Identification of phase change fronts by Bezier splines and BEM

This paper discusses the identification of the phase change front in continuous casting. The problem is formulated and solved as a geometry inverse problem. Sensitivity analysis and boundary element method are used for the estimation of the identified values. The interface between solid and liquid phases in the process is modelled by Bezier curves.The measured temperatures required to solve the problem are always affected by measurement errors. Extensive calculations allow us to determine the influence of measurement errors on the accuracy of the phase change front location.     

Solution of steady-state thermoelastic problems using a scaled boundary representation based on nonuniform rational B-splines

This work explores the application of isogeometric scaled boundary method in the two-dimensional thermoelastic problems of irregular geometry. The proposed method inherits the advantages of both isogeometric analysis and scaled boundary finite element method and overcomes their respective disadvantages. In the proposed approach, the boundaries of the problem domain are discretized with nonuniform rational B-splines (NURBS) basis functions, while the temperature distributions inside the domain are represented by a sequence of power functions in terms of radial coordinate within the framework of scaled boundary finite element method. The resulting solution of the stress in radial direction can be computed analytically for the temperature changes. The construction of tensor product structure is circumvented for the two-dimensional problems as only the boundary information of the problem domain is required. Hence, the flexibility to represent the complex geometry can be significantly improved in the proposed method. Numerical examples are presented to validate the performance of the proposed method where it is seen that superior accuracy, e?ciency, and convergence behavior can be achieved over the conventional scaled boundary finite element method.     

Shape identification for inverse geometry heat conduction problems by FEM without iteration

The boundary geometry shape is identified by the finite element method (FEM) without iteration and mesh reconstruction for two-dimensional (2-D) and three-dimensional (3-D) inverse heat conduction problems. First, the direct heat conduction problem with the exact domain is solved by the FEM and the temperatures of measurement points are obtained. Then, by introducing a virtual boundary, a virtual domain is formed. By minimizing the difference between the temperatures of measurement points in the exact domain and those in the virtual domain, the temperatures of the points on the virtual boundary are calculated based on the least square error method and the Tikhonov regularization. Finally, the objective geometry shape can be estimated by the method of searching the isothermal curve or isothermal surface for 2-D or 3-D problems, respectively. In the process, no iterative calculation is needed. The proposed method has a tremendous advantage in reducing the computational time for the inverse geometry problems. Numerical examples are presented to test the validity of the proposed approach. Meanwhile, the influences of measurement noise, virtual boundary, measurement point number, and measurement point position on the boundary geometry prediction are also investigated in the examples. The solutions show that the method is accurate and efficient to identify the unknown boundary geometry configurations for 2-D and 3-D heat conduction problems.     

The method of fundamental solutions for solving free boundary saturated seepage problem

The problem of seepage flow through a dam is free boundary problem that is more conveniently solved by a meshless method than a mesh-based method such as finite element method (FEM) and finite difference method (FDM). This paper presents method of fundamental solutions, which is one kind of meshless methods, to solve a dam problem using the fundamental solution to the Laplace's equation. Solutions on free boundary are determined by iteration and cubic spline interpolation. The numerical solutions then are compared with the boundary element method (BEM), FDM and FEM to display the performance of present method.     

Finite difference solution of one-dimensional Stefan problem with periodic boundary conditions

A finite difference method is used to solve the one-dimensional Stefan problem with periodic Dirichlet boundary condition. The temperature distribution, the position of the moving boundary and its velocity are evaluated. It is shown that, for given oscillation frequency, both the size of the domain and the oscillation amplitude of the periodically oscillating surface temperature, strongly influence the temperature distribution and the boundary movement. Furthermore, good agreement between the present finite difference results and numerical results obtained previously using the nodal integral method is seen.     

A transient three-dimensional inverse geometry problem in estimating the space and time-dependent irregular boundary shapes

A transient three-dimensional shape identification problem (inverse geometry problem) to determine the unknown irregular and moving boundary configurations by utilizing the steepest descend method (SDM) and a general purpose commercial code CFD-RC is successfully developed and examined in this study based on the simulated measured temperature distributions on the bottom surface by infrared thermography. The advantage of calling CFD-RC as a subroutine in the present inverse calculation lies in that its auto-mesh function enables the handling of this moving boundary problem. Results obtained by using the technique of SDM to solve the inverse geometry problem are justified based on the numerical experiments. Two test cases are performed to test the validity of the present algorithm by using different types of boundary shapes, initial guesses and measurement errors. Results show that reliable estimations on the unknown space and time-dependent boundary geometry can be obtained when the measurement errors are considered.     

边界元法在气冷涡轮叶栅气热耦合计算中的应用

将边界元计算方法应用到三维N-S方程求解程序中,流体部分采用有限差分法求解N-S方程,边界元法求解固体区域热传导方程。开发气热耦合计算程序对NASA-MarkII高压气冷涡轮叶栅热环境进行气热耦合分析。利用边界元法特有的优势(降维、解析与离散相结合的特点),避免了固体区域的网格生成和内部节点求解工作,提高了计算精度。结果表明,耦合计算程序能够高效、准确求解多场的气热耦合问题,计算结果与实验结果吻合的较好,二者平均误差为3%。     

考虑无限元边界的混凝土重力坝动力损伤分析

针对无限元边界模拟地基可解决使用截断地基时地震波在地基中的能量无法逸散的问题,以观音岩重力坝为例,采用混凝土塑性损伤模型和无限元人工边界,计算了重力坝在不同地震荷载作用下的动力时程响应,验证了无限元边界对地震波能量的逸散作用,比较了材料损伤引起的材料刚度降低对坝体响应的影响,获得了不同地震荷载下坝体损伤的发展规律。     

Periodic heat transfer by forced laminar boundary layer flow over a semi-infinite flat plate

The paper reports a study of periodic convection in a steady forced laminar boundary layer flow over a semi-infinite impermeable flat plate due to periodical variation of the wall heat flux. The Fourier transform based approach allows to obtain a transfer function for the boundary layer that can be used to solve also transient (non-periodic) heating problems, and examples are reported comparing with available studies in the open literature. The effect of periodic heating on the value of the the average heat transfer coefficient is analysed and it is found to be important for relatively high frequency fluctuations of the imposed heat flux, whereas fluctuation amplitude of the instantaneous heat transfer coefficient is non-negligible also for lower exciting frequency.     

改进的比例边界有限元法求解层状地基动力刚度矩阵

在传统比例边界有限元法基础上,提出了一种新的坐标变换关系来求解层状地基动力刚度矩阵即改进的比例有限元法。以一条相似轴代替传统比例边界有限元法的相似中心,并利用加权余量法推导得到层状地基位移及动力刚度矩阵方程,不仅在水平向保持了解析特性、满足了水平无穷远处的辐射边界条件,且克服了传统比例边界有限元法在求解侧边平行的层状问题时的不适应性。通过求解3个典型层状地基的动力刚度矩阵,验证了改进方法的求解效率及其对多层地基的广泛适应性。     

Identification of Geometric Boundary Configurations in Heat Transfer Problems Via Element-Free Galerkin and Level-Set Methods

This article presents a numerical model to solve inverse geometry heat transfer problems to determine an unknown boundary shape. The evolution of unknown shapes is described by the level-set method (LSM) and is controlled by a Hamilton–Jacobi equation which is solved by a finite-difference (FD) scheme. The element-free Galerkin method (EFGM) is employed to determine the temperature field in the process of boundary evolution via a slight adjustment of the position and number of nodes. The proposed numerical model is verified via an identification of a curvilinear boundary, and the effects of initial guess, number of probing points, measurement error, and density of EFGM nodes and the LSM FD grid are taken into account.     

Steady-state heat conduction analysis of solids with small open-ended tubular holes by BFM

In this work, the boundary face method (BFM) is applied to implement steady-state heat conduction analysis of solids containing a large number of open-ended tubular shaped holes of small diameters. A new meshing scheme is used to discretize the boundary integral equations (BIE) such that the holes can be modeled by a small number of surface elements while keeps the exact geometry, resulting in substantial savings in both modeling effort and computational cost. In the scheme, each tubular pipe surface is represented with a number of curvilinear tube elements similar to the ‘hole element’ proposed by P.K. Banerjee. To model the end faces that are intersected by the tubular holes, a special triangular element with negative parts is proposed. These elements are defined in the parametric space of the surface, and the exact geometry data can be directly available from CAD models of the solids. Numerical examples show that current implementation is very efficient in modeling of solids with many holes of arbitrary shape. The temperature and flux on the pipe surfaces or inside solids are obtained with high accuracy, even the local thermal concentration on and near the holes can be captured.     

A new numerical method to simulate the non-Fourier heat conduction in a single-phase medium

Many non-equilibrium heat conduction processes can be described by the macroscopic dual-phase lag model (DPL model). In this paper, a numerical method, which combines the dual reciprocity boundary element method (DRBEM) with Laplace transforms, is constructed to solve such mathematical equation. It is used to simulate the non-Fourier phenomenon of heat conduction in a single-phase medium, then numerically predict the differences between the thermal diffusion, the thermal wave and the non-Fourier heat conduction under different boundary conditions including pulse for one- and two-dimensional problems. In order to check this numerical method's reliability, the numerical solutions are still compared with two known analytical solutions.     

Radial integration boundary element method for heat conduction problems with convective heat transfer boundary

A new boundary domain integral equation with convective heat transfer boundary is presented to solve variable coefficient heat conduction problems. Green’s function for the Laplace equation is used to derive the basic integral equation with varying heat conductivities, and as a result, domain integrals are included in the derived integral equations. The existing domain integral is converted into an equivalent boundary integral using the radial integration method by expressing the normalized temperature as a series of radial basis functions. This treatment results in a pure boundary element analysis algorithm and requires no internal cells to evaluate the domain integral. Numerical examples are presented to demonstrate the accuracy and efficiency of the present method.     

STEADY THERMAL STRESS ANALYSIS BY IMPROVED MULTIPLE-RECIPROCITY BOUNDARY ELEMENT METHOD

Steady thermal stress analysis without heat generation can easily be solved by the boundary element method. However, for the case with heat generation, the domain integral is necessary. This paper shows that the problem of steady thermal stress with heat generation can be solved approximately without the domain integral using an improved multiple-reciprocity boundary element method. In this method, the domain integral in each step is divided into point, line, and area integrals in the case of a two-dimensional problem. In order to solve the problem, the contour lines of heat generation, which approximate the actual heat generation, are used.     

Study of shear thinning fluid flow through highly permeable porous media

The Brinkman equation is used to model the flow of power-law fluids in a highly permeable porous medium. Isothermal flow of shear thinning fluids in a porous medium between two impermeable parallel walls at different Darcy parameters (Da) and power-law index is studied. Both finite element method and analytical solutions are applied to solve the Brinkman equation. The analytical solution is based on a trial and error procedure. This solution reveals a channelling in the flow regime within the thin near walls boundary layer. Finite element solution is, in general, unstable but can be stabilised for a limited rang of Darcy parameter and power-law index.     

A new boundary meshfree method for calculating the multi-domain constant coefficient heat conduction with a heat source problem

A new high-precision boundary meshfree method, namely virtual boundary meshfree Galerkin method (VBMGM), for calculating the multi-domain constant coefficient heat conduction with a heat source problem is given. In the paper, the radial basis function interpolation is used to solve the virtual source function of virtual boundary and the heat source within each subdomain. Simultaneously, the equation of VBMGM for multi-domain constant coefficient heat conduction with a heat source problem is obtained by the Galerkin method. Therefore, the proposed method has common advantages of the boundary element method, meshfree method, and Galerkin method. Coefficient matrix of this specific expression is symmetrical and the specific expression of VBMGM for the multi-domain constant coefficient heat conduction with a heat source problem is given. Two numerical examples are given. The numerical results are also compared with other numerical methods. The accuracy and feasibility of the method for the multi-domain constant coefficient heat conduction with a heat source problem are proved.     

Effect of wavy plate geometry configurations on the temperature and flow distributions

As fluid flowing through the wavy plate, breaking and destabilizing in the thermal boundary layer are induced. In the present study, the numerical investigation on the heat transfer and flow distributions in the channel with various geometry configuration wavy plates under constant heat flux conditions is considered. A finite volume method with the structured uniform grid system is used to solve the turbulent model. Effects of geometry configuration of wavy plates, wavy plate arrangements, and air flow rates on the temperature and flow developments are considered. The sharp edge of wavy plate has a significant effect on the flow structure and heat transfer enhancement. The results of this study are expected to lead to guidelines that will allow the selected wavy plate geometry configuration for designing heat exchanger which increase thermal performance.     

HIGH-ORDER SCHEME IMPLEMENTATION USING NEWTON-KRYLOV SOLUTION METHODS

Implementation of high-order discretization for the connective transport terms in the inexact Newton method for a benchmark fluid flow and heat transfer problem using various solution configurations at two Reynolds numbers has been investigated. These configurations include fully consistent discretization of the Jacobian, preconditioner and residual of the Newton method, low-order preconditioning using a matrix-free method to approximate the action of the Jacobian, and defect correction or low-order Jacobian and preconditioning. The residual in each case employs high-order discretization to preserve the high-order solution. Two preconditioners, point incomplete lower-upper factorization ILU( k) and block incomplete lower-upper factorization BILUO(K) for k = 0,1,2 were applied. Also, “one-way” muitigrid-ing and capping the inner iterations was applied to determine the behavior of the solution performance. It was determined that, overall, the configuration using low-order preconditioning with ILU(1), BILU(1), or BILU(2), mesh sequencing, and inner linear solve iterations capped at the same value of the dimension n, used with the GMRES(n) iterative solver( i.e., no restarts), performed best for time, memory, and robustness considerations.     

Modified Collocation Trefftz Method for the Geometry Boundary Identification Problem of Heat Conduction

In this article, a meshless numerical algorithm is proposed for the boundary identification problem of heat conduction, one kind of inverse problem. In the geometry boundary identification problem, the Cauchy data is given for part of the boundary. The Neumann boundary condition is given for the other portion of the boundary, whose spatial position is unknown. In order to stably solve the inverse problem, the modified collocation Trefftz method, a promising boundary-type meshless method, is adopted for discretizing this problem. Since the spatial position for part of the boundary is unknown, the numerical discretization results in a system of nonlinear algebraic equations (NAEs). Then, the exponentially convergent scalar homotopy algorithm (ECSHA) is used to efficiently obtain the convergent solution of the system of NAEs. The ECSHA is insensitive to the initial guess of the evolutionary process. In addition, the efficiency of the computation is greatly improved, since calculation of the inverse of the Jacobian matrix can be avoided. Four numerical examples are provided to validate the proposed meshless scheme. In addition, some factors that might influence the performance of the proposed scheme are examined through a series of numerical experiments. The stability of the proposed scheme can be proven by adding some noise to the boundary conditions.