Fast Isogeometric Boundary Element Method based on Independent Field Approximation

An isogeometric boundary element method for problems in elasticity is presented, which is based on an independent approximation for the geometry, traction and displacement field. This enables a flexible choice of refinement strategies, permits an efficient evaluation of geometry related information, a mixed collocation scheme which deals with discontinuous tractions along non-smooth boundaries and a significant reduction of the right hand side of the system of equations for common boundary conditions. All these benefits are achieved without any loss of accuracy compared to conventional isogeometric formulations. The system matrices are approximated by means of hierarchical matrices to reduce the computational complexity for large scale analysis. For the required geometrical bisection of the domain, a strategy for the evaluation of bounding boxes containing the supports of NURBS basis functions is presented. The versatility and accuracy of the proposed methodology is demonstrated by convergence studies showing optimal rates and real world examples in two and three dimensions.Comment: 32 pages, 27 figure

Extended isogeometric boundary element method (XIBEM) for two-dimensional Helmholtz problems.

Isogeometric analysis is a topic of considerable interest in the field of numerical analysis. The boundary element method (BEM) requires only the bounding surface of geometries to be described; this makes non-uniform rational B-splines (NURBS), which commonly describe the bounding curve or surface of geometries in CAD software, appear to be a natural tool for the approach. This isogeometric analysis BEM (IGABEM) provides accuracy benefits over conventional BEM schemes due to the analytical geometry provided by NURBS. When applied to wave problems, it has been shown that enriching BEM approximations with a partition-of-unity basis, in what has become known as the PU-BEM, allows highly accurate solutions to be obtained with a much reduced number of degrees of freedom. This paper combines these approaches and presents an extended isogeometric BEM (XIBEM) which uses partition-of-unity enriched NURBS functions; this new approach provides benefits which surpass those of both the IGABEM and the PU-BEM. Two numerical examples are given: a single scattering cylinder and a multiple-scatterer made up of two capsules and a cylinder

Extended isogeometric boundary element method (XIBEM) for three-dimensional medium-wave acoustic scattering problems

A boundary element method (BEM), based on non-uniform rational B-splines (NURBS), is used to find solutions to three-dimensional wave scattering problems governed by the Helmholtz equation. The method is extended in a partition-of-unity sense, multiplying the NURBS functions by families of plane waves; this method is called the eXtended Isogeometric Boundary Element Method (XIBEM). In this paper, the collocation XIBEM formulation is described and numerical results are given. The numerical results are compared against closed-form or converged solutions. Comparisons are made against the conventional boundary element method and the non-enriched isogeometric BEM (IGABEM). When compared to non-enriched boundary element simulations, using XIBEM significantly reduces the number of degrees of freedom required to obtain a solution of a given error; thus, with a fixed computational resource, problems of a shorter wavelength can be solved

3D isogeometric indirect BEM solution based on virtual surface sources on the boundaries of Helmholtz acoustic problems

A solution for 3D Helmholtz acoustic problems is introduced based on an indirect boundary element method (indirect BEM) coupled with isogeometric analysis (IGA). The novelty of this work arises from using virtual surface sources placed directly on the scatterer boundaries,producing robust results. These virtual surface sources are discretized by the same Non-Uniform Rational B-Splines (NURBS) approximating the scatterer CAD model. This allows modelling of general irregular geometries. The proposed solution has the same features of BEM approaches, which do not need any domain discretization or truncation boundaries at the far-field. It shows an additional merit by arranging the linear system of equations directly depending on a single coefficient matrix, consuming less computational time compared to other BEM methods. A Greville abscissae collocation scheme is proposed with offsets at C0-continuities. This collocation scheme allows for easy evaluation for both free-terms and normals at the collocation points.The performance of the proposed solution is discussed on 3D numerical exterior problems and compared against other BEM methods. Then, the practical interior muffler problem with internal extended thin tubes is studied and the obtained results are compared against other numerical methods in addition to the available experimental data, showing the capability of the proposed solution in handling thin-walled geometries

Shape-optimization of 2D hydrofoils using an Isogeometric BEM solver

In this paper, an optimization procedure, based on an Isogeometric BEM solver for the potential ow, is developed and used for the shape optimization of hydrofoils. The formulation of the exterior potential- ow problem reduces to a Boundary-Integral Equation (BIE) for the associated velocity potential exploiting the null-pressure jump Kutta condition at the trailing edge. The numerical solution of the BIE is performed by an Isogeometric Boundary-Element Method (BEM) combining a generic B-splines parametric modeler for generating hydrofoil shapes, using a set of eight parameters, the very same basis of the geometric representation for representing the velocity potential and collocation at the Greville abscissas of the knot vector of the hydrofoil's B-splines representation. Furthermore, the optimization environment is developed based on the geometric parametric modeler for the hydrofoil, the Isogeometric BEM solver and an optimizer employing a controlled elitist genetic algorithm. Multi-objective hydrofoil shape optimization examples are demonstrated with respect to the criteria i) maximum lift coefficient and ii) minimum deviation of the hydrofoil area from a reference area

A BEM-ISOGEOMETRIC method for the ship wave-resistance problem

In the present work IsoGeometric Analysis is applied to the solution of the Boundary Integral Equation associated with the Neumann-Kelvin problem and the calculation of the wave resistance of ships. As opposed to low-order panel methods, where the body is represented by a large number of quadrilateral panels and the velocity potential is assumed to be piecewise constant (or approximated by low degree polynomials) on each panel, the isogeometric concept is based on exploiting the same NURBS basis, used for representing exactly the body geometry, for approximating the singularity distribution (and, in general, the dependent physical quantities). In order to examine the accuracy of the present method, numerical results obtained in the case of submerged and surface piercing bodies are * Corresponding author. Tel: (+30) 2107721138, Fax: (+30) 2107721397, e-mail: [email protected] 2 compared against analytical solutions, experimental data and predictions provided by the low-order panel or other similar methods appeared in the pertinent literature, illustrating the superior efficiency of the isogeometric approach. The present approach by applying Isogeometric Analysis and Boundary Element Method to the linear NK problem has the novelty of combining modern CAD systems for ship-hull design with computational hydrodynamics tools

Shape-optimization of 2D hydrofoils using an isogeometric BEM solver

In this paper, an optimization procedure, based on an Isogeometric BEM solver for the potential flow, is developed and used for the shape optimization of hydrofoils. The formulation of the exterior potential-flow problem reduces to a Boundary-Integral Equation (BIE) for the associated velocity potential exploiting the null-pressure jump Kutta condition at the trailing edge. The numerical solution of the BIE is performed by an Isogeometric Boundary-Element Method (BEM) combining a generic B-splines parametric modeler for generating hydrofoil shapes, using a set of eight parameters, the very same basis of the geometric representation for representing the velocity potential and collocation at the Greville abscissas of the knot vector of the hydrofoil's B-splines representation. Furthermore, the optimization environment is developed based on the geometric parametric modeler for the hydrofoil, the Isogeometric BEM solver and an optimizer employing a controlled elitist genetic algorithm. Multi-objective hydrofoil shape optimization examples are demonstrated with respect to the criteria (i) maximum lift coefficient and (ii) minimum deviation of the hydrofoil area from a reference area

Extended isogeometric boundary element method (XIBEM) for acoustic wave scattering problems

Isogeometric analysis is the concept of using the same functions that describe a geometry in computer-aided design to approximate unknown elds in numerical simulations. This has become a topic of considerable interest to the boundary integral methods community. This paper introduces an eXtended Isogeometric Boundary Element Method (XIBEM), in which isogeometric functions approximating wave potential are enriched using the partition-of-unity method. In this new method, the isogeometric basis is formed from a space of non-uniform rational B-spline (NURBS) functions multiplied by families of plane waves. Using numerical examples, it is shown that this reduces the total number of equations that need to be solved for a given frequency and geometry of problem; this improves the accuracy of and extends the supported frequency range of the boundary element method to include short wave diraction problems