Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the $p$-version

Plane wave discontinuous Galerkin (PWDG) methods are a class of Trefftz-type methods for the spatial discretization of boundary value problems for the Helmholtz operator $-\Delta-\omega^2$, $\omega>0$. They include the so-called ultra weak variational formulation from [O. Cessenat and B. Després, SIAM J. Numer. Anal., 35 (1998), pp. 255–299]. This paper is concerned with the a priori convergence analysis of PWDG in the case of $p$-refinement, that is, the study of the asymptotic behavior of relevant error norms as the number of plane wave directions in the local trial spaces is increased. For convex domains in two space dimensions, we derive convergence rates, employing mesh skeleton-based norms, duality techniques from [P. Monk and D. Wang, Comput. Methods Appl. Mech. Engrg., 175 (1999), pp. 121–136], and plane wave approximation theory.

Numerical-asymptotic boundary integral methods in high-frequency acoustic scattering

In this article we describe recent progress on the design, analysis and implementation of hybrid numerical-asymptotic boundary integral methods for boundary value problems for the Helmholtz equation that model time harmonic acoustic wave scattering in domains exterior to impenetrable obstacles. These hybrid methods combine conventional piecewise polynomial approximations with high-frequency asymptotics to build basis functions suitable for representing the oscillatory solutions. They have the potential to solve scattering problems accurately in a computation time that is (almost) independent of frequency and this has been realized for many model problems. The design and analysis of this class of methods requires new results on the analysis and numerical analysis of highly oscillatory boundary integral operators and on the high-frequency asymptotics of scattering problems. The implementation requires the development of appropriate quadrature rules for highly oscillatory integrals. This article contains a historical account of the development of this currently very active field, a detailed account of recent progress and, in addition, a number of original research results on the design, analysis and implementation of these methods.

A Posteriori analysis of discontinuous Galerkin schemes for systems of hyperbolic conservation laws

In this work we construct reliable a posteriori estimates for some semi- (spatially) discrete discontinuous Galerkin schemes applied to nonlinear systems of hyperbolic conservation laws. We make use of appropriate reconstructions of the discrete solution together with the relative entropy stability framework, which leads to error control in the case of smooth solutions. The methodology we use is quite general and allows for a posteriori control of discontinuous Galerkin schemes with standard flux choices which appear in the approximation of conservation laws. In addition to the analysis, we conduct some numerical benchmarking to test the robustness of the resultant estimator.

A priori error analysis of space–time Trefftz discontinuous Galerkin methods for wave problems

We present and analyse a space–time discontinuous Galerkin method for wave propagation problems. The special feature of the scheme is that it is a Trefftz method, namely that trial and test functions are solution of the partial differential equation to be discretised in each element of the (space–time) mesh. The method considered is a modification of the discontinuous Galerkin schemes of Kretzschmar et al. (2014) and of Monk & Richter (2005). For Maxwell’s equations in one space dimension, we prove stability of the method, quasi-optimality, best approximation estimates for polynomial Trefftz spaces and (fully explicit) error bounds with high order in the meshwidth and in the polynomial degree. The analysis framework also applies to scalar wave problems and Maxwell’s equations in higher space dimensions. Some numerical experiments demonstrate the theoretical results proved and the faster convergence compared to the non-Trefftz version of the scheme.

Reduced relative entropy techniques for a posteriori analysis of multiphase problems in elastodynamics

We give an a posteriori analysis of a semidiscrete discontinuous Galerkin scheme approximating solutions to a model of multiphase elastodynamics, which involves an energy density depending not only on the strain but also the strain gradient. A key component in the analysis is the reduced relative entropy stability framework developed in Giesselmann (2014, SIAM J. Math. Anal., 46, 3518–3539). This framework allows energy-type arguments to be applied to continuous functions. Since we advocate the use of discontinuous Galerkin methods we make use of two families of reconstructions, one set of discrete reconstructions and a set of elliptic reconstructions to apply the reduced relative entropy framework in this setting.

Two- and three-dimensional shape fabric analysis by the intercept method in grey levels

The count intercept is a robust method for the numerical analysis of fabrics Launeau and Robin (1996). It counts the number of intersections between a set of parallel scan lines and a mineral phase, which must be identified on a digital image. However, the method is only sensitive to boundaries and therefore supposes the user has some knowledge about their significance. The aim of this paper is to show that a proper grey level detection of boundaries along scan lines is sufficient to calculate the two-dimensional anisotropy of grain or crystal distributions without any particular image processing. Populations of grains and crystals usually display elliptical anisotropies in rocks. When confirmed by the intercept analysis, a combination of a minimum of 3 mean length intercept roses, taken on 3 more or less perpendicular sections, allows the calculation of 3-dimensional ellipsoids and the determination of their standard deviation with direction and intensity in 3 dimensions as well. The feasibility of this quick method is attested by numerous examples on theoretical objects deformed by active and passive deformation, on BSE images of synthetic magma flow, on drawing or direct analysis of thin section pictures of sandstones and on digital images of granites directly taken and measured in the field. (C) 2010 Elsevier B.V. All rights reserved.

Unemployment insurance an analysis of optimal mechanisms under aggregate shocks

The purpose of this work is to provide a brief overview of the literature on the optimal design of unemployment insurance systems by analyzing some of the most influential articles published over the last three decades on the subject and extend the main results to a multiple aggregate shocks environment. The properties of optimal contracts are discussed in light of the key assumptions commonly made in theoretical publications on the area. Moreover, the implications of relaxing each of these hypothesis is reckoned as well. The analysis of models of only one unemployment spell starts from the seminal work of Shavell and Weiss (1979). In a simple and common setting, unemployment benefits policies, wage taxes and search effort assignments are covered. Further, the idea that the UI distortion of the relative price of leisure and consumption is the only explanation for the marginal incentives to search for a job is discussed, putting into question the reduction in labor supply caused by social insurance, usually interpreted as solely an evidence of a dynamic moral hazard caused by a substitution effect. In addition, the paper presents one characterization of optimal unemployment insurance contracts in environments in which workers experience multiple unemployment spells. Finally, an extension to multiple aggregate shocks environment is considered. The paper ends with a numerical analysis of the implications of i.i.d. shocks to the optimal unemployment insurance mechanism.

A Higher order and stable method for the numerical integration of Random Differential Equations

Trabalho apresentado no Congresso Nacional de Matemática Aplicada à Indústria, 18 a 21 de novembro de 2014, Caldas Novas - Goiás

NUMERICAL SIMULATION of THE BEHAVIOR of THE BEHAVIOR of REINFORCED CONCRETE BEAMS and COLUMNS BY FEM CAASTEM 2000 PROGRAM

The objective of this paper is the numerical study of the behavior of reinforced concrete beams and columns by non-linear numerical simulations. The numerical analysis is based on the finite element method implemented in CASTEM 2000. This program uses the constitutive elastoplastic perfect model for the steel, the Drucker-Prager model for the concrete and the Newton-Raphson for the solution of non-linear systems. This work concentrates on the determination of equilibrium curves to the beams and force-strain curves to the columns. The numeric responses are confronted with experimental results found in the literature in order to check there liability of the numerical analyses.

Short Implant to Support Maxillary Restorations: Bone Stress Analysis Using Regular and Switching Platform

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

Regular and Platform Switching: Bone Stress Analysis Varying Implant Type

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

Numerical approximation of the Ginzburg-Landau equation with memory effects in the dynamics of phase transitions

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Influence of high insertion torque on implant placement - an anisotropic bone stress analysis

The aim of this study was to evaluate the influence of the high values of insertion torques on the stress and strain distribution in cortical and cancellous bones. Based on tomography imaging, a representative mathematical model of a partial maxilla was built using Mimics 11.11 and Solid Works 2010 softwares. Six models were built and each of them received an implant with one of the following insertion torques: 30, 40, 50, 60, 70 or 80 Ncm on the external hexagon. The cortical and cancellous bones were considered anisotropic. The bone/implant interface was considered perfectly bonded. The numerical analysis was carried out using Ansys Workbench 10.0. The convergence of analysis (6%) drove the mesh refinement. Maximum principal stress (σ max) and maximum principal strain (ε max) were obtained for cortical and cancellous bones around to implant. Pearson's correlation test was used to determine the correlation between insertion torque and stress concentration in the periimplant bone tissue, considering the significance level at 5%. The increase in the insertion torque generated an increase in the σ max and ε max values for cortical and cancellous bone. The σmax was smaller for the cancellous bone, with greater stress variation among the insertion torques. The ε max was higher in the cancellous bone in comparison to the cortical bone. According to the methodology used and the limits of this study, it can be concluded that higher insertion torques increased tensile and compressive stress concentrations in the periimplant bone tissue.

Influence of Platform and Abutment Angulation on Peri-Implant Bone. A Three-Dimensional Finite Element Stress Analysis

The aim of this study was to evaluate stress distribution on the pen-implant bone, simulating the influence of Nobel Select implants with straight or angulated abutments on regular and switching platform in the anterior maxilla, by means of 3-dimensional finite element analysis. Four mathematical models of a central incisor supported by external hexagon implant (13 mm x 5 mm) were created varying the platform (R, regular or S. switching) and the abutments (S, straight or A, angulated 15 degrees). The models were created by using Mimics 13 and Solid Works 2010 software programs. The numerical analysis was performed using ANSYS Workbench 10.0. Oblique forces (100 N) were applied to the palatine surface of the central incisor. The bone/implant interface was considered perfectly integrated. Maximum (sigma(max)) and minimum (sigma(min)) principal stress values were obtained. For the cortical bone the highest stress values (sigma(max)) were observed in the RA (regular platform and angulated abutment, 51 MPa), followed by SA (platform switching and angulated abutment, 44.8 MPa), RS (regular platform and straight abutment, 38.6 MPa) and SS (platform switching and straight abutment, 36.5 MPa). For the trabecular bone, the highest stress values (sigma(max)) were observed in the RA (6.55 MPa), followed by RS (5.88 MPa), SA (5.60 MPa), and SS (4.82 MPa). The regular platform generated higher stress in the cervical periimplant region on the cortical and trabecular bone than the platform switching, irrespective of the abutment used (straight or angulated).

Numerical modeling of flow through an industrial burner orifice

This paper presents numerical modeling of a turbulent natural gas flow through a non-premixed industrial burner of a slab reheating furnace. The furnace is equipped with diffusion side swirl burners capable of utilizing natural gas or coke oven gas alternatively through the same nozzles. The study is focused on one of the burners of the preheating zone. Computational Fluid Dynamics simulation has been used to predict the burner orifice turbulent flow. Flow rate and pressure at burner upstream were validated by experimental measurements. The outcomes of the numerical modeling are analyzed for the different turbulence models in terms of pressure drop, velocity profiles, and orifice discharge coefficient. The standard, RNG, and Realizable k-epsilon models and Reynolds Stress Model (RSM) have been used. The main purpose of the numerical investigation is to determine the turbulence model that more consistently reproduces the experimental results of the flow through an industrial non-premixed burner orifice. The comparisons between simulations indicate that all the models tested satisfactorily and represent the experimental conditions. However, the Realizable k-epsilon model seems to be the most appropriate turbulence model, since it provides results that are quite similar to the RSM and RNG k-epsilon models, requiring only slightly more computational power than the standard k-epsilon model. (C) 2014 Elsevier Ltd. All rights reserved.