Coupled Boundary Research Articles

Existence and uniqueness theorems for nonlinear coupled boundary value problem of the A B C fractional differential equation

Existence and uniqueness theorems for nonlinear coupled boundary value problem of the A B C fractional differential equation

Mechanism of Coupled Upper Troposphere and Boundary Layer Induces Two Types of Short‐Term Heavy Rainfall Along the Eastern Foothills of the Taihang Mountains

AbstractUsing 11 years of hourly merged rainfall records and ERA reanalysis data, this paper reveals two major circulation modes leading to two types of short‐term heavy rainfall (STHR) along the Taihang Mountains' eastern foothills, and further explains their mechanisms. One circulation mode has a distinct warm anomaly at 300 hPa covering most areas of North China, together with the boundary‐layer westerly anomaly occurring in North China and its southern region (UTWA‐BLWA). UTWA‐BLWA effectively contributes to the reinforcement of the upper‐level divergence and the low‐level moisture convergence by promoting the strengthening of upper anticyclonic and low‐level southwesterly anomalies. The combined effects of low‐level jet (LLJ) and topographic uplift form the central‐northern STHR pattern. The other circulation structure has a 300‐hPa warm anomaly located to the southeastern Russia and a prominent 300‐hPa cold anomaly covering the south, together with the boundary‐layer easterly anomaly occurring over the whole region of eastern China (UTCA‐BLEA). The southern STHR pattern is attributed to the exit of the boundary‐layer jet (BLJ) over lower elevations due to moist transport and dynamic uplift associated with the easterly anomaly. The results indicate that the locations of STHR are related to the direction, intensity, and height of the LLJ. The findings highlight that the upper‐tropospheric temperature anomalies (UTTA) and boundary‐layer easterly flow jointly modulate heavy rainfall. Analysis of the coupled upper troposphere and boundary layer could help understand and forecast heavy rainfall.

Coupled nodal integral-immersed boundary method (NI-IBM) for simulating convection-diffusion physics

The Nodal Integral Method (NIM), which is a coarse grid numerical method, works quite efficiently on orthogonal grids. However, there are many challenges to the implementation of this method in the complex shaped domains due to its intricate discretization process. In contrast, the Immersed Boundary Method (IBM) offers the advantage of implementing boundary conditions in an arbitrarily shaped domain using rectilinear, non-conformal body grids. This work presents the NIM, coupled with the IBM methodology, to solve convection-diffusion physics in complex shaped domains. IBM is utilized to enforce the boundary effect on the non-body-conformal Cartesian grid, while the NIM serves as a numerical technique to discretize the governing equations. The study describes the detailed implementation of coupled NI-IBM (Nodal Integral-Immersed Boundary Method) for convection-diffusion physics. The validation and verification of the NI-IBM methodology is done through the comparison of the numerical results with analytical solutions of the diffusion and convection-diffusion test cases. Results demonstrate that the scheme maintains its accuracy in the coarse grid structure as well as second-order accurate. Furthermore, the numerical results produced by NI-IBM are at par or more accurate with respect to the NIM based schemes for arbitrarily shaped domains and the FV-IBM (Finite Volume-Immersed Boundary Method) method for the convection-diffusion physics. The results also indicate that even with coarse grids and higher convection, the NI-IBM method preserves its nature to produce reasonably accurate numerical results.

Non-similar solution of Casson fluid flow over a curved stretching surface with viscous dissipation; Artificial neural network analysis

PurposeThe main focus is to provide a non-similar solution for the magnetohydrodynamic (MHD) flow of Casson fluid over a curved stretching surface through the novel technique of the artificial intelligence (AI)-based Lavenberg–Marquardt scheme of an artificial neural network (ANN). The effects of joule heating, viscous dissipation and non-linear thermal radiation are discussed in relation to the thermal behavior of Casson fluid.Design/methodology/approachThe non-linear coupled boundary layer equations are transformed into a non-linear dimensionless Partial Differential Equation (PDE) by using a non-similar transformation. The local non-similar technique is utilized to truncate the non-similar dimensionless system up to 2nd order, which is treated as coupled ordinary differential equations (ODEs). The coupled system of ODEs is solved numerically via bvp4c. The data sets are constructed numerically and then implemented by the ANN.FindingsThe results indicate that the non-linear radiation parameter increases the fluid temperature. The Casson parameter reduces the fluid velocity as well as the temperature. The mean squared error (MSE), regression plot, error histogram, error analysis of skin friction, and local Nusselt number are presented. Furthermore, the regression values of skin friction and local Nusselt number are obtained as 0.99993 and 0.99997, respectively. The ANN predicted values of skin friction and the local Nusselt number show stability and convergence with high accuracy.Originality/valueAI-based ANNs have not been applied to non-similar solutions of curved stretching surfaces with Casson fluid model, with viscous dissipation. Moreover, the authors of this study employed Levenberg–Marquardt supervised learning to investigate the non-similar solution of the MHD Casson fluid model over a curved stretching surface with non-linear thermal radiation and joule heating. The governing boundary layer equations are transformed into a non-linear, dimensionless PDE by using a non-similar transformation. The local non-similar technique is utilized to truncate the non-similar dimensionless system up to 2nd order, which is treated as coupled ODEs. The coupled system of ODEs is solved numerically via bvp4c. The data sets are constructed numerically and then implemented by the ANN.

Mixed Galerkin- Implicit Differences Methods for Solving Coupled Parabolic Boundary Value Problems with Variable Coefficients

Mixed Galerkin- Implicit Differences Methods for Solving Coupled Parabolic Boundary Value Problems with Variable Coefficients

On the Convergence of an Approximation Scheme of Fractional-Step Type, Associated to a Nonlinear Second-Order System with Coupled In-Homogeneous Dynamic Boundary Conditions

The paper concerns a nonlinear second-order system of coupled PDEs, having the principal part in divergence form and subject to in-homogeneous dynamic boundary conditions, for both θ(t,x) and φ(t,x). Two main topics are addressed here, as follows. First, under a certain hypothesis on the input data, f1, f2, w1, w2, α, ξ, θ0, α0, φ0, and ξ0, we prove the well-posedness of a solution θ,α,φ,ξ, which is θ(t,x),α(t,x)∈Wp1,2(Q)×Wp1,2(Σ), φ(t,x),ξ(t,x)∈Wν1,2(Q)×Wp1,2(Σ), ν=min{q,μ}. According to the new formulation of the problem, we extend the previous results, allowing the new mathematical model to be even more complete to describe the diversity of physical phenomena to which it can be applied: interface problems, image analysis, epidemics, etc. The main goal of the present paper is to develop an iterative scheme of fractional-step type in order to approximate the unique solution to the nonlinear second-order system. The convergence result is established for the new numerical method, and on the basis of this approach, a conceptual algorithm, alg-frac_sec-ord_u+varphi_dbc, is elaborated. The benefit brought by such a method consists of simplifying the computations so that the time required to approximate the solutions decreases significantly. Some conclusions are given as well as new research topics for the future.

Extension of finite particle method simulating thermal-viscoelastic flow and fluid–rigid body interactional process in weakly compressible smoothed particle hydrodynamics scheme

In computational non-Newtonian fluid dynamics, heat transfer has obvious effects on motions of viscoelastic fluids, mechanical mechanism of elasticity, and flow regimes. This study suggests an extended numerical scheme of smoothed particle hydrodynamics and finite particle method within density smoothing (SPH_DSFPM), which involves the discretization of smoothed particle hydrodynamics (SPH) and finite particle method (FPM) within density smoothing (DS) in the weakly compressible flow scheme. A corrected particle shifting technique is incorporated to eliminate tensile instability and inhomogeneity near solid boundaries. A corrected dynamic solid coupled boundary is introduced to deal with casting molding within high-pressure operations, which has a good compatibility between virtual particle method and repulsive force model. Numerical results show that the present scheme has the nearly lower relative error (0.5%) than conventional SPH (2.6%) in the case of evolutionary thermal-viscoelastic Poiseuille flow and heat effects have active influences on velocity, pressure variations for viscoelastic fluid flow around periodic circular cylinders. Three different printing modes of moving printers significantly generate into differentiated forming regimes through high-pressure extrusion. Adaptive particle distributions possess robust flow evolutions, by which the shocked jets can be tracked well and the sinking velocities of wedge entering into solutions can be numerically probed well considering different cuspidal biting angles. In the case of macroscopic fluid–rigid body interactions, the statistical degree of deviation on probed forces with experiments is relatively 4.35% and that is 12.5% for SPH. The proposed numerical scheme has a good performance on improved accuracy, convergence, and stability for simulating transient thermal-viscoelastic flows.

A Coupled Hybridizable Discontinuous Galerkin and Boundary Integral Method for Analyzing Electromagnetic Scattering

A Coupled Hybridizable Discontinuous Galerkin and Boundary Integral Method for Analyzing Electromagnetic Scattering

Coupled Nonlocal Boundary Value Problems for Fractional Integro-differential Langevin System via Variable Coefficient

In this paper, we aim to study a new coupled system of nonlinear fractional integro-differential Langevin equations with coupled multipoint boundary conditions. The existence and uniqueness of solution are investigated by using the Banach’s and Krasnoselskii’s fixed point theorems. The Ulam-Hyers stability of the mentioned equation is provided by applying the classical technique of functional analysis. Two examples are presented to verify our analysis.

Systems of Hilfer–Hadamard Fractional Differential Equations with Nonlocal Coupled Boundary Conditions

We study the existence and uniqueness of solutions for a system of Hilfer–Hadamard fractional differential equations. These equations are subject to coupled nonlocal boundary conditions that incorporate Riemann–Stieltjes integrals and a range of Hadamard fractional derivatives. To establish our key findings, we apply various fixed point theorems, notably including the Banach contraction mapping principle, the Krasnosel’skii fixed point theorem applied to the sum of two operators, the Schaefer fixed point theorem, and the Leray–Schauder nonlinear alternative.

Numerical Solution to a Free Boundary Problem for the Stokes Equation Using the Coupled Complex Boundary Method in Shape Optimization Setting

Numerical Solution to a Free Boundary Problem for the Stokes Equation Using the Coupled Complex Boundary Method in Shape Optimization Setting

NUMERICAL study of MAGNETO convective Buongiorno nanofluid flow in a rectangular enclosure under oblique magnetic field with heat generation/absorption and complex wall conditions

A mathematical model is presented to analyze the magnetohydrodynamic (MHD) convective flow in a rectangular enclosure. Earlier studies on the Tiwai-Das volume fraction nanofluid model did not consider the Buongiorno nanofluid model. This is the focus of the present analysis which examines the laminar mixed convection magnetohydrodynamic flow of a nanofluid in a differentially heated rectangular enclosure with complex boundary conditions under an inclined magnetic field. Buongiorno's two-component nanofluid model is employed, which incorporates the effects of Brownian motion and thermophoretic diffusion of nanoparticles. Magnetic nanofluids have considerable potential for enhancing transport processes in energy systems such as hybrid fuel cells. The study is essential with heat generation/absorption effects. Additionally, the work is highlighted by the general case of an oblique (inclined) magnetic field. The conservation equations for mass, primary and secondary momentum, energy, and nanoparticle concentration with wall boundary conditions are dimensionless using appropriate scaling transformations. A finite-difference computational scheme known as the Harlow-Welch Marker and Cell (MAC) method is employed to solve the dimensionless nonlinear coupled boundary value problem. A mesh independence study is included. Graphical plots are presented for the impact of key control parameters on streamline contours, isotherm contours, iso-concentration (nanoparticle mass) contours, and local Nusselt number. With heat sink (absorption), the Nusselt number is enhanced in magnitude whereas it is suppressed with heat generation since there is a heat reduction transmitted to the boundary. The configurations of streamlines, isotherms, and iso-concentrations are mostly invariant to magnetic field direction changes. The obtained results show interesting behaviors of the flow and thermal fields, which mainly involve the effect of Brownian motion and thermophoresis parameters, as well as unsteady regimes, depending on specific values of the Schmidt number, Richardson number, and Prandtl numbers. Increasing the Schmidt number induces a contraction in the central cooler zone in the enclosure and also reduces the iso-concentration magnitudes in the central region across the enclosure. The core region of the enclosure heats up as the thermophoresis and Brownian motion parameters rise, pushing the previously cooler top and bottom wall zones further away from the center. There is also a decrease in iso-concentration magnitudes in particular at the upper and lower boundaries at higher values of Brownian motion and thermophoresis parameters. At decreasing buoyancy ratios, the left vortex cell first decelerates while the right vortex cell accelerates. However, when the buoyancy ratio increases, the left vortex cell streamlines magnitudes increase with a contraction in vortex size, while the right cell develops. A Very minor alteration is observed in the isotherm and iso-concentration contours with an increasing buoyancy ratio. When the Richardson number increases, the vortex cell structures shift from a strong circulation cell on the left to a weaker cell on the right, resulting in reverse distribution. With the rising Richardson number, significant cooling is also caused in the core zone, as well as a drop in iso-concentrations, with the original dual low-concentration upper and lower zones merging into a single center zone. The original symmetric left and right vortex cells are gradually twisted diagonally towards the right wall as the magnetic field increases, yet the stronger right cell and the weaker left cell are maintained.

Numerical investigation on explosion hazards of lithium-ion battery vented gases and deflagration venting design in containerized energy storage system

Large-scale Energy Storage Systems (ESS) based on lithium-ion batteries (LIBs) are expanding rapidly across various regions worldwide. The accumulation of vented gases during LIBs thermal runaway in the confined space of ESS container can potentially lead to gas explosions, ignited by various electrical faults. However, a systematic simulation and assessment of the battery vented gases explosion under deflagration venting design still lack. In this work, a three-dimensional combustion model was developed within the frame of open source computational fluid dynamics code OpenFOAM based on a full-scale container, and the LIBs vented gases in realistic proportion were selected as the combustion gas. Coupled boundary conditions were introduced to enable the response of explosion vent doors and top deflagration vent panels on pressure. The internal and external overpressure, flame temperature, and wind velocity fields were employed to assess the gas explosion hazards to ESS container structure and surroundings. The results demonstrate that altering the vent door pressure, without the top vent panel, still leads to serious explosion accidents. There will be unacceptable overpressure for the container structure, as well as serious visible flames and high-speed airflow invading the external environment. Lower vented gas concentrations can reduce explosion hazards, and introducing the vent panel design aids to promote such reduction. The overpressure within the container is significantly decreased by guiding the top external secondary combustion through the vent panel, and the influence range for the environments is also significantly reduced on X-axis. The ignition location can affect the propagation of gas combustion within the container, and the proposed complete vent panel design minimizes the impact on the container structure and surroundings. The findings of this work can offer new references for the safety design of the LIBs ESS.

On a System of Hadamard Fractional Differential Equations with Nonlocal Boundary Conditions on an Infinite Interval

Our research focuses on investigating the existence of positive solutions for a system of nonlinear Hadamard fractional differential equations. These equations are defined on an infinite interval and involve non-negative nonlinear terms. Additionally, they are subject to nonlocal coupled boundary conditions, incorporating Riemann–Stieltjes integrals and Hadamard fractional derivatives. To establish the main theorems, we employ the Guo–Krasnosel’skii fixed point theorem and the Leggett–Williams fixed point theorem.

Lyapunov-Type Inequalities for Systems of Riemann-Liouville Fractional Differential Equations with Multi-Point Coupled Boundary Conditions

We consider a system of Riemann–Liouville fractional differential equations with multi-point coupled boundary conditions. Using some techniques from matrix analysis and the properties of the integral operator defined on two Banach spaces, we establish some Lyapunov-type inequalities for the problem considered. Moreover, the comparison between two Lyapunov-type inequalities is given under certain special conditions. The inequalities obtained compliment the existing results in the literature.

Stochastic Vibrations of a System of Plates Immersed in Fluid Using a Coupled Boundary Element, Finite Element, and Finite Difference Methods Approach.

The main objective of this work is to investigate the natural vibrations of a system of two thin (Kirchhoff-Love) plates surrounded by liquid in terms of the coupled Stochastic Boundary Element Method (SBEM), Stochastic Finite Element Method (SFEM), and Stochastic Finite Difference Method (SFDM) implemented using three different probabilistic approaches. The BEM, FEM, and FDM were used equally to describe plate deformation, and the BEM was applied to describe the dynamic interaction of water on a plate surface. The plate's inertial forces were expressed using a diagonal or consistent mass matrix. The inertial forces of water were described using the mass matrix, which was fully populated and derived using the theory of double-layer potential. The main aspect of this work is the simultaneous application of the BEM, FEM, and FDM to describe and model the problem of natural vibrations in a coupled problem in solid-liquid mechanics. The second very important novelty of this work is the application of the Bhattacharyya relative entropy apparatus to test the safety of such a system in terms of potential resonance. The presented concept is part of a solution to engineering problems in the field of structure and fluid dynamics and can also be successfully applied to the analysis of the dynamics of the control surfaces of ships or aircraft.

Efficient spectral coupled boundary element method for fully nonlinear wave–structure interaction simulation

Accurately analyzing wave–structure interactions is crucial for the design and operational safety of ships and marine structures. This paper presents a fully nonlinear potential-flow approach for simulating wave–structure interactions using the newly proposed spectral coupled boundary element method (SCBEM). The SCBEM efficiently models an extensive water body that encompasses structures by establishing a boundary element method (BEM) computational domain solely around the object of interest while accurately simulating the far-field broad water by a spectral layer. To further improve efficiency, graphics processing unit acceleration is hired during iterative solving of the boundary value problem in the already small-sized interior BEM domain. Simulations are conducted to validate the accuracy of the method on cases with strong nonlinear phenomena, including wave run-up on a single cylinder, diffraction of a four-cylinder array, near-trapped modes for closely spaced columns, and gap resonance that occurred in side-by-side offloading. The wave run-up, diffraction wave pattern, near-trapped mode, and gap resonance frequency obtained by the proposed method are in good agreement with data from experiments and published literature. The quite good accuracy and the exceptional computational efficiency of the SCBEM demonstrate its promising potential for more application in practical marine problems.

On the new coupled complex boundary method in shape optimization framework for solving stationary free boundary problems

<p style='text-indent:20px;'>We expose here a novel application of the so-called coupled complex boundary method – first put forward by Cheng et al. (2014) to deal with inverse source problems – in the framework of shape optimization for solving the exterior Bernoulli problem, a prototypical model of stationary free boundary problems. The idea of the method is to transform the overdetermined problem to a complex boundary value problem with a complex Robin boundary condition coupling the Dirichlet and Neumann boundary conditions on the free boundary. Then, we optimize the cost function constructed by the imaginary part of the solution in the whole domain in order to identify the free boundary. We also prove the existence of the shape derivative of the complex state with respect to the domain. Afterwards, we compute the shape gradient of the cost functional, and characterize its shape Hessian at the optimal domain under a strong, and then a mild regularity assumption on the domain. We then prove the ill-posedness of the proposed shape problem by showing that the latter expression is compact. Also, we devise an iterative algorithm based on a Sobolev gradient scheme via finite element method to solve the minimization problem. Finally, we illustrate the applicability of the method through several numerical examples, both in two and three spatial dimensions.</p>

Existence and uniqueness of solutions for fractional differential system with four-point coupled boundary conditions

Existence and uniqueness of solutions for fractional differential system with four-point coupled boundary conditions

Systems of Riemann–Liouville Fractional Differential Equations with ρ-Laplacian Operators and Nonlocal Coupled Boundary Conditions

In this paper, we study the existence of positive solutions for a system of fractional differential equations with ρ-Laplacian operators, Riemann–Liouville derivatives of diverse orders and general nonlinearities which depend on several fractional integrals of differing orders, supplemented with nonlocal coupled boundary conditions containing Riemann–Stieltjes integrals and varied fractional derivatives. The nonlinearities from the system are continuous nonnegative functions and they can be singular in the time variable. We write equivalently this problem as a system of integral equations, and then we associate an operator for which we are looking for its fixed points. The main results are based on the Guo–Krasnosel’skii fixed point theorem of cone expansion and compression of norm type.