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The purpose of this paper is to present a fast algorithm for the adaptive discretization of three-dimensional parametric curves.

The proposed algorithm computes the parametric increments of all segments to obtain the parametric coordinates of all discrete nodes. This process is recursively applied until the optimal discretization of curves is obtained. The parametric increment of a segment is inversely proportional to the number of sub-segments, which can be subdivided, and the sum of parametric increments of all segments is constant. Thus, a new expression for parametric increment of a segment can be obtained. In addition, the number of sub-segments, which a segment can be subdivided is calculated approximately, thus avoiding Gaussian integration.

The proposed method can use less CPU time to perform the optimal discretization of three-dimensional curves. The results of curves discretization can also meet requirements for mesh generation used in the preprocessing of numerical simulation.

Several numerical examples presented have verified the robustness and efficiency of the proposed algorithm. Compared with the conventional algorithm, the more complex the model, the more time the algorithm saves in the process of curve discretization.

The purpose of this paper is to develop a new finite-volume method of lines for one-dimensional reaction-diffusion equations that provides piece-wise analytical solutions in space and is conservative, compare it with other finite-difference discretizations and assess the effects of the nonlinear diffusion coefficient on wave propagation.

A conservative, finite-volume method of lines based on piecewise integration of the diffusion operator that provides a globally continuous approximate solution and is second-order accurate is presented. Numerical experiments that assess the accuracy of the method and the time required to achieve steady state, and the effects of the nonlinear diffusion coefficients on wave propagation and boundary values are reported.

The finite-volume method of lines presented here involves the nodal values and their first-order time derivatives at three adjacent grid points, is linearly stable for a first-order accurate Euler’s backward discretization of the time derivative and has a smaller amplification factor than a second-order accurate three-point centered discretization of the second-order spatial derivative. For a system of two nonlinearly-coupled, one-dimensional reaction-diffusion equations, the amplitude, speed and separation of wave fronts are found to be strong functions of the dependence of the nonlinear diffusion coefficients on the concentration and temperature.

A new finite-volume method of lines for one-dimensional reaction-diffusion equations based on piecewise analytical integration of the diffusion operator and the continuity of the dependent variables and their fluxes at the cell boundaries is presented. The method may be used to study heat and mass transfer in layered media.

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.

The axisymmetric steady‐state convective‐diffusive thermal field problem associated with direct‐chill, semi‐continuously cast billets has been solved using the dual reciprocity boundary element method. The solution is based on a formulation which incorporates the one‐phase physical model, Laplace equation fundamental solution weighting, and scaled augmented thin plate splines for transforming the domain integrals into a finite series of boundary integrals. Realistic non‐linear boundary conditions and temperature variation of all material properties are included. The solution is verified by comparison with the results of the classical finite volume method. Results for a 0.500[m] diameter Al 4.5 per cent Cu alloy billet at typical casting conditions are given.

A new‐type eigenvalue formulation of the two‐dimensional Helmholtz equation is presented in this paper. A boundary integral equation is derived using the T‐complete functions relevant to the Trefftz method, which is further transformed to the generalized eigenvalue problem. Boundary discretization and a standard eigenvalue computation routine, offered as a black box, are sufficient for the determination of the eigenvalues. The proposed method can reduce the users’ task in preprocessing and initial rough estimation when compared with the existing domain‐type solvers.

To present numerical approaches to the solution of physically coupled non‐linear problems, which frequently happen to be characterized by their multi‐domain character.

By adopting coupled solution strategies a considerable attention is devoted, in order to obtain a computationally efficient numerical algorithm, to the selection of appropriate space and time discretization, as well as to a proper discrete approximation method used.

Coupling of two methods, the finite element method and the boundary element method, respectively, has proved to be computationally exceedingly advantageous, particularly in case of moving domains.

As specific case studies computer simulation of an induction heating problem and a mushy‐state forming problem are considered. A thorough discussion on the coupling effects, characterizing the evolutions of respective physical quantities' fields, is given, and their impact on those evolutions is identified.

This paper presents efficient numerical strategies for the solution of a certain class of multi‐physics and multi‐domain problems.

This paper presents a modification of the classical boundary integral equation method (BIEM) for two‐dimensional potential boundary‐value problem. The proposed modification consists in describing the boundary geometry by means of Hermite curves. As a result of this analytical modification of the boundary integral equation (BIE), a new parametric integral equation system (PIES) is obtained. The kernels of these equations include the geometry of the boundary. This new PIES is no longer defined on the boundary, as in the case of the BIE, but on the straight line for any given domain. The solution of the new PIES does not require boundary discretization as it can be reduced merely to an approximation of boundary functions. To solve this PIES a pseudospectral method has been proposed and the results obtained compared with exact solutions.

We report on the progress in the development and application of a coupled boundary element/finite volume method temperature‐forward/flux‐back algorithm developed to solve conjugate heat transfer arising in 3D film‐cooled turbine blades. We adopt a loosely coupled strategy where each set of field equations is solved to provide boundary conditions for the other. Iteration is carried out until interfacial continuity of temperature and heat flux is enforced. The NASA‐Glenn explicit finite volume Navier‐Stokes code Glenn‐HT is coupled to a 3D BEM steady‐state heat conduction solver. Results from a CHT simulation of a 3D film‐cooled blade section are compared with those obtained from the standard two temperature model, revealing that a significant difference in the level and distribution of metal temperatures is found between the two. Finally, current developments of an iterative strategy accommodating large numbers of unknowns by a domain decomposition approach is presented. An iterative scheme is developed along with a physically‐based initial guess and a coarse grid solution to provide a good starting point for the iteration. Results from a 3D simulation show the process that converges efficiently and offers substantial computational and storage savings.

Direct and indirect time marching boundary element methods often become numerically unstable. Evidence of, and reasons for, these instabilities is provided in this paper. Two new time stepping schemes are presented, both of which are more stable than the existing standard schemes available. In particular, we introduce the Half‐step scheme, which is more accurate and far more stable than existing methods. This scheme, which is demonstrated on a simple crack problem for the displacement discontinuity method, can also be introduced into the direct boundary element method. Implementation of the Half‐step scheme into existing boundary element codes will allow researchers to attack more challenging problems than before.

The purpose of this paper is to present some modifications in the spline‐based differential quadrature method (DQM), in order to accelerate the convergence of the method. The improvements are explained and examined by the examples of the free vibration of conical shells. The composite laminated shell, as well as isotropic one, are taken under consideration.

To determine weighting coefficients for the DQM, the spline interpolation with non‐standard definitions of the end conditions is used. One of these definitions combines natural and not‐a‐knot end conditions, while the other one uses the boundary conditions for considered problem as the end conditions. The weighting coefficients can be determined by solving set of equations arising from spline interpolation.

It is shown that the proposed modifications significantly improve the convergence of the method, especially when the boundary conditions are introduced at the stage of the computation of the weighting coefficients. Unfortunately, the use of this approach is limited to some types of boundary conditions.

The paper describes development of the modified spline interpolation dedicated to DQM and examines the possibility of building boundary conditions into the weighting coefficients at the stage of the computation of these coefficients.