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A finite element method dealing with an open boundary condition for the analysis of long wave problem is presented. The key feature of the method is that spurious reflective waves which occurred for the initial transient state on the open boundary can be eliminated by introducing a subdomain technique. For the numerical outflow boundary condition, the progressive wave condition, based on the shallow water long wave theory, is successfully employed. This method is quite suitable for practical analysis because of its adaptability for the arbitrary configuration of the open boundary and shape of elements adjacent to the open boundary. This method is numerically verified for flow in a one dimensional channel and the two dimensional tidal current in Tokyo Bay. The numerical results are compared with analytical solutions and observed data obtained by field measurements. These results are all in close agreement.

Non‐linear reaction‐diffusion processes with cross‐diffusion in two‐dimensional, anisotropic media are analyzed by means of an implicit, iterative, time‐linearized approximate factorization technique as functions of the anisotropy of the heat and species diffusivity tensors, the Soret and Dufour cross‐diffusion effects, and five types of boundary conditions. It is shown that anisotropy and cross‐diffusion deform the reaction front and affect the front velocity, and the magnitude of these effects increases as the magnitude of the off‐diagonal components of the heat and species diffusivity tensors is increased. It is also shown that the five types of boundary conditions employed in this study produce similar results except when there is either strong anisotropy in the species or heat diffusivity tensors and there are no Soret and Dufour effects, or the species and heat diffusivity tensors are isotropic, but the anisotropy of the Soret and Dufour effects is important. If the species and heat diffusivity tensors are isotropic, the effects of either the Soret or the Dufour cross‐diffusion effects are small for the cases considered in this study. The time required to achieve steady state depends on the anisotropy of the heat and diffusivity tensors, the cross‐diffusion effects, and the boundary conditions.

A method has been developed for the solution of inverse heat diffusion problems to find the initial condition, boundary condition, and the source and sink function in the heat diffusion equation. The method has been used in the development of a source‐and‐sink method to find the boundary conditions in inverse Stefan problems. Green's functions have been used in the solution, and the problems are solved by using two approaches: a series solution approach, and a time incremental approach. Both can be used to find the boundary conditions without reliance on the flux information to be supplied at both sides of the interface. The methods are efficient in that they require less equations to be solved for the conditions. The numerical results have shown to be accurate, convergent, and stable. Most of all, the results do not degrade with time as in other time marching schemes reported in the literature. Algorithms can also be easily developed for the solution of the conditions.

The purpose of this paper is to investigate the effect of numerical boundary condition implementation on local error and convergence in L2-norm of a finite volume discretization of the transient heat conduction equation subject to several boundary conditions, and for cases with volumetric heat generation, using both fully implicit and Crank-Nicolson time discretizations. The goal is to determine which combination of numerical boundary condition implementation and time discretization produces the most accurate solutions with the least computational effort.

The paper studies several benchmark cases including constant temperature, convective heating, constant heat flux, time-varying heat flux, and volumetric heating, and compares the convergence rates and local to analytical or semi-analytical solutions.

The Crank-Nicolson method coupled with second-order expression for the boundary derivatives produces the most accurate solutions on the coarsest meshes with the least computation times. The Crank-Nicolson method allows up to 16X larger time step for similar accuracy, with nearly negligible additional computational effort compared with the implicit method.

The findings can be used by researchers writing similar codes for quantitative guidance concerning the effect of various numerical boundary condition approximations for a large class of boundary condition types for two common time discretization methods.

The paper provides a comprehensive study of accuracy and convergence of the finite volume discretization for a wide range of benchmark cases and common time discretization methods.

A mathematical analysis of the time‐dependent multi‐dimensional Hydrodynamic model is performed to determine the well‐posed boundary conditions for semiconductor device simulation. The number of independent boundary conditions that need to be specified at electrical contacts of a semi‐conductor device are derived. Using the classical energy method, a mathematical relation among the physical parameters is established to define the well‐posed boundary conditions for the problem. Several possible sets of boundary conditions are given to illustrate the proper boundary conditions. Natural boundary conditions that can be specified are obtained from the boundary integrals of the weak‐form finite element formulations. An example is included to illustrate the importance of well‐posedness of the boundary conditions for device simulation.

Numerical simulation of a fluid flow involves the specification of boundary conditions along all or part of the boundary. Designs a means of handling outflow boundary conditions for the incompressible Navier‐Stokes equations. Addresses through‐flow problems involving the specification of outflow conditions at the synthetic boundary. This outflow boundary condition is applicable to a developing flow problem. The underlying objectives behind designing the boundary condition at the truncated boundary are three‐fold, namely: matching with Navier‐Stokes equations inside the domain; taking both non‐linear and diffusive contributions into account; and ensuring the discrete divergence‐free condition. In order to meet these requirements, follows the concept of a free boundary condition by taking the outflow nodal values of u, v and p as unknowns, which are coupled with the interior unknowns through the surface integrals in the momentum equations. The computed solutions can be legitimately regarded as solutions to conservation equations under consideration when both components of the surface traction vector approach zero. With the convergent property accommodated in the present mixed finite element analysis, the task remains to simply improve the accuracy. Demonstrates the capability of the proposed non‐linear outflow boundary conditions through several benchmark tests.

This paper aims to propose a new boundary condition and a web-spline basis of finite element space approximation to remedy the problems of constraints due to homogeneous and non-homogeneous; Dirichlet boundary conditions. This paper considered the two-dimensional linear elasticity equation of Navier–Lamé with the condition CAB. The latter allows to have a total insertion of the essential boundary condition in the linear system obtained; without using a numerical method as Lagrange multiplier. This study have developed mixed finite element; method using the B-splines Web-spline space. These provide an exact implementation of the homogeneous; Dirichlet boundary conditions, which removes the constraints caused by the standard; conditions. This paper showed the existence and the uniqueness of the weak solution, as well as the convergence of the numerical solution for the quadratic case are proved. The weighted extended B-spline; approach have become a much more workmanlike solution.

In this paper, this study used the implementation of weighted finite element methods to solve the Navier–Lamé system with a new boundary condition CA, B (Koubaiti et al., 2020), that generalises the well-known basis, especially the Dirichlet and the Neumann conditions. The novel proposed boundary condition permits to use a single Matlab code, which summarises all kind of boundary conditions encountered in the system. By using this model is possible to save time and programming recourses while reap several programs in a single directory.

The results have shown that the Web-spline-based quadratic-linear finite elements satisfy the inf–sup condition, which is necessary for existence and uniqueness of the solution. It was demonstrated by the existence of the discrete solution. A full convergence was established using the numerical solution for the quadratic case. Due to limited regularity of the Navier–Lamé problem, it will not change by increasing the degree of the Web-spline. The computed relative errors and their rates indicate that they are of order 1/H. Thus, it was provided their theoretical validity for the numerical solution stability. The advantage of this problem that uses the CA, B boundary condition is associated to reduce Matlab programming complexity.

The mixed finite element method is a robust technique to solve difficult challenges from engineering and physical sciences using the partial differential equations. Some of the important applications include structural mechanics, fluid flow, thermodynamics and electromagnetic fields (Zienkiewicz and Taylor, 2000) that are mainly based on the approximation of Lagrange. However, this type of approximation has experienced a great restriction in the level of domain modelling, especially in the case of complicated boundaries such as that in the form of curvilinear graphs. Recently, the research community tried to develop a new way of approximation based on the so-called B-spline that seems to have superior results in solving the engineering problems.

This paper aims to present a computational framework to generate numeric enrichment functions for two-dimensional problems dealing with single/multiple local phenomenon/phenomena. The two-scale generalized/extended finite element method (G/XFEM) approach used here is based on the solution decomposition, having global- and local-scale components. This strategy allows the use of a coarse mesh even when the problem produces complex local phenomena. For this purpose, local problems can be defined where these local phenomena are observed and are solved separately by using fine meshes. The results of the local problems are used to enrich the global one improving the approximate solution.

The implementation of the two-scale G/XFEM formulation follows the object-oriented approach presented by the authors in a previous work, where it is possible to combine different kinds of elements and analyses models with the partition of unity enrichment scheme. Beside the extension of the G/XFEM implementation to enclose the global–local strategy, the imposition of different boundary conditions is also generalized.

The generalization done for boundary conditions is very important, as the global–local approach relies on the boundary information transferring process between the two scales of the analysis. The flexibility for the numerical analysis of the proposed framework is illustrated by several examples. Different analysis models, element formulations and enrichment functions are used, and the accuracy, robustness and computational efficiency are demonstrated.

This work shows a generalize imposition of different boundary conditions for global–local G/XFEM analysis through an object-oriented implementation. This generalization is very important, as the global–local approach relies on the boundary information transferring process between the two scales of the analysis. Also, solving multiple local problems simultaneously and solving plate problems using global–local G/XFEM are other contributions of this work.

This paper proposes a novel approach to impose the Neumann boundary condition for isogeometric analysis (IGA) of Euler–Bernoulli beam with 1-D formulation. The proposed method is for only IGA in which it is difficult to handle the Neumann boundary conditions. The control points of B-spline are equivalent to nodes in finite element method. With 1-D formulation, it is not possible to accommodate multiple degrees of freedom in IGA. This case arises in the analysis of beams. The paper aims to propose a way to work around this issue in a simple way.

Neumann boundary conditions, which are even-order derivatives (example: double derivative) of the primary variable, are inherently satisfied in the weak form. Boundary conditions with an odd number of derivatives (example: slope) are imposed with the introduction of a new penalty matrix.

The proposed method can impose a slope boundary condition for IGA of a beam using 1-D formulation.

From the literature, it can be observed that the beam is formulated in 1-D by considering it as either a rotation-free element or a 2-D formulation by considering shear strain along with the normal strain. The work represents 1-D formulation of a beam while considering the slope boundary condition, which is easy and effective to formulate, compared with the slope boundary conditions reported in previous works.

The boundary integral method (BIM) provides unparalleled computational efficiency for solving problems wherever it is applicable. For Stokes flows, the BIM in its current form can only be applied to a limited class of problems that generally comprises boundaries with either specified velocity or stress. This study aims to radically extend the applicability by developing a general method within the BIM framework that can handle periodic, symmetry, zero normal-velocity gradient and the specified pressure boundary conditions. This study is limited in scope to steady-state flows.

The proposed method introduces a set of points near the boundary for the symmetry, zero normal-velocity gradient and specified pressure boundary conditions. The formulation for the first two boundary conditions use a spatial discretization procedure within the BIM framework to arrive at a set of equations for the unknowns. The specified pressure boundary condition warrants the decomposition of the unknown traction term into simpler components before the discretization procedure can be executed. Though the new methodology is illustrated in detail for two-dimensional rectangular domains, it can be generalized to more complex three-dimensional cases. This will be the subject for future investigations.

The current endeavor has successfully demonstrated the incorporation of the above boundary conditions through simple Stokes flow problems like plane channel flow, flow through ribbed duct and plane wall jet. The predicted results matched adequately with either analytical solutions or with available literature data.

To the best of the author’s knowledge, this is the first time that the exit boundary conditions like zero normal-velocity gradient and specified pressure have been formulated within the BIM for Stokes flows. These boundary conditions are extremely powerful and the current research initiative has the potential to dramatically increase the range of applicability of the BIM for Stokes flow simulations.