A depth-integrated 2D coastal and estuarine model with conformal boundary-fitted mesh generation

Details are given of the development and application of a 2D depth-integrated, conformal boundary-fitted, curvilinear model for predicting the depth-mean velocity field and the spatial concentration distribution in estuarine and coastal waters. A numerical method for conformal mesh generation, based on a boundary integral equation formulation, has been developed. By this method a general polygonal region with curved edges can be mapped onto a regular polygonal region with the same number of horizontal and vertical straight edges and a multiply connected region can be mapped onto a regular region with the same connectivity. A stretching transformation on the conformally generated mesh has also been used to provide greater detail where it is needed close to the coast, with larger mesh sizes further offshore, thereby minimizing the computing effort whilst maximizing accuracy. The curvilinear hydrodynamic and solute model has been developed based on a robust rectilinear model. The hydrodynamic equations are approximated using the ADI finite difference scheme with a staggered grid and the solute transport equation is approximated using a modified QUICK scheme. Three numerical examples have been chosen to test the curvilinear model, with an emphasis placed on complex practical applications

Some uniform stability and convergence results for integral equations on the real line and projection methods for their solution

The paper considers second kind equations of the form (abbreviated x=y + K2x) in which and the factor z is bounded but otherwise arbitrary so that equations of Wiener-Hopf type are included as a special case. Conditions on a set are obtained such that a generalized Fredholm alternative is valid: if W satisfies these conditions and I − Kz, is injective for each z ε W then I − Kz is invertible for each z ε W and the operators (I − Kz)−1 are uniformly bounded. As a special case some classical results relating to Wiener-Hopf operators are reproduced. A finite section version of the above equation (with the range of integration reduced to [−a, a]) is considered, as are projection and iterated projection methods for its solution. The operators (where denotes the finite section version of Kz) are shown uniformly bounded (in z and a) for all a sufficiently large. Uniform stability and convergence results, for the projection and iterated projection methods, are obtained. The argument generalizes an idea in collectively compact operator theory. Some new results in this theory are obtained and applied to the analysis of projection methods for the above equation when z is compactly supported and k(s − t) replaced by the general kernel k(s,t). A boundary integral equation of the above type, which models outdoor sound propagation over inhomogeneous level terrain, illustrates the application of the theoretical results developed.

A high frequency $hp$ boundary element method for scattering by convex polygons

In this paper we propose and analyze a hybrid $hp$ boundary element method for the solution of problems of high frequency acoustic scattering by sound-soft convex polygons, in which the approximation space is enriched with oscillatory basis functions which efficiently capture the high frequency asymptotics of the solution. We demonstrate, both theoretically and via numerical examples, exponential convergence with respect to the order of the polynomials, moreover providing rigorous error estimates for our approximations to the solution and to the far field pattern, in which the dependence on the frequency of all constants is explicit. Importantly, these estimates prove that, to achieve any desired accuracy in the computation of these quantities, it is sufficient to increase the number of degrees of freedom in proportion to the logarithm of the frequency as the frequency increases, in contrast to the at least linear growth required by conventional methods.

Wind pressure on a single building immersed in a low-jet wind profile

This paper for the first time discuss the wind pressure distribution on the building surface immersed in wind profile of low-level jet rather than a logarithmic boundary-layer profile. Two types of building models are considered, low-rise and high-rise building, relative to the low-level jet height. CFD simulation is carried out. The simulation results show that the wind pressure distribution immersed in a low-jet wine profile is very different from the typical uniform and boundary-layer flow. For the low-rise building, the stagnation point is located at the upper level of windward façade for the low-level jet wind case, and the separation zone above the roof top is not as obvious as the uniform case. For the high-rise building model, the height of stagnation point is almost as high as the low-level jet height.

A high frequency boundary element method for scattering by a class of nonconvex obstacles

In this paper we propose and analyse a hybrid numerical-asymptotic boundary element method for the solution of problems of high frequency acoustic scattering by a class of sound-soft nonconvex polygons. The approximation space is enriched with carefully chosen oscillatory basis functions; these are selected via a study of the high frequency asymptotic behaviour of the solution. We demonstrate via a rigorous error analysis, supported by numerical examples, that to achieve any desired accuracy it is sufficient for the number of degrees of freedom to grow only in proportion to the logarithm of the frequency as the frequency increases, in contrast to the at least linear growth required by conventional methods. This appears to be the first such numerical analysis result for any problem of scattering by a nonconvex obstacle. Our analysis is based on new frequency-explicit bounds on the normal derivative of the solution on the boundary and on its analytic continuation into the complex plane.

Progress in observing and modelling the urban boundary layer

The urban boundary layer (UBL) is the part of the atmosphere in which most of the planet’s population now lives, and is one of the most complex and least understood microclimates. Given potential climate change impacts and the requirement to develop cities sustainably, the need for sound modelling and observational tools becomes pressing. This review paper considers progress made in studies of the UBL in terms of a conceptual framework spanning microscale to mesoscale determinants of UBL structure and evolution. Considerable progress in observing and modelling the urban surface energy balance has been made. The urban roughness sub-layer is an important region requiring attention as assumptions about atmospheric turbulence break down in this layer and it may dominate coupling of the surface to the UBL due to its considerable depth. The upper 90% of the UBL (mixed and residual layers) remains under-researched but new remote sensing methods and high resolution modelling tools now permit rapid progress. Surface heterogeneity dominates from neighbourhood to regional scales and should be more strongly considered in future studies. Specific research priorities include humidity within the UBL, high-rise urban canopies and the development of long-term, spatially extensive measurement networks coupled strongly to model development.

Plane wave discontinuous Galerkin methods: exponential convergence of the hp-version

We consider the two-dimensional Helmholtz equation with constant coefficients on a domain with piecewise analytic boundary, modelling the scattering of acoustic waves at a sound-soft obstacle. Our discretisation relies on the Trefftz-discontinuous Galerkin approach with plane wave basis functions on meshes with very general element shapes, geometrically graded towards domain corners. We prove exponential convergence of the discrete solution in terms of number of unknowns.

Monotone positive solutions for a fourth order equation with nonlinear boundary conditions

This work is concerned with the existence of monotone positive solutions for a class of beam equations with nonlinear boundary conditions. The results are obtained by using the monotone iteration method and they extend early works on beams with null boundary conditions. Numerical simulations are also presented. (C) 2009 Elsevier Ltd. All rights reserved.

Semilinear parabolic problems in thin domains with a highly oscillatory boundary

In this paper, we study the behavior of the solutions of nonlinear parabolic problems posed in a domain that degenerates into a line segment (thin domain) which has an oscillating boundary. We combine methods from linear homogenization theory for reticulated structures and from the theory on nonlinear dynamics of dissipative systems to obtain the limit problem for the elliptic and parabolic problems and analyze the convergence properties of the solutions and attractors of the evolutionary equations. (C) 2011 Elsevier Ltd. All rights reserved.

CONSTRUCTING QUANTUM OBSERVABLES AND SELF-ADJOINT EXTENSIONS OF SYMMETRIC OPERATORS. III. SELF-ADJOINT BOUNDARY CONDITIONS

This paper completes the review of the theory of self-adjoint extensions of symmetric operators for physicists as a basis for constructing quantum-mechanical observables. It contains a comparative presentation of the well-known methods and a newly proposed method for constructing ordinary self-adjoint differential operators associated with self-adjoint differential expressions in terms of self-adjoint boundary conditions. The new method has the advantage that it does not require explicitly evaluating deficient subspaces and deficiency indices (these latter are determined in passing) and that boundary conditions are of explicit character irrespective of the singularity of a differential expression. General assertions and constructions are illustrated by examples of well-known quantum-mechanical operators like momentum and Hamiltonian.

Boundary and internal conditions for adjoint fluid-flow problems

The ever-increasing robustness and reliability of flow-simulation methods have consolidated CFD as a major tool in virtually all branches of fluid mechanics. Traditionally, those methods have played a crucial role in the analysis of flow physics. In more recent years, though, the subject has broadened considerably, with the development of optimization and inverse design applications. Since then, the search for efficient ways to evaluate flow-sensitivity gradients has received the attention of numerous researchers. In this scenario, the adjoint method has emerged as, quite possibly, the most powerful tool for the job, which heightens the need for a clear understanding of its conceptual basis. Yet, some of its underlying aspects are still subject to debate in the literature, despite all the research that has been carried out on the method. Such is the case with the adjoint boundary and internal conditions, in particular. The present work aims to shed more light on that topic, with emphasis on the need for an internal shock condition. By following the path of previous authors, the quasi-1D Euler problem is used as a vehicle to explore those concepts. The results clearly indicate that the behavior of the adjoint solution through a shock wave ultimately depends upon the nature of the objective functional.

Existence of orthogonal geodesic chords on Riemannian manifolds with concave boundary and homeomorphic to the N-dimensional disk

In this paper we give a proof of the existence of an orthogonal geodesic chord on a Riemannian manifold homeomorphic to a closed disk and with concave boundary. This kind of study is motivated by the link (proved in Giambo et al. (2005) [8]) of the multiplicity problem with the famous Seifert conjecture (formulated in Seifert (1948) [1]) about multiple brake orbits for a class of Hamiltonian systems at a fixed energy level. (C) 2010 Elsevier Ltd. All rights reserved.

GENERICITY OF NONDEGENERATE GEODESICS WITH GENERAL BOUNDARY CONDITIONS

Let M be a possibly noncompact manifold. We prove, generically in the C(k)-topology (2 <= k <= infinity), that semi-Riemannian metrics of a given index on M do not possess any degenerate geodesics satisfying suitable boundary conditions. This extends a result of L. Biliotti, M. A. Javaloyes and P. Piccione [6] for geodesics with fixed endpoints to the case where endpoints lie on a compact submanifold P subset of M x M that satisfies an admissibility condition. Such condition holds, for example, when P is transversal to the diagonal Delta subset of M x M. Further aspects of these boundary conditions are discussed and general conditions under which metrics without degenerate geodesics are C(k)-generic are given.

Testing of combined heating systems for small houses: Improved procedures for whole system test methods : Deliverable 2.3

Dynamic system test methods for heating systems were developed and applied by the institutes SERC and SP from Sweden, INES from France and SPF from Switzerland already before the MacSheep project started. These test methods followed the same principle: a complete heating system – including heat generators, storage, control etc., is installed on the test rig; the test rig software and hardware simulates and emulates the heat load for space heating and domestic hot water of a single family house, while the unit under test has to act autonomously to cover the heat demand during a representative test cycle. Within the work package 2 of the MacSheep project these similar – but different – test methods were harmonized and improved. The work undertaken includes:  • Harmonization of the physical boundaries of the unit under test. • Harmonization of the boundary conditions of climate and load. • Definition of an approach to reach identical space heat load in combination with an autonomous control of the space heat distribution by the unit under test. • Derivation and validation of new six day and a twelve day test profiles for direct extrapolation of test results.   The new harmonized test method combines the advantages of the different methods that existed before the MacSheep project. The new method is a benchmark test, which means that the load for space heating and domestic hot water preparation will be identical for all tested systems, and that the result is representative for the performance of the system over a whole year. Thus, no modelling and simulation of the tested system is needed in order to obtain the benchmark results for a yearly cycle. The method is thus also applicable to products for which simulation models are not available yet. Some of the advantages of the new whole system test method and performance rating compared to the testing and energy rating of single components are:  • Interaction between the different components of a heating system, e.g. storage, solar collector circuit, heat pump, control, etc. are included and evaluated in this test. • Dynamic effects are included and influence the result just as they influence the annual performance in the field. • Heat losses are influencing the results in a more realistic way, since they are evaluated under "real installed" and representative part-load conditions rather than under single component steady state conditions.   The described method is also suited for the development process of new systems, where it replaces time-consuming and costly field testing with the advantage of a higher accuracy of the measured data (compared to the typically used measurement equipment in field tests) and identical, thus comparable boundary conditions. Thus, the method can be used for system optimization in the test bench under realistic operative conditions, i.e. under relevant operating environment in the lab.   This report describes the physical boundaries of the tested systems, as well as the test procedures and the requirements for both the unit under test and the test facility. The new six day and twelve day test profiles are also described as are the validation results.

Superconducting boundary conditions for mesoscopic circular samples

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)