Deformable Obstacle Research Articles

3D/1D 하이브리드 유한요소 모델을 이용한 동력 분산형 차세대 고속열차 전체차량의 충돌 해석

Abstract : In this paper, collision analysis of the full rake for the Next Generation High-speed EMU is conducted using a 3D/1D hybrid model, which combines 3-dimensional (3D) front-end structure of finite element model and 1-dimensional (1D) multi-body dynamics model in order to analyze train collision with a standard 3D deformable obstacle. The crush forces, passengers’ accelerations and energy absorptions of a full rake train can be easily obtained through a simulation of a 1D dynamics model composed of nonlinear springs, dampers and masses. Also the obtained simulation results are very similar to those of a 3D model if an overriding behavior does not occur during collision. The standard obstacle in TSI regulation has been changed from a rigid body to a deformable body, and therefore 3D collision simulations should be conducted because their simulation results depends on the front-end structure of a train. According to the obstacle collision analysis of this study, the obstacle collides with the driver’s upper structure after overriding over the front-end module. The 3D/1D hybrid model is effective to evaluate a main energy-absorbing module that is frequently changed during design process and reduce the need time of the modeling and analysis when compared to a 3D full car body.

A numerical study of the derailment caused by collision of a rail vehicle using a virtual testing model

This study investigated the wheel-lift and roll-over derailment mechanisms caused by train collisions using a precise virtual testing model (VTM) of a Korean high-speed train. The VTM was a complex, nonlinear finite element model composed of the shell, beam, solid, spring, and surface contact elements for the car body, bogies, suspensions, and wheel–rail interfaces. The VTM was validated by checking the errors in the total energy and the dynamic responses of the spring elements. To achieve a quick, dynamic relaxation of the dead weight of the VTM before the collision analysis, the artificial damping method and the artificial force method were introduced and numerically evaluated. The surface-to-surface contact model from commercial software, Ls-Dyna, was applied to the VTM in order to simulate the derailment mechanisms caused by collision accidents. The numerical analyses of the VTM colliding with a large deformable obstacle or a rigid wall revealed for the first time that a mixed slip/roll-over-type derailment mechanism generally occurs. Furthermore, the simulation results were consistent with the results from a simplified theoretical derailment model of a wheel set.

EU의 TSI 규정 및 국내 철도차량안전기준의 대형장애물 유한요소모델 개발과 분산형 고속열차의 충돌성능평가에 적용

본 논문은 유럽 TSI와 국내철도차량안전기준에 정의된 건널목 충돌사고시나리오에서 요구하는 성능의 대형 변형체 장애물의 유한요소 모델에 대하여 2가지 종류의 모델을 개발하여 한국형 분산형 고속철도 차량에 적용하고 평가하였다. 규정에서 요구하는 대형 장애물은 기존 강체모델에서 현재의 변형체모델로 변경되었으며 규정에 정의된 방법으로 변형체 강성 값이 검증되어야 한다. 여러 번의 시뮬레이션을 통해 기준을 만족하는 균일한 밀도와 강성의 솔리드 형 장애물 모델과 균일하지 않은 셸 형 장애물 모델 등 2가지를 개발하였다. 본 연구에서 개발된 대형장애물을 사용하여 분산형 고속열차를 대상으로 규정의 대형장애물 충돌시뮬레이션을 수행하였고 그 결과를 평가하였다. 셸 형과 솔리드 형 장애물은 열차와 충돌 후 거동에 상당한 차이가 있었고, 셸 모델이 더 가혹한 결과를 나타내었다. The purpose of this paper is to develop two kinds of finite element models for the heavy deformable obstacle defined in grade crossing collision scenario of the Europe TSI and the Korean rolling stock safety regulations and to apply the crashworthiness evaluation for the Korean high-speed EMU with the FE model. The numerical models of the heavy obstacle were changed from a past rigid one to a current deformable one whose stiffness requirement should be verified by a collision simulation defined in the regulations. Through several trial simulations, two types of numerical models for the heavy obstacle were developed, which satisfied physical properties specifies in the regulations. One is a solid-type obstacle with uniform density and the other is a shell-type. With the obstacles developed in this study, the grade crossing collision scenario for Korean high-speed EMU was simulated and evaluated for the two-type obstacle models. From the simulation results, the shell and solid-type obstacles showed quite different behaviors after collision, and the shell type model gave more severe results.

The control variational method for beams in contact with deformable obstacles

AbstractWe consider a mathematical model which describes the equilibrium of an elastic beam in contact with two obstacles. The contact is modeled with a normal compliance type condition in such a way that the penetration is allowed but is limited. We state the variational formulation of the problem and prove an existence and uniqueness result for the weak solution. Then, we provide an alternative approach to the model, based on the control variational method. Necessary and sufficient optimality conditions are derived, together with an approximation property. We also adapt our results to some versions of the model which describe the contact with a single obstacle. For this type of problems we present two methods of numerical approach, based on the iterative control variational method and the Newton method, respectively, and provide numerical simulations for the two algorithms.

A normal compliance contact problem in viscoelasticity: An a posteriori error analysis and computational experiments

In this work, the numerical approximation of a viscoelastic contact problem is studied. The classical Kelvin–Voigt constitutive law is employed, and contact is assumed with a deformable obstacle and modelled using the normal compliance condition. The variational formulation leads to a nonlinear parabolic variational equation. An existence and uniqueness result is recalled. Then, a fully discrete scheme is introduced, by using the finite element method to approximate the spatial variable and the implicit Euler scheme to discretize time derivatives. A priori error estimates recently proved for this problem are recalled. Then, an a posteriori error analysis is provided, extending some preliminary results obtained in the study of the heat equation and other parabolic equations. Upper and lower error bounds are proved. Finally, some numerical experiments are presented to demonstrate the accuracy and the numerical behaviour of the error estimates.

Numerical analysis of a contact problem including bone remodeling

In this work, a contact problem between an elastic body and a deformable obstacle is numerically studied. The bone remodeling of the material is also taken into account in the model and the contact is modeled using the normal compliance contact condition. The variational problem is written as a nonlinear variational equation for the displacement field, coupled with a first-order ordinary differential equation to describe the physiological process of bone remodeling. An existence and uniqueness result of weak solutions is stated. Then, fully discrete approximations are introduced based on the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives. Error estimates are obtained, from which the linear convergence of the algorithm is derived under suitable regularity conditions. Finally, some 2D numerical results are presented to demonstrate the behavior of the solution.

Numerical analysis of a quasistatic elasto-piezoelectric contact problem with damage

In this work, a contact problem between an elasto-piezoelectric body and a deformable obstacle is numerically studied. The damage of the material, caused by internal tension or compression, is also included into the model. The variational formulation leads to a coupled system composed of a nonlinear variational equation for the displacement field, a linear variational equation for the electric potential, and a nonlinear parabolic variational equation for the damage field. The existence of a unique weak solution is recalled. Then, a fully discrete scheme is introduced by using a finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives. Error estimates are derived on the approximate solutions, and the linear convergence of the algorithm is derived under suitable regularity conditions. Finally, a two-dimensional example is presented to demonstrate the behaviour of the solution.

A posteriori error analysis for the normal compliance problem

In this work, a contact problem between a linear elastic material and a deformable obstacle is numerically analyzed. The contact is modeled using the well-known normal compliance contact condition. The weak formulation leads to a nonlinear variational equation which is approximated by using the finite element method. A priori error estimates are recalled. Then, we define an a posteriori error estimator of residual type to evaluate the accuracy of the finite element approximation of the problem. Upper and lower bounds of the discretization error are proved for this estimator.

Numerical analysis of a frictionless viscoelastic piezoelectric contact problem

In this work, we consider the quasistatic frictionless contact problem between a viscoelastic piezoelectric body and a deformable obstacle. The linear electro-viscoelastic constitutive law is employed to model the piezoelectric material and the normal compliance condition is used to model the contact. The variational formulation is derived in a form of a coupled system for the displacement and electric potential fields. An existence and uniqueness result is recalled. Then, a fully discrete scheme is introduced based on the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives. Error estimates are derived on the approximative solutions and, as a consequence, the linear convergence of the algorithm is deduced under suitable regularity conditions. Finally, some two-dimensional examples are presented to demonstrate the performance of the algorithm.

An Elastic-viscoplastic Quasistatic Contact Problem: Existence and Uniqueness of a Weak Solution

In this paper, the contact between an elastic-viscoplastic body and a deformable obstacle is studied. The effect of the damage, due to internal tension or compression and caused by the opening and growth of micro-cracks and micro-cavities, is also considered. The variational formulation leads to a coupled system of evolutionary equations. An existence and uniqueness result is established by using approximate problems, the pseudomonotone operators theory, Schauder fixed-point theorem and a comparison result.

Numerical analysis of a dynamic piezoelectric contact problem arising in viscoelasticity

A dynamic frictionless contact problem between a viscoelastic piezoelectric body and a deformable obstacle is numerically studied in this paper. The contact is modelled using the normal compliance contact condition and the linear electro-viscoelastic constitutive law is employed to simulate the piezoelectric effects. The variational formulation is a coupled system composed of a parabolic nonlinear variational equation for the velocity field and a linear variational equation for the electric potential. An existence and uniqueness result is recalled. Then, a fully discrete scheme is introduced based on the finite element method to approximate the spatial variable and the backward Euler scheme to discretize the time derivatives. Error estimates are derived on the approximative solutions from which the linear convergence of the algorithm is deduced under suitable regularity conditions. Finally, some two-dimensional examples are presented to demonstrate the accuracy and the performance of the algorithm.

An elastic–viscoplastic quasistatic contact problem with damage

In this paper, the contact between an elastic–viscoplastic body and a deformable obstacle is studied. The effect of the damage, due to tension or compression, is also considered. The variational formulation leads to a coupled system of evolutionary equations for which an existence and uniqueness result is stated. Then, a fully discrete scheme is introduced using the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives. Error estimates are obtained and, as a consequence, the convergence of the algorithm is deduced. Finally, some numerical simulations are presented to demonstrate the accuracy of the algorithm.

Numerical analysis of an elastic contact problem with damage

AbstractIn this work, we consider a mathematical model for the quasistatic contact problem between an elastic body and a deformable obstacle, including the effect of the damage of the material, within the framework of the small deformation theory. The numerical analysis of the variational problem is provided using the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives. Finally, a two‐dimensional numerical problem is presented to show the performance of the method. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Dynamic frictional contact of a viscoelastic beam

In this paper, we study the dynamic frictional contact of a viscoelastic beam with a deformable obstacle. The beam is assumed to be situated horizontally and to move, in both horizontal and tangential directions, by the effect of applied forces. The left end of the beam is clamped and the right one is free. Its horizontal displacement is constrained because of the presence of a deformable obstacle, the so-called foundation, which is modelled by a normal compliance contact condition. The effect of the friction is included in the vertical motion of the free end, by using Tresca's law or Coulomb's law. In both cases, the variational formulation leads to a nonlinear variational equation for the horizontal displacement coupled with a nonlinear variational inequality for the vertical displacement. We recall an existence and uniqueness result. Then, by using the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives, a numerical scheme is proposed. Error estimates on the approximative solutions are derived. Numerical results demonstrate the application of the proposed algorithm.

Numerical analysis of dynamic thermoviscoelastic contact with damage of a rod

We describe and analyse a model for a problem of thermoviscoelastic dynamic contact which allows for the general evolution of the material damage. The effects on the mechanical properties of the material due to crack expansion are described by a damage field, which measures the decrease in the load-bearing capacity of the material. The damage process is assumed to be reversible and the microcracks which develop as a result of tension or compression may grow or disappear. The geometric setting is that of a 1D rod which may contact a deformable obstacle. The contact is modelled by the normal compliance condition and the stress–strain constitutive equation is of Kelvin–Voigt type. The model consists of a coupled system of energy–elasticity equations together with a non-linear parabolic inclusion for the damage field. The existence of a local weak solution is established using penalization, a finite element algorithm for the solution is constructed and analysed and the results of numerical simulations based on this algorithm are presented. The simulations illustrate how the size of the normal compliance coefficients, the damage rate coefficients and the applied force affect the character of the evolution of the damage. In particular, cycles of bonding and debonding can occur.

Numerical analysis of a quasistatic thermoviscoelastic frictional contact problem

We study a quasistatic frictional contact problem between a viscoelastic body and a deformable obstacle. A thin lubricant layer is assumed on the contact surface and, then, a normal damped response contact condition considered. Thermal and frictional effects are also taken into account. A fully discrete scheme is proposed, using the finite element method for the spatial approximation and the Euler scheme for discretizing the time derivatives. Error estimates on the solutions are derived and the linear convergence, under suitable regularity hypotheses, is obtained. The scheme was implemented and some numerical examples are included to show the performance of the method.

Global Convergence of an Elastic Mode Approach for a Class of Mathematical Programs with Complementarity Constraints

We prove that any accumulation point of an elastic mode approach, that approximately solves the relaxed subproblems, is a C-stationary point of the problem of optimizing a parametric mixed P variational inequality. If, in addition, the accumulation point satisfies the MPCC-LICQ constraint qualification, and if the solutions of the subproblem satisfy approximate second-order sufficient conditions, then the limiting point is an M-stationary point. Moreover, if the accumulation point satisfies the upper-level strict complementarity condition, the accumulation point will be a strongly stationary point. If we assume that the penalty function associated with the feasible set of the mathematical program with complementarity constraints has bounded level sets, and if the objective function is bounded below, we show that the algorithm will produce bounded iterates and will therefore have at least one accumulation point. We prove that the obstacle problem satisfies our assumptions for both a rigid and a deformable obstacle. The theoretical conclusions are validated by several numerical examples.

Modeling of the impact of a bubble-liquid droplet

Since a bubble liquid has a low wave resistance as compared to the resistance of a pure liquid and a strongly nonlinear compression-test diagram, the quantitative characteristics of interaction on impact will differ from the analogous ones for the case of the liquid without gas occlusions. The distinctive features of the phenomenon of impact of the droplets of a bubble liquid on rigid or deformable obstacles have been described; the calculations have been performed within the framework of the generalized-equilibrium model of the bubble liquid.

FORCED MOTION OF FRICTION OSCILLATORS LIMITED BY A RIGID OR DEFORMABLE OBSTACLE1*

The forced dynamics of nonsmooth oscillators have not yet been sufficiently investigated when damping is simultaneously due to friction and impact. Because of the theoretical and practical interest in this type of system, an effort is made in this article to determine the behavior of a single-degree-of-freedom oscillator colliding with an obstacle and excited by a harmonic driving force and by a moving base with constant velocity. The response of this system has been investigated under the assumptions of rigid stop and of Coulomb's friction law, with a static coefficient of friction included that is different from the kinetic one. The evolution through stable closed orbits and period-doubling routes to chaos are studied in terms of the clearance between the mass in the initial place and the obstacle. Periodic solutions exhibiting more than one stop and more than one collision per cycle as well as chaotic motions are investigated. An improvement of the friction-impact model is proposed that allows simulating an exponential velocity-dependent friction law and a deformable (hysteretic) obstacle. This model was tested via a sample application. *Communicated by E. V. Zakhariev.

Friction oscillator excited by moving base and colliding with a rigid or deformable obstacle

The dynamics of non-smooth oscillators has not yet sufficiently been investigated, when damping is simultaneously due to friction and impact. Because of the theoretical and practical interest of this type of systems, an effort is made in this paper to lighten the behaviour of a single-degree-of-freedom oscillator colliding with an obstacle and excited by a moving base, which transfers energy to the system via friction. The different nature of discontinuities arising in the combined problem of friction and impact has been recognized and discussed. Closed-form solutions are presented for both transient and steady-state response, assuming Coulomb's friction law and a rigid stop-limiting motion. Furthermore, a deformable (hysteretic) obstacle has been considered, and its influence on the response has been investigated.