The Wave Finite Element Method Applied to a One-Dimensional Linear Elastodynamic Problem With Unilateral Constraints

Abstract

The Wave Finite Element Method (WFEM) is implemented to accurately capture travelling waves propagating at a finite speed within a bouncing rod system and induced by unilateral contact collisions with a rigid foundation; friction is not accounted for. As opposed to the traditional Finite Element Method (FEM) within a time-stepping framework, potential discontinuous deformation, stress and velocity wave fronts are accurately described, which is critical for the problem of interest. A one-dimensional benchmark with an analytical solution is investigated. The WFEM is compared to two time-stepping solution methods formulated on a FEM semi-discretization in space: (1) an explicit technique involving Lagrange multipliers and (2) a non-smooth approach implemented in the Siconos package. Attention is paid to the Gibb’s phenomenon generated during and after contact occurrences together with the time evolution of the total energy of the system. It is numerically found that the WFEM outperforms the FEM and Siconos solution methods because it does not induce any spurious oscillations or dispersion and diffusion of the shock wave. Furthermore, energy is not dissipated over time. More importantly, the WFEM does not require any impact law to close the system of governing equations.

Nonsmooth modal analysis via the boundary element method for one-dimensional bar systems

“…As stated previously for the other boundary conditions, and to avoid the Gibb's phenomenon in the solution, other families of periodic functions could be considered for the test and trial functions in (29) and (31). The gain in how the Signorini conditions will be satisfied might be mitigated by the fact that other boundary conditions (Dirichlet or Robin) will not be exactly satisfied as with Fourier series: this has yet to be clarified.…”

“…Discretization comes into the proposed solution strategy when the Fourier expansions (20) are truncated to a finite number m of harmonics, such that we define u .m/ .1; t/ u.1; t/ and p .m/ .1; t/ p.1; t/. A second level of discretization lies in the computation of the integrals (29). It was found that the computed solutions were not sensitive to that numerical aspect.…”

“…The system of nonlinear equations in .a; b/, and .a; b; d; f/ for the two-bar system, is then solved numerically using a trust-region dogleg [18] solver, a built-in numerical solver of Matlab®, even though the equations are not expected to be sufficiently smooth, due to (29). For the Dirichlet-Signorini case, the system is formed by Equations ( 24) and ( 29).…”

“…The analysis is conducted on three main aspects: the accuracy of the contact force, the convergence of an energy residual and comparison to existing solutions, all with respect to m. Compared to known results [25,28] and as expected, FD-BEM exhibits residual penetration at the contact interface, as shown in Figure 4. This is explained by the fact that FD-BEM enforces the Signorini boundary condition in weighted residual sense only, through (29), in contrast with the TD-BEM and WFEM formulations where the complementary condition is enforced at every time step of the scheme, in a quasi-exact fashion up to a chosen tolerance. Another factor of error is the already mentioned Gibbs' phenomenon, also observed in Figure 4.…”

Nonlinear Modal Analysis of a One-Dimensional Bar Undergoing Unilateral Contact via the Time-Domain Boundary Element Method

“…The impact law should be purely elastic to preserve energy, making it difficult to describe lasting contact phases which are expected to exist in the continuous framework. The Wave Finite Element Method (WFEM), which appropriately combines space and time, has shown promising results for one-dimensional systems undergoing contact conditions [28].…”