Types Of Boundary Conditions Research Articles

Numerical solution of the initial-boundary value problem describing separation processes in a distillation column

A modification of the numerical method of characteristics is developed to solve the initial-boundary value problem that arises when modeling the rectification process in the column. The process is described by a system of first-order hyperbolic equations. A specific peculiarity of the model is in the boundary conditions of a special type. At each of the boundaries, boundary conditions are determined from a system of ordinary differential equations, which also includes unknown values of functions on another boundary. A characteristic difference grid is constructed on the base of a linear transformation of a classical rectangular grid. Implicit second-order difference schemes are used, taking into account the features of the problem at the boundaries. The advantage of this approach is in consideration of the specifics of the propagation of perturbations in hyperbolic equations. Numerical implementation of the method was carried out. An illustrative example shows the effectiveness of the proposed modification of the characterization method. This method is a base for further solution of optimal control problems of flows in columns.

FORCED OSCILLATIONS OF A PLATE CONSISTING OF ELLIPSES AND CIRCLES OF VARIOUS SIZES WITH COMPLEX CONTOURS ON MACHINE PARTS

The article considers forced oscillations of elastic plates in the multilinked area. The surface of the plate undergoes actions of the disturbing force directed along the normal. The task leads to solution of different kinds of differential equations satisfying to initial and boundary conditions of various type. These equations are solved by the Bessel functions and through the method of the theory of functions with complex variables. Thus, the article offers the general methodology to the solutions of the dynamics of the elastic plates that is under action of the disturbing force.

Effect of variable viscosity, porous walls and mixed thermal boundary condition on the onset of Rayleigh-Bénard convective instability

In this paper, we have studied the criteria for the onset of instability in the Rayleigh-Bénard convection problem, by considering a fluid layer confined between two infinitely extended rough boundaries, which are subjected to mixed-type thermal boundary conditions, physically representing the imperfectly conducting boundaries. The rough boundaries are assumed to be shallow porous layers with different pore size, properties, and permeabilities. Mathematically, it is given by the Saffman type boundary condition. Therefore, we have two generalized forms of boundary conditions, one for the flow field and the other for the thermal field. The two limiting cases of the mixed-type thermal boundary condition correspond to a perfectly conducting boundary and to a perfectly insulating boundary, whereas the two limiting cases of the hydrodynamic boundary condition correspond to a no-slip condition and to a free-surface condition, depending upon the limiting values of the parameters involved in these two generalized boundary conditions. The linear and nonlinear energy stability analyses are performed, and the existence of the region of subcritical instability is checked. Through the principle of exchange of stability analysis, it is found that the instability occurs only in the stationary mode. The adiabatic boundary condition is found to be more restrictive than the isothermal boundary condition. Under given conditions, the instability is observed to be occurring in the infinite wavelength mode for the case of adiabatic boundaries. In the present work, the effect of temperature and pressure dependent viscosity on the stability of the system has been shown and found to be of destabilizing nature. However, the roughness of the boundaries is found to be of stabilizing nature.

Influence of the Droplet Diameter on Compressor Rotor Performance under Constant Humidification

Aiming at the near-clogging boundary condition of a certain type of transonic compressor rotor, a numerical simulation method is used to systematically study the influence of different droplet diameters on its performance under the condition of moisture content of 3%. The results show that humidifying the intake air of the transonic compressor near the blockage boundary can reduce the exhaust temperature, the mass flow rate, and the total pressure ratio. In the axial direction, the mass fraction of water vapor increases slowly and linearly under the large droplet condition, while the mass fraction of water vapor under the medium and the small droplet conditions presents a two-stage parabolic growth. The mass fraction of water vapor under the three conditions increases by 0.002, 0.0055, and 0.0072 from inlet to outlet, respectively. On the radial section with the axial span = 0.7, the difference in water vapor mass fraction between the shroud and the hub under the small droplet condition is 0.00329, and that under the medium and the large droplet condition are 46.81% and 27.96% of the small droplet condition. The research results can provide theoretical guidance for the analysis of droplet evaporation at different positions in compressor near-plugging conditions.

GEOFILTRATION MODELING OF WATER FLOWS TO PRODUCED PRIMARY DIAMOND DEPOSITS AND INJECTION IN ASSOCIATED AREAS

Safe development of primary diamond deposits in Western Yakutia requires constant monitoring of the hydrogeological regime of the exposed aquifer complexes within the quarry and mine fields of the deposits, as well as in the adjacent areas of drainage water injection. Over the entire period of development of the Alakit-Markhinsky, Daldynsky, Mirninsky and Nakynsky kimberlitic fields, about 400 million m3 of highly mineralized drainage water from quarries and mines were involved in the pumping-injection process. Complex cryohydrogeological conditions of the territory, i.e., lithological-facial specifics, continuous distribution of permafrost, structural confinement of kimberlite fields, fault-block structure of individual pipes, influence the dynamics of occurring changes and make the cryohydrogeological conditions of each individual pipe unique and having no complete analogues. In order to predict successfully and implement subsequently technical solutions aimed at controlling inflows of all types formed within mine and quarry fields, the Yakutniproalmaz Institute elaborated a program for the development, constant maintenance and updating of hydrogeological “digital twins” of all key mining deposits. The developed models take into account the influence of both natural factors and the applied schemes for opening and draining deposits, which impose their own limitations. The filtration problem was solved out using the licensed program FEFLOW, which performs modeling of hydrogeological conditions by the finite element method in a multilayer strata for areas of arbitrary configuration with boundary conditions of type I, II, III changing according to the known law in the presence of filtration heterogeneities in plan and section, as well as vertical filtration. The creation and constant updating of permanent digital models made it possible not only to acquire a modern tool for forecasting water inflows, but also helped to improve the planning process in terms of drilling drainage and injection wells, purchasing pumping equipment, etc. Deviation of predicted values from those actually observed within the short-term forecast for the period of use 2021–2023 ranged from 5 to 10%.

Existence of non-negative periodic solutions for a degenerate anisotropic parabolic equation with strongly nonlinear source

Abstract The purpose of the present work is investigating a degenerate parabolic equation with anisotropic p-Laplacian operator and strongly nonlinear source under boundary conditions of Dirichlet type; the existence of a periodic non-negative solution is shown. The proof is based on the Leray–Schauder topological degree, which is tricky to work with in this kind of equations.

Entropy generation analysis on heat transfer characteristics of Twisted corrugated spiral heat exchanger utilized in solar pond

In the present study, heat transfer and entropy generation in the spiral corrugated heat exchanger used in the solar pond have been numerically studied. The thermal boundary condition of the third type has been selected for simulation and different geometric parameters have been studied to improve heat transfer and reduce entropy generation. New correlations based on experimental data have been used to validate the simulation. The results were obtained by changing the parameters such as the number of corrugations, the twist number of the corrugations, and the change of the Reynolds number. Also, dimensionless parameters have been defined to investigate the increase of heat transfer and decrease of entropy generation based on the first and second laws of thermodynamics, and finally, the optimal geometries have been introduced by the NH multi-criteria parameter. The simulation results showed that the corrugation creation on the tube will increase heat transfer and decrease entropy generation. Therefore, it was found that the twist number of the corrugation has a greater effect than the number of corrugations. In the best case, when the number of corrugations and their twist is high, the heat transfer improvement number (NH) can grow up to 89%, which will decrease with the increase of Reynolds number.

Parameter uniform higher order numerical treatment for singularly perturbed Robin type parabolic reaction diffusion multiple scale problems with large delay in time

In this paper, we address a class of boundary layer originated singularly perturbed parabolic reaction-diffusion problems with Robin boundary conditions having large time delay; for the higher order numerical analysis. The boundary layer originated nonuniform mesh, is generated by adaptive moving mesh approach based on equidistribution principle. The mesh is obtained by solving a nonlinear system; involving the discrete problem and the discrete nonlinear mesh generating functions. We provide the theoretical mesh structure. The difference scheme for Robin type boundary conditions having normalized flux; uses a combination of space and time discretizations so that the order of convergence in space is higher than the usual linear accurate upwind discretization in space. The present theoretical convergence analysis requires less regularity of the solution and proves to be exactly second order parameter uniform accurate in space. Numerical experiments highly validate the theoretical findings and also show that the present approach is better than the well-known upwind discretizations.

Well‐posedness and exponential stability for a nonlinear wave equation with acoustic boundary conditions

In this paper, we prove the well‐posedness of a nonlinear wave equation coupled with boundary conditions of Dirichlet and acoustic type imposed on disjoints open boundary subsets. The proposed nonlinear equation models small vertical vibrations of an elastic medium with weak internal damping and a general nonlinear term. We also prove the exponential decay of the energy associated with the problem. Our results extend the ones obtained in previous results to allow weak internal dampings and removing the dimensional restriction . The method we use is based on a finite‐dimensional approach by combining the Faedo‐Galerkin method with suitable energy estimates and multiplier techniques.

Splitting schemes for the semi-linear wave equation with dynamic boundary conditions

This paper introduces novel bulk–surface splitting schemes of first and second order for the wave equation with kinetic and acoustic boundary conditions of semi-linear type. For kinetic boundary conditions, we propose a reinterpretation of the system equations as a coupled system. This means that the bulk and surface dynamics are modeled separately and connected through a coupling constraint. This allows the implementation of splitting schemes, which show first-order convergence in numerical experiments. On the other hand, acoustic boundary conditions naturally separate bulk and surface dynamics. Here, Lie and Strang splitting schemes reach first- and second-order convergence, respectively, as we reveal numerically.

Uniform decay rate estimates for the 2D wave equation posed in an inhomogeneous medium with exponential growth source term, locally distributed nonlinear dissipation, and dynamic Cauchy–Ventcel–type boundary conditions

AbstractWe study the wellposedness and stabilization for a Cauchy–Ventcel problem in an inhomogeneous medium with dynamic boundary conditions subject to a exponential growth source term and a nonlinear damping distributed around a neighborhood ω of the boundary according to the geometric control condition. We, in particular, generalize substantially the work due to Almeida et al. (Commun. Contemp. Math. 23 (2021), no. 03, 1950072), in what concerns an exponential growth for source term instead of a polynomial one. We give a proof based on the truncation of a equivalent problem and passage to the limit in order to obtain in one shot, the energy identity as well as the observability inequality, which are the essential ingredients to obtain uniform decay rates of the energy. We show that the energy of the equivalent problem goes uniformly to zero, for all initial data of finite energy taken in bounded sets of finite energy phase space. One advantage of our proof is that the decay rate is independent of the nonlinearity.

On eigenfunctions and eigenvalues of some boundary value problems for a nonlocal biharmonic operator

In this note, the concept of a nonlocal biharmonic operator is introduced. When introducing this operator, mappings of the type of involution are used. Namely, in the differential expression of this operator, in addition to the variables 12( , ,..., )nxx xx, transformed arguments with mappings of the form 111,...,,,,...,,1jjjjjnS xxxpx xxj n and their multiplication also involved. Spectral problems with Dirichlet and Neumann-type boundary conditions are considered in an n-dimensional parallelepiped for a given nonlocal biharmonic operator. The eigenfunctions and eigenvalues of the problems under consideration are explicitly constructed. When constructing these elements, eigenfunctions and eigenvalues of the classical biharmonic operator with Dirichlet and Neumann type boundary conditions are essentially used. Theorems on the orthonomization and completeness of the systems of eigenfunctions of the problems under consideration are proved. Examples of the corresponding parameters for special cases involved in the problems under consideration are given. In addition, in the two-dimensional case for the corresponding nonlocal biharmonic operator, spectral issues of boundary value problems of the Samarsky-Ionkin type are also investigated. The proper and attached functions of the problem under consideration are found and theorems on the completeness ofthese systems are proved.

Observability and observer design for a class of hyperbolic PDEs with van de Pol type boundary conditions

This paper focuses on observability and observer design for nonlinear complex dynamical systems described by a class of hyperbolic partial differential equations (PDEs) with nonlinear van de Pol type boundary conditions. The systems exhibit complex dynamics due to its imbalance of energy flows. Both the exact observability and approximate observability of the systems with different boundary output measurements are shown by using methods of characteristics and boundary nonlinear reflections. Motivated by the approximate observability of the systems, a PDE state observer by using the boundary displacement measurement only is designed, and a sufficient condition to guarantee the estimation error systems to be exponentially stable is given. Theoretical results are proved rigorously, with some numerical simulations performed to validate the effect of the proposed observer.

Determination of one unknown coefficient in a two-phase free boundary problem in an angular domain with variable thermal conductivity and specific heat

Two different two-phase free boundary Stefan problems in an angular domain with temperature-dependent thermal coefficients are considered. Analytical similarity solutions are obtained imposing a Dirichlet or Neumann type boundary condition, respectively, by solving functional problems. Moreover, formulas are obtained for the determination of one unknown thermal coefficient in the overspecified problem that consists in adding a Neumann condition to the problem with a Dirichlet one if and only if some restrictions on data are verified.

The compressible Navier–Stokes equations with slip boundary conditions of friction type

We study a mathematical model of a viscous compressible fluid obeying the slip boundary condition of friction type. We present a notion of weak solutions to this model, in which the momentum equation and the associated energy inequality are combined into a single relation. Moreover, the slip boundary condition of friction type is incorporated into this relation by the use of a boundary integral. Our main result proves the existence of such weak solutions. The proof of this result combines the classical existence theory for the compressible Navier–Stokes equations with an approximation of the aforementioned boundary integral via a convex regularization of the absolute value function.

A deep learning framework for solving forward and inverse problems of power-law fluids

We for the first time leverage deep learning approaches to solve forward and inverse problems of two-dimensional laminar flows for power-law fluids. We propose a deep-learning framework, called Power-Law-Fluid-Net (PL-Net). We develop a surrogate model to solve the forward problems of the power-law fluids, and solve the inverse problems utilizing only a small set of measurement data under the assumption that boundary conditions (BCs) can be partially known. In the design of the methods, we incorporate the hard boundary condition constraints to accelerate the iteration of stochastic gradient descent methods for minimizing loss functions. For the forward problems, by incorporating the constitutive parameters into the input variables of neural networks, the PL-Net serves as a surrogate model for simulating the pressure-driven flows inside pipes having cross sections of varying shapes. We investigate the influences of the BC type, activation function type, and number of collocation points on the accuracy of numerical solutions. For the inverse problems, the PL-Net infers the physical quantities or constitutive parameters from a small number of measurements of flow field variables. The BCs of the inverse problems can even be partially known. We demonstrate the effects of BC type, number of sensors, and noise level on accuracy of inferred quantities. Computational examples indicate the high accuracy of the PL-Net in tackling both the forward and inverse problems of the power-law fluids.

The invariant manifold approach applied to global long-time dynamics of FitzHugh-Nagumo systems

We consider the FitzHugh-Nagumo system equipped with boundary condition of Dirichlet type on some two/three-dimensional domains. This system describes the signal transmission across the axonal membrane in neurophysiology. It is a semilinear parabolic PDE for the voltage variable coupled with a first-order ODE of space-time type for the recovery variable. We prove that there exists a finite-dimensional global manifold in the case of the fast recovery variable. Since the manifold is uniformly attracting, it gives geometric insight into the global long-time dynamics of the solutions. The proof is based on an abstract invariant manifold theorem for dynamical systems on a Banach space.

Comparison between periodic and non-periodic boundary conditions in the multi-particle finite element modelling of ductile powders

In recent years, the multi-particle finite element method (MPFEM) has demonstrated its suitability for the micromechanical modelling of granular systems made up of highly deformable particles. This method combines the advantages of the finite element method to compute particles’ deformations and of the discrete element method to simulate particles’ interactions. Being computationally expensive, its main drawback is the limitation of the number of particles which can be included in a simulation. However, using a multi-scale approach, it is possible to deduce mesoscopic material properties from the simulation of a relatively small number of particles within an elementary volume, through appropriate averaging techniques. Such simulations are significantly influenced by the boundary conditions (BCs) applied. Periodic BCs are commonly admitted to be the most accurate. However, they are technically rather difficult to implement in the MPFEM because of the multiple contacts which need to be handled in the model. As a result, simplified BCs involving for example, rigid walls, remain used. This study analyses the effect of different types of BCs on the simulated mechanical response of a numerical granular sample, under quasi-static loading. Two different techniques for implementing full or simplified periodic BCs, introduced by Schmidt et al. [1] and Loidolt et al. [2], respectively, were compared with simpler, non-periodic BCs. In the comparison, the same initial microstructure was used; it consisted of an assembly of 50 spherical particles with a periodic topology. The results suggest that considering the additional computational cost induced by the implementation of multi-point constraints, the advantage brought by periodic BCs is relatively limited. The study also highlights the differences between BC types and provides interpretations relative to the particularities of the numerical method used.

Regularity and stability of wave equations with variable coefficients and Wentzell type boundary conditions

We investigate regularity and stability of wave equations with variable coefficients and the frictional damping, where the damping effect is only on Wentzell boundary and there is no interior damping at all. Based on the spectral approach and the Reimannian geometry method, we present a new way of studying the challenging Wentzell-type problem concerning “the lack of interior damping” and the variable coefficients adiv(A(x)∇u). First, we establish a high-order regularity estimate for the solution of variable coefficient Wentzell problem and a spectral theorem for the operator derived from this problem. And then, after a series of specialized analysis and investigation on the special structure and basic properties of some auxiliary systems, the lower-order term and the resolvent of the operator associated to the variable coefficient Wentzell system by virtue of the Riemannian geometry method and other ideas, we prove eventually that the energies of the system decay with the rate 1/t. As can be seen, even for the constant coefficient case, the decay rate 1/t obtained in this paper is essentially better than the decay rate t−4/5 given in the previous work.

Modeling of Creep Deformation Behavior of DZ411 and Finite Element Simulation of Turbine Blade

Creep tests were conducted on DZ411 material at 930 °C and 850 °C, and creep curves were recorded and employed in normalization creep model building. A yield function suitable for directional solidification nickel-based materials was proposed in an ascending-order approach. Combined with the normalized creep model and the proposed function, a creep subroutine was compiled to simulate the creep deformation behavior of a turbine blade. The typical boundary conditions of the blade were determined and used for finite element analysis. According to the analysis results, the assessment positions for the actual application of a turbine blade were determined and checked for endurance intensity. The phenomenon of deviation angle between crystal axis and blade height direction in actual casting was further analyzed. Multiple angles and directional deviation angles were simulated for 10,000 h creep deformation. Considering the difficulties and challenges of the complex geometric structures of blades, it is necessary to conduct creep tests of DZ411 material and a simulation analysis of a real blade. Based on the above analysis and discussion, the present work sheds light on finite element analysis and has great potential for structural analyses in the engineering applications of complex high-temperature structures.