Finite Domain Research Articles

Weighted Composition Operators between the Bloch Type Space and the Hardy Space on the Unit Ball of Complex Banach Spaces

Let X be a finite or infinite dimensional complex Banach space. We characterize the bounded weighted composition operators between the Bloch type space and the Hardy space on the unit ball of X, extending several known results for finite dimensional domains.

Profile of a two-dimensional vortex condensate beyond the universal limit.

It is well known that an inverse turbulent cascade in a finite (2π×2π) two-dimensional periodic domain leads to the emergence of a system-sized coherent vortex dipole. We report a numerical hyperviscous study of the spatial vorticity profile inside one of the vortices. The exciting force was shortly correlated in time, random in space, and had a correlation length l_{f}=2π/k_{f} with k_{f} ranging from 100 to 12.5. Previously, it was found that in the asymptotic limit of small-scale forcing, the vorticity exhibits the power-law behavior Ω(r)=(3ε/α)^{1/2}r^{-1}, where r is the distance to the vortex center, α is the bottom friction coefficient, and ε is the inverse energy flux. Now we show that for a spatially homogeneous forcing with finite k_{f} the vorticity profile becomes steeper, with the difference increasing with the pumping scale but decreasing with the Reynolds number at the forcing scale. Qualitatively, this behavior is related to a decrease in the effective pumping of the coherent vortex with distance from its center. To support this statement, we perform an additional simulation with spatially localized forcing, in which the effective pumping of the coherent vortex, on the contrary, increases with r, and show that in this case the vorticity profile can be flatter than the asymptotic limit.

Three-dimensional anisotropy modelling and simulation of gas hydrate borehole-to-surface responses

Electromagnetic (EM) methods have provided an alternative means of detecting and estimating gas hydrate, especially in places where hydrates have been established to exist but the gas hydrate shows no bottom-simulating reflection. However, most EM numerical studies being done focused on surface/seafloor Controlled Source Electromagnetic (CSEM) studies with little consideration for borehole EM studies. The assessment and interpretation methods of data from EM surveys require fast and accurate forward modelling algorithms integrated with the complexity of the earth's geological formation. This study develops 2D and 3D anisotropy forward modelling codes using a high-order finite difference frequency domain (FDFD) method for borehole surface EM studies. The newly developed numerical codes can model the effects of anisotropy on borehole-surface earth responses using gas hydrate models. The simulation results revealed that the newly developed numerical codes are highly efficient and suitable for simulating borehole surface CSEM data, especially for marine gas hydrate exploration and other complex earth models. An increase in the coefficient of anisotropy resulted in an increase in the magnitude of the electric field at the surface for the x and z components. Changes in layer resistivity affect the EM field diffusion, recorded Borehole Surface Controlled Source Electromagnetic (BSCSEM) responses, and the overall data interpretation. Also, amplitude of the 3D simulation response increases in the anisotropic faulted earth systems. Lastly, the results imply that Borehole Surface Electromagnetic (BSEM) simulation results can assist with the location and extent of gas hydrates in subsurface.

RBFNN for Calculating the Stationary Response of SDOF Nonlinear Systems Excited by Poisson White Noise

Random perturbations in nature described by non-Gaussian excitation models are far more widely applied and development prospects than that of Gaussian excitation models in practice. However, the stochastic dynamics research of non-Gaussian excitation is still not very mature. In this work, radial-basis-function-neural-network (RBFNN) is applied for calculating the stationary response of single-degree-of-freedom (SDOF) nonlinear system excited by Poisson white noise. Specifically, the trial probability-density-function (PDF) solution of reduced generalized-Fokker–Plank–Kolmogorov (GFPK) equation is constructed by a suitable number of Gaussian basis functions (GBFs) with a fixed set of means and standard deviations. Subsequently, an approximate squared error of the GFPK equation in a finite domain is considered. Together with the normalization condition, the approximate squared error can be minimized by establishing a Lagrangian function, and then the optimal weight coefficients associated with the approximate PDF solution are solved from a system of linear algebraic equations. For demonstrating the effectiveness of the proposed procedure, two specific examples are presented. The corresponding reduced GFPK equation is truncated with higher order for the strong non-Gaussian case. The precision of the analytical solution is verified against the Monte Carlo simulation (MCS) data. In addition, all the results indicate that RBFNN shows fairly high efficiency under the premise of ensuring high precision in the whole computational procedure.

Ideal Membership Problem over 3-Element CSPs with Dual Discriminator Polymorphism

In this paper we examine polynomial ideals that are the vanishing ideals of solution sets of combinatorial problems encoded by constraint satisfaction problems over a finite language. We consider a 3-element domain and the dual discriminator polymorphism (constraints under this polymorphism are a generalization of the 2-satisfiability problem). Assuming the graded lexicographic ordering of monomials, we show that the reduced Gröbner basis of ideals whose varieties are closed under this polymorphism can be computed in polynomial time. This proves polynomial time solvability of the ideal membership problem (IMP) with restrictions on degree $d=O(1)$, which we call IMP$_d$, for these constrained problems. It is a first step toward the challenging long-term goal of identifying when IMP$_d$ is polynomial time solvable for a finite domain.

Effects of Gravity on Evaporation for Soil‐Limited Conditions

AbstractThis manuscript systematically investigates the effects of gravity during soil‐limited (Stage 2) evaporation from bare soil. Analytical and numerical solutions to the one‐dimensional Richards soil moisture flow equation are developed based on simplified hydraulic functions that depend exponentially on soil water matric potential. Semi‐infinite domains with either freely draining or hydrostatic bottom boundary conditions and finite domains with no‐flow bottom boundary conditions are considered. Cases with gravity are directly compared to those without. The simplified solutions are verified by comparison to numerical solutions with fully realistic soil hydraulic functions for two soils at the extreme ends of the range of possible soil textures. Cumulative evaporation versus time and soil volumetric water content versus depth at selected times are determined for each case from the simplified solutions. These results are reported in terms of non‐dimensional variables that are universal, spanning all soil textures and other conditions. For the semi‐infinite domain cases cumulative evaporation with gravity is reduced by several orders of magnitude for very coarse soils at times of practical interest while gravity has a negligible effect for the finest textures at those times. For the finite domain cases gravity is less important because cumulative evaporation is ultimately limited by the fixed amount of water in the profile.

On the Coupling of the Curved Virtual Element Method with the One-Equation Boundary Element Method for 2D Exterior Helmholtz Problems

We consider the Helmholtz equation defined in unbounded domains, external to 2D bounded ones, endowed with a Dirichlet condition on the boundary and the Sommerfeld radiation condition at infinity. To solve it, we reduce the infinite region, in which the solution is defined, to a bounded computational one, delimited by a curved smooth artificial boundary and we impose on this latter a non reflecting condition of boundary integral type. Then, we apply the curved virtual element method in the finite computational domain, combined with the one-equation boundary element method on the artificial boundary. We present the theoretical analysis of the proposed approach and we provide an optimal convergence error estimate in the energy norm. The numerical tests confirm the theoretical results and show the effectiveness of the new proposed approach.

Optimal searcher distribution for parallel random target searches.

We consider a problem of finding a target located in a finite d-dimensional domain, using N independent random walkers, when partial information about the target location is given as a probability distribution. When N is large, the first-passage time sensitively depends on the initial searcher distribution, which invokes the question of the optimal searcher distribution that minimizes the first-passage time. Here, we analytically derive the equationfor the optimal distribution and explore its limiting expressions. If the target volume can be ignored, the optimal distribution is proportional to the target distribution to the power of one third. If we consider a target of a finite volume and the probability of the initial overlapping of searchers with the target cannot be ignored in the large N limit, the optimal distribution has a weak dependence on the target distribution, with its variation being proportional to the logarithm of the target distribution. Using Langevin dynamics simulations, we numerically demonstrate our predictions in one and two dimensions.

Effects of geometric parameters on photoemission of AlGaN nanowire array photocathode

In recent years, with the continuous development of solar blind ultraviolet photodetectors, III-V compounds are widely used as semiconductor materials. The nanowire array structure has excellent ‘light trapping effect’. However, if the distance of nanowire is too close, the secondary absorption problem of the nanowire will occur. If the distance of nanowire is too far, the number of photocarriers generated in the nanowire array per unit volume will be reduced. Meanwhile, the absorption capacity of the nanowire structure with different shapes is different. Based on this background, we studied the influence of the period and geometry of AlGaN nanowires with different Al components on the optical response properties. The finite time domain difference (FDTD) method was used to compare the quantum efficiency and collection efficiency of AlGaN nanowires with different Al components, we found that the quantum efficiency of the hexagonal nanowire array with Al component of 0.42 is as high as 45%, which is the highest in our structure. At the same time, its cutoff wavelength is at 280 nm, which has excellent solar blindness. Therefore, the study in this paper can provide some theoretical reference for the experiment and preparation of AlGaN photocathode.

An investigation of space distributed-order models for simulating anomalous transport in a binary medium

Recent studies highlight that diffusion processes in highly heterogeneous, fractal-like media can exhibit anomalous transport phenomena, which motivates us to consider the use of generalised transport models based on fractional operators. In this work, we harness the properties of the distributed-order space fractional potential to provide a new perspective on dealing with boundary conditions for nonlocal operators on finite domains. Firstly, we consider a homogeneous space distributed-order model with Beta distribution weight. An a priori estimate based on the L2 norm is presented. Secondly, utilising finite Fourier and Laplace transform techniques, the analytical solution to the model is derived in terms of Kummer’s confluent hypergeometric function. Moreover, the finite volume method combined with Jacobi-Gauss quadrature is applied to derive the numerical solution, which demonstrates high accuracy even when the weight function shows near singularity. Finally, a one-dimensional two-layered problem involving the use of the fractional Laplacian operator and space distributed-order operator is developed and the correct form of boundary conditions to impose is analysed. Utilising the idea of ‘geometric reconstruction’, we introduce a ‘transition layer’ whereby the fractional operator index varies from fractional order to integer order across a fine layer at the boundary of the domain when transitioning from the complex internal structure to the external conditions exposed to the medium. An important observation is that for a fractional dominated case, the diffusion behaviour in the main layer is similar to fractional diffusion, while near the boundary the behaviour transitions to the case of classical diffusion.

A coarse-grained concurrent multiscale method for simulating brittle fracture

This paper proposed a concurrent multiscale method for modeling crack propagation in brittle solids. The method couples the continuum finite element domain with a recently-developed coarse-grained atomistic potential, which has been developed to simulate crack nucleation and propagation within the molecular dynamics domain. The main advantage of this potential is to suppress dislocation nucleation from the crack tip which significantly eases the coupling procedure. In this multiscale framework, complexities like large deformation, crack nucleation and propagation are simulated in the atomistic domain while the rest such as elastic wave propagation and boundary conditions are modeled in the continuum domain. As a result, the atomistic area will be limited around the cracking zone which increases the computational efficiency. Two examples of edge crack propagation and sliding surfaces in contact are presented and compared with the fully atomistic models to verify and check the efficiency of the proposed multiscale method. The comparison shows that the proposed method can accurately capture the crack formation while the computation cost reduces significantly.

Fluid-gel coexistence in lipid membranes under differential stress

A widely conserved property of many biological lipid bilayers is their asymmetry. In addition to having distinct compositions on its two sides, a membrane can also exhibit different tensions in its two leaflets, a state known as differential stress. Here, we examine how this stress can influence the phase behavior of the constituent lipid monolayers of a single-component membrane. For temperatures sufficiently close to, but still above, the main transition, molecular dynamics simulations show the emergence of finite gel domains within the compressed leaflet. We describe the thermodynamics of this phenomenon by adding two empirical single-leaflet free energies for the fluid-gel transition, each evaluated at its respective asymmetry-dependent lipid density. Finite size effects arising in simulation are included in the theory through a geometry-dependent interfacial term. Our model reproduces the phase coexistence observed in simulation. It could therefore be used to connect the “hidden variable” of differential stress to experimentally observable properties of the main phase transition. These ideas could be generalized to any first-order bilayer phase transition in the presence of asymmetry, including liquid-ordered/liquid-disordered phase separation.

Effect of choices of boundary conditions on the numerical efficiency of direct solutions of finite difference frequency domain systems with perfectly matched layers.

Direct solvers are a common method for solving finite difference frequency domain (FDFD) systems that arise in numerical solutions of Maxwell's equations. In a direct solver, one factorizes the system matrix. Since the system matrix is typically very sparse, the fill-in of these factors is the single most important computational consideration in terms of time complexity and memory requirements. As a result, it is of great interest to determine ways in which this fill-in can be systematically reduced. In this paper, we show that in the context of commonly used perfectly matched boundary layer methods, the choice of boundary condition behind the perfectly matched boundary layer can be exploited to reduce fill-in incurred during the factorization, leading to significant gains of up to 40% in the efficiency of the factorization procedure. We illustrate our findings by solving linear systems and eigenvalue problems associated with the FDFD method.

Finite frequency fault estimation and fault-tolerant control for dynamics of high-speed train based on descriptor systems

In this paper, novel fault estimation and fault-tolerant control methods are proposed for dynamics of high-speed train based on descriptor systems with uncertainties in finite frequency domain. Dynamics of high-speed train is established based on multi-particle model considering that basic resistance is seen as the coefficient of state variables, and additive resistance and the operating noise are seen as multi-source disturbance. Concurrent actuator, sensor faults, and wind gust are considered simultaneously; wind gust is modeled as a disturbance generated by the exogenous system, and an uncertain descriptor system with actuator fault and the exogenous disturbance is established by seeing the sensor fault of high-speed train as the state variables. A robust disturbance-observer-based fault estimation method is proposed to decouple the non-linearity of the descriptor system, so that the combining estimation of the fault and wind gust is implemented. This observer has an unknown input structure, and its gain matrices are formulated as linear matrix inequalities. The observer not only guarantees the augmented state estimation error is asymptotic stable but also the actuator fault estimation and wind gust estimation errors are robust to the multi-source disturbance and the uncertainties. Based on the estimation results, the fault-tolerant controller associated with the state estimation, faults estimation, and wind gust estimation results is proposed to implement a stable close-loop fault-tolerant control for dynamics of high-speed train. Simulation examples are given to illustrate the effectiveness of this method.

Thermally nonlinear thermoelasticity of a one-dimensional finite domain based on the finite strain concept

This research examines the application of finite strain theory to the generalized coupled thermoelasticity of isotropic 1D structures. The energy equation is derived based on the Lord-Shulman theorem to adapt to the temperature behavior of the structure due to rapid heating. Furthermore, the thermally nonlinear concept is utilized because the variation between the structure temperature and the reference temperature compared to the reference temperature cannot be overlooked. The nonlinear theory of finite strain is employed in the equation of motion, which is ideal for large deformations and rotations in the configuration. The second Piola-Kirchhoff stress and the Green-Lagrangian strains are applied in the formulation. Two nonlinear coupled partial differential equations of energy and dynamic are displayed in nondimensional version for better interpretation. Next, these equations are turned into discrete ordinary differential time-dependent equations through the generalized differential quadrature method. Newmark time marching integration methods and Picard iterative algorithm solve the resulting nonlinear equations. After validating the results with available papers, the effect of various nonlinearities, boundary conditions, and other parameters on the temporal evolution and wave propagation of deformation, temperature, and stress is investigated. It is demonstrated that the effect of nonlinear terms remarkably depends on the material type. The authors also show that utilizing the nonlinear formulation can change the wavefront speed of displacement, temperature, and stress in the domain.

A study of distributed‐order time fractional diffusion models with continuous distribution weight functions

AbstractThe distributed‐order time fractional diffusion model with Dirichlet nonhomogeneous boundary conditions on a finite domain is considered. Four choices of continuous distribution weight functions with mean μ and standard deviation σ are investigated to study their impact on both the short‐time and long‐time solution behavior. An implicit numerical method implemented on a graded mesh is proposed to solve the model and the stability and convergence analysis are presented. Semi‐analytic solutions are also derived for these distributions to assess the accuracy of the scheme. Numerical results highlight that the four continuous distribution weight functions produce a short‐time solution behavior that is consistent with those solutions from the classical time fractional partial differential equation with fractional order γ* = μ. There are however long‐time differences in the solution behavior that become more distinguishable as σ increases. In particular, we find a smaller value of σ produces more diffuse profiles and the diffusion rate slows as σ increases. Furthermore, the asymptotic behavior of the solution may be influenced by the time‐fractional orders ranging between the smallest nonzero weight order and mean μ for the continuous uniform and raised cosine distribution weight functions, respectively. Similar findings are also observed for the truncated normal and beta distributions.

Drainage of power-law fluids from fractured or porous finite domains

We develop a sharp-interface model that captures the coupled effect of spatial heterogeneity and fluid rheology on one-dimensional Newtonian and non-Newtonian buoyancy-driven flow spreading in fractured and porous media over a horizontal impermeable bed. We study the flow in three different geometries: (i) a constant uniform aperture, (ii) an aperture variable along the vertical axis, i.e. perpendicular to the direction of propagation and (iii) an aperture variable along the horizontal axis, i.e. parallel to the direction of propagation. The non-Newtonian rheology is described by the power-law equation of rheological index n and the aperture variation in both directions by a positive number r . The self-similar solutions of the flow obtained at late times allow the transformation of the nonlinear PDEs governing the spreading into nonlinear ODEs. The current shape is affected by the interplay between the rheological index and the spatial variability of the aperture. The residual liquid mass that remains in the fracture at any given time is computed from the current profiles, obtaining a negative power-law behavior in the time of exponent dependent on n and r . In addition, sensitivity analysis is performed to highlight the impact of the model parameters on the current profile and residual mass. The dimensionless analysis outcomes are compared to two real examples of flow within a uniform and a wedge-shaped aperture along the flow direction. The numerical results of the examples confirm that the proposed model can successfully capture the propagation of the gravity current, its profile, and drainage flow rate. • Drainage of one-dimensional non-Newtonian gravity-driven currents from an edge of a fractured finite domain is investigated. • Late-time similarity solutions are derived for different geometrical variations of fracture aperture. • The analogy between Hele-Shaw non-Newtonian flow in fractures and porous media flow is discussed. • Two realistic case studies illustrating the applicability of our results are extensively discussed.

Finitely star regular domains

Let R be an integral domain, S t a r ( R ) the set of all star operations on R and S t a r F C ( R ) the set of all star operations of finite type on R . Then R is said to be star regular if | S t a r ( T ) | ≤ | S t a r ( R ) | for every overring T of R . In this paper we introduce the notion of finitely star regular domain as an integral domain R such that | S t a r F C ( T ) | ≤ | S t a r F C ( R ) | for each overring T of R . First, we show that the notions of star regular and finitely star regular domains are completely different and do not imply each other. Next, we extend/generalize well-known results on star regularity in Noetherian and Prüfer contexts to finitely star regularity. Also we handle the finite star regular domains issued from classical pullback constructions to construct finitely star regular domains that are not star regular.

H − / L ∞ UIO‐based actuator and sensor fault diagnosis design in the finite frequency domain for discrete‐time Lipschitz nonlinear systems

This study investigates the problem of fault diagnosis observer design in the finite frequency domain for discrete-time Lipschitz nonlinear systems simultaneously subjected to actuator and sensor faults and unknown input disturbances. An augmented descriptor system is developed by constructing an augmented state consisting of system states and sensor faults. A novel H−/L∞ unknown input observer (UIO) is proposed in the finite frequency domain using the generalized Kalman–Yakubovich–Popov lemma to render the residual sensitive to actuator faults and robust against disturbances, which can achieve a better fault diagnosis result compared with existing observers in the full frequency domain. In the fault diagnosis scheme, a time-varying threshold is presented based on the L∞ performance index to detect actuator faults, and the sensor fault estimation is simultaneously achieved through an asymptotical estimation of the augmented state. Design conditions for the proposed H−/L∞ UIO are derived and converted into linear matrix inequalities to solve its design matrices together. The effectiveness of the developed H−/L∞ UIO and the fault diagnosis scheme are validated via simulations of an example of a single-link flexible joint robot.

Weighted Model Counting in FO2 with Cardinality Constraints and Counting Quantifiers: A Closed Form Formula

Weighted First-Order Model Counting (WFOMC) computes the weighted sum of the models of a first-order logic theory on a given finite domain. First-Order Logic theories that admit polynomial-time WFOMC w.r.t domain cardinality are called domain liftable. We introduce the concept of lifted interpretations as a tool for formulating closed forms for WFOMC. Using lifted interpretations, we reconstruct the closed-form formula for polynomial-time FOMC in the universally quantified fragment of FO2, earlier proposed by Beame et al. We then expand this closed-form to incorporate cardinality constraints, existential quantifiers, and counting quantifiers (a.k.a C2) without losing domain-liftability. Finally, we show that the obtained closed-form motivates a natural definition of a family of weight functions strictly larger than symmetric weight functions.