Simple Boundary Research Articles

On Non-Linear Differential Systems with Mixed Boundary Conditions

For the constructive analysis of locally Lipschitzian system of non-linear differential equations with mixed periodic and two-point non-linear boundary conditions, a numerical-analytic approach is developed, which allows one to study the solvability and construct approximations to the solution. The values of the unknown solution at the two extreme points of the given interval are considered as vector parameters whose dimension is the same as the dimension of the given differential equation. The original problem can be reduced to two auxiliary ones, with simple separable boundary conditions. To study these problems, we introduce two different types of parametrized successive approximations in analytic form. To prove the uniform convergence of these series, we use the appropriate technique to see that they form Cauchy sequences in the corresponding Banach spaces. The two parametrized limit functions and the given boundary conditions generate a system of algebraic equations of suitable dimensions, the so-called system of determining equations, which give the numerical values of the introduced unknown parameters. We prove that the system of determining equations define all possible solutions of the given boundary value problems in the domain of definition. We established also the existence of the solution based on the approximate determining system, which can always be produced in practice. The theory was presented in detail in the case of a system of differential equations consisting of two equations and having two different solutions.

Static stability of functionally graded porous nanoplates under uniform and non-uniform in-plane loads and various boundary conditions based on the nonlocal strain gradient theory

This work examines the buckling behavior of functionally graded porous nanoplates embedded in elastic media. Size effects are added to the nanoplate constitutive equations using nonlocal strain gradient theory. The four-variable refined plate theory is employed for nanoplate modeling. This theory assures stress-free conditions on both sides of the nanoplate and has less uncertainty than high-order shear deformation theories. It is postulated that the nanoplate experiences in-plane compressive loads, which may have both linear and nonlinear distributions. Additionally, uniform and non-uniform porosity distributions are considered. The governing partial differential equations are extracted using the notion of the minimal total potential energy. Following this, the Galerkin method is employed to solve these equations utilizing trigonometric shape functions. Simple, clamped, and combined boundary conditions for nanoplate edges are studied. Once the governing algebraic equations were extracted, the critical buckling load of the nanoplate is determined. To conduct a validation study, the obtained data are juxtaposed with the findings of previous studies, revealing a notable level of concurrence. After the critical buckling load has been ascertained, an inquiry is undertaken to assess the influence of various parameters including nonlocal and length scale parameters, boundary conditions, porosity distribution type, in-plane loading type, geometric dimensions of the nanoplate, and stiffness of the elastic environment, on the static stability of nanoplates.

A Mathematical Model for Collective Behaviors and Emergent Patterns Driven by Multiple Distinct Stimuli Produced by Multiple Species

Collective migration underlies key developmental and disease processes in vertebrates. Mathematical models describing collective migration can shed light on emergent patterns arising from simple mechanisms. In this paper, a mathematical model for collective migration is given for arbitrary numbers and types of individuals using principles outlined as a set of assumptions, such as the assumed preference for individuals to be “close but not too close" to others. The model is then specified to the case of two species with arbitrary numbers of individuals in each species. A particular form of signal response is used that may be parameterized based on experiments involving two or three agents. In its simplest form, the model describes two species of individuals that emit distinct signals, distinguishes between them, and exhibits responses unique to the type by moving according to signal gradients in various planar regions, a situation described as "mixotaxis". Beyond this simple form, initial conditions and boundary conditions are altered to simulate specific, additional in vitro as well as in vivo dynamics. The behaviors that were specifically accounted for include motility, directed migration, and a functional response to a signal. Ultimately, the paper’s results highlight the ability of a single framework for signal and response to account for patterns seen in multi-species systems, in particular the emergent self-organization seen in the embryonic development of placodal cells, which display chase-and-run behavior, flocking behavior, herding behavior, and the splitting of a herd, depending on initial conditions. Numerical experiments focus around the primary example of neural crest and placodal cell “chase-and-run” and “flocking” behaviors; the model reproduces the separation of placodal cells into distinct clumps, as described in the literature for neural crest and placodal cell development. This model was developed to describe a heterogeneous environment and can be expanded to capture other biological systems with one or more distinct species.

Numerical study on temperature and thermal stress behaviors in silicon carbide heating elements within high-temperature annealing furnaces

In response to global sustainability goals, the transition from fossil fuel-based to electric silicon carbide (SiC) heating elements in continuous annealing furnaces is critical for reducing emissions, enhancing safety, and improving operational environments in the steel industry. This study investigates the cross-sectional temperature profiles and the resultant thermal stress of hollow cylindrical SiC heating elements under varied design conditions. To independently analyze the complex effects of multiple factors, both theoretical and numerical analyses were conducted using simplified boundary conditions in two dimensions, which closely simulate the actual temperature profiles observed in heaters within a continuous annealing line. The results indicate that the radial temperature gradient and resulting thermal stress in these elements are directly proportional to the surface heating load, with the maximum axial tensile stress occurring at the outer surface of the heater. Scenarios involving closely spaced multiple heaters or the introduction of cold steel strips lead to increased axial thermal stress due to additional circumferential temperature gradients. Conversely, the application of shield tubes, despite increasing the heater’s temperature, reduces thermal stress by homogenizing radiation within the tube. Consequently, optimizing heater configuration and operational conditions is essential in furnace design to effectively manage heaters’ temperature and thermal stress, thereby ensuring both efficiency and service longevity of the heating elements.

Dynamic coefficient of flexural motion of beam experiencing simple support under successive moving loads

AbstractAnalytic expressions of the dynamic coefficient (DC) factor and vibrational behavior of a uniformly elastic isotropic beam with a simple boundary condition caused by accelerating masses with varying velocities are analyzed. The motion of this problem is described by a fourth‐order partial differential equation, which governs its behavior. The weighted residual method converts the governing equation into a sequence of linked second‐order differential equations to facilitate the analysis. A rewritten version of Struble's asymptotic method further simplifies the transformed governing equation. This modification aids reduction in the complexity of the equation. The closed‐form response is contrasted across three force motions: acceleration, deceleration, and uniform motion. The study thoroughly examines how different velocities and frequencies of the moving force affect the dynamic behavior of the beam. The study also examines the influence of load velocity on the DC of the beam subjected to pinned–pinned boundary conditions.

Three-Dimensional Axisymmetric Analysis of Annular One-Dimensional Hexagonal Piezoelectric Quasicrystal Actuator/Sensor with Different Configurations

The presented article is about the axisymmetric deformation of an annular one-dimensional hexagonal piezoelectric quasicrystal actuator/sensor with different configurations, analyzed by the three-dimensional theory of piezoelectricity coupled with phonon and phason fields. The state space method is utilized to recast the basic equations of one-dimensional hexagonal piezoelectric quasicrystals into the transfer matrix form, and the state space equations of a laminated annular piezoelectric quasicrystal actuator/sensor are obtained. By virtue of the finite Hankel transform, the ordinary differential equations with constant coefficients for an annular quasicrystal actuator/sensor with a generalized elastic simple support boundary condition are derived. Subsequently, the propagator matrix method and inverse Hankel transform are used together to achieve the exact axisymmetric solution for the annular one-dimensional hexagonal piezoelectric quasicrystal actuator/sensor. Numerical illustrations are presented to investigate the influences of the thickness-to-span ratio on a single-layer annular piezoelectric quasicrystal actuator/sensor subjected to different top surface loads, and the effect of material parameters is also presented. Afterward, the present model is applied to compare the performance of different piezoelectric quasicrystal actuator/sensor configurations: the quasicrystal multilayer, quasicrystal unimorph, and quasicrystal bimorph.

Extension of the local domain-free discretization method to large eddy simulation of compressible flows

Most of the flow problems encountered in practical engineering are wall-bounded turbulent flows at high Reynolds numbers. Wall-modeled large eddy simulation (WMLES) is one of the most viable approaches for predicting these realistic flows. Immersed boundary (IB) approach is an efficient computational technique to solve flow problems involving complex and/or moving geometries. This work extends a sharp-interface IB method, named the local domain-free discretion (DFD), to WMLES of compressible flows at high Reynolds numbers. An equilibrium wall model based on solving the simplified compressible turbulent boundary layer equations is utilized to alleviate the requirement of high near-wall mesh resolution. In conjunction with the approximate boundary conditions prescribed by the modeled wall shear stress and wall heat flux, the tangential velocity and temperature at an exterior dependent node are evaluated. Then, the closure of the discrete form of governing equations at an interior node in the immediate vicinity of the immersed wall is accomplished. A simple non-equilibrium correction of the wall shear stress provided by the equilibrium wall model is introduced explicitly. The WMLES/DFD method is applied to a supersonic zero-pressure-gradient turbulent boundary layer flow, a shock wave/flat-plate boundary layer interaction, a supersonic compression ramp flow and high-speed turbulent Couette flows with various thermal boundary conditions. The influence of grid resolution is investigated in the simulation of zero-pressure-gradient turbulent boundary layer flow. By comparing the computed results with the referenced experimental data and/or numerical results, the accuracy and ability of the WMLES/DFD method to simulate compressible turbulent flows are verified.

Simplified inverse distance weighting-immersed boundary method for simulation of fluid-structure interaction

When simulating fluid-structure interaction (FSI) problems involving moving objects, the implicit inverse distance weighting-immersed boundary method (IDW-IBM) developed by Du et al. [1] has to construct a large square correlation matrix and solve its inversion at each time step. In this work, a simplified inverse distance weighting-immersed boundary method (SIDW-IBM) is proposed to eliminate the intrinsic limitations in the original implicit IDW-IBM. Through error analysis using Taylor series expansion, a second order approximation can be derived, which allows us to approximate the large square correlation matrix into a diagonal matrix; thereby, we proposed the SIDW-IBM based on this second order approximation to explicitly evaluate the velocity corrections, where the needs to assemble the large correlation matrix and inverse it are circumvented. Owing to the fact that the inverse distance weighting interpolation removes the limitations in the Dirac delta function, the proposed SIDW-IBM has been successfully implemented on the non-uniform meshes to further improve the computational efficiency. The proposed SIDW-IBM is integrated with the reconstructed lattice Boltzmann flux solver (RLBFS) [2] to simulate some classic incompressible viscous flows, including flow past an in-line oscillating cylinder, flow past a heaving airfoil, and flow past a three-dimensional flexible plate. The good agreement between the present results and reference data demonstrates the capability and feasibility of the SIDW-IBM for simulating FSI problems with moving boundaries and large deformations.

A simple, effective and high-precision boundary meshfree method for solving 2D anisotropic heat conduction problems with complex boundaries

A simple, effective and high-precision boundary meshfree method called virtual boundary meshfree Galerkin method (VBMGM) is used to tackle 2D anisotropic heat conduction problems with complex boundaries. Temperature and heat flux are expressed by virtual boundary element method. The virtual source function is constructed through the utilization of radial basis function interpolation. Calculation model diagram and discrete model diagram of real boundaries, and schematic diagram of VBMGM are demonstrated. Using Galekin method and considering boundary conditions, the integral equation and the discrete formula of VBMGM are given in detail. The benefits of the Galerkin, meshfree, and boundary element methods are all presented in VBMGM. Seven numerical examples of general anisotropic heat conduction problems (including three numerical examples with complex boundaries and four numerical examples with mixed boundary conditions) are computed and contrasted with precise solutions and different numerical methods. The computation time of each example is given. The number of degrees of freedom used in many examples is half or less than that of the numerical method being compared. The suggested method has been demonstrated to be effective and high-precision for solving the 2D anisotropic heat conduction problems with complex boundaries.

Three-dimensional central-moment pseudopotential lattice Boltzmann model with improved discrete additional term

In this work, a three-dimensional central-moment pseudopotential lattice Boltzmann model is developed to simulate a two-phase flow and wetting phenomena. In this model, an improved discrete additional term is proposed to regulate the thermodynamic consistency and surface tension. Different from the discrete additional terms in previous models where only low-order terms are derived at the macroscopic Navier–Stokes equation level, high-order terms are correctly constructed at the mesoscopic lattice Boltzmann equation level in the present improved discrete additional term so that the high-order central moments can be modified in the collision step. With the improved discrete additional term, the simple relationship between the interaction force and the pseudopotential functions is well preserved. On this basis, a simplified wetting boundary scheme is further proposed, which eliminates the complex process for choosing proper characteristic vectors and interpolation. Numerical simulations demonstrate that the proposed model can achieve better performance in thermodynamic consistency, Galilean invariance, numerical stability and computational efficiency, and have great ability to simulate two-phase flow and wetting phenomena on realistic conditions.

Boundary conditions in flutter simulations of subsonic, transonic and supersonic blade cascades

Simulations of blade flutter are highly sensitive to undesired wave reflections at inlet and outlet boundaries. A careful treatment of boundary conditions is required to prevent the generation of perturbations. This study is motivated by the need to perform flutter analysis of low-pressure steam turbine blades, for which supersonic inflow conditions may occur in the near-tip region. The exact steady non-reflecting boundary condition (NRBC), the spectral NRBC and a simple isentropic boundary condition are implemented in a time-marching flow solver and applied to turbomachinery flutter simulations covering a wide range of operating conditions. For the first time, the spectral NRBC is applied to a blade flutter simulation with a supersonic inlet and its performance is analysed and compared with other boundary condition formulations. It is shown that an effective non-reflective treatment in the design of the boundary condition is essential for an accurate aeroelastic prediction at all operating conditions, including the subsonic flow regime. The limitation of the exact steady NRBC to spatial modes causes it to perform poorly in some unsteady flow simulations, whereas the spectral NRBC achieves a satisfactory suppression of undesired wave reflections in all investigated cases.

Total capacity and LOS estimation of innovative and conventional roundabouts through macroscopic fundamental diagrams

Traffic congestion remains a significant problem, especially in urban areas. To avoid congestion phenomena, several measures can be adopted, including the construction of roundabouts and the control of the measures of effectiveness (MOEs) at intersections. This article compares different types of roundabouts by estimating the deterministic fundamental diagram (MFD) and using microscopic traffic simulations. The deduction of MFD for different roundabout types (single and double-lane, flower and turbo-roundabout) is based on reasonably simplified traffic and boundary conditions. MFDs are estimated through several OD traffic matrices, considering as outputs the values of the macroscopic traffic variable deduced in time intervals of 5 min and 1 h. The findings from this research, in terms of total capacity, travel times, and level of service (LOS) estimation, may be useful for traffic and highway engineers in many realistic applications to partially face congestion phenomena in urban areas. MFDs allow us to determine the best roundabout type based on the traffic demand and represent a parsimonious method for monitoring and controlling traditional and unconventional or smart roundabouts, even in the presence of connected and automated vehicles (CAVs).

Dynamic Recrystallization of Olivine During Simple Shear: Evolution of Microstructure and Crystallographic Preferred Orientation From Full‐Field Numerical Simulations

AbstractUpper mantle deformation is mainly controlled by the mechanical behavior of olivine. Crystallographic preferred orientations (CPOs) develop in olivine due to crystal‐plastic deformation during mantle flow, where the a‐axes of olivine polycrystalline aggregates are aligned with the flow direction. Therefore, the observed CPO in olivine‐rich rocks is used as an indicator of the mantle flow direction. Experimental data show that olivine rheology is strongly controlled by the microstructure. While the influence of plastic deformation is in general well characterized, the role of dynamic recrystallization during deformation is not totally understood, limiting our ability to interpret the deformation history of naturally deformed rocks. This contribution presents microdynamic numerical simulations of olivine polycrystalline aggregates with different iron content (i.e., fayalite content) with the aim of exploring the CPO and grain size response to dynamic recrystallization. We use a full‐field approach with an explicit simulation of viscoplastic deformation and dynamic recrystallization processes under simple shear boundary conditions up to high strain. The simulations show that the CPOs are similar and practically reach the same maximum regardless of the iron content. CPOs are characterized by a single cluster of a‐axis and two‐clusters of b‐axis, reveling a joint activity of the easy glide [100](010) and the moderate strength [100](010) slip systems. High‐strain domains of our models are consistent with experimental results, showing an A‐type fabric with double maxima, and where the CPO is aligned with the shear direction. The model provides a deeper understanding of the dynamic recrystallization influence on olivine CPOs resulting from plastic deformation.

The asymptotic solution of the elasticity theory problem for a radially inhomogeneous cylinder

The elasticity theory problem for a radially inhomogeneous cylinder of small thickness, whose elastic moduli are arbitrary continuous functions depending on the radius of the cylinder, is considered. It is assumed that the side surface of the cylinder is stress-free, and the boundary conditions that keep the cylinder in equilibrium are given at its seats. Asymptotic solutions are constructed by the asymptotic integration method. It is shown that the asymptotic solution consists of the sum of the penetrating solution, simple boundary effect and boundary layer solutions. The character of the stress-strain state corresponding to the penetrating solution, simple boundary effect and boundary layer solutions is determined. Asymptotic formulas are obtained for displacements and stresses, which allow to calculate the stress-strain state of a cylinder.The problem of torsion of a radially inhomogeneous cylinder is studied, with its lateral surface free from stress and the boundary conditions keeping it in equilibrium at its seats. By applying the asymptotic integration method, it is determined that the asymptotic solution for the torsion problem consists of the sum of the penetrating solution and boundary layer solutions.Numerical analysis is performed and the effect of material inhomogeneity on the stress-strain state of the cylinder is evaluated.

Adaptive immune fuzzy quasi-sliding mode control for leader–follower formation of wheeled mobile robots under uncertainties and disturbances with obstacle avoidance

PurposeThis paper aims to propose a robust kinematic controller based on sliding mode theory designed to solve the trajectory tracking problem and also the formation control using the leader–follower strategy for nonholonomic differential-drive wheeled mobile robots with a PD dynamic controller.Design/methodology/approachTo deal with classical sliding mode control shortcomings, such as the chattering and the requirement of a priori knowledge of the limits of the effects of disturbances, an immune regulation mechanism-inspired approach is proposed to adjust the control effort magnitude adaptively. A simple fuzzy boundary layer method and an adaptation law for the immune portion gain online adjustment are also considered. An obstacle avoidance reactive strategy is proposed for the leader robot, given the importance of the leader in the formation control structure.FindingsTo verify the adaptability of the controller, obstacles are distributed along the reference trajectory, and the simulation and experimental results show the effectiveness of the proposed controller, which was capable of generating control signals avoiding chattering, compensating for disturbances and avoiding the obstacles.Originality/valueThe proposed design stands out for the ability to adapt in a case involving obstacle avoidance, trajectory tracking and leader–follower formation control by nonholonomic robots under the incidence of uncertainties and disturbances and also considering that the immune-based control provided chattering mitigation by adjusting the magnitude of the control effort, with adaptability improved by a simple integral-type adaptive law derived by Lyapunov stability analysis.

Necessary and sufficient conditions for avoiding Babuška’s paradox on simplicial meshes

Abstract It is shown that discretizations based on variational or weak formulations of the plate bending problem with simple support boundary conditions do not lead to the failure of convergence when polygonal domain approximations are used and the imposed boundary conditions are compatible with the nodal interpolation of the restriction of certain regular functions to approximating domains. It is further shown that this is optimal in the sense that a full realization of the boundary conditions leads to failure of convergence for conforming methods. The abstract conditions imply that standard nonconforming and discontinuous Galerkin methods converge correctly while conforming methods require a suitable relaxation of the boundary condition. The results are confirmed by numerical experiments.

Assessment of transient thermal stresses in gasketed plate heat exchangers

Thermal and mechanical stresses in stainless steel 316 L plates of gasketed plate heat exchangers (GPHEs) have been assessed with the aid of experiments. Stresses were indirectly determined with the aid of extensometers in critical plate areas. To the best of our knowledge, no experimental analysis of transient thermal loads on GPHE plates has been reported before. Experiments occurred with sudden and gradual heating processes with the aid of strain gauges. The former process implies that hot fluid enters the GPHE branch with the final target temperature, while the latter indicates that the hot fluid is progressively heated to the target temperature. Furthermore, combined mechanical and thermal stresses resulting from in-phase and out-of-phase loads are assessed in single and double operating conditions. Experiments were carried out with two plate thicknesses (0.5 and 0.7 mm). Stresses as obtained from experiments were compared to those provided by models containing simplified geometries and boundary conditions. Mechanical stresses promoted by the pressure difference between GPHE branches mostly affected the distribution area in single configuration. Thermal stresses at the porthole were higher than the ones found at the distribution zones, particularly for thicker plates. Besides, thermal stresses increased in double operation. Sudden heating processes with the system at rest promoted thermal peaking stress in a timescale of seconds, while the timescale to reach peak values by gradually heating the hot fluid is in the order of minutes. The most critical condition would be achieved at the porthole with in-phase loads and in double operation for the given settings.

A new type of simplified inverse Lax-Wendroff boundary treatment I: Hyperbolic conservation laws

In this paper, we design a new kind of high order inverse Lax-Wendroff (ILW) boundary treatment for solving hyperbolic conservation laws with finite difference method on a Cartesian mesh. This new ILW method decomposes the construction of ghost point values near inflow boundary into two steps: interpolation and extrapolation. At first, we impose values of some artificial auxiliary points through a polynomial interpolating the interior points near the boundary. Then, we will construct a Hermite extrapolation based on those auxiliary point values and the spatial derivatives at boundary obtained via the ILW procedure. This polynomial will give us the approximation to the ghost point value. By an appropriate selection of those artificial auxiliary points, high-order accuracy and stable results can be achieved. Moreover, theoretical analysis indicates that comparing with the original ILW method, especially for higher order accuracy, the new proposed one would require fewer terms using the relatively complicated ILW procedure and thus improve computational efficiency on the premise of maintaining accuracy and stability. We perform numerical experiments on several benchmarks, including one- and two-dimensional scalar equations and systems. The robustness and efficiency of the proposed scheme is numerically verified.

Assessing the impact of surface water and groundwater interactions for regional-scale simulations of water table elevation

Groundwater flow models are increasingly considered for the regional scale simulation of hydraulic heads and water table elevation. In the most complete configuration, models explicitly simulate two-way interactions between surface water (SW) and groundwater (GW) to reproduce and forecast both SW and GW water levels. In most regional scale groundwater models, however, SW-GW interactions are represented by simplified boundary conditions that only allow one-way interaction from SW to GW, neglecting most of the dynamic exchange fluxes between SW and GW. To evaluate the potential consequences of such simplifications on the simulation of regional GW levels, we compare two models on a 36,900 km2 regional aquifer system in Southern Quebec. One model explicitly simulates both SW and GW flow with two-way SW-GW feedback and the other model only simulates GW flow with a surface boundary flux to represent a one-way interaction with the land surface. Both models are developed with the same numerical code to ensure that the only differences are the representation of SW water flow and SW-GW feedback. The one-way model simulates overall deeper water tables because it removes all exfiltrated groundwater from the system once it exits the subsurface, therefore not allowing exfiltrating groundwater to re-infiltrate. This effect is most pronounced in areas where the water table is close to the surface and for low-flow periods. The inclusion of two-way feedback also reduces the sensitivity of simulated GW levels to the magnitude of the hydraulic conductivity. This result highlights the need for additional data on other system states to improve the calibration of regional scale models that explicitly simulate two-way SW-GW interactions.

An analytical solution for vertical infiltration in homogeneous bounded profiles

AbstractIn this study, we derive an analytical solution to address the problem of vertical infiltration within 1D homogeneous bounded profiles. Initially, we consider the Richards equation together with Dirichlet boundary conditions. We assume constant diffusivity and linear dependence between the conductivity and the water content, resulting to a linear partial differential equation of diffusion type. To solve the simplified initial boundary value problem over a finite interval, we apply the unified transform, commonly known as the Fokas method. Through this methodology, we obtain an integral representation of the solution that can be efficiently and directly computed numerically, yielding a convergent scheme. We examine various cases, and we compare our solution with well‐known approximate solutions. This work can be seen as a first step to derive analytical solutions for the far more difficult and complex problem of modelling water flow in heterogeneous layered soils.