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Journal ArticleDOI

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

TL;DR: In this article, 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.
Abstract: In 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.
Citations
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Journal ArticleDOI
TL;DR: In this article, a novel enriched finite element method (EFEM) for wave analysis is presented, where the original linear nodal shape functions are enriched by using the additional interpolation cover functions over patches of elements.

28 citations

Journal ArticleDOI
01 Oct 2022
TL;DR: In this article , two numerical investigations are performed using XIBEM for two-dimensional problems, and the number of plane waves is varied to find out the suitable enrichment scheme to achieve accurate results for higher frequency problems than those in the literature.
Abstract: Isogeometric analysis (IGA) in the framework of the boundary element method (BEM) – known as isogeometric boundary element analysis or IGABEM – has shown recently respectable performance in the field of acoustics and handling the time harmonic wave propagation equation of Helmholtz. However, IGABEM still requires fine meshes to handle the cases of very high frequencies due to the large number of elements needed to capture the oscillatory behaviour, leading to large computational costs. IGABEM can be enriched by the partition of unity expansion of plane waves in the framework of the eXtended IGABEM (XIBEM) which can be used to simulate high frequency problems with small-scale wavelengths using coarser meshes than those used in other numerical approaches. In this paper, two numerical investigations are performed using XIBEM for two-dimensional problems. First, the number of plane waves is varied to find out the suitable enrichment scheme to achieve accurate results for higher frequency problems than those in the literature. After that, XIBEM is coupled with a non-iterative topological-shape sensitivity inverse analysis and applied for the problem of scatterer shape reconstruction. Different distributions of the receptor points are checked for this investigation with varied initial scatterer shapes.

3 citations

Journal ArticleDOI
TL;DR: In this paper , a reduction of the On-Surface Radiation Condition (OSRC) formulation based on a plane wave ansatz is introduced, which enhances the efficiency of the OSRC methods.

1 citations

Journal ArticleDOI
TL;DR: In this article , an indirect boundary element method (BEM) based on isogeometric analysis (IGA) is proposed for 2D Helmholtz acoustic problems using virtual continuous sources placed directly on the problem boundary.
Abstract: An indirect boundary element method (BEM) based on isogeometric analysis (IGA) is proposed for 2D Helmholtz acoustic problems using virtual continuous sources placed directly on the problem boundary. The virtual sources form a virtual boundary identical to the main problem boundary. The proposed solution couples the CAD model with the analysis model by approximating both the CAD model and the virtual continuous sources with the same non-uniform rational B-spline functions (NURBS). Moreover, neither domain discretization nor truncation boundaries at the far-field are required. This solution creates only one coefficient matrix by directly arranging the linear system of equations. The solution follows a collocation scheme on the boundaries based on Greville abscissae with offsets wherever C0 continuity is encountered to permit an easy prediction for the normal directions and the free-terms at the collocation points required in the cases of Neumann and Robin boundary conditions. It allows us also to treat all integrals with standard Gauss quadrature points. Several numerical examples for exterior and interior acoustic problems are discussed to verify the proposed solution with comparisons to analytical solutions and previously published numerical results. These examples prove the robustness of the proposed solution even for high wavenumbers, in contrast to the previous attempts which implemented extensive investigations to find out the optimum place of the sources outside the domain producing minimum errors. Furthermore, no fictitious eigenfrequency problem is observed for exterior acoustic problems.

1 citations

References
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Book
01 Apr 2003
TL;DR: This chapter discusses methods related to the normal equations of linear algebra, and some of the techniques used in this chapter were derived from previous chapters of this book.
Abstract: Preface 1. Background in linear algebra 2. Discretization of partial differential equations 3. Sparse matrices 4. Basic iterative methods 5. Projection methods 6. Krylov subspace methods Part I 7. Krylov subspace methods Part II 8. Methods related to the normal equations 9. Preconditioned iterations 10. Preconditioning techniques 11. Parallel implementations 12. Parallel preconditioners 13. Multigrid methods 14. Domain decomposition methods Bibliography Index.

13,484 citations

Journal ArticleDOI
TL;DR: Numerical experiments and numerical comparisons show that the PML technique works better than the others in all cases; using it allows to obtain a higher accuracy in some problems and a release of computational requirements in some others.

9,875 citations

Journal ArticleDOI
TL;DR: In this article, the concept of isogeometric analysis is proposed and the basis functions generated from NURBS (Non-Uniform Rational B-Splines) are employed to construct an exact geometric model.

5,137 citations

Book
01 Jan 1989
TL;DR: Inverse Boundary Value Problems (IBV) as discussed by the authors, the heat equation is replaced by the Tikhonov regularization and regularization by Discretization (TBD) method.
Abstract: Normed Spaces.- Bounded and Compact Operators.- Riesz Theory.- Dual Systems and Fredholm Alternative.- Regularization in Dual Systems.- Potential Theory.- Singular Integral Equations.- Sobolev Spaces.- The Heat Equation.- Operator Approximations .-Degenerate Kernel Approximation.- Quadrature Methods.- Projection Methods.- Iterative Solution and Stability.- Equations of the First Kind.- Tikhonov Regularization.- Regularization by Discretization.- Inverse Boundary Value Problems.- References.- Index.

2,323 citations

Journal ArticleDOI
TL;DR: In this article, the authors present state-of-the-art numerical techniques to solve the wave equation in heterogeneous fluid-solid media and present a comprehensive and modern introduction to computational ocean acoustics accessible to students.
Abstract: Senior level/graduate level text/reference presenting state-of-the- art numerical techniques to solve the wave equation in heterogeneous fluid-solid media. Numerical models have become standard research tools in acoustic laboratories, and thus computational acoustics is becoming an increasingly important branch of ocean acoustic science. The first edition of this successful book, written by the recognized leaders of the field, was the first to present a comprehensive and modern introduction to computational ocean acoustics accessible to students. This revision, with 100 additional pages, completely updates the material in the first edition and includes new models based on current research. It includes problems and solutions in every chapter, making the book more useful in teaching (the first edition had a separate solutions manual). The book is intended for graduate and advanced undergraduate students of acoustics, geology and geophysics, applied mathematics, ocean engineering or as a reference in computational methods courses, as well as professionals in these fields, particularly those working in government (especially Navy) and industry labs engaged in the development or use of propagating models.

1,344 citations