Optimal relaxation of I/O electrical requirements under packaging uncertainty by stochastic methods

Stochastic collocation and stochastic Galerkin methods for linear differential algebraic equations

In this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms (input data of the model). The input data are assumed to depend on a finite number of random variables. The method consists in a Galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the Monte Carlo approach. It can be seen as a generalization of the stochastic Galerkin method proposed in [I. Babuška, R. Tempone, and G. E. Zouraris, SIAM J. Numer. Anal., 42 (2004), pp. 800–825] and allows one to treat easily a wider range of situations, such as input data that depend nonlinearly on the random variables, diffusivity coefficients with unbounded second moments, and random variables that are correlated or even unbounded. We provide a rigorous convergence analysis and demonstrate exponential convergence of the “probability error” with respect to the number of Gauss points in each direction in the probability space, under some regularity assumptions on the random input data. Numerical examples show the effectiveness of the method.

In this paper, the Stochastic Collocation Method is introduced into EMC uncertainty analysis. It is demonstrated to be accurate, with high computational efficiency, compared with the Method of Moments, the perturbation method and the Monte Carlo method. The Stochastic Collocation Method requires no modification to the original solver, and it is fit for being introduced into EMC simulation software.

To analysis the reliability of Dongjiang Bridge, a steel truss bridge stiffened with rigid cables, a new analysis method that combined with each advantage of some common reliability computing method was put forward, which could get the probability distribution and numeric attributes of complex bridge structure's response expediently, and then get the reliability index and failure probability. A stochastic finite element model was established to analyze this bridge, in which some parameters such as material, geometric dimensioning, loads, and so on, were simulated as stochastic variables. The analysis result show that the lowest reliability index of main member bar's stress is 5.10, more than the allowable value 4.7, and the corresponding failure probability is 1.71×10-7. The reliability index of mid-span deflection in serviceability limit states reach up to 6.99 and its failure probability is 1.53×10-7. All results indicate that the strength and stiffness of Dongjiang Bridge has higher reliability.

Methodical aspects of the reliability-based structural optimisation using stochastic quasigradient methods are considered. For an example of the simply supported reinforced concrete beam, the employment of the Lagrange multiplier method that belongs to the class of stochastic quasigradient methods is demonstrated. The classical optimum design goal to minimise structural cost or weight under the constraint on the structural failure probability is taken for consideration. Optimisation problems solved with the Lagrangemultiplier method are formulated in form of general stochastic programming problem. The mathematical expectation of the concrete volume reduced with respect to the in-place cost of the beam materials is taken as the objective function. Constraint function is the limitation placed on the beam failure probability. The beam is considered as a series structural system. Values of the prescribed allowable failure probability belongs to the interval in which the estimation of the failure probabilities by the simple Monte-Carlomethod is possible with an acceptable confidence. The time-independent case as well as the time-dependent one is considered in the optimisation problems. The generalisation on the time-dependent case is undertaken through the introduction into the constraint function of the quasi-linear distribution law of the random variables. In the time-dependent case, the objective function is associated with beginning and the constraint function with end of the service period. An expression of the stochastic gradient based on the differentiation under the integral sign is used for calculations with the Lagrange multiplier method. The stochastic gradient used is computationally more effective in comparison with stochastic finite-difference formulae usual in stochastic quasigradient methods because it requires only one computation of the structure in search iteration of the optimisation process. Three rules based on statistical argumentation are used for the stopping of the seat according to the procedure of the Lagrange multiplier method. The optimising of the beam shows that the Lagrange multiplier method is applicable for the optimal design of structures in that cases when the structural reliability can be estimated by means of the simple Monte-Carlo method. Additional research is needed for integration in the Lagrange multiplier method of statistical simulation techniques for the estimation of small structural failure probabilities.

Unraveling pine wilt disease: Comparative study of stochastic and deterministic model using spectral method

Attitude maneuvers are required for reorienting the satellite for meeting various mission objectives. A higher satellite slew rate during maneuvers is a desirable characteristic for any mission design. At the same time, a higher slew rate demands a higher actuator torque capacity which is always constrained due to the physical limits of the actuator. This paper discusses the satellite attitude maneuver problem under slew rate and actuator capacity constraints. A quaternion feedback controller with quaternion and angular rate feedback is implemented to achieve a rest to rest reorientation maneuver. The control gains are computed based on the slew rate limits while the control commands incorporate the actuator limits. Three different configurations of four reaction wheels, (a) pyramid configuration, (b) skew configuration with three wheels along the body axes and the fourth wheel along body diagonal and (c) skew configuration with skew angle of 45°, are studied. The paper further discusses single reaction wheel failure conditions for all the three configurations and demonstrates satellite rest to rest maneuver within the slew rate and actuator limits.

This paper proposes an approach for the robust averaged control of random vibrations for the Bernoulli–Euler beam equation under uncertainty in the flexural stiffness and in the initial conditions. The problem is formulated in the framework of optimal control theory and provides a functional setting, which is so general as to include different types of random variables and second-order random fields as sources of uncertainty. The second-order statistical moment of the random system response at the control time is incorporated in the cost functional as a measure of robustness. The numerical resolution method combines a classical descent method with an adaptive anisotropic stochastic collocation method for the numerical approximation of the statistics of interest. The direct and adjoint stochastic systems are uncoupled, which permits to exploit parallel computing architectures to solve the set of deterministic problem that arise from the stochastic collocation method. As a result, problems with a relative large number of random variables can be solved with a reasonable computational cost. Two numerical experiments illustrate both the performance of the proposed method and the significant differences that may occur when uncertainty is incorporated in this type of control problems.

We examined a portable cryostat utilizing liquid nitrogen with temperature control function for operation of high-temperature superconducting quantum interference devices (HTS-SQUIDs) to have a higher slew rate. To operate a non-destructive evaluation system with HTS-SQUID in city areas with serious environmental electromagnetic noise, we often need a higher slew rate. A glass Dewar is installed in an outer case made of aluminum, which is sealed hermetically. The temperature of liquid nitrogen is controlled by using a vacuum pump connected to a feedback circuit, which monitors the temperature of liquid nitrogen. The temperature of liquid nitrogen could be controlled in the range between 73 and 77.2 K with a 0.1 K step. For example, the temperature was kept at 75 K with precision of ±0.05 K for 9 hs and SQUID was operated stably for this period. The ΔV and the slew rate of a test SQUID were increased from 22 μV and 3.2 mT/s at 77.2 K to 37 μV and 5.9 mT/s at 73 K, respectively. The white noise was also improved from 48 fT/√Hz at 77.2 K to 32 fT/√Hz at 74 K. We believe that this technology is useful to extend application of HTS-SQUIDs in the fields.

Stochastic Modelling in Yearly Rainfall at Tirunelveli District, Tamil Nadu, India

Our ultimate goal is a topology optimization for a permanent-magnet (PM) machine, while including material uncertainties. The uncertainties in the output data are, e.g., due to measurement errors in the non-/linear material laws. In the resulting stochastic forward problem, these uncertainties are stochastically modeled by random fields. The solution of the underlying PDE, which describes magnetostatics, is represented using the generalized polynomial chaos expansion. As crucial ingredient we exploit the stochastic collocation method (SCM). Eventually, this leads to a random-dependent bi-objective cost functional, which is comprised of the expectation and the variance. Subject to the optimization of the PM machine are the shapes of the rotor poles, which are described by zero-level sets. Thus, the optimization will be done by redistributing the iron and magnet material over the design domain, which allows to attain an innovative low cogging torque design of an electric machine. For this purpose, the gradient directions are evaluated by using the continuous design sensitivity analysis in conjunction with the SCM. In the end, our numerical result for the optimization of a two-dimensional model demonstrates that the proposed approach is robust and effective.

Uncertainty investigations in nonlinear aeroelastic systems

There have been many challenges in studying multiscale kinetic models with uncertainties. Quantifying uncertainties in the models such as arisen from collision kernels, initial or boundary data, forcing terms, and developing efficient computational methods have become an important task in industrial applications. In this article, we will report some of the recent progress on multiscale kinetic models with random inputs. We will discuss from the aspects of both mathematical theory, by using the hypocoercivity of kinetic operators, and numerical computation. Two categories of intrusive and non-intrusive methods will be addressed, in particular the stochastic Galerkin method for problems with relatively low-dimensional random inputs and stochastic collocation method in a multi-fidelity setting for more challenging, higher-dimensional problems. Numerical experiments for several kinetic models such as the Boltzmann, bipolar Boltzmann-Poisson, linear transport and epidemic kinetic equations will be shown as examples.

In this work we focus on the numerical approximation of the solution u of a linear elliptic PDE with stochastic coefficients. The problem is rewritten as a parametric PDE and the functional dependence of the solution on the parameters is approximated by multivariate polynomials. We first consider the stochastic Galerkin method, and rely on sharp estimates for the decay of the Fourier coefficients of the spectral expansion of u on an orthogonal polynomial basis to build a sequence of polynomial subspaces that features better convergence properties, in terms of error versus number of degrees of freedom, than standard choices such as Total Degree or Tensor Product subspaces. We consider then the Stochastic Collocation method, and use the previous estimates to introduce a new class of Sparse Grids, based on the idea of selecting a priori the most profitable hierarchical surpluses, that, again, features better convergence properties compared to standard Smolyak or tensor product grids. Numerical results show the effectiveness of the newly introduced polynomial spaces and sparse grids.

This paper presents an uncertainty propagation analysis of viscoelastic sandwich laminated structures using a stochastic collocation method based on Smolyak’s sparse grids approach. Shear deformation in the viscoelastic layers is accurately captured by means of layer-wise plate finite elements. Modal properties are determined by solving nonlinear eigenvalue problems due to the frequency-dependent nature of the considered viscoelastic materials. Variations in material properties and layer thicknesses are both treated in a probabilistic way. The variabilities of natural frequencies and modal loss factors of two different viscoelastic laminated plate structures (plane and cylindrical) are studied. Results show that the present method converges quickly with small sample sizes when compared to the classical Monte-Carlo method. The influence of the variability of several parameters on viscoelastic structure modal properties is commented.