Dirichlet-to-Neumann Research Articles

Floquet Multipliers of Periodic Waveguides via Dirichlet-to-Neumann Maps

A new approach to calculating Floquet spectra of multilayered periodic waveguides is presented. The problem is formulated as an eigenvalue problem of the Helmholtz equation on an infinite strip with discontinuous wavenumber. The strip is decomposed into a rectangle and two semi-infinite domains, and the problem is reduced to a nonlinear eigenvalue problem involving Dirichlet-to-Neumann (DtN) operators on the interfaces of the domains. A solution scheme based on the Taylor expansion of the DtN operator with respect to the Floquet exponent, whose order of convergence can be made arbitrarily large, is derived. An application to a typical waveguide geometry demonstrates the efficiency and accuracy of the approach.

Variational formulations of transmission problems via FEM, BEM and DtN mappings

The purpose of this work is to review some of the main results on the weak solvability and Galerkin approximations of interior and exterior transmission problems arising in potential theory and elastostatics. Both linear and nonlinear model problems are included in our presentation. We consider variational formulations based on the combination of mixed finite elements with either boundary integral equations or Dirichlet-to-Neumann (DtN) mappings. We provide existence and uniqueness of solution for the continuous formulations and give general approximation estimates for the associated Galerkin schemes. Some open questions are also mentioned.

Inverse boundary value problem for ocean acoustics

For an ocean with constant depth and rigid bottom which contains compactly supported inhomogeneity of the water sound velocity, we prove uniqueness for the identification of the inhomogeneity from the Dirichlet-to-Neumann (DtN) map on the surface of a bounded region containing the inhomogeneity. The DtN map is the map which maps the pressure applied on the boundary of this region to the corresponding flux (displacement). In an analogous geometric configuration and with similar boundary conditions, the uniqueness for the inverse electroconductivity problem from the DtN map (i.e. voltage-to-current map) can be proved in the same framework. Copyright © 2001 John Wiley & Sons, Ltd.

Recent advances in the DtN FE Method

The Dirichlet-to-Neumann (DtN) Finite Element Method is a general technique for the solution of problems in unbounded domains, which arise in many fields of application. Its name comes from the fact that it involves the nonlocal Dirichlet-to-Neumann (DtN) map on an artificial boundary which encloses the computational domain. Originally the method has been developed for the solution of linear elliptic problems, such as wave scattering problems governed by the Helmholtz equation or by the equations of time-harmonic elasticity. Recently, the method has been extended in a number of directions, and further analyzed and improved, by the author's group and by others. This article is a state-of-the-art review of the method. In particular, it concentrates on two major recent advances: (a) the extension of the DtN finite element method tononlinear elliptic and hyperbolic problems; (b) procedures forlocalizing the nonlocal DtN map, which lead to a family of finite element schemes with local artificial boundary conditions. Possible future research directions and additional extensions are also discussed.

Layer Stripping for a Transversely Isotropic Elastic Medium

In this paper we develop a layer stripping algorithm for transversely isotropic elastic materials in three-dimensional space. We first prove that by making displacement and traction measurements at the boundary of these elastic materials one can determine the elastic tensor and all its derivatives at the boundary. The elastic measurements made at the boundary are encoded in the Dirichlet to Neumann (DN) map. Then we use the Riccati equation developed in [G. Nakamura and G. Uhlmann, A layer stripping algorithm in elastic impedance tomography, in Inverse Problems in Wave Propagation, IMA Vol. Math. Appl. 90, Springer-Verlag, New York, 1997, pp. 375-384] for the DN map in any anisotropic elastic medium to approximate the elastic tensor in the interior.

A Domain Decomposition Method for the Exterior Helmholtz Problem

A new domain decomposition method is presented for the exterior Helmholtz problem. The nonlocal Dirichlet-to-Neumann (DtN) map is used as a nonreflecting condition on the outer computational boundary. The computational domain is divided into nonoverlapping subdomains with Sommerfeld-type conditions on the adjacent subdomain boundaries to ensure uniqueness. An iterative scheme is developed, where independent subdomain boundary-value problems are obtained by applying the DtN operator to values from the previous iteration. The independent problems are then discretized with finite elements and can be solved concurrently. Numerical results are presented for a two-dimensional model problem, and both the solution accuracy and convergence rate are investigated.

Discrete Dirichlet-to-Neumann maps for unbounded domains

It is shown how to construct a discrete counterpart of the Dirichlet-to-Neumann (DtN) map for use on an artificial boundary B introduced in exterior boundary value problems, following the idea of Deakin and Dryden. This discrete map provides an approximate non-reflecting or absorbing boundary condition for use in formulating a problem in the finite computational domain Ω bounded by B . Different discrete maps are constructed for use with finite difference and finite element methods in Ω. The solution of some simple problems shows that use of the discrete Dirichlet-to-Neumann (DDtN) map yields nearly the same accuracy as that of using the continuous DtN map.

High modal density approximations for equipment in the time domain

This article considers modeling the effect of complex subsystems on the dynamics of the main structure to which they are attached. It is proposed that the substructure (say a piece of equipment) be replaced by an equivalent set of forces which react back on the main structure. These forces are given as time convolutions of the displacements at the equipment attachment points. The convolution integral, which represents a time domain DtN (Dirichlet-to-Neumann) map, is approximated in the high modal density limit with determined error bounds. This approximation leads to a family of equipment representations. The simplest requires few measured equipment properties, though more information can lead to greater accuracy. Our approximate DtNs are demonstrated numerically in finite element simulations.

On the implementation of the Dirichlet-to-Neumann radiation condition for iterative solution of the Helmholtz equation

The Helmholtz equation posed on an unbounded domain with the Sommerfeld condition prescribed at infinity is considered. The unbounded domain is eliminated by imposing a Dirichlet-to-Neumann (DtN) map or a modified DtN map on a truncating surface and the resulting bounded domain problem is modeled using the finite element method. The resulting system of linear equations is then solved using a Krylov subspace iterative method. New, efficient algorithms to compute matrix-vector products that are based on the structure of the DtN and the modified DtN map are presented. Connections between the DtN map and the discrete Fourier transform in two dimensions and discrete spherical transform in three dimensions are established, and are utilized to develop fast implementations of matrix-vector product algorithms. Also, an SSOR-type preconditioner that is based on a local radiation condition is considered for the modified DtN formulation. An efficient implementation is proposed by extending Eisenstat's trick for the standard SSOR preconditioner. Finally, numerical examples which illustrate the efficacy of the proposed algorithms are presented.

Optimal local non-reflecting boundary conditions

A class of numerical methods to solve problems in unbounded domains is based on truncating the infinite domain via an artificial boundary β and applying some boundary condition on β, which is called a Non-Reflecting Boundary Condition (NRBC). In this paper a systematic way to derive optimal local NRBCs of given order is developed in various configurations. The optimality is in the sense that the local NRBC best approximates the exact nonlocal Dirichlet-to-Neumann (DtN) boundary condition for C ∞ functions in the L 2 norm. The optimal NRBC may be of low order but still represent high-order modes in the solution. It is shown that the previously derived localized DtN conditions are special cases of the new optimal conditions. The performance of the first-order optimal NRBC is demonstrated via numerical examples, in conjunction with the finite element method.

Dirichlet-to-Neumann Maps for Unbounded Wave Guides

Dirichlet-to-Neumann (DtN) boundary conditions for unbounded wave guides in two and three dimensions are derived and analyzed, defining problems that are suitable for finite element analysis. In the most general cases considered wave numbers may vary in arbitrary cross sections. The full DtN operator, in the form of an infinite series, is exact. Nonunique solutions may occur when this operator is truncated. Simple criteria for the number of terms in the truncated operator that guarantee unique solutions are presented. A simple modification of the truncated operator leads to uniqueness for any number of terms. Numerical results validate the performance of DtN formulations for wave guides and confirm the criteria for uniqueness.

H-Adaptive finite element computation of time-harmonic exterior acoustics problems in two dimensions

In this paper we carry out adaptive finite element computations of the Helmholtz equation in two dimensions, in the context of time-harmonic exterior acoustics. The purpose is to demonstrate potentially significant cost savings engendered through adaptivity for propagating solutions at moderate wave numbers ( ka = 2 π, 4 π). The computations are performed on meshes of linear triangles, and are adapted to the solution by locally changing element sizes ( h-refinement). The adaptive procedure involves applying an explicit, residual-based a posteriori error estimator and an h-adaptive strategy, which were derived in a previous paper. The adaptive meshes are then obtained through global mesh regeneration using an advancing front mesh generator. Two different finite element formulations are considered: The Galerkin and Galerkin Least-Squares (GLS) methods. The infinite physical domain is truncated by an artificial exterior boundary, on which a fully coupled Dirichlet-to-Neumann (DtN) boundary condition is applied. Two different problems are computed: Plane wave scattering by a rigid infinite surface (circular and square cross sections), and nonuniform radiation from an infinite circular cylinder. Detailed cost studies with respect to an active column direct solver are performed. For the nonuniform radiation problem, the adaptive mesh is twenty times more cost effective than a uniform mesh (for Galerkin). When coupled with the GLS formulation, the adaptive mesh is forty times more efficient than Galerkin computations on a uniform mesh.

High-order boundary conditions and finite elements for infinite domains

A finite element method for the solution of linear elliptic problems in infinite domains is proposed. The two-dimensional Laplace, Helmholtz and modified Helmholtz equations outside an obstacle and in a semi-infinite strip, are considered in detail. In the proposed method, an artificial boundary B is first introduced, to make the computational domain Ω finite. Then the exact nonlocal Dirichlet-to-Neumann (DtN) boundary condition is derived on B . This condition is localized, and a sequence of local boundary conditions on B , of increasing order, is obtained. The problem in Ω, with a localized DtN boundary condition on B , is then solved using the finite element method. The numerical stability of the scheme is discussed. A hierarchy of special conforming finite elements is developed and used in the layer adjacent to B , in conjunction with the local high-order boundary condition applied on B . An error analysis is given for both nonlocal and local boundary conditions. Numerical experiments are presented to demonstrate the performance of the method.

A MATRIX-FREE INTERPRETATION OF THE NON-LOCAL DIRICHLET-TO-NEUMANN RADIATION BOUNDARY CONDITION

This communication describes an efficient implementation of the non-local Dirichlet-to-Neumann (DtN) radiation boundary condition which arises in the solution of exterior problems in acoustics. Exterior problems in acoustics involve unbounded fluid domains whose finite element solution requires the introduction of a truncation boundary in order to obtain a finite computational domain. The non-local (DtN) condition is an exact non-reflecting boundary condition which is imposed on this truncation boundary. Unfortunately, the discretization of the non-local (DtN) boundary condition results in a dense, fully populated matrix whose storage and factorization become increasingly expensive. We describe here a matrix-free interpretation of the non-local (DtN) map suitable for iterative solution methods, which allows the use of this exact boundary condition without any storage penalties related to its non-local nature.

A space-time finite element method for structural acoustics in infinite domains part 2: Exact time-dependent non-reflecting boundary conditions

In Part 1, a new space-time finite element method for transient structural acoustics in exterior domains was given. The formulation employs a finite computational fluid domain surrounding the structure and incorporates local time-dependent non-reflecting boundary conditions on the fluid truncation boundary. In this paper, new exact time-dependent non-reflecting boundary conditions are developed for solutions of the scalar wave equation in three space dimensions. These high-order accurate absorbing boundary conditions are based on the exact impedance relation for the acoustic fluid through the Dirichlet-to-Neumann (DtN) map in the frequency domain and are exact for solutions consisting of the first N spherical wave harmonics. Time-dependent boundary conditions are obtained through an inverse Fourier transform procedure. Two alternative sequences of boundary conditions are derived; the first involves both temporal and spatial derivatives (local in time and local in space version), and the second involves temporal derivatives and a spatial integral (local in time and non-local in space version). These non-reflecting boundary conditions are incorporated as ‘natural’ boundary conditions in the space-time variational equation, i.e. they are enforced weakly in both space and time. Several numerical examples involving transient radiation are presented to illustrate the high-order accuracy and efficiency achieved by the new space-time finite element formulation for transient structural acoustics with non-reflecting boundaries.

FE mode-matching schemes for the exterior Helmholtz problem and their relationship to the FE-DtN approach

Finite element (FE) mode-matching procedures for the solution of Helmholtz' equation on an unbounded domain are reviewed and a symmetric general formulation is presented. This is a formal restatement of procedures applied previously to computations involving scattering of shallow water waves, acoustic transmission in non-uniform ducts and acoustic radiation from prismatic sheet metal ducts. An essential feature of the method is the use of a Galerkin procedure, rather than collocation, to match a finite computational model to a truncated modal expansion with the desired radiation characteristics. The method produces a symmetric set of linear equations which can be solved to give the unknown nodal values of the dependent variable and the modal coefficients of an outer expansion. Either of these sets of variables can be eliminated prior to solution to yield a reduced set of equations in the remaining parameters. The reduced equations obtained by eliminating the modal coefficients are shown to be identical to those obtained by applying a truncated Dirichlet-to-Neumann (DtN) boundary condition. If applied in this form, mode-matching can therefore be regarded as an alternative to the DtN method for generating this common set of discrete equations while permitting simultaneous solution for the modal coefficients in the outer region.

Equipment representations as time‐domain fuzzy structures.

The presence of complex subsystems can dramatically affect the dynamics of the main structure to which they are attached. Exactly modeling these subsystems, however, is often impossible. Here, replacing the substructure (say a piece of equipment) by an equivalent set of forces which react back on the main structure is proposed. These forces are given as time convolutions of the displacements at the equipment attachment points. The convolution integral, which represents a time‐domain DtN (Dirichlet‐to‐Neumann) map, is approximated in the high modal density limit with determined error bounds. Local in time approximations to the convolution integral are obtained using Pade approximants. These yield a family of equipment representations. The simplest requires two measured equipment properties, though more information can lead to greater accuracy. Our approximate DtNs are validated numerically in finite‐element simulations. [Work supported by ONR.]

Diffraction from simple shapes by a hybrid asymptotic‐numerical method.

The application of a hybrid asymptotic/finite‐element method to the problem of scattering from prismatically shaped objects is considered. The hybrid method is based on patching a short wavelength asymptotic expansion of the scattered field to a finite‐element interpolation of the near field. In patching, the diffracted field shape functions with unknown amplitude are forced to agree smoothly with the solution in the near field along a curve at a prescribed distance from the diffraction points. An asymptotic DtN (Dirichlet‐to‐Neumann) map on this artificial boundary represents the effect of the outer domain on the solution within this new boundary. This allows us to replace the original boundary value problem with an asymptotically equivalent boundary value problem, the domain of which is small and efficiently discretized. The method is applied to diffraction by a blunted wedge, which in this context represents a degenerate prism. The hybrid scattering solution shall be compared to a complete analytic field representation found using matched asymptotic expansions. [Work supported by ONR.]

On Nonreflecting Boundary Conditions

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 (Dirichlet-to-Neumann) 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.

Finite element analysis of wave scattering from singularities

A combined analytic-finite element method is used for the efficient solution of the Helmholtz equation in the presence of geometrical singularities. In particular, time-harmonic waves in a membrane which contains one or more fixed-edge cracks (stringers) are investigated. The Dirichlet-to-Neumann (DtN) map is used in the procedure, to enable the replacement of the original singular problem by an equivalent regular problem, which is then solved by a finite element scheme. The method yields the solution in the entire membrane, as well as the dynamic “stress intensity factor.” Numerical results are presented for a circular membrane containing an edge stringer, two edge stringers and an internal stringer. The first few critical wave numbers of the membrane are also found.