Numerical Methods for Diffusion Phenomena in Building Physics A Practical Introduction

This book is the second edition of Numerical methods for diffusion phenomena in building physics: a practical introduction originally published by PUCPRESS (2016). It intends to stimulate research in simulation of diffusion problems in building physics, by providing an overview of mathematical models and numerical techniques such as the finite difference and finite-element methods traditionally used in building simulation tools. Nonconventional methods such as reduced order models, boundary integral approaches and spectral methods are presented, which might be considered in the next generation of building-energy-simulation tools. In this reviewed edition, an innovative way to simulate energy and hydrothermal performance are presented, bringing some light on innovative approaches in the field.

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 

Nathan Mendes

Nathan Mendes got his Ph.D. in Mechanical Engineering at the Federal University of Santa Catarina, Brazil, in collaboration with the Lawrence Berkeley National Laboratory, USA, in 1997. He is currently full professor and director of graduate studies at the Pontifical Catholic University of Paraná and coordinator of the institutional project of Excellence in Stricto Sensu. He received the Young Scientist Award in 2002, and he is a member of the editorial boards of the Journal of Building Physics and the Journal of Building Performance Simulation. He has more than 80 journal papers and 180 conference papers, 5 patent requirements and software registration and acts as a reviewer of more than 11 peer-reviewed journals and as a member of International Energy Agency Annexes (41, 55, 60 and 68). He coordinated the Post-Graduation Program in Mechanical Engineering at PUCPR from 2006 to 2011, was the president of the Brazilian regional IBPSA (International Association of Building Performance Simulation Association), Brazilian representative at the IBPSA-World since 2005 and the regional director of ASBRAV (2001-2008). He works in the area of Mechanical Engineering, with emphasis on energy, heat and moisture through porous building elements, modeling and experimental analysis of heating and air conditioning systems, energy efficiency of buildings and building simulation. He has participated in national and international advisory committees for evaluation of research projects.

Denys Dutykh

Dr. Denys Dutykh initially graduated in 2004 from the Faculty of Applied Mathematics, Dnipropetrovsk National University in Ukraine. Then, he moved to continue his education at Ecole Normale Supérieure de Cachan (ENS de Cachan), France, where he obtained in 2005 the Master degree (ex-Diplôme d’Etudes Approfondies) in Numerical Methods for Models of Continuous Media (MN2MC). The next logical step was to prepare the PhD thesis. It was done at the Centre de Mathématiques et Leurs Applications (CMLA UMR 8536) under the guidance of Prof. Frédéric Dias. The PhD thesis entitled “Mathematical modelling of tsunami waves” was defended on the 3^rd of December 2007 at ENS de Cachan. In October 2008 Denys started to work as CNRS Researcher at the University Savoie Mont Blanc, where he defended his Hanilitation thesis in December 2010 on the topic of “Mathematical Modelling in the Environment”. During the period of 2012 - 2013 he was on leave from CNRS to the Univerisity College Dublin, Ireland to participate in the ERC AdG MultiWave project. Since 2014 he continues to work at the Laboratory of Mathematics (LAMA UMR 5127) again as a CNRS Researcher on a variety topics ranging from Mathematics to its applications in the real world.

 

Marx Chhay

Ph.D. in Mechanical Engineering - Université de La Rochelle (2008). Researcher at the Optimization Laboratory of Design and Environmental Engineering (LOCIE) -University Savoie Mont-Blanc, France. He has experience in Mechanical Engineering, focusing on Thermal Engineering and especially acting on the following themes: Numerical methods, Transport equations, Hydrodynamics, Geometric integration, Variational systems, and Symmetry. 

 

Julien Berger

Graduation in Civil Engineering - Ecole Nationale des Travaux Publics de le Etat (2008), master's degree in MEGA - Université de Lyon (2008) and a Ph.D. - Université de Savoie, LOCIE (2014). Post-doctoral researcher at the Mechanical Engineering Graduate Program of the Pontifical Catholic University of Paraná - PUCPR, Curitiba, Brazil. He has experience in Civil Engineering, focusing on numerical computational methods and especially acting on the following themes: Proper generalized decomposition, Heat and mass transfers in materials, Proper orthogonal decomposition, and Model reduction technique