Different Types Of Boundary Conditions Research Articles

English

The paper deals with the Stokes problem, associated with a flow of a viscous incompressible fluid through a spatially periodic profile cascade. We use results from [32] (the maximum regularity property in the $L^2$-framework) and [33] (the weak solvability in $W^{1,r}$), and extend the findings on the maximum regularity property to the general $L^r$-framework (for $1<r<\infty$). Using the reduction to one spatial period $\Omega$, the problem is formulated by means of boundary conditions of three types: the conditions of periodicity on curves $\Gamma_0$ and $\Gamma_1$, the Dirichlet boundary conditions on $\Gamma_{\rm in}$ and $\Gamma_P$ and an artificial do nothing-type boundary condition on $\Gamma_{\rm out}$ (see Fig. 1). We show that, although domain $\Omega$ is not and different types of boundary conditions meet in the vertices of $\partial\Omega$, the considered problem has a strong solution with the maximum regularity property for smooth data. We explain the sense in which the do nothing boundary condition is satisfied for both weak and strong solutions.

A Novel Nonlinear Elasticity Approach for Analysis of Nonlinear and Hyperelastic Structures

There are many materials that the linear elasticity theory cannot predict their mechanical behavior against applied loads. This research proposes a comprehensive theoretical method to obtain the mechanical response of hyperelastic models (with nonlinear elastic deformations) such as polymers and rubbers. They are vital in the design phase of complicated engineering structures like engine mounts and structural bearings in aerospace and automotive industries. The presented theory is implemented in detail and has no limitations in analyzing geometrically and physically nonlinear materials. As a test case, the governing equations of a sheet made of nonlinear elastic material are derived within the framework of this new approach. The derived governing equations are completely nonlinear in all major directions. Consequently, results can be one of the most accurate mathematical simulation results for nonlinear elastic material structures. Then, the obtained equations are solved using a meshless solution method named the semi-analytical polynomial method. The stress and deformation results of the structure under uniform and non-uniform transverse external loadings are obtained for different types of boundary conditions, loading, and material properties. Consequently, this research can be widely used as a suitable essential reference for researchers studying the mechanical behavior of nonlinear elastic structures.

Exploration of energy‐based volumetric heating on ferrothermal porous convection: Effects of MFD viscosity and boundary conditions

AbstractThe impact of energy‐based volumetric internal heating and magnetic field‐dependent (MFD) viscosity on the onset of ferrothermal porous convection for different types of boundary conditions is investigated. The lower and upper boundaries are considered to be either rigid or stress‐free. The lower boundary is insulating, while at the upper boundary a Robin type of thermal condition is applied. Besides this, a general type of boundary conditions is invoked on the magnetic potential. The stability eigenvalue problem is solved numerically using the Galerkin‐type of weighted residuals technique with the Rayleigh number, as the eigenvalue. The results obtained for different boundary combinations are found to be qualitatively similar though not quantitatively. The effect of Biot number, MFD viscosity, and porous parameters is to hinder, while the influence of internal heat source strength, magnetic number, and the nonlinearity of fluid magnetization is to advance the onset of convection. The rigid and conducting boundaries offer a more stabilizing influence on the onset when compared to stress‐free and insulating boundaries. The convection cells get contracted the most when the boundaries are rigid and also when the upper boundary is conducting. The results delineated under the limiting cases are found to be in good agreement with those available in the literature.

Numerical study of generalized 2-D nonlinear Schrödinger equation using Kansa method

The present study is influenced by the wide applications of the Schrödinger equations. Its occurrence can be easily seen in electromagnetic wave propagation, quantum mechanics, plasma physics, nonlinear optics, underwater acoustics, etc. Solving equations of this type is always difficult. In the current paper, we have discussed a very easy numerical technique which is also known as the Kansa method along with polyharmonic radial basis function for the numerical study of generalized 2-D nonlinear Schrödinger equations. The stability analysis of the present method is discussed. The efficiency and accuracy of the present method are demonstrated by considering three numerical cases along with different types of boundary conditions.

Physics‐Informed Neural Networks (PINNs) for Wave Propagation and Full Waveform Inversions

AbstractWe propose a new approach to the solution of the wave propagation and full waveform inversions (FWIs) based on a recent advance in deep learning called physics‐informed neural networks (PINNs). In this study, we present an algorithm for PINNs applied to the acoustic wave equation and test the method with both forward models and FWI case studies. These synthetic case studies are designed to explore the ability of PINNs to handle varying degrees of structural complexity using both teleseismic plane waves and seismic point sources. PINNs' meshless formalism allows for a flexible implementation of the wave equation and different types of boundary conditions. For instance, our models demonstrate that PINN automatically satisfies absorbing boundary conditions, a serious computational challenge for common wave propagation solvers. Furthermore, a priori knowledge of the subsurface structure can be seamlessly encoded in PINNs' formulation. We find that the current state‐of‐the‐art PINNs provide good results for the forward model, even though spectral element or finite difference methods are more efficient and accurate. More importantly, our results demonstrate that PINNs yield excellent results for inversions on all cases considered and with limited computational complexity. We discuss the current limitations of the method with complex velocity models as well as strategies to overcome these challenges. Using PINNs as a geophysical inversion solver offers exciting perspectives, not only for the full waveform seismic inversions, but also when dealing with other geophysical datasets (e.g., MT, gravity) as well as joint inversions because of its robust framework and simple implementation.

Initial-boundary value problems on a half-strip for the generalized Kawahara–Zakharov–Kuznetsov equation

Initial-boundary value problems on a half-strip with different types of boundary conditions for the generalized Kawahara-Zakharov-Kuznetsov equation with nonlinearity of higher order are considered. In particular, nonlinearity can be quadratic and cubic. Results on global existence and uniqueness in classes of weak and strong solutions and large-time decay of small solutions are established. The solutions are considered in weighted at infinity Sobolev spaces. The use of weighted spaces is crucial for the study. To this end new interpolating inequalities in weighted anisotropic Sobolev spaces are established. Both exponential and power weights are admissible.

Size-Effect Analysis on Vibrational Response of Functionally Graded Annular Nano-Plate Based on Nonlocal Stress-Driven Method

Dynamic analysis of functionally graded size-dependent annular nano-plate is the main concern in this study. To obtain the vibrational behavior of this plate, the stress-driven nonlocal integral elasticity, as well the strain gradient theory were used in conjunction with the classical plate theory. The resulting equilibrium equations were solved using the generalized differential quadrature rule (GDQR) and the influences of various parameters such as; size-effect parameter, material heterogeneity index, the aspect ratio of the inner to outer radii, and the effects of different boundary conditions were investigated on the vibrational behavior of the nano-plate, based on different types of boundary conditions. Results indicate that the natural frequencies increase with an increase in the heterogeneity index [Formula: see text] and the increase in size-effect parameter shows a similar effect in both models. Additionally, for the simply supported and free-edge boundary conditions (for both edges), as well as the free and knife-edges, and simply supported-free edges, the strain gradient theory predicts higher values of frequency ratios as [Formula: see text] was increased. Similar results were obtained for the remaining types of boundary conditions, with a higher sensitivity to [Formula: see text], provided the stress-driven model is used. This behavior can be interpreted as the sensitivity of the nano-plate to [Formula: see text] that is manifested by the use of the stress-driven model for the prediction of vibrational behavior of the nano-plate.

On a comparison theorem for parabolic equations with nonlinear boundary conditions

Abstract In this article, a new type of comparison theorem for some second-order nonlinear parabolic systems with nonlinear boundary conditions is given, which can cover classical linear boundary conditions, such as the homogeneous Dirichlet or Neumann boundary condition. The advantage of our comparison theorem over the classical ones lies in the fact that it enables us to compare two solutions satisfying different types of boundary conditions. As an application of our comparison theorem, we can give some new results on the existence of blow-up solutions of some parabolic equations and systems with nonlinear boundary conditions.

Nonlocal and surface effects on nonlinear vibration response of a graded Timoshenko nanobeam

The free and forced vibration of a graded geometrically nonlinear Timoshenko nanobeam supported by on a nonlinear foundation is considered in this paper. The main contribution of this study is to propose a new formulation for the dynamic response of this beam by combining nonlocal and surface elasticity in addition to employing the physical neutral axis method which eliminates the quadratic nonlinearity from the equation of motion. Using the principle of virtual work, a fourth-order nonlinear partial differential equation is formulated and Galerkin technique is employed to yield a fourth-order ordinary differential equation with cubic nonlinearity in the temporal domain. The method of multiple scales is employed to obtain the analytical expression of the nonlinear frequency of the beam and its frequency response curve from a primary resonance analysis. To assess the accuracy of this analytical solution, it is compared with a numerical solution obtained using the differential quadrature method. The obtained analytical results are successfully validated for particular cases of the considered problem with results published by other authors. The effects of surface elasticity, nonlocality, the physical neutral axis, the beam aspect ratio, the power-law index and the elastic foundation coefficients on the free and forced vibration response of the graded Timoshenko nanobeam are thoroughly investigated for different types of boundary conditions .

On the circumferential wave responses of connected elliptical-cylindrical shell-like submerged structures strengthened by nano-reinforcer

This paper is devoted to estimate the dynamic performance related to Connected Elliptical-Cylindrical Shells (CECSs), which can be applicable in submerged structures. Conducive to progress the vibrational behavior associated with the homogenous CECS, nanocomposite materials including Graphene Nano Platelets (GNP) nano-reinforcer are used. It is obvious that the corresponding mechanical criteria of nanocomposite counting polymer and GNP nano-reinforcer are required for obtaining the governing formulation. Halpin-Tsai and the rule of mixture schemes are utilized for predicting the equivalent density, Poisson's ratio, and Young's modulus. Furthermore, various patterns are registered for the distribution of GNP nano-reinforcer along with the thickness of the CECSs. According to the obtained equivalent mechanical characteristics, by means of Hamilton's concept, the governing motion equations of CECSs are achieved as well. Moreover, the effect of shear deformation is considered subject to the First Order Shear Deformation Hypothesis (FOSDH). Afterward, a professional semi-analytical approach, licensed the Generalized Differential Quadrature (GDQ) scheme, is employed for discretizing the differential equations at different separator points throughout the longitudinal (for the cylindrical part) and meridional (for the elliptical section) direction of the structure. By solving an eigenvalue problem, the natural frequencies of CECSs having different types of Boundary Conditions (BCs) can be evaluated numerically. First, a validation study is performed to prove the correctness and accuracy of the proposed formulation. Then, several novel and complex problems are analyzed numerically and the effects of different parameters including boundary conditions, geometric properties, and material characteristics on the frequency parameter of CECSs are also investigated.

Solving second order two-point boundary value problems accurately by a third derivative hybrid block integrator

This article deals with the development of an optimized third-derivative hybrid block method for integrating general second order two-point boundary value problems (BVPs) subject to different types of boundary conditions (BCs) such as Dirichlet, Neumann or Robin. A purely interpolation and collocation approach has been used in order to develop the method. A constructive approach has been applied in the development of the method to consider two off-step optimal points among an infinite number of possible choices in a two-step block corresponding to a generic interval of the form [xn,xn+2]. The obtained method simultaneously produces an approximate solution over the entire integration interval. Some numerical experiments have been presented that show the good performance of the presented scheme.

An Advanced Galerkin Approach to Solve the Nonlinear \\[6pt]Reaction-Diffusion Equations With Different Boundary Conditions

This study proposed a scheme originated from the Galerkin finite element method (GFEM) for solving nonlinear parabolic partial differential equations (PDEs) numerically with initial and different types of boundary conditions. The scheme is applied generally handling the nonlinear terms in a simple way and throwing over restrictive assumptions. The convergence and stability analysis of the method are derived. The error of the method is estimated. In the series, eminent problems are solved, such as&amp;nbsp; Fisher&amp;#39;s equation, Newell-Whitehead-Segel equation, Burger&amp;#39;s equation, and&amp;nbsp; Burgers-Huxley equation to demonstrate the validity, efficiency, accuracy, simplicity and applicability of this scheme. In each example, the comparison results are presented both numerically and graphically

On the Neural Network Solution of One-Dimensional Wave Problem

Artificial neural network has become an emerging popular method to handle various problems, especially in case where it has deep multiple neural layers. In this study, we use a deep artificial neural network model to solve one-dimensional wave equation, without any external datasets. Different type of boundary conditions, i.e., Dirichlet, Neumann, and Robin, are used. We analyze the model learning capabilities in a set of settings, such as data setup and the model width and depth. We also present some discussions of advantages and disadvantages of the model in comparison with other matured existing techniques to solve wave equation.

Finite element analysis of the parameters of fracture in a piezoelectric bimaterial with interface crack under different types of boundary conditions on its faces

Finite element analysis of the parameters of fracture in a piezoelectric bimaterial with interface crack under different types of boundary conditions on its faces

FORCE DRIVEN VIBRATIONS OF NONLINEAR PLATES ON A VISCOELASTIC WINKLER FOUNDATION UNDER THE HARMONIC MOVING LOAD

In the present paper, the forced driven nonlinear vibrations of an elastic plate in a viscoelastic medium and resting on a viscoelastic Winkler-type foundation are studied. The damping features of the surrounding medium and foundation are described by the Kelvin-Voigt model and standard linear solid model with fractional derivatives, respectively. The dynamic response of the plate is described by the set of nonlinear differential equations with due account for the fact that the plate is being under the conditions of the internal resonance accompanied by the external resonance. The expressions for the stress function and nonlinear coefficients for different types of boundary conditions are presented.

Synthesis of a smoothing bicubic spline for differentiating experimental data of nonparametric identification

Over the past two decades, so-called Voltaire series have been used to describe the dynamics of nonlinear systems in terms of input-output. Nonparametric identification of models using Voltaire series consists in constructing estimates for impulse transition functions (IPFs) depending on two or more arguments, which naturally makes identification algorithms much more complicated than in the one-dimensional case. So, in order to identify the two-dimensional IPF (corresponding to the quadratic term of the Voltaire series), it is necessary to calculate the second-order mixed derivatives of the output two-dimensional signal of the system, when a series of rectangular pulses of different amplitudes at different times are fed to its input. Everyone knows, the problem of differentiation is an ill-posed problem and one of the manifestations of incorrectness is poor resistance to errors in the initial data. It is proposed to use two-dimensional smoothing cubic (bicubic) spline (abbreviated SBS) to overcome this problem. The two tasks that constitute SBS synthesis: assignment and implementation of different types of boundary conditions at the border of the rectangular region where SBS is determined; optimal values estimation of two smoothing parameters due to the different “smoothness” of IPF for different two arguments. An acceptable solution to this synthesis problem is proposed in the paper. Our performed computational experiment showed the efficiency of the proposed algorithm for calculating second-order mixed derivative from noisy initial data.

DeepBHCP: Deep neural network algorithm for solving backward heat conduction problems

This paper extends a deep neural network method, a semi-supervised one, to solve backward heat conduction problems which have been long-standing computational challenges due to being ill-posed. The effectiveness and robustness of the methodology are demonstrated through various problems including different types of boundary conditions, several types of time-dependent thermal diffusivity factors, and a variety of domains for two-dimensional tests. In spite of traditional methods, there is no need to apply any regularization technique. According to simulation results, this revolutionary strategy can efficiently and accurately extract the pattern of the solutions even when the noise corruption up to ten percent is imposed on input data. Moreover, when the final time is increased further, this approach is efficient in recovering the data at the initial time, which accentuates the method's robustness. To demonstrate the superiority of the semi-supervised neural network process, we make a comparison with the radial basis functions finite difference (RBF-FD) method which is a localized RBF method.

Analysis of the dynamical behaviour of spinning annular disks with various boundary conditions

This study investigates the vibration characteristics of spinning annular disks with various boundary conditions aiming to investigate the effect of radius ratio parameter and the rotation speed on natural frequencies of a spinning disk analytically for nine different types of boundary conditions. The stress distributions on the disks are obtained for all boundary conditions. The Galerkin method is employed with polynomial functions as approximate displacement functions. Analytical results are presented and compared to the results of the finite element analyses as well as experimental ones.

A FFT accelerated high order finite difference method for elliptic boundary value problems over irregular domains

For elliptic boundary value problems (BVPs) involving irregular domains and Robin boundary condition, no numerical method is known to deliver a fourth order convergence and O(Nlog⁡N) efficiency, where N stands for the total degree-of-freedom of the system. Based on the matched interface and boundary (MIB) and fast Fourier transform (FFT) schemes, a new finite difference method is introduced for such problems, which involves two main components. First, a ray-casting MIB scheme is proposed to handle different types of boundary conditions, including Dirichlet, Neumann, Robin, and their mix combinations. By enclosing the concerned irregular domain by a large enough cubic domain, the ray-casting MIB scheme generates necessary fictitious values outside the irregular domain by imposing boundary conditions along the normal direction of the boundary, so that a high order central difference discretization of the Laplacian can be formed. Second, an augmented MIB formulation is built, in which Cartesian derivative jumps are reconstructed on the boundary as auxiliary variables. By treating such variables as unknowns, the discrete Laplacian can be efficiently inverted by the FFT algorithm, in the Schur complement solution of the augmented system. The accuracy and efficiency of the proposed augmented MIB method are numerically examined by considering various elliptic BVPs in two and three dimensions. Numerical results indicate that the new algorithm not only achieves a fourth order of accuracy in treating irregular domains and complex boundary conditions, but also maintains the FFT efficiency.

On viscous incompressible flows of nonsymmetric fluids with mixed boundary conditions

In this paper we are concerned with the initial boundary value problem for the micropolar fluid system in nonsmooth domains with mixed boundary conditions. The considered boundary conditions are of two types: Navier’s slip conditions on solid surfaces and Neumann-type boundary conditions on free surfaces. The Dirichlet boundary condition for the microrotation of the fluid is commonly used in practice. However, the well-posedness of problems with different types of boundary conditions for microrotation are completely unexplored. The present paper is devoted to the proof of the existence, regularity and uniqueness of the solution in distribution spaces.