Rectangular Computational Domain Research Articles

Transparent boundary condition and its effectively local approximation for the Schrödinger equation on a rectangular computational domain

The transparent boundary condition for the free Schrödinger equation on a rectangular computational domain requires implementation of an operator of the form ∂t−i△Γ where △Γ is the Laplace-Beltrami operator. It is known that this operator is nonlocal in time as well as space which poses a significant challenge in developing an efficient numerical method of solution. The computational complexity of the existing methods scale with the number of time-steps which can be attributed to the nonlocal nature of the boundary operator. In this work, we report an effectively local approximation for the boundary operator such that the resulting complexity remains independent of number of time-steps. At the heart of this algorithm is a Padé approximant based rational approximation of certain fractional operators that handles corners of the domain adequately. For the spatial discretization, we use a Legendre-Galerkin spectral method with a new boundary adapted basis which ensures that the resulting linear system is banded. A compatible boundary-lifting procedure is also presented which accommodates the segments as well as the corners on the boundary. The proposed novel scheme can be implemented within the framework of any one-step time marching schemes. In particular, we demonstrate these ideas for two one-step methods, namely, the backward-differentiation formula of order 1 (BDF1) and the trapezoidal rule (TR). For the sake of comparison, we also present a convolution quadrature based scheme conforming to the one-step methods which is computationally expensive but serves as a golden standard. Finally, several numerical tests are presented to demonstrate the effectiveness of our novel method as well as to verify the order of convergence empirically.

Supervised learning and meshless methods for two-dimensional fractional PDEs on irregular domains

Recently, several numerical methods have been developed for solving time-fractional differential equations not only on rectangular computational domains but also on convex and non-convex non-rectangular computational geometries. On the other hand, due to the existence of integrals in the definition of space-fractional operators, there are few numerical schemes for solving space-fractional differential equations on irregular regions. In this paper, we develop a novel numerical solution based on the machine learning technique and a generalized moving least squares approximation for two-dimensional fractional PDEs on irregular domains. The scheme is constructed on the monomials, and this is the strength of this technique. Moreover, it will be used to approximate the space derivatives on convex and non-convex non-rectangular computational domains. The numerical results are extended to solve the fractional Bloch–Torrey equation, fractional Gray–Scott equation, and fractional Fitzhugh–Nagumo equation.

Nonlinear vibrations of a dielectric elastomer cantilever combined-stepped-plate actuator

A dielectric elastomer (DE) cantilever combined-stepped-plate actuator is proposed and its nonlinear vibration enhancement behaviors are investigated. The irregular geometry of the actuator is transformed into a rectangular computational domain with the ideal of the iso-parametric element transformation; then the governing equations describing the nonlinear vibration of the actuator are formulated by the variational method. The nonlinear periodic responses are solved through the harmonic balance method and the arc-length continuation method. The results indicate that the single side (upper or lower) voltage excitation is more efficient in the actuating performance. The asymmetrical combined plate presents a better motion performance comparing with the symmetrical one. Not only can more drive modes be achieved (For instance, both symmetrical flapping mode and asymmetrical torsional mode can be excited by changing the voltage frequency), but also higher responses can be obtained. Additionally, isolated responses are found for some particular dimension parameters, which provides a potential design strategy to regulate the frequency domain of significant responses. This research concludes that, the asymmetrical combined plate subjected to an asymmetrical voltage loading would show a more efficient locomotion performance, which can provide an insight into the structural design and the excitation strategy for DE actuators.

Boundary-to-solution mapping for groundwater flows in a Toth basin

In this paper, the authors propose a new approach to solving the groundwater flow equation in the Toth basin of arbitrary top and bottom topographies using deep learning. Instead of using traditional numerical solvers, they use a DeepONet to produce the boundary-to-solution mapping. This mapping takes the geometry of the physical domain along with the boundary conditions as inputs to output the steady state solution of the groundwater flow equation. To implement the DeepONet, the authors approximate the top and bottom boundaries using truncated Fourier series or piecewise linear representations. They present two different implementations of the DeepONet: one where the Toth basin is embedded in a rectangular computational domain, and another where the Toth basin with arbitrary top and bottom boundaries is mapped into a rectangular computational domain via a nonlinear transformation. They implement the DeepONet with respect to the Dirichlet and Robin boundary condition at the top and the Neumann boundary condition at the impervious bottom boundary, respectively. Using this deep-learning enabled tool, the authors investigate the impact of surface topography on the flow pattern by both the top surface and the bottom impervious boundary with arbitrary geometries. They discover that the average slope of the top surface promotes long-distance transport, while the local curvature controls localized circulations. Additionally, they find that the slope of the bottom impervious boundary can seriously impact the long-distance transport of groundwater flows. Overall, this paper presents a new and innovative approach to solving the groundwater flow equation using deep learning, which allows for the investigation of the impact of surface topography on groundwater flow patterns.

Generalized finite difference method (GFDM) based analysis for subsurface flow problems in anisotropic formation

This paper applies the meshless generalized finite difference method (GFDM) to analyze the subsurface flow problem in anisotropic formations for the first time, and develops the treatment methods of anisotropic permeability tensor in continuous function and discrete distribution expressions respectively. In particular, in case of the common discrete distribution expression of anisotropic permeability tensor in reservoir simulation, this paper suggests combining general difference operators and harmonic average scheme to simply and efficiently obtain the meshless discrete scheme of the anisotropic flow equation. Three numerical examples of steady and transient flow in the rectangular computational domain, transient flow with complex boundary, and steady flow with strongly discontinuous anisotropic permeability are used to verify the high accuracy and good convergence of GFDM to handle anisotropic flow problems. Compared with the traditional mesh-based reservoir numerical simulation method that replies on laborious gridding to discretize the reservoir domain and complicated multi-point flux approximation (MPFA) to handle discrete distribution anisotropic permeability tensor, the meshless GFDM can be more practically applied to the anisotropic flow problems in the reservoir domain with complex geometry and complex boundary conditions. Besides, it is found that the radius of the influence domain has little impact on the accuracy of steady flow. For the transient flow, it generally holds that the larger the radius of the influence domain, the lower the calculation accuracy in case of Cartesian collocation. In sum, this work provides an efficient and simple meshless solver to handle anisotropic flow problems in porous media under GFDM framework, which reveals the great application potential of GFDM in reservoir numerical simulation.

Numerical study of multi-dimensional hyperbolic telegraph equations arising in nuclear material science via an efficient local meshless method

Abstract This research work is to study the numerical solution of three-dimensional second-order hyperbolic telegraph equations using an efficient local meshless method based on radial basis function (RBF). The model equations are used in nuclear material science and in the modeling of vibrations of structures. The explicit time integration technique is utilized to semi-discretize the model in the time direction whereas the space derivatives of the model are discretized by the proposed local meshless procedure based on multiquadric RBF. Numerical experiments are performed with the proposed numerical scheme for rectangular and non-rectangular computational domains. The proposed method solutions are converging quickly in comparison with the different existing numerical methods in the recent literature.

Solution of Multi-Term Time-Fractional PDE Models Arising in Mathematical Biology and Physics by Local Meshless Method

Fractional differential equations depict nature sufficiently in light of the symmetry properties which describe biological and physical processes. This article is concerned with the numerical treatment of three-term time fractional-order multi-dimensional diffusion equations by using an efficient local meshless method. The space derivative of the models is discretized by the proposed meshless procedure based on the multiquadric radial basis function though the time-fractional part is discretized by Liouville–Caputo fractional derivative. The numerical results are obtained for one-, two- and three-dimensional cases on rectangular and non-rectangular computational domains which verify the validity, efficiency and accuracy of the method.

TSC simulation of vertical position control by the Bang-bang method on EAST

Vertical instability is a potentially serious hazard for elongated plasma. Many methods have been proposed to control the plasma vertical position. The Bang-bang method, which is designed for the inside coil (IC) voltage mode based on optimal control theory and the RZIp rigid plasma response model, has been successfully used for plasma vertical position control in the EAST 2014 experimental campaign (Wang et al 2016 Fusion Eng. Des. 112 692–8). The Tokamak Simulation Code (TSC) is a numerical model of tokamak plasma and the associated control systems. It studies the evolution of the magnetic field in a rectangular computational domain using the Maxwell magnetohydrodynamic equations for plasma coupled with the boundary conditions to the circuit equations for the poloidal field coils. In this paper, the numerical model of the Bang-bang method has been implemented in the TSC code, which is used to simulate shot 52444. Simulated trajectories of the vertical position, IC voltage, and current fit the experimental data well. On the basis of benchmarked calculation, a numerical investigation of the Bang-bang controller has been performed using the TSC. This controller was able to work with 8 mm random noise and dz = 41.8 mm displacement. As a predictive tool, the TSC code is used to investigate the the maximum controllable vertical displacement value for different maximum current, and evaluate the ability to control the vertical position by the Bang-bang method.

Nonlinear vibration of a nanocomposite laminated piezoelectric trapezoidal actuator in subsonic airflow under combined electrical and forcing excitations

The nonlinear dynamic and vibration behaviors of a cantilevered carbon nanotube-reinforced composite trapezoidal plate with two surface-bonded piezoelectric layers as an actuator in micro air vehicles are considered in this article. The plate is reinforced by single-walled carbon nanotubes and is exposed to subsonic airflow under combined parametric and external excitations. The large deflection von Karman plate assumptions and the classical laminated plate theory are applied to derive the governing equations of the motion of the piezoelectric nanocomposite laminated trapezoidal plate by using Hamilton's principle. The geometry of the trapezoidal plate is mapped into a rectangular computational domain. The Galerkin's approach is used for transforming the nonlinear partial differential equations of motion into nonlinear two-degrees-of-freedom ordinary differential equations of cubic nonlinearities. The case of 1:3 internal resonance and primary resonance is considered, and the multiple scales method is employed. The aerodynamic pressure distribution formula is modeled by linear potential flow theory. The frequency and time history responses and phase portrait in free forced vibrations are obtained to analyze the nonlinear dynamic behavior of the plate. The effects of different parameters such as the plate geometry, volume fraction of carbon nanotubes, and different excitations on the nonlinear vibration of the thin laminated plate are also discussed. A complex softening nonlinearity with two peaks in the higher mode is observed in frequency response curves. The influence of electrical excitation with several amplitudes and frequencies on dynamic stability is investigated using time response curves.

Application of local meshless method for the solution of two term time fractional-order multi-dimensional PDE arising in heat and mass transfer

In this article, we presented an efficient local meshless method for the numerical treatment of two term time fractional-order multi-dimensional diffusion PDE. The demand of meshless techniques increment because of its meshless nature and simplicity of usage in higher dimensions. This technique approximates the solu?tion on set of uniform and scattered nodes. The space derivatives of the models are discretized by the proposed meshless procedure though the time fractional part is discretized by Liouville-Caputo fractional derivative. The numerical re?sults are obtained for 1-, 2- and 3-D cases on rectangular and non-rectangular computational domains which verify the validity, efficiency and accuracy of the method.

Application of local meshless method for the solution of two term time fractional-order multi-dimensional PDE arising in heat and mass transfer

In this article, we presented an efficient local meshless method for the numerical treatment of two term time fractional-order multi-dimensional diffusion PDE. The demand of meshless techniques increment because of its meshless nature and simplicity of usage in higher dimensions. This technique approximates the solu?tion on set of uniform and scattered nodes. The space derivatives of the models are discretized by the proposed meshless procedure though the time fractional part is discretized by Liouville-Caputo fractional derivative. The numerical re?sults are obtained for 1-, 2- and 3-D cases on rectangular and non-rectangular computational domains which verify the validity, efficiency and accuracy of the method.

Studying the fractional derivative for natural convection in slanted cavity containing porous media

This paper studies the numerical solution of the unsteady free convention in a slanted cavity under effect of the fractional derivative containing a porous medium. To simplify all the computations, the main equations are mapped from the irregular physical domain into a regular polygon in a shape of a rectangular computational domain utilizing nonlinear axis transformations. Using the finite differences method, the fractional partial differential equations are solved. The primary results are teased in the ordinary case at $$\alpha = 1, Ra = 1000$$ and $$\varphi = 0.0$$ and those are found in an excellent agreement with the previous results from the open literatures. The obtained numerical data are represented in terms of the isotherms and streamlines contours as well as the local and average Nusselt numbers at the heated wall. Wide ranges of the key-parameters are considered i.e. orders of the fractional derivatives $$\alpha$$ and $$\beta$$ are varied from 1 to 0.7, the Rayleigh number Ra is varied from $$10^{2}$$ to $$10^{4}$$ and the inclination angle takes the values $$\varphi = 0^{^\circ } , 15^{^\circ } , 30^{^\circ }$$ and $$45^{^\circ }$$ . The results revealed that the decrease in order of the fractional derivatives enhances the fluid activity while both of the local and average Nusselt numbers are reduced regardless of the Rayleigh number values.

Galerkin proper orthogonal decomposition-reduced order method (POD-ROM) for solving generalized Swift-Hohenberg equation

PurposeThe current paper aims to develop a reduced order discontinuous Galerkin method for solving the generalized Swift–Hohenberg equation with application in biological science and mechanical engineering. The generalized Swift–Hohenberg equation is a fourth-order PDE; thus, this paper uses the local discontinuous Galerkin (LDG) method for it.Design/methodology/approachAt first, the spatial direction has been discretized by the LDG technique, as this process results in a nonlinear system of equations based on the time variable. Thus, to achieve more accurate outcomes, this paper uses an exponential time differencing scheme for solving the obtained system of ordinary differential equations. Finally, to decrease the used CPU time, this study combines the proper orthogonal decomposition approach with the LDG method and obtains a reduced order LDG method. The circular and rectangular computational domains have been selected to solve the generalized Swift–Hohenberg equation. Furthermore, the energy stability for the semi-discrete LDG scheme has been discussed.FindingsThe results show that the new numerical procedure has not only suitable and acceptable accuracy but also less computational cost compared to the local DG without the proper orthogonal decomposition (POD) approach.Originality/valueThe local DG technique is an efficient numerical procedure for solving models in the fluid flow. The current paper combines the POD approach and the local LDG technique to solve the generalized Swift–Hohenberg equation with application in the fluid mechanics. In the new technique, the computational cost and the used CPU time of the local DG have been reduced.

Pulsatile flow and heat transfer of blood in an overlapping vibrating atherosclerotic artery: A numerical study

The paper is devoted to a numerical investigation of the pulsatile flow of blood through a porous overlapping constricted artery under the influence of an externally imposed magnetic field and vibration environment that is originated from the body force. Blood is considered as micropolar fluid. The heat transfer phenomenon arising out of viscous dissipation is also studied. The problem is solved numerically by developing a Crank–Nicolson finite difference scheme after transforming the original governing equations from the physical domain to a rectangular computational domain. The computational results for the velocity and temperature distributions, fluid acceleration, skin friction and Nusselt number are presented graphically for different values of the physical parameters. The study shows that the Nusselt number increases with rise in Prandtl number and Brinkman number both and that owing to the dissipation of energy caused by blood viscoelasticity and magnetic field effect, during pulsatile flow of blood, the heat transfer rate at the wall of the artery is enhanced.

High-order discretization of a gyrokinetic Vlasov model in edge plasma geometry

We describe a new spatial discretization of a continuum gyrokinetic Vlasov model in axisymmetric tokamak edge plasma geometries. The geometries are represented using a multiblock decomposition in which logically distinct blocks are smoothly mapped from rectangular computational domains and are aligned with magnetic flux surfaces to accommodate strong anisotropy induced by the magnetic field. We employ a fourth-order, finite-volume discretization in mapped coordinates to mitigate the computational expense associated with discretization on 4D phase space grids. Applied to a conservative formulation of the gyrokinetic system, a finite-volume approach expresses local conservation discretely in a natural manner involving the calculation of normal fluxes at cell faces. In the approach presented here, the normal fluxes are computed in terms of face-averaged velocity normals in such a way that (i) the divergence-free property of the gyrokinetic velocity is preserved discretely to machine precision, (ii) the configuration space normal velocities are independent of mapping metrics, and (iii) the configuration space normal velocities are computed from exact pointwise evaluation of magnetic field data except for one term. The algorithms described in this paper form the foundation of a continuum gyrokinetic edge code named COGENT, which is used here to perform a convergence study verifying the accuracy of the spatial discretization.

Iterative Rankine HOBEM analysis of hull-form effects in forward-speed diffraction problem

A time-domain numerical algorithm based on the higher-order boundary element method and the iterative time-marching scheme is proposed for seakeeping analysis. The ship waves generated by a hull advancing at a constant forward speed in incident waves and the resultant diffraction forces acting on the hull are computed to investigate the hull-form effects on the hydrodynamic forces. A rectangular computational domain travelling at ship's speed is considered. An artificial damping beach for satisfying the radiation condition is installed at the outer portion of the free surface except the downstream side. An iterative time-marching scheme is employed for updating both kinematic and dynamic free-surface boundary conditions for numerical accuracy and stability. The boundary integral equation is solved by distributing higher-order boundary elements over the wetted body surface and the free surface. The hull-form effects on the naval hydrodynamics are investigated by comparing three different Wigley models. Finally, the corresponding unsteady wave patterns and the wave profiles around the hulls are illustrated and discussed.

The adaptive mesh method based on HOC difference scheme for convection diffusion equations with boundary layers

In this paper, an adaptive mesh method is proposed to generate orthogonal nonuniform mesh for 2D rectangular computational domain. The algorithm adopts the dimension split idea which redistribute the mesh on each direction sequently. This approach efficiently decreases the difficultly of algorithm and enables the adapted mesh to be orthogonal. The adaptive mesh method can redistribute the mesh based on the solution without specifying the location of the boundary layer or the grid distribution function to generate the nonuniform mesh in advance. To verify the effectiveness and robustness of the algorithm, the presented method combined with the HOC difference scheme on nonuniform grid is used to solve the 1D and 2D convection diffusion problems with boundary layers. Besides, the multigrid method is used to accelerate the iterative convergence speed. The numerical solutions of adaptive mesh is more flexible than uniform or nonuniform grids methods if the location of the boundary layers is unknown in advance.

Exact Boundary Condition for Semi-discretized Schrödinger Equation and Heat Equation in a Rectangular Domain

A convolution type exact/transparent boundary condition is proposed for simulating a semi-discretized linear Schrodinger equation on a rectangular computational domain. We calculate the kernel functions for a single source problem, and subsequently those over the rectangular domain. Approximate kernel functions are pre-computed numerically from discrete convolutionary equations. With a Crank---Nicolson scheme for time integration, the resulting approximate boundary conditions effectively suppress boundary reflections, and resolve the corner effect. The proposed boundary treatment, with a parameter modified, applies readily to a semi-discretized heat equation.

Numerical investigation on detonation velocity in rotating detonation engine chamber

A two-dimensional investigation is conducted to clarify the velocity decrease mechanism of a detonation wave propagating within a rotating detonation engine (RDE) chamber, focusing on the effect of injection conditions of the fuel and oxidizer. Two numerical studies are conducted; the first imitates the RDE chamber with injection ports in a rectangular computational domain, and the second simplifies the flow field in the RDE chamber based on observations of the first simulation. The injection is parameterized by the geometrical jet and the gas conditions (i.e., premixed or non-premixed). In the first simulation of the RDE chamber, the total length of the injection nozzles constitutes 20% of the length of the chamber's lower end. The number of nozzles, N, ranges from 10 to 500 under premixed and 25 and 125 under the non-premixed condition. As the nozzle interval increases, more burned gas exists in front of the detonation wave. Under thenon-premixedcondition for N=25, the propagation velocity decreases to 84% of the CJ velocity, and 82.4% of the injected C2H4 is burned by the detonation. The CJ velocity is theoretically calculated, based on the amount of burned C2H4, to be 2283m/s, which is not close to the propagation velocity of the simulation, 1999m/s. In the simplified model in the second study, C2H4, O2, and burned gas are quiescently placed rather than injected, and the role of the upper wall is also examined. Significant velocity decrease, down to 86% of the CJ velocity, is observed when C2H4 and O2 are placed separately with a large width of burned gas between them. The amount ofincompletely combusted C2H4 is inversely proportional to the propagation velocity except in the case with an upper wall. Therefore, burned gas has a larger effect on propagation velocity in the non-premixed condition.

High-Order Finite Difference Schemes for the First Derivative in Von Mises Coordinates

Third- and fourth-order accurate finite difference schemes for the first derivative of the square of the speed are developed, for both uniform and non-uniform grids, and applied in the study of a two-dimensional viscous fluid flow through an irregular domain. The von Mises transformation is used to transform the governing equations, and map the irregular domain onto a rectangular computational domain. Vorticity on the solid boundary is expressed in terms of the first partial derivative of the square of the speed of the flow in the computational domain, and the schemes are used to calculate the vorticity at the computational boundary grid points using up to five computational domain grid points. In all schemes developed, we study the effect of coordinate clustering on the computed results.