Fundamentals of the Finite Element Method for Heat and Fluid Flow
Fundamentals of the Finite Element Method for Heat and Fluid Flow
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DOI：
10.1002/0470014164
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发表时间：
2004-05
期刊：
影响因子：
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通讯作者：
K. N. Seetharamu;R. Lewis;P. Nithiarasu
中科院分区：
文献类型：
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作者：
K. N. Seetharamu;R. Lewis;P. Nithiarasu
前言。1绪论。1.1传热的重要性。1.2传热方式。1.3传热规律。1.4传热问题的公式。1.4.1暴露在太阳热流中的板的传热。1.4.2白炽灯。1.4.3具有相对运动和内部发热的系统。1.5热传导方程。1.6边界和初始条件。1.7求解方法。1.8总结。1.9习题。参考文献。2一些基本离散系统。2.1导论。2.2稳态问题。2.2.1复合板的热流。2.2.2流体流动网络。2.2.3散热器的换热（传导-对流复合）。2.2.4换热器分析。2.3瞬态传热问题（传播问题）。2.4总结。2.5练习。参考文献。3有限元法。3.1绪论。3.2单元与形状函数。3.2.1一维线性单元。3.2.2一维二次元。3.2.3二维线性三角形单元。3.2.4区域坐标。3.2.5二次三角元。3.2.6二维四边形元素。3.2.7等参元素。3.2.8三维元素。3.3配方（元素特性）。3.3.1 Ritz法（热平衡积分法- goodman法）。3.3.2 Rayleigh-Ritz法（变分法）。3.3.3加权残差法。3.3.4 Galerkin有限元法。3.4热传导方程的表达式。3.4.1变分法。3.4.2伽辽金法。3.5插值函数要求。3.6总结。3.7练习。参考文献。4一维稳态热传导。4.1简介。4.2平面壁面。4.2.1均匀壁面。4.2.2复合墙。4.2.3有限元离散化。4.2.4变截面积墙体。4.2.5带热源的平面壁面：线性元解。4.2.6带热源的平面壁面：二次元求解。4.2.7带热源的平面壁面：修正二次方程解（静态冷凝）。4.3气缸内的径向热流。4.3.1带热源的气缸。4.4传导-对流系统。4.5总结。4.6练习。参考文献5多维稳态热传导。5.1简介。5.2二维平面问题。5.2.1三角元。5.3矩形单元。5.4变厚板。5.5三维问题。5.6轴对称问题。5.6.1线性三角形轴对称单元的伽辽金方法。5.7总结。5.8练习。参考书目。6瞬态热传导分析。6.1简介。6.2集总热容量系统。6.3数值解。6.3.1瞬态控制方程及边界和初始条件。6.3.2伽辽金法。6.4.1利用有限差分法（FDM）进行时间离散化。6.4.2采用有限元法进行时间离散化。6.5稳定性。6.6多维瞬态热传导。6.7相变问题-凝固和熔化。6.7.1控制方程。6.7.2焓公式。6.8反热传导问题6.8.1一维热传导6.9总结。6.10练习。7.1导论。7.1.1流体运动辅助热传递的类型。7.2 Navier-Stokes方程7.2.1质量守恒或连续性方程7.2.2动量守恒。7.2.3能量方程。7.3控制方程的无量纲形式。7.3.1强制对流。7.3.2自然对流（浮力驱动对流）。7.3.3混合对流。7.4瞬态对流扩散问题。7.4.1对流扩散方程的有限元解。7.4.2多维扩展。7.5稳定性条件7.6基于特征的分割（CBS）方案7.6.1空间离散化7.6.2时间步长计算。7.6.3边界和初始条件。7.6.4稳态和瞬态解法。7.7人工压缩方案。7.8努塞尔数、拖拽和流函数。7.8.1努塞尔数。7.8.2阻力计算。7.8.3流函数。7.9网格收敛。7.10层流等温流动。7.10.1几何、边界和初始条件。7.10.2解7.11层流非等温流动7.11.1强制对流换热7.11.2浮力驱动对流换热。7.11.3混合对流换热。7.12紊流导论。7.12.1求解过程及结果。7.13轴对称问题的推广。7.14总结。7.15练习。参考书目。8多孔介质中的对流。8.1导论。8.2广义多孔介质流动方法。8.2.1无量纲尺度。8.2.2限制情况。8.3离散化过程。8.3.1时间离散化。8.3.2空间离散化。8.3.3半隐式和拟隐式形式。8.4非等温流动。8.5强制对流。8.6自然对流。8.6.1恒孔隙介质。8.7总结。8.8练习。参考书目。9流体流动和传热问题的一些例子。9.1导论。9.2等温流动问题。9.2.1稳态问题。9.2.2瞬态流。9.3非等温基准流动问题。9.3.1后向步进。9.4电子封装中的热传导。9.5热源的强制对流传热。9.6总结。9.7练习。参考书目。10计算机代码的实现。10.1简介。10.2预处理。10.2.1网格生成。10.2.2线性三角元数据。10.2.3元素大小计算。10.2.4形状函数及其导数。10.2.5边界正常计算。10.2.6质量矩阵和质量集总。10.2.7隐含压力或热传导矩阵。10.3.1时间步长计算。10.3.2元件循环和装配。10.3.3更新方案。10.3.4边界条件。10.3.5监控稳态。10.4后处理。10.4.1数据插值。10.5总结。参考书目。一个格林引理。B积分公式。B.1直线三角形。B.2线性四面体。C有限元装配程序。D . Navier-Stokes方程的简化形式。索引。
Preface. 1 Introduction. 1.1 Importance of Heat Transfer. 1.2 Heat Transfer Modes. 1.3 The Laws of Heat Transfer. 1.4 Formulation of Heat Transfer Problems. 1.4.1 Heat transfer from a plate exposed to solar heat flux. 1.4.2 Incandescent lamp. 1.4.3 Systems with a relative motion and internal heat generation. 1.5 Heat Conduction Equation. 1.6 Boundary and Initial Conditions. 1.7 Solution Methodology. 1.8 Summary. 1.9 Exercise. Bibliography. 2 Some Basic Discrete Systems. 2.1 Introduction. 2.2 Steady State Problems. 2.2.1 Heat flow in a composite slab. 2.2.2 Fluid flow network. 2.2.3 Heat transfer in heat sinks (combined conduction-convection). 2.2.4 Analysis of a heat exchanger. 2.3 Transient Heat Transfer Problem (Propagation Problem). 2.4 Summary. 2.5 Exercise. Bibliography. 3 The Finite Elemen t Method. 3.1 Introduction. 3.2 Elements and Shape Functions. 3.2.1 One-dimensional linear element. 3.2.2 One-dimensional quadratic element. 3.2.3 Two-dimensional linear triangular elements. 3.2.4 Area coordinates. 3.2.5 Quadratic triangular elements. 3.2.6 Two-dimensional quadrilateral elements. 3.2.7 Isoparametric elements. 3.2.8 Three-dimensional elements. 3.3 Formulation (Element Characteristics). 3.3.1 Ritz method (Heat balance integral method-Goodman's method). 3.3.2 Rayleigh-Ritz method (Variational method). 3.3.3 The method of weighted residuals. 3.3.4 Galerkin finite element method. 3.4 Formulation for the Heat Conduction Equation. 3.4.1 Variational approach. 3.4.2 The Galerkin method. 3.5 Requirements for Interpolation Functions. 3.6 Summary. 3.7 Exercise. Bibliography. 4 Steady State Heat Conduction in One Dimension. 4.1 Introduction. 4.2 Plane Walls. 4.2.1 Homogeneous wall. 4.2.2 Composite wall. 4.2.3 Finite element discretization. 4.2.4 Wall with varying cross-sectional area. 4.2.5 Plane wall with a heat source: solution by linear elements. 4.2.6 Plane wall with a heat source: solution by quadratic elements. 4.2.7 Plane wall with a heat source: solution by modified quadratic equations (static condensation). 4.3 Radial Heat Flow in a Cylinder. 4.3.1 Cylinder with heat source. 4.4 Conduction-Convection Systems. 4.5 Summary. 4.6 Exercise. Bibliography. 5 Steady State Heat Conduction in Multi-dimensions. 5.1 Introduction. 5.2 Two-dimensional Plane Problems. 5.2.1 Triangular elements. 5.3 Rectangular Elements. 5.4 Plate with Variable Thickness. 5.5 Three-dimensional Problems. 5.6 Axisymmetric Problems. 5.6.1 Galerkin's method for linear triangular axisymmetric elements. 5.7 Summary. 5.8 Exercise. Bibliography. 6 Transient Heat Conduction Analysis. 6.1 Introduction. 6.2 Lumped Heat Capacity System. 6.3 Numerical Solution. 6.3.1 Transient governing equations and boundary and initial conditions. 6.3.2 The Galerkin method. 6.4 One-dimensional Transient State Problem. 6.4.1 Time discretization using the Finite Difference Method (FDM). 6.4.2 Time discretization using the Finite Element Method (FEM). 6.5 Stability. 6.6 Multi-dimensional Transient Heat Conduction. 6.7 Phase Change Problems-Solidification and Melting. 6.7.1 The governing equations. 6.7.2 Enthalpy formulation. 6.8 Inverse Heat Conduction Problems. 6.8.1 One-dimensional heat conduction. 6.9 Summary. 6.10 Exercise. Bibliography. 7 Convection Heat Transfer 173 7.1 Introduction. 7.1.1 Types of fluid-motion-assisted heat transport. 7.2 Navier-Stokes Equations. 7.2.1 Conservation of mass or continuity equation. 7.2.2 Conservation of momentum. 7.2.3 Energy equation. 7.3 Non-dimensional Form of the Governing Equations. 7.3.1 Forced convection. 7.3.2 Natural convection (Buoyancy-driven convection). 7.3.3 Mixed convection. 7.4 The Transient Convection-diffusion Problem. 7.4.1 Finite element solution to convection-diffusion equation. 7.4.2 Extension to multi-dimensions. 7.5 Stability Conditions. 7.6 Characteristic-based Split (CBS) Scheme. 7.6.1 Spatial discretization. 7.6.2 Time-step calculation. 7.6.3 Boundary and initial conditions. 7.6.4 Steady and transient solution methods. 7.7 Artificial Compressibility Scheme. 7.8 Nusselt Number, Drag and Stream Function. 7.8.1 Nusselt number. 7.8.2 Drag calculation. 7.8.3 Stream function. 7.9 Mesh Convergence. 7.10 Laminar Isothermal Flow. 7.10.1 Geometry, boundary and initial conditions. 7.10.2 Solution. 7.11 Laminar Non-isothermal Flow. 7.11.1 Forced convection heat transfer. 7.11.2 Buoyancy-driven convection heat transfer. 7.11.3 Mixed convection heat transfer. 7.12 Introduction to Turbulent Flow. 7.12.1 Solution procedure and result. 7.13 Extension to Axisymmetric Problems. 7.14 Summary. 7.15 Exercise. Bibliography. 8 Convection in Porous Media. 8.1 Introduction. 8.2 Generalized Porous Medium Flow Approach. 8.2.1 Non-dimensional scales. 8.2.2 Limiting cases. 8.3 Discretization Procedure. 8.3.1 Temporal discretization. 8.3.2 Spatial discretization. 8.3.3 Semi- and quasi-implicit forms. 8.4 Non-isothermal Flows. 8.5 Forced Convection. 8.6 Natural Convection. 8.6.1 Constant porosity medium. 8.7 Summary. 8.8 Exercise. Bibliography. 9 Some Examples of Fluid Flow and Heat Transfer Problems. 9.1 Introduction. 9.2 Isothermal Flow Problems. 9.2.1 Steady state problems. 9.2.2 Transient flow. 9.3 Non-isothermal Benchmark Flow Problem. 9.3.1 Backward-facing step. 9.4 Thermal Conduction in an Electronic Package. 9.5 Forced Convection Heat Transfer From Heat Sources. 9.6 Summary. 9.7 Exercise. Bibliography. 10 Implementation of Computer Code. 10.1 Introduction. 10.2 Preprocessing. 10.2.1 Mesh generation. 10.2.2 Linear triangular element data. 10.2.3 Element size calculation. 10.2.4 Shape functions and their derivatives. 10.2.5 Boundary normal calculation. 10.2.6 Mass matrix and mass lumping. 10.2.7 Implicit pressure or heat conduction matrix. 10.3 Main Unit. 10.3.1 Time-step calculation. 10.3.2 Element loop and assembly. 10.3.3 Updating solution. 10.3.4 Boundary conditions. 10.3.5 Monitoring steady state. 10.4 Postprocessing. 10.4.1 Interpolation of data. 10.5 Summary. Bibliography. A Green's Lemma. B Integration Formulae. B.1 Linear Triangles. B.2 Linear Tetrahedron. C Finite Element Assembly Procedure. D Simplified Form of the Navier-Stokes Equations. Index.