scholarly journals The Meyers Estimates for Domains Perforated along the Boundary

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 3015
Author(s):  
Gregory A. Chechkin
Keyword(s):  
Elliptic Problem ◽  
Outer Boundary ◽  
Dirichlet Condition ◽  
Neumann Condition ◽  
Higher Integrability

In this paper, we consider an elliptic problem in a domain perforated along the boundary. By setting a homogeneous Dirichlet condition on the boundary of the cavities and a homogeneous Neumann condition on the outer boundary of the domain, we prove higher integrability of the gradient of the solution to the problem.

Topological sensitivity analysis revisited for time-harmonic wave scattering problems. Part I: the free space case

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Frédérique Le Louër ◽  
María-Luisa Rapún
Keyword(s):  
Free Space ◽  
Order Term ◽  
Harmonic Wave ◽  
Dirichlet Condition ◽  
Point Sources ◽  
Neumann Condition ◽  
Scattering Problems ◽  
Content Type ◽  
Time Harmonic ◽  
Topological Gradient

PurposeIn this paper, the authors revisit the computation of closed-form expressions of the topological indicator function for a one step imaging algorithm of two- and three-dimensional sound-soft (Dirichlet condition), sound-hard (Neumann condition) and isotropic inclusions (transmission conditions) in the free space.Design/methodology/approachFrom the addition theorem for translated harmonics, explicit expressions of the scattered waves by infinitesimal circular (and spherical) holes subject to an incident plane wave or a compactly supported distribution of point sources are available. Then the authors derive the first-order term in the asymptotic expansion of the Dirichlet and Neumann traces and their surface derivatives on the boundary of the singular medium perturbation.FindingsAs the shape gradient of shape functionals are expressed in terms of boundary integrals involving the boundary traces of the state and the associated adjoint field, then the topological gradient formulae follow readily.Originality/valueThe authors exhibit singular perturbation asymptotics that can be reused in the derivation of the topological gradient function that generates initial guesses in the iterated numerical solution of any shape optimization problem or imaging problems relying on time-harmonic acoustic wave propagation.

Error analysis of PML-FEM approximations for the Helmholtz equation in waveguides

2019 ◽  
Vol 53 (4) ◽  
pp. 1191-1222 ◽  
Author(s):  
Seungil Kim
Keyword(s):  
Finite Element ◽  
Helmholtz Equation ◽  
Piecewise Linear ◽  
A Priori ◽  
Approximate Solutions ◽  
Dirichlet Condition ◽  
Neumann Condition ◽  
Element Analysis ◽  
The Finite Element Analysis ◽  
Two Parameters

In this paper, we study finite element approximate solutions to the Helmholtz equation in waveguides by using a perfectly matched layer (PML). The PML is defined in terms of a piecewise linear coordinate stretching function with two parameters for absorbing propagating and evanescent components respectively, and truncated with a Neumann condition on an artificial boundary rather than a Dirichlet condition for cutoff modes that waveguides may allow. In the finite element analysis for the PML problem, we have to deal with two difficulties arising from the lack of full regularity of PML solutions and the anisotropic nature of the PML problem with, in particular, large PML damping parameters. Anisotropic finite element meshes in the PML regions depending on the damping parameters are used to handle anisotropy of the PML problem. As a main goal, we establish quasi-optimal a priori error estimates, that does not depend on anisotropy of the PML problem (when no cutoff mode is involved), including the exponentially convergent PML error with respect to the width and the strength of PML. The numerical experiments that confirm the convergence analysis will be presented.

scholarly journals On inverse problem for tree of Stieltjes strings

2021 ◽  
Vol 14 (1) ◽  
pp. 1-18
Author(s):  
Анастасія Ігорівна Дудко ◽  
Vyacheslav Pivovarchik
Keyword(s):  
Inverse Problem ◽  
Spectral Problem ◽  
Dirichlet Condition ◽  
Neumann Condition ◽  
Metric Tree

For a given metric tree and two strictly interlacing sequences of numbers there exits a distribution of point masses on the edges (which are Stieltjes strings) such that one of the sequences is the spectrum of the   spectral problem  with the Neumann condition at the root of the tree while the second sequence is the spectrum of the   spectral problem with the Dirichlet condition at the root.

scholarly journals Three spectra problem for Stieltjes string equation and Neumann conditions

2019 ◽  
Vol 12 (1) ◽  
pp. 41-55 ◽  
Author(s):  
Anastasia Dudko ◽  
Vyacheslav Pivovarchik
Keyword(s):  
Dirichlet Problem ◽  
Neumann Problem ◽  
Dirichlet Condition ◽  
The Other ◽  
Neumann Condition ◽  
String Equation ◽  
Dirichlet Problems ◽  
Transversal Vibrations ◽  
The Right ◽  
The Masses

Spectral problems are considered which appear in description of small transversal vibrations of Stieltjes strings. It is shown that the eigenvalues of the Neumann-Neumann problem, i.e. the problem with the Neumann conditions at both ends of the string interlace with the union of the spectra of the Neumann-Dirichlet problems, i.e. problems with the Neumann condition at one end and Dirichlet condition at the other end on two parts of the string. It is shown that the spectrum of Neumann-Neumann problem on the whole string, the spectrum of Neumann-Dirichlet problem on the left part of the string, all but one eigenvalues of the Neumann-Dirichlet problem on the right part of the string and total masses of the parts uniquely determine the masses and the intervals between them.

SCALING PROPERTIES OF THE SPREAD HARMONIC MEASURES

Fractals ◽  
2006 ◽  
Vol 14 (03) ◽  
pp. 231-243 ◽  
Author(s):  
DENIS S. GREBENKOV
Keyword(s):  
Mixed Boundary ◽  
Dirichlet Condition ◽  
Transport Processes ◽  
Neumann Condition ◽  
Mixed Boundary Value Problem ◽  
Reflected Brownian Motion ◽  
Continuous Transition ◽  
Scaling Properties ◽  
Harmonic Measures ◽  
Frequency Behavior

A family of the spread harmonic measures is naturally generated by partially reflected Brownian motion. Their relation to the mixed boundary value problem makes them important to characterize the transfer capacity of irregular interfaces in Laplacian transport processes. This family presents a continuous transition between the harmonic measure (Dirichlet condition) and the Hausdorff measure (Neumann condition). It is found that the scaling properties of the spread harmonic measures on prefractal boundaries are characterized by a set of multifractal exponent functions depending on the only scaling parameter. A conjectural extension of the spread harmonic measures to fractal boundaries is proposed. The developed concepts are applied to give a new explanation of the anomalous constant phase angle frequency behavior in electrochemistry.

Thermal blow-up in a subdiffusive medium due to a nonlinear boundary flux

2014 ◽  
Vol 17 (1) ◽  
Author(s):  
Colleen Kirk ◽  
W. Olmstead
Keyword(s):  
Nonlinear Boundary ◽  
Volterra Equation ◽  
Blow Up ◽  
Dirichlet Condition ◽  
Neumann Condition ◽  
Nonlinear Boundary Flux ◽  
One Dimensional ◽  
Fractional Heat Equation ◽  
The Right ◽  
Boundary Flux

AbstractA fractional heat equation is used to model thermal diffusion in a one-dimensional bar that exhibits subdiffusive behavior. The left end of the bar is subjected to a nonlinear influx of heat. For the boundary constraint at the right end of the bar, two cases are considered, namely a homogeneous Neumann condition and a homogeneous Dirichlet condition. By reducing both cases to a nonlinear Volterra equation, it is shown that a blow-up always occurs. The asymptotic behavior near the blow-up is determined for both cases. It is also shown that the solution for the Neumann case dominates that of the Dirichlet case.

scholarly journals Influence of boundary conditions on the optimization of a geothermal heat pump studied using a thermodynamic model

2022 ◽  
Vol 960 (1) ◽  
pp. 012003
Author(s):  
A Arz ◽  
A Minghini ◽  
M Feidt ◽  
M Costea ◽  
C Moyne
Keyword(s):  
Boundary Conditions ◽  
Heat Pump ◽  
Thermal Contact ◽  
Dirichlet Condition ◽  
Thermal Capacity ◽  
Coefficient Of Performance ◽  
Neumann Condition ◽  
Carnot Cycle ◽  
Geothermal Heat ◽  
Geothermal Heat Pump

Abstract This paper is the logical follow-up to a work [1] whose results were presented at the 28th French Thermal Congress which was to be held in Belfort in 2020. The model developed at that time is completed in this proposal to consider the specificity of the geothermal heat pump. This is a machine operating upon a mechanical vapor compression cycle, the limit of which is an inverse Carnot cycle. Its specificity consists of a cold loop at the source with the geothermal exchanger and the evaporator, then a hot loop at the sink with the condenser and a floor heat exchanger in the application considered here. We are particularly concerned with the optimal sizing of these heat exchangers through their effectiveness. The parametric sensitivity of this distribution to various boundary conditions is studied, especially by focusing on different conditions at the source: (1) imposed soil temperature, corresponding to a Dirichlet condition, (2) imposed heat flux (including adiabatic case), corresponding to a Neumann condition, (3) imposed mechanical power consumed by the heat pump, and (4) imposed coefficient of performance COP, to all cases being associated a finite thermal capacity in thermal contact with the geothermal exchanger operating in steady-state conditions.

Diffraction of a plane wave by a half plane under mixed boundary conditions

1981 ◽  
Vol 59 (8) ◽  
pp. 974-984 ◽  
Author(s):  
T. C. Kaladhar Rao
Keyword(s):  
Plane Wave ◽  
Boundary Conditions ◽  
Half Plane ◽  
Infinite Series ◽  
Series Solution ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Dirichlet Condition ◽  
Neumann Condition ◽  
Transform Method

The problem of diffraction of a plane wave by a semi-infinite half plane with mixed boundary conditions (Dirichlet condition on one face and Neumann condition on the other) is solved by a direct and rather straightforward method. The infinite series solution and the far field are in agreement with the previous solutions obtained by the Lebedev–Kontorovich transform method as expected, as the two methods are basically equivalent. An alternate representation of the infinite series solution is presented which is valid for any type of incident field including cylindrical and spherical fields. This representation facilitates easy analysis of transient problems and the special case of an incident plane unit step function on the half plane is given as an example.

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