Higher order weak linearizations of stochastically driven nonlinear oscillators

Saha, Nilanjan; Roy, Debasish

doi:10.1098/rspa.2007.1852

Cited by 12 publications

(8 citation statements)

References 21 publications

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“…Following the concept of local linearization (Roy, 2001), the linearization point (such that U* :¼ U(t*)) could be chosen anywhere in the closed interval without affecting the formal error order in terms of the step size h i :¼ t i+1 À t i . While our present choice of t* = t i yields the explicit phase space linearization (PSL) (Roy, 2000), choosing t* = t i+1 results in the implicit locally transversal linearization (LTL) (Roy, 2004;Saha and Roy, 2007).…”

Section: Numerical Implementationmentioning

confidence: 99%

New computational approaches for wrinkled and slack membranes

Pimprikar

Banerjee

Roy

et al. 2010

International Journal of Solids and Structures

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a b s t r a c tThe static response of thin, wrinkled membranes is studied using both a tension field approximation based on plane stress conditions and a 3D nonlinear elasticity formulation, discretized through 8-noded Cosserat point elements. While the tension field approach only obtains the wrinkled/slack regions and at best a measure of the extent of wrinkliness, the 3D elasticity solution provides, in principle, the deformed shape of a wrinkled/slack membrane. However, since membranes barely resist compression, the discretized and linearized system equations via both the approaches are ill-conditioned and solutions could thus be sensitive to discretizations errors as well as other sources of noises/imperfections. We propose a regularized, pseudo-dynamical recursion scheme that provides a sequence of updates, which are almost insensitive to the regularizing term as well as the time step size used for integrating the pseudo-dynamical form. This is borne out through several numerical examples wherein the relative performance of the proposed recursion scheme vis-à-vis a regularized Newton strategy is compared. The pseudo-time marching strategy, when implemented using 3D Cosserat point elements, also provides a computationally cheaper, numerically accurate and simpler alternative to that using geometrically exact shell theories for computing large deformations of membranes in the presence of wrinkles.

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Section: Numerical Implementationmentioning

confidence: 99%

New computational approaches for wrinkled and slack membranes

Pimprikar

Banerjee

Roy

et al. 2010

International Journal of Solids and Structures

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show abstract

“…Following the concept of local linearization [25], the linearization point t * (such that l * := l(t * )) could be chosen anywhere in the closed interval [t, t + t] without affecting the formal error order. While choosing t * = t yields the explicit phase space linearization (PSL) [26], t * = t + t results in the implicit locally transversal linearization (LTL) [27,28]. Denoting h = t k+1 −t k to be the time step and l k := l(t k ), the explicit PSL map corresponding to the continuous update (5) is written as…”

Section: A Deterministic Pseudo-dynamical Schemementioning

confidence: 99%

Self‐regularized pseudo time‐marching schemes for structural system identification with static measurements

Banerjee

Roy

Vasu

2009

Numerical Meth Engineering

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SUMMARYWe explore the application of pseudo time marching schemes, involving either deterministic integration or stochastic filtering, to solve the inverse problem of parameter identification of large dimensional structural systems from partial and noisy measurements of strictly static response. Solutions of such non-linear inverse problems could provide useful local stiffness variations and do not have to confront modeling uncertainties in damping, an important, yet inadequately understood, aspect in dynamic system identification problems. The usual method of least-square solution is through a regularized Gauss-Newton method (GNM) whose results are known to be sensitively dependent on the regularization parameter and data noise intensity. Finite time, recursive integration of the pseudo-dynamical GNM (PD-GNM) update equation addresses the major numerical difficulty associated with the near-zero singular values of the linearized operator and gives results that are not sensitive to the time step of integration. Therefore, we also propose a pseudo-dynamic stochastic filtering approach for the same problem using a parsimonious representation of states and specifically solve the linearized filtering equations through a pseudo-dynamic ensemble Kalman filter (PD-EnKF). For multiple sets of measurements involving various load cases, we expedite the speed of the PD-EnKF by proposing an inner iteration within every time step. Results using the pseudo-dynamic strategy obtained through PD-EnKF and recursive integration are compared with those from the conventional GNM, which prove that the PD-EnKF is the best performer showing little sensitivity to process noise covariance and yielding reconstructions with less artifacts even when the ensemble size is small.

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“…The concept of local (or conditional) statistical linearization has been used by various authors for developing numerical schemes in order to calculate the response pdf of SDEs [61,62]. Local linearization has been more widely used in solving nonlinear deterministic and SDEs in the time domain (see [63,64] and references therein). To complete the formulation of the LSL problem (5.2) at any point (α 0 , β 0 ) of the RE phase space, we have to calculate the coefficients A 0 , B 0 .…”

Section: (B) Localized Statistical Linearizationmentioning

confidence: 99%

Beyond the Markovian assumption: response–excitation probabilistic solution to random nonlinear differential equations in the long time

2015

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Uncertainty quantification for dynamical systems under non-white excitation is a difficult problem encountered across many scientific and engineering disciplines. Difficulties originate from the lack of Markovian character of system responses. The response–excitation (RE) theory, recently introduced by Sapsis & Athanassoulis (Sapsis & Athanassoulis 2008 Probabilistic Eng. Mech. 23, 289–306 ( doi:10.1016/j.probengmech.2007.12.028 )) and further studied by Venturi et al. (Venturi et al. 2012 Proc. R. Soc. A 468, 759–783 ( doi:10.1098/rspa.2011.0186 )), is a new approach, based on a simple differential constraint which is exact but non-closed. The evolution equation obtained for the RE probability density function (pdf) has the form of a generalized Liouville equation, with the excitation time frozen in the time-derivative term. In this work, the missing information of the RE differential constraint is identified and a closure scheme is developed for the long-time, stationary, limit-state of scalar nonlinear random differential equations (RDEs) under coloured excitation. The closure scheme does not alter the RE evolution equation, but collects the missing information through the solution of local statistically linearized versions of the nonlinear RDE, and interposes it into the solution scheme. Numerical results are presented for two examples, and compared with Monte Carlo simulations.

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Higher order weak linearizations of stochastically driven nonlinear oscillators

Cited by 12 publications

References 21 publications

New computational approaches for wrinkled and slack membranes

New computational approaches for wrinkled and slack membranes

Self‐regularized pseudo time‐marching schemes for structural system identification with static measurements

Beyond the Markovian assumption: response–excitation probabilistic solution to random nonlinear differential equations in the long time

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