In this paper, a novel approach is proposed for solving the nonlinear problems based on the collocation and energy balance methods (EBMs). Rational approximation is employed as an initial guess and then it is combined with EBM and collocation method for solving nonlinear oscillators with cubic term. Obtained frequency amplitude relationship is compared with exact numerical solution and subsequently, a very excellent accuracy will be revealed. According to the numerical comparisons, this method provides high accuracy with 0.03% relative error for Duffing equation with strong nonlinearity in the second-order of approximation. Furthermore, achieved results are compared with other types of modified EBMs and the second-order of harmonic balance method. It is demonstrated that the new proposed method has the highest accuracy in comparison with different approaches such as modified EBMs and the second-order of harmonic balance method.

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The feasibility of reduced order modeling for turbulent flows using Proper Orthogonal Decomposition (POD) based Surrogate modeling and Galerkin Projection is demonstrated for use in the hydrodynamic modeling of the Very High Temperature Reactor (VHTR) lower plenum. The lower plenum of the Helium-cooled VHTR consists of vertical cylinder arrays subjected to turbulent jetting and cross-flow. Unsteady Reynolds-Averaged Navier-Stokes (RANS) Computational Fluid Dynamics (CFD) simulations are used to acquire an ensemble of possible solution fields for flow around a circular cylinder in an open domain. Numerical results are validated to prior published literature. From the resultant data ensemble are extracted the coherent structures to create an optimal basis. POD is used to extract the coherent structures as this technique has been demonstrated to provide a basis of a chosen dimension such that the average L2-error is minimized for the best approximation of the basis to the data ensemble. The resultant optimal basis is used to construct accurate reduced order models. The computational effectiveness and insights revealed by this reduced order modeling approach are discussed for both the Surrogate modeling approach and Galerkin Projection.

(2016) 'Adaptive reduced-basis generation for reduced-order modeling for the solution of stochastic nondestructive evaluation problems.', Computer methods in applied mechanics and engineering., 310. pp. 172-188. Further information on publisher's website: http://dx. Additional information: Use policy The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that: • a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders. Please consult the full DRO policy for further details. Abstract A novel algorithm for creating a computationally efficient approximation of a system response that is defined by a boundary value problem is presented. More specifically, the approach presented is focused on substantially reducing the computational expense required to approximate the solution of a stochas-tic partial differential equation, particularly for the purpose of estimating the solution to an associated nondestructive evaluation problem with significant system uncertainty. In order to achieve this computational efficiency, the approach combines reduced-basis reduced-order modeling with a sparse grid col-location surrogate modeling technique to estimate the response of the system of interest with respect to any designated unknown parameters, provided the distributions are known. The reduced-order modeling component includes a novel algorithm for adaptive generation of a data ensemble based on a nested grid technique, to then create the reduced-order basis. The capabilities and potential applicability of the approach presented are displayed through two simulated case studies regarding inverse characterization of material properties for two different physical systems involving some amount of significant uncertainty. The first case study considered characterization of an unknown localized reduction in stiffness of a structure from simulated frequency response function based nondestructive testing. Then, the second case study considered characterization of an unknown temperature-dependent thermal conductivity of a solid from simulated thermal testing. Overall, the surrogate modeling approach was shown through both simulated examples to provide accurate solution estimates to inverse problems for systems represented by stochastic partial differential equations with a fraction of the typical computational cost.