Advanced numerical methods for modelling of elastic wave propagation in solids and contact-impact problems

Abstract

This habilitation thesis summarizes relevant advancements in numerical modelling of elastic wave propagation in solids and contact-impact problems. The advanced numerical methods for accurate numerical wave modelling in solids and contact-impact modelling are presented to improve cardinal phenomena which pollute the numerical results as spurious oscillations of the stress eld in the wave propagation, improve convergence rate for free vibration tasks, analysis of dispersion errors in the nite element modelling in explicit time schemes and avoid the stability trouble in the penalty-based dynamic contact-impact modelling with stabilization of contact force oscillations. On that topics, the author of the thesis worked for twelve years to improving standard numerical technologies in FEM which can be implemented into commercial software for dynamics, wave propagation and contactimpact problems. The rst Section presents the analysis of the dispersive properties of the nite element method in elastic wave propagation. The dispersive properties of plane square bilinear and biquadratic FEs are analyzed for the full spatial/grid disctretization and also for the temporal-spatial semidiscretization with the explicit and implicit time integration. The higher-order FEM discretization via the spline-based shape functions is studied with intention to describe dispersion and e ects of order and smoothness of the spline shape function are discussed with respect to the the dispersion behaviour. The second Section summaries the activities in the area of the advanced time stepping for the nite element discretized equations of motions, the analysis of its accuracy and stability. The main attention is paid to diminishing the dispersion errors and spurious stress oscillations of the numerical results. The partitioned longitudinal and shear wave algorithm was found, implemented and tested. The spline based FEM together with several time explicit and implicit schemes are studied with respect to the spurious oscillations. Finally, the heterogenous time stepping algorithm for elastic wave propagation is presented. It is based on the localized Lagrange multilpliers and solving of the interface problems without dissipation of the energy on the interfaces. The third Section focuses on applications of the nite element method in elastic wave propagation in layered and heterogeneous media. The veri cation of the numerical results by the experimental and analytical approaches are presented. The applications of the FEM for time reversal tasks and in nondestructive testing of materials and structures are mentioned. The review of elastic wave propagation in 3D printed structures and identi cation of the residual stresses in bodies is introduced. In the fourth Section, the advanced numerical methods based on higher-order FEM are presented. Mainly, the isogeometric analysis applied into the free vibration problems is shown with the discussion on convergence rate in eigen-value computations. The direct inversion mass matrix - called reciprocal mass matrix - in free vibration problems is mentioned and discussed. The fth Section focuses on the advanced numerical methods for modelling of the contact-impact problems based on the bipenalty method as an extension of the classical penalty method. On the numerical tests, the accuracy and stability of the nominated methods for contact-impact are summarized and commented for one-dimensional and two-dimensional problems. The thesis nishes with the summary and comments of possible future works and today challenging directions for the studies in the areas of the thesis.