Symplectic Analysis of Wrinkles in Elastic Layers With Graded Stiffnesses

2018 ◽  
Vol 86 (1) ◽  
Author(s):  
Jianjun Sui ◽  
Junbo Chen ◽  
Xiaoxiao Zhang ◽  
Guohua Nie ◽  
Teng Zhang
Keyword(s):  
Finite Element ◽  
Hamiltonian System ◽  
Eigenvalue Problem ◽  
Time Variable ◽  
Time Varying ◽  
Thickness Direction ◽  
Varying Coefficients ◽  
Elastic Layers ◽  
Critical Strains ◽  
Graded Structures

Wrinkles in layered neo-Hookean structures were recently formulated as a Hamiltonian system by taking the thickness direction as a pseudo-time variable. This enabled an efficient and accurate numerical method to solve the eigenvalue problem for onset wrinkles. Here, we show that wrinkles in graded elastic layers can also be described as a time-varying Hamiltonian system. The connection between wrinkles and the Hamiltonian system is established through an energy method. Within the Hamiltonian framework, the eigenvalue problem of predicting wrinkles is defined by a series of ordinary differential equations with varying coefficients. By modifying the boundary conditions at the top surface, the eigenvalue problem can be efficiently and accurately solved with numerical solvers of boundary value problems. We demonstrated the accuracy of the symplectic analysis by comparing the theoretically predicted displacement eigenfunctions, critical strains, and wavelengths of wrinkles in two typical graded structures with finite element simulations.

scholarly journals A Mathematical Analysis of the Strip-Element Method for the Computation of Dispersion Curves of Guided Waves in Anisotropic Layered Media

2010 ◽  
Vol 2010 ◽  
pp. 1-17 ◽  
Author(s):  
F. Schöpfer ◽  
F. Binder ◽  
A. Wöstehoff ◽  
T. Schuster
Keyword(s):  
Finite Element ◽  
Eigenvalue Problem ◽  
Guided Waves ◽  
Dispersion Curves ◽  
Thickness Direction ◽  
Finite Element Discretization ◽  
Element Discretization ◽  
Infinite Dimensional ◽  
Mathematical Background ◽  
Element Method

Dispersion curves of elastic guided waves in plates can be efficiently computed by the Strip-Element Method. This method is based on a finite-element discretization in the thickness direction of the plate and leads to an eigenvalue problem relating frequencies to wavenumbers of the wave modes. In this paper we present a rigorous mathematical background of the Strip-Element Method for anisotropic media including a thorough analysis of the corresponding infinite-dimensional eigenvalue problem as well as a proof of the existence of eigenvalues.

scholarly journals A State Space Method for Surface Instability of Elastic Layers With Material Properties Varying in Thickness Direction

2014 ◽  
Vol 81 (8) ◽  
Author(s):  
Zhigen Wu ◽  
Jixiang Meng ◽  
Yihua Liu ◽  
Hao Li ◽  
Rui Huang
Keyword(s):  
Eigenvalue Problem ◽  
State Space ◽  
Elastic Properties ◽  
Linear Elasticity ◽  
Surface Instability ◽  
The State ◽  
State Space Method ◽  
Thickness Direction ◽  
Elastic Layers ◽  
Space Method

A state space method is proposed for analyzing surface instability of elastic layers with elastic properties varying in the thickness direction. By assuming linear elasticity with nonlinear kinematics, the governing equations for the incremental stress field from a fundamental state are derived for arbitrarily graded elastic layers subject to plane-strain compression, which lead to an eigenvalue problem. By discretizing the elastic properties into piecewise constant functions with homogeneous sublayers, a state space method is developed to solve the eigenvalue problem and predict the critical condition for onset of surface instability. Results are presented for homogeneous layers, bilayers, and continuously graded elastic layers. The state space solutions for elastic bilayers are in close agreement with the analytical solution for thin film wrinkling within the limit of linear elasticity. Numerical solutions for continuously graded elastic layers are compared to finite element results in a previous study (Lee et al., 2008, J. Mech. Phys. Solids, 56, pp. 858–868). As a semi-analytical approach, the state space method is computationally efficient for graded elastic layers, especially for laminated multilayers.

scholarly journals The VIT Transform Approach to Discrete-Time Signals and Linear Time-Varying Systems

Eng ◽  
2021 ◽  
Vol 2 (1) ◽  
pp. 99-125
Author(s):  
Edward W. Kamen
Keyword(s):  
Discrete Time ◽  
Initial Conditions ◽  
Linear Time ◽  
Order Term ◽  
Initial Time ◽  
Time Varying ◽  
Varying Coefficients ◽  
Time Signals ◽  
Time Varying Systems ◽  
Time Varying Coefficients

A transform approach based on a variable initial time (VIT) formulation is developed for discrete-time signals and linear time-varying discrete-time systems or digital filters. The VIT transform is a formal power series in z−1, which converts functions given by linear time-varying difference equations into left polynomial fractions with variable coefficients, and with initial conditions incorporated into the framework. It is shown that the transform satisfies a number of properties that are analogous to those of the ordinary z-transform, and that it is possible to do scaling of z−i by time functions, which results in left-fraction forms for the transform of a large class of functions including sinusoids with general time-varying amplitudes and frequencies. Using the extended right Euclidean algorithm in a skew polynomial ring with time-varying coefficients, it is shown that a sum of left polynomial fractions can be written as a single fraction, which results in linear time-varying recursions for the inverse transform of the combined fraction. The extraction of a first-order term from a given polynomial fraction is carried out in terms of the evaluation of zi at time functions. In the application to linear time-varying systems, it is proved that the VIT transform of the system output is equal to the product of the VIT transform of the input and the VIT transform of the unit-pulse response function. For systems given by a time-varying moving average or an autoregressive model, the transform framework is used to determine the steady-state output response resulting from various signal inputs such as the step and cosine functions.

scholarly journals Estimating latent group structure in time-varying coefficient panel data models

2019 ◽  
Author(s):  
Jia Chen
Keyword(s):  
Panel Data ◽  
Data Models ◽  
Panel Data Models ◽  
Time Varying ◽  
Clustering Method ◽  
Varying Coefficient ◽  
Varying Coefficients ◽  
Time Varying Coefficients ◽  
Latent Group ◽  
Group Structures

Summary This paper studies the estimation of latent group structures in heterogeneous time-varying coefficient panel data models. While allowing the coefficient functions to vary over cross-sections provides a good way to model cross-sectional heterogeneity, it reduces the degree of freedom and leads to poor estimation accuracy when the time-series length is short. On the other hand, in a lot of empirical studies, it is not uncommon to find that heterogeneous coefficients exhibit group structures where coefficients belonging to the same group are similar or identical. This paper aims to provide an easy and straightforward approach for estimating the underlying latent groups. This approach is based on the hierarchical agglomerative clustering (HAC) of kernel estimates of the heterogeneous time-varying coefficients when the number of groups is known. We establish the consistency of this clustering method and also propose a generalised information criterion for estimating the number of groups when it is unknown. Simulation studies are carried out to examine the finite-sample properties of the proposed clustering method as well as the post-clustering estimation of the group-specific time-varying coefficients. The simulation results show that our methods give comparable performance to the penalised-sieve-estimation-based classifier-LASSO approach by Su et al. (2018), but are computationally easier. An application to a panel study of economic growth is also provided.

scholarly journals Finite Element Analysis of Thermoelastic Contact Stability

1994 ◽  
Vol 61 (4) ◽  
pp. 919-922 ◽  
Author(s):  
Taein Yeo ◽  
J. R. Barber
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Steady State ◽  
Eigenvalue Problem ◽  
Linear Perturbation ◽  
Dimensional System ◽  
The Finite Element Method ◽  
One Dimensional ◽  
Thermoelastic Contact ◽  
Element Method

When heat is conducted across an interface between two dissimilar materials, theimoelastic distortion affects the contact pressure distribution. The existence of a pressure-sensitive thermal contact resistance at the interface can cause such systems to be unstable in the steady-state. Stability analysis for thermoelastic contact has been conducted by linear perturbation methods for one-dimensional and simple two-dimensional geometries, but analytical solutions become very complicated for finite geometries. A method is therefore proposed in which the finite element method is used to reduce the stability problem to an eigenvalue problem. The linearity of the underlying perturbation problem enables us to conclude that solutions can be obtained in separated-variable form with exponential variation in time. This factor can therefore be removed from the governing equations and the finite element method is used to obtain a time-independent set of homogeneous equations in which the exponential growth rate appears as a linear parameter. We therefore obtain a linear eigenvalue problem and stability of the system requires that all the resulting eigenvalues should have negative real part. The method is discussed in application to the simple one-dimensional system of two contacting rods. The results show good agreement with previous analytical investigations and give additional information about the migration of eigenvalues in the complex plane as the steady-state heat flux is varied.

Integrity Assessment of a Free-Fall Lifeboat Launched From a FPSO

2015 ◽  
Author(s):  
Guomin Ji ◽  
Nabila Berchiche ◽  
Sébastien Fouques ◽  
Thomas Sauder ◽  
Svein-Arne Reinholdtsen
Keyword(s):  
Finite Element ◽  
Pressure Distribution ◽  
Structural Integrity ◽  
Dynamic Effect ◽  
Sensitivity Study ◽  
Computational Time ◽  
Load Factor ◽  
Time Varying ◽  
Integrity Assessment

The paper addresses the structural integrity assessment of lifeboat launched from floating production, storage and offloading (FPSO) vessels. The study is based on long-term drop lifeboat simulations accounting for more than 50 years of hindcast data of metocean conditions and corresponding FPSO motions. Selection of the load cases and strength analyses with high computational time is a challenge. The load cases analyzed are those corresponding to the 99th percentile of long term distribution of indicators for large slamming loads (CARXZ) or large submergence (Imaxsub). For six selected cases, the time-varying pressure distribution on the lifeboat hull during and after water impact is calculated by CFD simulations using StarCCM+. The finite element model (FEM) of the composite structure of the lifeboat is modelled by ABAQUS. Quasi-static finite element (FE) analyses are performed for the selected load cases. The structural integrity is assessed by the maximum stress and Tsai-Wu failure measure. In the present study, the load and resistance factors are combined and applied to the response. A sensitivity study is performed to investigate the non-linear load/response effects when the load factor is applied to the load. In addition, dynamic analysis is performed with the time-varying pressure distribution for selected case and the dynamic effect is investigated.

Influence of Axial Excitations and of Boundary Conditions on the Parametric Instability of a Beam

1995 ◽  
Author(s):  
Régis Dufour ◽  
Alain Berlioz ◽  
Thomas Streule
Keyword(s):  
Finite Element ◽  
Boundary Conditions ◽  
Ritz Method ◽  
Parametric Instability ◽  
Experimental Tests ◽  
Time Varying ◽  
Periodic Force ◽  
Modal Reduction ◽  
Time Varying Parameter ◽  
The Stability

Abstract In this paper the stability of the lateral dynamic behavior of a pinned-pinned, clamped-pinned and clamped-clamped beam under axial periodic force or torque is studied. The time-varying parameter equations are derived using the Rayleigh-Ritz method. The stability analysis of the solution is based on Floquet’s theory and investigated in detail. The Rayleigh-Ritz results are compared to those of a finite element modal reduction. It shows that the lateral instabilities of the beam depend on the forcing frequency, the type of excitation and the boundary conditions. Several experimental tests enable the validation of the numerical results.

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