Seismic Waves

The importance of seismic wave research lies not only in our ability to understand and predict earthquakes and tsunamis, it also reveals information on the Earth's composition and features in much the same way as it led to the discovery of Mohorovicic's discontinuity. As our theoretical understanding of the physics behind seismic waves has grown, physical and numerical modeling have greatly advanced and now augment applied seismology for better prediction and engineering practices. This has led to some novel applications such as using artificially-induced shocks for exploration of the Earth's subsurface and seismic stimulation for increasing the productivity of oil wells. This book demonstrates the latest techniques and advances in seismic wave analysis from theoretical approach, data acquisition and interpretation, to analyses and numerical simulations, as well as research applications. A review process was conducted in cooperation with sincere support by Drs. Hiroshi Takenaka, Yoshio Murai, Jun Matsushima, and Genti Toyokuni

Symmetric Galerkin boundary element method.

This review concerns a methodology for solving numerically, to engineering purposes, boundary and initial-boundary value roblems by a peculiar approach characterized by the following features: the continuous formulation is centered on integral equations based on the combined use of single-layer and double-layer sources, so that the integral operator turns out to be symmetric with respect to a suitable bilinear form; the discretization is performed either on a variational basis or by a Galerkin weighted residual procedure, the interpolation and weight functions being chosen so that the variables in the approximate formulation are generalized variables in Prager's sense. As main consequences of the above provisions, symmetry is exhibited by matrices with a key role in the algebraized versions, some quadratic forms have a clear energy meaning, variational properties characterize the solutions and other results, invalid in traditional boundary element methods, enrich the theory underlying the computational applications. The present survey outlines recent theoretical and computational developments of the title methodology with particular reference to linear elasticity, elastoplasticity, fracture mechanics, time-dependent problems, variational approaches, singular integrals, approximation issues, sensitivity analysis, coupling of boundary and finite elements, computer implementations. Areas and aspects which at present require further research are dentified and comparative assessments are attempted with respect to traditional boundary integral-element methods

Virtual Element based formulations for computational materials micro-mechanics and homogenization

In this thesis, a computational framework for microstructural modelling of transverse behaviour of heterogeneous materials is presented. The context of this research is part of the broad and active field of Computational Micromechanics, which has emerged as an effective tool both to understand the influence of complex microstructure on the macro-mechanical response of engineering materials and to tailor-design innovative materials for specific applications through a proper modification of their microstructure. While the classical continuum approximation does not account for microstructural details within the material, computational micromechanics allows detailed modelling of a heterogeneous material's internal structural arrangement by treating each constituent as a continuum. Such an approach requires modelling a certain material microstructure by considering most of the microstructure's morphological features. The most common numerical technique used in computational micromechanics analysis is the Finite Element Method (FEM). Its use has been driven by the development of mesh generation programs, which lead to the quasi-automatic discretisation of the artificial microstructure domain and the possibility of implementing appropriate constitutive equations for the different phases and their interfaces. In FEM's applications to computational micromechanics, the phase arrangements are discretised using continuum elements. The mesh is created so that element boundaries and, wherever required, special interface elements are located at all interfaces between material's constituents. This approach can be effective in modelling many microstructures, and it is readily available in commercial codes. However, the need to accurately resolve the kinematic and stress fields related to complex material behaviours may lead to very large models that may need prohibitive processing time despite the increasing modern computers' performance. When rather complex microstructure's morphologies are considered, the quasi-automatic discretisation process stated before might fail to generate high-quality meshes. Time-consuming mesh regularisation techniques, both automatic and operator-driven, may be needed to obtain accurate numeric results. Indeed, the preparation of high-quality meshes is today one of the steps requiring more attention, and time, from the analyst. In this respect, the development of computational techniques to deal with complex and evolving geometries and meshes with accuracy, effectiveness, and robustness attracts relevant interest. The computational framework presented in this thesis is based on the Virtual Element Method (VEM), a recently developed numerical technique that has proven to provide robust numerical results even with highly-distorted mesh. These peculiar features have been exploited to analyse two-dimensional representations of heterogeneous materials' microstructures. Ad-hoc polygonal multi-domain meshing strategies have been developed and tested to exploit the discretisation freedom that VEM allows. To further simplify the preprocessing stage of the analysis and reduce the total computational cost, a novel hybrid formulation for analysing multi-domain problems has been developed by combining the Virtual Element Method with the well-known Boundary Element Method (BEM). The hybrid approach has been used to study both composite material's transverse behaviour in the presence of inclusions with complex geometries and damage and crack propagation in the matrix phase. Numerical results are presented that demonstrate the potential of the developed framework

A statistical approach for fracture property realization and macroscopic failure analysis of brittle materials

Lacking the energy dissipative mechanics such as plastic deformation to rebalance localized stresses, similar to their ductile counterparts, brittle material fracture mechanics is associated with catastrophic failure of purely brittle and quasi-brittle materials at immeasurable and measurable deformation scales respectively. This failure, in the form macroscale sharp cracks, is highly dependent on the composition of the material microstructure. Further, the complexity of this relationship and the resulting crack patterns is exacerbated under highly dynamic loading conditions. A robust brittle material model must account for the multiscale inhomogeneity as well as the probabilistic distribution of the constituents which cause material heterogeneity and influence the complex mechanisms of dynamic fracture responses of the material. Continuum-based homogenization is carried out via finite element-based micromechanical analysis of a material neighbor which gives is geometrically described as a sampling windows (i.e., statistical volume elements). These volume elements are well-defined such that they are representative of the material while propagating material randomness from the inherent microscale defects. Homogenization yields spatially defined elastic and fracture related effective properties, utilized to statistically characterize the material in terms of these properties. This spatial characterization is made possible by performing homogenization at prescribed spatial locations which collectively comprise a non-uniform spatial grid which allows the mapping of each effective material properties to an associated spatial location. Through stochastic decomposition of the derived empirical covariance of the sampled effective material property, the Karhunen-Loeve method is used to generate realizations of a continuous and spatially-correlated random field approximation that preserve the statistics of the material from which it is derived. Aspects of modeling both isotropic and anisotropic brittle materials, from a statistical viewpoint, are investigated to determine how each influences the macroscale fracture response of these materials under highly dynamic conditions. The effects of modeling a material both explicitly by representations of discrete multiscale constituents and/or implicitly by continuum representation of material properties is studies to determine how each model influences the resulting material fracture response. For the implicit material representations, both a statistical white noise (i.e., Weibull-based spatially-uncorrelated) and colored noise (i.e., Karhunen-Loeve spatially-correlated model) random fields are employed herein

Adaptive numerical simulation of contact problems : Resolving local effects at the contact boundary in space and time

This thesis is concerned with the space discretization of static and the space and time discretization of dynamic contact problems. In particular, we derive a new efficient and reliable residual-type a posteriori error estimator for static contact problems and a new space-time connecting discretization scheme for dynamic contact problems in linear elasticity. The methods enable the efficient resolution of local effects at the contact boundary in space and time. Firstly, we prove efficiency and reliability of the new residual-type a posteriori error estimator for the case of simplicial meshes. Several numerical examples in the two- and three-dimensional case show the performance of the residual-type a posteriori error estimator for simplicial and even for non-simplicial meshes. Secondly, for the discretization in time, we present a new method which allows to implicitly compute the local impact times of each node without decreasing the time step size. As it turns out this method gives rise to a generalization of the Newmark scheme which takes into account the local impact times without additional computational effort

Eulerian finite element methods for interface problems and fluid-structure interactions

In this thesis, we develop an accurate and robust numerical framework for interface problems involving moving interfaces. In particular, we are interested in the simulation of fluid-structure interaction problems in Eulerian coordinates. Our numerical model for fluid-structure interactions (FSI) is based on the monolithic "Fully Eulerian" approach. With this approach we can handle both strongly-coupled problems and large structural displacements up to contact. We introduce modified discretisation schemes of second order for both space and time discretisation. The basic concept of both schemes is to resolve the interface locally within the discretisation. For spatial discretisation, we present a locally modified finite element scheme that is based on a fixed patch mesh and a local resolution of the interface within each patch. It does neither require any remeshing nor the introduction of additional degrees of freedom. For discretisation in time, we use a modified continuous Galerkin scheme. Instead of polynomials in direction of time, we define polynomial functions on space-time trajectories that do not cross the interface. Furthermore, we introduce a pressure stabilisation technique based on "Continuous Interior Penalty" method and a simple stabilisation technique for the structural equation that increases the robustness of the Fully Eulerian approach considerably. We give a detailed convergence analysis for all proposed discretisation and stabilisation schemes and test the methods with numerical examples. In the final part of the thesis, we apply the numerical framework to different FSI applications. First, we validate the approach with the help of established numerical benchmarks. Second, we investigate its capabilities in the context of contact problems and large deformations. We study contact problems of a falling elastic ball with the ground, an inclined plane and some stairs including the subsequent bouncing. For the case that no fluid layer remains between ball and ground, we use a simple contact algorithm. Furthermore, we study plaque growth in blood vessels up to a complete clogging of the vessel. Therefore, we use a monolithic mechano-chemical fluid-structure-interaction model and include the fast pulsating flow dynamics by means of a temporal two-scale scheme. We present detailed numerical studies for all three applications including a numerical convergence analysis in space and time, as well as an investigation of the influence of different material parameters

A hp-adaptive discontinuous Galerkin finite element method for accurate configurational force brittle crack propagation

Engineers require accurate determination of the configurational force at the crack tip for fracture fatigue analysis and accurate crack propagation. However, obtain- ing highly accurate crack tip configuration force values is challenging with numer- ical methods requiring knowledge of the stress field around the crack tip a priori. In this thesis, the symmetric interior penalty discontinuous Galerkin finite element method is combined with a residual based a posteriori error estimator which drives a hp-adaptive mesh refinement scheme to determine accurate solutions of the stress field about about the crack. This facilitates the development of a novel method to calculate the crack tip configurational force that is accurate, requires no a priori knowledge of the stress field about the crack tip with, its error bound by an error estimator which is calculated a posteriori. Benchmark values of the crack tip con- figurational force are presented for problems containing multiple mixed mode cracks in both isotropic and anisotropic materials. Additionally, the hp-adaptivity is com- bined with a mathematical analysis of the stress field at the crack tip to critique the convergence and limitations of other methods in the literature to calculate the crack tip configurational force. Two methods for staggered quasi-static crack prop- agation are also presented. An rp-adaptive method which is simple to implement and computationally inexpensive, element edges aligned with the crack propagation path with the exploitation of the discontinuous Galerkin edge sti↵ness terms exist- ing along element interfaces to propagate a crack. The second method is denoted the hpr-adaptive method which combines the accurate computation of the crack tip configuration force with r-adaptivity to produce a computationally expensive but accurate method to propagate multiple cracks simultaneously. Further, for indeter- minate systems, an average boundary condition that restrains rigid body motion and rotation is introduced to make the system determinate