Roland W. Lewis

Fundamentals of the Finite Element Method for Heat and Fluid Flow: Lewis/Finite Element Method for Heat and Fluid Flow

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-Goodmans 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 Galerkins 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 Greens 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.

Finite element simulation of metal casting

Finite element algorithms are presented for the entire casting process from the mould filling stage to the prediction of the final distorted shape. The various algorithms available in the literature for solidification modelling are discussed in detail. Special emphasis is given to the coupling of the solidification analysis based on the heat conduction equation with fluid flow or thermal stress analysis. Finally, some results are presented to demonstrate the capabilities of the numerical models. Copyright

A plasticity model for metal powder forming processes

Abstract In this paper, a double-surface plasticity model, based on a combination of a convex yield surface consisting of a failure envelope, such as a Mohr–Coulomb yield surface and, a hardening cap model, is developed for the nonlinear behaviour of powder materials in the concept of a generalized plasticity formulation for the description of cyclic loading. This model reflects the yielding, frictional and densification characteristics of powder along with strain and geometrical hardening which occur during the compaction process. The solution yields details on the powder displacement from which it is possible to establish the stress state in the powder and the densification is derived from consideration of the elemental volumetric strain. A hardening rule is used to define the dependence of the yield surface on the degree of plastic straining. Finally, an adaptive finite element model (FEM) analysis is employed by the updated Lagrangian formulation to simulate the compaction of a set of complex powder forming processes.

Finite element simulation for dynamic large elastoplastic deformation in metal powder forming

Abstract In this paper, a transient dynamic analysis of the powder compaction process is simulated by a large displacement finite element method based on a total and updated Lagrangian formulation. A combination of the Mohr–Coulomb and elliptical yield cap model, which reflects the stress state and degree of densification, is applied to describe the constitutive model of powder materials. A Coulomb friction law and a plasticity theory of friction in the context of an interface element formulation are employed in the constitutive modelling of the frictional behaviour between the die and powder. Finally, the powder behaviour during the compaction of a plain bush, a rotational flanged and a shaped tip component are analysed numerically. It is shown that the updated Lagrangian formulation, using a combination of the Mohr–Coulomb and elliptical cap model, can be effective in simulating metal powder compaction.

Numerical modelling of large deformation in metal powder forming

Abstract In this paper, the transient dynamic analysis of metal powder during the cold compaction process is simulated by the finite element method based on a ‘Total’ and ‘Updated’ Lagrangian formulation. Since the compaction process involves a very large reduction in volume, the behaviour of the powders is assumed to be that of a rate-independent elasto-plastic material. The process is therefore described by a large displacement finite element formulation for the spatial discretization. A generalized Newmark scheme is used for the time domain discretization and then the final nonlinear equations are solved by a Newton-Raphson procedure. A combination of a Mohr-Coulomb and elliptical cap yield model is utilised as a constitutive model to describe the nonlinear behaviour of powder materials. An incremental elasto-plastic material model is used to simulate the compaction process and a plasticity theory for friction is employed in the treatment of the powder-tooling interface. The interfacial behaviour between the die and powder is modelled by using an ‘interface’ element mesh. Finally, the powder behaviour during the compaction of a plane bush, a cutting tool and a rotational flanged component is analysed numerically. The predictive compaction forces at different displacements, the variation with time of the displacement, relative density and stress contours are obtained. It is shown that the proposed large displacement elasto-plastic finite element approach is capable of simulating the metal powder during compaction.

A novel finite element double porosity model for multiphase flow through deformable fractured porous media

Based on the theory of double-porosity, a novel mathematical model for multiphase fluid flow in a deforming fractured reservoir is developed. The present formulation, consisting of both the equilibrium and continuity equations, accounts for the significant influence of coupling between fluid flow and solid deformation, usually ignored in the reservoir simulation literature. A Galerkin-based finite element method is applied to discretize the governing equations both in the space and time domain. Throughout the derived set of equations the solid displacements as well as the fluid pressure values are considered as the primary unknowns and may be used to determine other reservoir parameters such as stresses, saturations, etc. The final set of equations represents a highly non-linear system as the elements of the coefficient matrices are updated during each iteration in terms of the independent variables. The model is employed to solve a field scale example where the results are compared to those of ten other uncoupled models. The results illustrate a significantly different behaviour for the case of a reservoir where the impact of coupling is also considered.

Three-dimensional finite element simulation of three-phase flow in a deforming fissured reservoir

The development of a capacity to predict the exploitation of structurally complicated and fractured oil reservoirs is essential for the rational use of investment capital. A poor understanding of how the reservoir behaves during production may lead to inept, costly and inefficient development schemes. The mathematical formulation of a three-phase, three-dimensional fluid flow and rock deformation in fractured reservoirs is hence presented. The present formulation, consisting of both the equilibrium and multiphase mass conservation equations, accounts for the significant influence of coupling between the fluid flow and solid deformation, an aspect usually ignored in the reservoir simulation literature. A Galerkin-based finite element method is applied to discretise the governing equations in space and a finite difference scheme is used to march the solution in time. The final set of equations, which contain the additional cross coupling terms as compared to similar existing models, are highly non-linear and the elements of the coefficient matrices are updated implicitly during each iteration in terms of the independent variables. A field scale example is employed as an alpha case to test the validity and robustness of the currently formulation and numerical scheme. The results illustrate a significantly different behaviour for the case of a reservoir where the impact of coupling is also considered.

Three-dimensional unstructured mesh generation: Part 2. Surface meshes

Abstract This paper deals with surface patches and surface meshing. Triangular and quadrilateral patches in linear and quadratic forms, and Non-Uniform Rational B-Spline (NURBS) patch have been used to define surface geometry. From the real application of view, data conversion between the mesh generator and some existing CAD packages has been considered. As a result of this study, converters from a graphics data standard and several CAD data formats have been implemented. The quality improvement of surface meshes has been discussed in terms of parametric plane stretching, diagonal swapping and smoothing procedures. Furthermore, a scheme of visual representation is introduced to utilize colour effect in validating the geometry and its surface meshes.

A coupled finite element model for the consolidation of nonisothermal elastoplastic porous media

A coupled finite element model for the analysis of the deformation of elastoplastic porous media due to fluid and heat flow is presented. A displacement-pressure temperature formulation is used for this purpose. This formulation results in an unsymmetric coefficient matrix, even in the case of associated plasticity. A partitioned solution procedure is applied to restore the symmetry of the coefficient matrix. The partitioning procedure is an algebraic one which is carried out after integration in the time domain. For this integration, a two-point recurrence scheme is used. The finite element model is applied to the investigation of nonisothermal consolidation in various situations.

A mixed Lagrangian–Eulerian approach to modelling fluid flow during mould filling

An updated free surface Lagrangian-Eulerian finite element kinematic description is used to simulate free surface flow problems associated with mould filling. The method proposed results in an accurate determination of the front, making it ideal for problems in which free surface boundary conditions play an important role. Significant saving in CPU time can be obtained over other fixed mesh approaches by virtue of the air domain being ignored. Assuming a laminar regime for the flow field, a mixed interpolation formulation is used to approximate the discretized governing equations for elimination. Of particular interest is the method implementation to restrict the number of remeshing operations and track the moving free surface within an arbitrary domain, either with or without internal obstacles. The method used to automatically assign boundary conditions to the changing domain is described. A dam break problem is modelled numerically and compared against experimentally derived data in order to validate the model. A further numerical example demonstrates the capabilities of the algorithm developed to model the filling of an industrial casting.