In this paper the formulation of hybrid-Trefftz displacement finite elements for transient problems in three-dimensional elastic media is derived. The mathematical model is derived from the classical theory of elasticity. The governing domain and boundary equations are discretized in time using a wavelet basis expansion to yield a series of spectral problems which only depend on space. Displacements are the main approximation in the domain of the element (displacement model) and tractions are approximated on the essential boundaries (hybrid formulation). The displacement trial functions are constrained to satisfy exactly the domain equations (Trefftz constraint), and consist of one family of compression waves and two families of shear waves. The problem is reduced to solving an algebraic system whose unknowns are the weights of the displacement bases. Strains and stresses are derived from the displacement approximations through the compatibility and elasticity equations. The formulation is implemented as a new module in FreeHyTE, an open-source and user-friendly Trefftz platform. Numerical tests have been carried out with the new 3D FreeHyTE module implemented with the functions derived in this paper and the results are satisfactory.

The hybrid-Trefftz displacement element formulation for poroelastodyamics in three-dimensional saturated media is presented in this article. The governing equations are integrated in time using a wavelet decomposition which expands the time-dependent problem into several spectral problems. Displacements are discretized in each finite element and tractions are approximated on each essential element boundary (displacement model). The displacement approximation functions satisfy the governing equations of the media (Trefftz constraint) and are not linked to the nodes. Trefftz bases are derived solving the governing equations and are composed of four types of modes: two compression waves and two shear waves. The weak enforcement of the Navier equation and boundary conditions yields a solving system where the unknown are the weights of the approximation bases. Numerical tests have been carried out to assess the convergence of the models. The results of the simulations are satisfactory: the numerical tests converge to known analytical solutions and the simulation of a propagation of a vertical shock wave is consistent to the one obtained with a 2D plane simulation under the same conditions. The results of a bender element test simulation in three dimensions are also presented.

A novel, local and frequency-dependent absorbing boundary condition for unsaturated porous media is presented. It relates the far-field tractions and displacements produced by a wave which travels through an unsaturated porous medium described by the theory of mixtures with interfaces. The absorbing boundary condition is valid for all types of waves that may occur in unsaturated media, namely three types of compression waves (one for each phase) and a single shear wave, propagating through the solid phase. Although it is only exact for absorbing boundaries situated infinitely far from the source of excitation and orthogonal incident waves, an efficiency study, measuring the ratio between the (spurious) reflected energy and the energy of the incident wave, shows that the absorbing boundary condition is highly efficient even when situated at close range and oblique in respect to the incident waves. It is also easy to embed in conventional finite element and finite difference software, as it amounts to a linear Robin boundary condition. A numerical application involving the propagation of a shock wave through a multi-layered semi-unbounded unsaturated media is also presented.

Hybrid-Trefftz finite elements are well suited for modeling the response of materials under highly transient loading. Their approximation bases are built using functions that satisfy exactly the differential equations governing the problem. This option embeds relevant physical information into the approximation basis and removes the well-known sensitivity of the conventional finite elements to high solution gradients and short wavelength excitations. Despite such advantages, no public software using hybrid-Trefftz finite elements to model wave propagation through solid and porous media exists to date. This paper covers the formulation and implementation of hybrid-Trefftz finite elements for single-phase, biphasic and triphasic media, subjected to dynamic loads. The formulation is cast in a unified framework, valid for the three types of materials alike, and independent of the nature (harmonic, periodic or transient) of the applied load. Displacement, traction, elastic and absorbing boundary conditions are accommodated. The implementation is made in three novel, open-source and user-friendly computational modules which are freely distributed online.

FreeHyTE Direct Boundary Method Toolbox is a new computational framework for the solution of interior and exterior boundary value problems in two dimensions using three classes of direct methods: the Boundary Element Method, the Method of Fundamental Solutions and the Trefftz-Herrera Method. The toolbox, currently including solvers for Laplace and Helmholtz boundary value problems, is straightforward to use, featuring a simple graphical user interface and automatic mesh generators, and amenable to extension, as it provides modular computational procedures, directly applicable to other types of boundary elements and differential equations. The toolbox supports the definition of simply or multiply-connected domains, boundary elements of any order, complex wavenumbers, and Dirichlet, Neumann and Robin boundary conditions. FreeHyTE Direct Boundary Method Toolbox is freely distributed under the GNU General Public License and supported by manuals to quickly get new users started.

Bender elements are shear wave transducers, used for the computation of small strain shear moduli of geomaterials. However, the distortion of the output signal caused by residual compression waves may lead to important errors in the shear modulus estimates. We present a novel procedure for the optimisation of the location of the receiver bender element, to avoid regions where the distortion of the output signal is strong, without compromising the strength of the shear wave signal. The procedure is based on a computational technique naturally able to distinguish between the compression and shear waves present in the seismic response of geomaterials. This property enables the construction of compression and shear amplitude maps, that can be used to decide the best location for the receiver prior to running the experiment. The experimental validation of the procedure confirms that it leads to output signals which are easier to interpret than those obtained with the transmitter and receiver in the conventional, tip-to-tip configuration.

A new hybrid-Trefftz finite element for the solution of transient, non-homogeneous parabolic problems is formulated. The governing equations are first discretized in time and reduced to a series of non-homogeneous elliptic problems in space. The complementary and particular solutions of each elliptic problem are approximated independently. The complementary solution is expanded in Trefftz bases, designed to satisfy exactly the homogeneous form of the problem. Trefftz bases are regular, and defined independently for each finite element, using arbitrary orders. A novel dual reciprocity method is used for the approximation of the particular solution, to avoid domain integration. The same, regular basis is used for the expansions of the source function and particular solution, avoiding the cumbersome expressions of the latter that typify conventional dual reciprocity techniques. Moreover, the bases of the complementary and particular solutions are defined by the same expressions, with different wave numbers. The finite element formulation is obtained by enforcing weakly the domain equations and boundary conditions. To enhance the reproducibility of this work, the formulation is implemented in the computational platform FreeHyTE, where it takes advantage of the pre-programmed numerical procedures and graphical user interfaces. The resulting software is open-source, user-friendly and freely distributed to the scientific community through the FreeHyTE web page.

Hybrid-Trefftz finite elements combine favourable features of the Finite and Boundary Element methods. The domain of the problem is divided into finite elements, where the unknown quantities are approximated using bases that satisfy exactly the homogeneous form of the governing differential equation. The enforcement of the governing equations leads to sparse and Hermitian solving systems (as typical to Finite Element Method), with coefficients defined by boundary integrals (as typical to Boundary Element Method). Moreover, the physical information contained in the approximation bases renders hybrid-Trefftz elements insensitive to gross mesh distortion, nearly-incompressible materials, high frequency oscillations and large solution gradients. A unified formulation of hybrid-Trefftz finite elements for dynamic problems is presented in this chapter. The formulation reduces all types of dynamic problems to formally identical series of spectral equations, regardless of their nature (parabolic or hyperbolic) and method of time discretization (Fourier series, time-stepping procedures, or weighted residual methods). For non-homogeneous problems, two novel methods for approximating the particular solution are presented. The unified formulation supports the implementation of hybrid-Trefftz finite elements for a wide range of physical applications in the same computational framework.

The solution of transient parabolic problems using a novel hybrid-Trefftz finite element is presented in this chapter. The governing equations are first discretized in time and reduced to a series of non-homogeneous elliptic problems in space variables only. The complementary and particular solutions of each elliptic problem are approximated independently. The complementary solution is expanded in Trefftz bases, designed to satisfy exactly the homogeneous form of the problem. Trefftz bases include regular functions of arbitrary orders and are defined independently for each finite element. A novel dual reciprocity method is used for the approximation of the particular solution, to avoid domain integration. The same regular basis is used for the expansions of the source function and particular solution, avoiding the cumbersome expressions of the latter that typify conventional dual reciprocity techniques. Moreover, the bases of the complementary and particular solutions are defined by the same expressions, with different wave numbers. The finite element formulation is obtained by enforcing weakly the domain equations and boundary conditions. The formulation is implemented in the computational platform FreeHyTE, where it takes advantage of the pre-programmed numerical procedures and graphical user interfaces. The resulting software is open-source, user-friendly and freely distributed to the scientific community.

Hybrid-Trefftz finite elements are well suited for modelling the response of materials under highly transient loading. These finite elements approximate not only the displacement in the domain, but also the stresses on its essential boundaries. The domain approximation functions are tailored for each kind of physical problem that is being solved and are rich in physical information on the problem.

Their approximation basis is also built using functions that satisfy exactly the differential equation governing the problem. This option embeds relevant physical information into the approximation basis and removes the well-known sensitivity of the conventional finite elements to high solution gradients and short wavelength excitations.

This dissertation covers the formulation and implementation of hybrid-Trefftz finite elements for problems defined on solid media subjected to transient loads. The formulation is cast in a unified framework, and independent of the nature (harmonic, periodic or transient) of the applied load.

Dirichlet, Neumann, Robin and absorbing boundary conditions are accommodated. The implementation is made in an open-source and user-friendly computational platform, FreeHyTE, which is freely distributed online.

The performance of the hybrid Trefftz finite elements is illustrated through the solution of the problems involving the propagation of a shock wave in bounded and in semi-infinite media.