This chapter introduces the finite element method (FEM) as a tool for solution of classical electromagnetic problems. Although we discuss the main points in the application of the finite element method to electromagnetic design, including formulation and implementation, those who seek deeper understanding of the finite element method should consult some of the works listed in the bibliography section.

Introduction to the Finite Element Method

Workbook of atmospheric dispersion estimates

Membrane fractionation of milk: state of the art and challenges

We present a new family of stabilized methods for the Stokes problem. The focus of the paper is on the lowest order velocity-pressure pairs. While not LBB compliant, their simplicity and attractive computational properties make these pairs a popular choice in engineering practice. Our stabilization approach is motivated by terms that characterize the LBB 'deficiency' of the unstable spaces. The stabilized methods are defined by using these terms to modify the saddle-point Lagrangian associated with the Stokes equations. The new stabilized methods offer a number of attractive computational properties. In contrast to other stabilization procedures, they are parameter free, do not require calculation of higher order derivatives or edge-based data structures, and always lead to symmetric linear systems. Furthermore, the new methods are unconditionally stable, achieve optimal accuracy with respect to solution regularity, and have simple and straightforward implementations. We present numerical results in two and three dimensions that showcase the excellent stability and accuracy of the new methods.

Stabilization of low-order mixed finite elements for the stokes equations.

http://coastal.usc.edu/plynett/publications/Lynett%20-%20Modeling%20Wave%20Runup%202002%20CE.pdf

Modeling wave runup with depth-integrated equations

Modelling of combined Navier–Stokes and Darcy flows in crossflow membrane filtration

Free flow channel confined by porous walls is a feature of many of the natural and industrial settings. Viscous flows adjacent to saturated porous medium occur in cross-flow and dead-end filtrations employed primarily in pharmaceutical and chemical industries for solid–liquid or gas–solid separations. Various mathematical models have been put forward to describe the conjugate flow dynamics based on theoretical grounds and experimental evidence. Despite this fact, there still exists a wide scope for extensive research in numerical solutions of these coupled models when applied to problems with industrial relevance. The present work aims towards the numerical analysis of coupled free/porous flow dynamics in the context of industrial filtration systems. The free flow dynamics has been expressed by the Stokes equations for the creeping, laminar flow regime whereas the flow behaviour in very low permeability porous media has been represented by the conventional Darcy equation. The combined free/porous fluid dynamical behaviour has been simulated using a mixed finite element formulation based on the standard Galerkin technique. A nodal replacement technique has been developed for the direct linking of Stokes and Darcy flow regimes which alleviates specification of any additional constraint at the free/porous interface. The simulated flow and pressure fields have been found for flow domains with different geometries which represent prototypes of actual industrial filtration equipment. Results have been obtained for varying values of permeability of the porous medium for generalised Newtonian fluids obeying the power law model. A series of numerical experiments has been performed in order to validate the coupled flow model. The developed model has been examined for its flexibility in dealing with complex geometrical domains and found to be generic in delivering convergent, stable and theoretically consistent results. The validity and accuracy of the simulated results has been affirmed by comparing with available experimental data.

Numerical Analysis of Coupled Stokes/Darcy Flows in Industrial Filtrations

Despite the widespread use of the Darcy equation to model porous flow, it is well known that this equation is inconsistent with commonly prescribed no slip conditions at flow domain walls or interfaces between different sections. Therefore, in cases where the wall effects on the flow regime are expected to be significant, the Darcy equation which is only consistent with perfect slip at solid boundaries, cannot predict velocity and pressure profiles properly and alternative models such as the Brinkman equation need to be considered. This paper is devoted to the study of the flow of a Newtonian fluid in a porous medium between two impermeable parallel walls at different Darcy parameters (Da). The flow regime is considered to be isothermal and steady. Three different flow regimes can be considered using the Brinkman equation: free flow (Da > 1), porous flow (high permeability, 1 > Da > 10−6) and porous flow (low permeability Da < 10−6). In the present work the described bench mark problem is used to study the effects of solid walls for a range of low to high Darcy parameters. Both no-slip and slip conditions are considered and the results of these two cases are compared. The range of the applicability of the Brinkman equation and simulated results for different cases are shown.

Finite Element Modelling of Flow Through a Porous Medium Between Two Parallel Plates Using The Brinkman Equation

1. The Basic Equations of Non-Newtonian Fluid Mechanics. 1.1 Governing Equations of Non-Newtonian Fluid Mechanics 1.2 Classification of Inelastic Time-Independent Fluids 1.3 Inelatic Time-Dependent Fluids 1.4 Viscoelastic Fluids 2. Weighted Residual Finite Element Methods - An Outline. 2.1 Finite Element Approximation 2.2 Numerical Solutions of Differential Equations by the Weighted Residual Method 3. Finite Element Modelling of Polymeric Flow Processes. 3.1 Solution of the Equations of Continuity and Motion 3.2 Modelling of Viscoelastic Flow 3.3 Solution of the Energy Equation 3.4 Imposition of Boundary Conditions in Polymeric Processing Models 3.5 Free Surface and Moving Boundary Problems 4. Working Equations of the Finite Element Schemes. 4.1 Modelling of Steady State Stokes Flow of a Generalized Newtonian Fluid 4.2 Variations of Viscosity 4.3 Modelling of Steady State Viscometric Flow - Working Equations of the Continuous Penalty Scheme in Cartesian Coordinate Systems 4.4 Modelling of Thermal Energy Balance 4.5 Modelling of Transient Stokes Flow of Generalized Newtonian and Non-Newtonian Fluids 5. Rational Approximations and Illustrative Examples. 5.1 Models based on Simplified Domain Geometry 5.2 Models based on Simplified Governing Equations 5.3 Models representing Selected Segments of a Large Domain 5.4 Models based on Decoupled Flow Equations - Simulation of the Flow inside a Cone-and-Plate Rheometer 5.5 Models based on Thin Layer Approximation 5.6 Stiffness Analysis of Solid Polymeric Materials 6. Finite Element Software - Main Components. 6.1 General Consideration to Finite Element Mesh Generation 6.2 Main Components of Finite Element Processor Programs 6.3 Numerical Solution of the Global Systems of Algebraic Equations 6.4 Solutions Algorithms based on the Gaussian Elimination Method 6.5 Computation Errors 7. Computer Simulations - Finite Element Program. 7.1 Program Structure and Algorithm 7.2 Program Specifications 7.3 Input Data File 7.4 Extension of PPVN.f to Axisymmetric Problems 7.5 Circulatory Flow in a Rectangular Domain 7.6 Source Code of PPVN.f References 8. Appendix - Summary of Vector and Tensor Analysis. 8.1 Vector Algebra 8.2 Some Vector Calculua Relations 8.3 Tensor Algebra 8.4 Some Tensor Calculus Relations Author Index. Subject Index.

Practical Aspects of Finite Element Modelling of Polymer Processing

A simulation code that can model fluid hydrodynamics in permeable media and pleated filter cartridges has been developed and used to evaluate the performance and design of pleated cartridge filter elements. The effects of medium compression, pleat deformation and pleat crowding are analysed. A permeability model that interprets medium compression effects and which is based on data obtained from flat sheets of the filter material used in the fabrication of filter cartridges is incorporated into the main model. During pleating, bending and creasing of the medium deforms it and leads to a loss of permeability and/or effective filtering area. The combined effects of compression and reduction in filtration area cause deviations from Darcy's law. The present work is directed towards the prediction of accurate values of compression factors and percentage losses in the filtration area, which assist in designing pleated cartridge filter elements. The simulated results have been compared against the experimental data for purposes of validation.

Vahid Nassehi

Papers

Modelling of combined Navier–Stokes and Darcy flows in crossflow membrane filtration

Numerical Analysis of Coupled Stokes/Darcy Flows in Industrial Filtrations

Finite Element Modelling of Flow Through a Porous Medium Between Two Parallel Plates Using The Brinkman Equation

Practical Aspects of Finite Element Modelling of Polymer Processing

Analysis of Pleat Crowding and Medium Compression in Pleated Cartridge Filters