Numerical methods for diffusion phenomena in building physics

Foreword......Page 6
Preface......Page 8
Acknowledgements......Page 10
Contents......Page 11
Acronyms......Page 15
1 A Brief History of Diffusion in Physics......Page 17
Part I Basics of Numerical Methods for Diffusion Phenomena in Building Physics......Page 22
 2.1 A Brief Historical......Page 23
 2.2 Heat and Mass Diffusion Models......Page 27
 2.3 Boundary Conditions......Page 38
 2.4 Discretization......Page 40
 2.5 Stability Conditions......Page 43
 2.6 Linearization of Boundary Conditions or Source Terms......Page 44
 2.7 Numerical Algorithms......Page 45
 2.8 Multitridiagonal-Matrix Algorithm......Page 46
 2.9 Mathematical Model for a Room Air Domain......Page 48
 2.10 Hygrothermal Models Used in Some Available Simulation Tools......Page 50
 2.11 Final Remarks......Page 56
3 Finite-Difference Method......Page 58
  3.1.1 An Introductory Example......Page 59
  3.1.2 Generalization......Page 65
  3.1.3 Systems of ODEs......Page 73
  3.1.4 Exercises......Page 77
 3.2 Parabolic PDE......Page 79
  3.2.1 The Heat Equation in 1D......Page 80
  3.2.2 Nonlinear Case......Page 88
  3.2.3 Applications in Engineering......Page 90
  3.2.4 Heat Equation in Two and Three Space Dimensions......Page 94
  3.2.5 Exercises......Page 97
 4.1 Heat Equation......Page 101
  4.1.1 Weak Formulation and Test Functions......Page 102
  4.1.2 Finite Element Representation......Page 103
  4.1.3 Finite Element Approximation......Page 106
  4.2.1 Reference Element......Page 108
  4.2.4 Final Remarks......Page 111
Part II Advanced Numerical Methods......Page 113
5 Explicit Schemes with Improved CFL Condition......Page 114
 5.1 Some Healthy Criticism......Page 115
 5.2 Classical Numerical Schemes......Page 116
  5.2.1 The Explicit Scheme......Page 117
  5.2.2 The Implicit Scheme......Page 118
  5.2.3 The Leap-Frog Scheme......Page 119
  5.2.4 The Crank–Nicolson Scheme......Page 120
  5.2.5 Information Propagation Speed......Page 121
  5.3.1 Dufort–Frankel Method......Page 122
  5.3.2 Saulyev Method......Page 124
  5.3.3 Hyperbolization Method......Page 127
 5.4 Discussion......Page 130
  6.1.1 Physical Problem and Large Original Model......Page 132
  6.1.2 Model Reduction Methods for Building Physics Application......Page 133
  6.2.1 Formulation of the ROM......Page 135
  6.2.3 Building the ROM......Page 136
  6.2.5 Application and Exercise......Page 137
  6.2.6 Remarks on the Use of Balanced Truncation......Page 140
  6.3.1 Formulation of the ROM......Page 141
  6.3.2 Identification Process......Page 142
  6.3.4 Application and Exercise......Page 143
  6.3.5 Some Remarks on the Use of the MIM......Page 149
  6.4.1 Basics......Page 150
  6.4.3 Building the POD Model......Page 151
  6.4.5 Application and Exercise......Page 152
  6.4.6 Remarks on the Use of the POD......Page 156
  6.5.2 Iterative Solution......Page 157
  6.5.3 Computing the Modes......Page 158
  6.5.5 Synthesis of the Algorithm......Page 159
  6.5.6 Application and Exercise......Page 160
 6.6 Final Remarks......Page 162
7 Boundary Integral Approaches......Page 164
  7.1.1 Domain and Boundary Integral Expressions......Page 165
  7.1.2 Green Function and Boundary Integral Formulation......Page 167
  7.1.3 Numerical Formulation......Page 168
 7.2 Trefftz Method......Page 170
  7.2.1 Trefftz Indirect Method......Page 171
  7.2.2 Method of Fundamental Solutions......Page 174
  7.2.3 Trefftz Direct Method......Page 175
  7.2.4 Final Remarks......Page 177
 8.1 Introduction to Spectral Methods......Page 178
  8.1.1 Choice of the Basis......Page 180
  8.1.2 Determining Expansion Coefficients......Page 187
 8.2 Aliasing, Interpolation and Truncation......Page 189
  8.2.1 Example of a Second Order Boundary Value Problem......Page 191
  8.3.1 An Elementary Example......Page 193
  8.3.2 A Less Elementary Example......Page 195
  8.3.3 A Real-Life Example......Page 196
 8.4 Indications for Further Reading......Page 199
 8.5 Appendix 1: Some Identities Involving Tchebyshev Polynomials......Page 201
  8.5.1 Compositions of Tchebyshev Polynomials......Page 204
 8.6 Appendix 2: Trefftz Method......Page 205
 8.7 Appendix 3: Monte–Carlo Approach to the Diffusion Simulation......Page 209
  8.7.1 Brownian Motion Generation......Page 212
 8.8 Appendix 4: An Exact Non-periodic Solution to the 1D Heat Equation......Page 213
 8.9 Some Popular Numerical Schemes for ODEs......Page 215
  8.9.1 Existence and Unicity of Solutions......Page 218
Part III Praxis......Page 221
  9.1.1 Treatment of the Boundary Conditions......Page 222
  9.1.2 Numerical Solution......Page 224
 9.2 Heat and Mass Diffusion: Numerical Solution......Page 227
  9.3.1 Heat Transfer......Page 231
  9.3.2 Moisture Transfer......Page 233
 9.4 Heat and Mass Diffusion: Analysis of the Physical Behavior......Page 234
10 Conclusions......Page 237
 BookmarkTitle:......Page 241
Index......Page 251