Adaptive time discontinuous Galerkin method for numerical modelling of wave propagation in shell and 3D structures

A unified variational framework for the space discontinuous Galerkin method for elastic wave propagation in anisotropic and piecewise homogeneous media

“…The time dG solver used in this work discretizes the space with a finite element mesh combined with one linear element in time for each space-time slab. It has been validated by our previous studies [18][19][20] and also by other authors [17,21].…”

“…According to the definition of the Jacobian operator (10), the eigenmodes (18) and the definition (15) and the property (14) of the Christoffel tensor Γ n , we obtain the following equations by considering separately ℘ vect (Eqn. (74)) and n • ℘ tens (Eqn.…”

“…We recall that the basic idea of the time discontinuous space-time Galerkin method is to subdivide the studied space-time domain Ω×]0, T [ into a series of space-time slabs Ω×]t n , t n+1 [ and to write in each space-time slab a variational formulation for the displacement and velocity fields simultaneously in space and in time. Within each space-time slab, continuous finite elements are used, but between two successive slabs, both displacement and velocity fields are discontinuous [15][16][17][18]. When space-time FE basis functions are chosen equal to the product of FE basis functions in space and FE basis functions in time, it can be shown that time dG methods often lead to unconditionally stable implicit time-stepping schemes.…”

Some comparisons and analyses of time or space discontinuous Galerkin methods applied to elastic wave propagation in anisotropic and heterogeneous media

“…Otherwise, we recall that space-time meshes generally used for the time dG method are composed of a classical finite element mesh in space and one linear element in time in each space-time slab and so an implicit solver is actually obtained [4][5][6][7][8]17].…”

“…By choosing specific spacetime elements, the time dG method can be finally written as a time-stepping scheme and results in an implicit solver. As one important advantage, the method provides an appropriate framework to develop adaptive computing as it remains unconditionally stable even if the space discretization changes over time [4][5][6][7][8]. From one time step to another, the energies that are dissipated in the jumps in time of both displacement and velocity fields guarantee the unconditional stability.…”