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19
On Nonreflecting Boundary Conditions
 J. COMPUT. PHYS
, 1995
"... Improvements are made in nonreflecting boundary conditions at artificial boundaries for use with the Helmholtz equation. First, it is shown how to remove the difficulties that arise when the exact DtN (DirichlettoNeumann) condition is truncated for use in computation, by modifying the truncated ..."
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Cited by 219 (4 self)
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Improvements are made in nonreflecting boundary conditions at artificial boundaries for use with the Helmholtz equation. First, it is shown how to remove the difficulties that arise when the exact DtN (DirichlettoNeumann) condition is truncated for use in computation, by modifying the truncated condition. Second, the exact DtN boundary condition is derived for elliptic and spheroidal coordinates. Third, approximate local boundary conditions are derived for these coordinates. Fourth, the truncated DtN condition in elliptic and spheroidal coordinates is modified to remove difficulties. Fifth, a sequence of new and more accurate local boundary conditions is derived for polar coordinates in two dimensions. Numerical results are presented to demonstrate the usefulness of these improvements.
The Partition of Unity Method
 International Journal of Numerical Methods in Engineering
, 1996
"... A new finite element method is presented that features the ability to include in the finite element space knowledge about the partial differential equation being solved. This new method can therefore be more efficient than the usual finite element methods. An additional feature of the partitionofu ..."
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Cited by 211 (2 self)
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A new finite element method is presented that features the ability to include in the finite element space knowledge about the partial differential equation being solved. This new method can therefore be more efficient than the usual finite element methods. An additional feature of the partitionofunity method is that finite element spaces of any desired regularity can be constructed very easily. This paper includes a convergence proof of this method and illustrates its efficiency by an application to the Helmholtz equation for high wave numbers. The basic estimates for aposteriori error estimation for this new method are also proved. Key words: Finite element method, meshless finite element method, finite element methods for highly oscillatory solutions TICAM, The University of Texas at Austin, Austin, TX 78712. Research was partially supported by US Office of Naval Research under grant N0001490J1030 y Seminar for Applied Mathematics, ETH Zurich, CH8092 Zurich, Switzerland....
Chebyshev rational spectral and pseudospectral methods on a semiinfinite interval
 INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING
, 2002
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An application of shape optimization in the solution of inverse acoustic scattering problems
 Inverse Problems
"... We consider the problem of determining the shape of an object immersed in an acoustic medium from measurements obtained at a distance from the object. We recast this problem as a shape optimization problem where we search for the domain that minimizes a cost function that quantifies the difference b ..."
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We consider the problem of determining the shape of an object immersed in an acoustic medium from measurements obtained at a distance from the object. We recast this problem as a shape optimization problem where we search for the domain that minimizes a cost function that quantifies the difference between the measured and expected signals. The measured and expected signals are assumed to satisfy a boundaryvalue problem given by the Helmholtz equation with the Sommerfeld condition imposed at infinity. Gradientbased algorithms are used to solve this optimization problem. At every step of the algorithm the derivative of the cost function with respect to the parameters that describe the shape of the object is calculated. We develop an efficient method based on the adjoint equations to calculate the derivative and show how this method is implemented in a finite element setting. The predominant cost of each step of the algorithm is equal to one forward solution and one adjoint solution and therefore is independent of the number of parameters used to describe the shape of the object. Numerical examples showing the efficacy of the proposed methodology are presented. (Some figures in this article are in colour only in the electronic version) M This article features online multimedia enhancements 4 Author to whom any correspondence should be addressed.
Advanced Computational Techniques for the Analysis of 3D FluidStructure Interaction
, 2002
"... The final copy of this thesis has been examined by the signatories, and we find that both the content and the form meet acceptable presentation standards of scholarly work in the above mentioned discipline. iii ..."
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Cited by 4 (1 self)
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The final copy of this thesis has been examined by the signatories, and we find that both the content and the form meet acceptable presentation standards of scholarly work in the above mentioned discipline. iii
A few new (?) facts about infinite elements
, 2006
"... We discuss the (conjugated) Bubnov–Galerkin and Petrov–Galerkin infinite element (IE) discretizations to Helmholtz equation including the use of elements of locally variable order, optimal choice of IE shape functions, calculation of Echo Area (EA), and automatic hpadaptivity. The discussion is ill ..."
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Cited by 2 (1 self)
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We discuss the (conjugated) Bubnov–Galerkin and Petrov–Galerkin infinite element (IE) discretizations to Helmholtz equation including the use of elements of locally variable order, optimal choice of IE shape functions, calculation of Echo Area (EA), and automatic hpadaptivity. The discussion is illustrated with 2D numerical experiments.
Multiscale Finite Elements for Acoustics: Continuous, Discontinuous, and Stabilized Methods
 INTERNATIONAL JOURNAL FOR MULTISCALE COMPUTATIONAL ENGINEERING, 6(5&6)
, 2008
"... This work describes two perspectives for understanding the numerical difficulties that arise in the solution of wave problems, and various advances in the development of efficient discretization schemes for acoustics. Standard, loworder, continuous Galerkin finite element methods are unable to cop ..."
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Cited by 1 (0 self)
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This work describes two perspectives for understanding the numerical difficulties that arise in the solution of wave problems, and various advances in the development of efficient discretization schemes for acoustics. Standard, loworder, continuous Galerkin finite element methods are unable to cope with wave phenomena at short wave lengths because the computational effort required to resolve the waves and control numerical dispersion errors becomes prohibitive. The failure to adequately represent subgrid scales misses not only the finescale part of the solution, but often causes severe pollution of the solution on the resolved scale as well. Since computation naturally separates the scales of a problem according to the mesh size, multiscale considerations provide a useful framework for viewing these difficulties and developing methods to counter them. The Galerkin/least squares method arises in multiscale settings, and its stability parameter is defined by dispersion considerations. Bubble enriched methods employ auxiliary functions that are usually expressed in the form of infinite series. Dispersion analysis provides guidelines for the implementation of the series representation in practice. In the discontinuous enrichment method, the fine scales are spanned by freespace homogeneous solutions of the governing equations. These auxiliary functions may be discontinuous across element boundaries, and continuity is enforced weakly by Lagrange multipliers.
A FullWave Helmholtz Model for ContinuousWave Ultrasound Transmission
"... A fullwave Helmholtz model of continuouswave (CW) ultrasound fields may offer several attractive features over widely used partialwave approximations. For example, many fullwave techniques can be easily adjusted for complex geometries and multiple reflections of sound are automatically taken int ..."
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A fullwave Helmholtz model of continuouswave (CW) ultrasound fields may offer several attractive features over widely used partialwave approximations. For example, many fullwave techniques can be easily adjusted for complex geometries and multiple reflections of sound are automatically taken into account in the model. To date, however, the fullwave modeling of CW fields in general 3D geometries has been avoided due to large computational cost associated with the numerical approximation of the Helmholtz equation. Recent developments in computing capacity together with improvements in finite element type modeling techniques are making possible wave simulations in 3D geometries which reach over tens of wavelengths. The aim of this study is to investigate the feasibility of a fullwave solution of the 3D Helmholtz equation for modeling of continuouswave ultrasound fields in an inhomogeneous medium. The numerical approximation of the Helmholtz equation is computed using the ultra weak variational formulation (UWVF) method. In addition, an inverse problem technique is utilized to reconstruct the velocity distribution on the transducer which is used to model the sound source in the UWVF scheme. The modeling method is verified by comparing simulated and measured fields in the case of transmission of 531 kHz CW fields through layered plastic plates. The comparison shows a reasonable agreement between simulations and measurements at low angles of incidence but due to mode conversion, the Helmholtz model becomes insufficient for simulating ultrasound fields in plates at large angles of incidence. I.
On Performance of 3D Infinite Elements for HighFrequency
"... The infinite elements for edge based finite element methods (FEM) have been shown effective for open boundary problems. In the infinite elements, electromagnetic fields are expressed in terms of radially decaying basis functions. On the other hand, the perfect matched layer has widely been used for ..."
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The infinite elements for edge based finite element methods (FEM) have been shown effective for open boundary problems. In the infinite elements, electromagnetic fields are expressed in terms of radially decaying basis functions. On the other hand, the perfect matched layer has widely been used for FEM for highfrequency problems. In this paper, numerical performance of both methods is comparably discussed. The numerical experiments show that the former has higher computational efficiency. Index Terms—Finite element method, infinite element, perfect matched layer, highfrequency problem.
L.X. Li A Generalized Infinite Element for Acoustic Radiation
"... A generalized infinite element is presented by combining following aspects: (1) The geometry mapping in the Cartesian ..."
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A generalized infinite element is presented by combining following aspects: (1) The geometry mapping in the Cartesian