Types Of Boundary Conditions Research Articles

Twenty Vertex Model and Domino Tilings of the Aztec Triangle

We show that the number of configurations of the 20 Vertex model on certain domains with domain wall type boundary conditions is equal to the number of domino tilings of Aztec-like triangles, proving a conjecture of the author and Guitter. The result is based on the integrability of the 20 Vertex model and uses a connection to the U-turn boundary 6 Vertex model to re-express the number of 20 Vertex configurations as a simple determinant, which is then related to a Lindström-Gessel-Viennot determinant for the domino tiling problem. The common number of configurations is conjectured to be $2^{n(n-1)/2}\prod_{j=0}^{n-1}\frac{(4j+2)!}{(n+2j+1)!}=1, 4, 60, 3328, 678912...$ The enumeration result is extended to include refinements of both numbers.

Method of lines for solving linear equations of mathematical physics with the third and first types boundary conditions

The use of the method of lines in solving multidimensional problems of mathematical physics makes it possible to eliminate the discrepancies caused by the use of the sweep method in certain coordinates. As a result, the solution of the Poisson equation, for example, is obtained without using the relaxation method. In the article, the problem on the eigenvalues and vectors of the transition matrix is solved for boundary conditions of the third and first types, used to solve a one-dimensional equation of parabolic type by the method of lines. Due to the features of boundary conditions of the third type for determining the eigenvalues, a mixed method was proposed based on the Vieta theorem and the representation of the characteristic equation in trigonometric form typical for the method of lines. To solve the eigenvector problem, a simple sweep method was used with the algebraic compliments to the transition matrix. Discontinuous solutions of a one-dimensional parabolic equation were presented for various values of complex 1 -αl; the method for solving the characteristic equation was selected based on these values. The calculation results are in good agreement with the analytical solution.

ON ROOT FUNCTIONS OF NONLOCAL DIFFERENTIAL SECOND-ORDER OPERATOR WITH BOUNDARY CONDITIONS OF PERIODIC TYPE

In this paper we consider one class of spectral problems for a nonlocal ordinary differential operator (with involution in the main part) with nonlocal boundary conditions of periodic type. Such problems arise when solving by the method of separation of variables for a nonlocal heat equation. We investigate spectral properties of the problem for the nonlocal ordinary differential equation Ly (x) ≡ −y 00 (x) + εy00 (−x) = λy (x), −1 < x < 1. Here λ is a spectral parameter, |ε| < 1. Such equations are called nonlocal because they have a term y 00 (−x) with involutional argument deviation. Boundary conditions are nonlocal y 0 (−1) + ay0 (1) = 0, y (−1) − y (1) = 0. Earlier this problem has been investigated for the special case a = −1. We consider the case a 6= −1. A criterion for simplicity of eigenvalues of the problem is proved: the eigenvalues will be simple if and only if the number r = sqrt{p (1 − ε) / (1 + ε)} is irrational. We show that if the number r is irrational, then all the eigenvalues of the problem are simple, and the system of eigenfunctions of the problem is complete and minimal but does not form an unconditional basis in L2(−1, 1). For the case of rational numbers r, it is proved that a (chosen in a special way) system of eigen- and associated functions forms an unconditional basis in L2(−1, 1).

Boundary Conditions in a Lattice Boltzmann Method For Plane Strain Problems

AbstractThe Lattice Boltzmann Method (LBM), e.g. in [1] and [2], can be interpreted as an alternative method for the numerical solution of certain partial differential equations that is not restricted to its origin in computational fluid mechanics. The interpretation of the LBM as a general numerical tool allows to extend the LBM to solid mechanics as well, see e.g. [3], which is concerned with the simulation of elastic solids under simplified deformation assumptions, and [4] as well as [5] which propose LBMs for the general plane strain case. In previous works on a LBM for plain strain such as [5], the treatment of practically relevant boundary conditions like Neumann and Dirichlet type boundary conditions is not the main focus and thus periodic conditions or absorbing layers are specified to simulate numerical examples. In this work, we show how Neumann and Dirichlet type boundary conditions are implemented in our LBM for plane strain from [4].

Closed-form solution for optimization of buckling column

The optimization problem for a column, loaded by compression forces is studied in this article. The direction of the applied forces coincides until buckling moment with the axis of the column. The critical values of buckling are equal among all competitive designs of the columns. The dimensional analysis eases the mathematical technique for the optimization problem. The dimensional analysis introduces two dimensionless factors, one for the total material volume and one for the total stiffness of the columns. With the method of dimensional analysis, the solution of the nonlinear algebraic equations for the Lagrange multiplier will be superfluous. The closed-form solutions for Sturm-Liouville and mixed types boundary conditions are derived. The solutions are expressed in terms of the higher transcendental functions. The principal results are the closed-form solution in terms of the hypergeometric and elliptic functions, the analysis of single- and bimodal regimes, and the exact bounds for the masses of the optimal columns. The isoperimetric inequality was formulated as the strict inequality sign, because the optimal solution could not be attained for any finite seting of the design parameter. The additional restriction on the minimal area of the cross-section regularizes the optimization problem and leads to the definite attainable shape of the optimal column.

The more envious the consumer, the more impulsive? The moderating role of self-monitoring and product type

Purpose This paper aimed to explore the influence of envy on impulsive consumption from aspects of the internal psychological mechanism and boundary conditions.Design/methodology/approach Based on social comparison theory, four studies were conducted in this research: The first study explored the effect of envy on impulsive consumption; the second study explored the moderating effect of self-monitoring and the mediating effect of materialism; the third study explored the moderating effect of product type and the fourth study explored the effectiveness of social comparison contexts on the arousal of envy.Findings Study 1 showed that envy could significantly trigger consumers' impulsive consumption. Study 2 indicated that participants experiencing self-monitoring had a higher level of materialism and a stronger propensity to consume impulsively once the emotion of envy emerged. Study 3 suggested that when participants were more envious, their levels of materialism increased with more impulsivity to buy material products. Study 4 revealed that upward comparisons led to a higher level of envy and re-validated the mediating role of materialism between envy and impulsive consumption.Research limitations/implications This study provides evidence for the association between envy and consumer behaviour and clarifies the underlying mechanisms of the relationship between envy and impulsive consumption.Practical implications Marketers could take advantage of consumers' envy after social comparisons without damaging brand image.Originality/value First, this study extended the effects of envy on consumer decisions, suggesting that envy stimulates impulsive consumption by increasing consumers' materialism. Second, this study revealed the boundary condition of product type, namely, material and experiential.

Analysis of Stokes system with solution-dependent subdifferential boundary conditions

We study the Stokes problem for the incompressible fluid with mixed nonlinear boundary conditions of subdifferential type. The latter involve a unilateral boundary condition, the Navier slip condition, a nonmonotone version of the nonlinear Navier–Fujita slip condition, and the threshold slip and leak condition of frictional type. The weak form of the problem leads to a new class of variational–hemivariational inequalities on convex sets for the velocity field. Solution existence and the weak compactness of the solution set to the inequality problem are established based on the Schauder fixed point theorem.

On second-order and fourth-order elliptic systems consisting of bulk and surface PDEs: Well-posedness, regularity theory and eigenvalue problems

In this paper, we study second-order and fourth-order elliptic problems which include not only a Poisson equation in the bulk but also an inhomogeneous Laplace–Beltrami equation on the boundary of the domain. The bulk and the surface PDE are coupled by a boundary condition that is either of Dirichlet or Robin type. We point out that both the Dirichlet and the Robin type boundary condition can be handled simultaneously through our formalism without having to change the framework. Moreover, we investigate the eigenvalue problems associated with these second-order and fourth-order elliptic systems.We further discuss the relation between these elliptic problems and certain parabolic problems, especially the Allen–Cahn equation and the Cahn–Hilliard equation with dynamic boundary conditions.

Inverse problem for cracked inhomogeneous Kirchhoff\u2013Love plate with two hinged rigid inclusions

We consider a family of variational problems on the equilibrium of a composite Kirchhoff–Love plate containing two flat rectilinear rigid inclusions, which are connected in a hinged manner. It is assumed that both inclusions are delaminated from an elastic matrix, thus forming an interfacial crack between the inclusions and the surrounding elastic media. Displacement boundary conditions of an inequality type are set on the crack faces that ensure a mutual nonpenetration of opposite crack faces. The problems of the family depend on a parameter specifying the coordinate of a connection point of the inclusions. For the considered family of problems, we formulate a new inverse problem of finding unknown coordinates of a hinge joint point. The continuity of solutions of the problems on this parameter is proved. The solvability of this inverse problem has been established. Using a passage to the limit, a qualitative connection between the problems for plates with flat and bulk hinged inclusions is shown.

Analytical Solution of Non-Newtonian Nanofluid Flows Within Circular Duct under Convective Boundary Condition

At the outset, this work aims to carried out an analytical investigation of forced convection by establishing a laminar flow into circular duct under convective boundary conditions of the third type for non-Newtonian nanofluid, the fluid containing TiO2 uniformly dispersed in aqueous solution with 0.5wt% of CMC solutions (Carboxymethyl Cellulose) is used as working fluid. The viscous dissipation effects are taken into account, the employed methodology is based on a combination of the Ritz variational approach with the Laplace transformation technique, so the power-law fluids flow model is used to describe the non-Newtonian fluid behavior. The effect of dimensionless parameters such as Biot (Bi), Brinkman (Br), Peclet (Pe) numbers, power-law index (n), and nanoparticles concentration (φ) on the temperature distribution contours and on the examined local Nusselt number. Our results have been compared with those found in the literature in particular the cases of base fluids (φ=0) with and without viscous dissipation effects.

Stabilization of the extended finite element method for stiff embedded interfaces and inclusions

AbstractThis article investigates the performance of the XFEM for stiff embedded interfaces and inclusions. It is known that the XFEM may lead to ill‐conditioned stiffness matrices and oscillations in the interface traction field, the severity of which depends on the underlying basis functions, as well as on the orientation of the interface. The jumps at the discontinuity are shown to contain quadratic bubble residuals that can introduce oscillatory behavior. Those residuals are worsened by the ill‐conditioning of the system with very stiff interfaces. A variationally consistent method is proposed to overcome the oscillatory behavior and ill‐conditioning, in which the assumed strain method is developed to directly eliminate bubble residuals and deploy Legendre polynomials and explore their orthogonality properties to improve the conditioning of the stiffness matrices. Numerical examples illustrate the efficiency and generality of the proposed approach at both element and structural levels. The new approach is shown to be robust for imposing Dirichlet type boundary conditions at the crack interface, such as crack closure and initially rigid cohesive laws. The effects of numerical oscillations on the prediction of effective properties of composites in XFEM‐based computational homogenization procedures are also discussed.

On an Inverse Problem for the Stationary Equation with a Boundary Condition of the Third Type

The identification of an unknown coefficient in the lower term of elliptic second-order differential equation Mu + ku = f with the boundary condition of the third type is considered. The identification of the coefficient is based on integral boundary data. The local existence and uniqueness of the strong solution for the inverse problem is proved

Surface manipulation of elastic half-space to suppress Rayleigh wave propagation: A step towards boundary condition-based meta-surface design

A boundary condition (BC)-based design methodology for metasurfaces was recently proposed to control low-frequency Lamb waves in a plate. This study highlights the significance of a type of Cauchy BCs, called Mindlin BCs, for controlling Rayleigh wave propagation and provides motivation for a BC-based metamaterial design for surface wave control. Analytical study on an elastic half-space with Mindlin BCs reveals no possible surface wave solutions. The frequency-domain and time-domain simulations by imposing Mindlin BCs on the surface in the path of Rayleigh wave propagation are consistent with analytical predictions exhibiting no surface wave transmission. The simulation results reveal mode conversions from Rayleigh wave to bulk waves directed into the half-space and a low amplitude Rayleigh wave reflection. For a finite-sized Mindlin BC patch, a portion of mode-converted bulk waves near the surface converts back to Rayleigh waves at the end of the BC patch suggesting the requirement of a minimum Mindlin BC patch length for effective suppression of surface waves. Finally, we show that frequency-dependent Mindlin BCs are imposed by an array of rod-like resonators resulting in a surface wave bandgap. These findings provide new insights into designing novel metasurfaces that provide the necessary coupling for desired surface wave control.

Modeling and Numerical Solution of the Fate of Multicomponent Substrate in Porous Media

A mathematical model was proposed to appropriately describe the fate of multicomponent substrates in porous media especially soil. The model utilised appropriate biodegradation kinetic expressions that better describe the consumption or degradation rate of the substrate. The Equation, with the second and third type boundary conditions in non-dimensionalised form was solved using the Finite Volume method and simulated in the Matlab environment. An experiment, using a 5 cm (inside diameter) x 60 cm (height) glass column packed with severally autoclaved soil spiked with 2 % substrate (a mixture of hexadecane, heneicosane, 1-methylnaphthalene, 2- methylnaphthalene and 1, 3-dimethylnapthalene) and a consortium of organisms (Providential rettgeri, Streptococcus salivarius, Trichoderma harzianum, Aspergillus flavipes, and Candida famata) was set up to validate the model. The result showed that the model describes the fate of each component within the multicomponent substrate. It also indicates that both Peclet and Thiele numbers affect the biodegradation of the substrate. It was observed that small Peclet number should be allowed for effective biodegradation of the substrate. The model was validated with data obtained for an experiment where a mixture of hydrocarbons was degraded with a mixed culture of microorganisms. The results of the experiment were well described by the model indicating that the model can be used to predict compositions of components of a mixture during biodegradation.

Estimates of characteristics of a micropolar flow passing through an axially symmetric cell

We study a model for the filtration of micropolar fluid in the framework of a cell model technique. A porous medium is presented as an assemblage of axially symmetric cells of an arbitrarily geometry. Each cell consists of a solid core, porous layer and liquid shell. The influence of the neighboring cells is taken into account via Cunningham's-type boundary condition. We derive a priori estimates for flow characteristics which show the behavior of the velocity filed. The boundedness of velocity filed is justified by the derived estimates.
 For more information see https://ejde.math.txstate.edu/Volumes/2021/74/abstr.html

Mathematical modeling and algorithm for calculation of thermocatalytic process of producing nanomaterial

<p>At present, when constructing a mathematical description of the pyrolysis reactor, partial differential equations for the components of the gas phase and the catalyst phase are used. In the well-known works on modeling pyrolysis, the obtained models are applicable only for a narrow range of changes in the process parameters, the geometric dimensions are considered constant. The article poses the task of creating a complex mathematical model with additional terms, taking into account nonlinear effects, where the geometric dimensions of the apparatus and operating characteristics vary over a wide range. An analytical method has been developed for the implementation of a mathematical model of catalytic pyrolysis of methane for the production of nanomaterials in a continuous mode. The differential equation for gaseous components with initial and boundary conditions of the third type is reduced to a dimensionless form with a small value of the peclet criterion with a form factor. It is shown that the laplace transform method is mainly suitable for this case, which is applicable both for differential equations for solid-phase components and calculation in a periodic mode. The adequacy of the model results with the known experimental data is checked.</p>

Nonlinear quasi-static poroelasticity

We analyze a quasi-static Biot system of poroelasticity for both compressible and incompressible constituents. The main feature of this model is a nonlinear coupling of pressure and dilation through the system's permeability tensor. Such a model has been analyzed previously from the point of view of constructing weak solutions through a fully discretized approach. In this treatment, we consider simplified Dirichlet type boundary conditions in both the elastic displacement and pressure variables and give a full treatment of weak solutions. Our construction of weak solutions for the nonlinear problem is based on a priori estimates, a requisite feature in addressing the nonlinearity. We utilize a spatial semi-discretization and employ a multi-valued fixed point argument for a clear construction of weak solutions. We also provide regularity criteria for uniqueness of solutions.

Galilean-Invariant Characteristic-Based Volume Penalization Method for Supersonic Flows with Moving Boundaries

This work extends the characteristic-based volume penalization method, originally developed and demonstrated for compressible subsonic viscous flows in (J. Comput. Phys. 262, 2014), to a hyperbolic system of partial differential equations involving complex domains with moving boundaries. The proposed methodology is shown to be Galilean-invariant and can be used to impose either homogeneous or inhomogeneous Dirichlet, Neumann, and Robin type boundary conditions on immersed boundaries. Both integrated and non-integrated variables can be treated in a systematic manner that parallels the prescription of exact boundary conditions with the approximation error rigorously controlled through an a priori penalization parameter. The proposed approach is well suited for use with adaptive mesh refinement, which allows adequate resolution of the geometry without over-resolving flow structures and minimizing the number of grid points inside the solid obstacle. The extended Galilean-invariant characteristic-based volume penalization method, while being generally applicable to both compressible Navier–Stokes and Euler equations across all speed regimes, is demonstrated for a number of supersonic benchmark flows around both stationary and moving obstacles of arbitrary shape.

The effects of fins number and bypassed energy fraction on the solidification process of a phase change material

Nowadays, utilizing latent heat thermal energy storage systems containing phase change materials (PCMs) has been considered one of the best ways for integrating and optimizing thermal energy systems. Although using fins has been suggested by many researchers for the heat transfer enhancement in these systems, this technique is not sufficient in some cases. So, finding its reasons is the main purpose of this paper. For this aim, four different cylindrical containers, one without fins and the others with radial fins, were intended. Then, solidification of the PCM inside the containers under the constant and time-dependent first, second, and third type boundary conditions was modeled by the finite volume method based on the enthalpy formulation. To analyze fins performance, bypassed energy fraction (β) criterion was proposed. The results show that under the first type boundary condition, the value of β is zero, and the fins have a major effect on the heat transfer enhancement, but under the second and third type boundary conditions, the value of this parameter is significant, and the fins have a lower effect. Furthermore, it is found that by increasing the fins number, the value of β and complete freezing time decreased. The results of this study were compared and validated with the results of the analytical approach.

Experimental and Numerical Analysis of the Transient Behavior of the Oil Decongealing Process in an Aero Fuel-Cooled Oil Cooler Under Low-Temperature Conditions

Abstract The fuel-cooled oil cooler (FCOC) in the lubrication circuit plays a critical role in the aero gas-turbine engine's aerothermal management. However, the low temperature of the operating environment can congeal the oil and reduce the FCOC efficiency. The oil bypass valve (OBV) installed on the FCOC prevents pressure loss. Its failure may cause overheating, requiring preemptive performance prediction. Experimental and numerical analyses were used to evaluate the cooler's decongealing performance under typical boundary conditions of pressure and temperature, OBV configurations, and rerouting of feed oil and fuel flow paths. The temporal variation of oil and fuel mass flow rates, temperature, and pressure of the feed oil and fuel provided an insight into the decongealing process and duration. The experimental data were used to develop a one-dimensional (1D) flow and thermal network analysis model based on the effectiveness (ε)-number of thermal units (NTU) method to predict the transient oil decongealing performance of the FCOC. The customized commercial code predicted the decongealing phenomena using empirical correlations with property correction schemes, showing good agreement with the experiment. The findings revealed various ways to enhance the decongealing performance of the FCOC. The study results showed that the operating boundary conditions, OBV location and status, and flow arrangements affect decongealing behavior and time. The present numerical model provides results quickly and can effectively predict experimentally costly and complicated cases. The attempted estimates of steady heat rejection and detailed methodology could guide future studies and practical applications.