Search results

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 purpose of this paper is to present two efficient immersed boundary methods (IBM) for simulation of thermal flow problems. One method is for given temperature condition (Dirichlet type), while the other is for given heat flux condition (Neumann type). The methods are applied to simulate natural and mixed convection problems to check their performance. The comparison of present results with available data in the literature shows that the present methods can obtain accurate numerical results efficiently.

The paper presents two efficient IBM solvers, in which the effect of thermal boundary to its surrounding fluid is considered through the introduction of a heat source/sink term into the energy equation. One is the temperature correction‐based IBM developed for problems with given temperature on the wall. The other is heat flux correction‐based IBM for problems with given heat flux on the wall. Note that in this solver, the offset of derivative condition is directly used to correct the temperature field.

As compared with existing solvers, the temperature correction‐based IBM determines the heat source/sink implicitly instead of pre‐calculated explicitly, so that the boundary condition for temperature is accurately satisfied. To the best of the authors' knowledge, the work of heat flux correction‐based IBM is the first endeavour for application of IBM to solve thermal flow problems with Neumann (heat flux) boundary condition. It was found that both methods presented in this work can efficiently obtain accurate numerical results for thermal flow problems.

The two methods presented in this paper are novel. They can effectively solve thermal flow problems with Dirichlet and Neumann boundary conditions.

In this paper, the authors revisit the computation of closed-form expressions of the topological indicator function for a one step imaging algorithm of two- and three-dimensional sound-soft (Dirichlet condition), sound-hard (Neumann condition) and isotropic inclusions (transmission conditions) in the free space.

From the addition theorem for translated harmonics, explicit expressions of the scattered waves by infinitesimal circular (and spherical) holes subject to an incident plane wave or a compactly supported distribution of point sources are available. Then the authors derive the first-order term in the asymptotic expansion of the Dirichlet and Neumann traces and their surface derivatives on the boundary of the singular medium perturbation.

As the shape gradient of shape functionals are expressed in terms of boundary integrals involving the boundary traces of the state and the associated adjoint field, then the topological gradient formulae follow readily.

The authors exhibit singular perturbation asymptotics that can be reused in the derivation of the topological gradient function that generates initial guesses in the iterated numerical solution of any shape optimization problem or imaging problems relying on time-harmonic acoustic wave propagation.

The purpose of the current paper is to develop a numerical methodology, based on the immersed boundary-lattice Boltzmann computational framework, for the Neumann and Dirichlet boundary conditions in problems involving natural and forced convection heat transfer.

The direct forcing immersed boundary method is extended to study the heat transfer by incompressible flow within the thermal lattice Boltzmann method (LBM) computational framework. The direct forcing and heating immersed boundary-LBM introduces a heat source term to the thermal LBM to account for the heat transfer occurring at the immersed boundary. New numerical treatments for the Neumann type of boundary condition and for the calculation of the local Nusselt number are developed. The developed methodologies have been applied to flows around immersed bodies with natural and forced convection, including steady as well as unsteady flows.

Numerical experiments involving immersed bodies in natural and forced convection have been performed in order to assess the validity of the direct heating IB-LBM. The flow cases studied also include steady and transient flow phenomena. Flow velocity field and isotherms have been used for qualitative comparisons with existing, published results. The surface averaged Nusselt number, Strouhal number, and lift coefficient (for the unsteady flow cases) have been used for quantitative comparison with published results. The results show that there are satisfactory agreements, qualitatively and quantitatively, between the results obtained by using the present method and those previously published.

Limited application of immersed boundary to thermal flows within the LBM has been studied by researchers; the few past studies were limited to Dirichlet boundary conditions and/or using of feedback forcing and heating approaches. In the current paper, the direct forcing and heating approach was used which helps to eliminate the arbitrary constants used in the feedback approaches. The developed new numerical treatments for the Neumann type of boundary condition and for the calculation of the local Nusselt number eliminate the need to determine surface normal and temperature gradient in the normal direction for heat transfer calculation, which is particularly beneficial in cases with deforming or changing boundaries.

The paper has the purpose of proposing a new open boundary condition to be used in conjunction with the finite integration technique (FIT) for the modelling of passive on‐chip components.

This boundary condition is ensured by using a virtual layer that surrounds the computational domain.

The paper proves which are the optimal material properties of the equivalent layer of open boundary.

When modelling passive on‐chip components with FIT, the method proposed is more efficient than the strategic dual image technique.

The paper shows the advantage of this approach – that the analysis algorithm remains unchanged, while saving the field‐circuit compatibility properties, such as current conservation.

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 discusses an electrostatic, homogeneous field in a uniform two‐dimensional domain with Neumann's boundary conditions. The boundary conditions are known only at some segments of the boundary. The synthesis is understood as the computation of the remaining boundary conditions which would ensure the required potential distribution in some subdomains within the boundary. The introduction of a single‐layer potential leads to Fredholm's equation of the second order. Stepwise approximation of the source distribution along the boundary rearranges Fredholm's equation and the requirements concerning the single layer potential distribution. It leads to a matrix equation with a rectangular coefficient matrix. In order to solve approximately this equation, in the sense of the least squares minimization, the singular value decomposition (SVD) method is used. The choice of subdomains with determined potential distribution influences significantly the conditioning of the equation. Easy selection of an acceptable solution among all possible solutions proves the suitability of the SVD method in the above problem. The numerical experiments reported in the paper are a good illustration of this.

The purpose of this paper is to examine the transient heat conduction in a two-dimensional anisotropic substrate coated with a thin layer of thermal barrier coating (TBC). Nowadays, materials with anisotropic properties have been extensively applied in various engineering applications for enhanced strength. However, under an extreme operating environment of high temperature, the strength of the materials may largely decline. As a common practice in engineering, TBC are usually applied to thermally insulate the substrates so as to allow for higher operating temperature. This research provides engineers a numerical approach for properly designing the TBC to protect the anisotropic substrate.

For this investigation, a finite difference scheme using the domain mapping technique, transforming the anisotropic domain into isotropic one, is employed. The analysis considers three respective boundary conditions, namely Dirichelete condition, Neumann condition, and also forced convection, and studies the effect of various variables on the heat conduction in the coated system. Additionally, formulas for the steady-state temperature drop across the coating layer at the center are analytically derived. By comparing the numerical results with the analytical solutions, the veracity of the formulas is verified.

A few interesting phenomena are observed from the numerical results. First, the rotation of the substrate's principal axes affects the temperature on the TBC front surface in a more obvious manner for the Neumann condition than that for convection. Second, the temperature profile of the Dirichelete condition rises faster than the other cases, although all their profiles present a similar pattern. Third, the transient temperature drop across the TBC under the convection condition presents a complicated pattern, depending on the TBC thickness. Finally, the increase of TBC thickness under the Dirichelete condition may provide better insulation than the other cases. In this paper, approximate analytical formulations for the steady-state temperature drop across the TBC are also presented. Numerical results by the finite difference method indicate excellent agreements with the analytical solutions.

In the past, the finite element method (FEM) is usually applied for analyzing the heat conduction problem of TBC. However, one serious deficiency of applying the FEM to the TBC problem lies in the demand for a vast amount of elements (or cells) when the TBC thickness is far smaller than the substrate dimension. For ultra-thin coating, an enormous amount of elements are required that may lead to an extremely heavy computational burden. The paper presents an innovative finite difference approach that can be applied to analyze the heat conduction across the TBC coated on an anisotropic substrate. On the interface between the TBC and the substrate, a special heat equilibrium condition and the compatibility condition of identical temperature on the adjacent materials are used to propose three new models to predict the temperature drop across the TBC.

Discrete differential operators of periodic base functions have been examined to solve boundary-value problems. This paper aims to identify the difficulties of using those operators to solve ordinary linear and nonlinear differential equations with Dirichlet and Neumann boundary conditions.

This paper presents a promising approach for solving two-dimensional (2D) boundary problems of elliptic differential equations. To create finite differential equations, specially developed discrete partial differential operators are used to replace the partial derivatives in the differential equations. These operators relate the value of the partial derivatives at each point to the value of the function at all points evenly distributed over the area where the solution is being sought. Exemplary 2D elliptic equations are solved for two types of boundary conditions: the Dirichlet and the Neumann.

An alternative method has been proposed to create finite-difference equations and an effective method to determine the leakage flux in the transformer window.

The proposed approach can be classified as an extension of the finite-difference method based on the new formulas approximating the derivatives. This method can be extended to the 3D or time-periodic 2D cases.

This paper presents a methodology for calculations of the self- and mutual-leakage inductances for windings arbitrarily located in the transformer window, which is needed for special transformers or in any case of the internal asymmetry of windings.

The presented methodology allows us to obtain the magnetic vector potential distribution in the transformer window only, for example, to omit the magnetic core of the transformer from calculations.

The interaction of a global model (GM) and a local (regional) model (LM) of heat flow is considered under the framework of so‐called “one‐way nesting”. In this framework, the GM is constructed in a large domain with coarse discretization in space and time, while the LM is set in a small subdomain with fine discretization.

The GM is solved first, and its results are then used via some boundary transfer operator (BTO) on the GM–LM interface in order to solve the LM. Past experience in various fields of application has shown that one has to be careful in the choice of BTO to be used on the GM–LM interface, since this choice affects both the stability and accuracy of the computational scheme. Here the problem is first theoretically analyzed for the linear heat equation, and stable BTOs are identified. Then numerical experiments are performed with one‐way nesting in a two‐dimensional channel for heat flow with and without radiation emission and linear reaction, using four different BTOs.

Among other conclusions, it is shown that the “negative Robin” BTO is unstable, whereas the Dirichlet, Neumann and “positive Robin” BTO are all stable. It is also shown that in terms of accuracy, the Neumann and “positive Robin” BTOs should be preferred over the Dirichlet BTO.

This study may be the first step in analyzing BTO accuracy and stability for more general atmospheric systems.