Kirchhoff Condition Research Articles

Well posedness of linear parabolic partial differential equations posed on a star-shaped network with local time Kirchhoff's boundary condition at the vertex

The main purpose of this work is to provide an existence and uniqueness result for the solution of a linear parabolic system posed on a star-shaped network, which presents a new type of Kirchhoff's boundary transmission condition at the junction. This new type of Kirchhoff's condition - that we decide to call here local-time Kirchhoff's condition - induces a dynamical behavior with respect to an external variable that may be interpreted as a local time parameter, designed to drive the system only at the singular point of the network. The seeds of this study point towards a theoretical inquiry of a particular generalization of Walsh's random spider motions, whose spinning measures would select the available directions according to the local time of the motion at the junction of the network.

Flow-coupled cohesive interface framework for simulating heterogeneous crack morphology and resolving numerical divergence in hydraulic fracture

A user-friendly and generic finite element framework for simulating hydraulic fracture with a complex crack network in unconventional oil and gas exploitation is developed in this work with the following unique features. While the cohesive zone model (CZM) removes crack singularities, its finite element simulations suffer numerical convergence problem during crack nucleation and growth, which can be regularized by a fictitious viscosity approach. The decoupling of cohesive-cracked solid and fluid into separate free body diagrams allows the development of a weak-form finite element formulation for the former and a finite-difference approach for the transport analysis in the latter. Enforcing the Kirchhoff condition in polycrystalline geomaterials allows the study of the fluid-driven complex fracture propagation process. Our method has been verified by analytical solutions, and then employed to simulate the synergistic roles of confining pressure and grain boundary anisotropy on the fracking morphology. Numerical implementation into a user-defined element (UEL) subroutine in ABAQUS provides easy adaptation and further development for the research community.

Instability of ground states for the NLS equation with potential on the star graph

We study the nonlinear Schrödinger equation with an arbitrary real potential $$V(x)\in (L^1+L^\infty )(\Gamma )$$ on a star graph $$\Gamma $$ . At the vertex an interaction occurs described by the generalized Kirchhoff condition with strength $$-\gamma <0$$ . We show the existence of ground states $$\varphi _{\omega }(x)$$ as minimizers of the action functional on the Nehari manifold under additional negativity and decay conditions on V(x). Moreover, for $$V(x)=-\dfrac{\beta }{x^{\alpha }}$$ , in the supercritical case, we prove that the standing waves $$e^{i\omega t}\varphi _{\omega }(x)$$ are orbitally unstable in $$H^{1}(\Gamma )$$ when $$\omega $$ is large enough. Analogous result holds for an arbitrary $$\gamma \in {\mathbb {R}}$$ when the standing waves have symmetric profile.

Spectrum of a non-selfadjoint quantum star graph

We study the spectrum of a quantum star graph with a non-selfadjoint Robin condition at the central vertex. We first prove that, in the high frequency limit, the spectrum of the Robin Laplacian is close to the usual spectrum corresponding to the Kirchhoff condition. Then, we describe more precisely the asymptotics of the difference in terms of the Barra–Gaspard measure of the graph. This measure depends on the arithmetic properties of the lengths of the edges. As a by-product, this analysis provides a Weyl law for non-selfadjoint quantum star graphs and it gives the asymptotic behaviour of the imaginary parts of the eigenvalues.

Model of time-dependent geometric graph for dynamical Casimir effect

Geometric graph model is suggested for the dynamical Casimir effect. The wave equation is considered at the graph edges and the Kirchhoff condition at the internal vertex. It is assumed that the edge lengths depend on time. Photon generation is considered in the framework of the Dodonov–Klimov model.

On the discrete spectrum of the Dirac operator on bent chain quantum graph

We study Dirac operators on an infinite quantum graph of a bent chain form which consists of identical rings connected at the touching points by δ-couplings with a parameter α ∈ ℝ. We are interested in the discrete spectrum of the corresponding Hamiltonian. It can be non-empty due to a local (geometrical perturbation of the corresponding infinite chain of rings. The quantum graph of analogous geometry with the Schrodinger operator on the edges was considered by Duclos, Exner and Turek in 2008. They showed that the absence of δ-couplings at vertices (i.e. the Kirchhoff condition at the vertices) lead to the absence of eigenvalues. We consider the relativistic particle (the Dirac operator instead of the Schrodinger one) but the result is analogous. Quantum graphs of such type are suitable for description of grapheme-based nanostructures. It is established that the negativity of α is the necessary and sufficient condition for the existence of eigenvalues of the Dirac operator (i.e. the discrete spectrum of the Hamiltonian in this case is not empty). The continuous spectrum of the Hamiltonian for bent chain graph coincides with that for the corresponding straight infinite chain. Conditions for appearance of more than one eigenvalue are obtained. It is related to the bending angle. The investigation is based on the transfer-matrix approach. It allows one to reduce the problem to an algebraic task. δ-couplings was introduced by the operator extensions theory method.

Drape simulation using solid-shell elements and adaptive mesh subdivision

In this paper, 4-node quadrilateral and 3-node triangular solid-shell elements are applied to drape simulations. With locking issues alleviated by the assumed natural strain method and plane-stress enforcement, static and dynamic drape problems are attempted by the quadrilateral element. If the drape is deep and the mesh density is inadequate, non-realistic sharp folds are predicted due to the non-physical interpenetration of top and bottom element surfaces. To avoid the interpenetration, a reversible adaptive subdivision based on the 1–4 splitting method is developed. To ensure displacement compatibility among elements at different subdivision levels, macro-transition elements are formed by quadrilateral and triangular solid-shell elements. To reduce the dynamic oscillation induced by newly inserted nodes, the discrete Kirchhoff condition is employed to determine the related nodal variables. Dynamic drape examples using adaptive meshing are presented. It can be seen that the predictions look realistic and deep drapes can be predicted with the interpenetration avoided yet the required number of nodes can be kept relatively small.

Random Motion on Simple Graphs

Consider a stochastic process that lives on n-semiaxes joined at the origin. On each ray it behaves as one dimensional Brownian Motion and at the origin it chooses a ray uniformly at random (Kirchhoff condition). The principal results are the computation of the exit probabilities and certain other probabilistic quantities regarding exit and occupation times.

The Eigenvalues of the Laplacian on Locally Finite Networks Under Generalized Node Transition

We consider the canonical continuous Laplacian on an infinite locally finite network with different edge lengths under natural transition conditions as continuity at the ramification nodes and Kirchhoff flow conditions at all vertices that can be consistent with the Laplacian or not. It is shown that the eigenvalues of this Laplacian in a L∞-setting are closely related to those of a row–stochastic operator of the network resulting from the length adjacency operator and the weights in the Kirchhoff condition. In this way the point spectrum in L∞ is determined completely in terms of combinatorial and metrical quantities of the underlying graph as in the finite case [2] and as in the equal length case [10], and, in addition, in terms of the coefficients in the transition condition. Though the multiplicity formulae are formally the same as in the cited cases, the multiplicities can change strongly. Another main concern is the real or nonreal character of the eigenvalues.

Krein formula with compensated singularities for the ND-mapping and the generalized Kirchhoff condition at the Neumann Schrödinger junction

The Neumann Schrodinger operator \( \mathcal{L} \) is considered on a thin 2D star-shaped junction, composed of a vertex domain Ωint and a few semi-infinite straight leads ωm, m = 1, 2, ..., M, of width δ, δ ≪ diam Ωint, attached to Ωint at Γ ⊂ ∂Ωint. The potential of the Schrodinger operator lω on the leads vanishes, hence there are only a finite number of eigenvalues of the Neumann Schrodinger operator Lint on Ωint embedded into the open spectral branches of lω with oscillating solutions χ±(x, p) = \( e^{ \pm iK_ + x} e_m \) of lωχ± = p2χ±. The exponent of the open channels in the wires is $$ K_ + (\lambda ) = p\sum\limits_{m = 1}^M {e^m } \rangle \langle e^m = \sqrt \lambda P_ + $$ , with constant em, on a relatively small essential spectral interval Δ ⊂ [0, π2δ−2). The scattering matrix of the junction is represented on Δ in terms of the ND mapping $$ \mathcal{N} = \frac{{\partial P_ + \Psi }} {{\partial x}}(0,\lambda )\left| {_\Gamma \to P_ + \Psi _ + (0,\lambda )} \right|_\Gamma $$ as $$ S(\lambda ) = (ip\mathcal{N} + I_ + )^{ - 1} (ip\mathcal{N} - I_ + ), I_ + = \sum\limits_{m = 1}^M {e^m } \rangle \langle e^m = P_ + $$ . We derive an approximate formula for \( \mathcal{N} \) in terms of the Neumann-to-Dirichlet mapping \( \mathcal{N}_{\operatorname{int} } \) of Lint and the exponent K− of the closed channels of lω. If there is only one simple eigenvalue λ0 ∈ Δ, Lintφ0 = λ0φ0 then, for a thin junction, \( \mathcal{N} \approx |\vec \phi _0 |^2 P_0 (\lambda _0 - \lambda )^{ - 1} \) with $$ \vec \phi _0 = P_ + \phi _0 = (\delta ^{ - 1} \int_{\Gamma _1 } {\phi _0 (\gamma )} d\gamma ,\delta ^{ - 1} \int_{\Gamma _2 } {\phi _0 (\gamma )} d\gamma , \ldots \delta ^{ - 1} \int_{\Gamma _M } {\phi _0 (\gamma )} d\gamma ) $$ and \( P_0 = \vec \phi _0 \rangle |\vec \phi _0 |^{ - 2} \langle \vec \phi _0 \), $$ S(\lambda ) \approx \frac{{ip|\vec \phi _0 |^2 P_0 (\lambda _0 - \lambda )^{ - 1} - I_ + }} {{ip|\vec \phi _0 |^2 P_0 (\lambda _0 - \lambda )^{ - 1} + I_ + }} = :S_{appr} (\lambda ) $$ . The related boundary condition for the components P+Ψ(0) and P+Ψ′(0) of the scattering Ansatz in the open channel \( P_ + \Psi (0) = (\bar \Psi _1 ,\bar \Psi _2 , \ldots ,\bar \Psi _M ), P_ + \Psi '(0) = (\bar \Psi '_1 , \bar \Psi '_2 , \ldots , \bar \Psi '_M ) \) includes the weighted continuity (1) of the scattering Ansatz Ψ at the vertex and the weighted balance of the currents (2), where $$ \frac{{\bar \Psi _m }} {{\bar \phi _0^m }} = \frac{{\delta \sum\nolimits_{t = 1}^M { \bar \Psi _t \bar \phi _0^t } }} {{|\vec \phi _0 |^2 }} = \frac{{\bar \Psi _r }} {{\bar \phi _0^r }} = :\bar \Psi (0)/\bar \phi (0), 1 \leqslant m,r \leqslant M $$ (1) , $$ \sum\limits_{m = 1}^M {\bar \Psi '_m } \bar \phi _0^m + \delta ^{ - 1} (\lambda - \lambda _0 )\bar \Psi /\bar \phi (0) = 0 $$ (1) . Conditions (1) and (2) constitute the generalized Kirchhoff boundary condition at the vertex for the Schrodinger operator on a thin junction and remain valid for the corresponding 1D model. We compare this with the previous result by Kuchment and Zeng obtained by the variational technique for the Neumann Laplacian on a shrinking quantum network.

Excitation of an Infinite Microstrip Line With a Vertical Coaxial Feed

An efficient method is presented for the analysis of a vertical coaxial probe excitation of an infinite microstrip line. The novel feature of the method is that it uses a semianalytical Green's function that is derived for current sources in the presence of the infinite microstrip line. Hence, in a method of moments approach, unknown currents need only be placed on the conducting probe feed and not on the infinite strip surface. The method also utilizes an attachment mode at the contact point between the probe and line so that the correct Kirchhoff condition is automatically satisfied. Once the surface current density on the probe is known, the surface current density on the strip conductor can be readily obtained using the Green's function of the background grounded substrate. The method is valid even at high frequency, where simple transmission line theory fails to account for effects such as the continuous-spectrum current that is excited on the line. After validating the method with various commercial software simulation packages, results are presented to study the fundamental behavior of the input impedance, probe current, and current launched on the microstrip line, and to examine the high-frequency behavior of these currents.

A meshfree thin shell method for non‐linear dynamic fracture

AbstractA meshfree method for thin shells with finite strains and arbitrary evolving cracks is described. The C1 displacement continuity requirement is met by the approximation, so no special treatments for fulfilling the Kirchhoff condition are necessary. Membrane locking is eliminated by the use of a cubic or quartic polynomial basis. The shell is tested for several elastic and elasto‐plastic examples and shows good results. The shell is subsequently extended to modelling cracks. Since no discretization of the director field is needed, the incorporation of discontinuities is easy to implement and straightforward. Copyright © 2007 John Wiley &amp; Sons, Ltd.

Gemischte und nicht-standard Finite-Elemente-Methoden mit Anwendungen

Mixed finite element methods (MFEM) form a general mathematical framework for the spatial discretisation of partial differential equations, mainly applied to elliptic equations of second order. They become increasingly important for the solution of nonlinear problems. In contrast to standard finite element schemes the mixed finite element discretisation of problems in divergence form, i.e. f +{\rm div} \, \sigma = 0 where \sigma = A (\nabla \, u) , \sigma \in L and u \in H , allows more flexibility in the design of the discrete approximation spaces contained in L and H , i.e. in the spaces for the direct variables and the Lagrange multipliers. The workshop focuses on new developments in the field of mixed and non-standard finite element methods. The main points are The workshop aimed at bridging the gap between the computational engineering community and applied mathematicians and in consequence to unify the scientific language and foster later collaboration. Nonlinear mixed schemes were of particular concern for problems in elasticity and plasticity, but electromagnetics and mathematically related topics were also included. Mixed finite element methods for elliptic problems are based on a variational description in saddle-point form. Side conditions such as divergence free velocity fields in incompressible fluid dynamics are usually treated in this framework. The appearance of ‘soft’ side conditions is typical for structural mechanics as is the case with nearly incompressible materials or plates and shells with small thickness parameters. We also mention materials which almost satisfy the Kirchhoff condition, i.e. problems with a high but finite shear stiffness. In such cases, which are by no means ‘soft’ from the mathematical point of view, mixed methods lead to a more robust discretisation. The arising stability conditions and computational techniques cannot be understood fully by intuitive mechanical principles; however, from the mathematician's point of view their reasoning is natural, clear and insightful. Mixed and non-standard finite element methods gain increasing prominence in the prevention of locking phenomena. We highlight a topic which is currently actively investigated: the development of stable and efficient plate and shell elements with regard to shear locking, which is more intricate than volume locking. Here it is important to understand how techniques based on heuristic ideas are consistent with more modern mathematical methods. Availability of fast solvers is decisive for the competitiveness of numerical techniques. For a variety of applications, multigrid methods are crucial for the efficiency of the implementation. Methods have been proposed which do not appear plausible if one wants to deduce the algorithms directly from the physical model. The advanced methods depend on rigorous error estimators in order to guarantee that the numerical solutions represent the exact solutions of the physical model.

Computer reconstruction of a periodic surface by means of thescattered and transmitted acoustic fields

The profile of a periodic surface for which an analytical expressionin Fourier series exists is computer reconstructed. The surface is consideredas a sum of sinusoidal gratings because of the fact that each surface can bepresented in Fourier series. A periodic surface is chosen for testing of themathematical model based on the solution of the Helmholtz integral equation inthe case for which the Kirchhoff condition is realized for each harmonic. Thesurface profile is reconstructed using the acoustic field scattered from thesurface as well as the transmitted acoustic field. In both cases all acousticfields are registered (to the left and to the right relative to the centralpeak of the mirror reflected field) by a piezoelectric transducer moved by acomputer controlled table. The fields are registered at a frequency of theradiating transducer of 1.1 MHz (Λ = 1.3636 mm at acoustic velocity1500 m s-1 in water). The surface profile is reconstructed from thefields registered to the left as well as to the right according to themathematical model based on Kirchhoff's inequality. In both cases thereconstructed profile has the same form, period and distance from peak tovalley as the real surface profile.

Dynamical interface transition in ramified media with diffusion

We consider interaction problems in ramified spaces of a rather general type with a distinguished interface transition in form of a dynamical Kirchhoff condition. These conditions stem from applica...

General formulation of C� bending models based on the rational w-? constraint

With regard to the C° plate bending elements, the constraints resulting from the Kirchhoff condition are separated into two sets. The first set is called the zero ϑ constraint and should be released so that there is no shear locking. The second set is called the w-ϑ constraint and should be imposed to determine the element shear strain. We take the line segment integral of the Kirchhoff condition to establish the rational version of w-ϑ constraint which ensures the resulting element to pass the patch test. In such a way, the 4, 8 and 9-node C° bending models are presented as a general formulation. The resulting elements, in particular the 9-node one, show excellent numerical performance.

Polarizationally Non-sensitive Pseudo-holographic Computer Reconstruction of Random Rough Surface

Abstract The function, describing a profile of a random rough surface (RRS) is expanded in a Fourier series, i.e. the surface is considered as a composition of sinusoidal gratings. The total diffracted optical field from this RRS is a sum of the fields due to all harmonic gratings, since Kirchhoff's condition for ‘locally flat surface’ is realized for each harmonic grating at a given light wavelength and at an appropriate choice of the basic grating period. The registered s and p components of the diffracted (+1 diffraction orders of each harmonic gratings), incident and mixed optic fields are separated with an optical analyser. These fields are experimentally measured and from these values the phase and the amplitude of each grating are determined. The profile of the surface is reconstructed for s and p polarization of the light scattered field, when the electric vector of the incident light concludes an arbitrary angle with the incidence plane. The mean roughness is determined in both cases. It is shown...

Iterative Solution Methods for Plate Bending Problems: Multigrid and Preconditionedcgalgorithm

The numerical solution of the linear equations arising from a discretization of the plate bending problem based on the Zienkiewicz triangle is studied. The finite element scheme proceeds from the Mindlin–Reissner formulation with modified shear energy. However, the Kirchhoff condition is imposed on discrete points. A multigrid algorithm is analyzed and numerical examples are provided. Furthermore, a suitable preconditioning is established for the use of conjugate gradients.

Direct measurement of surface-scatter channel coherence by impulse probing

A method of studying the statistical properties of the random time-varying surface-scatter channel using ultrasonic scale modelling and using a novel digital-probing system is discussed. Measurements of the coherent transfer function of such channels are presented and are shown to be in minor disagreement with existing theoretical predictions. Various mechanisms to explain this disagreement are explored leading to the tentative conclusion that a violation of the Kirchhoff condition is probably responsible for the effect observed.

Sudden appearance of a crack in a bent plate

This paper is concerned with the determination of the transient stress state at the site in a bent plate where a through crack of finite length has appeared suddenly. Such a phenomenon can be easily visualized as the type of damage caused by projectile penetration. Mindlin's equations of flexural motion are used so that the three natural boundary conditions can be satisfied on the crack surfaces whereas the approximate Kirchhoff condition in the classical theory leads to unrealistic results near the crack. A set of dual integral equations is derived for this problem and solved numerically. The moment intensity factor is found to increase monotonically with time and is always lower than the static limit. Its amplitude is a function of the plate thickness to crack length ratio. This is different from the in-plane stretching case where the dynamic stress-intensity factor rises above the static limit very quickly before it decays.