A BEM based on the Bézier/Bernstein polynomial for acoustic waveguide modelization

42nd International Conference on Boundary Elements and other Mesh Reduction Methods, BEM/MRM 2019; ITeCons-University of CoimbraCoimbra; Portugal; 2 July 2019 through 4 July 2019; Code 155806. Publicado en WIT Transactions on Engineering Sciences, Vol 126This paper proposes a novel boundary element approach formulated on the Bézier–Bernstein basis to yield a geometry-independent field approximation. The proposed method is geometrically based on both computer aided design (CAD) and isogeometric analysis (IGA), but field variables are independently approximated from the geometry. This approach allows the appropriate approximation functions for the geometry and variable field to be chosen. We use the Bézier–Bernstein form of a polynomial as an approximation basis to represent both geometry and field variables. The solution of the element interpolation problem in the Bézier–Bernstein space defines generalised Lagrange interpolation functions that are used as element shape functions. The resulting Bernstein–Vandermonde matrix related to the Bézier–Bernstein interpolation problem is inverted using the Newton–Bernstein algorithm. The applicability of the proposed method is demonstrated by solving the Helmholtz equation over an unbounded region in a two-and-a-half dimensional (2.5D) domain.Ministerio de Economía y Competitividad BIA2016-75042-C2-1-

New hybrid quadrature schemes for weakly singular kernels applied to isogeometric boundary elements for 3D Stokes flow

This work proposes four novel hybrid quadrature schemes for the efficient and accurate evaluation of weakly singular boundary integrals (1/r kernel) on arbitrary smooth surfaces. Such integrals appear in boundary element analysis for several partial differential equations including the Stokes equation for viscous flow and the Helmholtz equation for acoustics. The proposed quadrature schemes apply a Duffy transform-based quadrature rule to surface elements containing the singularity and classical Gaussian quadrature to the remaining elements. Two of the four schemes additionally consider a special treatment for elements near to the singularity, where refined Gaussian quadrature and a new moment-fitting quadrature rule are used. The hybrid quadrature schemes are systematically studied on flat B-spline patches and on NURBS spheres considering two different sphere discretizations: An exact single-patch sphere with degenerate control points at the poles and an approximate discretization that consist of six patches with regular elements. The efficiency of the quadrature schemes is further demonstrated in boundary element analysis for Stokes flow, where steady problems with rotating and translating curved objects are investigated in convergence studies for both, mesh and quadrature refinement. Much higher convergence rates are observed for the proposed new schemes in comparison to classical schemes

Design of quadrature rules for Müntz and Müntz-logarithmic polynomials using monomial transformation

A method for constructing the exact quadratures for Müntz and Müntz-logarithmic polynomials is presented. The algorithm does permit to anticipate the precision (machine precision) of the numerical integration of Müntz-logarithmic polynomials in terms of the number of Gauss-Legendre (GL) quadrature samples and monomial transformation order. To investigate in depth the properties of classical GL quadrature, we present new optimal asymptotic estimates for the remainder. In boundary element integrals this quadrature rule can be applied to evaluate singular functions with end-point singularity, singular kernel as well as smooth functions. The method is numerically stable, efficient, easy to be implemented. The rule has been fully tested and several numerical examples are included. The proposed quadrature method is more efficient in run-time evaluation than the existing methods for Müntz polynomial

Enrichment of the Boundary Element Method through the Partition of Unity Method for Fracture Analysis using Local and Global Formulations

The present thesis proposes an innovative technique of applying enrichment to the Boundary Element Method to allow accurate analysis of 2D crack problems. An overview of fracture mechanics is given, with particular emphasis given to numerical methods and the techniques used to extract the highly important stress intensity factors - a measure of the singularity of a crack tip. The Boundary Element Method framework is described and later, the implementation of the new technique of enrichment is defined in detail. Finally, several crack problems are used to verify the accuracy of the method where the results are shown to compare very favourably with other well-established numerical methods

Plane waves with weak singularities

We study a class of time dependent solutions of the vacuum Einstein equations which are plane waves with weak null singularities. This singularity is weak in the sense that though the tidal forces diverge at the singularity, the rate of divergence is such that the distortion suffered by a freely falling observer remains finite. Among such weak singular plane waves there is a sub-class which do not exhibit large back reaction in the presence of test scalar probes. String propagation in these backgrounds is smooth and there is a natural way to continue the metric beyond the singularity. This continued metric admits string propagation without the string becoming infinitely excited. We construct a one parameter family of smooth metrics which are at a finite distance in the space of metrics from the extended metric and a well defined operator in the string sigma model which resolves the singularity.Comment: 22 pages, Added references and clarifying comment

Extended Boundary Element Method approach for Direct and Accurate Evaluation of Stress Intensity Factors

This thesis introduces an alternative method to evaluate Stress Intensity Factors (SIFs) in computational fracture mechanics directly, using the Extended Dual Boundary Element Method (XBEM) for 2D problems. A novel auxiliary equation introduced which enforces displacement continuity at the crack tip to yield a square system. Additionally, the enrichment method has been extended to 3D, so that the J-integral with XBEM and a direct technique are used to evaluate SIFs. This includes a complete description of the formulation of enrichment functions, a substitution of the enriched form of displacement into boundary integral equations, treatment of singular integrals, assembly of system matrices and the introduction of auxiliary equations to solve the system directly. The enrichment approach utilizes the Williams expansions to enrich crack surface elements for accurate evaluation of stress intensity factors. Similar to other enrichment methods, the new approach can yield accurate results on coarse discretisations, and the enrichment increases the 2D problem size by only two degrees of freedom per crack tip. In the case of 3D, the number of the new degrees of freedom depends on the desired number of crack front points where SIFs need to be evaluated. The auxiliary equations required to yield a square system are derived by enforcing continuity of displacement at the crack front. The enrichment approach provides the values of singular coefficients KI, KII and KIII directly in the solution vector; without any need for postprocessing such as the J-integral. Numerical examples are used to compare the accuracy of these directly computed SIFs to the J-integral processing of both conventional and XBEM approximations

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

Numerical methods for computing Casimir interactions

We review several different approaches for computing Casimir forces and related fluctuation-induced interactions between bodies of arbitrary shapes and materials. The relationships between this problem and well known computational techniques from classical electromagnetism are emphasized. We also review the basic principles of standard computational methods, categorizing them according to three criteria---choice of problem, basis, and solution technique---that can be used to classify proposals for the Casimir problem as well. In this way, mature classical methods can be exploited to model Casimir physics, with a few important modifications.Comment: 46 pages, 142 references, 5 figures. To appear in upcoming Lecture Notes in Physics book on Casimir Physic