Time-Dependent Numerical Method with Boundary-Conforming Curvilinear Coordinates Applied to Wave Interactions with Prototypical Antennas

Villamizar, Vianey; Rojas, Otilio

doi:10.1006/jcph.2001.6987

Cited by 8 publications

(10 citation statements)

References 42 publications

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“…They also generalize to the situation of nonzero forcing. Clearly these time-stepping schemes can also be combined with finite difference methods on highly stretched grids, such as body-fitted grids [42], if the underlying finite difference discretization leads to a symmetric stiffness matrix. In the presence of hierarchical mesh refinement, each local time step in the fine region can itself include further local time steps inside a smaller subregion with an even higher degree of local mesh refinement.…”

Section: Two-dimensional Examplementioning

confidence: 99%

Energy Conserving Explicit Local Time Stepping for Second-Order Wave Equations

Diaz¹,

Grote²

2009

SIAM J. Sci. Comput.

106

175

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Locally refined meshes impose severe stability constraints on explicit time-stepping methods for the numerical simulation of time dependent wave phenomena. To overcome that stability restriction, local time-stepping methods are developed, which allow arbitrarily small time steps precisely where small elements in the mesh are located. When combined with a symmetric finite element discretization in space with an essentially diagonal mass matrix, the resulting discrete numerical scheme is explicit, is inherently parallel, and exactly conserves a discrete energy. Starting from the standard second-order "leap-frog" scheme, time-stepping methods of arbitrary order of accuracy are derived. Numerical experiments illustrate the efficiency and usefulness of these methods and validate the theory. Introduction.The efficient and accurate numerical solution of the wave equation is of fundamental importance for the simulation of time dependent acoustic, electromagnetic, or elastic wave phenomena. Finite difference methods are commonly used for the simulation of time dependent waves because of their simplicity and their efficiency on structured Cartesian meshes [27,28,34]. However, in the presence of complex geometry or small geometric features that require locally refined meshes, their usefulness is somewhat limited. In contrast, finite element methods (FEMs) easily handle locally refined unstructured meshes; moreover, their extension to high order is straightforward, even in the presence of curved boundaries or material interfaces.The finite element Galerkin discretization of second-order hyperbolic problems typically leads to a second-order system of ordinary differential equations. Even if explicit time stepping is employed, the mass matrix arising from the spatial discretization by standard continuous finite elements must be inverted at each time step-a major drawback in terms of efficiency. To overcome that problem, various "mass lumping" techniques have been proposed, which effectively replace the mass matrix by a diagonal approximation. While straightforward for piecewise linear elements [8,32], mass lumping techniques require particular quadrature or cubature rules at higher order to preserve the accuracy and guarantee numerical stability [14,22].Alternatively, discontinuous Galerkin (DG) methods offer even greater flexibility for local mesh refinement by accommodating nonconforming grids and hanging nodes.

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Section: Two-dimensional Examplementioning

confidence: 99%

Energy Conserving Explicit Local Time Stepping for Second-Order Wave Equations

Diaz¹,

Grote²

2009

SIAM J. Sci. Comput.

106

175

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“…On the other hand, most of the techniques and applications of 3D grid generation have been, and continue to be, devised initially for two-dimensional configurations. This statement is supported by a considerable number of recent publications in 2D grid generation techniques [1][2][3]8,23], and also by recent publications in applied areas that use 2D grids [3,7,13,14,20,22].…”

Section: Discussionmentioning

confidence: 80%

“…In contrast, for complex geometries such as the astroid and the epicycloid, stretching the distance between neighboring grid lines may be required to avoid self-overlapping near sharp corners or cusps. Also, adjusting the distance between neighbor- ing grid lines is especially important when implementing an explicit finite-difference marching in time scheme for wave propagation problems [4,21,20]. As discussed in Section 7, the cell sizes of the grids used in the computation should satisfy the CFL stability condition.…”

Section: Impact Of Bcgc Grids In Numerical Simulationmentioning

confidence: 99%

Generation of curvilinear coordinates on multiply connected regions with boundary singularities

Villamizar

Rojas

Mabey

2007

Journal of Computational Physics

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“…Wave scattering emerges in many contexts. For example, it occurs when an electromagnetic wave encounters a receiving antenna [1], when a sound pressure wave impinges on an obstacle in an unbounded medium [2,3], or when an elastic wave propagating underground hits a crack saturated by oil in sedimentary rocks [4,5]. These important physical problems are usually modelled by involved vector equations coupled through nontrivial boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

“…These important physical problems are usually modelled by involved vector equations coupled through nontrivial boundary conditions. For instance, Maxwell's equations are the primary model for the antenna problem [1]. However, in some important cases, the physical conditions allow further simplification.…”

Section: Introductionmentioning

confidence: 99%

Boundary‐conforming coordinates with grid line control for acoustic scattering from complexly shaped obstacles

Villamizar

Weber

2007

Numerical Methods Partial

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The current work sets forth a practical approach to numerically solve two-dimensional direct acoustic scattering problems from complexly shaped scatterers with severe singularities, such as corners and cusps. First, boundary conforming coordinates are generated. This generation is performed through an elliptic grid generator algorithm, including control of the coordinate lines. The grid line control solely depends on the initial distribution of grid points. Following the grid generation process, the initial boundary value problem, modelling the scattering phenomenon, is formulated in terms of the new curvilinear coordinates, and a finite-difference time domain method is implemented. The presence of the boundary singularities causes instability of the numerical method. However, by appropriately controlling the distance between grid lines in the vicinity of these singularities, stability and convergence are achieved. A semianalytical formula for the differential scattering cross-section is obtained from the discrete Fourier transform of the computed scattered pressure field. The method is successfully applied to several interesting scatterers of various shapes.

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Time-Dependent Numerical Method with Boundary-Conforming Curvilinear Coordinates Applied to Wave Interactions with Prototypical Antennas

Cited by 8 publications

References 42 publications

Energy Conserving Explicit Local Time Stepping for Second-Order Wave Equations

Energy Conserving Explicit Local Time Stepping for Second-Order Wave Equations

Generation of curvilinear coordinates on multiply connected regions with boundary singularities

Boundary‐conforming coordinates with grid line control for acoustic scattering from complexly shaped obstacles

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