Numerical Methods for Diffusion Phenomena in Building Physics: A Practical Introduction

A brief history of diffusion in physics

Part I Basics of numerical methods for diffusion phenomena in building physics

 

2. Heat and Mass Diffusion in Porous Building Elements

2.1 A brief historical

2.2 Heat and mass diffusion models

2.3 Boundary conditions

2.4 Discretization

2.5 Stability conditions

2.6 Linearization of boundary conditions or source terms

2.7 Numerical algorithms

2.8 Multitridiagonal-matrix algorithm

2.9 Mathematical model for a room air domain

2.10 Hygrothermal models used in some available simulation tools

2.11 Final remarks

 

3. Finite-Difference Method

3.1 Numerical methods for time evolution: ODE

3.1.1 An introductory example

3.1.2 Generalization

3.1.3 Systems of ODEs

3.1.4 Exercises

3.2 Parabolic PDE

3.2.1 The heat equation in 1D

3.2.2 Nonlinear case

3.2.3 Applications in engineering

3.2.4 Heat equation in two and three space dimensions

3.2.5 Exercises

 

4. Basics in Practical Finite-Element Method

4.1 Heat Equation

4.1.1 Weak formulation and test functions

4.1.2 Finite element representation

4.1.3 Finite element approximation

4.2 Finite element approach revisited

4.2.1 Reference element

4.2.2 Connectivity table

4.2.3 Stiffness matrix construction

4.2.4 Final remarks

Part II Advanced numerical methods

 

5 Explicit schemes with improved CFL condition

5.0.1 Some healthy criticism

5.1 Classical numerical schemes

5.1.1 The Explicit scheme

5.1.2 The Implicit scheme

5.1.3 The Leap-frog scheme

5.1.4 The Crank-Nicholson scheme

5.1.5 Information propagation speed

5.2 Improved explicit schemes

5.2.1 Dufort-Frankel method

5.2.2 Saulyev method

5.2.3 Hyperbolization method

5.3 Discussion

 

6 Reduced Order Methods

6.1 Introduction

6.1.1 Physical problem and Large Original Model

6.1.2 Model reduction methods for Building physics application

6.2 Balanced truncation

6.2.1 Formulation of the ROM

6.2.2 Marshall truncation Method

6.2.3 Building the ROM

6.2.4 Synthesis of the algorithm

6.2.5 Application and exercise

6.2.6 Remarks on the use of balanced truncation

6.3 Modal Identification

6.3.1 Formulation of the ROM

6.3.2 Identification process

6.3.3 Synthesis of the algorithm

6.3.4 Application and exercise

6.3.5 Some remarks on the use of the MIM

6.4 Proper Orthogonal Decomposition Basics

6.4.2 Capturing the main information

6.4.3 Building the POD model

6.4.4 Synthesis of the algorithm

6.4.5 Application and Exercise

6.4.6 Remarks on the use of the POD

6.5 Proper Generalized Decomposition

6.5.1 Basics

6.5.2 Iterative solution

6.5.3 Computing the modes

6.5.4 Convergence of global enrichment

6.5.5 Synthesis of the algorithm

6.5.6 Application and Exercise

6.5.7 Remarks on the use of the PGD

6.6 Final remarks

 

7. Boundary Integral Approaches

7.1 Basic BIEM

7.1.1 Domain and boundary integral expressions

7.1.2 Green function and boundary integral formulation

7.1.3 Numerical formulation

7.2 Trefftz method

7.2.1 Trefftz indirect method

7.2.2 Method of fundamental solutions

7.2.3 Trefftz direct method

7.2.4 Final remarks

 

8. Spectral Methods

8.1 Introduction to spectral methods

8.1.1 Choice of the basis

8.1.2 Determining expansion coefficients

8.2 Aliasing, interpolation and truncation

8.2.1 Example of a second order boundary value problem

8.3 Application to heat conduction

8.3.1 An elementary example

8.3.2 A less elementary example

8.3.3 A real-life example

8.4 Indications for further reading

8.5 Appendix 1: Some identities involving Tchebyshev polynomials

8.5.1 Compositions of Tchebyshev polynomials

8.6 Appendix 2: Trefftz method

8.7 Appendix 3: Monte-Carlo approach to the diffusion simulation

8.7.1 Brownian motion generation

8.8 Appendix 4: An exact non-periodic solution to the 1D heat equation

8.9 Some popular numerical schemes for ODEs

8.9.1 Existence and unicity of solutions

Part III Exercises and problems

 

9. Exercises and Problems

9.1 Discretization of Diffusion Equations

9.1.1 Treatment of the boundary conditions

9.1.2 Numerical solution

9.2 Heat and mass diffusion: Numerical solution

9.3 Whole building energy simulation

9.4 Heat and mass diffusion: Analysis of the physical behavior

 

10. Conclusions

11. References

References

Index