The KW-boundary hybrid digital waveguide mesh for room acoustics applications

The digital waveguide mesh is a discrete-time simulation used to model acoustic wave propagation through a bounded medium. It can be applied to the simulation of the acoustics of rooms through the generation of impulse responses suitable for auralization purposes. However, large-scale three-dimensional mesh structures are required for high quality results. These structures must therefore be efficient and also capable of flexible boundary implementation in terms of both geometrical layout and the possibility for improved mesh termination algorithms. The general one-dimensional N-port boundary termination is investigated, where N depends on the geometry of the modeled domain and the mesh topology used. The equivalence between physical variable Kirchoff-model, and scattering-based wave-model boundary formulations is proved. This leads to the KW-hybrid one-dimensional N-port boundary-node termination, which is shown to be equivalent to the Kirchoff- and wave-model cases. The KW-hybrid boundary-node is implemented as part of a new hybrid two-dimensional triangular digital waveguide mesh. This is shown to offer the possibility for large-scale, computationally efficient mesh structures for more complex shapes. It proves more accurate than a similar rectilinear mesh in terms of geometrical fit, and offers significant savings in processing time and memory use over a standard wave-based model. The new hybrid mesh also has the potential for improved real-world room boundary simulations through the inclusion of additional mixed modeling algorithms

Curvature and topological effects on dynamical symmetry breaking in a four- and eight-fermion interaction model

A dynamical mechanism for symmetry breaking is investigated under the circumstances with the finite curvature, finite size and non-trivial topology. A four- and eight-fermion interaction model is considered as a prototype model which induces symmetry breaking at GUT era. Evaluating the effective potential in the leading order of the 1/N-expansion by using the dimensional regularization, we explicitly calculate the phase boundary which divides the symmetric and the broken phase in a weakly curved space-time and a flat space-time with non-trivial topology, $R^{D-1} \otimes S^1$.Comment: 20 pages, 21 figure

Topological boundary conditions, the BPS bound, and elimination of ambiguities in the quantum mass of solitons

We fix the long-standing ambiguity in the 1-loop contribution to the mass of a 1+1-dimensional supersymmetric soliton by adopting a set of boundary conditions which follow from the symmetries of the action and which depend only on the topology of the sector considered, and by invoking a physical principle that ought to hold generally in quantum field theories with a topological sector: for vanishing mass and other dimensionful constants, the vacuum energies in the trivial and topological sectors have to become equal. In the two-dimensional N=1 supersymmetric case we find a result which for the supersymmetric sine-Gordon model agrees with the known exact solution of the S-matrix but seems to violate the BPS bound. We analyze the nontrivial relation between the quantum soliton mass and the quantum BPS bound and find a resolution. For N=2 supersymmetric theories, there are no one-loop corrections to the soliton mass and to the central charge (and also no ambiguities) so that the BPS bound is always saturated. Beyond 1-loop there are no ambiguities in any theory, which we explicitly check by a 2-loop calculation in the sine-Gordon model.Comment: 38 page

Transition Amplitudes in 3D Quantum Gravity: Boundaries and Holography in the Coloured Boulatov Model

We consider transition amplitudes in the coloured simplicial Boulatov model for three-dimensional Riemannian quantum gravity. First, we discuss aspects of the topology of coloured graphs with non-empty boundaries. Using a modification of the standard rooting procedure of coloured tensor models, we then write transition amplitudes systematically as topological expansions. We analyse the transition amplitudes for the simplest boundary topology, the 2-sphere, and prove that they factorize into a sum entirely given by the combinatorics of the boundary spin network state and that the leading order is given by graphs representing the closed 3-ball in the large N limit. This is the first step towards a more detailed study of the holographic nature of coloured Boulatov-type GFT models for topological field theories and quantum gravity.Comment: 42+15 pages, 28+14 figures; revised version matching article published in Annales Henri Poincar\'

Quantum Mechanics of the Vacuum State in Two-Dimensional QCD with Adjoint Fermions

A study of two-dimensional QCD on a spatial circle with Majorana fermions in the adjoint representation of the gauge groups SU(2) and SU(3) has been performed. The main emphasis is put on the symmetry properties related to the homotopically non-trivial gauge transformations and the discrete axial symmetry of this model. Within a gauge fixed canonical framework, the delicate interplay of topology on the one hand and Jacobians and boundary conditions arising in the course of resolving Gauss's law on the other hand is exhibited. As a result, a consistent description of the residual $Z_N$ gauge symmetry (for SU(N)) and the ``axial anomaly" emerges. For illustrative purposes, the vacuum of the model is determined analytically in the limit of a small circle. There, the Born-Oppenheimer approximation is justified and reduces the vacuum problem to simple quantum mechanics. The issue of fermion condensates is addressed and residual discrepancies with other approaches are pointed out.Comment: 44 pages; for hardcopies of figures, contact [email protected]

Topology determines force distributions in one-dimensional random spring networks

Networks of elastic fibers are ubiquitous in biological systems and often provide mechanical stability to cells and tissues. Fiber-reinforced materials are also common in technology. An important characteristic of such materials is their resistance to failure under load. Rupture occurs when fibers break under excessive force and when that failure propagates. Therefore, it is crucial to understand force distributions. Force distributions within such networks are typically highly inhomogeneous and are not well understood. Here we construct a simple one-dimensional model system with periodic boundary conditions by randomly placing linear springs on a circle. We consider ensembles of such networks that consist of N nodes and have an average degree of connectivity z but vary in topology. Using a graph-theoretical approach that accounts for the full topology of each network in the ensemble, we show that, surprisingly, the force distributions can be fully characterized in terms of the parameters (N,z). Despite the universal properties of such (N,z) ensembles, our analysis further reveals that a classical mean-field approach fails to capture force distributions correctly. We demonstrate that network topology is a crucial determinant of force distributions in elastic spring networks

Isogeometric topology optimization for auxetic metamaterials and structures

University of Technology Sydney. Faculty of Engineering and Information Technology.It is known that topology optimization is located at the conceptual design phase, which can effectively determine the numbers, connectivity and existence of holes in the structural design domain and evolve design elements to improve the concerned performance. General speaking, topology optimization works as an important tool to seek for the optimal material distribution, which has been identified as one of the most promising sub-field of structural optimization due to its superior features occurring in the conceptual design stage without prior knowledge of the design domain. In the current work, the main intention is to propose a novel numerical method for the topology optimization with more effectiveness and efficiency for the single-material structures and structures with multiple materials. Meanwhile, the proposed topology optimization method is also applied to implement the rational design of auxetic metamaterials and auxetic composites. In Chapter 1, we provide a brief description for the main intention of the current work. In Chapter 2, the comprehensive review about the developments of topology optimization, isogeometric topology optimization and the rational design of auxetic materials is provided. In Chapter 3, a more effective and efficient topology optimization method using isogeometric analysis is proposed for continuum structures using an enhanced density distribution function (DDF). The construction of the DDF mainly involves two steps: (1) the smoothness of nodal densities is improved by the Shepard function; (2) the higher-order NURBS basis functions are combined with the smoothed nodal densities to construct the DDF with the continuity. A topology optimization formulation to minimize the structural mean compliance is developed using the DDF and isogeometric analysis (IGA) is applied to solve structural responses. An integration of the geometry parametrization and numerical analysis offer several benefits for the optimization. The Chapter 4 intends to develop a Multi-material Isogeometric Topology Optimization (M-ITO) method. Firstly, a new Multi-material Interpolation model is established with the use of NURBS (Non-uniform Rational B-splines), termed by the “N-MMI” model, which mainly involves three components: (1) Multiple Fields of Design Variables (DVFs); (2) Multiple Fields of Topology Variables (TVFs); (3) Multi-material interpolation. Two different M-ITO formulations are developed using the N-MMI model to address the problems with multiple volume constraints and the total mass constraint, respectively. The decoupled expression and serial evolving of the DVFs and TVFs can effectively eliminate numerical difficulties in the multi-material problems. In Chapter 5, the proposed ITO method is applied for the systematic design of both 2D and 3D auxetic metamaterials. An energy-based homogenization method (EBHM) to evaluate the macroscopic effective properties is numerically implemented by IGA, with the imposing of periodic boundary formulation on material microstructure. An ITO formulation for 2D and 3D auxetic metamaterials is developed using the DDF, where the objective function is defined as a combination of the homogenized elastic tensor. A relaxed optimality criteria (OC) method is used to update the design variables, due to the non-monotonic property of the problem. In Chapter 6, the proposed M-ITO method is applied for the systematic design of both 2D and 3D auxetic composites. The homogenization, that evaluates macroscopic effective properties of auxetic composites, is numerically implemented by IGA, with the imposing of the periodic boundary formulation on composite microstructures. The developed N-MMI model is applied to describe the multi-material topology and evaluate the multi-material properties. A topology optimization formulation for the design of both two-dimensional (2D) and three-dimensional (3D) auxetic composites is developed. Finite element simulations of auxetic composites are discussed using the ANSYS to show different deformation mechanisms. Finally, conclusions and prospects are given in Chapter 7