Jeongho Ahn

An Euler--Bernoulli Beam with Dynamic Contact: Discretization, Convergence, and Numerical Results

In this paper, we formulate a time-discretization using the implicit Euler method for contact conditions and the midpoint rule for the elastic part of the equations. The energy functional is defined, and convergence for the time-discretization is investigated. Our time-discretization leads to energy dissipation. Using this time discretization and the finite element method with B-spline basis functions, we compute numerical solutions. We show that there is a converging subsequence, and the limit of any such converging subsequence is a solution of the dynamic impact problem. In order to solve the linear complementarity problem that arises in the numerical method, we use a smoothed guarded Newton method. We also investigate numerically the question of whether the numerical solutions converge strongly to their limit and if energy is conserved for the limit. Our numerical results give some evidence that this is so.

A viscoelastic Timoshenko beam with Coulomb law of friction

Abstract This work focuses on the analysis and numerical simulations of dynamic frictional contact between a Timoshenko beam and a stationary rigid obstacle. The beam is assumed to be viscoelastic, and is clamped at its left end while it is free at its right end. When the beam moves horizontally and its right end contacts the rigid obstacle, contact forces arise and then the vertical motion of its right end is considered with friction. Thus the right end of the beam satisfies two contact conditions: the Signorini unilateral contact condition and Coulomb law of dry friction. Moreover, the slip rate dependence of the coefficient of friction is taken into account. The existence of weak solutions to the problem is shown by using finite time stepping and the necessary a priori estimates that allow us to vanish the time step in the limiting process. The energy balance in the system is investigated both theoretically and numerically. A fully discrete numerical scheme for the variational formulation of the problem is implemented and numerical results are presented.

An Euler–Bernoulli Beam with Dynamic Frictionless Contact: Penalty Approximation and Existence

In this work, we consider the dynamic frictionless Euler–Bernoulli equation with the Signorini contact conditions along the length of a thin beam. The existence of solutions is proved based on the penalty method. Employing energy functional with the penalty method, we bound integral of contact forces over space and time. Hölder continuity of the fundamental solution plays an important role in the convergence theory.

Dynamic frictionless contact of a nonlinear beam with two stops

In this work, we consider mathematical and numerical approaches to a dynamic contact problem with a highly nonlinear beam, the so-called Gao beam. Its left end is rigidly attached to a supporting device, whereas the other end is constrained to move between two perfectly rigid stops. Thus, the Signorini contact conditions are imposed to its right end and are interpreted as a pair of complementarity conditions. We formulate a time discretization based on a truncated variational formulation. We prove the convergence of numerical trajectories and also derive a new form of energy balance. A fully discrete numerical scheme is implemented to present numerical results.

C0 interior penalty methods for a dynamic nonlinear beam model

Abstract In this work, we aim to develop efficient numerical schemes for a nonlinear fourth-order partial differential equation arising from the so-called dynamic Gao beam model. We use C 0 interior penalty finite element methods over the spatial domain to set up the semi-discrete formulations. Convergence results for the semi-discrete case are shown, based on a truncated variational formulation and its equivalent abstract formulations. We combine time discretizations to derive fully discrete numerical formulations. Newton’s method is applied to compute one time step numerical solutions of a nonlinear system. Two numerical examples are provided: one supports our theoretical results and the other presents a buckling state of the Gao beams.

A Simplified Model of Impact

The standard Signorini contact condition is integrated against a given function ψ over the boundary to obtain a simplified model of contact suitable for impact problems. An implicit method (implicit mid-point rule for elasticity, and implicit Euler for the contact conditions) is proposed to numerically solve the simplified model, and some properties of the solution are obtained. The results are only partial at this stage, but they seem to indicate that contact forces in elastic impacts are considerably more regular than general measures.