A survey of Trefftz methods for the Helmholtz equation

Trefftz methods are finite element-type schemes whose test and trial functions are (locally) solutions of the targeted differential equation. They are particularly popular for time-harmonic wave problems, as their trial spaces contain oscillating basis functions and may achieve better approximation properties than classical piecewise-polynomial spaces. We review the construction and properties of several Trefftz variational formulations developed for the Helmholtz equation, including least squares, discontinuous Galerkin, ultra weak variational formulation, variational theory of complex rays and wave based methods. The most common discrete Trefftz spaces used for this equation employ generalised harmonic polynomials (circular and spherical waves), plane and evanescent waves, fundamental solutions and multipoles as basis functions; we describe theoretical and computational aspects of these spaces, focusing in particular on their approximation properties. One of the most promising, but not yet well developed, features of Trefftz methods is the use of adaptivity in the choice of the propagation directions for the basis functions. The main difficulties encountered in the implementation are the assembly and the ill-conditioning of linear systems, we briefly survey some strategies that have been proposed to cope with these problems.Comment: 41 pages, 2 figures, to appear as a chapter in Springer Lecture Notes in Computational Science and Engineering. Differences from v1: added a few sentences in Sections 2.1, 2.2.2 and 2.3.1; inserted small correction

Coupling of hybridisable discontinuous Galerkin and finite volumes for transient compressible flows

Fast, high-fidelity solution workflows for transient flow phenomena is an important challenge in the computational fluid dynamics (CFD) community. Current low-order methodologies suffer from large dissipation and dispersion errors and require large mesh sizes for unsteady flow simulations. Recently, on the other hand, high-order methods have gained popularity offering high solution accuracy. But they suffer from the lack of robust, curvilinear mesh generators.A novel methodology that combines the advantages of the classical vertex-centred finite volume (FV) method and high-order hybridisable discontinuous Galerkin (HDG) method is presented for the simulation of transient inviscid compressible flows. The resulting method is capable of simulating the transient effects on coarse, unstructured meshes that are suitable to perform steady simulations with traditional low-order methods. In the vicinity of the aerodynamic shapes, FVs are used whereas in regions where the size of the element is too large for finite volumes to provide an accurate answer, the high-order HDG approach is employed with a non-uniform degree of approximation. The proposed method circumvents the need to produce tailored meshes for transient simulations, as required in a low-order context, and also the need to produce high-order curvilinear meshes, as required by high-order methods.FV and HDG methods for compressible inviscid flows with an implicit time-stepping method and capable of handling flow discontinuities is developed. A two-way coupling of the methods in a monolithic manner was achieved by the consistent application of the so-called transmission conditions at the FV-HDG interface. Numerical tests highlight the optimal convergence properties of the coupled HDG-FV scheme. Numeri-cal examples demonstrate the potential and suitability of the developed methodology for unsteady 2D and 3D flows in the context of simulating the wind gust effect on aerodynamic shapes

High-order methods for computational fluid dynamics

2010/2011In the past two decades, the growing interest in the study of fluid flows involving discontinuities, such as shocks or high gradients, where a quadratic-convergent method may not provide a satisfactory solution, gave a notable impulse to the employment of high-order techniques. The present dissertation comprises the analysis and numerical testing of two high-order methods. The first one, belonging to the discontinuous finite-element class, is the discontinuous control-volume/finite-element method (DCVFEM) for the advection/ diffusion equation. The second method refers to the high-order finite-difference class, and is the mixed weighted non-oscillatory scheme (MWCS) for the solution of the compressible Euler equations. The methods are described from a formal point of view, a Fourier analysis is used to assess the dispersion and dissipation errors, and numerical simulations are conducted to confirm the theoretical results.XXIV Ciclo198

SOLID-SHELL FINITE ELEMENT MODELS FOR EXPLICIT SIMULATIONS OF CRACK PROPAGATION IN THIN STRUCTURES

Crack propagation in thin shell structures due to cutting is conveniently simulated using explicit finite element approaches, in view of the high nonlinearity of the problem. Solidshell elements are usually preferred for the discretization in the presence of complex material behavior and degradation phenomena such as delamination, since they allow for a correct representation of the thickness geometry. However, in solid-shell elements the small thickness leads to a very high maximum eigenfrequency, which imply very small stable time-steps. A new selective mass scaling technique is proposed to increase the time-step size without affecting accuracy. New ”directional” cohesive interface elements are used in conjunction with selective mass scaling to account for the interaction with a sharp blade in cutting processes of thin ductile shells