A node-based smoothed point interpolation method (NS-PIM) for three-dimensional heat transfer problems

A node-based smoothed point interpolation method (NS-PIM) is formulated for three-dimensional (3D) heat transfer problems with complex geometries and complicated boundary conditions. Shape functions constructed here through PIM possess the delta function property and hence allow the straightforward enforcement of essential boundary conditions. The smoothed Galerkin weak form is employed to create discretized system equations, and the node-based smoothing domains are used to perform the smoothing operation and the numerical integration. The accuracy and efficiency of the NS-PIM solutions are studied through detailed analyses of actual 3D heat transfer problems. It is found that the NS-PIM can provide higher accuracy in temperature and its gradient than the reference approach does, in which very fine meshes are used in standard FEM code available with homogeneous essential boundary conditions. More importantly, the upper bound property of the NS-PIM is obtained using the same tetrahedral mesh. Together with the FEM, we now have a simple means to obtain both upper and lower bounds of the exact solution to heat transfer using the same type of mesh.     

An edge-based smoothed point interpolation method (ES-PIM) for heat transfer analysis of rapid manufacturing system

This paper formulates an edge-based smoothed point interpolation method (ES-PIM) for analyzing 2D and 3D transient heat transfer problems with mixed boundary conditions and complicated geometries. In the ES-PIM, shape functions are constructed using the polynomial PIM with the Delta function property for easy treatment of essential boundary conditions. A generalized smoothing technique is used to reconstruct the temperature gradient field within the edge-based smoothing domains. The generalized smoothed Galerkin weak form is then used to establish the discretized system equations. Our results show that the ES-PIM can provide more close-to-exact stiffness compared with the “overly-stiff” finite element method (FEM) and the “overly-soft” node-based smoothed point interpolation method (NS-PIM). Owing to this important property, the present ES-PIM provides more accurate solutions than standard FEM using the same mesh. As an example, a practical cooling system of the rapid direct plasma deposition dieless manufacturing is studied in detail using the present ES-PIM, and a set of “optional” processing parameters of fluid velocity and temperature are found.     

基于无网格NS-PIM与有限元法的边坡应变能及位移上下限解研究

为尽可能精确求解边坡应变能及位移的上下限解的问题,结合无网格NS-PIM方法与有限元法,求解出了离散精度不同的边坡模型的应变能及位移,快速确定了精确解的分布范围,并依据工程精度要求,进一步细化模型以使计算结果逼近精确解。实例计算表明,NS-PIM方法、有限元方法可分别得到应变能和总体位移的上、下限解,为位移监测分析和边坡稳定性评价提供了良好思路。     

A Node-based Smoothed Point Interpolation Method (NS-PIM) for Three-dimensional Thermoelastic Problems

A node-based smoothed point interpolation method (NS-PIM) is formulated to analyze 3-D steady-state thermoelastic problems subjected to complicated thermal and mechanical loads. Gradient smoothing technique with node-based smoothing domains is utilized to modify the gradient fields and to perform the numerical integration required in the weak form formulation. Numerical results show that NS-PIM can achieve more accurate solutions even when the 4-node tetrahedral mesh is used compared to the finite-element method (FEM) using the same mesh, especially for strains and hence stresses. Most importantly, it can produce an upper bound solution of the exact solution in energy norm for both temperature and stress fields when a reasonably fine mesh is used. Together with FEM, we now for the first time have a simple means to obtain both upper and lower bounds of the exact solution to complex thermoelastic problems.     

Analysis of elastic–plastic problems using edge-based smoothed finite element method

In this paper, an edge-based smoothed finite element method (ES-FEM) is formulated for stress field determination of elastic–plastic problems using triangular meshes, in which smoothing domains associated with the edges of the triangles are used for smoothing operations to improve the accuracy and the convergence rate of the method. The smoothed Galerkin weak form is adopted to obtain the discretized system equations, and the numerical integration becomes a simple summation over the edge-based smoothing domains. The pseudo-elastic method is employed for the determination of stress field and Hencky's total deformation theory is used to define effective elastic material parameters, which are treated as field variables and considered as functions of the final state of stress fields. The effective elastic material parameters are then obtained in an iterative manner based on the strain controlled projection method from the uniaxial material curve. Some numerical examples are investigated and excellent results have been obtained demonstrating the effectivity of the present method.     

Investigation of Several Interpolation Functions for Unstructured Meshes in Conjunction with Compositional Reservoir Simulation

One of the key parameters in numerical simulation of fluid flow problems is the interpolation function used for adjusting the numerical value from the position where it is evaluated to the interface of the control volumes; for instance, if the finite-volume method is employed. In this article, we present an investigation of several interpolation functions in conjunction with the element based finite-volume method (EbFVM) using unstructured triangular meshes. We investigate the mass weighted upwind (MWU) and a modified version of this method, the upwind scheme, a streamline based upwind scheme, and a limited second order upwind scheme. These interpolation functions are implemented in a compositional simulator using vertex cell unstructured grids. The accuracy of these interpolation functions are evaluated for several case studies. For some of them, we also compare the results with the available analytical solutions.     

Unified finite element approach for generalized coupled thermoelastic analysis of 3D beam-type structures,part 1: Equations and formulation

An innovative 1D finite element (FE) approach is developed to analyze the 3D static, transient, and dynamic problems in the coupled and uncoupled thermoelasticity for the nonhomogeneous anisotropic materials. The Galerkin method is directly applied to the governing equations to obtain a weak formulation of the thermoelasticity problems with arbitrary loads and boundary conditions. To surmount the restrictions of the classical beam theories, a 1D FE procedure is proposed in the context of the Carrera Unified Formulation (CUF). Since coupled thermoelastic analyses are computationally demanding, the proposed 1D FE approach can be used as a powerful means to simulate the generalized coupled thermoelastic behavior of structures. This methodology, indeed, reduces the 3D problems to 1D models with 3D-like accuracies and very low computational costs. The Lord-Shulman and the Green-Lindsay models are considered as the generalized theories of thermoelasticity. Furthermore, as simplified cases, the classical coupled, dynamic uncoupled, quasi-static uncoupled and steady-state uncoupled theories of thermoelasticity may be derived from the formulation. Moreover, effects of the structural damping can be taken into account in the present formulation. The accuracy of the formulation has been evaluated through numerical simulations and comparisons, which have been presented in a companion article (Part 2).     

三维有限元网格的面切割自动生成方法

讨论了三维有限元网格被平面、三角形网、地形面切割时的各种情况,针对六面体单元和三棱柱单元的切割都提出了相应的算法,并用程序予以实现。这种切割方法可以细分有限元网格,生成复杂模型。通过实际例子验证了该法的可行性。     

Stability analysis of generalized mass formulation in dynamic heat transfer

ABSTRACT

In this article, the generalized mass formulation is developed in an explicit analysis of transient transport problems. It has been well known that the time step is typically smaller in explicit analysis than in implicit analysis when the same size mesh is used. Further, the over-stiffness of conventional finite-element model may result in poor accuracy with linear triangular or tetrahedral elements. In order to improve the computational efficiency and numerical accuracy, this article proposes a generalized mass formulation by matching the mass matrix to the smoothed stiffness matrix using linear triangular elements in 2-D problems. The proposed mass matrix can be obtained by simply shifting the integration points from the conventional locations. Without loss of generality, several 2-D examples, including conduction, convection, and radiation heat transfer problems, are presented to demonstrate that the generalized mass formulation allows a larger time step in explicit analysis compared with the lumped and consistent mass matrices. In addition, it is found that the maximum allowable time step is proportional to the softened effect of the discretized model in an explicit analysis.     

A Novel Boundary-Integral Based Finite Element Method for 2D and 3D Thermo-Elasticity Problems

In this article a new hybrid boundary integral-based (HBI) finite element method (FEM) is presented for analyzing two-dimensional (2D) and three-dimensional (3D) thermoelastic problems with arbitrary distribution of body force and temperature changes. The method of particular solution is used to decompose the displacement field into homogeneous part and particular part. The homogeneous solution is obtained by using the HBI-FEM with fundamental solutions, yet the particular solution related to the body force and temperature change is approximated by radial basis function (RBF). The detailed formulation for both 2D and 3D HBI-FEM for thermoelastic problems are given, and two different approaches for treating the inhomogenous terms are presented and compared. Five numerical examples are presented to demonstrate the accuracy and performance of the proposed method. When compared with the existing analytical solutions or ABAQUS results, it is found that the proposed method works well for thermoelastic problems and also when using a very coarse mesh, results with satisfactory accuracy can be obtained.     

边界点法在传热问题数值分析中的应用

将一种新的数值分析方法-边界点法应用于传热问题的研究，对无内热源稳态热传导问题，通过传统边界元法将边界积分方程离散化，发现可以不直接求解影响系数矩阵，而是通过对偶关系，由域外虚源构造方程组的特解场形成边界已知和未知温度，热流密度的系数矩阵，而且域内温度和热流密度的求解将不依赖于边界未知参数的求解，对于有内热源的问题，可以将非齐次方程的解转换为齐次方程的解和某一确定解的叠加，对于非线性问题，可以通过基尔霍夫变换，将非线性问题转化为线性问题求解，这种边界点方法不但具有边界元法降维的优势，而且不须求解奇异积分，大大节约了计算时间，计算精度极高，以有内热源非线性稳态热传导问题的实例印证了这种方法的高效性。     

A two-dimensional finite element recursion relation for the transport equation using nine-diagonal solvers

Abstract

A Galerkin-based finite element recursion relation is used to solve the heat transport equation in two-dimensions. The finite element method (FEM) is a powerful technique that is commonly used for solving complex engineering problems. However, the implementation of the FEM in multi-dimensional problems can be computationally expensive. A finite element recursion algorithm based on bilinear triangular, bilinear quadrilateral and quadratic Lagrangian approximations are employed to discretize the 2-D advection-diffusion equation. This algorithm is an extension of the 1-D Chapeau (linear element) technique, which employed a tridiagonal recursion expression common to the classical central finite-difference approach. The global matrix is nine-diagonal (for 2-D) and is solved using a modified strongly implicit procedure and a left-to-right sweep method.     

IMPORTANCE OF INERTIA TERM IN DYNAMIC CRACK PROBLEMS CONSIDERING LORD–SHULMAN THEORY OF THERMOELASTICITY

A numerical technique is presented for the accurate calculation of stress intensity factors as a function of time for generalized coupled thermoelastic problems. In this task, the effect of the inertia term is investigated, considering different theories of thermoelasticity, and its importance is shown.

A boundary element method using the Laplace transform in time domain is developed for the analysis of fracture mechanics; dynamic coupled thermoelasticity problems with relaxation time are considered in the two-dimensional finite domain. The Laplace transform method is applied to the time domain and the resulting equations in the transformed field are discretized using the boundary element method. Actual physical quantities in the time domain are obtained using the numerical inversion of the Laplace transform method.

The singular behavior of the temperature and displacement fields in the vicinity of the crack tip is modeled by quarter-point elements. The thermal dynamic stress intensity factor for mode I is evaluated using the J-integral method. The accuracy of the method is investigated through comparison of the results with the data available in literature.

The J integral, which represents the dynamic energy release rate for propagating cracks, contains a boundary integral and a domain integral. The boundary integral contains strain energy, tractions, and strains whereas the domain integral contains inertia and strains. The J-integral method allows these two terms to be calculated separately. In this way, the importance of each term may be investigated by considering different theories of dynamic thermoelasticity.     

插值算法在内燃机瞬时转速测量中的应用研究

提出了采样信号的插值计算公式 ;考虑到测量信号中调制现象的影响 ,提出了插值法在瞬时转速计算中的误差公式 ;并针对 6 - 135 G柴油机最高转速对两点插值和三点插值的精度进行了计算 ;最后用实测数据进行了验证。结果表明这是提高瞬时转速测量精度的一种有效方法 ,在 5 0 k Hz的低采样率下即可得到较平滑的瞬时转速波形。     

Comparative Study on Triangular and Quadrilateral Meshes by a Finite-Volume Method with a Central Difference Scheme

In this article, comparative studies on computational accuracies and convergence rates of triangular and quadrilateral meshes are carried out in the frame work of the finite-volume method. By theoretical analysis, we conclude that the number of triangular cells needs to be 4/3 times that of quadrilateral cells to obtain similar accuracy. The conclusion is verified by a number of numerical examples. In addition, the convergence rates of the triangular meshes are found to be slower than those of the quadrilateral meshes when the same accuracy is obtained with these two mesh types.     

Applied theories of thermoelasticity of micropolar thin beams

In the present article, a system of rather general hypotheses is developed, on the basis of which with the account of transverse shears the applied theories of thermoelasticity of bending of micropolar thin beams are introduced: (a) With free fields of displacements and rotations; (b) with constrained rotation; (c) the reduced model. On the basis of the constructed models, specific problems of thermoelastic bending of micropolar beams are studied. Numerical analysis of the studied problems states the effectiveness of the micropolar material from the point of view of the beam rigidity compared with that of the classical case.     

General fully implicit discretization of the diffusion term for the finite volume method

In this paper, a fully implicit method for the discretization of the diffusion term is presented in the context of the cell-centered finite volume method. The newly developed fully implicit method is denoted by the modified implicit nonlinear diffusion (MIND) scheme. The method is used to solve several isotropic and anisotropic diffusion problems in two-dimensional domains covered with structured (quadrilateral elements) and unstructured (triangular elements) grid systems. The comparison of generated results with similar ones obtained using the semi-implicit scheme demonstrates the superior robustness and accuracy of the MIND scheme and its good convergence characteristics for all types of meshes.     

A High Order Control Volume Finite Element Method for Transient Heat Conduction Analysis of Multilayer Functionally Graded Materials with Mixed Grids

This paper describes a new two-dimensional(2-D) control volume finite element method(CV-FEM) for transient heat conduction in multilayer functionally graded materials(FGMs). To deal with the mixed-grid problem, 9-node quadrilateral grids and 6-node triangular grids are used. The unknown temperature and material properties are stored at the node. By using quadratic triangular grids and quadratic quadrilateral grids, the present method offers greater geometric flexibility and the potential for hig...     

Numerical study of a high-hydrogen micromix model burner using flamelet-generated manifold

The flamelet-generated manifolds (FGM) method was adopted in this study to consider the preferential diffusion in a high-hydrogen micro-mixing model burner. That is, when solving the FGM flamelet, accurate diffusion rate was obtained from two methods: multicomponent formulation and constant detailed Lewis numbers assumption. Then a new method of filling the thermochemical state and the source term in the mixture fraction and the process variable space also was proposed, namely the linear triangular dissection interpolation method, to predict the position of the hydrogen-rich micro-mixing flame front. Compared with the Fluent approach to establish the diffusion FGM flamelet, the results showed that the two FGMs have similar flame predictions in high hydrogen content fuels, and both can accurately capture the location of the internal and external shear layer boundaries of the micro-mixing multi-jet flame in the steady state, while the Fluent approach based on the uniform Lewis number assumption predicts results that deviated significantly from the experimental results. However, for the internal shear layer, both methods have large predicted OH gradients compared to the experimental results due to the lack of effective Lewis number correction for the control variable transport equation. The results using linear triangular dissection interpolation maybe superior to the method with linear interpolation of the process variable quenching boundary toward zero, which leads to flashback due to overestimation of the process variable source term in the region below the diffusion FGM quenching boundary.     

A Numerical Study of Weight Functions,Scaling, and Penalty Parameters for Heat Transfer Applications

ABSTRACT

This article describes a numerical study of weight functions, scaling, and penalty parameters for heat transfer problems. The numerical analysis is carried out using a meshless element-free Galerkin (EFG) method, which utilizes moving least-square (MLS) approximants to approximate the unknown function of temperature. These MLS approximants are constructed by using a weight function, a basis function, and a set of coefficients that depend on position. Lagrange multiplier and penalty methods are used to enforce the essential boundary conditions. MATLAB software is developed to obtain the EFG results. A new rational weight function is proposed. Comparisons are made among the results obtained using cubic spline, quartic spline, Gaussian, quadratic, hyperbolic, rational, exponential and, cosine weight functions in one-dimensional (1-D), two-dimensional (2-D), and three-dimensional (3-D) heat transfer problems. The L₂ error norm and rate of convergence are evaluated for different EFG weight functions and the finite-element method (FEM). The effect of scaling and penalty parameters on EFG results is discussed in detail. The results obtained by the EFG method are compared with those obtained by finite-element and analytical methods.