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On some applications of integral equations in elasticity
Published in C Constanda, J Saranen, S Seikkala, Integral methods in science and engineering, 2020
The splitting of a vector field into the sum of a gradient and the curl of a solenoidal vector field is often called the Helmholtz decomposition. In paper [9] this decomposition is considered when the domain Ω is a bounded subset of ℝ3 : υ(x)=∇φ(x)+curlA(x),x∈Ω,
Surface waves in layered thermoelastic medium with double porosity structure: Rayleigh and Stoneley waves
Published in Mechanics of Advanced Materials and Structures, 2022
Davinder Kumar, Dilbag Singh, S. K. Tomar
Introducing the scalar potential and vector potential through Helmholtz decomposition of vector, we have where Plugging (9) into (1)-(4), we obtain where is the specific heat capacity at constant volume per unit volume.
A Trefftz Discontinuous Galerkin method for time-harmonic waves with a generalized impedance boundary condition
Published in Applicable Analysis, 2020
Shelvean Kapita, Peter Monk, Virginia Selgas
Our analysis of the discrete problem follows the pattern of the analysis of finite element methods for approximating the standard problem of scattering by an impenetrable scatterer using the Dirichlet-to-Neumann boundary condition from [21]. We first show that the GIBC can be discretized leaving the displacement equation continuous. Then we show that this semidiscrete problem can also be discretized successfully. The analysis of the error in the TDG part of the problem is motivated by the analysis of TDG for Maxwell's equations in [22] and uses the Helmholtz decomposition of the vector field satisfying (2) as a critical tool.
A note on stable Helmholtz decompositions in 3D
Published in Applicable Analysis, 2020
Now, with regards to an eventual stable Helmholtz decomposition for , we first recall that once one finds a bounded operator A for which is divergence-free (usually through an auxiliary boundary value problem), the existence of an operator B completing the verification of (1) only is already well established in 2D and 3D (see, e.g. [1, Chapter I, Theorems 3.1 and 3.4]). In turn, the required boundedness of B is always guaranteed in 2D thanks to the fact that the corresponding operator , which is defined in that case as , satisfies , whereas in the 3D case, in which the foregoing identity does not hold, additional geometric or regularity properties of the domain are needed to arrive at the same conclusion. In particular, the stability of a Helmholtz decomposition for has already been established for convex Lipschitz-continuous polyhedral regions in 3D (see, e.g. [2, Proposition 4.52]) by using [3, Theorems 2.17 and 3.12]. Moreover, the latter work by Amrouche et al. is actually a classical reference providing existence, uniqueness and regularity results concerning vector potentials associated with a divergence-free function in a bounded three-dimensional domain. Indeed, another consequence that follows also from [3] refers to the aforementioned stability for the case of arbitrary (not necessarily convex) domains of class . In the present paper we make further use of some results from [3] to extend [2, Proposition 4.52] to non-convex Lipschitz-continuous polyhedral regions in 3D and to a special case of homogeneous Neumann boundary conditions on a part of the boundary. While the first aforementioned extension is already available in [4, Lemma 4.3], we include it here mainly for sake of completeness since similar arguments to those employed in its proof will be utilized to prove the second extension. In addition, the fact that both results are gathered in the present work guarantees a greater visibility of them and hence increases the chances of extending their applicability to other situations different from the ones to be mentioned below (3D version of [5, 6]). Furthermore, we highlight in advance that, although all our results are established for tensors, they are also valid for vector fields since the arguments utilized works equally row wise. Additionally, we remark that related and very similar results already exist in the literature (see [7, Lemmas 3.7 and 3.8] and [8, Theorem 3.2]), but under alternative, and possibly more restrictive, assumptions on the domain. We refer in more details to them at appropriate places along the paper.