The method of finite spheres in acoustic wave propagation through nonhomogeneous media: Inf-sup stability conditions

Nicomedes, Williams L.; Bathe, Klaus‐Jürgen; Moreira, Fernando J. S.; Mesquita, Renato C.

doi:10.15625/0866-7136/15336

Cited by 4 publications

(7 citation statements)

References 45 publications

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“…From now on we assume p r : Ω r ⟶ ℂ . For all r = 1, ⋯, 4, we look for weak solutions p r regular enough so that p r ∈ H 1 (Ω r ), and assume material properties regular enough to satisfy

false(1 / ρ_{r, rel} false) \in C false({\overset{true‾}{normalΩ}}_{r} false)

and

false(K_{r, italicrel} false) \in C false({\overset{true‾}{normalΩ}}_{r} false) .

72 More details about the regularity of weak solutions to the Helmholtz equation can be found in the literature 28,73,81 . We next introduce the spaces

𝒳

and

𝒴

defined as:

𝒳 \overset{def}{=} H^{1} false(Ω_{1} false) \times H^{1} false(Ω_{2} false) \times H^{1} false(Ω_{3} false) \times H^{1} false(Ω_{4} false),

𝒴 \overset{def}{=} H^{- 1 / 2} false(Γ_{1, 2} false) \times H^{- 1 / 2} false(Γ_{2, 4} false) \times H^{- 1 / 2} false(Γ_{3, 4} false) .

It is useful to represent vectors from the spaces

𝒳

and

𝒴

by lowercase boldface letters, like:

v <...>

…”

Section: Weak Formsmentioning

confidence: 99%

“…The problem in weak form consists in finding a vector of pressure fields

p \overset{def}{=} false(p_{1} p_{2} p_{3} p_{4} false) \in 𝒳

, and a vector of Lagrange multiplier fields

λ \overset{def}{=} false(λ_{1, 2} λ_{2, 4} λ_{3, 4} false) \in 𝒴

. After a lengthy reasoning, 72 and considering geometrical domains in which the assumptions made in Section 2.1 hold true, it can be shown that the weak problem derived from (5), (6a), and the interface conditions discussed in Section 2.4 is given by

Find false(boldp, boldλ false) \in 𝒳 \times 𝒴 such that

\begin{array}{l} a (boldp, boldv) + b (v, λ) = ⟨ Q^{'} ∣ {boldv ⟩}_{𝒳^{⋆}, script𝒳}, for normalany boldv \in script𝒳, \\ b (p, μ) = 0, for normalany boldμ \in script𝒴 . \end{array}

In (12a), the bilinear form

a : 𝒳 \times 𝒳 ⟶ ℂ

is given by:

a false(w, v false) \overset{def}{=} \sum_{r = 1}^{4}

…”

Section: Weak Formsmentioning

confidence: 99%

“…The kernel (or nullspace) of B h is denoted by ker B h , and

{script𝒴}_{h}^{⋆}

is the dual space of

{script𝒴}_{h} .

40,41,80,82 These conditions are discussed in detail in another paper 72 . In that work, we touch on subjects like the non‐coercivity (or non‐ellipticity 87 ) of the bilinear form a derived from the Helmholtz equation, 88 and also on ways to overcome the difficulty of dealing with norms in the fractional Sobolev space H −1/2 , which are embedded in the norm of the space

𝒴

, according to (8b), (10b), and (11b).…”

Section: The Inf‐sup Conditionsmentioning

confidence: 99%

“…Then we finally derive 72 stronger and simplified inf‐sup conditions, in which only real‐valued numbers (i.e., real‐valued vectors and matrices) are used (we no longer use complex numbers). These are given by:

1 . There is a constant α_{h} > 0 such that \inf_{\begin{matrix} subarray \\ \overset{true˜}{bold-frakturw} \in ℝ^{2 K} \\ \overset{true˜}{bold-frakturw} \neq 0 \end{matrix}} \sup_{\begin{matrix} subarray \\ \overset{true˜}{bold-frakturv} \in ℝ^{2 K} \\ \overset{true˜}{bold-frakturv} \neq 0 \end{matrix}} \frac{{\overset{w}{true}}^{T} \overset{true‾}{double-struckA} \overset{true˜}{bold-frakturv}}{\sqrt{{\overset{w}{true}}^{T} \overset{true‾}{double-struckD} \overset{true˜}{bold-frakturw}} \sqrt{{\overset{v}{true}}^{T} \overset{true‾}{double-struckD} \overset{true˜}{bold-frakturv}}} \geq α_{h},

2 . There is a constant β_{h} > 0 such that \inf_{\begin{matrix} subarray \\ \overset{true˜}{bold-frakturu} \in ℝ^{2 \dim {script𝒴}_{h}} \\ \overset{true˜}{bold-frakturu} \neq 0 \end{matrix}} \sup_{\begin{matrix} subarray \\ \overset{true˜}{bold-frakturv} \in ℝ^{2 \dim {script𝒳}_{h}} \\ \overset{true˜}{bold-frakturv} \neq 0 \end{matrix}} {\overset{u}{true}}^{T}

…”

Section: The Inf‐sup Conditionsmentioning

confidence: 99%

“…A rigorous derivation of the mixed problem associated with the Helmholtz equation, and specialized to the particular geometrical setting considered here, is presented in one of our works, 72 which is a companion paper to this one. In that work we show that the Lagrange multipliers arise naturally in the formulation, thus rendering this approach more straightforward to implement (since there are no ‘amendments’ like jump functions and tunable penalty factors).…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Acoustic scattering in nonhomogeneous media and the problem of discontinuous gradients: Analysis and inf‐sup stability in the method of finite spheres

Nicomedes

Bathe

Moreira

et al. 2021

Numerical Meth Engineering

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In this paper we focus on a meshfree formulation for the solution of time‐harmonic acoustic scattering problems and verify the stability of the procedure. The sound waves propagate in nonhomogeneous media, giving rise to discontinuities in the gradients of the pressure field across the interfaces between regions of different material properties. Meshfree methods usually do not reproduce accurately the discontinuities in a numerical solution. We overcome this issue by introducing Lagrange multiplier fields defined at the interfaces in order to treat the discontinuities in the gradients of the pressure field. The method does not depend on any kind of adjustable parameter. We show by a numerical study of the applicable inf‐sup conditions that the resulting mixed formulation leads to well‐posed problems. The use of the proposed method is illustrated in the solution of a number of problems of wave propagation through nonhomogeneous media.

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false(1 / ρ_{r, rel} false) \in C false({\overset{true‾}{normalΩ}}_{r} false)

and

false(K_{r, italicrel} false) \in C false({\overset{true‾}{normalΩ}}_{r} false) .

72 More details about the regularity of weak solutions to the Helmholtz equation can be found in the literature 28,73,81 . We next introduce the spaces