Sensitivity, probabilistic and stochastic analysis of the thermo-piezoelectric phenomena in solids by the stochastic perturbation technique

Abstract

The main aim here is to present the application of the generalized stochastic perturbation technique to thermo-piezoelectric analysis of solid continua. The general nth order Taylor series representation for all random input parameters and the state functions is employed to formulate the coupled thermo-electro-elasticity equilibrium equations of the additional order; a determination of any probabilistic moments and characteristics is described; the discretization of the problem in terms of the Stochastic perturbation-based Finite Element Method is also provided. Since this expansion includes the lowest order partial derivatives, the structural sensitivity analysis using direct differentiation is performed at the same time with probabilistic modeling contrasted with the Monte-Carlo simulation results. The probabilistic approach is extended here towards an accounting for the stochastic ageing processes, which appear frequently in aggressive external environments and under dynamic excitation. The two parametric stochastic process with Gaussian initial value and ageing velocity is tested for this purpose. The entire procedure is tested on the example of the thermo-electro-elastic pulsation of the beam modeled analytically using the symbolic software MAPLE, where polynomial approximations, design sensitivities, probabilistic moments and their histories are computed and visualized.

Numerical solution of the Duffing equation with random coefficients

“…3, 4 and 5 determined by using of the stochastic tenth order perturbation scheme (left series) and, independently, via the Monte-Carlo simulation scheme (right series). The most apparent difference to the well documented previous models available in linear elasticity [11] is an enormous increase of the resulting extreme coefficient of variation for displacements (Fig. 3) which is close to 1.5 and this means 15 times larger than the input value of this parameter.…”

“…It is important to notice that partial derivatives of the structural dynamic response with respect to the given input random parameter are calculated at its mean value in the traditional deterministic manner. Since an analytical interrelation of this dynamic response with respect to the chosen input random parameter is usually implicit, we apply the Weighted Least Squares Method (WLSM) [10] here, to approximate this function numerically [11]. Inserting this expansion into the definitions (3-4) brings for the Gaussian distributions the following results for the expectations and variances of the same function:…”

“…This strategy is tested on two different cases-the first one concerns damped vibrations of a linear oscillator and it serves rather for a comparison with the Monte-Carlo simulation scheme, the second one concerns a case study devoted to the forced vibration of a micro-beam exhibiting stochastic damping. An application of this strategy to coupled field Finite Element Method analysis has been provided before in [11]. It should be mentioned that an application of the perturbation technique itself in the deterministic context is well known from the solution of various problems in dynamics.…”

The effect of uncertain material properties on free vibrations of thin periodic plates

“…A non-stationary stochastic excitation process is used to a flexural stiffness or eigenvalue frequency identification of a linear structure by Jarczewska et al [52], where the dynamical problem is transformed into a static one by integrating the input and the output signals. The generalized stochastic perturbation technique is applied to thermo-piezoelectric analysis of solid continua by Kamiński and Corigliano [53], where the discretization is made of the stochastic perturbation-based finite element method. The effect of the prestress on the overall mechanical properties of the random elastic composite with residual stresses is considered by Dal Corso and Deseri [54].…”

Bifurcation Control of an Electrostatically-Actuated MEMS Actuator with Time-Delay Feedback

“…The perturbation method plays an important role in the nonlinear dynamic analysis of MEMS resonators [ 14 ]. Kaminski et al [ 29 , 30 ] studied stochastic nonlinear dynamic behaviors of a MEMS device using the generalized stochastic perturbation technique.…”