Effective Transmission Conditions Research Articles

An energy preserving time scheme based on the mortar element method for effective transmission conditions between fluid and solid domains in transient wave propagation problems

AbstractWe are interested in providing an efficient numerical scheme to represent wave propagation in time domain for configurations where a fluid domain and a solid domain are separated by a thin coating layer. Various computational challenges and bottlenecks occur in this context: incorporating fluid–solid coupling, enabling non‐conform space discretization when wavelengths greatly differ from the fluid and the solid domain, and rendering robust time discretization w.r.t. the thin layer thickness. In order to address these issues, the proposed approach combines the mortar element method with so‐called effective transmission conditions, where the effect of the coating layer is incorporated through spring—mass coefficients at the interface between the fluid and solid domain. Using discrete energy arguments, we are able to prove that the associated fully discrete scheme is stable upon a stability condition independent of the thin coating layer. Moreover, we provide an efficient time marching algorithm leading to significantly improved performances, illustrated in 2D and 3D configurations linked to ultrasonic testing experiments modeling.

Scaling effects on the periodic homogenization of a reaction-diffusion-convection problem posed in homogeneous domains connected by a thin composite layer

We study the question of periodic homogenization of a variably scaled reaction-diffusion problem with non-linear drift posed for a domain crossed by a flat composite thin layer. The structure of the non-linearity in the drift was obtained in earlier works as hydrodynamic limit of a totally asymmetric simple exclusion process (TASEP) for a population of interacting particles crossing a domain with obstacle.Using energy-type estimates as well as concepts like thin-layer convergence and two-scale convergence, we derive the homogenized evolution equation and the corresponding effective model parameters for a regularized problem. Special attention is paid to the derivation of the effective transmission conditions across the separating limit interface in essentially two different situations: (i) finitely thin layer and (ii) infinitely thin layer.This study should be seen as a preliminary step needed for the investigation of averaging fast non-linear drifts across material interfaces—a topic with direct applications in the design of thin composite materials meant to be impenetrable to high-velocity impacts.

On meso-scale approximations for vibrations of membranes with lower-dimensional clusters of inertial inclusions

Formal asymptotic algorithms are considered for a class of meso-scale approximations for problems of vibration of elastic membranes that contain clusters of small inertial inclusions distributed along contours of predefined smooth shapes. Effective transmission conditions have been identified for inertial structured interfaces, and approximations to solutions of eigenvalue problems have been derived for domains containing lower-dimensional clusters of inclusions.

Analysis of Boundary Layer Effects Due to Usual Boundary Conditions or Geometrical Defects in Elastic Plates Under Bending: An Improvement of the Love-Kirchhoff Model

We propose a model of flexural elastic plates accounting for boundary layer effects due to the most usual boundary conditions or to geometrical defects, constructed via matched asymptotic expansions. In particular, considering a rectangular plate clamped at two opposite edges while the other two are free, we derive the effective boundary conditions or effective transmission conditions that the two first terms of the outer expansion must satisfy. The new boundary value problems thus obtained are studied and compared with the classical Love-Kirchhoff plate model. Two examples of application illustrate the results.

A stable, unified model for resonant Faraday cages.

We study some effective transmission conditions able to reproduce the effect of a periodic array of Dirichlet wires on wave propagation, in particular when the array delimits an acoustic Faraday cage able to resonate. In the study of Hewett & Hewitt (2016 Proc. R. Soc. A 472, 20160062 (doi:10.1098/rspa.2016.0062)) different transmission conditions emerge from the asymptotic analysis whose validity depends on the frequency, specifically the distance to a resonance frequency of the cage. In practice, dealing with such conditions is difficult, especially if the problem is set in the time domain. In the present study, we demonstrate the validity of a simpler unified model derived in Marigo & Maurel (2016 Proc. R. Soc. A 472, 20160068 (doi:10.1098/rspa.2016.0068)), where unified means valid whatever the distance to the resonance frequencies. The effectiveness of the model is discussed in the harmonic regime owing to explicit solutions. It is also exemplified in the time domain, where a formulation guaranteeing the stability of the numerical scheme has been implemented.

Effective transmission conditions for second-order elliptic equations on networks in the limit of thin domains

We consider star-shaped tubular domains consisting of a number of non intersecting semi-infinite strips of small thickness that are connected by a central region of diameter proportional to the thickness of the strips. At the thin-domain limit, the region reduces to a network of half-lines with the same end point (junction). We show that the solutions of uniformly elliptic partial differential equations set on the domain with Neumann boundary conditions converge, in the thin-domain limit, to the unique solution of a second-order partial differential equation on the network satisfying an effective Kirchhoff-type transmission condition at the junction. The latter is found by solving an “ergodic”-type problem at infinity obtained after a first-order blow up at the junction.

Mathematical modelling of plasmonic strain sensors

Abstract We provide a mathematical analysis for a metasurface constructed of plasmonic nanoparticles mounted periodically on the surface of a microcapsule. We derive an effective transmission condition, which exhibits resonances depending on the inter-particle distance. When the microcapsule is deformed, the resonances are shifted. We fully characterize the dependence of these resonances on the deformation of the microcapsule, enabling the detection of strains at the microscale level. We present numerical simulations to validate our results.

Effective transmission conditions for reaction–diffusion processes in domains separated by thin channels

ABSTRACT We consider a reaction–diffusion equation in a domain consisting of two bulk regions connected via small channels periodically distributed within a thin layer. The height and the thickness of the channels are of order ε, and the equation inside the layer depends on the parameter ε and an additional parameter , which describes the size of the diffusion in the layer. We derive effective models for the limit , when the channel-domain is replaced by an interface between the two bulk-domains.

Numerical modeling of wave propagation in anisotropic viscoelastic laminated materials in transient regime: Application to modeling ultrasonic testing of composite structures

SummaryComposite laminate structures remain an important family of materials used in cutting‐edge industrial areas. Building efficient numerical modeling tools for high‐frequency wave propagation in order to represent ultrasonic testing experiments of these materials remains a major challenge. In particular, incorporating attenuation phenomena within anisotropic plies, and thin intermediate isotropic layers between the plies often represent significant obstacles for standard numerical approaches. In our work, we address both issues by proposing a systematic study of the fully discrete propagators associated to the Kelvin‐Voigt, Maxwell, and Zener models, and by incorporating effective transmission conditions between plies using the mortar element method. We illustrate the soundness of our approach by proposing intermediate one‐dimensional and two‐dimensional numerical evidence, and we apply it to a more realistic configuration of a curved laminate composite structure in a three‐dimensional setting.

Homogenization of a transmission problem with Hamilton–Jacobi equations and a two-scale interface. Effective transmission conditions

We consider a family of optimal control problems in the plane with dynamics and running costs possibly discontinuous across a two-scale oscillatory interface. Typically, the amplitude of the oscillations is of the order of ε while the period is of the order of ε2. As ε→0, the interfaces tend to a straight line Γ. We study the asymptotic behavior of the value function as ε→0. We prove that the value function tends to the solution of Hamilton–Jacobi equations in the two half-planes limited by Γ, with an effective transmission condition on Γ keeping track of the oscillations.

Derivation of effective transmission conditions for domains separated by a membrane for different scaling of membrane diffusivity

We consider a system of non-linear reaction-diffusion equations in a domain consisting of two bulk regions separated by a thin layer with periodic structure. The thickness of the layer is of order \begin{document}$\epsilon$\end{document} , and the equations inside the layer depend on the parameter \begin{document}$\epsilon$\end{document} and an additional parameter \begin{document}$\gamma \in [-1,1)$\end{document} , which describes the size of the diffusion in the layer. We derive effective models for the limit \begin{document}$\epsilon \to 0 $\end{document} , when the layer reduces to an interface \begin{document}$\Sigma$\end{document} between the two bulk domains. The effective solution is continuous across \begin{document}$\Sigma$\end{document} for all \begin{document}$\gamma \in [-1,1)$\end{document} . For \begin{document}$\gamma \in (-1,1)$\end{document} , the jump in the normal flux is given by a non-linear ordinary differential equation on \begin{document}$\Sigma$\end{document} . In the critical case \begin{document}$\gamma = -1$\end{document} , a dynamic transmission condition of Wentzell-type arises at the interface \begin{document}$\Sigma$\end{document} .

Two-scale homogenization to determine effective parameters of thin metallic-structured films.

We present a homogenization method based on matched asymptotic expansion technique to derive effective transmission conditions of thin structured films. The method leads unambiguously to effective parameters of the interface which define jump conditions or boundary conditions at an equivalent zero thickness interface. The homogenized interface model is presented in the context of electromagnetic waves for metallic inclusions associated with Neumann or Dirichlet boundary conditions for transverse electric or transverse magnetic wave polarization. By comparison with full-wave simulations, the model is shown to be valid for thin interfaces up to thicknesses close to the wavelength. We also compare our effective conditions with the two-sided impedance conditions obtained in transmission line theory and to the so-called generalized sheet transition conditions.

Transmission Conditions for the Helmholtz-Equation in Perforated Domains

We study the Helmholtz equation in a perforated domain Ω e . The domain Ω e is obtained from an open set $\phantom {\dot {i}\!}{\Omega }\subset \mathbb {R}^{3}$ by removing small obstacles of typical size e>0, the obstacles are located along a 2-dimensional manifold Γ0⊂Ω. We derive effective transmission conditions across Γ0 that characterize solutions in the limit e→0. We obtain that, to leading order O(e 0), the perforation is invisible. On the other hand, at order O(e 1), inhomogeneous jump conditions for the pressure and the flux appear. The form of the jump conditions is derived.

Effective transmission conditions for Hamilton–Jacobi equations defined on two domains separated by an oscillatory interface

We consider a family of optimal control problems in the plane with dynamics and running costs possibly discontinuous across an oscillatory interface Γε. The oscillations of the interface have small period and amplitude, both of the order of ε, and the interfaces Γε tend to a straight line Γ. We study the asymptotic behavior as ε→0. We prove that the value function tends to the solution of Hamilton–Jacobi equations in the two half-planes limited by Γ, with an effective transmission condition on Γ keeping track of the oscillations of Γε.

Spectral Element Method and Domain Decomposition for Low-Frequency Subsurface EM Simulation

Low-frequency subsurface electromagnetic measurements are important tools for characterizing natural resources and environmental wastes. Rapid simulations of low-frequency subsurface electromagnetic measurements are still a challenge because of the large computational domain and low-frequency breakdown phenomenon. We develop an effective method to simulate these low-frequency subsurface electromagnetic measurements by using the spectral element method together with a domain decomposition method (DDM). A specific mesh has been designed based on the traveling wave nature in the air and the diffusion field nature in the underground space to greatly reduce the number of unknowns. The frequency-domain version of the Riemann solver (upwind flux) is used as an effective transmission condition to simulate the interactions between neighboring subdomains in DDM. Several numerical examples demonstrate the efficiency of the proposed approach in low-frequency subsurface electromagnetics simulations.

Dopamine disruption increases negotiation for cooperative interactions in a fish.

Humans and other animals use previous experiences to make behavioural decisions, balancing the probabilities of receiving rewards or punishments with alternative actions. The dopaminergic system plays a key role in this assessment: for instance, a decrease in dopamine transmission, which is signalled by the failure of an expected reward, may elicit a distinct behavioural response. Here, we tested the effect of exogenously administered dopaminergic compounds on a cooperative vertebrate’s decision-making process, in a natural setting. We show, in the Indo-Pacific bluestreak cleaner wrasse Labroides dimidiatus, that blocking dopamine receptors in the wild induces cleaners to initiate more interactions with and to provide greater amounts of physical contact to their client fish partners. This costly form of tactile stimulation using their fins is typically used to prolong interactions and to reconcile with clients after cheating. Interestingly, client jolt rate, a correlate of cheating by cleaners, remained unaffected. Thus, in low effective dopaminergic transmission conditions cleaners may renegotiate the occurrence and duration of the interaction with a costly offer. Our results provide first evidence for a prominent role of the dopaminergic system in decision-making in the context of cooperation in fish.

Optimized Schwarz Methods for Model Problems with Continuously Variable Coefficients

Optimized Schwarz methods perform better than classical Schwarz methods because they use more effective transmission conditions between subdomains. These transmission conditions are determined by optimizing the convergence factor, which is obtained by Fourier analysis for simple two subdomain model problems. Such optimizations have been performed for many different types of partial differential equations, but almost exclusively based on the assumption of constant coefficients, because only then Fourier analysis can be applied. We use in this paper the technique of separation of variables to study optimized Schwarz methods for a model problem with a continuously variable reaction term, and a similar analysis could be performed as well for many other problems with variable coefficients. We obtain several new interesting results: first, we show that the technique of separation of variables can successfully decouple the spatial variables and give the convergence factor of subdomain iterations as a function of the eigenvalues of a certain Sturm--Liouville problem that contains the variable coefficient. Second, we introduce a new natural transmission condition involving second order derivatives along the interface, which turns the corresponding optimization problem into a well-studied problem, from which the optimized transmission parameters follow. Finally, we find that for variable coefficient problems, the most important information that enters into the optimized transmission conditions is described by the smallest eigenvalue of the corresponding Sturm--Liouville problem. We illustrate our results with extensive numerical experiments.

Optimized Schwarz waveform relaxation for advection reaction diffusion equations in two dimensions

Optimized Schwarz waveform relaxation methods have been developed over the last decade for the parallel solution of evolution problems. They are based on a decomposition in space and an iteration, where only subproblems in space-time need to be solved. Each subproblem can be simulated using an adapted numerical method, for example with local time stepping, or one can even use a different model in different subdomains, which makes these methods very suitable also from a modeling point of view. For rapid convergence however, it is important to use effective transmission conditions between the space-time subdomains, and for best performance, these transmission conditions need to take the physics of the underlying evolution problem into account. The optimization of these transmission conditions leads to mathematically hard best approximation problems of homographic functions. We study in this paper in detail the best approximation problem for the case of linear advection reaction diffusion equations in two spatial dimensions. We prove comprehensively best approximation results for transmission conditions of Robin and Ventcel (higher order) type, which can also be used in the various limits for example for the heat equation, since we include in our analysis a positive low frequency limiter both in space and time. We give for each case closed form asymptotic values for the parameters which can directly be used in implementations of these algorithms, and which guarantee asymptotically best performance of the iterative methods. We finally show extensive numerical experiments, including cases not covered by our analysis, for example decompositions with cross points. We use Q1 finite element discretizations in space and Forward and Backward Euler discretizations in time (other discretization could also have been considered, since all our analysis is at the continuous level), and in all cases, we measure performance corresponding to our analysis.

Effective transmission conditions for thin-layer transmission problems in elastodynamics. The case of a planar layer model

This article is concerned with the design, analysis, numerical approximation and implementation of effective transmission conditions (ETCs) for the propagation of elastic waves through a thin planar elastic layer with small uniform thickness η which is embedded in a reference elastic medium, under transient conditions, with both materials assumed to have isotropic properties. A family of ETCs of order k (i.e. whose approximation error is of expected order O(η k+1)) is formulated by deriving and exploiting a formal asymptotic expansion in powers of η of the transmission solution inside the layer. The second-order ETCs are then retained as the main focus for the remainder of the article, and given a full justification in terms of both the stability of the resulting transient elastodynamic problem and the error analysis. The latter is performed by establishing and justifying asymptotic expansions for the solutions of both the exact transmission problem and its approximation based on the second-order ETCs. As a result, the error (in energy norm) between those two solutions is shown to be, as expected, of order O(η 3). Finally, the numerical approximation of the proposed second-order ETC within the framework of spectral element methods is studied, with special attention devoted to the selection of a robust time-stepping scheme that is mostly explicit (and conditionally stable). Among these, a scheme that is implicit only for the interfacial degrees of freedom, termed semi-implicit, is shown to be stable under the same stability condition as for the layer-less configuration. The main theoretical results of this work are illustrated and validated by 2D and 3D numerical experiments under transient elastodynamic conditions.

Axisymmetric eddy current inspection of highly conducting thin layers via asymptotic models

Thin copper deposits covering the steam generator tubes can blind eddy current probes in non-destructive testings of problematic faults and it is therefore important that they are identified. Existing methods based on shape reconstruction using eddy current signals encounter difficulties of high numerical costs due to the layer’s small thickness and high conductivity. In this article, we approximate the axisymmetric eddy current problem with some appropriate asymptotic models using effective transmission conditions representing the thin deposits. In these models, the geometrical information related to the deposit is transformed into parameter coefficients on a fictitious interface. A standard iterative inversion algorithm is then applied to the asymptotic models to reconstruct the thickness of the thin copper layers. Numerical tests both validating the asymptotic model and showing the benefits of the inversion procedure are provided.