DtN Operator Research Articles

Monotonicity Principle for Tomography in Nonlinear Conducting Materials

In this paper, we deal with an inverse electrical conductivity problem which considers the reconstruction of nonlinear electrical conductivity in steady currents operations using boundary measurements. In the current set up, we establish a monotonic relation between the unknown material property to the (measured) Dirichlet-to-Neumann operator (DtN). It is in fact the Monotonicity Principle which is the base of a class of non-iterative and real-time imaging methods and algorithms. To be more precise, we indicate the issues appear in our nonlinear case to transfer this Monotonicity result from the Dirichlet Energy to the DtN operator which is the fundamental huddle in comparison to linear and p-Laplacian cases. Finally, we introduce a new Average DtN operator which is different from the existing ones and resolves complications produced by non-linearity in our problem.

Monotonicity Principle in tomography of nonlinear conducting materials *

We treat an inverse electrical conductivity problem which deals with the reconstruction of nonlinear electrical conductivity starting from boundary measurements in steady currents operations. In this framework, a key role is played by the Monotonicity Principle, which establishes a monotonic relation connecting the unknown material property to the (measured) Dirichlet-to-Neumann operator (DtN). Monotonicity Principles are the foundation for a class of non-iterative and real-time imaging methods and algorithms. In this article, we prove that the monotonicity principle for the Dirichlet Energy in nonlinear problems holds under mild assumptions. Then, we show that apart from linear and p-Laplacian cases, it is impossible to transfer this monotonicity result from the Dirichlet Energy to the DtN operator. To overcome this issue, we introduce a new boundary operator, identified as an average DtN operator.

On the existence and the uniqueness of the solution of a fluid–structure interaction scattering problem

The existence and uniqueness of the solution of a fluid–structure interaction problem is investigated. The proposed analysis distinguishes itself from previous studies by employing a weighted Sobolev space framework, the DtN operator properties, and the Fredholm theory. The proposed approach allows to extend the range of validity of the standard existence and uniqueness results to the case where the elastic scatterer is assumed to be only Lipschitz continuous, which is of more practical interest.

Wave propagation in locally perturbed periodic media (case with absorption): Numerical aspects

We are interested in the numerical simulation of wave propagation in media which are a local perturbation of an infinite periodic one. The question of finding artificial boundary conditions to reduce the actual numerical computations to a neighborhood of the perturbation via a DtN operator was already developed in [1] at the continuous level. We deal in this article with the numerical aspects associated to the discretization of the problem. In particular, we describe the construction of discrete DtN operators that relies on the numerical solution of local cell problems, non stationary Ricatti equations and the discretization of non standard integral equations in Floquet variables.

Parallel iterative solution for the Helmholtz equation with exact non-reflecting boundary conditions

Efficient and scalable parallel solution methods are presented for the Helmholtz equation with global non-reflecting DtN boundary conditions. The symmetric outer-product structure of the DtN operator is exploited to significantly reduce inter-processor communication required by the non-locality of the DtN to one collective communication per iteration with a vector size equal to the number of harmonics included in the DtN series expansion, independent of the grain size. Numerical studies show that in the context of iterative equation solvers, and for the same accuracy, the global DtN applied to a tightly fitting spheroidal boundary and implemented as a low-rank update with the multiplicative split is more cost-effective (both in wall-clock times and memory) compared to local approximate boundary conditions.

A mixed-type finite element approximation for radiation problems using fictitious domain method

In the finite element approximation of the exterior Helmholtz problem, we propose an approximation method to implement the DtN mapping formulated as a pseudo-differential operator on a computational artificial boundary. The method is then combined with the fictitious domain method. Our method directly gives an approximation matrix for the sesqui-linear form for the DtN mapping. The eigenvalues of the approximation matrix are simplified to a closed form and can be computed efficiently by using a continued fraction formula. Solution outside the computational domain and the far-field solution can also be computed efficiently by expressing them as operations of pseudo-differential operators. An inner artificial DtN boundary condition is also implemented by our method. We prove the convergence of the solution of our method and compare the performance with the standard finite element approximation based on the Fourier series expansion of the DtN operator. The efficiency of our method is demonstrated through numerical examples.

An Iterative FEM for Solving Wave Problems of a Halfspace

An iterative finite element method for solving wave problems of a halfspace is presented in this paper. The halfspace is first truncated by introducing a fictive finite boundary on which some fictive boundary conditions must be imposed. A finite computational domain is in each iteration subjected to actual boundary conditions on real boundary and to fictive Dirichlet or Neumann boundary conditions on the fictive boundary. The radiation condition is satisfied by using DtN operator. The DtN operator is not introduce in the finite element formulation on the fictive boundary so any finite elements can be used. The method is simple and specially useful for computing higher harmonics.

Analysis of a coupled finite-infinite element method for exterior Helmholtz problems

This analysis of convergence of a coupled FEM-IEM is based on our previous work on the FEM and the IEM for exterior Helmholtz problems. The key idea is to represent both the exact and the numerical solution by the Dirichlet-to-Neumann operators that they induce on the coupling hypersurface in the exterior of an obstacle. The investigation of convergence can then be related to a spectral analysis of these DtN operators. We give a general outline of our method and then proceed to a detailed investigation of the case that the coupling surface is a sphere. Our main goal is to explore the convergence mechanism. In this context, we show well-posedness of both the continuous and the discrete models. We further show that the discrete inf-sup constants have a positive lower bound that does not depend on the number of DOF of the IEM. The proofs are based on lemmas on the spectra of the continuous and the discrete DtN operators, where the spectral characterization of the discrete DtN operator is given as a conjecture from numerical experiments. In our convergence analysis, we show algebraic (in terms of N) convergence of arbitrary order and generalize this result to exponential convergence.

On non-reflecting boundary conditions in unbounded elastic solids

Problems in unbounded domains can be solved using domain-based computation by introducing an artificial boundary, and then selecting appropriate boundary conditions. The DtN method, which specifies such boundary conditions, is investigated in this work for wave problems in elastic solids. The DtN method defines an exact relation between the displacement field and its normal and tangential tractions on an artificial boundary. This relation is expressed in terms of an infinite series. The DtN boundary conditions are shown to be non-reflective, thus uniqueness of the solution is guaranteed. For practical purposes the full DtN operator is truncated. The truncated DtN operator fails to completely inhibit reflections of higher modes, resulting in loss of uniqueness at characteristic wave numbers of higher harmonics. Guidelines for determining a sufficient number of terms in the truncated operator to retain uniqueness of the solution at any given wave number are derived. The validity of these guidelines is examined and verified by numerical examples. Local DtN boundary conditions are also investigated, and it is shown that local boundary conditions guarantee uniqueness of the solution for all wave numbers, regardless of the number of terms in the operator. This property is used here to modify the truncated DtN operator and to enhance its capability to retain uniqueness of solutions. A modified DtN operator, combining the truncated operator with the local one, is introduced. The modified DtN operator is shown to retain uniqueness of solutions regardless of the number of terms and regardless of the wave number.

Studies of domain‐based formulations for computing exterior problems of acoustics

AbstractThe propagation and decay of acoustic waves in exterior domains in an essential ingredient in the study of fluid–structure interaction. A strategy must be devised to compute solutions over domains which are unbounded. Exact impedance conditions at an artificial external boundary are specified by the DtN method, yielding an equivalent problem that is suitable for domain‐based computation. The DtN boundary condition is non‐reflective, giving rise to exact (and thereby unique) solutions. The truncated DtN operator, which is employed in practice, fails to inhibit the reflection of higher modes, so that non‐unique solutions may occur at their harmonics. Simple expressions determine a sufficient number of terms in the truncated operator for unique solutions at any given wave number. There are three characteristic length scales in the computational problem: the radius of the artificial boundary, the geometry of the body (represented by the internal boundary) and the mesh size. Numerical studies examine the dependence of the conditioning of finite element coefficient matrices on the number of terms in the truncated DtN operator vs. the wave number non‐dimensionalized by each of the length scales. Analytic results regarding the number of terms sufficient for unique solutions are confirmed. As long as this criterion is respected, no upper limit on the allowable wave number is detected. A local approximation of the boundary conditions restores uniqueness for all wave numbers.

Analysis of continuous formulations underlying the computation of time-harmonic acoustics in exterior domains

Potential non-uniqueness of boundary representations of the Helmholtz equation underscores the importance of investigating continuous boundary-based and domain-based formulations, which is the main purpose of this work. Uniqueness properties of the solutions of boundary integral equations are reviewed. We analyze formulations for domain-based computation that are derived by the DtN method, which imposes a relation between the function and its normal derivative on an artificial boundary. The DtN formulation is shown to possess non-reflective boundary conditions and to give rise to exact (and thereby unique) solutions. In practical implementation the DtN map is often truncated. The truncated DtN operator fails to completely inhibit reflection of higher modes, resulting in loss of uniqueness at characteristic wave numbers of higher harmonics. However, simple expressions that determine a sufficient number of terms in the operator for unique solutions at any given wave number are derived. We prove that a local approximation of the boundary conditions restores uniqueness for all wave numbers, and derive its three-dimensional version.