The Theory of Quasiconformal Mappings in Higher Dimensions, I

We present a survey of the many and various elements of the modern higher-dimensional theory of quasiconformal mappings and their wide and varied application. It is unified (and limited) by the theme of the author's interests. Thus we will discuss the basic theory as it developed in the 1960s in the early work of F.W. Gehring and Yu G. Reshetnyak and subsequently explore the connections with geometric function theory, nonlinear partial differential equations, differential and geometric topology and dynamics as they ensued over the following decades. We give few proofs as we try to outline the major results of the area and current research themes. We do not strive to present these results in maximal generality, as to achieve this considerable technical knowledge would be necessary of the reader. We have tried to give a feel of where the area is, what are the central ideas and problems and where are the major current interactions with researchers in other areas. We have also added a bit of history here and there. We have not been able to cover the many recent advances generalising the theory to mappings of finite distortion and to degenerate elliptic Beltrami systems which connects the theory closely with the calculus of variations and nonlinear elasticity, nonlinear Hodge theory and related areas, although the reader may see shadows of this aspect in parts

Convergent Interpolation to Cauchy Integrals over Analytic Arcs with Jacobi-Type Weights

We design convergent multipoint Pade interpolation schemes to Cauchy transforms of non-vanishing complex densities with respect to Jacobi-type weights on analytic arcs, under mild smoothness assumptions on the density. We rely on our earlier work for the choice of the interpolation points, and dwell on the Riemann-Hilbert approach to asymptotics of orthogonal polynomials introduced by Kuijlaars, McLaughlin, Van Assche, and Vanlessen in the case of a segment. We also elaborate on the $\bar\partial$-extension of the Riemann-Hilbert technique, initiated by McLaughlin and Miller on the line to relax analyticity assumptions. This yields strong asymptotics for the denominator polynomials of the multipoint Pade interpolants, from which convergence follows.Comment: 42 pages, 3 figure

Funktionentheorie

[no abstract available

Multiscale modelling and an experimental investigation on size-scale effects in concrete

Includes abstract.Includes bibliographical references (leaves 82-90).Classical continuum mechanics assumes that constitutive parameters are associated with a so-called Representative Volume Element (RVE) and are a statistical average. This concept is based on the presumption that the specimen size is much larger than the size of its constituents, so that the behaviour of a single constituent can be neglected. This presumption does not hold true if the considered problem domain is smaller than the RVE. The size of material constituents in relation to the dimension of the specimen can then not be considered negligible and the interaction between the constituents needs to be addressed. In this context, so-called generalised continuum formulations have proven to provide a remedy

Critical measures, quadratic differentials, and weak limits of zeros of Stieltjes polynomials

We investigate the asymptotic zero distribution of Heine-Stieltjes polynomials - polynomial solutions of a second order differential equations with complex polynomial coefficients. In the case when all zeros of the leading coefficients are all real, zeros of the Heine-Stieltjes polynomials were interpreted by Stieltjes as discrete distributions minimizing an energy functional. In a general complex situation one deals instead with a critical point of the energy. We introduce the notion of discrete and continuous critical measures (saddle points of the weighted logarithmic energy on the plane), and prove that a weak-* limit of a sequence of discrete critical measures is a continuous critical measure. Thus, the limit zero distributions of the Heine-Stieltjes polynomials are given by continuous critical measures. We give a detailed description of such measures, showing their connections with quadratic differentials. In doing that, we obtain some results on the global structure of rational quadratic differentials on the Riemann sphere that have an independent interest.Comment: 70 pages, 14 figures. Minor corrections, to appear in Comm. Math. Physic

A collocated C0 finite element method: Reduced quadrature perspective, cost comparison with standard finite elements, and explicit structural dynamics

We demonstrate the potential of collocation methods for efficient higher-order analysis on standard nodal finite element meshes. We focus on a collocation method that is variationally consistent and geometrically flexible, converges optimally, embraces concepts of reduced quadrature, and leads to symmetric stiffness and diagonal consistent mass matrices. At the same time, it minimizes the evaluation cost per quadrature point, thus reducing formation and assembly effort significantly with respect to standard Galerkin finite element methods. We provide a detailed review of all components of the technology in the context of elastodynamics, that is, weighted residual formulation, nodal basis functions on Gauss–Lobatto quadrature points, and symmetrization by averaging with the ultra-weak formulation. We quantify potential gains by comparing the computational efficiency of collocated and standard finite elements in terms of basic operation counts and timings. Our results show that collocation is significantly less expensive for problems dominated by the formation and assembly effort, such as higher-order elastostatic analysis. Furthermore, we illustrate the potential of collocation for efficient higher-order explicit dynamics. Throughout this work, we advocate a straightforward implementation based on simple modifications of standard finite element codes. We also point out the close connection to spectral element methods, where many of the key ideas are already established

Free and forced propagation of Bloch waves in viscoelastic beam lattices

Beam lattice materials can be characterized by a periodic microstructure realizing a geometrically regular pattern of elementary cells. Within this framework, governing the free and forced wave propagation by means of spectral design techniques and/or energy dissipation mechanisms is a major issue of theoretical interest with applications in aerospace, chemical, naval, biomedical engineering. The first part of the Thesis addresses the free propagation of Bloch waves in non-dissipative microstructured cellular materials. Focus is on the alternative formulations suited to describe the wave propagation in the bidimensional infinite material domain, according to the classic canons of linear solid or structural mechanics. Adopting the centrosymmetric tetrachiral cell as prototypical periodic topology, the frequency dispersion spectrum is obtained by applying the Floquet-Bloch theory. The dispersion spectrum resulting from a synthetic Lagrangian beam lattice formulation is compared with its counterpart derived from different continuous models (high-fidelity first-order heterogeneous and equivalent homogenized micropolar continua). Asymptotic perturbation-based approximations and numerical spectral solutions are compared and cross-validated. Adopting the low-frequency band gaps of the dispersion spectrum as functional targets, parametric analyses are carried out to highlight the descriptive limits of the synthetic models and to explore the enlarged parameter space described by high-fidelity models. The microstructural design or tuning of the mechanical properties of the cellular microstructure is employed to successfully verify the wave filtering functionality of the tetrachiral material. Alternatively, band gaps in the material spectrum can be opened at target center frequencies by using metamaterials with inertial resonators. Based on these motivations, in the second part of the Thesis, a general dynamic formulation is presented for determining the dispersion properties of viscoelastic metamaterials, equipped with local dissipative resonators. The linear mechanism of local resonance is realized by tuning periodic auxiliary masses, viscoelastically coupled with the beam lattice microstructure. As peculiar aspect, the viscoelastic coupling is derived by a mechanical formulation based on the Boltzmann superposition integral, whose kernel is approximated by a Prony series. Consequently, the free propagation of damped Bloch waves is governed by a linear homogeneous system of integro-differential equations of motion. Therefore, differential equations of motion with frequency-dependent coefficients are obtained by applying the bilateral Laplace transform. The corresponding complex-valued branches characterizing the dispersion spectrum are determined and parametrically analyzed. Particularly, the spectra corresponding to Taylor series approximations of the equation coefficients are investigated. The standard dynamic equations with linear viscous damping are recovered at the first order approximation. Increasing approximation orders determine non-negligible spectral effects, including the occurrence of pure damping spectral branches. Finally, the forced response to harmonic single frequency external forces in the frequency and the time domains is investigated. The response in the time domain is obtained by applying the inverse bilateral Laplace transform. The metamaterial responses to non-resonant, resonant and quasi-resonant external forces are compared and discussed from a qualitative and quantitative viewpoint