A nodal discontinuous Galerkin finite element method for nonlinear elastic wave propagation

Abstract

A nodal discontinuous Galerkin finite element method (DG-FEM) to solve the linear and nonlinear elastic wave equation in heterogeneous media with arbitrary high order accuracy in space on unstructured triangular or quadrilateral meshes is presented. This DG-FEM method combines the geometrical flexibility of the finite element method, and the high parallelization potentiality and strongly nonlinear wave phenomena simulation capability of the finite volume method, required for nonlinear elastodynamics simulations. In order to facilitate the implementation based on a numerical scheme developed for electromagnetic applications, the equations of nonlinear elastodynamics have been written in a conservative form. The adopted formalism allows the introduction of different kinds of elastic nonlinearities, such as the classical quadratic and cubic nonlinearities, or the quadratic hysteretic nonlinearities. Absorbing layers perfectly matched to the calculation domain of the nearly perfectly matched layers type have been introduced to simulate, when needed, semi-infinite or infinite media. The developed DG-FEM scheme has been verified by means of a comparison with analytical solutions and numerical results already published in the literature for simple geometrical configurations: Lamb's problem and plane wave nonlinear propagation.

Plane-strain waves in nonlinear elastic solids with softening

“…The numerical method is well-suited to the computation of nonlinear waves in the Lagrangian framework, and it can be used for various hyperelastic material models (cf. the related study [19] and references therein).…”

“…The spectral radius ̺ f ′ (q) of f ′ (q) corresponds to c P (expression detailed in the Appendix A), ditto the spectral radius ̺ g ′ (q) of g ′ (q). The stability of the scheme (19) is also restricted by the spectral radius of the Jacobian matrix r ′ (q). As in 1D [17], the stability limits imply that the scheme (19) is stable under the classical CFL condition (22).…”

“…The stability of the scheme (19) is also restricted by the spectral radius of the Jacobian matrix r ′ (q). As in 1D [17], the stability limits imply that the scheme (19) is stable under the classical CFL condition (22). Hence, given a spatial discretization and a Courant number Co 1, the value of the time step ∆t is imposed by (22).…”

Nonlinear elastic wave tomography for the imaging of corrosion damage

“…micro-cracks) is negligible. Nonlinear elastic effects of damaged materials such as higher and sub-harmonics generation can be assessed with both numerical multiscale materials models [8][9][10] and experimental nonlinear elastic wave spectroscopy (NEWS) methods [11][12][13]. Particularly, NEWS techniques measure nonlinear classical and non-classical phenomena in the kHz and MHz range and they have shown an extreme sensitivity in diagnosing manufacturing defects such as porosity, undesired component assembly contact conditions, and incipient damage in the form of micro-cracks, delaminations, clapping areas, and adhesive bond weakening [14].…”

Nonlinear Waves in Solid Continua with Finite Deformation

“…Vibrations and wave propagation in thick FGM cylinders with temperature dependent material properties are investigated in reference [15]. A nodal discontinuous Galerkin finite element method is considered for nonlinear elastic wave propagation in reference [16]. Nonlinear transient stress wave propagation in thick FGM cylinder using a unified generalized thermoelasticity theory is considered in reference [17].…”