Table of Contents
1 Weighted Residual Finite Element Method 1
1.1 Basic Concept 1
1.1.1 Practical procedure 2
1.1.2 Finite element approximations 4
1.2 Numerical Integration 6
1.2.1 Mapping of irregular and curved elements onto master elements 7
1.3 Steps Used to Obtain a Finite Element Solution for a Field Problem 13
1.3.1 Example 1 15
1.3.2 Example 2 21
References 27
2 Shape Functions and Fundamental Properties of Finite Elements 29
2.1 Interpolation Polynomials 30
2.2 Shape Functions of Simplex Elements 31
2.2.1 One-dimensional linear element 31
2.2.2 Two-dimensional simplex element 32
2.2.3 Three-dimensional simplex element 35
2.3 Some Examples of Complex and Multiplex Elements 37
2.3.1 Quadratic and cubic one-dimensional elements 37
2.3.2 Four-node two-dimensional element 40
2.4 Convergence of Finite Element Approximations 42
2.5 Continuity Conditions 44
2.6 Solved Examples 44
2.6.1 Example 1: Limitation of the standard Galerkin procedure 44
2.6.2 Example 2: A further problem associated with the use of the standard Galerkin finite element method 48
2.6.3 Bubble function enriched elements 51
References 53
3 Basic Concepts of Multiscale Finite Element Modeling 55
3.1 Stabilization of Finite Element Schemes 55
3.1.1 Upwinded finite element schemes 56
3.1.2 Upwinding techniques 57
3.2 Multiscale Approach 59
3.2.1 Variational multiscale method 60
3.2.2 (RFB) function method 62
3.3 Practical Implementation of the RFB Function Method 64
3.3.1 Multiscale finite element solution of the DR equation using RFB function method 65
3.3.2 Multiscale finite element solution of the CD equation using RFB function method 87
3.3.3 Multiscale finite element solution of the CDR equation using RFB function method 94
3.4 Practical Implementation of the STC Method 106
3.4.1 Multiscale finite element solution of the DR equation using STC method 108
3.4.2 Multiscale finite element solution of the CD equation using STC method 113
3.4.3 Multiscale finite element solution of the CDR equation using STC method 118
References 122
4 Simulation of Multiscale Transport Phenomena in Multidimensional Domains 125
4.1 Two-dimensional Multiscale Finite Element Technique 125
4.2 Selection of Bubble Functions and Calculation of Bubble Coefficients 129
4.3 RFB Functions Corresponding to the Two-Dimensional DR Equation 131
4.3.1 Derivation of polynomial bubble function coefficients for the DR equation using the STC method 133
4.3.2 Elimination of the boundary integrals 133
4.3.3 Solution of a benchmark two-dimensional DR problem 134
4.4 Solution of Two-Dimensional Convection-Diffusion (CD) Equation 140
4.4.1 Governing CD equation and boundary conditions 140
4.4.2 RFB functions corresponding to the two-dimensional CD equation 141
4.4.3 Two-dimensional bubble functions for bilinear elements 141
4.4.4 Elimination of the boundary integrals 142
4.4.5 Solution of a benchmark two-dimensional CD problem 143
4.5 Solution of Convection-Diffusion-Reaction (CDR) Problems 143
4.5.1 Governing CDR equation and boundary conditions 144
4.5.2 Derivation of RFB functions for the CDR equation 146
4.5.3 Solution of a benchmark two-dimensional CDR equation 146
4.6 Solution of Transport Equations Using Bubble Function-Enriched Triangular Elements 149
4.7 Multiscale Finite Element Methods for Time-Dependent Problems 152
4.7.1 Multiscale space-time finite element discretization 153
4.7.2 Multiscale space-time finite element modeling 154
4.7.3 Two-dimensional space-time bubble functions for enrichment of bilinear elements 155
4.7.4 Elimination of the boundary integrals 156
4.7.5 Transient CD problem 156
References 158
5 Application of Multiscale Finite Element Schemes to Fluid Flow Problems 161
5.1 Two-Dimensional and Axisymmetric Flow Regimes - Governing Equations 161
5.2 Modeling of Isothermal Brinkman Flow of a Newtonian Fluid in a Two-Dimensional Domain Using Multiscale Finite Element Schemes 164
5.2.1 Continuous penalty scheme 167
5.2.2 Bubble-enriched shape functions used in conjunction with the continuous penalty method 172
5.2.3 Solved example - unidirectional flow 172
5.3 Solution of the Energy Equation 174
5.3.1 Finite element scheme used to solve the energy equation 175
5.3.2 Bubble-enriched shape functions used in the solution of the energy equation 176
5.3.3 Solved example - Non-isothermal unidirectional flow 176
5.3.4 Higher-order elements and mesh refinement 179
5.4 Multidirectional, Nonisothermal Porous Flow 180
5.5 Inclusion of Inertia Effects in Porous Flow Models 181
5.5.1 Derivation of bubble functions for Brinkman-Hazen-Dupuit-Darcy equation 184
5.5.2 Solved example - High velocity unidirectional porous flow 184
5.5.3 Solved example - Multidirectional, high-velocity porous flow 185
5.6 Solution of Axisymmetric Brinkman Equation 185
5.6.1 Derivation of the continuous penalty finite element scheme for axisymmetric Brinkman equation 188
5.6.2 Derivation of bubble functions for axisymmetric Brinkman equation 189
References 191
6 Computer Program 193
6.1 Program Structure 193
6.2 Source Code 194
6.3 Input File Structure 220
6.4 Output Files 225
6.4.1 DR equation-dissipation case 225
6.4.2 DR equation-production case 228
6.4.3 CD equation 230
6.4.4 CDR equation 233
Appendices 237
Author Index 245
Subject Index 247
