Symmetric Boundary Conditions Research Articles

Development of thermal-hydraulic coupling model for deep-geological disposal of high-level radioactive wastes

The buffer plays an important role in a engineered barrier system for the deep geological repository of high-level radioactive waste (HLW). High temperature may result in degradation of buffer performance. It is crucial to investigate the buffer temperature behavior so that its peak temperature is not higher than the criterion of 100 °C. In the present work, the thermal-hydraulic (T-H) coupling model is developed to simulate the histories of temperature and saturation in the disposal repository. This model is validated with the previous simulation works in the DECOVALEX 2015 Task B2 in order to simulate the geological repository of HLW with confidence. The whole repository of HLW in the present works can be reasonably simplified as a single hole with the symmetric boundary conditions on its peripheral sides. Effects of tunnel spacing and hole spacing of repository on the histories of buffer peak temperature are considered since the layout with various spacing is strongly related to the peak temperature of buffer. Based on the comparison of simulation results from the T-H coupling model and T-only model for the different layout of whole HLW repository, the maximum value of peak temperature in the buffer predicted by the T-H coupling model is lower than that from the T-only model. In other words, less area is needed for the selection of repository, rendering that the present T-H coupling model can assist in the design of HLW repository.

Variational approach to a symmetric boundary value problem generated by a system of equations and separated boundary conditions

This work provides some information on the eigenvalues and eigenfunctions of a problem which is constructed by a system of equations and symmetric boundary conditions that includes the ordinary second‐order Sturm–Liouville boundary value problem. In particular, we show that the problem has an infinite number of discrete eigenvalues with a greatest lower bound and the corresponding eigenfunctions are complete in mean and energy. We introduce the results using the variational approach that enables us to consider only continuous pair functions instead of absolutely continuous pair functions.

Examining the Applicability of Direct Analytic Method (DAM) to Normal Modes of Poloidal Oscillations Under Symmetric and Asymmetric Boundary Conditions

AbstractA working Direct Analytic Method (DAM) model is envisaged to explain the normal modes of poloidal Alfven waves in the Earth's magnetosphere. The model solves the ideal, cold, magnetohydrodynamic (MHD) equations associated with transverse components of the magnetic perturbations in a dipolar magnetic field. DAM model is used to study the transverse poloidal waves in different regions of magnetosphere characterized by their L‐value and different plasma variability. The plasma density distribution is assumed to be governed by the standard power law, 1/rm, where r is the geocentric distance of any point of interest on the field line and m is the density index. The eigen frequencies and spatial structures are obtained analytically under different ideal ionospheric boundary conditions and the results are compared with the numerical solutions to establish the validity of the model. DAM, being an analytic model, is used to explain the distinctive structural features of transverse poloidal waves which are obtained under different boundary conditions, for different density indices. Furthermore, the application of the analytic model in the computation of eigen frequency as well as plasma density is demonstrated under different observational scenario.

Using the method of differential equations by quadratic to solve the free vibrations of columns under the effect of axial load and column weight

This paper investigates various techniques used to solve the differential equations governing the free vibration of columns. The present work focuses on the study of the free vibration of Euler’s Bernoulli column of equal strength in compression, considering its own weight and the axial load in compression and tension while subjected to symmetrical boundary conditions. The investigation utilizes the differential quadrature method to examine the fifth natural frequency parameters of the column in different states of column boundary conditions and varying geometric section shapes, including pin-pin and clamp-clamp configurations. The results of this work contribute valuable insights for informed decisions on selecting the cross-section types and appropriate boundary conditions for ensuring the stability of such columns in civil constructions.

Influence of Design Parameters of Liquid Crystal Cell on the Optical Threshold Voltage Value of Electro-Optical Effect

The influence of liquid-crystal (LC) cell design parameters on the voltage of the optical threshold of the interference effect of polarized beams controlled by an electric field has been studied by computer simulation. It is shown that under strict boundary conditions (both symmetrical and antisymmetrical), the voltage value of the optical threshold decreases rapidly depending on the cell thickness in an exponential manner and reaches a certain constant value. Under non-strict boundary conditions (both symmetrical and antisymmetrical), the voltage dependence of the optical threshold on the LC layer thickness follows a hyperbolic pattern. At the same time, under strict symmetric boundary conditions, the voltage magnitude of the optical threshold effect decreases exponentially with an increase in the pretilt angle of LC molecules on the orienting surfaces. If the boundary conditions are rigid and antisymmetric, an increase in the pretilt angle on the orienting surfaces results in a linear decrease of the optical threshold voltage of the effect. The voltage magnitude of the optical threshold effect increases linearly with an increase in the twist angle of the LC structure in the cell, for both symmetric and antisymmetric boundary conditions. A decrease in the anchoring energy of LC molecules on a substrate leads to a decrease in the voltage of the optical threshold of the effect, and this regularity has an exponential character depending on the reciprocal value of the anchoring energy of LC molecules on orienting substrates.

Analysis of the Burgers–Huxley Equation Using the Nondimensionalisation Technique: Universal Solution for Dirichlet and Symmetry Boundary Conditions

The Burgers–Huxley equation is important because it involves the phenomena of accumulation, drag, diffusion, and the generation or decay of species, which are common in various problems in science and engineering, such as heat transmission, the diffusion of atmospheric contaminants, etc. On the other hand, the mathematical technique of nondimensionalisation has proven to be very useful in the appropriate grouping of the variables involved in a physical–chemical phenomenon and in obtaining universal solutions to different complex engineering problems. Therefore, a deep analysis using this technique of the Burgers–Huxley equation and its possible boundary conditions can facilitate a common understanding of these problems through the appropriate grouping of variables and propose common universal solutions. Thus, in this case, the technique is applied to obtain a universal solution for Dirichlet and symmetric boundary conditions. The validation of the methodology is carried out by comparing different cases, where the coefficients or the value of the boundary condition are varied, with the results obtained through a numerical simulation. Furthermore, one of the cases presented presents a boundary condition that changes at a certain time. Finally, after applying the technique, it is studied which phenomenon is predominant, concluding that from a certain value diffusion predominates, with the rest being practically negligible.

Analysis of effects of material anisotropy on ductile damage using microscopic unit‐cell model

AbstractIt is experimentally observed that the failure in ductile metals is mainly due to the nucleation, growth, and coalescence of micro‐voids as well as micro‐shear‐cracks. Furthermore, plastic anisotropy has significant role in damage and failure behavior of ductile metals. Finite element simulations of unit cell provide a basis to understand different mechanisms on micro‐scale, for example, changes in shape and size of single voids and defects as well as localization of plastic strains. This contribution deals with the numerical analysis of unit cell containing spherical void subjected to symmetrical boundary conditions taking material anisotropy into account. Elastic isotropic behavior is described by Hooke's law while Hoffman yield criterion considering the strength‐differential effect is used to model the anisotropic plastic behavior. Generalized anisotropic stress invariants, generalized stress triaxiality, and generalized Lode parameter are introduced to characterize the stress state in the anisotropic ductile metal. The effect of plastic anisotropy on the damage behavior of the aluminum alloy EN AW‐2017A is studied in detail by performing a series of numerical simulations covering a wide range of stress triaxialites and Lode parameters. Stress triaxility and Lode parameter are controlled and kept constant during the entire loading process. The numerical results are then used to discuss general mechanisms of damage and failure process in ductile metals.

Elastic graphs with clamped boundary and length constraints

AbstractWe study two minimization problems concerning the elastic energy on curves given by graphs subject to symmetric clamped boundary conditions. In the first, the inextensible problem, we fix the length of the curves while in the second, the extensible problem, we add a term penalizing the length. This can be considered as a one-dimensional version of the Helfrich energy. In both cases, we prove existence, uniqueness and qualitative properties of the minimizers. A key ingredient in our analysis is the use of Noether identities valid for critical points of the energy and derived from the invariance of the energy functional with respect to translations. These identities allow us also to prove curvature bounds and ordering of the minimizers even though the problem is of fourth order and hence in general does not allow for comparison principles.

An Improved Lax-Wendroff Scheme for Two-Dimensional Transient Thermal Simulation

The stability and accuracy of explicit high-order finite difference (HOFD) algorithms have been research hotspots in different fields. To improve the stability and accuracy of the HOFD algorithms in thermal simulations, we present a Lax-Wendroff high-order finite difference (LHOFD) algorithm to solve the 2D transient heat transfer equation in this paper and develop an improved LHOFD (IHOFD) algorithm to improve the stability of the LHOFD algorithm. The formulas of the general high-order central FD (HOCFD) coefficients and the truncation error coefficient as well as the high-order non-central FD (HONFD) coefficients and the truncation error coefficient of the fourth-order spatial derivative are derived concisely in a different way. Furthermore, a unified analytical formula of the general HOCFD and HONFD coefficients, which can calculate the spatial derivative of any integer order, is derived. A new strategy of combination with the HOCFD and HONFD approximations under the same high-order accuracy as the internal computational domain is proposed to calculate the mixed derivatives of the boundary domains with high accuracy, no additional computational cost, and easy implementation. Then, the accuracy analysis, stability analysis, and comparative analysis of numerical simulation results obtained by the LHOFD and IHOFD algorithms with the exact solution show the correctness and validity of the proposed algorithms and their stability formulas, and the advantages of the proposed algorithms. The proposed algorithms are valid under both symmetric and asymmetric boundary conditions. The stability factor of the LHOFD algorithm is slightly higher than that of the conventional algorithm. The stability factor of the IHOFD algorithm is twice that of the conventional algorithm, and the maximum absolute error of the thermal simulation is within 0.015 (°C).

Development of computational methodology in simulating thermal responses of spent fuel in deep geological disposal

Deep geological repository with multiple barriers has been recognized as the safety management of nuclear wastes in the world. The decay heat from the spent fuel can elevate the temperature values of canisters, buffers, backfills, and host rock, which may challenge repository integrity. It is crucial to investigate the thermal response of in order to maintain the performance and the integrity of repository system. A gap of millimeter order in general exists between the canister/buffer interface. This gap effect would strongly influence the temperature distribution in the buffer and its peak temperature. A large amount of mesh size and computing power is needed to realistically model this small gap for numerically simulating the temperature responses in the entire disposal site. A simplified computational methodology with the effective conductivity of buffer (λeff) is proposed in this paper, which can reasonably predict the peak buffer temperature with dramatic reduction in the mesh size and numerical computation. In addition, under the uniform heat source distribution in the whole repository system, a geometric simplified model with 1/4 deposited hole + 4-side symmetric boundary conditions can also precisely capture the peak buffer temperature that is close to the value predicted using the entire disposal-site simulation. In the engineering point of view, the λeff model and the geometric simplified model can be applied in the numerical simulation of temperature responses for the whole repository system.

Methodology for Solving Engineering Problems of Burgers–Huxley Coupled with Symmetric Boundary Conditions by Means of the Network Simulation Method

The Burgers–Huxley equation is a partial differential equation which is based on the Burgers equation, involving diffusion, accumulation, drag, and species generation or sink phenomena. This equation is commonly used in fluid mechanics, air pollutant emissions, chloride diffusion in concrete, non-linear acoustics, and other areas. A general methodology is proposed in this work to solve the mentioned equation or coupled systems formed by it using the network simulation method. Additionally, the implementation of the most common possible boundary conditions in different engineering problems is indicated, including the Neumann condition that enables symmetry to be applied to the problem, reducing computation times. The method consists mainly of establishing an analogy between the variables of the differential equations and the electrical voltage at a central node. The methodology is also explained in detail, facilitating its implementation to similar engineering problems, since the equivalence, for example, between the different types of spatial and time derivatives and its correspondence with the electrical device is detailed. As an example, several cases of both the equation and a coupled system are solved by varying the boundary conditions on one side and applying symmetry on the other.

Loading-unloading of spherical and cylindrical cavities in cohesive-frictional materials with arbitrary radially symmetric boundary conditions

Many scenarios in geotechnical engineering involve cavity unloading from a loaded state under different boundary conditions, especially for thick-walled cylinder tests in laboratory conditions. A novel analytical cavity expansion-contraction solution is proposed for modeling thick-walled cylinder tests during a complete loading-unloading process under arbitrary symmetric boundary conditions. The proposed solution has potential implication in investigating ambiguous boundary effects for problems involving cavity expansion-contraction. The mechanical model considers two concentric layers of materials around the cavity, where the inner layer is considered as cohesive-frictional and the circumambient infinite layer is assumed as elastic to simulate the arbitrary radially symmetric boundary condition, including two extreme scenarios with constant-pressure and zero-displacement boundaries. The derived solution provides the evolutions of cavity pressure and distributions of stresses and displacements in both layers during the loading-unloading process, which is then rigorously validated against finite element results. The effects of boundary conditions and size of the cohesive-frictional layer have been comprehensively investigated by analyzing the cavity pressure and the developments of plastic zones during expansion and contraction. The developed solution has been proved to be applicable in geotechnical engineering, with comparisons between analytical results and experimental data for pressuremeter tests in calibration chambers and pile installation tests in a rigid container. The proposed solution might also be helpful in the analysis of autofrettaged pressure vessels.

Numerical Calculation of Thermal Radiative Boundary Layer Nanofluid Flow across an Extending Inclined Cylinder

This research presents the numerical analysis of the fluid flow containing the micro gyrotactic organism with heat and mass transfer. The flow is allowed to pass through an inclined stretching cylinder with the effects of heat generation/a heat source and activation energy subject to the symmetric boundary conditions at the cylinder walls. Similarity transformation is employed in the system of PDEs (partial differential equations) to transform them into non-dimensional ODEs (ordinary differential equations). The solution to the proposed problem is obtained by using the bvp4c (numerical scheme). The graphical results are plotted for various flow parameters in order to show their impact on the flow, mass, energy, and motile microorganism profiles. Moreover, the angle of inclination disturbs the flow within an inclined cylinder and slows down the fluid motion, while it elevates the energy of the fluid inside an inclined cylinder. Similarly, the curvature effect is also highlighted in the dynamics of fluid velocity, temperature, and the motile microorganism profile. From the obtained results, it is elucidated that growing values of the curvature factor accelerate the temperature, velocity, and motile microbes’ profiles. Finally, some engineering quantities are calculated in terms of skin friction, the Nusselt and Sherwood number, and the density of motile microbes. The acquired results are also displayed in tabular form.

Automatic boundary fitting framework of boundary dependent physics-informed neural network solving partial differential equation with complex boundary conditions

The physics-informed neural network (PINN) has received much attention in the field of partial differential equation (PDE) solving due to its adaptability to different governing equations. Boundary dependent PINN (BDPINN) has much higher precision than PINN, but requires to manually design complex form of a trial function to fit boundary conditions (BCs). In this paper, an automatic boundary fitting framework is introduced into BDPINN for the automatic construction of a trial function satisfying BCs to solve PDE problems with complex geometry, and radial basis function (RBF) and neural network function (NNF) are used to construct boundary fitting functions, respectively. The proposed automatic boundary fitting framework can successfully construct a trial function of BDPINN for common BCs and their combinations, including initial conditions, Dirichlet BCs, symmetric BCs and periodic BCs. With this framework, BDPINN is applied in to 4 cases. The corresponding governing equations include diffusion, advection, and advection–diffusion equations, with dimensions ranging from 2D to 4D, and the corresponding geometric boundaries include random, irregular, and distorted boundaries. The calculation results show that the BDPINN with this framework succeeds in various governing equations, in various dimensions, and in various BCs with high accuracy, and that BDPINN with RBF is more accurate than that with NNF.

Progressive failure analysis of anisogrid cylindrical composite lattice structure with manufacturing defects

This study performs a compression test of a cylindrical composite lattice structure and numerically investigates the failure process and behavior. Prior to the test and analysis, the distributions of the fiber tow, matrix, and manufacturing defects were observed and measured in cross-sectional images. A progressive failure analysis that considers different matrix pocket lengths and void contents was performed to examine the effects of manufacturing defects on the compressive strength. The intersecting fiber tow and matrix pocket was modeled using a continuum element and the geometry measured in the cross-sectional image. A cyclically symmetric boundary condition was applied to the symmetric model. Fracture energy-based damage evolution for the fiber tow and cohesive zone modeling for the interface failure were employed. The failure behaviors of the different damage models were compared, and the progressive failure analysis model was verified by comparing it with the test results. The results of the progressive failure analysis, considering manufacturing defects, were similar to the test results.

Rayleigh–Bénard magnetoconvection with asymmetric boundary condition and comparison of results with those of symmetric boundary condition

Rayleigh–Bénard magnetoconvection with asymmetric boundary condition and comparison of results with those of symmetric boundary condition

Comparison of the effect of suction-injection-combination on Rayleigh–Bénard convection in the case of asymmetric boundaries with those of symmetric ones

The effect of suction-injection-combination (SIC) on the linear and weakly nonlinear stability of Rayleigh–Bénard convection is considered in the paper for the cases of symmetric and asymmetric boundary conditions. Using the Maclaurin series with an appropriate number of terms, expression for eigenfunctions is obtained. The linear theory corroborates the results obtained using the chosen eigenfunctions in the limiting case of the no-SIC effect by matching accurately with the exact values concerning the critical Rayleigh number (Rac) and the wave number (αc). It is found that the effect of SIC is to stabilize the system in the case of symmetric boundaries irrespective of SIC being pro-gravity or anti-gravity. However, the effect of SIC is to stabilize/destabilize the system depending on SIC being pro-gravity or anti-gravity in the case of the asymmetric boundaries. We also noted a similar effect in the case of αc wherein a maximum error of order 10−4 was observed. The main novelty of the present work is studying the influence of SIC on the nonlinear dynamics of the considered problem. It is shown that the effect of SIC is to hasten the onset of chaos. Using various indicators (the largest Lyapunov exponent, the time series solution, the amplitude spectrum, and the phase-space plots), the dynamical behavior of the system is analyzed and the influence of SIC on the dynamics is recorded. The change due to the boundary effect and the SIC on the size of convection rolls and the trapping region where the dynamical system evolves within a bound is highlighted in the paper.

Three-dimensional electro-convective flows of dielectric liquids between concentric cylinders

This study numerically investigates the flow structures and bifurcations of three-dimensional (3D) electro-convection (EC) between concentric cylinders. The flow motion is driven by the volumetric Coulomb force exerting on the free ions introduced by a strong unipolar injection. Three different sets of cases with periodic or symmetric boundary conditions are carried out. The finite volume method is used to numerically resolve the model problem. The co-effect of 2D roll mode and 3D polygon mode at the initial period of the EC flow is identified. The EC flows are made up of different types of regular or irregular centrally downflowing polygonal cells and are characterized by the central charge density cores surrounded by the charge void regions. The symmetry, periodicity, and staggered distribution characteristics of polygonal cells are identified. The cell patterns of EC flow are strongly influenced by the computational domain and the electric Rayleigh number (T). The subcritical bifurcation of linear instability together with a hysteresis loop is observed. In addition, the stability of the triangular flow pattern is analyzed at a large range of T and found that the annular EC flow is more likely to be unstable in three dimensions than in two dimensions.

On the proofs of orthogonality of eigenfunctions for heat conduction, wave propagation, and advection-diffusion problems

The orthogonality of eigenfunctions in problems of unsteady heat conduction in an infinite slab with symmetric and nonsymmetric convective boundary conditions are demonstrated by performing actual integration of the products of the derived forms of eigenfunctions and by implementing the corresponding eigenvalue conditions. The analysis also applies to longitudinal vibrations of an elastic rod attached at its ends to linear elastic springs, and to advection-diffusion problems under appropriate boundary conditions. The same form of eigenfunctions and the same type of eigenvalue condition apply in all three considered cases. The proofs are compared with the well-known general proof of orthogonality of eigenfunctions, which circumvents the actual integration. The presented analysis is pedagogically appealing for use in engineering and applied physics education.

Systems of Sequential ψ1-Hilfer and ψ2-Caputo Fractional Differential Equations with Fractional Integro-Differential Nonlocal Boundary Conditions

In this paper, we introduce and study a new class of coupled and uncoupled systems, consisting of mixed-type ψ1-Hilfer and ψ2-Caputo fractional differential equations supplemented with asymmetric and symmetric integro-differential nonlocal boundary conditions (systems (2) and (13), respectively). As far as we know, this combination of ψ1-Hilfer and ψ2-Caputo fractional derivatives in coupled systems is new in the literature. The uniqueness result is achieved via the Banach contraction mapping principle, while the existence result is established by applying the Leray–Schauder alternative. Numerical examples illustrating the obtained results are also presented.