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Two variational formulations of the eddy‐current problem in a bounded domain are considered. The first is based on the scalar magnetic potential and the vector potential of eddy currents; the second on the scalar and vector magnetic potentials. For both formulations existence and uniqueness of exact and approximate (finite element‐Galerkin's) solutions are proved. The correlation between the error of approximation and the error of the finite element solution is established.

Gives a bibliographical review of the error estimates and adaptive finite element methods from the theoretical as well as the application point of view. The bibliography at the end contains 2,177 references to papers, conference proceedings and theses/dissertations dealing with the subjects that were published in 1990‐2000.

The purpose of this article is to derive an implicit-explicit local differential transform method (IELDTM) in dealing with the spatial approximation of the stiff advection-diffusion-reaction (ADR) equations.

A direction-free numerical approach based on local Taylor series representations is designed for the ADR equations. The differential equations are directly used for determining the local Taylor coefficients and the required degrees of freedom is minimized. The complete system of algebraic equations is constructed with explicit/implicit continuity relations with respect to direction parameter. Time integration of the ADR equations is continuously utilized with the Chebyshev spectral collocation method.

The IELDTM is proven to be a robust, high order, stability preserved and versatile numerical technique for spatial discretization of the stiff partial differential equations (PDEs). It is here theoretically and numerically shown that the order refinement (p-refinement) procedure of the IELDTM does not affect the degrees of freedom, and thus the IELDTM is an optimum numerical method. A priori error analysis of the proposed algorithm is done, and the order conditions are determined with respect to the direction parameter.

The IELDTM overcomes the known disadvantages of the differential transform-based methods by providing reliable convergence properties. The IELDTM is not only improving the existing Taylor series-based formulations but also provides several advantages over the finite element method (FEM) and finite difference method (FDM). The IELDTM offers better accuracy, even when using far less degrees of freedom, than the FEM and FDM. It is proven that the IELDTM produces solutions for the advection-dominated cases with the optimum degrees of freedom without producing an undesirable oscillation.

This paper aims to present a computationally efficient finite element model for the simulation of isothermal immiscible two-phase flow in a rigid porous media with a particular application to CO₂ sequestration in underground formations. Focus is placed on developing a numerical procedure, which is effectively mesh-independent and suitable to problems at regional scales.

The averaging theory is utilized to describe the governing equations of the involved unsaturated multiphase flow. The level-set (LS) method and the extended finite element method (XFEM) are utilized to simulate flow of the CO₂ plume. The LS is employed to trace the plume front. A streamline upwind Petrov-Galerkin method is adopted to stabilize possible occurrence of spurious oscillations due to advection. The XFEM is utilized to model the high gradient in the saturation field front, where the LS function is used for enhancing the weighting and the shape functions.

The capability of the proposed model and its features are evaluated by numerical examples, demonstrating its accuracy, stability and convergence, as well as its advantages over standard and upwind techniques. The study showed that a good combination between a mathematical model and a numerical model enables the simulation of complicated processes occurring in complicated and large geometry using minimal computational efforts.

A new computational model for two-phase flow in porous media is introduced with basic requirements for accuracy, stability, and convergence, which are met using relatively coarse meshes.

This paper presents a novel framework for solving elliptic partial differential equations (PDEs) over irregularly spaced meshes on bounded domains.

Second‐generation wavelet construction gives rise to a powerful generalization of the traditional hierarchical basis (HB) finite element method (FEM). A framework based on piecewise polynomial Lagrangian multiwavelets is used to generate customized multiresolution bases that have not only HB properties but also additional qualities.

For the 1D Poisson problem, we propose – for any given order of approximation – a compact closed‐form wavelet basis that block‐diagonalizes the stiffness matrix. With this wavelet choice, all coupling between the coarse scale and detail scales in the matrix is eliminated. In contrast, traditional higher‐order (n>1) HB do not exhibit this property. We also achieve full scale‐decoupling for the 2D Poisson problem on an irregular mesh. No traditional HB has this quality in 2D.

Similar techniques may be applied to scale‐decouple the multiresolution finite element (FE) matrices associated with more general elliptic PDEs.

By decoupling scales in the FE matrix, the wavelet formulation lends itself particularly well to adaptive refinement schemes.

The paper explains second‐generation wavelet construction in a Lagrangian FE context. For 1D higher‐order and 2D first‐order bases, we propose a particular choice of wavelet, customized to the Poisson problem. The approach generalizes to other elliptic PDE problems.

Vorticity formulations for the incompressible Navier‐Stokes equations have certain advantages over primitive‐variable formulations including the fact that the number of equations to be solved is reduced through the elimination of the pressure variable, identical satisfaction of the incompressibility constraint and the continuity equation, and an implicitly higher‐order approximation of the velocity components. For the most part, vorticity methods have been used to solve exterior isothermal problems. In this research, a vorticity formulation is used to study the natural convection flows in differentially‐heated enclosures. The numerical algorithm is divided into three steps: two kinematic steps and one kinetic step. The kinematics are governed by the generalized Helmholtz decomposition (GHD) which is solved using a boundary element method (BEM) whereas the kinetics are governed by the vorticity equation which is solved using a finite element method (FEM). In the first kinematic step, vortex sheet strengths are determined from a novel Galerkin implementation of the GHD. These vortex sheet strengths are used to determine Neumann boundary conditions for the vorticity equation. (The thermal boundary conditions are already known.) In the second kinematic step, the interior velocity field is determined using the regular (non‐Galerkin) form of the GHD. This step, in a sense, linearizes the convective acceleration terms in both the vorticity and energy equations. In the third kinetic step, the coupled vorticity and energy equations are solved using a Galerkin FEM to determine the updated values of the vorticity and thermal fields. Two benchmark problems are considered to show the robustness and versatility of this formulation including natural convection in an 8×1 differentially‐heated enclosure at a near critical Rayleigh number.

The Galerkin finite element method (FEM) based on the characteristic-based split (CBS) scheme is applied to simulate the nanofluid flow and thermal fields inside an inclined geometry filled by a heat-generating hydrodynamically and thermally anisotropic non-Darcy porous medium using the local thermal non-equilibrium model (LTNEM). Property of the hydrodynamic anisotropy is taken in both the Forchheimer coefficient and permeability and these tools are considered as functions of inclination of the principal axes. Also, the thermal conductivity for the porous phase is assumed to be anisotropic.

The Galerkin FEM based on the CBS scheme is applied to solve the partial differential equations governing the flow and thermal fields.

It is noted that the net rate of the heat transfer between the nanofluid and solid phases are influenced by variations of the anisotropic properties. Also, the system is reached to the thermal equilibrium state at H > 100. Further, the maximum nanofluid temperature is reduced by 12.27% when the nanoparticles volume fraction is varied from 0% to 4%.

This paper aims to study the nanofluid flow and heat transfer characteristics inside an inclined enclosure filled with a heat-generating, hydrodynamically and thermally anisotropic porous medium using the CBS scheme. The LTNEM is considered between the nanofluid and porous phases while the local thermal equilibrium model (LTEM) between the base fluid (water) and the nanoparticles (alumina) is taken into account. The Galerkin FEM is introduced to discretize the governing system of equations. Also, examine influences of the anisotropic properties (permeability, Forchheimer terms and thermal conductivity of the porous medium), inclination angle and nanoparticles volume fraction on the net rate of the heat transfer between the nanofluid and porous phases and on the local thermal non-equilibrium state is one of the concerns of this paper.

In this study, a railway superstructure is modeled with a new approach called locally continuous supporting, and its behavior under the effect of moving load is analyzed by using analytical and numerical techniques. The purpose of the study is to demonstrate the success of the new modeling technique.

In the railway superstructure, the support zones are not modeled with discrete spring-damping elements. Instead of this, it is considered to be a continuous viscoelastic structure in the local areas. To model this approach, the governing partial differential equations are derived by Hamilton’s principle and spatially discretized by the Galerkin’s method, and the time integration of the resulting ordinary differential equation system is carried out by the Newmark–Beta method.

Both the proposed model and the solution technique are verified against conventional one-dimensional and three-dimensional finite element models for a specific case, and a very good agreement between the results is observed. The effects of geometric, structural, and loading parameters such as rail-pad length, rail-pad stiffness, rail-pad damping ratio, the gap between rail pads and vehicle speed on the dynamic response of railway superstructure are investigated in detail.

There are mainly two approaches to the modeling of rail pads. The first approach considers them as a single spring-damper connected in parallel located at the centroid of the rail pad. The second one divides the rail pad into several parts, with each of part represented by an equivalent spring-damper system. To obtain realistic results with minimum CPU time for the dynamic response of railway superstructure, the rail pads are modeled as continuous linearly viscoelastic local supports. The mechanical model of viscoelastic material is considered as a spring and damper connected in parallel.

A finite element method for the analysis of combined radiative and conductive heat transport in a finite axisymmetric configuration is presented. The appropriate integro‐differential governing equations for a grey and non‐scattering medium with grey and diffuse walls are developed and solved for several model problems. We consider axisymmetric, cylindrical geometries with top and bottom boundaries of arbitrary convex shape. The method is accurate for media of any optical thickness and is capable of handling a wide array of axisymmetric geometries and boundary conditions. Several techniques are presented to reduce computational overhead, such as employing a Swartz‐Wendroff approximation and cut‐off criteria for evaluating radiation integrals. The method is successfully tested against several cases from the literature and is applied to some additional example problems to demonstrate its versatility. Solution of a free‐boundary, combined‐mode heat transfer problem representing the solidification of a semitransparent material, the Bridgman growth of an yttrium aluminium garnet (YAG) crystal, demonstrates the utility of this method for analysis of a complex materials processing system. The method is suitable for application to other research areas, such as the study of glass processing and the design of combustion furnace systems.

The purpose of this paper is to study the flow and heat transfer characteristics of a phase transition, melting problem. In this problem, phase transition between solid and liquid takes place within a square enclosure in the presence of natural convection.

The physical problem, described with non-linear partial differential equations, is simulated using a hybrid finite element and element free Galerkin method (FEM/EFGM) approach. In energy conservation equation, the fixed-domain, effective heat capacity method is used to take into account the latent heat of phase change. The governing partial differential equations are solved with a meshfree, EFGM near the phase transition front while in the region away from the front with uniform nodal distribution; problem is simulated with traditional FEM.

A sensitivity analysis of characteristic dimensionless numbers Rayleigh number (Ra), Prandtl number (Pr), Stefan number (ste) is presented in order to investigate their impact on thermal and flow fields. Typically computational times of EFGM are higher than that of FEM. Therefore, by using EFGM only in that portion of physical problem where phase transition occurs, the hybrid FEM/EFGM strategy employed in present paper could reduce the computational time of EFGM while still retaining its accuracy. Also, the consistent performance of the results obtained with this hybrid approach is validated with those already available in literature for some special cases.

The hybrid methodology adopted in this paper, is quite new for solving such type of phase transition problem.