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In this paper, a novel robust vibration control scheme, namely, one degree-of-freedom fuzzy active force control (1DOF-FAFC) is applied to a nonlinear electromagnetic-actuated flexible plate system. First, the flexible plate with clamped-free-clamped-free (CFCF) boundary conditions is modeled and simulated. Then, the validity of the simulation platform is evaluated through experiment. A nonlinear electromagnetic actuator is developed and experimentally modeled through a parametric system identification scheme. Next, the obtained nonlinear model of the actuator is applied to the simulation platform and performance of the proposed control technique in suppressing unwanted vibrations is investigated via simulation. A fuzzy controller is applied to the robust 1DOF control scheme to tune the controller gain using acceleration feedback. Consequently, an intelligent self-tuning vibration control strategy based on an inexpensive acceleration sensor is proposed in the paper. Furthermore, it is demonstrated that the proposed acceleration-based control technique owns the benefits of the conventional velocity feedback controllers. Finally, an experimental rig is developed to investigate the effectiveness of the 1DOF-FAFC scheme. It is found that the first, second, and third resonant modes of the flexible system are attenuated up to 74%, 81%, and 90% respectively through which the effectiveness of the proposed control scheme is affirmed. [ABSTRACT FROM AUTHOR]

In this paper we continue to deal with the initial–boundary value problem for the coupled Keller–Segel–Stokes system { n t + u ⋅ ∇ n = Δ n − ∇ ⋅ ( n S ( x , n , c ) ⋅ ∇ c ) , ( x , t ) ∈ Ω × ( 0 , T ) , c t + u ⋅ ∇ c = Δ c − c + n , ( x , t ) ∈ Ω × ( 0 , T ) , u t + ∇ P = Δ u + n ∇ ϕ , ( x , t ) ∈ Ω × ( 0 , T ) , ∇ ⋅ u = 0 , ( x , t ) ∈ Ω × ( 0 , T ) , where Ω ⊂ R d is a bounded domain with smooth boundary and the chemotactic sensitivity S is not a scalar function but rather attains values in R d × d , and satisfies | S ( x , n , c ) | ≤ C S ( 1 + n ) − α with some C S > 0 and α > 0 . When d = 2 , our previous work (J. Differential Equations, 2015) has established the existence of global bounded classical solutions under the subcritical assumption α > 0 , which is consistent with the corresponding results of the fluid-free system, but the method seems to be invalid in the three-dimensional setting. In this paper, for the case d = 3 , we develop a new method to establish the existence and boundedness of global classical solutions for arbitrarily large initial data under the assumption α > 1 2 , which is slightly stronger than the corresponding subcritical assumption α > 1 3 on the fluid-free system, where such an assumption is essentially necessary and sufficient for the existence of global bounded solutions. The key idea here is to establish the general L p regularity of u from a rather low L p regularity of n , which will be obtained by a new combinational functional. [ABSTRACT FROM AUTHOR]

This paper presents the theoretical framework of the virtual acoustic volume concept and two related inverse Patch Transfer Functions (iPTF) identification methods (called u-iPTF and m-iPTF depending on the chosen boundary conditions for the virtual volume). They are based on the application of Green׳s identity on an arbitrary closed virtual volume defined around the source. The reconstruction of sound source fields combines discrete acoustic measurements performed at accessible positions around the source with the modal behavior of the chosen virtual acoustic volume. The mode shapes of the virtual volume can be computed by a Finite Element solver to handle the geometrical complexity of the source. As a result, it is possible to identify all the acoustic source fields at the real surface of an irregularly shaped structure and irrespective of its acoustic environment. The m-iPTF method is introduced for the first time in this paper. Conversely to the already published u-iPTF method, the m-iPTF method needs only acoustic pressure and avoids particle velocity measurements. This paper is focused on its validation, both with numerical computations and by experiments on a baffled oil pan. [ABSTRACT FROM AUTHOR]

This paper deals with the two-species chemotaxis system { u t = Δ u − ∇ ⋅ ( u χ 1 ( w ) ∇ w ) + μ 1 u ( 1 − u ) in Ω × ( 0 , ∞ ) , v t = Δ v − ∇ ⋅ ( v χ 2 ( w ) ∇ w ) + μ 2 v ( 1 − v ) in Ω × ( 0 , ∞ ) , w t = d Δ w + h ( u , v , w ) in Ω × ( 0 , ∞ ) , where Ω is a bounded domain in R n with smooth boundary ∂Ω, n ∈ N ; h , χ i are functions satisfying some conditions. Negreanu–Tello (2014, 2015) [12,13] proved global existence and asymptotic behavior of solutions to the above system when 0 ≤ d < 1 . The main purpose of the present paper is to remove the restriction of 0 ≤ d < 1 . [ABSTRACT FROM AUTHOR]

This paper estimates the blow-up time for the heat equation u t = Δ u with a local nonlinear Neumann boundary condition: The normal derivative ∂ u / ∂ n = u q on Γ 1 , one piece of the boundary, while on the rest part of the boundary, ∂ u / ∂ n = 0 . The motivation of the study is the partial damage to the insulation on the surface of space shuttles caused by high speed flying subjects. We show the finite time blow-up of the solution and estimate both upper and lower bounds of the blow-up time in terms of the area of Γ 1 . In many other work, they need the convexity of the domain Ω and only consider the problem with Γ 1 = ∂ Ω . In this paper, we remove the convexity condition and only require ∂Ω to be C 2 . In addition, we deal with the local nonlinearity, namely Γ 1 can be just part of ∂Ω. [ABSTRACT FROM AUTHOR]

The generalized Debye source representation of time-harmonic electromagnetic fields yields well-conditioned second-kind integral equations for a variety of boundary value problems, including the problems of scattering from perfect electric conductors and dielectric bodies. Furthermore, these representations, and resulting integral equations, are fully stable in the static limit as ω → 0 in multiply connected geometries. In this paper, we present the first high-order accurate solver based on this representation for bodies of revolution. The resulting solver uses a Nyström discretization of a one-dimensional generating curve and high-order integral equation methods for applying and inverting surface differentials. The accuracy and speed of the solvers are demonstrated in several numerical examples. • Efficient solver for wideband electromagnetic scattering from bodies of revolution. • Thorough discussion of the evaluation of modal Green's functions. • Computation of high-fidelity radar cross sections. • Numerical examples demonstrating that high-order accuracy is basically required for high-frequency calculations. [ABSTRACT FROM AUTHOR]

Abstract We prove a well-posedness result for stochastic Allen–Cahn type equations in a bounded domain coupled with generic boundary conditions. The (nonlinear) flux at the boundary aims at describing the interactions with the hard walls and is motivated by some recent literature in physics. The singular character of the drift part allows for a large class of maximal monotone operators, generalizing the usual double-well potentials. One of the main novelties of the paper is the absence of any growth condition on the drift term of the evolution, neither on the domain nor on the boundary. A well-posedness result for variational solutions of the system is presented using a priori estimates as well as monotonicity and compactness techniques. A vanishing viscosity argument for the dynamic on the boundary is also presented. [ABSTRACT FROM AUTHOR]

Abstract In this paper we consider the quasilinear critical problem (P λ) { − Δ p u = λ u q − 1 + u p ⁎ − 1 in Ω u > 0 in Ω u = 0 on ∂ Ω where Ω is a regular bounded domain in R N , N ≥ p 2 , 1 < p < 2 , p ≤ q < p ⁎ , p ⁎ = N p / (N − p) , λ > 0 is a parameter. In spite of the lack of C 2 regularity of the energy functional associated to (P λ) , we employ new Morse techniques to derive a multiplicity result of solutions. We show that there exists λ ⁎ > 0 such that, for each λ ∈ (0 , λ ⁎) , either (P λ) has P 1 (Ω) distinct solutions or there exists a sequence of quasilinear problems approximating (P λ) , each of them having at least P 1 (Ω) distinct solutions. These results complete those obtained in [23] for the case p ≥ 2. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we present a novel method for the identification of the local bending stiffness of beams. We use shearography to capture measurements of vibrating beams, so the input data for the identification is the modal slope – the differential of the modal shape. The modal slope is fitted by two Fourier-series functions, one of which is derived from a thin-beam model. The local bending stiffness is identified as the one corresponding with the best match between the measured and the two fitted modal slopes. This identification method, which we call simultaneous Fourier-series fitting, is demonstrated on numerically-generated inputs, as well as on experimental measurements. We use a flat, concave and convex beam, as well as beams with locally varying bending stiffness mimicking local damage to verify the method. It is shown that the method gives accurate results and is robust to noise. Additionally, it has advantageous properties that make it useful and practical: using this method, it is possible to perform the identification from only a sub-region of a beam and even without specifying the boundary conditions. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study the stability of supersonic contact discontinuity for the two-dimensional steady compressible Euler flows in a finitely long nozzle of varying cross-sections. We formulate the problem as an initial–boundary value problem with the contact discontinuity as a free boundary. To deal with the free boundary value problem, we employ the Lagrangian transformation to straighten the contact discontinuity and then the free boundary value problem becomes a fixed boundary value problem. We develop an iteration scheme and establish some novel estimates of solutions for the first order of hyperbolic equations on a cornered domain. Finally, by using the inverse Lagrangian transformation and under the assumption that the incoming flows and the nozzle walls are smooth perturbations of the background state, we prove that the original free boundary problem admits a unique weak solution which is a small perturbation of the background state and the solution consists of two smooth supersonic flows separated by a smooth contact discontinuity. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we prove the uniqueness of stationary standing wave solutions of an optical model generated by Type II Second Harmonic Generation (SHG) with behaviors tending to zero at infinity under certain conditions on parameters. In addition, we provide the same issues for the Dirichlet boundary value problems on the ball centered at the origin. A classification of solutions for radial case is also established. [ABSTRACT FROM AUTHOR]

Abstract Let α : G ↷ X be a minimal free continuous action of an infinite countable amenable group on an infinite compact metrizable space. In this paper, under the hypothesis that the invariant ergodic probability Borel measure space E G (X) is compact and zero-dimensional, we show that the action α has the small boundary property. This partially answers an open problem in dynamical systems that asks whether a minimal free action of an amenable group has the small boundary property if its space M G (X) of invariant Borel probability measures forms a Bauer simplex. In addition, under the same hypothesis, we show that dynamical comparison implies almost finiteness, which was shown by Kerr to imply that the crossed product is Z -stable. Finally, we discuss some rank properties and provide two classifiability results for crossed products, one of which is based on the work of Elliott and Niu. [ABSTRACT FROM AUTHOR]

Abstract In the present paper, the reducibility is derived for the wave equations with finitely smooth and time-quasi-periodic potential subject to periodic boundary conditions. More exactly, the linear wave equation u t t − u x x + M u + ε (V 0 (ω t) u x x + V (ω t , x) u) = 0 , x ∈ R / 2 π Z can be reduced to a linear Hamiltonian system with a constant coefficient operator which is of pure imaginary point spectrum set, where V is finitely smooth in (t , x) , quasi-periodic in time t with Diophantine frequency ω ∈ R n , and V 0 is finitely smooth and quasi-periodic in time t with Diophantine frequency ω ∈ R n. Moreover, it is proved that the corresponding wave operator possesses the property of pure point spectra and zero Lyapunov exponent. [ABSTRACT FROM AUTHOR]

Abstract In this paper, an efficient, high-order accurate, level set reinitialisation method is proposed, based on the elliptic reinitialisation method (Basting and Kuzmin, 2013 [1]), which is discretised spatially using the discontinuous Galerkin (DG) symmetric interior penalty method (SIPG). In order to achieve this a number of improvements have been made to the elliptic reinitialisation method including; reformulation of the underlying minimisation problem driving the solution; adoption of a Lagrange multiplier approach for enforcing a Dirichlet boundary condition on the implicit level set interface; and adoption of a narrow band approach. Numerical examples confirm the high-order accuracy of the resultant method by demonstrating experimental orders of convergence congruent with optimal convergence rates for the SIPG method, that is h p + 1 and h p in the L 2 and DG norms respectively. Furthermore, the degree to which the level set function satisfies the Eikonal equation improves proportionally to h p , and the often ignored homogeneous Dirichlet boundary condition on the interface is shown to be satisfied accurately with a rate of convergence of at least h 2 for all polynomial orders. [ABSTRACT FROM AUTHOR]

Abstract An analytical vibration response in the time domain for an axially translating and laterally vibrating string with mixed boundary conditions is considered in this paper. The domain of the string is a constant, dependent upon the general initial conditions. The translating tensioned strings possess different types of mixed boundary conditions, such as fixed_dashpot, fixed_spring-dashpot, fixed_mass-spring-dashpot. An analytical solution using a reflected wave superposition method is presented for a finite translating string. Firstly, the cycle of boundary reflection for strings is provided, which is dependent upon the string length. Each cycle is divided into three time intervals according to the travelling speed and direction of the string. Applying D'Alembert's principle and the reflection properties, expressions for the reflected waves under three different non-classical boundary conditions are derived. Then, the vibrational response of the axially translating string is solved for three time intervals by using a reflected wave superposition method. The accuracy and efficiency of the proposed method are confirmed numerically by comparison to simulations produced using a Newmark- β method solution. The energy expressions for a travelling string with a fixed_dashpot boundary condition is obtained and the time domain curves for the total energy and the change of energy at the boundaries are given. [ABSTRACT FROM AUTHOR]

Abstract Geometric gradient flows for elastic energies of Willmore type play an important role in mathematics and in many applications. The evolution of elastic curves has been studied in detail both for closed as well as for open curves. Although elastic flows for networks also have many interesting features, they have not been studied so far from the point of view of mathematical analysis. So far it was not even clear what are appropriate boundary conditions at junctions. In this paper we give a well-posedness result for Willmore flow of networks in different geometric settings and hence lay a foundation for further mathematical analysis. A main point in the proof is to check whether different proposed boundary conditions lead to a well posed problem. In this context one has to check the Lopatinskii–Shapiro condition in order to apply the Solonnikov theory for linear parabolic systems in Hölder spaces which is needed in a fixed point argument. We also show that the solution we get is unique in a purely geometric sense. [ABSTRACT FROM AUTHOR]

Abstract This paper deals with the Neumann problem for a fully parabolic chemotaxis–haptotaxis model of cancer invasion given by { u t = Δ u − χ ∇ ⋅ (u ∇ v) − ξ ∇ ⋅ (u ∇ w) + u (a − μ u r − 1 − λ w) , x ∈ Ω , t > 0 , τ v t = Δ v − v + u , x ∈ Ω , t > 0 , w t = − v w , x ∈ Ω , t > 0. Here, Ω ⊂ R N (N ≥ 1) is a bounded domain with smooth boundary and τ > 0 , r > 1 , λ ≥ 0 , a ∈ R , μ , ξ and χ are positive constants. It is shown that the corresponding initial–boundary value problem possesses a unique global bounded classical solution in the cases r > 2 or r = 2 , with μ > μ ⁎ = (N − 2) + N (χ + C β) C N 2 + 1 1 N 2 + 1 for some positive constants C β and C N 2 + 1. Furthermore, the large time behavior of solutions to the problem is also investigated. Specially speaking, when a is appropriately large, the corresponding solution of the system exponentially decays to ((a μ) 1 r − 1 , (a μ) 1 r − 1 , 0) if μ is large enough. This result improves or extends previous results of several authors. [ABSTRACT FROM AUTHOR]

Abstract This paper presents an immersed boundary method (IBM) to model the behaviour of solid–fluid interfaces in incompressible flows on non-conforming, unstructured meshes. Based on a smooth-interface direct forcing formulation, the proposed forcing method directly and implicitly uses the discretised momentum equations to evaluate the source terms required to enforce a no-slip condition on the boundary. The IBM is derived for a finite-volume framework with an implicit pressure-velocity coupling and a colocated variable arrangement, to enable application in complex domains. The novel framework is applied to flows with stationary and moving IBs on both Cartesian and tetrahedral meshes, and the results are compared to and validated with findings reported in the literature. The results demonstrate that the proposed formulation results in an accurate enforcement of the no-slip condition on the IB at every time-step, even for flows with strong transient behaviour and high velocity and pressure gradients. The results are similar to or better than those obtained with other methods that can only be applied on Cartesian meshes. The current method is shown to accurately preserve local continuity in the vicinity of the IB, ensuring local and global mass conservation in addition to satisfying the local no-slip condition. Moreover, the proposed forcing framework shows good results for unstructured meshes, with a similar accuracy as obtained on Cartesian meshes. Highlights • Development of Immersed Boundary Method for unstructured meshes with a co- located variable arrangement. • Derivation using the analytical form of the discretised momentum equations. • Suitable for fully-coupled pressure-velocity solver framework. • Consistent treatment of spreading and interpolation, leading to the exact enforcement of the no-slip condition. • Comprehensive analysis, comparison, and validation using representative test cases. [ABSTRACT FROM AUTHOR]

Abstract Conventionally, the dynamic modeling of H-type gantry stage(HGS) assumes that the stiffness of mid-beam is extremely high so that its deformation is ignorable. This assumption apparently simplifies the study, but unavoidably compromises the modeling precision as it is not consistent with the real working status especially at high frequency motion. Comparing with similar studies on moving mass problems, the beam vibration of HGS is excited not only by the moving mass and system acceleration but also by the non-synchronized motion of the two moving ends of mid-beam. Therefore, the objective of this paper is to propose an assumed mode method based on simplified mode shapes(AMM-SMS) by simultaneously considering the lateral flexibility and the motion of the ends of mid-beam. The proposed AMM-SMS adopts the mode shapes of a simplified system, which has the same geometry boundary condition(BC) with the investigated system, to discretize the nonlinear kinematic equations of the proposed system. This method improves the modeling precision with low complexity, and can be applied in different BCs of FHGS. The proposed method is verified by various comparisons with the finite element method. Based on the AMM-SMS, the first three natural frequencies and mode shapes are studied. The open-loop responses of the system are obtained to analyze the effects of different BCs, of the motion of end-effector and of the external exciting force on the vibratory behavior. [ABSTRACT FROM AUTHOR]

Abstract The suppression of expansion in thin clamped panels subjected to elevated thermal loading often results in buckling. However, a number of possible post-buckled equilibrium configurations typically exist, and which shape ensues depends on a number of factors including the role of symmetry, boundary conditions, aspect ratio, and the effect of small geometric imperfections associated with the initial shape. It is possible to force the panel to go between different buckled shapes, given a sufficiently large perturbation. Sometimes the panel will spontaneously jump, or snap, when the temperature is gradually increased or decreased (mode jumping). The extent to which these features occur when the thermal loading is applied locally is also investigated. This paper describes some interesting nonlinear (essentially buckling) behavior in thermally-loaded panels from a primarily experimental perspective, with an additional focus on non-uniform heating. The full force of stereo 3D digital image correlation and forward-looking infrared cameras are exploited to provide a relatively complete picture of this behavior. [ABSTRACT FROM AUTHOR]

In this paper, vibrational behavior of functionally graded piezoelectric material (FGPM) in a nano circular plate subjected to rotational and thermal loads is studied. Thermo-electro–elastic characteristics of FGPM nano circular plate are changed continuously along the thickness with the power-law model. This analysis is based on nonlocal strain gradient theory (NSGT), strain gradient theory (SGT) and nonlocal Eringen theory (NET) within the first order shear deformation theory (FSDT). Via Hamilton’s principle, the governing equations of FGPM nano circular plate obtained and solved using the differential quadrature method (DQM). It is seen that the vibrational behavior of FGPM nano circular plate is considerably affected by temperature rises, angular velocity, external voltage, FG power index, nonlocal parameter and material length scale for clamped and hinged boundary conditions. [ABSTRACT FROM AUTHOR]

Abstract This paper concerns with the problem of how to running an insurance company to maximize his total discounted expected dividends. In our model, the dividend rate is limited in [ 0 , M ] and the company is allowed to transfer any proportion of risk by reinsuring. So there are two strategies which we call dividend strategy and reinsurance strategy. The objective function and the corresponding optimal two strategies are the solution and the two free boundaries of the following Barenblatt parabolic equation v t − max 0 ≤ a ≤ 1 ⁡ (1 2 a 2 σ 2 v x x + a μ v x) + c v − max 0 ≤ l ≤ M ⁡ [ (1 − v x) l ] = 0 under certain boundary conditions on an angular domain Q T = { (x , t) | 0 < x < M t , 0 < t ≤ T }. The main effort is to analyze the properties of the solution and the free boundaries to show the optimal decision for the insurance company. [ABSTRACT FROM AUTHOR]

Abstract This paper is concerned with the uniqueness, existence, partial smoothing effect, comparison principle and long-time behavior of solutions to the initial-boundary value problem for a unidirectional diffusion equation. The unidirectional evolution often appears in Damage Mechanics due to the strong irreversibility of crack propagation or damage evolution. The existence of solutions is proved in an L 2 -framework by employing a backward Euler scheme and by introducing a new method of a priori estimates based on a reduction of discretized equations to variational inequalities of obstacle type and by developing a regularity theory for such obstacle problems. The novel discretization argument will be also applied to prove the comparison principle as well as to investigate the long-time behavior of solutions. [ABSTRACT FROM AUTHOR]

Abstract A multiscale Galerkin method (MGM) was proposed recently by the same authors in order to solve second-order boundary value problems of Fredholm integro-differential equation. Although, the numerical solution of MGM is always stable because of the multiscale bases properties, obligatory of considerable computational cost and huge memory for achieving great approximation accuracy, are the main draw backs. To overcome MGM problems, in this paper, a new multilevel augmentation method (MAM) in order to solve discrete linear system is proposed. Applying the special matrix splitting techniques, approximate solution is obtained by (1) solving a linear system only at an initial lower level; (2) compensating the error by directly computing the product of matrices and vectors at the higher level without any iterations. Theoretical and experimental results approve that MAM and MGM have similar and optimum convergence orders, though MAM is more efficient than MGM. [ABSTRACT FROM AUTHOR]

Abstract A special type of fluid–structure interaction (FSI) problems are problems with periodic boundary conditions like in turbomachinery. The steady state FSI response of these problems is usually calculated with similar techniques as used for transient FSI analyses. This means that, when the fluid and structure problem are not simultaneously solved with a monolithic approach, the problem is partitioned into a fluid and structural part and that each time step coupling iterations are performed to account for strong interactions between the two sub-domains. This paper shows that a time-partitioned FSI computation can be very inefficient to compute the steady state FSI response of periodic problems. A new approach is introduced in which coupling iterations are performed on periodic level instead of per time step. The convergence behaviour can be significantly improved by implementing existing partitioned solution methods as used for time step coupling (TSC) algorithms in the time periodic coupling (TPC) framework. The new algorithm has been evaluated by comparing the convergence behaviour to TSC algorithms. It is shown that the number of fluid–structure evaluations can be considerably reduced when a TPC algorithm is applied instead of a TSC. One of the most appealing advantages of the TPC approach is that the structural problem can be solved in the frequency domain resulting in a very efficient algorithm for computing steady state FSI responses. Highlights • A time periodic coupling (TPC) is introduced for computing the FSI response of periodic problems. • A quasi-Newton inverse least squares (QN-ILS) method is included in the TPC for stabilization. • The QN-ILS TPC method converges for any case, irrespective of the amount of damping and added mass. • Computation time can be significantly reduced with the QN-ILS TPC approach. [ABSTRACT FROM AUTHOR]

Highlights • Two-sided fractional diffusion equations are written in conservation form. • Mass-preserving, reflecting boundary conditions for these diffusion equations are a combination of fractional derivatives. • Stable explicit and implicit Euler schemes for two-sided fractional diffusion equations with any combination of absorbing and reflecting boundary conditions are presented. • Closed-form, steady-state solutions are derived. • Numerical experiments verify that the explicit and implicit Euler schemes converge to the analytical steady-state solution for large time. Abstract This paper develops appropriate boundary conditions for the two-sided fractional diffusion equation, where the usual second derivative in space is replaced by a weighted average of positive (left) and negative (right) fractional derivatives. Mass preserving, reflecting boundary conditions for two-sided fractional diffusion involve a balance of left and right fractional derivatives at the boundary. Stable, consistent explicit and implicit Euler methods are detailed, and steady state solutions are derived. Steady state solutions for two-sided fractional diffusion equations using both Riemann–Liouville and Caputo flux are computed. For Riemann–Liouville flux and reflecting boundary conditions, the steady-state solution is singular at one or both of the end-points. For Caputo flux and reflecting boundary conditions, the steady-state solution is a constant function. Numerical experiments illustrate the convergence of these numerical methods. Finally, the influence of the reflecting boundary on the steady-state behavior subject to both the Riemann–Liouville and Caputo fluxes is discussed. [ABSTRACT FROM AUTHOR]

Abstract Local artificial boundary conditions (ABCs) for the numerical simulation of waves have been successfully used for decades (most notably, the boundary conditions due to Engquist & Majda, Bayliss, Gunzburger & Turkel, and Higdon). The basic idea behind these boundary conditions is that they cancel several leading terms in an expansion of the solution. The larger the number of terms canceled, the higher the order of the boundary condition and, in turn, the smaller the reflection error due to truncation of the original unbounded domain by an artificial outer boundary. In practice, however, the use of local ABCs has been limited to low orders (first and second), because higher order boundary conditions involve higher order derivatives of the solution, which may harm well-posedness and cause numerical instabilities. They are also difficult to implement especially in finite elements. A prominent exception is the development of local high order ABCs based on auxiliary variables. In the current paper, we implement high order Bayliss–Turkel ABCs directly — with no auxiliary variables yet no discrete approximation of the constituent high order derivatives either. Instead, we represent the solution at the boundary as an expansion with respect to a continuous basis. For the spherical artificial boundary, the basis consists of eigenfunctions of the Beltrami operator (spherical harmonics), which enable replacing the high order derivatives in the ABCs with powers of the corresponding eigenvalues. The continuous representation at the boundary is coupled to higher order compact finite differences inside the domain by the method of difference potentials (MDP). It maintains high order accuracy even when the boundary is not aligned with the discretization grid. Highlights • Direct implementation of high order Bayliss–Gunzburger–Turkel (BGT) ABCs. • No auxiliary variables; no discretization of high order derivatives on the grid either. • Spectral representation of ABCs coupled with high order finite differences in the interior. • The method of difference potentials (MDP) renders the coupling most efficiently. • Numerical simulations fully corroborate the theoretical developments and expectations. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider an initial-boundary value problem of the p-Laplacian parabolic equation { u t (x , t) = div (| ∇ u (x , t) | p − 2 ∇ u (x , t)) + f (u (x , t)) , (x , t) ∈ Ω × (0 , + ∞) , u (x , t) = 0 , (x , t) ∈ ∂ Ω × [ 0 , + ∞) , u (x , 0) = u 0 ≥ 0 , x ∈ Ω ‾ , where p ≥ 2 and Ω is a bounded domain of R N (N ≥ 1) with smooth boundary ∂Ω. The main contribution of this work is to introduce a new condition (C p) α ∫ 0 u f (s) d s ≤ u f (u) + β u p + γ , u > 0 for some α , β , γ > 0 with 0 < β ≤ (α − p) λ 1 , p p , where λ 1 , p is the first eigenvalue of p-Laplacian Δ p , and we use the concavity method to obtain the blow-up solutions to the above equations. In fact, it will be seen that the condition (C p) improves the conditions ever known so far. [ABSTRACT FROM AUTHOR]

Abstract In this paper we investigate a mathematical model arising from volcanology describing surface deformation effects generated by a magma chamber embedded into Earth's interior and exerting on it a uniform hydrostatic pressure. The modeling assumptions translate mathematically into a Neumann boundary value problem for the classical Lamé system in a half-space with an embedded pressurized cavity. We establish well-posedness of the problem in suitable weighted Sobolev spaces and analyse the inverse problem of determining the pressurized cavity from partial measurements of the displacement field proving uniqueness and stability estimates. [ABSTRACT FROM AUTHOR]

Abstract This paper compares and contrasts higher-order, semi-implicit temporal integration strategies for the incompressible Navier–Stokes methods based on spectral deferred corrections applied to certain gauge or auxiliary variable formulations of the equations. Particular focus is placed on the imposition of boundary conditions in the semi-implicit formulation, the accurate treatment of the pressure term, and the smoothness of the numerical solution for different formulations. The main result presented here is the formulation and numerical validation of a single-step, semi-implicit method called spectral deferred pressure corrections , which can in theory obtain arbitrary formal order of accuracy in time. Numerical results demonstrate up to eighth-order accuracy, scenarios where optimal temporal accuracy is attained, and scenarios where order reduction is observed due to time-dependent boundary conditions. Highlights • High-order accurate temporal integration methods for the incompressible Navier–Stokes equations are discussed. • Comparison between primary variable, gauge, and auxiliary variable formulations. • Design of a single-step, spectral deferred pressure correction scheme with arbitrary formal order of accuracy. • Numerical tests demonstrating schemes of up to eighth-order accuracy in square and curved domains. • Discussion of order reduction owing to time-dependent boundary conditions and in gauge formulations. [ABSTRACT FROM AUTHOR]

Abstract Large computation time is widely considered to be the most important issue in scientific research especially in solving structural evolution problems. Recent developments in this domain have shown that the use of non-incremental schemes through Model Order Reduction led to important results in saving time. Yet, the question arises here how to obtain more time-saving. This paper examines an approach based on a collection of significant modes given by Proper Generalized Decomposition (PGD) solution for different time scales in order to save more computation time. The dictionary of the significant modes allows to construct an accurate solution for different characteristic times and different boundary problems compared to the full solution with a relative error rate less than 5% and with a large time saving of order 50 compared to Finite Element Method (FEM). The ability of the approach with respect to cycle time is discussed. Highlights • A new approach is proposed to decrease computational time for cyclic problems involving different time scales. • The approach is based on the use of pre-computed space–time modes which are further collected in a dictionary. • For a physical time belonging to the range of the dictionary, the solution can be accurately predicted. • For a given cycle time, the dictionary leads to an accurate if it contains the modes associated with this specific time. [ABSTRACT FROM AUTHOR]

Abstract A semi-analytical method is proposed for calculating the response of a linear cascade with spanwise mean flow subject to oblique gusts. It is developed based on the boundary value problems defined by Lloyd and Peake (AIAA paper 2008–2840). A gust strength parameter is introduced and the correct three-dimensional (3-D) response is obtained. The classic similarity rules are extended; the approach can be used to extend any 2-D methods to account for oblique gusts and 3-D mean flows. It is validated against analytical approximations for single-airfoil and cascade responses. The method is used to investigate the effect of gust angle α g on the unsteady lift and the sound field. It is found that as α g increases, the 2-D equivalent response varies slightly. However, the 3-D lift is amplified by factor 1 / cos α g , and the spanwise phase variation increases. Cascade effects are also studied. The inter-blade phase angle (IBPA) is important even for very low solidity. As the solidity increases, the chordwise distribution of lift is no longer leading-edge dominant. Cascade effects are small only when the cascade blade count is lower than a limit. A statistics analysis reveals that Mach number is the most important parameter for determining this blade count limit, and frequency is the least important. [ABSTRACT FROM AUTHOR]

Abstract The motion of moderately thick spray can be modeled by a coupled system of equations consisting of the incompressible Navier–Stokes equations and the Vlasov–Boltzmann equation. In this paper, we study the initial value problem for the Navier–Stokes–Vlasov–Boltzmann equations. The existence of global weak solutions is established by the weak convergence method. [ABSTRACT FROM AUTHOR]

Abstract We study the phenomenon of strengthening rate of polynomial approximation near boundary points of a convex domain. The main result of the paper presents this type of approximation near conic boundary points of these domains while Gopengauz's counter-example implies impossibility of such approximation near other points of the boundary. [ABSTRACT FROM AUTHOR]

Abstract The purpose of this paper is to investigate a parabolic problem with coupled reaction terms and free boundary, including the existence, uniqueness, regularity and long-time behavior of positive solutions. Firstly, by the contraction mapping theorem, we establish the (local) existence and uniqueness of positive solutions. Then, we prove the solution blows up in finite time with large initial data by comparison principle, and give more details for these large initial data by exhibiting an energy condition. The simultaneous blow-up result of the maximum of u and v is obtained, and the blow-up set of blow-up solutions is a compact subset of [ 0 , h 0 ]. Furthermore, there is a global and fast solution, which decays uniformly at an exponential rate if the initial datum is small, while there is a global and slow solution provided that the initial datum is suitably large. Finally, for initial data σ (φ (x) , ψ (x)) , we obtain a trichotomy conclusion by considering the size of parameter σ. [ABSTRACT FROM AUTHOR]

Highlights • We derive that Cahn–Hilliard equation possesses a local energy dissipation law (LEDL). • Based on the observation, three LEDL schemes for the CH equation are derived. • Three schemes are independent of boundary conditions. • Such schemes are proven to conserve the LEDL in any local region. • Our schemes hold the total energy stable laws and the mass laws. Abstract In this paper, we show that the Cahn–Hilliard equation possesses a local energy dissipation law, which is independent of boundary conditions and produces much more information of the original problem. To inherit the intrinsic property, we derive three novel local structure-preserving algorithms for the 2D Cahn–Hilliard equation by the concatenating method. In particular, when the nonlinear bulk potential f (ϕ) in the equation is chosen as the Ginzburg–Landau double-well potential, the method discussed by Zhang and Qiao (2012) [50] is a special case of our scheme II. Thanks to the Leibnitz rules and properties of operators, the three schemes are rigorously proven to conserve the discrete local energy dissipation law in any local time–space region. Under periodic boundary conditions, the schemes are proven to possess the discrete mass conservation and total energy dissipation laws. Numerical experiments are conducted to show the performance of the proposed schemes. [ABSTRACT FROM AUTHOR]

Abstract The discretization by the method of moments (MoM) of integral equations in the electromagnetic scattering analysis most often relies on divergence-conforming basis functions, such as the Rao–Wilton–Glisson (RWG) set, which preserve the normal continuity of the expanded currents across the edges arising from the discretization of the target boundary. Although for such schemes the boundary integrals become free from hypersingular kernel-contributions, which is numerically advantageous, their practical implementation in real-life scenarios becomes particularly cumbersome. Indeed, the application of the normal continuity condition on composite objects becomes elaborate and convoluted at junction-edges, where several regions intersect. Also, such edge-based schemes cannot even be applied to nonconformal meshes, where adjacent facets may not share single matching edges. In this paper, we present nonconforming schemes of discretization for the scattering analysis of complex objects based on the expansion of the boundary unknowns, electric or magnetic currents, with the facet-based monopolar-RWG set. We show with examples how these schemes exhibit great flexibility when handling composite piecewise homogeneous objects with junctions or targets modeled with nonconformal meshes. Furthermore, these schemes offer improved near- and far-field accuracy in the scattering analysis of electrically small single sharp-edged dielectric targets with moderate or high dielectric contrasts. Highlights • Development of facet-based implementations of the EFIE and the PMCHWT formulation. • Improved near- and far-field accuracies when compared with standard implementations. • Faster or similar decrease in error when compared with conventional implementations. • Versatile and flexible tools in the EM scattering analysis of composite objects. • Versatile and flexible tools in handling nonconformal meshes. [ABSTRACT FROM AUTHOR]

In this paper, we consider a domain Ω ε ⊂ R N , N ≥ 2 , with a very rough boundary depending on ε . For instance, if N = 3 Ω ε has the form of a brush with an ε -periodic distribution of thin cylindrical teeth with fixed height and a small diameter of order ε . In Ω ε we consider a nonlinear monotone problem with nonlinear Signorini boundary conditions, depending on ε , on the lateral boundary of the teeth. We study the asymptotic behavior of this problem, as ε vanishes, i.e. when the number of thin attached cylinders increases unboundedly, while their cross sections tend to zero. We identify the limit problem which is a nonstandard homogenized problem. Namely, in the region filled up by the thin cylinders the limit problem is given by a variational inequality coupled to an algebraic system. [ABSTRACT FROM AUTHOR]

This paper deals with the solvability of planar complex vector fields with homogeneous degeneracies. Hölder continuous solutions are obtained via a Cauchy type integral operator associated to the vector field. An associated boundary value problem of Riemann–Hilbert type is also considered. [ABSTRACT FROM AUTHOR]

This paper deals with an attraction–repulsion chemotaxis system { u t = ∇ ⋅ ( D ( u ) ∇ u ) − χ ∇ ⋅ ( u ∇ v ) + ξ ∇ ⋅ ( u ∇ w ) , x ∈ Ω , t > 0 , τ 1 v t = Δ v + α u − β v , x ∈ Ω , t > 0 , τ 2 w t = Δ w + γ u − δ w , x ∈ Ω , t > 0 under homogeneous Neumann boundary conditions in a smooth bounded domain Ω ⊂ R N ( N ≥ 2 ), where parameters τ i ( i = 1 , 2 ) , χ , ξ , α , β , γ and δ are positive, and diffusion coefficient D ( u ) ∈ C 2 ( 0 , + ∞ ) satisfies D ( u ) > 0 for u ≥ 0 , D ( u ) ≥ d u m − 1 with d > 0 and m ≥ 1 for all u > 0 . It is proved that the corresponding initial–boundary value problem possesses a unique global bounded classical solution for m > 2 − 2 N . In particular in the case τ 1 = τ 2 and χ α = ξ γ , the solution is globally bounded if m > 2 − 2 N − N + 2 N 2 − N + 2 . Therefore, due to the inhibition of repulsion to the attraction, the range of m > 2 − 2 N of boundedness is enlarged and the results of [21] is thus extended to the higher dimensional attraction–repulsion chemotaxis system with nonlinear diffusion. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we mainly investigate the critical points associated to solutions u of a quasilinear elliptic equation with nonhomogeneous Dirichlet boundary conditions in a connected domain Ω in R 2. Based on the fine analysis about the distribution of connected components of a super-level set { x ∈ Ω : u (x) > t } for any min ∂ Ω u (x) < t < max ∂ Ω u (x) , we obtain the geometric structure of interior critical points of u. Precisely, when Ω is simply connected, we develop a new method to prove Σ i = 1 k m i + 1 = N , where m 1 , ⋯ , m k are the respective multiplicities of interior critical points x 1 , ⋯ , x k of u and N is the number of global maximal points of u on ∂Ω. When Ω is an annular domain with the interior boundary γ I and the external boundary γ E , where u | γ I = H , u | γ E = ψ (x) and ψ (x) has N local (global) maximal points on γ E. For the case ψ (x) ≥ H or ψ (x) ≤ H or min γ E ψ (x) < H < max γ E ψ (x) , we show that Σ i = 1 k m i ≤ N (either Σ i = 1 k m i = N or Σ i = 1 k m i + 1 = N). [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider the zero viscosity limit of the anisotropic incompressible Navier–Stokes equations with no-slip boundary condition (see (1.1)) in R + 2. We prove that there exist T independently on ε such that the strong solutions of (1.1) convergence to the solution of (1.2) away from the boundary and to solution of (1.3) near the boundary in L ∞ ((0 , T) , L 2 ∩ L ∞ (R + 2)) when the vertical viscosity vanish, provided that the initial velocity is regular enough and we obtain the optimal convergence rate. [ABSTRACT FROM AUTHOR]

This paper is concerned with the evolution of the periodic boundary value problem and the mixed boundary value problem for a compressible mixture of binary fluids modeled by the Navier–Stokes–Cahn–Hilliard system in one dimensional space. The global existence and the large time behavior of the strong solutions for these two systems are studied. The solutions are proved to be asymptotically stable even for the large initial disturbance of the density and the large velocity data. We show that the average concentration difference for the two components of the initial state determines the long time behavior of the diffusive interface for the two-phase flow. [ABSTRACT FROM AUTHOR]

This paper presents a new approach, referred to here as the two-dimensional spectral-Tchebychev (2D-ST) technique, to predict the dynamics of thick plates having arbitrary geometries under different boundary conditions. The integral boundary value problem governing the dynamics of plate-like structures is obtained using the Mindlin plate theory and following an energy-based approach. To solve the boundary value problem numerically, a spectral-Tchebychev based solution technique is developed. To simplify the calculation of integral and derivative operations and thus to increase the numerical efficiency of the solution approach, a one-to-one coordinate mapping technique is used to map the arbitrary geometry onto an equivalent rectangular in-plane shape of the plate. The proposed solution technique is applied to various different plate problems to assess the accuracy and show the applicability of the technique. In each case, the convergence of the solution is analyzed, and the predicted (non-dimensional) natural frequencies are compared to those found in the literature or to those found using finite element modeling. It is shown that the calculated natural frequencies converge exponentially with increasing number of Tchebychev polynomials used and are in excellent agreement with those found in the literature and found form a finite elements solution. Therefore, it is concluded that the presented spectral-Tchebychev solution technique can accurately and efficiently capture the dynamics of thick plates having arbitrary geometries. Furthermore, the utility of the 2D-ST is demonstrated by comparing the results obtained using a three-dimensional solution approach. [ABSTRACT FROM AUTHOR]

We are concerned with the nonlinear problem u t = u x x + f ( u ) , where f is of combustion type, coupled with the Stefan-type free boundary h ( t ) . According to [4,5] , for some critical initial data, the transition solution u locally uniformly converges to θ , which is the ignition temperature of f , and the free boundary satisfies h ( t ) = C t + o ( 1 ) t for some positive constant C and all large time t . In this paper, making use of two different approaches, we establish more accurate upper and lower bound estimates on h ( t ) for the transition solution, which suggest that the nonlinearity f can essentially influence the propagation speed. [ABSTRACT FROM AUTHOR]

The spectral properties of non-self-adjoint extensions A [ B ] of a symmetric operator in a Hilbert space are studied with the help of ordinary and quasi boundary triples and the corresponding Weyl functions. These extensions are given in terms of abstract boundary conditions involving an (in general non-symmetric) boundary operator B . In the abstract part of this paper, sufficient conditions for sectoriality and m-sectoriality as well as sufficient conditions for A [ B ] to have a non-empty resolvent set are provided in terms of the parameter B and the Weyl function. Special attention is paid to Weyl functions that decay along the negative real line or inside some sector in the complex plane, and spectral enclosures for A [ B ] are proved in this situation. The abstract results are applied to elliptic differential operators with local and non-local Robin boundary conditions on unbounded domains, to Schrödinger operators with δ -potentials of complex strengths supported on unbounded hypersurfaces or infinitely many points on the real line, and to quantum graphs with non-self-adjoint vertex couplings. [ABSTRACT FROM AUTHOR]

In this paper, a convected frequency-domain approach for acoustic scattering prediction is suggested, for sources and scattering surfaces in a uniform constant flow. Frequency-domain moving-medium formulations are used for prediction of the incident acoustic pressure and acoustic pressure gradient. The latter is required for evaluation of the hardwall boundary condition in a moving medium and allows a convected definition of the equivalent sources. The scattering approach is validated by the analytical case of scattering of a pulsating monopole source by a rigid sphere. The applicability of the methodology to moving-medium problems is demonstrated for rotating and pulsating monopole point sources in a uniform flow, located near an infinite flat scattering surface. The suggested approach allows frequency-domain scattering predictions for sources in a uniform constant flow of any subsonic velocity, enabling direct inclusion of convection effects on incident and scattered acoustics. The hardwall boundary condition is thus evaluated directly in a moving medium. The need for a Lorentz transform is obviated, overcoming the complexity it introduces and the limitations it imposes. [ABSTRACT FROM AUTHOR]

The coupled plates are commonly seen in flight vehicle structures, which are always subjected to a severe combination of thermal, aerodynamic and mechanical loads during the cruise. In this paper, a unified solution is derived for the aero-thermo-elastic flutter problems of the coupled plate structure with general boundary conditions, in which the classical and elastic boundary conditions can be dealt with. The Mindlin plate theory in conjunction with the supersonic piston theory to consider the supersonic flow effect is adopted to formulate the theoretical model of the heated coupled plate structure subjected to supersonic flow. Each displacement component of the coupled plate is constructed as a two-dimensional Fourier series supplemented with auxiliary functions in order to satisfy the possible boundary conditions. The flutter analysis of a single flat plate, to which great efforts have been devoted, can be considered as a special case in the proposed model by setting the coupling angle of the coupled plate equal to zero. A considerable number of numerical cases are presented to show the accuracy and efficiency of the proposed method. The effects of the aerodynamic pressure, boundary condition, coupling spring and temperature change on the flutter characteristics of the coupled plate are also discussed in detail. The present formulation permits to analyze the flutter problems of the coupled plate having arbitrary coupling angles and boundary conditions. [ABSTRACT FROM AUTHOR]

This paper presents experiments and numerical simulations of a nonlinear clamped-clamped beam subjected to Harmonic excitations and epistemic uncertainties. These uncertainties are propagated in order to calculate the dynamic response of the nonlinear structure via a coupling between the Harmonic Balance Method (HBM) and a non-intrusive Polynomial Chaos Expansion (PCE). The system studied is a clamped-clamped steel beam. First of all, increasing and decreasing swept sine experiments are performed in order to show the hardening effect in the vicinity of the primary resonance, and to extract the experimental multi-Harmonic frequency response of the structure. Secondly, the Harmonic Balance Method (HBM) is used alongside a continuation process to simulate the deterministic response of the nonlinear clamped-clamped beam. Good correlations were observed with the experiments, on the condition of updating the model for each excitation level. Finally, the effects of the epistemic uncertainties on the variability of the nonlinear response are investigated using a non-intrusive Polynomial Chaos Expansion (PCE) alongside the Harmonic Balance Method (HBM). A new methodology based on a phase criterion was developed in order to allow the PCE analysis to be performed despite the presence of bifurcations in the nonlinear response. The efficiency and robustness of the proposed methodology is demonstrated by comparison with Monte Carlo simulations. Then, the stochastic numerical results are shown to envelope the experimental responses for each excitation level without the need for model updating, validating the nonlinear stochastic methodology as a whole. [ABSTRACT FROM AUTHOR]

We discuss a free boundary problem for a reaction–diffusion equation with Dirichlet boundary conditions on both fixed and free boundaries of a one-dimensional interval. The problem was proposed by Du and Lin (2010) to model the spreading of an invasive or new species by putting Neumann boundary condition on the fixed boundary. Asymptotic properties of spreading solutions for such problems have been investigated in detail by Du and Lou (2015) and Du, Matsuzawa and Zhou (2014). The authors (2011) studied a free boundary problem with Dirichlet boundary condition. In this paper we will derive sharp asymptotic properties of spreading solutions to the free boundary problem in the Dirichlet case under general conditions on f . It will be shown that the spreading speed is asymptotically constant and determined by a semi-wave problem and that the solution converges to a semi-wave near the spreading front as t → ∞ provided that the semi-wave problem has a unique solution. [ABSTRACT FROM AUTHOR]