Several Types Of Boundary Conditions Research Articles

Stability properties for a problem of light scattering in a dispersive metallic domain

<p style='text-indent:20px;'>In this work, we study the well-posedness and some stability properties of a PDE system that models the propagation of light in a metallic domain with a hole. This model takes into account the dispersive properties of the metal. It consists of a linear coupling between Maxwell's equations and a wave type system. We prove that the problem is well posed for several types of boundary conditions. Furthermore, we show that it is polynomially stable and that the exponential stability is conditional on the exponential stability of the Maxwell system.</p>

СРАВНИТЕЛЬНЫЙ АНАЛИЗ ТОЧНОСТИ РЕЗУЛЬТАТОВ РАСЧЕТА ТОНКОЙ ИЗГИБАЕМОЙ ПЛАСТИНКИ С ИСПОЛЬЗОВАНИЕМ РАЗЛИЧНЫХ ФОРМ МКЭ

The article presents the results of a comparative analysis of the accuracy of a thin bending plate according to the FEM in the form of a classical mixed method (CSM) and based on the FEM in dis placements using the SP LIRA SAPR. The analysis of calculation results for plates with several types of boundary conditions and representation in the form of an ensemble of rectangular finite elements with 3 unknowns at each node is performed. Based on the results of calculations, a conclusion was made about the accuracy and correctness of calculations using two forms of the FEM. Graphs of convergence of calculation results are given.

TLBfind: a Thermal Lattice Boltzmann code for concentrated emulsions with FINite-size Droplets

In this paper, we present TLBfind, a GPU code for simulating the hydrodynamics of droplets along with a dynamic temperature field. TLBfind hinges on a two-dimensional multi-component lattice Boltzmann (LB) model simulating a concentrated emulsion with finite-size droplets evolving in a thermal convective state, just above the transition from conduction to convection. The droplet concentration of the emulsion system is tunable and at the core of the code lies the possibility to measure a large number of physical observables characterising the flow and droplets. Furthermore, TLBfind includes a parallel implementation on GPU of the Delaunay triangulation useful for the detection of droplets' plastic rearrangements, and several types of boundary conditions, supporting simulations of channels with structured rough walls. Program summaryProgram Title: TLBfindCPC Library link to program files:https://doi.org/10.17632/hbk45696nf.1Developer's repository link:https://github.com/FrancescaPelusi/TLBfindLicensing provisions: MITProgramming language: CUDA-CNature of problem: Hydrodynamics of concentrated emulsions with finite-size droplets in a thermal convective state.Solution method: Single relaxation time Lattice Boltzmann method to solve Navier-Stokes equations for fluids, coupled with the temperature field dynamics. The output describes the dynamics of finite-size droplets of concentrated emulsions in presence of a temperature field. The temperature field obeys the advection-diffusion equation.Additional comments including restrictions and unusual features: Plastic rearrangements of droplets are detected via the parallel implementation of the Delaunay triangulation, and boundary conditions are tunable.

Geometrically nonlinear dynamic analysis of laminated composite plate using a nonpolynomial shear deformation theory

A computationally efficient C0 finite element model in conjunction with the nonpolynomial shear deformation theory (NPSDT) is extended to examine the free and forced vibration behavior of laminated composite plates. The employed NPSDT assumes the nonlinear distribution of in-plane displacements which qualify the requirement of traction free boundary conditions at the top and bottom surfaces. The present formulation utilizes both von Kármán and Green–Lagrange type of strain–displacement relations to model the geometric nonlinearity. Using Hamilton’s principle, the nonlinear governing equation of motion is derived and then discretized based on the nine-noded Lagrange element. The obtained equations are solved by utilizing unconditionally stable Newmark’s scheme in conjunction with Newton–Raphson method. A damping effect in the transient analysis has been introduced in the framework of the Rayleigh damping model. The steady state forced vibration analysis has also been carried out by employing harmonic force with excitation frequency around the natural frequency. The arc-length continuation method is applied to obtained the frequency response. The present model has been validated for a wide range of problems and a detailed numerical study has been carried out for several types of boundary conditions under various types of loading with different magnitude of the load.

Second-order entropy satisfying BGK-FVS schemes for incompressible Navier-Stokes equations

Kinetic BGK numerical schemes for the approximation of incompressible Navier-Stokes equations are derived via classical discrete velocity vector BGK approximations, but applied to an inviscid compressible gas dynamics system with small Mach number parameter, according to the approach of Carfora and Natalini (2008). As the Mach number, the grid size and the timestep tend to zero, the low Mach number limit and the time-space convergence of the scheme are achieved simultaneously, and the numerical viscosity tends to the physical viscosity of the Navier-Stokes system. The method is analyzed and formulated as an explicit finite volume/difference flux vector splitting (FVS) scheme over a Cartesian mesh. It is close in spirit to lattice Boltzmann schemes, but it has several advantages. The first is that the scheme is expressed only in terms of momentum and mass compressible variables. It is therefore very easy to implement, and several types of boundary conditions are straightforward to apply. The second advantage is that the scheme satisfies a discrete entropy inequality, under a CFL condition of parabolic type and a subcharacteristic stability condition involving a cell Reynolds number that ensures that diffusion dominates advection at the level of the grid size. This ensures the robustness of the method, with explicit uniform bounds on the approximate solution. Moreover the scheme is proved to be second-order accurate in space if the parameters are well chosen, this is the case in particular for the Lax-Friedrichs scheme with Mach number proportional to the grid size. The scheme falls then into the class of artificial compressibility methods, the novelty being its exceptionally good theoretical properties. We show the efficiency of the method in terms of accuracy and robustness on a variety of classical two-dimensional benchmark tests. The method is finally applied in three dimensions to compute the permeability of a porous medium defined by a complex idealized Kelvin-like cell. Relations between our scheme and compressible low Mach number schemes are discussed.

Numerical investigation on the effect of boundary conditions on the scaling of spontaneous imbibition

We present a numerical validation of the scaling group presented by Schmid and Geiger ((2012) Water Resour. Res. 48, 3) for Spontaneous Imbibition (SI) through simulating a core sample bounded by the wetting fluid. We combine the results of the simulations with the semi-analytical model for counter-current spontaneous imbibition presented by Schmid et al. ((2011) Water Resour. Res. 47, 2) to validate the upscaling of laboratory experiments to field dimensions using dimensionless time. We then present a detailed parametric study on the effect of Boundary Conditions (BC) and characteristic length to compare imbibition assisted oil recovery with several types of boundary conditions. We demonstrate that oil recovery was the fastest and most efficient when all faces are open to flow. We also demonstrate that all cases scale with the non-dimensionless time suggested by Schmid and Geiger ((2012) Water Resour. Res. 48, 3) and show a close match to the numerical simulation and the semi-analytical solution. Moreover, we discuss how the effect of constructing a model with varying grid sizes and dimensions affects the accuracy of the results through comparing the results of the 2-D and 3-D models. We observe that the 3-D model proved superior in the accuracy of the results to simulate simple counter-current SI. However, we deduce that 2-D models yield satisfying enough results in a timely manner in the One End Open (OEO) and Two Ends Open (TEO) cases, compared to 3-D models which are time-consuming. We finally conclude that the non-dimensionless time of Schmid and Geiger ((2012) Water Resour. Res. 48, 3) works well with counter-current SI cases regardless of the boundary condition imposed on the core.

An efficient preconditioner for the fast simulation of a 2D stokes flow in porous media

SummaryWe consider an efficient preconditioner for a boundary integral equation (BIE) formulation of the two‐dimensional Stokes equations in porous media. While BIEs are well‐suited for resolving the complex porous geometry, they lead to a dense linear system of equations that is computationally expensive to solve for large problems. This expense is further amplified when a significant number of iterations is required in an iterative Krylov solver such as generalized minimial residual method (GMRES). In this paper, we apply a fast inexact direct solver, the inverse fast multipole method, as an efficient preconditioner for GMRES. This solver is based on the framework of ‐matrices and uses low‐rank compressions to approximate certain matrix blocks. It has a tunable accuracy ε and a computational cost that scales as . We discuss various numerical benchmarks that validate the accuracy and confirm the efficiency of the proposed method. We demonstrate with several types of boundary conditions that the preconditioner is capable of significantly accelerating the convergence of GMRES when compared to a simple block‐diagonal preconditioner, especially for pipe flow problems involving many pores.

A functional-type a posteriori error estimate of approximate solutions for Reissner-Mindlin plates and its implementation

Paper is devoted to a numerical justification of the recent a posteriori error estimate for Reissner-Mindlin plates. This majorant provides a reliable control of accuracy of any conforming approximate solution of the problem including solutions obtained with commercial software for mechanical engineering. The estimate is developed on the basis of the functional approach and is applicable to several types of boundary conditions. To verify the approach, numerical examples with mesh refinements are provided.

Functional evolution of scalar fields in bounded one-dimensional regions

We discuss the unitarity of the quantum evolution between arbitrary Cauchy surfaces of a 1 + 1 dimensional free scalar field defined on a bounded spatial region and subject to several types of boundary conditions including Dirichlet, Neumann and Robin.

Initial-boundary value problems for the equations of the global atmospheric electric circuit

Time-dependent and steady-state problems for the equations of the global atmospheric electric circuit are studied in the quasi-stationary approximation. Maxwell's equations in an anisotropic medium are written in terms of the electric potential; several types of boundary conditions (motivated by applications) are considered, including the analogues of the classical Dirichlet, Neumann and mixed conditions, as well as a non-standard condition relating the potential and normal current density at symmetric points on one of the boundary components. In each case the variational formulation of the problem is derived and is shown to be well-posed. Also, certain stabilisation theorems are proved for the solution of time-dependent problem in case the main parameters do not depend on time.

Immersed boundary method implemented for the simulation of an external flow on unstructured meshes

A method is developed for the numerical simulation of an external flow around solid bodies of unstructured meshes. Several types of boundary conditions are set on the surface of the body, which are simulated by different modifications of the immersed boundary method. The method efficiency is demonstrated on 2D model problems on the computation of subsonic and supersonic flows around a cylinder.

Bending moment calculation for single lap joints with composite laminate adherends including bending-extensional coupling

This work deals with the analytical determination of section forces and moments in adhesively bonded single lap joints with composite laminate adherends including bending-extensional coupling. The analysis is also valid for unbalanced joints and the adhesive thickness is taken into account in the model. Several types of boundary conditions can be applied, e.g. simply supported ends, fixed ends, bonded doubler and single strap joints. Various configurations are studied and the results are compared to finite element analyses and other analytical approaches. Good agreement is achieved over a wide range of configurations. A closed form analytical solution is given for the section forces and moments at the overlap ends.

A steady flow through a plane cascade of profiles with an arbitrarily large inflow_The mathematical model, existence of a weak solution

The paper deals with a mathematical model of a viscous stationary incompressible flow through a cascade of profiles. The problem for the Navier–Stokes system is formulated in a domain corresponding to one spatial period of the cascade. We consider several types of boundary conditions on various parts of the boundary (the inflow, the artificial lower and upper boundaries, the profile, the outflow). We solve the problem with an arbitrarily large inflow into the turbine. We formulate the “artificial” boundary condition on the outflow (the modification of the so called do-nothing condition) which enables us to prove the existence of a weak solution for any inflow.

Torsion of cracked nanorods using a nonlocal elasticity model

This paper presents a nonlocal cracked-rod model from which we have analysed the torsional vibrations of a carbon nanotube with a circumferential crack. Several types of boundary conditions, including the consideration of a buckyball at the end of the nanotube, have been studied. The nonlocal Eringen elasticity theory is used to formulate the problem. The cracked rod is modelled by dividing the cracked element into two segments connected by a torsional linear spring whose stiffness is related to the crack severity. The effect of the nonlocal small-scale parameter, crack severity, cracked section position, different boundary conditions and attached mass are examined in this work.

The homogenization of elliptic partial differential systems on rugous domains with variable boundary conditions

This paper is devoted to studying the asymptotic behaviour of a sequence of elliptic systems posed in a sequence of rough domains Ωn. The solutions un are assumed to satisfy un(x) ϵ Vn(x), where Vn(x) is a vectorial space depending on $\smash{x\in\bar\varOmega_n}$. This enables one to consider several types of boundary conditions posed in variable sets of the boundary. For some choices of the vectorial spaces Vn(x), our study provides, in particular, some classical results for the homogenization of Dirichlet elliptic problems in varying domains.

Unified magnetomechanical homogenization framework with application to magnetorheological elastomers

The aim of this work is to present a general homogenization framework with application to magnetorheological elastomers under large deformation processes. The macroscale and microscale magnetomechanical responses of the composite in the material and spatial description are presented and the conditions for a well-established homogenization problem in Lagrangian description are identified. The connection between the macroscopic magnetomechanical field variables and the volume averaging of the corresponding microscopic variables in the Eulerian description is examined for several types of boundary conditions. It is shown that the use of kinematic and magnetic field potentials instead of kinetic field and magnetic induction potentials provides a more appropriate homogenization process.

Mixed Boundary Conditions for Two-Dimensional Transient Heat Transfer Conduction under Lattice Boltzmann Simulations

In this paper, lattice Boltzmann implementations of several types of boundary conditions are introduced and numerically demonstrated. A thermal lattice BGK model is used to simulate thermal fields for flows. The unknown thermal distribution functions at the boundary are subjected to the bounce back concept which is determined consistently with Dirichlet and/or Neumann and/or convective boundary conditions. A consistency analysis using heat transfer conduction is given and the algorithms are numerically tested in two space dimensions with respect to accuracy, numbers of iterations and CPU time. The method is used to simulate conduction transfer problems; numerical results and reference's solutions are found in satisfactory agreement for thermal fields.

On the similarity between Dirichlet–Neumann with interface artificial compressibility and Robin–Neumann schemes for the solution of fluid-structure interaction problems

In this note, the similarity between two implicit, partitioned solution techniques for fluidstructure interaction (FSI) problems is analyzed using finite volume discretization of the flow equations. Fluid-structure interaction refers to the mutual interaction between a fluid flow and a flexible structure. Partitioned solution techniques solve the flow equations and the structural equations separately. These techniques are classified as implicit (or strongly coupled) if they satisfy the interaction conditions on the fluid-structure interface in each time step and as explicit (or loosely coupled) if they do not. Both techniques analyzed in this note use block Gauss-Seidel (GS) iterations, meaning that the flow equations and the structural equations are solved consecutively within a time step until some convergence tolerance is reached. As the flow and structural equations are solved separately, the interaction conditions on the fluid-structure interface have to be converted into boundary conditions on the common boundary of the fluid and structure subdomains. Several types of boundary conditions can be applied, resulting in different decompositions. In the case of Dirichlet-Neumann (DN) decomposition, the flow equations are solved with a Dirichlet boundary condition (given velocity) on the fluid-structure interface, while the structural equations are solved with a Neumann boundary condition (given stress) on the interface. Conversely, RobinNeumann (RN) decomposition, introduced in [1], denotes a Robin boundary condition on the fluid side of the interface and a Neumann boundary condition on the structure side. The first technique in this comparison is block Gauss-Seidel iterations applied to the monolithic system previously multiplied by a suitable permutation matrix, leading to a Robin-Neumann decomposition (GS-RN). This first technique includes a simplified version of the structural model in the flow equations by means of a Robin boundary condition to accelerate the convergence of the GS iterations [1, 2]. The second technique is block Gauss-Seidel iterations with DirichletNeumann decomposition and Interface Artificial Compressibility (GS-DN-IAC). This second

Optimal topologies in structural design of micropolar materials

The authors developed the finite element method (FEM) for micropolar elastic solids and applied it to the problem of topology optimization. The weighted residual method and 8-node isoparameteric elements were employed for the FEM analysis. The solid isotropic material with penalization (SIMP) method and the gravity control function were employed in the optimization procedure. For the designing area, rectangular plates under several types of boundary conditions were analysed. In this analysis, the effects of the engineering material constants for micropolar elasticity, the coupling number N, which is related to stresses, and the characteristic length l, which is related to couple stresses, are researched. The optimal structures of micropolar materials are strongly influenced by material properties of bending. Though the optimal topologies of classical elastic solids are generally truss frames, those of micropolar materials are rigid frames, which are simple topologies. At the joint of members loaded moments, holes are more apt to appear in the optimization process for micropolar solids than classical solids, that is, optimal topologies becomes finer, in micropolar elasticity. Obtained results are determined by the tradeoff between simplification and refinement of structures. The proposed method could potentially solve the optimization problems of bio-materials such as bone tissue.

Free axisymmetric vibrations of solid cylinders: numerical problem solving

The problem of free vibrations of a solid cylinder with different boundary conditions is solved using the three-dimensional theory of elasticity and a numerical analytic approach. The spline-approximation and collocation methods are used to reduce the partial differential equations of elasticity to systems of ordinary differential equations of high order with respect to the radial coordinate. These equations are solved by stable numerical discrete orthogonalization and incremental search. Calculated results are presented for transversely isotropic and inhomogeneous materials of the cylinder and for several types of boundary conditions at its ends