Linear Elastic Systems Research Articles

A Generalized Structure-Dependent Semi-Explicit Method for Structural Dynamics

In this paper, a novel generalized structure-dependent semi-explicit method is presented for solving dynamical problems. Some existing algorithms with the same displacement and velocity update formulas are included as the special cases, such as three Chang algorithms. In general, the proposed method is shown to be second-order accurate and unconditionally stable for linear elastic and stiffness softening systems. The comprehensive stability and accuracy analysis, including numerical dispersion, energy dissipation, and the overshoot behavior, are carried out in order to gain insight into the numerical characteristics of the proposed method. Some numerical examples are presented to show the suitable capability and efficiency of the proposed method by comparing with other existing algorithms, including three Chang algorithms and Newmark explicit method (NEM). The unconditional stability and second-order accuracy make the novel methods take a larger time-step, and the explicitness of displacement at each time-step succeeds in avoiding nonlinear iterations for solving nonlinear stiffness systems.

Torsional Response Induced by Lateral Displacement and Inertial Force

Eccentric distribution of stiffness, damping, and mass of a structure, and spatially non-uniform ground motion input to a long or large base mat of a structure are well-known causes of torsional response. We have discovered that the torque generated by horizontal displacement and perpendicular inertial force, which we call the Q- effect, can be a cause of torsional response. We formulated the equation of motion of a single finite-size mass-linear elastic shear and torsion spring model and clarified the resonance condition of the torsional response to sinusoidal ground acceleration. Time-history response analysis verified that the torsional response forms beat and the maximum torsional response of the simulation result agrees with that theoretically predicted. Further time-history response analysis conducted of white noise ground acceleration showed that even one-directional white noise ground acceleration can induce torsional response in a linear elastic system without any structural eccentricity.

Eigenvalue convergence on perturbed Lipschitz domains for elliptic systems with mixed general decompositions of the boundary

We consider eigenvalues of an elliptic operatorLu=−∂∂xj(Aij∂u∂xi) where u=(u1,...,um)T is a vector valued function and the coefficients Aij are m×m matrices whose elements aijαβ are bounded and symmetric. We perturb our domain Ω0 by adding a set of small measure, Tε to form the domain Ωε. We prescribe mixed boundary conditions on quite general decompositions of the boundary and look at the behavior of the eigenvalues of Ωε as Tε shrinks to zero. We look at systems which satisfy either a strong ellipticity condition, a Legendre–Hadamard condition, or in particular, the system of linear elasticity.

Elastic and Inelastic Near-Fault Input Energy Spectra

The main purpose of this study is to develop a reliable model for predicting the input energy spectra of near-fault ground motions for linear elastic and inelastic systems, and to evaluate the effect of damping and lateral strength on energy dissipation demands. An attenuation model has been developed through one-stage nonlinear regression analysis. Comparative results revealed that near-fault ground motions have significantly larger energy dissipation demands, which are very sensitive to earthquake magnitude and soil type. The effect of damping on elastic and inelastic near-fault input energy spectra is insignificant. Near-fault input energy spectra for inelastic systems is dependent on lateral strength ratio R for short period systems, however, there is almost no dependency on lateral strength for intermediate and long period systems, recalling an equal energy rule. This is a significant advantage for an energy-based design approach.

Dynamical identification of a space varying coefficient in a system with persistent memory

In this paper we solve the problem of the identification of a coefficient which appears in the equation of a distributed system with persistent memory. This system is encountered for example when modelling the interconnection of linear elastic and viscoelastic systems. The additional data used in the identification are subsumed in the input output map from the deformation to the traction on the boundary. We extend a dynamical approach to identification introduced by Belishev in the case of purely elastic (memoryless) bodies and based on a special equation due to Blagoveshchenskiı̌. So, in particular, we extend Blagoveshchenskiı̌ equation to our class of systems with persistent memory.

Domain decomposition preconditioners for multiscale problems in linear elasticity

SummaryWe analyze two‐level overlapping Schwarz domain decomposition methods for vector‐valued piecewise linear finite element discretizations of the PDE system of linear elasticity. The focus of our study lies in the application to compressible, particle‐reinforced composites in 3D with large jumps in their material coefficients. We present coefficient‐explicit bounds for the condition number of the two‐level additive Schwarz preconditioned linear system. Thereby, we do not require that the coefficients are resolved by the coarse mesh. The bounds show a dependence of the condition number on the energy of the coarse basis functions, the coarse mesh, and the overlap parameters, as well as the coefficient variation. Similar estimates have been developed for scalar elliptic PDEs by Graham et al. The coarse spaces to which they apply here are assumed to contain the rigid body modes and can be considered as generalizations of the space of piecewise linear vector‐valued functions on a coarse triangulation. The developed estimates provide a concept for the construction of coarse spaces, which can lead to preconditioners that are robust with respect to high contrasts in Young's modulus and the Poisson ratio of the underlying composite. To confirm the sharpness of the theoretical findings, we present numerical results in 3D using vector‐valued linear, multiscale finite element and energy‐minimizing coarse spaces. The theory is not restricted to the isotropic (Lamé) case, extends to the full‐tensor case, and allows applications to multiphase materials with anisotropic constituents in two and three spatial dimensions. However, the bounds will depend on the ratio of largest to smallest eigenvalue of the elasticity tensor.

Study of the Regularity of Solutions for a Elasticity System with Integrable Data with Respect to the Distance Function to the Boundary

In this article, we are interested in the existence, uniqueness and regularity of the solution of the linear elasticity system. More precisely, the quasi-static elasticity system. In the first part, we study the existence of a weak solution and the regularity in the space $$W^{1, p}_0(\Omega ),\ \forall p \in ]1, +\infty [$$ for a p-integrable source function. In the second part, the very weak solution is introduced which can be considered when the second member is a function with a very weak solution, for example, a locally integrable function. Such source functions lead to a lack of regularity for the solution in the fact that existence in classical spaces is no longer assured. So, to overcome this difficulty, the strategy consists in approaching it by another more regular problem “converging” towards the initial problem “in a direction to be specified”.

An inverse problem in elastography involving Lamé systems

Abstract This paper deals with some inverse problems for the linear elasticity system with origin in elastography: we try to identify the material coefficients from some extra information on (a part of) the boundary. In our main result, we assume that the total variation of the coefficient matrix is a priori bounded. We reformulate the problem as the minimization of a function in an appropriate constraint set. We prove that this extremal problem possesses at least one solution with the help of some regularity results. Two crucial ingredients are a Meyers-like theorem that holds in the context of linear elasticity and a nonlinear interpolation result by Luc Tartar. We also perform some numerical experiments that provide satisfactory results. To this end, we apply the Augmented Lagrangian algorithm, completed with a limited-memory BFGS subalgorithm. Finally, on the basis of these experiments, we illustrate the influence of the starting guess and the errors in the data on the behavior of the iterates.

An approach for maximizing the smallest eigenfrequency of structure vibration based on piecewise constant level set method

Maximization of the smallest eigenfrequency of the linearized elasticity system with area constraint is investigated. The elasticity system is extended into a large background domain, but the void is vacuum and not filled with ersatz material. The piecewise constant level set (PCLS) method is applied to present two regions, the original material region and the void region. A quadratic PCLS function is proposed to represent the characteristic function. Consequently, the functional derivative of the smallest eigenfrequency with respect to PCLS function takes nonzero value in the original material region and zero in the void region. A penalty gradient algorithm is proposed, which initializes the whole background domain with the original material and decreases the area of original material region till the area constraint is satisfied. 2D and 3D numerical examples are presented, illustrating the validity of the proposed algorithm.

Boundary estimates in homogenization of systems of elasticity on Lipschitz domains: Lp theory

The primary purpose of this paper is to investigate a family of elliptic systems of linear elasticity with rapidly oscillating periodic coefficients, arising in the theory of homogenization in Lipschitz domains. As a consequence, for d≥4, we prove that the Lp Neumann and Lp Dirichlet boundary value problems for systems of second order linear elasticity are uniquely solvable for 2(d−1)d+1−δ<p<2+δ and 2−δ<p<2(d−1)d−3+δ respectively.

Proposition of crankshaft stiffness matrix modeling methodology

The paper presents the proposed methodology of crankshaft stiffness matrix modeling. Geometry and linear-elastic shaft properties are one of common occurrences of torsionally rigid vibration coupling in crank systems. Model and experimental studies require the stiffness of the object in question. In the case of linear-elastic systems it is identical with the definition of its stiffness matrix. In the introduction, the current state of knowledge in the field of identification and modeling of crankshaft dynamics was reviewed. The next chapter presents the research stand for the study of dynamic properties of crank systems. The measurement path was described, its parameters and preliminary results were presented. The next part concerns the dynamic model of the crank system used by the authors and the need for a methodology to determine the stiffness of the shaft in question. The following chapters describe the mathematical description of the proposed methodology and the results of the calculations. The article was finalized with a summary of conclusions drawn from the work carried out.

A new stochastic simulation algorithm for updating robust reliability of linear structural dynamic systems subjected to future Gaussian excitations

In this paper, we are interested in using system response data to update the robust failure probability that any particular response of a linear structural dynamic system exceeds a specified threshold during the time when the system is subjected to future Gaussian dynamic excitations. Computation of the robust reliability takes into account uncertainties from structural modeling in addition to the modeling of the uncertain excitations that the structure will experience during its lifetime. In partial, modal data from the structure are used as the data for the updating. By exploiting the properties of linear dynamics, a new approach based on stochastic simulation methods is proposed to update the robust reliability of the structure. The proposed approach integrates the Gibbs sampler for Bayesian model updating and Subset Simulation for failure probability computation. A new efficient approach for conditional sampling called ‘Constrained Metropolis within Gibbs sampling’ algorithm is developed by the authors. It is robust to the number of uncertain parameters and random variables and the dimension of modal data involved in the problem. The effectiveness and efficiency of the proposed approach are illustrated by two numerical examples involving linear elastic dynamic systems.

Effect of the damper property variability on the seismic reliability of linear systems equipped with viscous dampers

Viscous dampers are dissipation devices widely employed for seismic structural control. To date, the performance of systems equipped with viscous dampers has been extensively analysed only by employing deterministic approaches. However, these approaches neglect the response dispersion due to the uncertainties in the input as well as the variability of the system properties. Some recent works have highlighted the important role of these seismic input uncertainties in the seismic performance of linear and nonlinear viscous dampers. This study analyses the effect of the variability of damper properties on the probabilistic system response and risk. In particular, the paper aims at evaluating the impact of the tolerance allowed in devices’ quality control and production tests in terms of variation of the exceedance probabilities of the Engineering Demand Parameters (EDPs) which are most relevant for the seismic performance. A preliminary study is carried out to relate the variability of the constitutive damper characteristics to the tolerance limit allowed in tests and to evaluate the consequences on the device’s dissipation properties. In the subsequent part of the study, the sensitivity of the dynamic response is analysed by harmonic analysis. Finally, the seismic response sensitivity is studied by evaluating the influence of the allowed variability of the constitutive damper characteristics on the response hazard curves, providing the exceedance probability per year of EDPs. A set of linear elastic systems with different dynamic properties, equipped with linear and nonlinear dampers, are considered in the analyses, and subset simulation is employed together with the Markov Chain Monte Carlo method to achieve a confident estimate of small exceedance probabilities.

Homogenization in perforated domains and interior Lipschitz estimates

We establish interior Lipschitz estimates at the macroscopic scale for solutions to systems of linear elasticity with rapidly oscillating periodic coefficients and mixed boundary conditions in domains periodically perforated at a microscopic scale ε by establishing H1-convergence rates for such solutions. The interior estimates are derived directly without the use of compactness via an argument presented in [3] that was adapted for elliptic equations in [2] and [11]. As a consequence, we derive a Liouville type estimate for solutions to the systems of linear elasticity in unbounded periodically perforated domains.

Boundary estimates in elliptic homogenization

For a family of systems of linear elasticity with rapidly oscillating periodic coefficients, we establish sharp boundary estimates with either Dirichlet or Neumann conditions, uniform down to the microscopic scale, without smoothness assumptions on the coefficients. Under additional smoothness conditions, these estimates, combined with the corresponding local estimates, lead to the full Rellich type estimates in Lipschitz domains and Lipschitz estimates in $C^{1, \alpha}$ domains. The $C^\alpha$, $W^{1,p}$, and $L^p$ estimates in $C^1$ domains for systems with VMO coefficients are also studied. The approach is based on certain estimates on convergence rates. As a bi-product, we obtain sharp $O(\varepsilon)$ error estimates in $L^q(\Omega)$ for $q=\frac{2d}{d-1}$ and a Lipschitz domain $\Omega$, with no smoothness assumption on the coefficients.

Negative stiffness device for seismic protection of smart base isolated benchmark building

This paper presents a novel negative stiffness device (NSD) for the seismic protection of the base isolated benchmark building in near fault earthquake. The benchmark structure has eight stories and irregular plan, and the superstructure is considered to be a linear elastic system with lateral torsional behavior. Furthermore, it is equipped with low damping elastomeric rubber bearings and viscous dampers for seismic protection. The proposed NSD can change the stiffness of composite structure-device assembly, emulate “apparent yielding” of the structure system without permanent deformations. It consists of a highly compressed spring, a wheel and a curved template that the wheel can roll on. The device can achieve any force-displacement behavior within its designed range. Experimental results are presented to verify the analytical model of the NSD. Numerical simulation results under seven earthquake inputs show that the proposed device is effective in reducing the response of the base and superstructure in comparison with a sample clipped optimal controller and an H2/LQG sample controller.

Equivalent damping of bilinear hysteretic SDOF system considering the influence of initial elastic damping

The premise of equivalent linearization method is that the peak response of an inelastic system can be estimated as the peak response of a linear elastic system having reduced stiffness and increased damping.Different approaches have been used to determine the properties of the equivalent linear system, in particular the equivalent damping ratio. In general, equivalent damping is specified as the sum of elastic and hysteretic component, where the former is assumed to be constant, and the latter depends on the ductility and hysteresis model. It is found that, however, many studies only focus on the definition of hysteretic damping and omit the influence of elastic damping on the prediction accuracy of equivalent linearization method. Motivated by this limitation, comprehensive parametric analysis is performed in this paper based on bilinear hysteretic SDOF systems to identify the influence of elastic damping. Results show that elastic damping has significant influence on the estimation accuracy of equivalent linearization method. To improve the estimation accuracy of equivalent linearization method, improved method of equivalent damping is proposed by considering the influence of elastic damping. It is found that the proposed method is able to provide accurate and conservative values of equivalent damping for practical design of base isolation systems.

Boundary Korn Inequality and Neumann Problems in Homogenization of Systems of Elasticity

This paper concerns with a family of elliptic systems of linear elasticity with rapidly oscillating periodic coefficients, arising in the theory of homogenization. We establish uniform optimal regularity estimates for solutions of Neumann problems in a bounded Lipschitz domain with $L^2$ boundary data. The proof relies on a boundary Korn inequality for solutions of systems of linear elasticity and uses a large-scale Rellich estimate obtained in \cite{Shen-2016}.

A mixed-contact formulation for a dynamics simulation of flexible systems: An integration with model-reduction techniques

A new numerical procedure for efficient dynamics simulations of linear-elastic systems with unilateral contacts is proposed. The method is based on the event-driven integration of a contact problem with a combination of single- and set-valued force laws together with classical model-reduction techniques. According to the contact state, the developed event-driven integration enables the formulation of reduced system matrices. Moreover, to enable the transition among different reduced spaces the formulation of the initial conditions is also presented. The method has been developed separately for each of the four most popular model-reduction techniques (Craig–Bampton, MacNeal, Rubin and dual Craig–Bampton). The applicability of the newly presented method is demonstrated on a simple clamped-beam structure with a unilateral contact, which is excited with a harmonic force at the free end.

Hybrid and Multiplicative Overlapping Schwarz Algorithms with Standard Coarse Spaces for Mixed Linear Elasticity and Stokes Problems

AbstractThe goal of this work is to construct and study hybrid and multiplicative two-level overlapping Schwarz algorithms with standard coarse spaces for the almost incompressible linear elasticity and Stokes systems, discretized by mixed finite and spectral element methods with discontinuous pressures. Two different approaches are considered to solve the resulting saddle point systems: a) a preconditioned conjugate gradient (PCG) method applied to the symmetric positive definite reformulation of the almost incompressible linear elasticity system obtained by eliminating the pressure unknowns; b) a GMRES method with indefinite overlapping Schwarz preconditioner applied directly to the saddle point formulation of both the elasticity and Stokes systems. Condition number estimates and convergence properties of the proposed hybrid and multiplicative overlapping Schwarz algorithms are proven for the positive definite reformulation of almost incompressible elasticity. These results are based on our previous study [8] where only additive Schwarz preconditioners were considered for almost incompressible elasticity. Extensive numerical experiments with both finite and spectral elements show that the proposed overlapping Schwarz preconditioners are scalable, quasi-optimal in the number of unknowns across individual subdomains and robust with respect to discontinuities of the material parameters across subdomains interfaces. The results indicate that the proposed preconditioners retain a good performance also when the quasi-monotonicity assumption, required by the available theory, does not hold.