Spectral methods for exterior elliptic problems

Spectral approximations for exterior elliptic problems in two dimensions are discussed. As in the conventional finite difference or finite element methods, the accuracy of the numerical solutions is limited by the order of the numerical farfield conditions. A spectral boundary treatment is introduced at infinity which is compatible with the infinite order interior spectral scheme. Computational results are presented to demonstrate the spectral accuracy attainable. Although a simple Laplace problem is examined, the analysis covers more complex and general cases

Localized Orthogonal Decomposition for two-scale Helmholtz-type problems

In this paper, we present a Localized Orthogonal Decomposition (LOD) in Petrov-Galerkin formulation for a two-scale Helmholtz-type problem. The two-scale problem is, for instance, motivated from the homogenization of the Helmholtz equation with high contrast, studied together with a corresponding multiscale method in (Ohlberger, Verf\"urth. A new Heterogeneous Multiscale Method for the Helmholtz equation with high contrast, arXiv:1605.03400, 2016). There, an unavoidable resolution condition on the mesh sizes in terms of the wave number has been observed, which is known as "pollution effect" in the finite element literature. Following ideas of (Gallistl, Peterseim. Comput. Methods Appl. Mech. Engrg. 295:1-17, 2015), we use standard finite element functions for the trial space, whereas the test functions are enriched by solutions of subscale problems (solved on a finer grid) on local patches. Provided that the oversampling parameter $m$, which indicates the size of the patches, is coupled logarithmically to the wave number, we obtain a quasi-optimal method under a reasonable resolution of a few degrees of freedom per wave length, thus overcoming the pollution effect. In the two-scale setting, the main challenges for the LOD lie in the coupling of the function spaces and in the periodic boundary conditions.Comment: 20 page

PARTITION OF UNITY BOUNDARY ELEMENT AND FINITE ELEMENT METHOD: OVERCOMING NONUNIQUENESS AND COUPLING FOR ACOUSTIC SCATTERING IN HETEROGENEOUS MEDIA

The understanding of complex wave phenomenon, such as multiple scattering in heterogeneous media, is often hindered by lack of equations modelling the exact physics. Use of approximate numerical methods, such as Finite Element Method (FEM) and Boundary Element Method (BEM), is therefore needed to understand these complex wave problems. FEM is known for its ability to accurately model the physics of the problem but requires truncating the computational domain. On the other hand, BEM can accurately model waves in unbounded region but is suitable for homogeneous media only. Coupling FEM and BEM therefore is a natural way to solve problems involving a relatively small heterogeneity (to be modelled with FEM) surrounded by an unbounded homogeneous medium (to be modelled with BEM). The use of a classical FEM-BEM coupling can become computationally demanding due to high mesh density requirement at high frequencies. Secondly, BEM is an integral equation based technique and suffers from the problem of non-uniqueness. To overcome the requirement of high mesh density for high frequencies, a technique known as the ‘Partition of Unity’ (PU) method has been developed by previous researchers. The work presented in this thesis extends the concept of PU to BEM (PUBEM) while effectively treating the problem of non-uniqueness. Two of the well-known methods, namely CHIEF and Burton-Miller approaches, to overcome the non-uniqueness problem, are compared for PUBEM. It is shown that the CHIEF method is relatively easy to implement and results in at least one order of magnitude of improvement in the accuracy. A modified ‘PU’ concept is presented to solve the heterogeneous problems with the PU based FEM (PUFEM). It is shown that use of PUFEM results in close to two orders of magnitude improvement over FEM despite using a much coarser mesh. The two methods, namely PUBEM and PUFEM, are then coupled to solve the heterogeneous wave problems in unbounded media. Compared to PUFEM, the coupled PUFEM-PUBEM apporach is shown to result between 30-40% savings in the total degress of freedom required to achieve similar accuracy

High order methods for acoustic scattering: Coupling Farfield Expansions ABC with Deferred-Correction methods

Arbitrary high order numerical methods for time-harmonic acoustic scattering problems originally defined on unbounded domains are constructed. This is done by coupling recently developed high order local absorbing boundary conditions (ABCs) with finite difference methods for the Helmholtz equation. These ABCs are based on exact representations of the outgoing waves by means of farfield expansions. The finite difference methods, which are constructed from a deferred-correction (DC) technique, approximate the Helmholtz equation and the ABCs, with the appropriate number of terms, to any desired order. As a result, high order numerical methods with an overall order of convergence equal to the order of the DC schemes are obtained. A detailed construction of these DC finite difference schemes is presented. Additionally, a rigorous proof of the consistency of the DC schemes with the Helmholtz equation and the ABCs in polar coordinates is also given. The results of several numerical experiments corroborate the high order convergence of the novel method.Comment: 36 pages, 20 figure

Non Uniform Rational B-Splines and Lagrange approximations for time-harmonic acoustic scattering: accuracy and absorbing boundary conditions

International audienceIn this paper, the performance of the finite element method based on Lagrange basis functions and the Non Uniform Rational B-Splines (NURBS) based Iso-Geometric Analysis (IGA) are systematically studied for solving time-harmonic acoustic scattering problems. To assess their performance, the numerical examples are presented with truncated absorbing boundary conditions. In the first two examples , we eliminate the domain truncation error by applying second-order Bayliss-Gunzburger-Turkel (BGT-2) Absorbing Boundary Condition (ABC) and modifying the exact solution. Hence, the calculated error is an indicator of the numerical accuracy in the bounded computational domain with no artificial domain truncation error. Next, we apply a higher order local ABC based on the Karp's and Wilcox's far-field expansions for 2D and 3D problems, respectively. The performance of both methods in solving exterior problems is compared. The introduced auxiliary surface functions are also estimated using the corresponding basis functions. The influence of various parameters, viz., order of the approximating polynomial, number of degrees of freedom, wave number and the boundary conditions (BGT-2 and number of terms in the far-field expansions) on the accuracy and convergence rate is systematically studied. It is inferred that, irrespective of the order of the polynomial, IGA yields higher accuracy per degree of freedom when compared to the conventional finite element method with Lagrange basis

Une méthode de couplage éléments finis-conditions absorbantes de type-padé pour les problèmes de diffraction acoustique

Nous nous intéressons aux problèmes harmoniques de diffraction acoustique en milieu infini régis par l'équation de Helmholtz. La simulation numérique de ces phénomènes est complexe notamment lorsqu'il est question de fréquences élevées et d'obstacles de forme allongée tel qu'un sous-marin. Les codes éléments finis commerciaux sont incapables de cerner tous les aspects liés à ce type de problèmes. De plus, ce genre d'applications fait appel à de grandes ressources de calcul. En effet, la taille du système d'équations à résoudre (plusieurs millions de ddl) engendre souvent l'épuisement des ressources des calculateurs traditionnels. Notre objectif est de solutionner ce type de problèmes avec une précision pratique en utilisant le minimum de ressources. Nous proposons ainsi une méthode de couplage éléments finis de type Lagrange et à base d'ondes planes avec les conditions absorbantes d'ordre élevé basées sur les approximants complexes de Padé. A travers une série d'expériences numériques, nous montrons l'efficacité de ces conditions absorbantes en comparaison avec les conditions absorbantes de Bayliss-Gunzburger-Turkel d'ordre deux implémentées dans les codes commerciaux. La méthodologie proposée permet non seulement une réduction de la taille du domaine de calcul sans dégradation de la précision mais conduit également à la résolution de systèmes d'équations de taille relativement réduite

Leaf segmentation in plant phenotyping: a collation study

Image-based plant phenotyping is a growing application area of computer vision in agriculture. A key task is the segmentation of all individual leaves in images. Here we focus on the most common rosette model plants, Arabidopsis and young tobacco. Although leaves do share appearance and shape characteristics, the presence of occlusions and variability in leaf shape and pose, as well as imaging conditions, render this problem challenging. The aim of this paper is to compare several leaf segmentation solutions on a unique and first-of-its-kind dataset containing images from typical phenotyping experiments. In particular, we report and discuss methods and findings of a collection of submissions for the first Leaf Segmentation Challenge of the Computer Vision Problems in Plant Phenotyping workshop in 2014. Four methods are presented: three segment leaves by processing the distance transform in an unsupervised fashion, and the other via optimal template selection and Chamfer matching. Overall, we find that although separating plant from background can be accomplished with satisfactory accuracy (>>90 % Dice score), individual leaf segmentation and counting remain challenging when leaves overlap. Additionally, accuracy is lower for younger leaves. We find also that variability in datasets does affect outcomes. Our findings motivate further investigations and development of specialized algorithms for this particular application, and that challenges of this form are ideally suited for advancing the state of the art. Data are publicly available (online at http://​www.​plant-phenotyping.​org/​datasets) to support future challenges beyond segmentation within this application domain

Assessing the Feasibility of Nutrient Trading Between Point Sources and Nonpoint Sources in the Chao Lake Basin Final

This pilot project will determine the Feasibility of an effective point-nonpoint source nutrient trading program could be established in the Lake Chao Basin, Program's potential benefits, Framework and necessary elements for such a program