametrized linear partial differential equations. Our objective is to efficiently construct a reduced basis approximation of the solution as a function of the spatial coordinates, parameter space, and time. The proposed approach involves decomposing the solution in terms of undetermined spatial and parametrized temporal basis functions. The unknown basis functions in the decomposition are estimated using an alternating iterative Galerkin projection scheme. Numerical studies on the time‐dependent randomly parametrized diffusion equation are presented to demonstrate that the proposed approach provides good accuracy at significantly lower computational cost compared with polynomial chaos‐based Galerkin projection schemes. Comparison studies are also made against Nouy's generalized spectral decomposition scheme to demonstrate that the proposed approach provides a number of computational advantages. Copyright © 2012 John Wiley & Sons, Ltd.     

A two‐dimensional stochastic algorithm for the solution of the non‐linear Poisson–Boltzmann equation: validation with finite‐difference benchmarks

This paper presents a two‐dimensional floating random walk (FRW) algorithm for the solution of the non‐linear Poisson–Boltzmann (NPB) equation. In the past, the FRW method has not been applied to the solution of the NPB equation which can be attributed to the absence of analytical expressions for volumetric Green's functions. Previous studies using the FRW method have examined only the linearized Poisson–Boltzmann equation. No such linearization is needed for the present approach. Approximate volumetric Green's functions have been derived with the help of perturbation theory, and these expressions have been incorporated within the FRW framework. A unique advantage of this algorithm is that it requires no discretization of either the volume or the surface of the problem domains. Furthermore, each random walk is independent, so that the computational procedure is highly parallelizable. In our previous work, we have presented preliminary calculations for one‐dimensional and quasi‐one‐dimensional benchmark problems. In this paper, we present the detailed formulation of a two‐dimensional algorithm, along with extensive finite‐difference validation on fully two‐dimensional benchmark problems. The solution of the NPB equation has many interesting applications, including the modelling of plasma discharges, semiconductor device modelling and the modelling of biomolecular structures and dynamics. Copyright © 2005 John Wiley & Sons, Ltd.     

Uniformly convergent non‐standard finite difference methods for singularly perturbed differential‐difference equations with delay and advance

A new class of fitted operator finite difference methods are constructed via non‐standard finite difference methods ((NSFDM)s) for the numerical solution of singularly perturbed differential difference equations having both delay and advance arguments. The main idea behind the construction of our method(s) is to replace the denominator function of the classical second‐order derivative with a positive function derived systematically in such a way that it captures significant properties of the governing differential equation and thus provides the reliable numerical results. Unlike other FOFDMs constructed in standard ways, the methods that we present in this paper are fairly simple to construct (and thus enrich the class of fitted operator methods by adding these new methods). These methods are shown to be ε‐uniformly convergent with order two which is the highest possible order of convergence obtained via any fitted operator method for the problems under consideration. This paper further clarifies several doubts, e.g. why a particular scheme is not suitable for the whole range of values of the associated parameters and what could be the possible remedies. Finally, we provide some numerical examples which illustrate the theoretical findings. Copyright © 2005 John Wiley & Sons, Ltd.     

A method to estimate plastic loads for elbows with non‐uniform thicknesses

This paper provides a method to estimate plastic loads [defined by the twice‐elastic‐slope (TES)] for elbows with non‐uniform thicknesses under in‐plane bending and under internal pressure, based on systematic FE limit analyses using elastic‐perfectly plastic materials. The intrados thickness is assumed to be up to 30% thicker than the straight pipe thickness, whereas the extrados thickness up to 30% thinner. The FE plastic loads are compared with estimated ones using closed‐form approximations for elbows with uniform thicknesses, to provide guidance on the choice of a representative thickness to predict plastic loads for elbows with non‐uniform thicknesses. Results suggest that the use of the straight pipe thickness gives overall satisfactory predictions. Estimated plastic loads are conservative, and differ from FE results by less than 10% for all cases considered. Despite its simplicity, results look very promising, and thus the use of the straight pipe thickness can be recommended to estimate plastic loads for elbows with non‐uniform thicknesses in practice.     

Dynamic stiffness for piecewise non‐uniform Timoshenko column by power series—part II: Follower force

A follower force is an applied force whose direction changes according to the deformed shape during the course of deformation. The dynamic stiffness matrix of a non‐uniform Timoshenko column under follower force is formed by the power‐series method. The dynamic stiffness matrix is unsymmetrical due to the non‐conservative nature of the follower force. The frequency‐dependent mass matrix is still symmetrical and positive definite according to the extended Leung theorem. An arc length continuation method is introduced to find the influence of a concentrated follower force, distributed follower force, end mass and stiffness, slenderness, and taper ratio on the natural frequency and stability. It is found that the power‐series method can handle a very wide class of dynamic stiffness problem. Copyright © 2001 John Wiley & Sons, Ltd.     

A polynomial collocation method for the numerical solution of weakly singular and singular integral equations on non‐smooth boundaries

The aim of this paper is to show the efficiency of the use of smoothing changes of variable in the numerical treatment of 1D and 2D weakly singular and singular integral equations. The introduction of a smoothing transformation, besides smoothing the solution, allows also the use of a very simple and efficient collocation method based on Chebyshev polynomials of the first kind and their zeros. Further, we propose proper smoothing changes of variable also for the numerical approximation of those collocation matrix elements, which are given by weakly singular, singular or nearly singular integrals. Several numerical tests are given to point out the efficiency of the numerical approach we propose. Copyright © 2003 John Wiley & Sons, Ltd.     

Dynamic stiffness for piecewise non‐uniform Timoshenko column by power series—part I: Conservative axial force

The dynamic stiffness method uses the solutions of the governing equations as shape functions in a harmonic vibration analysis. One element can predict many modes exactly in the classical sense. The disadvantages lie in the transcendental nature and in the need to solve a non‐linear eigenproblem for the natural modes, which can be solved by the Wittrick–William algorithm and the Leung theorem. Another practical problem is to solve the governing equations exactly for the shape functions, non‐uniform members in particular. It is proposed to use power series for the purpose. Dynamic stiffness matrices for non‐uniform Timoshenko column are taken as examples. The shape functions can be found easily by symbolic programming. Step beam structures can be treated without difficulty. The new contributions of the paper include a general formulation, an extended Leung's theorem and its application to parametric study. Copyright © 2001 John Wiley & Sons, Ltd.     

Linear random vibration by stochastic reduced‐order models

A practical method is developed for calculating statistics of the states of linear dynamic systems with deterministic properties subjected to non‐Gaussian noise and systems with uncertain properties subjected to Gaussian and non‐Gaussian noise. These classes of problems are relevant as most systems have uncertain properties, physical noise is rarely Gaussian, and the classical theory of linear random vibration applies to deterministic systems and can only deliver the first two moments of a system state if the noise is non‐Gaussian. The method (1) is based on approximate representations of all or some of the random elements in the definition of linear random vibration problems by stochastic reduced‐order models (SROMs), that is, simple random elements having a finite number of outcomes of unequal probabilities, (2) can be used to calculate statistics of a system state beyond its first two moments, and (3) establishes bounds on the discrepancy between exact and SROM‐based solutions of linear random vibration problems. The implementation of the method has required to integrate existing and new numerical algorithms. Examples are presented to illustrate the application of the proposed method and assess its accuracy. Copyright © 2009 John Wiley & Sons, Ltd.     

A dual algorithm for the topology optimization of non‐linear elastic structures

Dual algorithms are ideally suited for the purpose of topology optimization since they work in the space of Lagrange multipliers associated with the constraints. To date, dual algorithms have been applied only for linear structures. Here we extend this methodology to the case of non‐linear structures. The perimeter constraint is used to make the topology problem well‐posed. We show that the proposed algorithm yields a value of perimeter that is close to that specified by the user. We also address the issue of manufacturability of these designs, by proposing a variant of the standard dual algorithm, which generates designs that are two‐dimensional although the loading and the geometry are three‐dimensional. Copyright © 2008 John Wiley & Sons, Ltd.     

Numerical method for non‐linear wave and diffusion equations by the variational iteration method

This paper applies He's variational iteration to the wave equations in an infinite one‐dimensional medium and some non‐linear diffusion equations. A suitable choice of an initial solution can lead to the needed exact solution by a few iterations. Copyright © 2007 John Wiley & Sons, Ltd.     

An efficient solution method for incompressible N‐S equations using non‐orthogonal collocated grid

An efficient strategy for the solution of N‐S Equations using collocated, non‐orthogonal grids is presented. The governing equations have been discretized in the physical plane itself without co‐ordinate transformation, thereby retaining the lucidity of the basic finite volume method. The non‐orthogonal terms and QUICK type corrections for the convective terms in the momentum equations are treated explicitly, while the other terms are taken in implicit form. In the pressure correction equation, the non‐orthogonal terms have been dropped altogether. The discretized equations have been solved by the preconditioned conjugate gradient square method. The specific combination of above steps has resulted in better convergence properties as compared to those of existing algorithms, even for highly skewed grids. The scheme has been validated against benchmark solutions such as lid‐driven flow in square and skewed cavities and experi mental results of flow over a single cylinder. Its applicability has also been illustrated for flow through a bank of staggered cylinders, with anti‐symmetric inlet and outlet boundary conditions. Copyright © 1999 John Wiley & Sons, Ltd.     

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.