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Topology optimization of nonlinear structural dynamics with invariant manifold-based reduced order models

structural AND MULTIDISCIPLINARY OPTIMIZATION 2025年第4期68卷 1-15页

作者： Pozzi, Matteo Marconi, Jacopo Jain, Shobhit Li, Mingwu Braghin, Francesco Politecn Milan Dept Mech Engn Via G La Masa 1 I-20156 Milan MI Italy Delft Univ Technol Delft Inst Appl Math Mekelweg 4 NL-2628CD Delft Netherlands Southern Univ Sci & Technol Dept Mech & Aerosp Engn Shenzhen Peoples R China

We present a structural topology optimization method to tailor the hardening/softening dynamic response of nonlinear mechanical systems. The coefficient that controls this behavior is computed analytically using the third-order normal-form parametrization of the Lyapunov subcenter manifold, which eliminates the need for expensive full-order simulations and numerical continuation to approximate the so-called backbone curve of the system. The method further leverages the adjoint method for efficiently computing sensitivities of the objective function and constraints, while the explicit formulation of nonlinear internal elastic forces through tensor notation simplifies these evaluations. Notably, this tensorial approach is computationally efficient, especially when applied to a regular grid of elements. Consequently, the proposed approach offers a robust and efficient framework for optimizing the dynamic performance of nonlinear mechanical structures modeled with high-dimensional finite element models. The findings are corroborated through examples of two geometrically nonlinear systems, a Messerschmitt-B & ouml;lkow-Blohm (MBB) beam and a microelectro-mechanical system (MEMS) inertial resonator.

关键词： Topology optimization nonlinear structural dynamics Lyapunov subcenter manifold Reduced order modeling

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nonlinear structural dynamics Analysis Using Weighted Residual Integration

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MECHANICS OF ADVANCED MATERIALS AND STRUCTURES 2013年第3期20卷 199-216页

作者： Gholampour, A. A. Ghassemieh, M. Univ Tehran Sch Civil Engn Tehran Iran

A new time integration method is proposed for solving a differential equation of motion of structural dynamics problems with nonlinear stiffness. In this method, it is assumed that order of variation of acceleration is quadratic in each time step and, therefore, variation of displacement in each time step is a fourth-order polynomial that has five coefficients. Other than implementing initial conditions and satisfying equation of motions at both ends, weighted residual integral is also used in order to evaluate the unknown coefficients. By increasing order of acceleration, more terms of Taylor series are used, which were expected to have better responses than the other classical methods. The proposed method is non-dissipative and its numerical dispersion showed to be clearly less than the other methods. By studying stability of the method, it reveals that critical time step duration for the proposed method can be larger than the critical time step duration obtained from other classical methods;also, computational time for analysis with critical time step duration for the proposed method becomes less in comparison with linear acceleration and central difference methods. The order of accuracy of the proposed method is four, which showed to be very accurate in almost all practical problems.

关键词： nonlinear structural dynamics weighted residual numerical stability numerical accuracy dissipation dispersion

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nonlinear structural dynamics for assessing aircraft impact at nuclear power plants 9

Nonlinear structural dynamics for assessing aircraft impact ...

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9th International Conference on structural dynamics (EURODYN)

作者： James, Randy J. Wiesner, Jeremy ANATECH Corp San Diego CA 92121 USA

ISBN: (纸本)9789727521654

The next generation of nuclear power plants is being designed worldwide with consideration for the effects of impact from a large commercial aircraft as a malevolent act. Detailed, realistic nonlinear structural analyses are required to assess the physical damage caused by an impact in order to meet regulatory requirements. In the U. S., industry developed methodology for conducting aircraft impact analyses have been accepted by the U. S. Nuclear Regulatory Commission (NRC) and requires advanced constitutive modeling of reinforced concrete, detailed and realistic structural modeling, appropriate material failure criteria, and realistic application of the impact load using robust explicit dynamics analysis software. The most common approach for load application is the Riera method that uses a force time history applied over a representative contact area as a pressure load. Sample studies demonstrate that the Riera method offers good analytical results for structural configurations that can fully resist the impact. For structural configurations that are insufficient to withstand the impact forces, the Riera method is useful for assessing the failure mode of the impacted wall, but the method fails to quantify the consequences of aircraft debris that penetrates beyond the initial point of impact. The sample studies also demonstrate that for some cases the design-basis structure can adequately meet aircraft impact requirements. For other cases where reinforcement is minimal or the span of the impacted wall is large, design enhancements can be made to offer the necessary resistance to limit damage to the exterior wall. If the impacted wall fails, the consequences of debris entering the plant can be determined by an alternative approach for load application called the missile-target interaction method. Further sample studies demonstrate that with this approach, the aircraft can be explicitly modeled and used to investigate a realistic extent of damage, which may be accepta

关键词： Aircraft Impact Nuclear Power Plants nonlinear structural dynamics Advanced Concrete Material Modeling

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Unscented Kalman filtering for nonlinear structural dynamics

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nonlinear dynamics 2007年第1-2期49卷 131-150页

作者： Mariani, Stefano Ghisi, Aldo Politecn Milan Dipartimento Ingn Strutturale I-20133 Milan Italy

Joint estimation of unknown model parameters and unobserved state components for stochastic, nonlinear dynamic systems is customarily pursued via the extended Kalman filter (EKF). However, in the presence of severe nonlinearities in the equations governing system evolution, the EKF can become unstable and accuracy of the estimates gets poor. To improve the results, in this paper we account for recent developments in the field of statistical linearization and propose an unscented Kalman filtering procedure. In the case of softening single degree-of-freedom structural systems, we show that the performance of the unscented Kalman filter (UKF), in terms of state tracking and model calibration, is significantly superior to that of the EKF.

关键词： Kalman filter nonlinear structural dynamics parameter identification statistical linearization

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Adjoint sensitivities of time-periodic nonlinear structural dynamics via model reduction

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COMPUTERS & STRUCTURES 2010年第19-20期88卷 1110-1123页

作者： Stanford, Bret Beran, Philip Kurdi, Mohammad USAF Res Lab Air Vehicles Directorate Wright Patterson AFB OH 45433 USA

This work details a comparative analysis of six methods for computing the transient system response and adjoint design derivatives of a nonlinear structure under a periodic external actuation. Time marching via implicit integration, a time-periodic spectral element method, and a time-periodic finite difference method are all considered, each with or without POD-based model reduction of the system of equations. Details of each method, and the concomitant adjoint sensitivities, are provided in terms of accuracy, stability, and Jacobian sparsity patterns. Each method is used for a gradient-based optimization of a nonlinear planar beam, periodically actuated at its root, subject to a large number of structural design parameters. The method with the lowest computational design cost is found to be a strong function of the harmonic content and the nonlinearity of the transient response. Published by Elsevier Ltd.

关键词： nonlinear structural dynamics Adjoint derivatives Model reduction Spectral elements

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An efficient and unconditionally stable numerical algorithm for nonlinear structural dynamics

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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING 2020年第20期121卷 4614-4629页

作者： Xu, Junjie Huang, Yuli Qu, Zhe China Earthquake Adm Key Lab Earthquake Engn & Engn Vibrat Inst Engn Mech Sanhe Peoples R China Univ Buffalo Dept Civil Struct & Environm Engn Buffalo NY USA Arup San Francisco CA 94105 USA

This article proposes an algorithm for express solutions in nonlinear structural dynamics. Our strategy is to adopt a typical time integrator and accept the solution after a constant number of iterations using a constant Jacobian matrix. Its success may not be initially obvious, but we demonstrate that the proposed algorithm not only is fully operational but also inherits the advantages of the host time integrators such as the unconditional stability, the order of accuracy, and the numerical dissipation that helps suppress the spurious higher mode oscillation. The use of a constant Jacobian matrix plays the key role in minimizing the computational expense associated with matrix operations. We first study the optimization of the number of iterations, then present the consistency and stability analysis followed by some examples verifying these features, and conclude by showing the exponential efficiency improvement in a response history analysis of a high-rise building fully equipped with nonlinearities.

关键词： constant number of iterations constant Jacobian matrix nonlinear structural dynamics

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A New Unconditionally Stable Time Integration Method for Analysis of nonlinear structural dynamics

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JOURNAL OF APPLIED MECHANICS-TRANSACTIONS OF THE ASME 2013年第2期80卷 021024-021024-12页

作者： Gholampour, Ali Akbar Ghassemieh, Mehdi Karimi-Rad, Mahdi Univ Tehran Sch Civil Engn Tehran 14174 Iran

A new time integration scheme is presented for solving the differential equation of motion with nonlinear stiffness. In this new implicit method, it is assumed that the acceleration varies quadratically within each time step. By increasing the order of acceleration, more terms of the Taylor series are used, which are expected to have responses with better accuracy than the classical methods. By considering this assumption and employing two parameters delta and alpha, a new family of unconditionally stable schemes is obtained. The order of accuracy, numerical dissipation, and numerical dispersion are used to measure the accuracy of the proposed method. Second order accuracy is achieved for all values of delta and alpha. The proposed method presents less dissipation at the lower modes in comparison with Newmark's average acceleration, Wilson-theta, and generalized-alpha methods. Moreover, this second order accurate method can control numerical damping in the higher modes. The numerical dispersion of the proposed method is compared with three unconditionally stable methods, namely, Newmark's average acceleration, Wilson-theta, and generalized-alpha methods. Furthermore, the overshooting effect of the proposed method is compared with these methods. By evaluating the computational time for analysis with similar time step duration, the proposed method is shown to be faster in comparison with the other methods.

关键词： nonlinear structural dynamics implicit method unconditionally stable method second order accuracy overshooting effect computational time

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Approximation and calibration of nonlinear structural dynamics

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nonlinear dynamics 2005年第1-2期39卷 113-128页

作者： Zimmerman, DC Hasselman, T Anderson, M Univ Houston Dept Mech Engn Houston TX 77204 USA ACTA Inc Torrance CA 90505 USA

In this paper, a methodology for the calibration of nonlinear structural dynamic models is presented. Calibration of nonlinear structural dynamics offers several additional challenges beyond that of linear dynamics. Even with advanced computational power, exact nonlinear finite element simulations often take several hours to complete on engineering workstations. Thus, the proposed model calibration method utilizes an approximate structural model. This approximate analysis is embedded in the outer loop, which utilizes an exact finite element analysis to verify the validity of the approximate model. If the approximate model is shown to be invalid at that point in parameter space, then the new exact analysis is used to develop an improved approximate model and the inner loop is executed again. Specifically, this paper will focus on the two key aspects of the inner loop, namely the development of an approximate model, and the parameter identification using the approximate model.

关键词： blast dynamics genetic algorithms neural networks nonlinear structural dynamics principle component analysis

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Input-ouput approximation for nonlinear structural dynamics

Input-ouput approximation for nonlinear structural dynamics

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作者： Beaver, Stefanie Rene’ Texas A&M University

Input¬output approximation of spacecraft motion is convenient and necessary in many situations. For a rigid¬body spacecraft, this process is simple because the system is governed by a set of equations that is linear in the system parameters. However, the combination of a ﬂexible appendage and a rigid hub adds complexity by increasing the degrees of freedom and by introducing nonlinear coupling between the hub and appendage. Assumed Modes is one technique for modeling ﬂexible body motion. Traditional Assumed Modes uses a set of linear assumed modes, but when dealing with rotating ﬂexible systems, a modiﬁcation of this method allows for the use of quadratic assumed modes. The quadratic assumed model provides an increased level of sophistication, but the derivation still neglects a set of higher¬order terms. This work develops the nonlinear equations of motion that arise from retaining these nonlinear, higher¬order terms. Simulation results reveal that the inclusion of these terms noticeably changes the motion of the system. Once the equations of motion have been developed, focus turns to the input¬output mapping of a system that is simulated using this set of equations. Approxi¬mating linear systems is straightforward, and many methods exist that can success¬fully perform this function. On the other hand, few approximation methods exist for nonlinear systems. Researchers at Texas A&M University recently developed one such method that obtains a linear estimation and then uses an adaptive polynomial estimation method to compensate for the disparity between that estimate and the true measurements. This research includes an in¬depth investigation of this nonlinear approximation technique. Finally, these two major research thrusts are combined, and input¬output approx¬imation is performed on the nonlinear rotational spacecraft model developed herein. The results of this simulation show that the nonlinear method holds a signiﬁcant advantage over a traditional linear meth

关键词： input-output approximation nonlinear structural dynamics Book Thesis

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An Efficient nonlinear structural dynamics Solver for Use in Computational Aeroelastic Analysis

An Efficient Nonlinear Structural Dynamics Solver for Use in...

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作者： Freno, Brian Andrew Texas A&M University

Aerospace structures with large aspect ratio, such as airplane wings, rotorcraft blades, wind turbine blades, and jet engine fan and compressor blades, are particularly susceptible to aeroelastic phenomena. Finite element analysis provides an effective and generalized method to model these structures; however, it is computationally expensive. Fortunately, these structures have a length appreciably larger than the largest cross-sectional diameter. This characteristic is exploitable as these potential aeroelastically unstable structures can be modeled as cantilevered beams, drastically reducing computational time. In this thesis, the nonlinear equations of motion are derived for an inextensional, non-uniform cantilevered beam with a straight elastic axis. Along the elastic axis, the cross-sectional center of mass can be o set in both dimensions, and the principal bending and centroidal axes can each be rotated uniquely. The Galerkin method is used, permitting arbitrary and abrupt variations along the length that require no knowledge of the spatial derivatives of the beam properties. Additionally, these equations consistently retain all third-order nonlinearities that account for flexural-flexural-torsional coupling and extend the validity of the equations for large deformations. Furthermore, linearly independent shape functions are substituted into these equations, providing an efficient method to determine the natural frequencies and mode shapes of the beam and to solve for time-varying deformation. This method is validated using finite element analysis and is extended to swept wings. The importance of retaining cubic terms, in addition to quadratic terms, for nonlinear analysis is demonstrated for several examples. Ultimately, these equations are coupled with a fluid dynamics solver to provide a structurally efficient aeroelastic program.

关键词： aeroelasticity nonlinear structural dynamics computational aeroelasticity elasticity aerodynamics Thesis

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