Abstract
In this paper, we numerically study a dynamic viscoelastic problem. The variational formulation is written as a linear parabolic variational equation for the velocity field. An existence and uniqueness result is recalled. Then, fully discrete approximations are introduced using the implicit Euler scheme and the finite element method, for which some a priori error estimates are derived, leading to the linear convergence of the algorithm under suitable additional regularity conditions. Finally, some one- and three-dimensional numerical simulations are presented to show the accuracy of the algorithm and the behaviour of the solution, including a comparison with an experimental study.
Highlights
We introduce an efficient algorithm for solving a linear case, proving error estimates which allow to show its convergence
Dynamic and static problems for viscoelastic or elastic materials have been studied in numerous publications
The numerical approximation of these problems were done. These viscoelastic materials have been utilized in many engineering applications since they can be customized to meet a desired performance while maintaining low cost
Summary
Dynamic and static problems for viscoelastic or elastic materials have been studied in numerous publications. The body is assumed linearly viscoelastic and it satisfies the following constitutive law (see, for instance, Duvaut & Lions, 1976), σ(x, t) = ε(u (x, t)) + ε(u(x, t)), x ∈ Ω, t ∈ (0, T), where = (aijkl) and = (bijkl) are the fourth-order viscous and elastic tensors, respectively. Ρ > 0 is the density of the material (which is assumed constant for simplicity), and u0 and v0 are initial conditions for the displacement and velocity fields, respectively. In order to obtain the variational formulation of Problem P, let us denote by H = [L2(Ω)]d, and define the variational spaces V and Q as follows,. ∀w ∈ V, Plugging (1) into (2) and using the previous boundary conditions, applying Green’s formula, we derive the following variational formulation of Problem P, written in terms of the velocity field v(t) = u (t). The solution satisfies v ∈ C([0, T];V) ∩ C1([0, T]; H)
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
