Derivation Of Boundary Conditions Research Articles

Research on the Separation Technology of Kelp and Shellfish Box Based on Shellfish–Kelp Mixed Culture Mode

Aiming at the problem of floating shellfish boxes interfering with kelp harvesting when mechanized kelp harvesting is based on shellfish–kelp mixed culture mode, this paper combines the structural characteristics of the shellfish box itself, designs the kelp harvesting unit test bench and develops a shellfish box separator device. The key factors affecting the box separator’s separation effect were derived through the theoretical analysis. The process of separation of a shellfish box by the box separator is simulated and optimized under the derived boundary conditions. The single-factor test for the separating effect of the box separator was conducted with ADAMS kinematics simulation software. The test showed the optimal utility intervals for the key factors under consideration. Further orthogonal tests were conducted for the three key factors, which were ranked in descending order of importance as box separator separation angle θ, box separator taper angle β and box separator placement depth h. The optimal parameter combination is the box separator separation angle of 31°, the box separator taper angle of 30° and the box separator placement depth of 550 mm. Verification experiments have shown that both indicators, the farthest horizontal distance of the shellfish box and the angle of the shellfish box deviating from the box separator, meet the actual production requirements. In summary, the separator can effectively separate the shellfish box from the kelp, and the device is simple in design, quick in operation, and accomplished separation without disturbing shellfish. This study can provide a theoretical basis for the separation technology of kelp and shellfish box under shellfish–kelp mixed culture mode.

A numerical approach to overcome the very-low Reynolds number limitation of the artificial compressibility for incompressible flows

We propose a numerical approach to solve a long-standing challenge which is the applicability of the artificial compressibility (AC) formulation for solving the incompressible Navier—Stokes equations at very-low Reynolds numbers. A wide range of engineering applications involves very-low Reynolds number flows in Micro-ElectroMechanical Systems (MEMS) and in the fields of chemical-, agricultural- and biomedical engineering. It is known that the already existing numerical methods using the AC approach fail to provide physically correct results at very-low Reynolds numbers (Re ≤ 1). To overcome the limitation of the AC method for these engineering applications, we propose a higher-order Neumann-type pressure outflow boundary condition treatment along with their up to fourth-order numerical approximations. We found that the numerical treatment of the pressure at the outlet boundary plays the main role in overcoming the limitation of the AC method at very-low Reynolds numbers (Re << 1). Therefore, we provide numerical evidence on the accuracy of the AC method beyond its previously reported limitations, e.g., the low Reynolds number Oseen flow (Re << 1) is first presented in this work. A third-order explicit total-variation diminishing (TVD) Runge–Kutta scheme has been employed with standard finite difference spatial discretisation schemes for improving the accuracy of the numerical solution. For modelling strongly viscous flows, the Reynolds number ranges from 10−1 to 10−4. Overall, we found that the accuracy limitation of the AC method below Re < 1 can be overcome with an accurate numerical treatment of the outlet pressure boundary condition instead of using high-order schemes in the governing equations. For the investigated Reynolds number range (10−1 ≤ Re ≤ 10−4), the obtained results show that the relative errors were smaller than 1% for the numerical simulations performed on the configurations of both the two- and three-dimensional, straight microfluidic channels. The imposition of high-order derivative Neumann-type pressure outflow boundary conditions reduced the maximum relative errors of the numerical solutions from 85% and 95% to below than 1% at the outlet section of the two- and three-dimensional, straight microfluidic channel flows, respectively. Taking the advantage of the numerical approach proposed here, two- and three-dimensional benchmark problems employed in the current investigation in comparison with analytical solutions available in the literature, clearly demonstrate that the artificial compressibility can be used beyond its previously known constraints for very-low Reynolds number incompressible flows.

Solvability of a Hadamard fractional boundary value problem with multi‐term integral and Hadamard fractional derivative boundary conditions

In the present paper, we construct the existence of nontrivial solutions to a new kind of Hadamard fractional boundary value problem on an unbounded domain. With the contribution of some fixed point theorems in cone and the corresponding Green function, we ensure sufficient conditions for the Hadamard fractional boundary value problem. Also, the paper concludes with two examples to demonstrate our results.

Analysis of Solutions for Nonlinear ψ-Caputo Fractional Differential Equations with Fractional Derivative Boundary Conditions in Banach Algebra

This article explores the solutions of nonlinear implicit ψ-Caputo fractional-order ordinary differential equations (NLIFDEs) with two-point fractional derivatives boundary conditions in Banach algebra. The research aims to establish the existence and uniqueness of solutions for this complex class of differential equations. Utilizing Banach’s and Krasnoselskii’s fixed point theorems, the study conducts a rigorous analysis of the solutions, ensuring their existence and uniqueness. This comprehensive investigation contributes to enhancing the understanding of the behavior of solutions of nonlinear fractional differentials within a challenging mathematical framework.

Meshfree RBF–FD methods for numerical simulation of PDE problems

The meshfree RBF–FD method is becoming an increasingly popular discretization method for challenging PDE problems due to its flexibility with respect to geometry and its ease of implementation. Using a combination of polyharmonic spline basis functions and polynomial basis functions has proven to be particularly successful with a convergence rate depending on the polynomial degree and good behavior at boundaries due to the contribution of the spline part. A challenge that has been observed is that derivative boundary conditions can give rise to large errors. A known remedy for this problem is to introduce one or more layers of ghost points, which improves the boundary approximations. In a recent paper, Tominec, Larsson, Heryudono (2021), it was shown that using oversampling is another effective way to handle this problem. Oversampling also allows for theoretical analysis of the method, which can then be viewed as a discretization of a continuous least-squares projection. By looking deeper into the error estimates we gain an understanding about how to best implement the method.

Optimizing ODE solutions: application of Nelder-Mead algorithm for solving mixed boundary value problems

This abstract introduces a novel application of the Nelder-Mead algorithm in optimizing solutions to Ordinary Differential Equations (ODEs) with derivative boundary conditions. The study presents a refined methodology that leverages the Nelder-Mead algorithm’s adaptability to tackle ODEs characterized by intricate derivative constraints. With the aim of enhancing the toolkit for solving complex ODEs, the objective of this research is to showcase the algorithm’s efficacy in optimizing solutions under derivative boundary conditions. The methodology involves adapting the Nelder-Mead algorithm to navigate the param-eter space while satisfying both the ODEs and their derivative constraints. Experimental results demonstrate the algorithm’s capability to identify solutions that meet these stringent requirements, marking a significant advancement in addressing ODEs with derivative boundary conditions. The study concludes by emphasizing the algorithm’s potential for advancing ODE-solving techniques, particularly in scenarios where gradient-based methods struggle, thus widening the scope of applications across various scientific and engineering domains.

The Maximum Principles of Differential Equations in Mathematical Physics—Elliptic Equations with Oblique Derivative Boundary Condition

The Maximum Principles of Differential Equations in Mathematical Physics—Elliptic Equations with Oblique Derivative Boundary Condition

Convergence of smooth solutions to parabolic equations with an oblique derivative boundary condition

&lt;abstract&gt;&lt;p&gt;In this paper, the parabolic equation with oblique derivative boundary condition is considered. The long time behavior of the solution is derived by selecting the appropriate auxiliary functions and making priori estimates. Through blow up analysis, time-dependent gradient estimates are obtained, followed by second-order derivative estimates. Then, the convergence of smooth solution to parabolic equations with the oblique derivative boundary condition is obtained using standard theory.&lt;/p&gt;&lt;/abstract&gt;

Green functions for stationary Stokes systems with conormal derivative boundary condition in two dimensions

AbstractWe construct Green functions of conormal derivative problems for the stationary Stokes system with measurable coefficients in a two‐dimensional Reifenberg flat domain.

A provably stable and high-order accurate finite difference approximation for the incompressible boundary layer equations

In this article we develop a high order accurate method to solve the incompressible boundary layer equations in a provably stable manner. We first derive continuous energy estimates, and then proceed to the discrete setting. We formulate the discrete approximation using high-order finite difference methods on summation-by-parts form and implement the boundary conditions weakly using the simultaneous approximation term method. By applying the discrete energy method and imitating the continuous analysis, the discrete estimate that resembles the continuous counterpart is obtained proving stability. We also show that these newly derived boundary conditions removes the singularities associated with the null-space of the nonlinear discrete spatial operator. Numerical experiments that verifies the high-order accuracy of the scheme and coincides with the theoretical results are presented. The numerical results are compared with the well-known Blasius similarity solution as well as that resulting from the solution of the incompressible Navier–Stokes equations.

Thermophoresis and uniform flow in rarefied polyatomic gases: The role of constitutive relations and boundary conditions

Recently, Rana and Barve [“A second-order constitutive theory for polyatomic gases: Theory and applications,” J. Fluid Mech. 958, A23 (2023)] developed a second-order coupled constitutive relations (CCR) for polyatomic gases that include quadratic nonlinearities in the entropy flux and apply the second law. However, in that work, the boundary conditions were heuristically obtained to match the drag coefficient on a sphere and may not be accurate in situations where thermal transpiration and thermal stress are significant factors, as indicated by their asymptotic analysis. This article presents a systematic approach for deriving thermodynamically admissible boundary conditions for the CCR model. We also propose a set of higher-order boundary conditions based on an asymptotic analysis of the solutions for drag on flow past a sphere and thermophoretic drag. The goal of deriving these boundary conditions is to improve the accuracy of the CCR model when applied to external flows, such as slow flow past particles and thermophoretic motion of a spherical particle and doublet. The results of the study demonstrate that the combination of the newly derived boundary conditions in conjunction with the CCR equations shows excellent agreement with both theoretical predictions and experimental data over a wide range of Knudsen numbers. The study suggests that the approach presented in this article can be used to improve the accuracy of the CCR model in a variety of external flow applications.

The Role of Pickup Ions in the Interaction of the Solar Wind with the Local Interstellar Medium. I. Importance of Kinetic Processes at the Heliospheric Termination Shock

The role of pickup ions (PUIs) in the solar wind interaction with the local interstellar medium is investigated with 3D, multifluid simulations. The flow of the mixture of all charged particles is described by the ideal MHD equations, with the source terms responsible for charge exchange between ions and neutral atoms. The thermodynamically distinct populations of neutrals are governed by individual sets of gas dynamics Euler equations. PUIs are treated as a separate, comoving fluid. Because the anisotropic behavior of PUIs at the heliospheric termination shocks is not described by the standard conservation laws (a.k.a. the Rankine–Hugoniot relations), we derived boundary conditions for them, which are obtained from the dedicated kinetic simulations of collisionless shocks. It is demonstrated that this approach to treating PUIs makes the computation results more consistent with observational data. In particular, the PUI pressure in the inner heliosheath (IHS) becomes higher by ∼40%–50% in the new model, as compared with the solutions where no special boundary conditions are applied. Hotter PUIs eventually lead to charge-exchange-driven cooling of the IHS plasma, which reduces the IHS width by ∼15% (∼8–10 au) in the upwind direction, and even more in the other directions. The density of secondary neutral atoms born in the IHS decreases by ∼30%, while their temperature increases by ∼60%. Simulation results are validated with New Horizons data at distances between 11 and 47 au.

Solving the fixed gravimetric boundary value problem by the finite element method using mapped infinite elements.

The numerical approach for solving the fixed gravimetric boundary value problem (FGBVP) based on the finite element method (FEM) with mapped infinite elements is developed and implemented. In this approach, the 3D semi-infinite domain outside the Earth is bounded by the triangular discretization of the whole Earth’s surface and extends to infinity. Then the FGBVP consists of the Laplace equation for unknown disturbing potential which holds in the domain, the oblique derivative boundary condition (BC) given directly at computational nodes on the Earth’s surface, and regularity of the disturbing potential at infinity. In this way, it differs from previous FEM approaches, since the numerical solution is not fixed by the Dirichlet BC on some part of the boundary of the computational domain. As a numerical method, the FEM with finite and mapped infinite triangular prisms has been derived and implemented. In experiments, at first, a convergence of the proposed numerical scheme to the exact solution is tested. Afterwards, a numerical study is focused on a reconstruction of the harmonic function (EGM2008) above the Earth’s topography. Here, a special discretization of the Earth’s surface which is able to fulfil the conditions that arise from correct geometrical properties of finite elements, and it is suitable for parallel computing is implemented. The obtained solutions at nodes on the Earth’s surface as well as nodes that lie approximately at the altitude of the GOCE satellite mission have been tested.

System of Nonlinear Second-Order Parabolic Partial Differential Equations with Interconnected Obstacles and Oblique Derivative Boundary Conditions on Non-Smooth Time-Dependent Domains

System of Nonlinear Second-Order Parabolic Partial Differential Equations with Interconnected Obstacles and Oblique Derivative Boundary Conditions on Non-Smooth Time-Dependent Domains

Generalized boundary conditions in closed cosmologies

Considering a generalization of the Gibbons-Hawking-York covariant boundary action that depends on both the extrinsic and the intrinsic geometry of the boundary, we derive boundary conditions for the cosmological background and tensor perturbations in a closed universe with space-like boundaries. We also give a general method to reconstruct the covariant boundary action starting from a given set of boundary conditions for the cosmological background. These results may be of special relevance in the context of the path-integral formulation of quantum cosmology, where boundary terms contain essential physical information of the system.

Semi-analytical solution to steady-state and transient heat transfer problem with variable conductivity properties of the domain

Introduction. Present paper studies the case of heat transfer problem where the conductivity property of the domain is not a constant value but varies depending on its temperature. In particular, a conductivity that linearly dependent onthe temperature is reviewed. A semi-analytical formulation is developed and proposed to solve steady-state and transient types of these kinds of heat transfer problems.&#x0D; Materials and methods. The domain is mathematically modeled by using finite element method where non-linearity property of the problem has been incorporated. For time-dependent (transient) case, the temporal solution has been achieved by analytical methods wherefore all necessary formulation has been derived. Non-linear equations have been solved by both Newton and Picard methods.&#x0D; Results. In order to proof check, the proposed calculations a wall with three layers has been calculated by using the proposed transient problem formulation. The derivative boundary conditions at the faces of the wall are given as the functions of time. Non-linearity has been solved by two different methods that gave the identical results. Having solved the non-linearity by these two methods, allowed to compare the efficiency of Newton method versus Picard method. Both methods reached the solution with the same number of iterations. The paper proposes the algorithms for solving the problems. The authors have used MatLab environment to implement those algorithms.&#x0D; Conclusions. Proposed formulation solves coupled heat transfer problems in multi-layered exterior walls. Multi-layered calculation ability allows to cover all non-homogeneous cases of the computation domain. The formulation solves the heat loss problems through the multi-layered walls with due and reliable accuracy.

A detailed quasigeoid model of the Hong Kong territories computed by applying a finite-element method of solving the oblique derivative boundary-value problem

Abstract New gravity and precise levelling measurements have been performed throughout the Hong Kong territories to modernize a vertical geodetic datum that is currently realized by heights of levelling benchmarks defined in the Hong Kong Principal Datum (HKPD). Modernization of the HKPD involved delivering various products, including new detailed geoid and quasigeoid models and newly determined orthometric and normal heights of levelling benchmarks. In this study, we present the result of gravimetric quasigeoid modelling. The method used to compute a detailed gravimetric quasigeoid model is based on the finite-element method to solve the geodetic boundary-value problem with oblique derivative boundary conditions considered directly at computational nodes on the discretized Earth’s topography. The result of a gravimetric quasigeoid modelling shows a good agreement with a geometric quasigeoid model at the Global Navigation Satellite System (GNSS)-levelling benchmarks. The standard deviation of differences between the gravimetric and geometric quasigeoid heights of ±3.3 cm is compatible with the expected accuracy of gravity, levelling, and GNSS measurements.

Functional Impulsive Fractional Differential Equations Involving the Caputo-Hadamard Derivative and Integral Boundary Conditions

In this paper, we investigate the existence and uniqueness of solutions for functional impulsive fractional differential equations and integral boundary conditions. Our results are based on some fixed point theorems. Finally, we provide an example to illustrate the validity of our main results.

A novel local Hermite radial basis function‐based differential quadrature method for solving two‐dimensional variable‐order time fractional advection–diffusion equation with Neumann boundary conditions

AbstractA novel Hermite radial basis function‐based differential quadrature (H‐RBF‐DQ) method is presented in this paper based on 2D variable order time fractional advection–diffusion equations with Neumann boundary conditions. The proposed method is designed to treat accurately for derivative boundary conditions, which considerably improve the approximation results and extend the range of applicability for the method of RBF‐DQ. The advantage of the present method is that the Hermite interpolation coefficients are only dependent of the point distribution yielding a substantially better imposition of boundary conditions, even for time evolution. The proposed algorithm is thoroughly validated and is demonstrated to handle the fractional calculus problems with both Dirichlet and Neumann boundaries very well.

On (k,ψ)-Hilfer Fractional Differential Equations and Inclusions with Mixed (k,ψ)-Derivative and Integral Boundary Conditions

In this paper we study single-valued and multi-valued (k,ψ)-Hilfer-type boundary value problems of fractional order in (1,2], subject to nonlocal boundary conditions involving (k,ψ)-Hilfer-type derivative and integral operators. The results for single-valued case are established by using Banach and Krasnosel’skiĭ fixed point theorems as well as Leray–Schauder nonlinear alternative. In the multi-valued case, we establish an existence result for the convex valued right-hand side of the inclusion via Leray–Schauder nonlinear alternative for multi-valued maps, while the second one when the right-hand side has non-convex values is obtained by applying Covitz–Nadler fixed point theorem for multi-valued contractions. Numerical examples illustrating the obtained theoretical results are also presented.