When is the error in the $$h$$ h -BEM for solving the Helmholtz equation bounded independently of $$k$$ k ?

Graham, Ivan G.; Löhndorf, Maike; Melenk, Jens Markus; Spence, Euan A.

doi:10.1007/s10543-014-0501-5

Cited by 31 publications

(44 citation statements)

References 35 publications

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“…(2) A single injection was made to the mid-belly of the target muscle instead of using multiple injection points [1,15,56,59]. However, although they are not capable of showing whether an effective distribution of toxin is achieved, the present results strongly suggest that a considerable paralysis did occur within the target muscle.…”

Section: Discussionmentioning

confidence: 67%

“…Although it binds with high affinity to local targets within the muscle, a larger volume, single injection may cause that site to be saturated and thus allows the spread of toxin to neighboring structures [56]. Therefore, the present injection protocol may have promoted toxin leakage to the adjacent EDL and EHL muscles.…”

Section: Discussionmentioning

confidence: 97%

“…Note that the present BTX-A injection protocol differs from common clinical practice: (1) the injected dose (0.1 U i.e., approximately 0.32 U/kg) was less than that used in patients for lower limb muscles (3-6 U/kg), including children with cerebral palsy [56][57][58]. (2) A single injection was made to the mid-belly of the target muscle instead of using multiple injection points [1,15,56,59].…”

Section: Discussionmentioning

confidence: 99%

“…Yaraskavitch et al explained their results with leakage of BTX-A into the noninjected muscle [14]; additionally, it is possible that partial paralyzation of these muscles is responsible for the presently measured force decreases. Note that the spread of BTX-A beyond the injection site is considered to be a side effect [16,56] and has been argued to occur not only after localized injections [14,15], but may be determined by the dose and concentration of the injection [56]. New studies are indicated to test for the role of different injection protocols beyond the injection site within an intact compartment.…”

Section: Discussionmentioning

confidence: 99%

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BTX-A Administration to the Target Muscle Affects Forces of All Muscles Within an Intact Compartment and Epimuscular Myofascial Force Transmission

Yücesoy

Arikan

Ateş

2012

Journal of Biomechanical Engineering

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Measurement of forces of mono- and bi-articular muscles of an entire intact muscle compartment can allow for a comprehensive assessment of the effects of Botulinum toxin type A (BTX-A) both at and beyond the injection site, and in conditions close to those in vivo. The goal was to test the hypotheses that BTX-A affects (1) the forces of not only the injected but also the noninjected muscles of the compartment, and (2) epimuscular myofascial force transmission (EMFT). Two groups of Wistar rats were tested: Control (no BTX-A injected) and BTX (0.1 units of BTX-A were injected exclusively to the mid-belly of TA). Isometric forces were measured simultaneously at the distal tendons of the tibialis anterior (TA) at different lengths, the restrained extensor digitorum longus (EDL) and the extensor hallucis longus (EHL) muscles and at the proximal tendon of EDL. Five days post-injection, BTX-A did affect the total forces of all muscles significantly: (1) The TA force decreased differentially (by 46.6%-55.9%) for most lengths such that a significant negative correlation was found between force reductions and increased muscle length. The maximum TA force decreased by 47.3%. However, the muscle's length range of force production did not change significantly. (2) Distal and proximal EDL forces decreased (on average by 67.8% and 62.9%, respectively). (3) The EHL force also decreased (on average by 9.2%). The passive forces of only the TA showed a significant increase at higher lengths. EMFT effects were shown for the control group: (1) at the shortest TA lengths, the EDL proximo-distal force differences were in favor of the distal force, which was reversed at higher lengths. (2) the EHL force measured at the shortest TA length decreased (by 34%) as a function of TA lengthening. After BTX-A exposure, such EMFT effects disappeared for the EDL, whereas they remained as profound for the EHL. Exposure to BTX-A does affect forces of all muscles operating in an intact compartment. For the BTX-A injected muscle, the reduction in muscle force becomes less pronounced at higher muscle lengths. BTX-A also has effects on EMFT, however, these effects are not uniform within the anterior crural compartment. Decreased forces of the noninjected synergistic muscles suggest the presence of unintended additional effects of BTX-A both for the targeted distal joint and for the nontargeted proximal joint.

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Section: Discussionmentioning

confidence: 67%

Section: Discussionmentioning

confidence: 97%

Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

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BTX-A Administration to the Target Muscle Affects Forces of All Muscles Within an Intact Compartment and Epimuscular Myofascial Force Transmission

Yücesoy

Arikan

Ateş

2012

Journal of Biomechanical Engineering

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“…Regarding Point 2: S k is compact when ∂Ω is Lipschitz (since S k : L 2 (∂Ω) → H 1 (∂Ω) in this case [56, Theorem 1.6]), and D k and D ′ k are compact when ∂Ω is C 1 [23, Theorem 1.2(c)]. Regarding Points 3 and 4:[29] performed a k-explicit version of the classic compactperturbation argument appearing in, e.g.,[3, Chapter 3]. The two k-explicit ingredients were the L 2 (∂Ω) → H 1 (∂Ω) bounds on S k , D k , and D ′…”

mentioning

confidence: 99%

Wavenumber-Explicit Regularity Estimates on the Acoustic Single- and Double-Layer Operators

Galkowski

Spence

2019

Integr. Equ. Oper. Theory

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We prove new, sharp, wavenumber-explicit bounds on the norms of the Helmholtz singleand double-layer boundary-integral operators as mappings from L 2 (∂Ω) → H 1 (∂Ω) (where ∂Ω is the boundary of the obstacle). The new bounds are obtained using estimates on the restriction to the boundary of quasimodes of the Laplacian, building on recent work by the first author and collaborators.Our main motivation for considering these operators is that they appear in the standard second-kind boundary-integral formulations, posed in L 2 (∂Ω), of the exterior Dirichlet problem for the Helmholtz equation. Our new wavenumber-explicit L 2 (∂Ω) → H 1 (∂Ω) bounds can then be used in a wavenumber-explicit version of the classic compact-perturbation analysis of Galerkin discretisations of these second-kind equations; this is done in the companion paper [Galkowski, Müller, Spence, arXiv 1608.01035]. Definition 1.1 (Smooth hypersurface)We say that Γ ⊂ R d is a smooth hypersurface if there exists Γ a compact embedded smooth d − 1 dimensional submanifold of R d , possibly with boundary, such that Γ is an open subset of Γ, with Γ strictly away from ∂ Γ, and the boundary of Γ can be written as a disjoint unionwhere each Y ℓ is an open, relatively compact, smooth embedded manifold of dimension d − 2 in Γ, Γ lies locally on one side of Y ℓ , and Σ is closed set with d − 2 measure 0 and Σ ⊂ n l=1 Y l . We then refer to the manifold Γ as an extension of Γ.For example, when d = 3, the interior of a 2-d polygon is a smooth hypersurface, with Y i the edges and Σ the set of corner points.Definition 1.2 (Curved) We say a smooth hypersurface is curved if there is a choice of normal so that the second fundamental form of the hypersurface is everywhere positive definite.Recall that the principal curvatures are the eigenvalues of the matrix of the second fundamental form in an orthonormal basis of the tangent space, and thus "curved" is equivalent to the principal curvatures being everywhere strictly positive (or everywhere strictly negative, depending on the choice of the normal). Definition 1.3 (Piecewise smooth) We say that a hypersurfaceDefinition 1.4 (Piecewise curved) We say that a piecewise smooth hypersurface Γ is piecewise curved if Γ is as in Definition 1.3 and each Γ j is curved.The main results of this paper are contained in the following theorem. We use the notation that a b if there exists a C > 0, independent of k, such that a ≤ Cb. Theorem 1.5 (Bounds on S k L 2 (∂Ω)→H 1 (∂Ω) , D k L 2 (∂Ω)→H 1 (∂Ω) , D ′ k L 2 (∂Ω)→H 1 (∂Ω) ) Let Ω be a bounded Lipschitz open set such that the open complement Ω + := R d \ Ω is connected. (a) If ∂Ω is a piecewise smooth hypersurface (in the sense of Definition 1.3), then, given k 0 > 1, S k L 2 (∂Ω)→H 1 (∂Ω) k 1/2 log k, (1.4)for all k ≥ k 0 . Moreover, if ∂Ω is piecewise curved (in the sense of Definition 1.4), then, given k 0 > 1, the following stronger estimate holds for all k ≥ k 0 S k L 2 (∂Ω)→H 1 (∂Ω) k 1/3 log k.(1.5) (b) If ∂Ω is a piecewise smooth, C 2,α hypersurface, for some α > 0, then, given ...

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For Most Frequencies, Strong Trapping Has a Weak Effect in Frequency‐Domain Scattering

Lafontaine

Spence

Wunsch

2020

Comm Pure Appl Math

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It is well‐known that when the geometry and/or coefficients allow stable trapped rays, the outgoing solution operator of the Helmholtz equation grows exponentially through a sequence of real frequencies tending to infinity. In this paper we show that, even in the presence of the strongest possible trapping, if a set of frequencies of arbitrarily small measure is excluded, the Helmholtz solution operator grows at most polynomially as the frequency tends to infinity. One significant application of this result is in the convergence analysis of several numerical methods for solving the Helmholtz equation at high frequency that are based on a polynomial‐growth assumption on the solution operator (e.g. hp‐finite elements, hp‐boundary elements, and certain multiscale methods). The result of this paper shows that this assumption holds, even in the presence of the strongest possible trapping, for most frequencies. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.

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When is the error in the $$h$$ h -BEM for solving the Helmholtz equation bounded independently of $$k$$ k ?

Cited by 31 publications

References 35 publications

BTX-A Administration to the Target Muscle Affects Forces of All Muscles Within an Intact Compartment and Epimuscular Myofascial Force Transmission

BTX-A Administration to the Target Muscle Affects Forces of All Muscles Within an Intact Compartment and Epimuscular Myofascial Force Transmission

Wavenumber-Explicit Regularity Estimates on the Acoustic Single- and Double-Layer Operators

For Most Frequencies, Strong Trapping Has a Weak Effect in Frequency‐Domain Scattering

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