170,020 research outputs found

    Triplen harmonics: myths and reality

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    In a symmetrical and balanced three-phase system with distorted waveforms, a well-known rule states that each harmonic order corresponds to a specified sequence (positive, negative or zero). In this ideal case, the current in the neutral conductor (or more generally in the return path) contains only triplen harmonics. However, this rule is no longer valid in practical distribution systems subject to unbalance and waveform distortion, in which phase and neutral currents at any sequence generally contain components of any harmonic order. Possible improper extension of the ideal case concepts to general situations may create a sort of myth, to be removed by providing tutorial and practical examples. This paper provides a direct quantification of the extent to which non-triplen harmonics are present in the zero-sequence current components and triplen harmonics are present in the positive and negative sequence current components. An original set of indicators, built on the basis of the theoretical symmetrical component-based framework developed by the authors, is introduced for assessing the specific impact of the triplen harmonics at the different sequences. Some classical myths based on the ideal case are illustrated and discussed on specific examples including theoretical cases and experimental analyses, quantifying the actual role played by the triplen harmonics in these applications

    Energy release rate and stress intensity factor in antiplane elasticity

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    In the setting of antiplane linearized elasticity, we show the existence of the stress intensity factor and its relation with the energy release rate when the crack path is a C(1.1) curve. Finally, we show that the energy release rate is continuous with respect to the Hausdorff convergence in a class of admissible cracks

    Limits of Dirichlet problems in perforated domains: a new formulation

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    Sia A un operatore ellittico lineare del secondo ordine con coefficienti misurabili e limitati su un aperto limitato Ω\Omega di Rn\mathbf{R}^{\textrm{n}} , sia K={wϵH01(Ω):Aw1inD(Ω), K*=\{w*\epsilon H_{0}^{1}\left(\Omega\right):A*w*\leq1\, in\,\mathcal{D}'\left(\Omega\right)\qquad, e,w0a.e.inΩ}, e,\, w*\geq0\, a.e.\, in\,\Omega\}\qquad, e sia Ωh\Omega_{h} un'arbitraria successione di sottoinsiemi aperti di Ω\Omega. Dimostriamo il seguente risultato di compattezza: esistono una sottosuccessione, che indichiamo ancora con Ωh\Omega_{h} ed una funzione w{*} ϵ\epsilon K{*} tali che, per ogni f ϵL(Ω)\epsilon L^{\infty}\left(\Omega\right) , le soluzioni uhϵH01(Ωh)_{h}\epsilon H_{0}^{1}\left(\Omega_{h}\right) delle equazioni Auh_{h} = f in Ωh\Omega_{h} , estese a zero su Ω/Ωh\Omega/\Omega_{h}, convergano debolmente in H01(Ω)H_{0}^{1}\left(\Omega\right) all'unica soluzione u del problema. (){uϵH01(Ω)L(Ω)Au,wφAw,uφ+1,uφ=f,wφφϵC0(Ω) \left(*\right)\begin{cases} \begin{array}{c} u\epsilon H_{0}^{1}\left(\Omega\right)\cap L^{\infty}\left(\Omega\right)\\ \left\langle Au,\, w*\varphi\right\rangle -\left\langle A*w*,\, u\varphi\right\rangle +\left\langle 1,u\varphi\right\rangle =\left\langle f,w*\varphi\right\rangle \:\forall\varphi\epsilon C_{0}^{\infty}\left(\Omega\right) \end{array}\end{cases} Studiamo inoltre in maniera sistematica le proprietà delle soluzioni di tale equazione. Dimostriamo infine il seguente risultato di densità: per ogni w{*}ϵ\epsilonK{*} esiste una successione Ωh\Omega_{h} di sottoinsiemi aperti di Ω\Omega tali che per ogni f ϵL(Ω)\epsilon L^{\infty}\left(\Omega\right) le soluzioni uhϵH01(Ωh)_{h}\epsilon H_{0}^{1}\left(\Omega_{h}\right) dell'equazione Auh_{h}=f in Ωh\Omega_{h}, estese a zero Ω/Ωh\Omega/\Omega_{h} convergano debolmente in H01(Ω)H_{0}^{1}\left(\Omega\right)alla soluzione di ({*}).Let A be a linear elliptic operator of the second order with bounded measurable coefficients on a bounded open set Ω\Omega of Rn\mathbf{R}^{\textrm{n}} , let K={wϵH01(Ω):Aw1inD(Ω), K*=\{w*\epsilon H_{0}^{1}\left(\Omega\right):A*w*\leq1\, in\,\mathcal{D}'\left(\Omega\right)\qquad, e,w0a.e.inΩ}, e,\, w*\geq0\, a.e.\, in\,\Omega\}\qquad, and let Ωh\Omega_{h} be an arbitrary sequence of open subsets of Ω\Omega. We prove the following compactness result: there exist a subsequence, still denoted by Ωh\Omega_{h} and a function w{*} ϵ\epsilon K{*} such that, for every f ϵL(Ω)\epsilon L^{\infty}\left(\Omega\right) , the solutions uhϵH01(Ωh)_{h}\epsilon H_{0}^{1}\left(\Omega_{h}\right) of the equation Auh_{h} = f in Ωh\Omega_{h} , extended by zero on Ω/Ωh\Omega/\Omega_{h}, converge weakly in H01(Ω)H_{0}^{1}\left(\Omega\right) to the unique solution u of the problem. (){uϵH01(Ω)L(Ω)Au,wφAw,uφ+1,uφ=f,wφφϵC0(Ω) \left(*\right)\begin{cases} \begin{array}{c} u\epsilon H_{0}^{1}\left(\Omega\right)\cap L^{\infty}\left(\Omega\right)\\ \left\langle Au,\, w*\varphi\right\rangle -\left\langle A*w*,\, u\varphi\right\rangle +\left\langle 1,u\varphi\right\rangle =\left\langle f,w*\varphi\right\rangle \:\forall\varphi\epsilon C_{0}^{\infty}\left(\Omega\right) \end{array}\end{cases} We provide a self-contained study of the properties of the solutions of ({*}). We prove also the following density result: for any w{*}ϵ\epsilonK{*} there exists a sequence Ωh\Omega_{h} of open subsets of Ω\Omega such that for every f ϵL(Ω)\epsilon L^{\infty}\left(\Omega\right) the solutions uhϵH01(Ωh)_{h}\epsilon H_{0}^{1}\left(\Omega_{h}\right) of the equation Auh_{h}=f in Ωh\Omega_{h}, extended by zero on Ω/Ωh\Omega/\Omega_{h} converge weakly in H01(Ω)H_{0}^{1}\left(\Omega\right)to the solution of ({*})
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