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xi

Contents

1 Modeling and Simulation of Macroscopic PedestrianFlow Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Naveen Kumar Mahato, Axel Klar, and Sudarshan Tiwari1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Optimal control problem of pedestrian flow from [2] . . . . . . . 21.3 Relation to the classical Hughes model [5] . . . . . . . . . . . . . . . . 41.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Modeling bimaterial 3D printing using galvanometermirror scanners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Daniel Bandeira and Marta Pascoal2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Emitters location problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 Emitters assignment problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4 Computational experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 Mathematical Modeling for Laser-InducedThermotherapy in Liver Tissue . . . . . . . . . . . . . . . . . . . . . . . . . . 21N. Siedow and C. Leithauser3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Mathematical Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2.1 The Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2.2 The Radiative Transfer . . . . . . . . . . . . . . . . . . . . . . . . . 233.2.3 The Damage Function . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

xiii

xiv Contents

4 Sparse Multiple Data Assimilation with K-SVD for theHistory Matching of Reservoirs . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Clement Etienam, Rossmary Villegas Velasquez, Oliver Dorn4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.2 Ensemble-Smoother Multiple Data Assimilation (ES-MDA) . 294.3 Dictionary Learning and OMP . . . . . . . . . . . . . . . . . . . . . . . . . . 294.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5 Hypercomplex Fourier Transforms in the Analysis ofMultidimensional Linear Time-Invariant Systems . . . . . . . . . 35 Lukasz B laszczyk5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.2 Algebras of octonions and quadruple-complex numbers . . . . . 365.3 Octonion Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.4 Recent results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.5 Multidimensional linear time-invariant systems . . . . . . . . . . . . 405.6 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

6 Local time stepping method for district heating networks . 43Matthias Eimer, Raul Borsche and Norbert Siedow6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446.3 Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

6.3.1 Local Time Stepping . . . . . . . . . . . . . . . . . . . . . . . . . . . 466.3.2 High Order Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . 476.3.3 Source Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

6.4 Results and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

7 Modelling Time-of-Flight transient currents withtime-fractional diffusion equations . . . . . . . . . . . . . . . . . . . . . . . . 51M. Luısa Morgado and Luıs F. Morgado7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517.2 Numerical method and results . . . . . . . . . . . . . . . . . . . . . . . . . . 53References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

8 Damage detection in thin plates via time-harmonicinfrared thermography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59Manuel Pena and Marıa-Luisa Rapun8.1 Statement of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598.2 Topological derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618.3 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Contents xv

9 Reverse Logistics Modelling of Assets Acquisition in aLiquefied Petroleum Gas Company . . . . . . . . . . . . . . . . . . . . . . . 67Cristina Lopes, Aldina Correia, Eliana Costa e Silva, MagdaMonteiro, Rui Borges Lopes9.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 679.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

9.2.1 Inventory models in literature with reverse flows . . . 699.3 Inventory models developed for the company based on

continuous replenishment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 709.3.1 Model D - Deterministic continuous returns . . . . . . . 719.3.2 Model R - Deterministic without purchases . . . . . . . . 739.3.3 Model S - Stochastic inventory model . . . . . . . . . . . . . 74

9.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

10 Stochastic Order Relations in a Gambling-typeEnvironment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77Sandor Guzmics10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7710.2 The discrete setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7810.3 The continuous setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8010.4 Stochastic order relations in the Lorenz order and in the

increasing convex order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8110.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

Chapter 1

Modeling and Simulation ofMacroscopic Pedestrian Flow Models

Naveen Kumar Mahato, Axel Klar, and Sudarshan Tiwari

1.1 Introduction

Mathematical modeling and numerical simulation of human crowd motionhave become a major subject of research with a wide field of applications.A variety of models for pedestrian behavior have been proposed on differ-ent levels of description in recent years. Macroscopic pedestrian flow modelinvolving equations for density and mean velocity of the flow is derived inRefs. [1, 2, 5, 7, 8].

In this work, we analyze numerically some macroscopic models of pedes-trian motion such as the classical Hughes model [5] and a mean field gamewith nonlinear mobilities [2], modeling fast exit scenarios in pedestriancrowds. A model introduced by Hughes consists of a non-linear conserva-tion law for the density of pedestrians coupled with an Eikonal equation fora potential modeling the common sense of the task. Mean field game withnonlinear mobilities is obtained by an optimal control approach, where themotion of every pedestrian is determined by minimizing a cost functional,which depends on the position, velocity, exit time and the overall density ofpeople. We consider a parabolic optimal control problem of nonlinear mobil-ity in pedestrian dynamics, which leads to a mean field game structure. Weshow how optimal control problem related to the Hughes model for pedes-

Naveen Kumar Mahato

Technische Universitat Kaiserslautern, Department of Mathematics, Erwin-Schrodinger-Straße, 67663 Kaiserslautern, Germany, e-mail: [email protected]

Axel Klar

Technische Universitat Kaiserslautern, Department of Mathematics, Erwin-Schrodinger-Straße, 67663 Kaiserslautern, Germany, e-mail: [email protected]

Sudarshan TiwariTechnische Universitat Kaiserslautern, Department of Mathematics, Erwin-Schrodinger-

Straße, 67663 Kaiserslautern, Germany, e-mail: [email protected]

1

2 Naveen Kumar Mahato, Axel Klar, and Sudarshan Tiwari

trian motion. Furthermore, we provide several numerical results which relateboth models in one and two dimensions.

1.2 Optimal control problem of pedestrian flow from [2]

For completeness of the presentation up to higher dimensions, we reviewthe macroscopic optimal control problem for pedestrian flow, see Refs. [2–4]. There, denoting the (normalized) density function of the pedestrians byρ(t, x) and the momentum (or the flux density) by m = F (ρ)v at positionx ∈ Ω, velocity v ∈ Ω and time t, where the function F (ρ(t, x)) describing thenon-linear mobility of the pedestrians (or the costs created by large densities)and Ω ∈ Rd, d = 1, 2 is a bounded domain representing the pedestrian area.We assume the boundary ∂Ω is split into a Neumann part ΓN ⊆ ∂Ω modelingwalls or obstacles, ΓE ⊆ ∂Ω modeling the exits such that ∂Ω = ΓN

⋃ΓE and

ΓN⋂ΓE = φ. If we denote the rate of passing the exit by β, then we have an

outflow proportional to βρ. Hence, for a stochastic particle and a final timeT sufficiently large, the minimization functional is given by the followingparabolic optimal control problem:

min(ρ,m)

IT (ρ,m) = min(ρ,m)

1

2

∫ T

0

∫Ω

|m(t, x)|2

F (ρ(t, x))dxdt+

α

2

∫ T

0

∫Ω

ρ(t, x)dxdt,

(1.1a)subject to

∂tρ+∇ ·m =σ2

2∆ρ, in Ω × (0, T ), (1.1b)(

m− σ2

2∇ρ)· n = 0, on ΓN × (0, T ), (1.1c)(

m− σ2

2∇ρ)· n = βρ, on ΓE × (0, T ), (1.1d)

ρ(0, x) = ρ0(x), in Ω. (1.1e)

This optimality system can be seen as the mean field game structure, seeRef. [6]. We start by defining the Lagrangian with dual variable Φ = Φ(t, x)as

1 Modeling and Simulation of Macroscopic Pedestrian Flow Models 3

LT (ρ,m,Φ) = IT (ρ,m) +

∫ T

0

∫Ω

(∂tρ+∇ ·m− σ2

2∆ρ)Φdxdt

= IT (ρ,m) +

[∫Ω

ρΦdx

]T0

+

∫ T

0

∫Ω

[ρ

(−∂tΦ−

σ2

2∆Φ

)−m · ∇Φ

]dxdt

+

∫ T

0

∫∂ΓE

−σ2

2∇ρ · nΦ+m · nΦ︸︷︷︸

βρΦ

+σ2

2ρ∇Φ · n

dsdt+

∫ T

0

∫∂ΓN

σ2

2ρ∇Φ · ndsdt.

The optimality condition with respect to m and ρ, yields the following equa-tions

0 = ∂mLT (ρ,m,Φ) =m(t, x)

F (ρ(t, x))−∇Φ,

0 = ∂ρLT (ρ,m,Φ) = −1

2

|m(t, x)|2F ′(ρ)

F 2(ρ)+α

2− ∂tΦ−

σ2

2∆Φ.

Inserting m = F (ρ(t, x))∇Φ we obtain the following system of equations

∂tρ+∇ · (F (ρ)∇Φ)− σ2

2∆ρ = 0, in Ω × (0, T ),

(1.2a)

∂tΦ+F ′(ρ)

2|∇Φ|2 +

σ2

2∆Φ =

α

2, in Ω × (0, T ),

(1.2b)(F (ρ)∇Φ− σ2

2∇ρ)· n = 0,

σ2

2∇Φ · n = 0, on ΓN × (0, T ),

(1.2c)(F (ρ)∇Φ− σ2

2∇ρ)· n = βρ,

σ2

2∇Φ · n+ βρ = 0, on ΓE × (0, T ),

(1.2d)

ρ(0, x) = ρ0(x), Φ(T, x) = 0, in Ω. (1.2e)

System (1.2) has the structure of a mean field game for pedestrian dynamics,which contains the Fokker-Planck equation (1.2a) has to be solved forward intime and the Hamilton-Jacobi equation (1.2b) that has to be solved backwardin time.


1.3 Relation to the classical Hughes model [5]

In this Section we discuss the relation which shows that for vanishing viscosityσ = 0 of the optimality system (1.1) has a similar structure as the classicalHughes model for pedestrian flow. Hughes proposed that pedestrians seek thefastest path to the exit, but at the same time try to avoid congested areas,for details see Ref. [5]. Let us consider the governing equations of Hughesmodel for pedestrian flow,

∂tρ−∇ · (ρf2(ρ)∇Φ)− σ2

2∆ρ = 0, in Ω × (0, T ), (1.3a)

|∇Φ| = 1

f(ρ), in Ω × (0, T ), (1.3b)(

ρf2(ρ)∇Φ− σ2

2∇ρ)· n = 0, Φ =∞, on ΓN × (0, T ),

(1.3c)(ρf2(ρ)∇Φ− σ2

2∇ρ)· n = βρ, Φ = 0, on ΓE × (0, T ),

(1.3d)

ρ(0, x) = ρ0(x), in Ω, (1.3e)

where the function f(ρ) = ρmax − ρ with ρmax denote the maximum densityand models how pedestrians change their direction and velocity due to thesurrounding density, i.e. provides a weighting or cost with respect to highdensities. Saturation effects are included via the function f(ρ) for ρ −→ ρmax.

On the other hand, if we choose the mobility/penalization function forhigh densities such as F (ρ) = ρf(ρ)2, then the optimality system (1.2) forvanishing viscosity can be written as

∂tρ+∇ · (ρf(ρ)2∇Φ) = 0, in Ω × (0, T ), (1.4a)

∂tΦ+f(ρ)

2(f(ρ) + 2ρf ′(ρ))|∇Φ|2 =

α

2, in Ω × (0, T ), (1.4b)

where the initial, terminal and boundary conditions are same as in system(1.2). Furthermore, one can expect the equilibration of Φ backward in timefor large T . Then for time t of order one the limiting model becomes

∂tρ+∇ · (ρf(ρ)2∇Φ) = 0, in Ω × (0, T ), (1.5a)

(f(ρ) + 2ρf ′(ρ))|∇Φ|2 =α

f(ρ), in Ω × (0, T ). (1.5b)

Hence, if we set α = 1, the system (1.5) is almost equivalent to the Hughesmodel (1.3) for vanishing viscosity. Note that the sign difference in equations(1.3a) and (1.5a) is not an actual, since due to the signs in the backward


equation we shall obtain Φ as the negative of the distance function used inthe Hughes model.

1.4 Numerical results

In this Section we present a series of numerical experiments for the equationsfrom both proposed models. We compare the relation between the modelsfor different parameters in one and two dimensions. We use finite differencescheme for solving the classical Hughes model, where central difference inspace and the forward difference in time, i.e. forward time centered space(FTCS) scheme for the nonlinear conservation law and an upwind Godunovscheme for the Eikonal equation. We follow the steepest descent algorithmfrom Ref. [2], to solve the mean field game structure, in which we use FTCSfinite difference scheme to solve both forward and backward equations.

We consider a one-dimensional domain Ω = [−1, 1] with exits located atx = ±1 for the numerical simulation as a configuration defined in Ref. [2].We choose the maximum density ρmax, the weighting parameter α and theflow rate parameter β as 1. Furthermore, we consider the time interval ast ∈ [0, 3]. The time step is set to ∆t = 10−4 for Hughes and ∆t = 10−3 forMFG. We use the spatial discretization h = 10−2, the diffusion coefficientσ2

2 = h and the initial density ρ0 = 13 in both models.

-1 -0.5 0 0.5 1

x-axis

0

0.5

1

1.5

Eik

onal

T = 0.1

T = 0.5

T = 0.7

T = 1.0

T = 1.3

T = 1.5

T = 2.0

-1 -0.5 0 0.5 1

x-axis

0

0.1

0.2

0.3

0.4

density

T = 0.1

T = 0.5

T = 0.7

T = 1.0

T = 1.3

T = 1.5

T = 2.0

-1 -0.5 0 0.5 1

x-axis

0

0.5

1

1.5

Eik

onal

T = 0.1

T = 0.5

T = 0.7

T = 1.0

T = 1.3

T = 1.5

T = 2.0

T = 2.5

T = 3.0

-1 -0.5 0 0.5 1

x-axis

0

0.1

0.2

0.3

0.4

density

T = 0.1

T = 0.5

T = 0.7

T = 1.0

T = 1.3

T = 1.5

T = 2.0

Fig. 1.1: Evolution of solutions at different times for the Hughes model (top) and for the

MFG structure (bottom).


Figure 1.1 shows the evolution of solutions at different times for bothmean field type structures. One observes that the non-stationary Eikonalsolution of the MFG structure has a similar behavior as the stationary Eikonalsolution of the classical Hughes model until the density is not zero, as weexpected equilibration of Φ in the equation (1.4b). One also observes fromthe density solution that both models have similar behavior as pedestriansstart in immediate vacuum formation at the center x = 0. Although themodels have a very similar structure, pedestrians wait for a little while at thecenter and then start to move at a higher speed in the case of the mean fieldgame compare to the Hughes model.

The extension of the above method into higher dimensions is straight for-ward. Here, we restrict ourselves to two-dimensional problem. Suppose thegeometry for numerical experiment is taken as Ω = [−1, 1] × [−1, 1] withexits located at (±1,±1). Furthermore, we choose all parameters as for onedimension. Figure 1.3 shows the evolution of solutions through the center

Fig. 1.2: Evolution of the density solution at different times for the Hughes model (top)and the MFG structure (bottom).

-1 -0.5 0 0.5 1

x-axis

0

0.5

1

1.5

Eik

onal

T = 0.1

T = 0.5

T = 0.7

T = 1.0

T = 1.3

T = 1.5

T = 2.0

-1 -0.5 0 0.5 1

x-axis

0

0.1

0.2

0.3

0.4

density

T = 0.1

T = 0.5

T = 0.7

T = 1.0

T = 1.3

T = 1.5

T = 2.0

-1 -0.5 0 0.5 1

x-axis

0

0.5

1

1.5

Eik

onal

T = 0.1

T = 0.5

T = 0.7

T = 1.0

T = 1.3

T = 1.5

T = 2.0

T = 2.5

T = 3.0

-1 -0.5 0 0.5 1

x-axis

0

0.1

0.2

0.3

0.4

density

T = 0.1

T = 0.5

T = 0.7

T = 1.0

T = 1.3

T = 1.5

T = 2.0

Fig. 1.3: Evolution of solutions through the center along the x-axis at different times for

the Hughes model (top) and for the MFG structure (bottom).


along the x-axis for both models at different times. One observes that thesolutions in two dimensional cases have a similar behavior as the solutions inone dimensional case, see Figure 1.1.

Acknowledgements This work is supported by the German research foundation, DFG

grant KL 1105/27-1, by RTG GrK 1932 “Stochastic Models for Innovations in the Engi-

neering Sciences”, project area P1 and by the DAAD PhD program MIC ”Mathematics inIndustry and Commerce”.

References

1. Bellomo, N., Dogbe, C.: On the modeling crowd dynamics from scaling to hyperbolic

macroscopic models, Math Models Methods Appl Sci, 18, 1317-1345 (2008)2. Burger M., Di Francesco M., Markowich P.A., Wolfram M-T.: Mean field games with

nonlinear mobilities in pedestrian dynamics, Discrete and Continuous Dynamical Sys-tems. Series B. A Journal Bridging Mathematics and Sciences, 19, 1311–1333 (2014)

3. Burger M., Markowich P.A., Pietschmann J-F.: Continuous limit of a crowd motion

and herding model: analysis and numerical simulations, Kinetic and Related Models,4, 1025–1047 (2011)

4. Burger M., Schlake B., Wolfram M-T.: Nonlinear Poisson-Nernst-Planck equations

for ion flux through confined geometries, Nonlinearity, 25, 961–990 (2012)5. Hughes R.L.: A continuum theory for the flow of pedestrians, Transportation Research

Part B: Methodological, 36, 507-535 (2000)

6. Lasry J-M., Lions P-L.: Mean field games, Japanese Journal of Mathematics, 2, 229–260 (2007)

7. Mahato N.K., Klar A., Tiwari S.: Particle methods for multi-group pedestrian flow,

Applied Mathematical Modelling, 53, 447-461 (2018)8. Mahato N.K., Klar A., Tiwari S.: A meshfree particle method for a vision-based

macroscopic pedestrian model, Int J Adv Eng Sci Appl Math, 10, 41-53 (2018)

Chapter 2

Modeling bimaterial 3D printing usinggalvanometer mirror scanners

Daniel Bandeira and Marta Pascoal

Abstract Three-dimensional printing is a process for building new parts witha specified shape. Despite its increasing popularity, printers capable of work-ing with more than one material are yet unavailable. In this work we modelthe design and the operation of an apparatus for printing with two materials,namely printing a component which includes a previously constructed innerstructure. The structure that supports the second material brings difficulties,resulting from the possible “shaded” areas on the printing surface. The prob-lem is addressed assuming the installation of galvanometer mirror scannersas additional light sources on the walls of the printer, and it is modeled intwo steps: finding the least number of emitters to use, so that the whole partcan be constructed, as well as their position; and assigning them with eachcell of the part to be reached. The first step is formulated as a set coveringproblem. The second is formulated as a linear integer problem and aiming atoptimizing two objectives: the number of emitters activated per layer and thequality of the printed part. Methods for solving the problems are describedand tested.

Daniel Bandeira

Department of Mathematics, University of Coimbra, 3001-501 Coimbra, Portugal, e-mail:[email protected]

Marta PascoalCMUC, Department of Mathematics, University of Coimbra, 3001-501 Coimbra, Portugal,

andInstitute for Systems Engineering and Computers – Coimbra, rua Sılvio Lima, Polo II,

3030-290 Coimbra, Portugal e-mail: [email protected]

9

10 Daniel Bandeira and Marta Pascoal

2.1 Introduction

Three-dimensional printing, or 3D printing, is an additive process for rapidfree form manufacturing, where the final object (known as part) is createdby the addition of successive thin layers of material. Each layer correspondsto a cross-section of the part to be constructed, and the printer draws eachlayer as if it was a 2D printing [2,6]. Printings are made of a single material,which can vary from resin to ceramics or metal (among others). One of thetechnologies used for 3D printing is stereolithography (SL). In this case, eachlayer is added using liquid resin exposed to a laser light, usually fixed at thetop of the printer and able to reach the printing platform. Only the zone ofthe resin that is reached by the laser beam is cured. Then the platform thatsupports the model moves to get ready for printing the next layer. This typeof 3D printing is fairly standard nowadays, and has become quite populardue to its ability to produce new parts quickly and at a low cost.

The present work focuses on a process analogous to SL, but with thegoal of printing an object in which the resin covers a previously constructed3D grid structure of a different material, like metal. This type of compo-nents has application to custom orthotics, intelligent components, complexor fragile parts where over-injection or others options are not feasible or noteconomically sustainable. In this case the metal structure may block the laserlight, thus preventing the cure of shaded areas. This work addresses the ques-tion of placing additional galvanometer mirror scanners on the walls of theprinter to overcome this issue. Their positions depend on the part to printand are fixed from the beginning of the printing process. However, the laserbeam reflected by each galvanometer scanner can be oriented with the goalof reaching the shaded areas. Hereafter galvanometer scanners and laser aresometimes simply referred to as emitters.

As explained before, the part is divided into layers, each one evenly parti-tioned in squares, called voxels. Assuming that both the voxels to cure andthe possible locations for the emitters are known, the problem is modeled intwo parts:

• Emitters location problem (ELP): The goal of which is to find the emitters’position that minimizes the number of emitters required to complete theprinting.

• Emitters assignment problem (EAP): Using the solution of the ELP, itis then necessary to determine the voxels of each layer that each emittershould reach.

The rest of the text is organized as follows. In Sections 2.2 and 2.3 inte-ger linear optimization models are presented for these two problems. Theformulations are empirically tested for a case study in Section 2.4. Finally,concluding remarks are discussed.

2 Modeling bimaterial 3D printing using galvanometer mirror scanners 11

2.2 Emitters location problem

The goal of the ELP is to find the minimum number of emitters that allowsto print a given part, as well as their location. To do this, let us considerthat there are m voxels that the laser light needs to cure and n possiblepositions for the laser emitters. The emitters coverage matrix is defined asA = [aij ]i=1,...,m;j=1,...,n, such that

aij =

1 if the emitter at position j can reach the voxel i0 otherwise

, i = 1, . . . ,m, j = 1, . . . , n.

The matrix A can be calculated by geometric arguments, as shown in [1]. Letalso xj be binary decision variables, such that

xj =

1 if the emitter at position j is installed0 otherwise

, j = 1, . . . , n.

The objective function of the ELP is the total number of emitters to use,which is to be minimized. This is given by

n∑j=1

xj .

We say that the emitter j covers the voxel i, or that i is covered by j, if it isable to reach it by means of a laser beam, for any j = 1, . . . , n, i = 1, . . . ,m.At least one emitter needs to cover each voxel, in order to cure the material.Therefore, a solution for the ELP is feasible if any voxel i, can be reached byat least one emitter, that is, if

n∑j=1

aijxj ≥ 1, i = 1, . . . ,m.

Thus, the ELP can be formulated as the set covering problem below,

minimizen∑j=1

xj

subject to Ax ≥ 1x ∈ 0, 1n

(2.1)

The optimal value of problem (2.1) is the number of emitters required toensure the complete printing of the part. Its optimal solution provides thepositions where the emitters should be installed, which corresponds to theindices j such that xj = 1, j = 1, . . . , n. The set covering problem is aclassical combinatorial optimization problem which has been shown to be


NP-complete [3,7], therefore, exact methods may be limited to solve it as thesize of the problem grows.

2.3 Emitters assignment problem

Assume now that n2 emitters have been installed in the positions determinedby the ELP. The goal of the EAP is then to select the emitter to assign toeach voxel. Also consider that p layers of the part need to be printed, eachone with mk voxels to cure, k = 1, . . . , p.

Let k be a layer to print, k = 1, . . . , p, xj be variables similar to those usedin the ELP, and yij be binary variables defined by

yij =

1 if the emitter j is activated to cure voxel i0 otherwise

, i = 1, . . . ,mk, j = 1, . . . , n2.

Two aspects are taken into account for defining the objective functions,the operability of the system and the quality of the printing. The first isexpressed by the number of emitters used on each layer of the part, and thesecond as the distortion of the laser light when it reaches the layer, denotedby z1 and z2, respectively. With this respect it should be noted that the lightbeam has the shape of a circle at its origin, but the circle is distorted as anellipse when reaching the layer, every time its incidence angle is not exactly90º. Therefore, similarly to the ELP, the first criteria, to minimize, can beexpressed by

z1(x, y) =

n2∑j=1

xj .

The second depends on the emitter that reaches each voxel, yij , and thebeam’s angle of incidence, θij ∈ [0, π2 ], i = 1, . . . ,mk, j = 1, . . . , n2, whichcan be calculated by a procedure similar to the emitters coverage matrix [1] –see Figure 2.5. Thus, minimizing the distortion of the laser light correspondsto maximizing the function

z2(x, y) =

mk∑i=1

n2∑j=1

θijyij .

The choice of the emitter used to reach a particular voxel must take twoaspects into account: the uniqueness of this solution and its viability. Assign-ment constraints can be used to ensure the first one

n2∑j=1

yij = 1, i = 1, . . . ,mk, (2.2)


whereas the second depends on the constraints

n2∑j=1

aijyij = 1, i = 1, . . . ,mk, (2.3)

where A = [aij ]i=1,...,mk;j=1,...,n2 is the submatrix of the emitters coveragematrix, restricted to the set of voxels to cure at layer k and the emittersinstalled in the printer. The other aspect to consider is the emitters that areactivated to print the k-th layer. For the emitters used in each layer, thecovering conditions introduced in Section 2.2 can be used,

n2∑j=1

aijxj ≥ 1, i = 1, . . . ,mk. (2.4)

The variables yij and xj are related, because when the emitter j is activatedto reach a voxel i, it is activated for the entire layer. This corresponds toimposing the constraints

yij ≤ xj , i = 1, . . . ,mk, j = 1, . . . , n2,

and, by aggregating these conditions, we can derive the equivalent constraints

mk∑i=1

yij ≤ mkxj , j = 1, . . . , n2. (2.5)

Finally, it should be noted that when (2.3) and (2.5) hold, the constraints(2.4) are satisfied as well. Combining all the information, we obtain the fol-lowing biobjective linear integer problem,

minimize z1(x, y)

maximize z2(x, y)

subject to (2.2), (2.3), (2.5)

yij ∈ 0, 1, i = 1, . . . ,mk, j = 1, . . . , n2 (2.6a)

xj ∈ 0, 1, j = 1, . . . , n2. (2.6b)

In general, the objective functions z1 and z2 are conflicting, which meansthat there is no feasible solution that optimizes both simultaneously. Solvingthe problem considering only z1 or z2 provides an idea of how much the twoobjective functions may range, but as an alternative to the usual conceptof optimality in single-objective problems, in these cases we usually seek forefficient solutions. A solution is said to be efficient if there is no other whichis strictly better than the first with respect to the two objective functionssimultaneously. The approaches for finding the efficient solutions of biobjec-tive integer problems can be classified into a priori, interactive or a posteriori,


depending on whether how one efficient solution is chosen among the all set.In the first case the decision maker (DM) knows how the relative importanceof the two objective functions, which can be aggregated accordingly beforeone solution is found. In the second case, partial efficient solutions are shownto the DM, who then guides the search for an acceptable solution. The lastcase consists of computing all the efficient solutions and then let the DMexpress the preferences with respect to that whole set. A single solution mustbe chosen before printing a given part, however it is not clear in advancehow the two objective functions are related, thus an a posteriori approach ismore indicated for the EAP. Several methods have been proposed to find theefficient solutions of biobjective integer problems like the EAP. For instance,two-phase methods or the ε-constrained method, among others [4, 5]. Thistopic will be the subject of future research.

2.4 Computational experiments

The formulations presented above were tested for a case study consisting ofconstructing the cube with 5 faces shown in Figure 2.1. The thickness of themetal grid is considered equal to the thickness of the resin layers, that is, 1.This is also the value used as the width of the voxels. The parameters forprinting the cube are:

• The length of each segment of the metal grid, lM .• The thickness of the resin added on each side of the metal grid, lP = 1.• The number of divisions of the metal grid, which is assumed to be uniform,nM .

• The distance between the cube to print and the side walls of the printer,where emitters can be installed, h, which depends on the size of the part,but ensuring that the printing platform is of size 1250× 1250 units.

• The height of the printing area, fixed to hV = 1250 units.

The used length unit corresponds to 0.2 millimeters, the length of the side ofthe voxels. Each layer contains nV ×nV voxels. The remaining characteristicsof the problems solved are summarized in Figure 2.2. The linear problemswere solved using CPLEX 12.7, whereas MATLAB R2016b was used for theremaining calculations. The presented run times are mean values obtainedfor 30 repetitions on an IntelR i7-6700 Quadcore of 3.4 GHz, with 8Mb ofcache and 16 Gb of RAM.

For the ELP it was assumed that a laser is already fixed at the centerof the top of the printer. Additionally, 80 possible locations are consideredfor other emitters on the side walls. The solutions of problem (2.1) givenby CPLEX are presented in Table 2.1. According to the results, between 2and 4 emitters besides the top one are required for completing the printing.


4cm

h

hV

lM

Fig. 2.1: Printing area and object to print

4cm

Test nV nM h

T1 200 5 525

T2 200 10 525

T3 200 20 525

T4 300 5 475

T5 300 10 475

T6 300 20 475

T7 500 5 375

T8 500 10 375

T9 500 20 375

Fig. 2.2: Test parameters

Fig. 2.3: Case study

Although most problems were solved in less than 3 minutes, one of themrequired almost half an hour, which reflects the hardness of the problem.

Table 2.1: ELP solutions and run times

Test Emitters’ positions Time (s)

T1 (1, 1000, 250) and (1250,1,250) 5.03

T2 (1, 1000, 1000), (1, 251,250) and (1250, 1250, 250) 19.13

T3 (1250, 1, 1000), (1, 1, 250), (1, 1250, 250) and (1250, 1250, 250) 140.03

T4 (1, 1, 750) and (1250, 1250, 250) 9.89

T5 (1, 750, 1000), (1, 251, 500) and (1250, 1000, 500) 34.98

T6 (1, 1, 500), (1250, 1250, 500), (1, 750, 250) and (1250, 501, 250) 1323.36

T7 (1, 1, 750) and (1250, 1250, 500) 27.52

T8 (1, 1, 500), (1250, 1250, 500) and (1250, 1, 250) 40.70T9 (1, 1, 500), (1250, 1250, 500) and (1250, 1, 250) 144.05


Using the solutions in Table 2.1, the EAP was considered when optimizingone objective function at a time. The approach that optimizes zi is repre-sented below by Ci, i = 1, 2. Figure 2.4 shows the mean results for the numberof non fixed emitters required for printing each layer, µ1, the mean value ofθ, µ2, and the run times regarding printing the whole part for each method.In terms of solutions the approach C1 always finds a way to print the partusing 2 or 3 emitters per layer besides the top one, while this only happenswith C2 when a broader grid is considered. For the remaining cases apply-ing C2 implies using 3 or 4 emitters. The average angles of incidence of thebeam over the voxels are between 60º and 80º when using approach C2 andbetween 45º and 80º when using C1. The results are worse, i.e., the angleof incidence is smaller, when the grid are denser. The approach C1 is moresensitive to this change than C2. As explained next, small angles of incidencemay lead to a distorted final part. In general in average the two approachesrun fast, a few seconds, and in approximately the same CPU time. However,the tests T3 and T8 were harder to solve using the approach C1 than usingC2, around 30 and 5 minutes, respectively.

•4 •

4

•

4

•4 •

4

•

4

•4 •

4

•

4

1 2 3 4 5 6 7 8 9

2

3

4

Test

µ1

•4

•

4

•

4

•4•4

•

4

•4•4

•

4

1 2 3 4 5 6 7 8 945

55

65

75

Test

µ2(o

)

•4 •4

•

4 •4 •4 •4 •4•4 •4

1 2 3 4 5 6 7 8 90

600

1200

1800

Test

Tim

e(s)

• C1

4 C2

Fig. 2.4: EAP solutions and run times

A measure of the quality of the produced part should also be taken intoaccount. As mentioned earlier, in general, the laser beam reaches the printing


surface as an ellipse because of angle θ. We have considered that the centersof the laser and of the voxel are aligned, thus, two situations may affect thequality of the part: a region beyond the target voxel may be cured, leadingto an outer area Aout, and part of the target voxel may lack the cure, leadingto an inner area Ain. Both are illustrated in Figure 2.5.

4cm

x

y

• Aout

• Ain

Fig. 2.5: Laser distortion on a voxel

5cm

(a)t

• C1 4 C2

•

•

•

••• •••4

44444 444

65 70 75 80

1

2

3

4

5

6

7

8

9

µin (%)

µout(%

)

Fig. 2.6: Distortion areas

Fig. 2.7: Quality of the printed part

The mean values of Ain and Aout were calculated for the same case study.Standard lasers for stereolithography have a radius of 0.05 millimeters, so,taking into account the considered unit of measurement, the laser has radius0.25 units. Figure 2.6 shows the mean value of the percentage relative to thevoxel area of Ain and Aout, respectively µin and µout. In all cases an areaof voxels is left to cure and for some of them there is also an area reachedoutside the voxels. The mean value of Ain was above 60% for all tests. Themain reasons are the assumption that the laser reaches only the center ofvoxels and considering voxels whose sides are twice the diameter of the laserbeam. Working with smaller voxels would result in a reduction of this area,


but would increase the values of Aout. The area Aout is almost null for mostof the cases. The instance with the highest values of Aout corresponds toapproach C1 when applied to test T3.

The polymer at a given voxel may be affected by a beam pointing at neigh-bor voxels. Likewise, only the outer area of voxels in the border is relevant forthe quality of the part. Therefore, the expressions of Ain and Aout are onlyestimate measures for the printing quality. Additionally, current 3D printingprocesses include a post-printing finishing phase where all part is exposed toUV light to cure any liquid resin left. This can reduce the theoretical valuesof Ain and allows to improve the produced part.

2.5 Conclusions

This work addressed the bimaterial 3D printing problem based on the in-stalation of galvanometer scanners on the walls of a printer. The problemis treated as locating the emitters and assigning them with the voxels of agiven part, which were formulated and tested for a case study. The softwareCPLEX was able to find exact solutions for the considered instances. Never-theless, these are computationally hard problems, thus heuristics should bedesigned to prevent cases for which this does not happen or when no com-mercial specialized software is available. Moreover, the unexpectedly curedarea of the extreme solutions of the EAP was relatively small, while the un-cured area of the part seems fairly high. In practice this latter issue can beaddressed with a post-printing finishing phase, a standard 3D printing pro-cedure. Our model can also restrict the emitter positions having in mind toreduce the laser distortion, although that may compromise the full printingof the part. Finally, the presented approach can still be used to print partswith more than 2 materials, by handling the product of the first print as theinner structure of the next one.

Acknowledgements This work was developed within the Project PT2020-POCI-SII &

DT 17963: NEXT.Parts, Next-Generation of Advanced Hybrid Parts, from the program

Portugal 2020, through COMPETE 2020 – POCI. The work of MP was partially sup-ported by the Portuguese Foundation for Science and Technology under project grants

UID/MAT/00324/2013 and UID/MULTI/00308/2013.

References

1. D. Bandeira, M. Pascoal, A. Mateus, and M. Reis Silva. Multi-material 3d printing

using stereolithography: an optimization approach. Submitted for publication, 2017.

Available at http://www.mat.uc.pt/∼marta/NextG/BandeiraEtAl17.pdf.


2. M. Burns. Automated fabrication: improving productivity in manufacturing. Prentice-

Hall, Inc., Boston, 1993.3. A. Caprara, P. Toth, and M. Fischetti. Algorithms for the set covering problem. Annals

of Operations Research 98:353–371, (2000)

4. M. Ehrgott. Multicriteria optimization. Springer Science & Business Media, Berlin,Heidelberg, 2006.

5. M. Ehrgott and X. Gandibleux. Multiobjective Combinatorial Optimization – Theory,

Methodology, and Applications, pages 369–444. Springer US, Boston, MA, 2002.6. I. Gibson, D. Rosen, and B. Stucker. Additive manufacturing technologies. Springer,

New York, 2015.7. R. M. Karp. Reducibility among combinatorial problems. In Complexity of computer

computations, pages 85–103. Springer, 1972.

Chapter 3

Mathematical Modeling forLaser-Induced Thermotherapy in LiverTissue

N. Siedow and C. Leithauser

Abstract Laser-induced thermotherapy (LITT) plays an important role inoncology to treat human liver tumors. LITT is a minimally invasive methodcausing tumor destruction due to heat ablation and coagulative effects ofthe tissue. Tumor tissue is much more sensitive to heat than normal healthytissue. The big advantage of the LITT compared to other minimally invasiveprocedures, such as microwave ablation or radiofrequency therapy, is that thetreatment primarily takes place under MRI control, such that patients areexposed to a small radiation dose. The present paper describes the mathemat-ical modeling of laser-induced thermotherapy and shows simulation resultsfor porcine liver.

3.1 Introduction

The aim of thermal ablation methods is to destroy cancer tissue by generatingcytotoxic temperature for a short time without damaging vital tissue. Thesemethods are minimally invasive and used for treating for example lunge, liver,and prostate cancer, when surgical resection is not possible or too dangerousfor the patient. Tumor tissue is much more sensitive to heat than normalhealthy tissue. Most proteins denature at 40C - 42C. Irreversible coagula-tion necrosis occurs in the temperature range of 60C to 100C. Temperaturesabove 150C result in vaporization and carbonation. This leads to an unde-sirable reduced thermal conductivity, and heat can not penetrate further intothe tissue.

The most popular thermal ablation methods are the radiofrequency abla-tion technique, the LITT, and the microwave ablation.

N. Siedow and C. LeithauserFaunhofer-Institute for Industrial Mathematics Kaiserslautern, Germany,

e-mail: norbert.siedow, [email protected]

21

22 N. Siedow and C. Leithauser

The principle of LITT is based on the local introduction of energy via anoptical fiber directly into the cancerous tissue. The laser fiber is located in awater-cooled, MR-compatible applicator. The introduction of the applicatorinto the tumor is done under the CT, while the actual treatment takes placeunder MRI control. Thus, the patient is exposed to only a small dose ofradiation during LITT. An additional advantage is the possibility to use MR-Thermometry for active image control. MR-Thermometry methods are basedon MR measured parameters depending on temperature like the longitudinalrelaxation time (T1), the diffusion coefficient (D), or the proton resonancefrequency (PRF ) of tissue water. The linear temperature dependence of theproton resonance frequency and its near-independence with respect to tissuetype make the PRF-based methods the preferred choice for many application.For a more deeper understanding to MR-Thermometry we refer to the reviewpaper [1].

In the following we discuss the mathematical modeling of the LITT, andcompare simulation results with temperature maps from MR-Thermometry.

3.2 Mathematical Modeling

Let Ω ⊂ R3 denote the geometry of the liver, which is obtained from MRIthrough segmentation. The boundary Γ of Ω consists of the radiating partof the adjacent applicator Γrad, which is not part of the liver, the cooled partof the applicator Γcool, and the surface of the liver Γamb (see Fig. 3.1). Themathematical model is described by a system of partial differential equationfor the heat transfer inside the liver, the radiative transfer from the applicatorinto the liver tissue, and a damage function. ( [2], [3], [4])

Fig. 3.1: Water cooled applicator with radiating laser fiber

3 Mathematical Modeling for Laser-Induced Thermotherapy in Liver Tissue 23

3.2.1 The Temperature

The heat transfer in the liver is modeled by the well-known bio-heat equation

cpρ∂T∂t (x, t) = ∇ · (kh∇T (x, t)) + ξb(Tb − T (x, t)) +Qrad,

T (x, 0) = Tinit(x), (3.1)

where T (x, t) denotes the temperature depending on the three-dimensionalposition x and the time t. cp is the thermal conductivity, ρ the density, khthe thermal conductivity, and ξb the perfusion rate due to the blood flux. Tbdenote the blood temperature and Qrad the energy source term due to theirradiation of the laser fiber. The initial temperature is assumed to be knownT0(x).

For the heat transfer between liver and applicator and with the surround-ings Robin type boundary conditions are used.

kh∂T∂n = αcool(Tcool − T ), x ∈ Γrad ∪ Γcool,kh

∂T∂n = αamb(Tamb − T ), x ∈ Γamb. (3.2)

Here n is the outer normal vector, αcool and αamb the heat transfer coefficientswith the cooling part of the applicator and the surroundings of the liver,respectively. Tcool denotes the cooling temperature and Tamb the ambienttemperature.

The source term in (3.1) is given by

Qrad =µa4π

∫S2

I(s, x)ds = µaφ(x), (3.3)

where µa is the absorption coefficient and φ(x) the radiative energy definedas the integral of the radiative intensity I(s, x) over all directions s of thewhole sphere S2.

3.2.2 The Radiative Transfer

The irradiation of laser light is described by the radiative transfer equation

s · ∇I(s, x) + (µa + µs) I(s, x) =µs4π

∫S2

P (s · s′)I(s′, x)ds′, (3.4)

with the absorption and scattering coefficients, µa and µs, the scatteringphase function P (s · s′) given by the Henyey-Greenstein term


P (s · s′) =1− g2

(1 + g2 − 2g(s · s′))3/2.

g is the so-called anisotropy factor. g = 0 describes the isotropic and g = 1the anisotropic scattering. The boundary condition is given by

I(s, x) = F, x ∈ Γrad, I(s, x) = 0, x /∈ Γrad, (3.5)

where F is an energy density defined later.Because of the high dimension of the radiative transfer equation (3.4) we

use the so-called P1-approximation to approximate (3.4). Introducing theansatz

I(s, x) = φ(x) + 3s · q(x),

where q(x) = 14π

∫S2

I(s, x)sds is radiative flux vector, one obtains the much

simpler three-dimensional diffusion equation

−∇ · (D∇φ(x)) + µaφ(x) = 0, D =1

3(µa + (1− g)µs). (3.6)

To approximate the boundary condition (3.5) we use the Marshak’s proce-dure described for instance in [5]. We obtain Robin type boundary conditions

D∂φ

∂n(x) =

qappAΓrad

, x ∈ Γrad, D∂φ

∂n(x) + bφ(x) = 0, x /∈ Γrad, (3.7)

where qapp is the energy delivered by the laser and AΓradthe surface area of

the radiating part of the fiber. The parameter b = 0.5 for x ∈ Γamb and b = 0for x ∈ Γcool. From the numerical point of view (3.6), (3.7) is much easier tocompute than the original system (3.4), (3.5).

3.2.3 The Damage Function

The damage of the liver/cancer tissue will be described by the so-called dam-age function. It is common (see [2], [3]) to use the Arrhenius law

w(x, t) =

t∫0

Ae−Ea/RT (x,τ)dτ, (3.8)

with so-call frequency factor A, activating energy Ea, and ideal gas constantR to describe the change of material properties due to coagulation. Thesubscript n stands for native tissue and c for the properties of the coagulatedtissue. For the optical parameters we obtain

3 Mathematical Modeling for Laser-Induced Thermotherapy in Liver Tissue 25

µa = µan + (1− e−w)(µac − µan),

µs = µsn + (1− e−w)(µsc − µsn), (3.9)

g = gn + (1− e−w)(gc − gn).

3.3 Results

The mathematical model described above was used to simulate the heating ofpig porcine. The liver geometry and applicator position were obtained fromsegmented MR-images. The computational geometry was generated usingOpen Cascade (Open Cascade SAS, Guyancourt, France) and the mesh usingthe software code Gmesh. The differential equations, including boundary con-ditions, were discretised and solved using GetDP (P. Dular and C. Geuzian,University of Liege). More details can be found in the co-work [6]. The usedheat and optical parameters are listed in [4]. The time-depending simula-tion results were compared with data from MR-Thermometry (see Fig. 3.2)and thermocouples placed at different positions around the laser-applicator.The simulation as well as the MR-Thermometry are in good agreement withmeasured data. Looking at Fig. 3.2 one can see the typical shape of thetemperature distribution around the radiating part of the applicator.

Fig. 3.2: Temperature simulated (left) and taken from MR-Thermometry (right)


3.4 Conclusions

LITT is a minimal-invasive method in the field of interventional oncology usedfor treating liver cancer. Mathematical modeling and computer simulationare important features for treatment planning and imaging the necrosis ofthe tissue. The numerical simulation is in good agreement with the MR-Thermometry and temperature measurements for porcine liver. For futurework blood perfusion has to be taken into account. The blood flux of vesselsand tissue has a cooling effect, which is very important for treating humansand depending on the physiology of the patient. To model these effects moreresearch is needed.

Acknowledgements The authors acknowledge the financial support by the Federal Min-

istry of Education and Research of Germany in the framework of the project proMT: Prog-nostische modellbasierte online MR-Thermometrie bei minimalinvasiver Thermoablation

zur Behandlung von Lebertumoren (Forderkennzeichen: 05M16AMA).

References

1. de Senneville, B.D., Quesson, B., Moonen, C.: Magnetic Resonance Temperature

Imaging. International Journal of Hyperthermia. Taylor Francis, 21:6 (2005)

2. Mohammed, Y., Verhey, J.F.: A finite method model to simulate laser interstitial ther-mos therapy in anatomical inhomogeneous regions. BioMedical Engineering OnLine,

4:2 (2005)

3. Fasano, A., Homberg, D., Naumov, D.: On a mathematical model for laser-inducedthermotherapy. Applied Mathematical Modelling, 34:12 (2010)

4. Hubner, F., Leithauser, C., Bazrafshan, B., Siedow, N., Vogl, T.J.: Validation of amathematical model for laser-induced thermotherapy in liver tissue. Lasers in Medical

Science, 32:6 (2017)

5. Modest, M.F.: Radiative Heat Transfer. Academic Press, San Diego (2003)6. Leithauser, C., Hubner, F., Bazrafshan, B., Siedow, N., Vogl, T.J.: Experimental

Validation of a Mathematical Model for Laser-Induced Thermotherapy. In: European

Consortium for Mathematics in Industry. Springer (2018)

Chapter 4

Sparse Multiple Data Assimilationwith K-SVD for the History Matchingof Reservoirs

Clement Etienam, Rossmary Villegas Velasquez, Oliver Dorn

Abstract Calibrating subsurface reservoir models with historical well obser-vations leads to a severely ill-posed inverse problem known as history match-ing. The recently proposed Ensemble Smoother with Multiple Data Assimila-tion (ES-MDA) method has proven to be a successful stochastic technique forsolving this inverse problem, but its computational cost can be high in realis-tic scenarios and it remains challenging to incorporate certain non-Gaussiantypes of a-priori information into it. In this work we combine the ES-MDAmethod with Multiple-Point Statistics (MPS) and the K-SVD technique forbuilding sparse dictionaries in order to obtain a novel sparsity-based historymatching scheme that preserves non-Gaussian structural prior informationand at the same time reduces computational cost. We present numericalexperiments in 3D on a modified SPE10 benchmark reservoir model thatdemonstrate the performance of this new technique.

4.1 Introduction

The reconstruction of subsurface geological features from production datadefines an inverse problem related to data assimilation, which has long beena challenge in the reservoir engineering community due to the small number

Clement Etienam

School of Chemical Engineering and Analytical Science, the University of Manchester,Manchester, M13 9PL, UK, e-mail: [email protected]

Rossmary Villegas Velasquez

Mathematics and Natural Sciences, Prince Mohammad Bin Fahd University, P.O. Box1664, Al Khobar 31952, Kingdom of Saudi Arabia, e-mail: [email protected]

Oliver DornSchool of Mathematics, the University of Manchester, Manchester, M13 9PL, UK, e-mail:

[email protected]

27

28 Clement Etienam, Rossmary Villegas Velasquez, Oliver Dorn

of observations available [6]. Recently the Ensemble Smoother with MultipleData Assimilation (ES-MDA) method has become popular for this task [5].However, the conventional ES-MDA framework fails to accurately capturenon-Gaussian spatial distributions, for example in channelized reservoirs. Inthose cases, novel image processing techniques based on sparsity representa-tions provide an interesting tool for enabling us to incorporate prior informa-tion in the assimilation task and thereby improve final results [6, 7]. In thework presented here we will reformulate this inverse problem as a sparse fieldrecovery task which will then be solved, in contrast to previous work, by usinga combination of ES-MDA and sparsity enhancing techniques, in particularK-SVD (an acronym for K-means and Singular Value Decomposition) andOrthogonal Matching Pursuit (OMP).

Our forward problem consists of a three-phase flow model of water, oiland gas fully derived using the combination of Darcy’s law and continuityequations [3]. It consists of a system of coupled non-linear partial differentialequations describing the evolution of the dynamic state variables inside theporous medium which change over time. These are in particular the water,oil and gas saturation levels Sγ , γ ∈ w, o, g, (where the subscripts w, o, grefer to the corresponding three different phases water, oil and gas that aresimultaneously present in the reservoir), and the associated pressure levelspγ . For a more detailed description of the underlying fluid flow model werefer to [3, 8].

There are also static parameters involved in the description of the fluidflow problem that do not change over time. These are in particular the effec-tive porosity ϕ and the absolute permeability K (all other static parametersof the fluid flow model are assumed known here) [3,8]. Classically, those staticparameters are the primary unknowns of the underlying inverse problem thatneed to be estimated from production data. Plugging those estimates into asimulator (e.g. [8]) will then (ideally) reproduce the correct dynamical statevariables. Notice that those static parameters are related to the lithologi-cal structure of the reservoir of which some a-priori knowledge is usuallyavailable from independent investigations. Those are encoded in our trainingimages used for generating the initial ensemble for the ES-MDA algorithmvia Multiple-Point Statistics (MPS), as well as defining the dictionary con-ditioning our sparsity-based data assimilation procedure later on. Thereby,in our algorithm this information will be incorporated throughout the dataassimilation algorithm to provide final reconstructions better satisfying thisprior structural information. In this sense, our sparsity approach does notonly speed up the reconstruction process, but also has a regularization effecton the final results.

4 Sparse Multiple Data Assimilation with K-SVD for History Matching 29

4.2 Ensemble-Smoother Multiple Data Assimilation(ES-MDA)

ES-MDA is a Monte Carlo approach to the underlying data assimilation prob-lem which was proposed in [5]. In ES-MDA, each given data set is assimi-lated multiple times as outlined in the following. The underlying statisticalproperties of the reservoir are represented by choosing an initial ensemble ofsize Ne of equi-probable parameter distributions. Let mj denote the staticpetro-physical properties to be estimated, where j indicates the ensemblemember (j = 1, . . . , Ne), and let in particular mf

j denote its current esti-mation in a given step of ES-MDA. Following the notation and the generalapproach outlined in [5], we denote the (perturbed) observed data by duc,jand the predicted data, running our simulator on mf

j , by dfj . Denote byNa the total number of ES-MDA iteration steps taken [5]. The parameterupdate/assimilation step is then carried out Na times as an iteration withupdate rule

maj = mf

j + CfMD

(CfDD + αCD

)−1× (duc,j − dfj ) (4.1)

for j = 1, . . . , Ne. In (4.1), CfMD is the cross-covariance matrix between the

prior vector of model parameters, mfj , and the vector of predicted data, dfj .

Furthermore, CfDD is the auto-covariance matrix of predicted data, and CDis the inflated data error covariance matrix. α is the data error covarianceinflation factor at the given data assimilation step which is selected priorto the history matching loop for all iteration steps, see [5] for details on itschoice and theoretical justifications.

We need an initial ensemble for starting the ES-MDA procedure. In thiswork, we use MPS [10] for the creation of this initial ensemble targetinga typical channelized model. Thereby the ensemble is conditioned on theinformation at the well locations and the corresponding analysis of statisticalproperties. Notice that, when introducing sparsity in this framework furtherbelow in this paper, we will then be able to replace some of the quantitiesthat occur in (4.1) by the corresponding sparse representations, potentiallyleading to a significantly reduced computational cost in each iteration.

4.3 Dictionary Learning and OMP

A key factor of sparse coding is the identification of a basis (also often calleddictionary or frame in this context) in which the field or signal under consid-eration permits a sparse representation [2,7]. A classical approach is to use ageneral-purpose basis (dictionary) for this, for example involving wavelets or


the Discrete Cosine Transform (DCT) [6]. The disadvantage of that choice isthat the basis might not be optimal for the particular ensemble to be repre-sented. Therefore, in this work we follow a different approach which employsa special dictionary learning algorithm, namely K-SVD, for determining asuitable basis [4]. To start with, we use MPS for generating a set of trainingsignals yi, i = 1, . . . , Nr, each of them having length Ny (the dimension of thepermeability field to be estimated), and arrange them as columns of a matrixY of size Ny×Nr. The K-SVD dictionary learning algorithm constructs nowsimultaneously a dictionary matrix D of size Ny×Ns, whose columns consistof the “dictionary atoms”, and an Ns×Nr matrix X, whose columns consistof the representation vectors xi, i = 1, . . . , Nr, by the (joint) minimizationtask

X,D = argminX,D

∥∥∥Y − DX∥∥∥2F

subject to ∀i. ‖xi‖0 ≤ T0 . (4.2)

In (4.2), the parameter T0 indicates the sparsity level imposed on the algo-rithm and ‖ ‖F denotes the Frobenius norm. Notice that, upon completion,(4.2) provides a sparse representation yi ≈ Dxi, i = 1, . . . , Nr. Practically,the K-SVD algorithm alternates between updates for D and updates for Xfor a given training set until optimality is reached. For more details of thisalgorithm, including a pseudo-code describing its individual stages, we referto [1]. The creation of this over-complete dictionary is done off-line and onlyonce before the history matching process starts. In our work, it is used totransform the permeability and porosity fields from a full spatial domain toa sparse domain and back.

Notice that the part of the K-SVD algorithm which finds an optimal sparserepresentation X for a given training set Y , given the current iterate for thedictionary D, requires us to choose a specific sparsity promoting algorithms,for which we select here the well-known Orthogonal Matching Pursuit (OMP)algorithm [9]. We refer the reader to [1, 9] for further information on OMP.Notice that, in addition to its use inside this off-line K-SVD step, the OMPalgorithm will also be used throughout our proposed ES-MDA algorithmwhenever permeabilities and porosities need to be mapped from full to sparserepresentations.

The sparsity enhanced ES-MDA algorithm described above is summarizedin the following Algorithm 1.

4.4 Numerical results

In order to demonstrate the performance of our SEOM algorithm, we runit on a modified version of the popular benchmark SPE10 model. Here wechoose Nr = 2000, Ns = 1500 and Ne = 100. Typical values for Na in ES-


Algorithm 1 Sparsity-Ensemble Optimization Method (SEOM)

1: procedure SEOM2: Generate Nr realizations of different permeability/porosity profiles using MPS.

Learn dictionary D consisting of Ns atoms tailored to these realizations using K-SVD,eqn. (4.2).

3: Independently select Ne initial ensemble realizations of permeability/porosity using

again MPS for the channelized test case.4: Choose number of assimilation iterations Na.

5: for k = 1 to Na do

6: Progress the Ne realizations over time (using non-sparse representation,7: here with simulator ECLIPSE 100 [8])

8: for j = 1 to Ne do

9: Find sparse representation of all mj using OMP with dictionary D.10: Carry out the ES-MDA analysis step (4.1) on sparse representation.

11: Transform mj back to the (non-sparse) spatial domain.

12: STOP

MDA and SEOM are Na ∈ 2, 4, 6, 8. The true model consists of five layersand can be seen in a 3D view on the top right side of Figure 4.1. The in-dividual five layers (permeability) can be seen in the first row on the left ofthe same figure. The second and third rows of Figure 4.1 display two (of theNs = 1500) members of the generated over-complete dictionary representingthis reservoir. Row four of this figure shows the ensemble-mean starting pro-file used in the ES-MDA and SEOM algorithms for this test case. Row fivedisplays the ensemble-mean result of standard ES-MDA (without sparsity),and row six shows the ensemble-mean result of the SEOM algorithm incor-porating sparsity with respect to the learned dictionary. Visually, the SEOMestimate looks more ‘channelized’ compared to the ES-MDA reconstructiondue to the incorporated a-priori information on the true model encoded inthe dictionary. For evaluating further the performance of the techniques com-pared here, the estimated final porosity and permeability profiles are used forrunning the simulator from the initial time step to the final time step Nk. Forthe obtained data d for all 100 ensemble members we calculate the root-mean-square error (RMSE) for each ensemble member j, denoted by RMSE(j),which is defined as

RMSE(j) =

1

NkNd

Nk∑k=1

Nd∑ν=1

(dνuc,j(k)− df,νj (k)

σν

)2 1

2

. (4.3)

Here the index ν runs over the Nd observed well data components, k indicatesthe Nk physical time steps, j is realization number, and σν is the error stan-dard deviation of data type ν. Table 1 shows and compares RMSE values forthree different realizations (j ∈ 13, 56, 92) which we have chosen (somehowarbitrarily) to represent the entire ensemble. Considering for example real-ization j = 56, the convergence for the standard ES-MDA history matching


was achieved in 8 iterations with an RMS error of 12.8. Compared to that,the proposed method SEOM converged in 4 iterations with an RMS error of1.34.

Table 1: RMSE values RMSE(j)Realization j Initial Final (ES-MDA) Final (SEOM)

13 49.33 24.21 4.3256 43.83 12.8 1.3492 56.87 24.58 6.44

Layer 1 Layer 2 Layer 3 Layer 4 Layer 5

Layer(1) Layer(2) Layer(3) Layer(4) Layer(5)

Layer(1) Layer(2) Layer(3) Layer(4) Layer(5)




X

I1

P1

P2

P4

I2

I4

Y

P3

I3

Z

Fig. 4.1: On the left, all rows show from left to right layers 1 to 5 of the SPE10 reservoiras specified in the following. First row: true permeability profile; second and third rows:two (out of Ns = 1500) members of the over-complete dictionary; fourth row: ensemble-

mean initial guess; fifth row: ensemble-mean final result of standard ES-MDA; sixth row:

ensemble-mean final result of the proposed SEOM algorithm. On the right a 3D view of thereservoir is provided in the top image, where the bottom image shows the corresponding

colour bar that applies to this figure.


References

1. Aharon M., Elad M. and Bruckstein A., K-SVD: An algorithm for designing over com-

plete dictionaries for sparse representation, IEEE Transactions on Signal Processing

54 (11), 4311-4322, 2006.2. Candes, E. J., Romberg, J. and Tao, T., 2006. Robust uncertainty principles: Ex-

act signal reconstruction from high incomplete frequency information. Information

Theory, IEEE Transactions on, 52(2), pp. 489-509.3. Crichlow, H. B., Modern Reservoir Engineering-A Simulation Approach. 1st ed. Ok-

lahoma: Prentice-Hall, Inc. Englewood Cliffs, New Jersey 07632, 1977.

4. Elsheikh, A., Wheeler, M. and Hoteit, I., 2013. Sparse calibration of subsurface flowmodels using nonlinear orthogonal matching pursuit and an iterative stochastic en-

semble method. Advances in Water Resources, 56(1), pp. 14-26.

5. Emerick A.A. and Reynolds A.C., Ensemble smoother with multiple data assimilation.Computers & Geosciences, Vol. 55, 3-15, 2013.

6. Jafarpour, B., Wavelet reconstruction of geologic facies from nonlinear dynamic flowmeasurements. IEEE Transactions Geosciences and Remote sensing, 49(5), pp. 1520-

1535, 2011.

7. Khaninezhad, M. M., Jafarpour, B. and Li, L., 2012. Sparse geologic dictionaries forsubsurface flow model calibration: Part 1, Inversion formulation. Advances in Water

Resources, 39(1), pp. 106-121.

8. Schlumberger GeoQuest, ECLIPSE 100 (Black Oil): Reference Manual and TechnicalDescription. Houston, 2014.

9. Tropp J.A. and Gilert A.C. Signal Recovery from random measurements via orthog-

onal matching pursuit IEEE Transactions on Information Theory 53 (12), 4655-4666,2007.

10. Wu, J., Boucher, A. and Zhang, T., A SGeMS code for pattern simulation of con-

tinuous and categorical variables: FILTERSIM. Computers & Geosciences, Vol 34(12),1863-1876, 2008.

Chapter 5

Hypercomplex Fourier Transformsin the Analysis of MultidimensionalLinear Time-Invariant Systems

Lukasz B laszczyk

5.1 Introduction

The classical signal theory deals with R- or C-valued functions and theirC-valued spectra. However, in some practical applications, signals tend tobe represented by hypercomplex algebras [4]. Hypercomplex Fourier trans-forms deserve special attention in this considerations. Quaternion Fouriertransform (QFT) allows us to analyze two dimensions of the sampling gridindependently, while the complex transform mixes those two dimensions. Itenables us to use the Fourier transform in the analysis of some 2-D lin-ear time-invariant (LTI) systems described by some linear partial differentialequations (PDEs) [3].

In [1] we presented some preliminary results concerning the octonionFourier transform (OFT). We showed that the OFT is well defined for R-valued functions and proved some basic properties of the OFT, analogousto the properties of the classical FT and QFT. Our research follows previ-ous results of Hahn and Snopek [6]. It should be noted that octonion signalprocessing have already found practical applications [5, 7], including imagesplicing detection [9] and neural networks [8].

In this paper, we introduce the most recent results, associating OFT (in-troduced in Section 5.3) with 3-D LTI systems of linear PDEs with constantcoefficients. Properties of the OFT in context of signal-domain operationssuch as derivation and convolution of R-valued functions are stated in Sec-tion 5.4. There are known results for QFT (see [3]), but they use the notionof other hypercomplex algebra, i.e. double-complex numbers. Results pre-sented here require defining other higher-order hypercomplex structure, i.e.quadruple-complex numbers defined in Section 5.2. This hypercomplex gen-

Lukasz B laszczykFaculty of Mathematics and Information Science, Warsaw University of Technology,

ul. Koszykowa 75, 00-662 Warszawa, Poland, e-mail: [email protected]

35

36 Lukasz B laszczyk

eralization of the Fourier transformation provides an excellent tool for theanalysis of 3-D LTI systems which is presented in Section 5.5. The paper isconcluded in Section 5.6 with short discussion of those results.

5.2 Algebras of octonions and quadruple-complexnumbers

Octonions (O) are an example of Cayley-Dickson hypercomplex algebra [1,6].Its elements are of the form

o = x0+x1 e1+x2 e2+x3 e3+x4 e4+x5 e5+x6 e6+x7 e7, x0, x1, . . . , x7 ∈ R,

where e1, e2, . . . , e7 are seven imaginary units satisfying appropriate mul-tiplication rules (presented in Tab. 5.1). Octonions form a non-associative,non-commutative (but alternative) composition and division algebra O of or-der 8 over the field of real numbers IR. Octonion algebra is endowed with thestandard norm

|o| =√o · o∗ =

√x20 + x21 + . . .+ x27,

where o∗ = x0 − x1e1 − . . .− x7e7 is the octonion conjugate of o.We define the octonion exponential function in a classical way – as the

infinite sum eo :=∑∞k=0

ok

k! . Due to the fact, that octonion multiplication isnon-commutative, for any o1, o2 ∈ O we have eo1+o2 = eo1 · eo2 if and only ifo1 · o2 = o2 · o1.

Due to non-associativity and non-commutativity of octonion multiplica-tion, many formulas concerning the Fourier transforms are quite complicated(see Sec. 5.4). To improve that, inspired by [3], we introduce the algebra ofquadruple-complex numbers F, which elements can be written as

p = (p0 + p1e1)︸︷︷︸=s0∈C

+ (p2 + p3e1)︸︷︷︸=s1∈C

e2 + (p4 + p5e1)︸︷︷︸=s2∈C

e4 + (p6 + p7e1)︸︷︷︸=s3∈C

e2e4.

Table 5.1: Multiplication rules in octonion algebra.

· 1 e1 e2 e3 e4 e5 e6 e7

1 1 e1 e2 e3 e4 e5 e6 e7

e1 e1 −1 e3 −e2 e5 −e4 −e7 e6

e2 e2 −e3 −1 e1 e6 e7 −e4 −e5

e3 e3 e2 −e1 −1 e7 −e6 e5 −e4

e4 e4 −e5 −e6 −e7 −1 e1 e2 e3

e5 e5 e4 −e7 e6 −e1 −1 −e3 e2

e6 e6 e7 e4 −e5 −e2 e3 −1 −e1

e7 e7 −e6 e5 e4 −e3 −e2 e1 −1

5 Hypercomplex Fourier Transforms... 37

Therefore, the algebra F consists of quadruples (s0, s1, s2, s3) ∈ C4 of complexnumbers. Multiplication in F is given by the formula

(s0, s1, s2, s3) (t0, t1, t2, t3) = (s0t0 − s1t1 − s2t2 + s3t3, s0t1 + s1t0 − s2t3 − s3t2,s0t2 + s2t0 − s1t3 − s3t1, s0t3 + s3t0 + s1t2 + s2t1),

where (s0, s1, s2, s3), (t0, t1, t2, t3) ∈ F. It is easy to check that multiplication is associative and commutative, but not all nonzero elements of F areinvertible with respect to , e.g. (1, 0, 0, 1) = 1 + e6 ∈ F doesn’t have an-inverse.

5.3 Octonion Fourier transform

Let u : R3 → R. The octonion Fourier transform (OFT) of u is defined by

U(f) =

∫R3

u(x) · e−2πe1f1x1 · e−2πe2f2x2 · e−2πe4f3x3 dx,

where x = (x1, x2, x3), f = (f1, f2, f3) and multiplication is done from left toright. Choice and order of imaginary units in the exponents is not accidental(see [1,6]). Conditions of existence (and invertibility) are the same as for theclassical (complex) Fourier transform. Let us recall the result from [1], wherethe inverse OFT formula was proved.

Theorem 5.1. Let u : R3 → R be a continuous and square-integrable. Then

u(x) =

∫R3

U(f) · e2πe4f3x3 · e2πe2f2x2 · e2πe1f1x1 df ,

where multiplication is done from left to right.

In fact, the above-mentioned theorem holds for the general case of O-valued functions (see [2]), but in this paper we will consider only the R-valued functions. In [1] we derived basic properties of the OFT, analogous tothe properties of the classical Fourier transform. Let us recall some of thoseresults.

Let U be the OFT of the R-valued function u and let αi(o) = −ei · (o ·ei),where is standard function composition. We have the following octonionanalogue of Hermitian symmetry :

U(−f1, f2, f3) = (α6 α4 α2)(U(f1, f2, f3)),

U( f1,−f2, f3) = (α5 α4 α1)(U(f1, f2, f3)),

U( f1, f2,−f3) = (α3 α2 α1)(U(f1, f2, f3)).


Moreover, if Uα, Uβ and Uγ denote the OFTs of functions u(x1 −α, x2, x3),u(x1, x2 − β, x3) and u(x1, x2, x3 − γ), respectively, then

Uα(f1, f2, f3) = cos(2πf1α) U(f1, f2, f3)− sin(2πf1α) U(f1,−f2,−f3) · e1,

Uβ(f1, f2, f3) = cos(2πf2β) U(f1, f2, f3)− sin(2πf2β) U(f1, f2,−f3) · e2,

Uγ(f1, f2, f3) = cos(2πf3γ) U(f1, f2, f3)− sin(2πf3γ) U(f1, f2, f3) · e4,

which is the octonion version of shift theorem. We also have the Planchereland Rayleigh theorems:∫R3

u(x)·v∗(x) dx =

∫R3

U(f)·V ∗(f) df , Rightarrow∫R3

|u(x)|2 dx =

∫R3

|U(f)|2 df ,

where V is the OFT of the R-valued function v. The above-presented theo-rems form the basis of the octonion signal theory and are the starting pointfor further research.

5.4 Recent results

We will now present properties that are a key element in the analysis ofmultidimensional LTI systems described by a system of PDEs. In theoremsstated below, we will denote the OFTs of the R-valued functions u and v byU and V , respectively.

Theorem 5.2 (OFTs of partial derivatives). Let Ux1 , Ux2 and Ux3 de-note the OFTs of ∂u

∂x1, ∂u∂x2

and ∂u∂x3

, respectively. Then

Ux1(f1, f2, f3) = U(f1,−f2,−f3) · (2πf1e1) = U(f1, f2, f3) (2πf1e1),

Ux2(f1, f2, f3) = U(f1, f2,−f3) · (2πf2e2) = U(f1, f2, f3) (2πf2e2),

Ux3(f1, f2, f3) = U(f1, f2, f3) · (2πf3e4) = U(f1, f2, f3) (2πf3e4).

Proof of this result follows from straightforward calculations and we leavedetails to the reader. It is worth noting, however, that the idea of this proof isto express the OFT of the derivative of u as a sum of components of differentparity, i.e.

Ux` = Uxelleee − Ux`

oeee1 − Uxèoee2 + Ux`

ooee3 − Uxèeoe4 + Ux`

oeoe5 + Uxèooe6 − Ux`

oooe7,(5.1)

where

Ux`

ijk(f) =

∫R3

∂u

∂x`· Fi(2πf1x1) · Fj(2πf2x2) · Fk(2πf3x3) dx (5.2)


and Fi(y) = cos(y) if i = e, and Fi(y) = sin(y) if i = o [1,6]. The claim of thetheorem follows from the integration by parts. Notice that treating octonionsas elements of F and using the multiplication , we get the same formulasas in classical theory.

The next result concerns function convolution. The convolution-multiplicationduality is one of the key properties used in the frequency analysis of LTI sys-tems [3]. Recall that the convolution of u, v : R3 → R is given by the formula

(u ∗ v)(x) =

∫R3

u(y) · v(x− y) dy.

Convolution of functions is commutative and associative while the multipli-cation of octonions is not, hence the octonion version of duality theorem willhave to differ significantly from its classical equivalent.

Theorem 5.3 (Convolution-multiplication duality). Let FOFTu ∗ vdenote the OFT of the convolution of u and v, i.e. u ∗ v. Then

FOFTu ∗ v(f) = V ( f1, f2, f3) · ( Ueee(f) − Ueeo(f) e4)

+ V ( f1,−f2,−f3) · (−Uoee(f) e1 + Uooe(f) e3)

+ V ( f1, f2,−f3) · (−Ueoe(f) e2 + Uoeo(f) e5)

+ V (−f1, f2,−f3) · ( Ueoo(f) e6 − Uooo(f) e7), (5.3)

where

U = Ueee − Uoeee1 − Ueoee2 + Uooee3 − Ueeoe4 + Uoeoe5 + Ueooe6 − Uoooe7

is a sum of 8 terms with different parity w.r.t. x1, x2, and x3, similar to (5.1)-(5.2).

As in the previous theorem, this result follow from expressing the OFTas a sum of components of different parity. For details of such formulationsee [1, 6]. Similar formulas concerning quaternion Fourier transform can befound in literature [3]. Notice that, as in the OFT of derivatives theorem,using the notion of quadruple-complex numbers we can improve the above-mentioned formulas.

Corollary 5.1. Using the F-multiplication we can rewrite formula (5.3) insimple form:

FOFTu ∗ v(f) = U(f) V (f).

Theorem 5.3 and Corollary 5.1 enable us to define the octonion frequencyresponse of a system as the OFT of impulse response. It is worth mentioningthat the notion of multiplication in F can be used to reduce parallel, cascadeand feedback connections of linear systems into simple algebraic equations,as in classical system theory.


5.5 Multidimensional linear time-invariant systems

It is a well-known fact that the Fourier transform converts differential equa-tions into algebraic equations. While the advantages of this approach in the1-D case are obvious, in the case of partial derivatives the classic approachhas some limitations.

Consider a function u : R3 → R that is even w.r.t. all variables (makingboth classical FT and OFT R-valued functions). The classical Fourier trans-form of ux1x2

is −U(f)·(2πf1)(2πf2), which is a R-valued function. Therefore,we loose the information that the function u was differentiated at all. On theother hand, the OFT of ux1x2 is U(f1,−f2,−f3) · (2πf1)(2πf2)e3, which isO-valued (purely imaginary). This information indicates that the functionhas been differentiated by x1 and x2.

As a direct consequence of Theorem 5.2, every linear PDE with constantcoefficients (i.e. every 3-D LTI system of PDEs) can be reduced to algebraicequation (with respect to multiplication in F). Consider the heat equation in2-D, i.e.

ut(t, x1, x2) = ux1x1(t, x1, x2) + ux2x2(t, x1, x2) + f(t, x1, x2),

where we get((2πf1)2 + (2πf2)2 + (2πτ)e1

) U(τ, f1, f2) = F (τ, f1, f2).

It is easy to show that((2πf1)2 + (2πf2)2 + (2πτ)e1

)−1exists if and only if

(τ, f1, f2) 6= (0, 0, 0) and is equal to

((2πf1)2 + (2πf2)2 + (2πτ)e1

)−1=

(2πf1)2 + (2πf2)2 − (2πτ)e1((2πf1)2 + (2πf2)2

)2+ (2πτ)2

.

Hence

U(τ, f1, f2) =(2πf1)2 + (2πf2)2 − (2πτ)e1((2πf1)2 + (2πf2)2

)2+ (2πτ)2

F (τ, f1, f2).

We have thus obtained a simple formula for the system’s response to thegiven stimulation. What’s more, it wouldn’t be possible using multiplicationin O.

5.6 Final remarks

Presented results further develop the foundation of octonion-based signal andsystem theory. At the moment we are left to find real-life applications of the


discussed theory. The results published in recent articles suggest that this isfeasible, e.g. in the field of multispectral image processing [5,7,9]. However, itwould be necessary to focus on the implementation of numerical algorithmsfor this purpose. It seems that extending octonion-based signal theory todiscrete-variable signals may also be achieved by methods used so far.

Acknowledgements The research conducted by the author was supported by NationalScience Centre (Poland) grant No. 2016/23/N/ST7/00131.

References

1. B laszczyk, L., Snopek, K.M.: Octonion Fourier Transform of real-valued functions of

three variables – selected properties and examples. Signal Process. 136, 29–37 (2017)2. B laszczyk, L.: Octonion Spectrum of 3D Octonion-Valued Signals – Properties and

Possible Applications, Proc. 2018 26th European Signal Processing Conference (EU-

SIPCO), 509–513 (2018)3. Ell, T.A.: Quaternion-Fourier transforms for analysis of 2-dimensional linear time-

invariant partial-differential systems. Proc. 32nd IEEE Conf. on Decision and Controll1–4, 1830–1841 (1993)

4. Ell, T.A., Le Bihan N., Sangwine S.J.: Quaternion Fourier Transforms for Signal and

Image Processing. Wiley-ISTE (2014)5. Grigoryan, A.M., Agaian, S.S.: Quaternion and Octonion Color Image Processing with

MATLAB. SPIE (2018)

6. Hahn, S.L., Snopek, K.M.: The Unified Theory of Complex and Hypercomplex Ana-lytic Signals. Bull. Polish Ac. Sci. Tech. Sci. 59 (2), 167–181 (2011)

7. Lazendic, S., De Bie, H., Pizurica, A.: Octonion Sparse Representation for Color

and Multispectral Image Processing. Proc. 2018 26th European Signal ProcessingConference (EUSIPCO), 608–612 (2018)

8. Popa, C. A.: Global exponential stability of octonion-valued neural networks with

leakage delay and mixed delays. Neural Networks 105, 277–293 (2018)9. Sheng, H., Shen, X., Lyu, Y., Shi, Z., Ma, S.: Image splicing detection based on

Markov features in discrete octonion cosine transform domain. IET Image Processing12 (10), 1815–1823 (2018)

Chapter 6

Local time stepping method fordistrict heating networks

Matthias Eimer, Raul Borsche and Norbert Siedow

Abstract In this article, we present a numerical solver for simulating districtheating networks. The method applies a local time stepping to networks oflinear advection equations. Numerical diffusion as well as the computationaleffort on each edge is reduced significantly. The combination with high ordercoupling and reconstruction techniques leads to a very efficient scheme.

6.1 Introduction

District heating is an efficient alternative to conventional heating systems, es-pecially in urban regions. The transport medium water is heated in a centralplant and distributed to the consumers through a network of pipes. Thereare systems for any common energy source e.g. fossil fuel, biomass and solarenergy. In combination with a power generator in so called CHPs (combinedheat and power) the systems have much higher energy efficiency and lesspollution than local boilers. In order to find an optimal control for such sys-tems, fast and accurate simulations are needed. In the following we presenta local time stepping scheme for district heating networks. In section 6.2 thefull model for the system is presented. After restricting ourselves to the com-putation of the energy transport, we present the new scheme in chapter 6.3.Additional insight is given to some special cases such as high order extension

Matthias Eimer

TU Kaiserslautern, Erwin-Schrodinger-Str. 1, 67663 Kaiserslautern,

e-mail: [email protected]

Raul BorscheTU Kaiserslautern, Erwin-Schrodinger-Str. 1, 67663 Kaiserslautern,

Norbert Siedow

Fraunhofer Institute for Industrial Mathematics ITWM

43

44 Matthias Eimer, Raul Borsche and Norbert Siedow

and the incorporation of source terms. In chapter 6.4 we discuss the resultsof the new scheme compared to a high order ADER scheme [7] [1].

6.2 Model

The behavior of density ρ, velocity v and energy density e of the transportmedium water inside a pipe is described by the Euler equations. Since thepressure level always keeps the water in its liquid phase, we can assumeincompressibility. The remaining system then reads

∂xv = 0

∂tv +1

ρ∂xp = − λ

2dv|v| − g∂xh

∂te+ v∂xe = −4k

d(T (e)− T∞) .

(6.1)

Here, p is the pressure inside the pipe, d its diameter and λ the frictionfactor according to the Darcy-Weisbach friction law. The term g(∂xh) takesthe vertical elevation h into account with the gravitational acceleration g.The right hand side of the third equation models the energy loss to theenvironment, where k is the heat transmission coefficient of the pipe, T (e) isthe fluid temperature depending on its energy density e and T∞ is the externaltemperature. In a district heating network, the water is distributed through asystem of pipes. This is modeled by connecting above equations via suitablecoupling conditions. They state the conservation of mass and energy in everynode of the network. Furthermore, we assume a perfect mixture of incomingflows in a node resulting in all outflows to have the same temperature. Finally,the pressure level in a node has to coincide for all adjacent pipes. In order toclose the model, as boundary conditions we set inflow temperature T0(t) andpressure level p0(t) at the CHP. The full system then has the following form

6 Local time stepping method for district heating networks 45

HYD

ODE

∂xv

i = 0

∂tvi +

1

ρ∂xp

i = − λ

2divi|vi| − g(∆bi)

CC

∑j∈J

Ajvjρj = 0

pi = pj , for i, j adjacent

BC p(0) = p0(t)

(6.2)

CONSUMERAkvk(e(T k)− e(Tout) = Qk(t) (6.3)

ENERGY

PDE

∂te

i + vi∂xei = −4k

di(T (ei)− T∞

)CC

∑j∈J

Ajvjej = 0

T (ei) = T (ej), for i, j outgoing flows

BCT (e1(x = 0, t)) = T0(t)

(6.4)

Equation (6.2) describes the hydraulics of the system. The first equationstates the incompressibility, the second one is the balance of momentum.The coupling conditions (CC) state conservation of mass in every node aswell as equality of pressure for all connected edges to a node. The superscripti ∈ I indicates the specific edge, the quantities belong to, where I is theset of all edges. As boundary condition (BC) the pressure at the CHP isprescribed. Equation (6.3) describes the coupling between the hydraulics andthe energy transport at consumer site, where A is the pipe’s cross section,Tout is a fixed temperature level to which the water is cooled down and Qk(t)is the power demand of the consumer k. Equation (6.4) formulates the energytransport in the network with the advection PDE. The coupling conditionsstate conservation of energy inside the nodes and perfect mixture of energiestherein. As boundary condition, the temperature at the inflow is prescribed.

6.3 Numerical Method

The full model for district heating networks (6.2)-(6.4) is a complex systemof algebraic and partial differential equations. For its numerical solution, weuse a splitting algorithm, i.e. for a given time t we first compute the flowwith (6.2) and (6.3) for fixed temperature in time. Afterwards we update theenergy using the new velocities. Such splitting reduces the accuracy of thefull system to first order. By exploiting the special structure of the network,the flow can be solved efficiently. In the following, we focus on solving theenergy equation (6.4). When a flow solver is needed, the method of [4] is used.


The time step of the splitting is chosen according to the fastest waves in theenergy model.

The evolution of energy density in the network is described by a networkof linear scalar balance laws. For solving this kind of problems the Godunovscheme is commonly used, which in the linear case coincides with the Upwindscheme. Furthermore, there are some recent extensions to higher order meth-ods for network of hyperbolic conservation laws [1]. All these classical schemeshave in common, that they use a global time step for the whole system. Thistime step is determined by the minimal CFL bound on all edges

∆tnet = mini∆ti (6.5)

where ∆ti is the maximal time step of edge i. However, since the error of theschemes scales with the local CFL number, and the relative flow velocitiesbetween different pipes can strongly vary this may lead to large numericaldiffusion.

6.3.1 Local Time Stepping

Motivated by above consideration we construct a Upwind-like scheme thatdecouples the time steps of every edge, such that the locally optimal timesteps can be chosen [2] [6]. Note that an optimal CFL number can also beachieved with adaptive spacial discretization, however the remeshing andinterpolation would be very costly, especially in the context of high ordermethods. Furthermore, we restrict ourselves to the homogeneous case, theextension to source terms is treated in section 6.3.3.The time step of an edge is chose according to∫ t+∆t

t

v(τ)dτ = ∆x . (6.6)

Therefore the CFL number is equal to 1 for every local time step. Note thatthe velocity is piecewise constant in the considered time interval due to thedifferent time step of the splitting. As with this definition the solution travelsexactly one cell each time step, no computation for inner cells is needed.However, the update can only be performed, if the fluxes over the edge’sboundaries are known for this time interval. In other words, the future timelevel of the current pipe can not exceed the future time levels of all adjacentedges, or to be more precise, current edge ei has to fulfill

ti +∆ti ≤ tk +∆tk (6.7)

for all neighboring edges ek. Whenever a pipe fulfills this condition, the nu-merical flux over its boundaries have to be computed. Note that the ad-


jacent edges have already computed parts of the flux for some subinterval[ti, tk] ⊆ [ti, ti + ∆ti]. In this case we store the flux (or its polynomial coef-ficients in the high order case, see section 6.3.2) in memory variables. Thecurrent pipe then just computes the remaining flux for the interval [tk, ti+∆ti]and the cell values are shifted in the direction of the flow, while the memoryvariable is emptied into the first cell. This procedure is continued, until a finaltime is reached. Since in general, the time steps do not exactly add up to thefinal time a classical upwind is performed in the last step. The procedure isschematically shown in Fig. 6.1 for a node connecting 3 edges.

e1e2

e3

t

e1e2

e3

t

e1e2

e3

t

e1e2

e3

t

Fig. 6.1: Illustration of the local time steps at a single node.

6.3.2 High Order Coupling

We can further increase the accuracy by incorporating a high order couplingat the nodes of the network. Therefore, we use a high order WENO recon-struction [5] and instead of cell means, we store the polynomial coefficients.The coupling conditions are formulated such that not only the energy density,but also its moments are conserved as well as the equality of all moments ofthe outgoing temperature up to a given order. In an update step, we then shiftthe polynomial coefficients instead of means. When the memory variables arecleared into the first cell of a pipe, we need to get a single polynomial repre-sentation out of several piecewise polynomials. This is done by solving a leastsquares problem for the new coefficients under the condition that the totalmass must be contained [3]. In most cases during the computation only few(1-3) piecewise polynomials have to be combined, such that the numericaleffort is lower than for storing means and using WENO reconstructions ineach step.


6.3.3 Source Term

When incorporating the source term, we can exploit the fact that its coef-ficients are constant in space. When tracing a characteristic of the energyevolution, the change of the energy follows an ODE of the form eτ = S(e).The evaluation of the source term does not necessarily be performed in everylocal update, but only when a given cell leaves the edge. We therefore keeptrack of the timespan each cell spends inside the given pipe (by integratingv) and solve the ODE for this whole interval just once.

10−3 10−2 10−1

10−9

10−7

10−5

10−3

10−1

∆x

e

ADER1ADER2ADER3ADER4ADER5LT1LT2LT3LT4LT5

10−1 100 101 102 103 104

10−9

10−7

10−5

10−3

10−1

t

e

ADER1ADER2ADER3ADER4ADER5

LT1LT2LT3LT4LT5

Fig. 6.2: Convergence plot for different schemes.

6.4 Results and Conclusion

In several simulations, the local time stepping scheme has been comparedto a high order ADER scheme with global time steps (see Fig. 6.2). Whenthe velocity is set constant, the expected convergence rates of the schemesare reached. The overall error of the local time stepping scheme is lowerbecause there is no numerical diffusion inside a pipe. The error only consistsof the part arising from the coupling of different pipes and the part thatdepends on the resolution of the input signal. When comparing computationtimes, we notice a large difference between the two schemes. The advantageof the local time stepping is that in one pipe update, only O(1) operationsare needed, while for classical finite volume schemes you need O(n), for thenumber of cells n. The overall computation time of the local time steppingtherefore only scales linearly with the number of cells vs. quadratically forthe ADER scheme. For more realistic settings, where a flow solver is involved


only first order convergence can be expected. The errors of the local timestepping scheme are below those of the ADER scheme, but the gain in termsof computation time is not as big since the flow solver, which is identical forboth schemes takes significant amount of time. To conclude, we constructeda numerical scheme that applies local time steps on the edges of the network.The advection inside the pipes is solved exactly which results in increasingaccuracy and computational efficiency.

Acknowledgements This research was supported by Verbundprojekt 05M2018 - EiFer:

Energieeffizienz durch intelligente Fernwarmenetze. 05M18AMB - 810303892568

References

1. Borsche, R., Kall, J.:ADER schemes and high order coupling on networks of hyperbolic

conservation laws. Journal of Computational Physics 273, 658–670 (2014)

2. Dumbser, M., Kaser, M., Toro, E.F.:An arbitrary high-order Discontinuous Galerkinmethod for elastic waves on unstructured meshes - V. Local time stepping and p-

adaptivity. In: Geophysical Journal International 171 695–717 (2007)3. Dumbser, M., Zanotti, O., Loubere, R., Diot, S.: A posteriori subcell limiting of the

discontinuous Galerkin finite element method for hyperbolic conservation laws. In:

Journal of Computational Physics 278 47–75 (2014)4. Jansen, L., Pade, J.:Global unique solvability for a quasi-stationary water network

model. In: Preprint series: Institut fur Mathematik, Humboldt-Universitat zu Berlin

(2013).https://www.mathematik.hu-berlin.de/de/forschung/pub/P-13-11

5. Jiang, G., Shu, C.: Efficient Implementation of Weighted ENO Schemes. In: Journal

of Computational Physics 126, 202–228 (1996)6. Muller, L.O., Blanco, P.J., Watanabe, S.M, Feijoo, R.A.:A high-order local time step-

ping finite volume solver for one-dimensional blood flow simulations: application to

the ADAN model. In:International Journal for Numerical Methods in Biomedical En-gineering 32 e02761, 36 (2016)

7. Toro, E.F., Millington, R.C., Nejad, L.A.M.: Towards very high order Godunov

schemes. Godunov methods (Oxford, 1999) 907–940, Kluwer/Plenum, New York(2001)

Chapter 7

Modelling Time-of-Flight transientcurrents with time-fractional diffusionequations

M. Luısa Morgado and Luıs F. Morgado

Abstract In this work we explore the use of tempered fractional derivativesin the modelling of transient currents in disordered materials. We partic-ularly focus on the numerical approximation of the involved problems. Asit is known, the solutions of fractional differential equations usually exhibitsingularities in the origin in time, and therefore, a decreasing of the con-vergence order of standard numerical schemes may be expected. In order toovercome this, we propose a finite difference scheme on a time graded mesh,in which the grading exponent can be properly chosen, taking into accountthe singularity type. Numerical results are presented and discussed.

7.1 Introduction

Since the 1960’s there is an increasing interest and research on organicsemiconductors, due to their particular characteristics (transparency, flexi-bility, low cost), as a material for the fabrication of optoelectronic devices,such as organic solar cells, light emitting diodes and light emitting transis-tors. The charge carrier mobility µ of these materials is one of the mainproperties of interest and the Time-of-Flight (TOF) technique is one of thepreferred methods to estimate it. In the TOF experiment, a transient cur-

M. Luısa Morgado

Center for Computational and Stochastic Mathematics (CEMAT), Lisbon and Departa-mento de Matematica, Universidade de Tras-os-Montes e Alto Douro, UTAD, Quinta de

Prados 5001-801, Vila Real, Portugal, e-mail: [email protected]

Luıs F. Morgado

Instituto de Telecomunicacoes, Lisboa, and Departamento de Fısica, Universidade de Tras-os-Montes e Alto Douro, UTAD, Quinta de Prados 5001-801, Vila Real, Portugal, e-mail:

[email protected]

51

52 M. Luısa Morgado and Luıs F. Morgado

rent I(t) through a thin layer of material sandwiched between two parallelelectrodes is obtained, as a result of the motion of excess charge carriers gen-erated by a laser or voltage pulse, under the influence of an externally appliedelectric field E directed normally to the electrodes. These transient currentsusually exhibit an anomalous dispersive character ( [5]) with two regions withpower-law behavior, separated by the transit time ttr : ∼ t−1+α, if t < ttrand ∼ t−1−α, if t > ttr with 0 < α < 1. An estimate for µ is calculated fromttr, the instant when the two power-law curves intersect. Such behavior isattributed to the trapping of carriers, in localized states distributed in themobility gap, for times τ , or waiting times, determined by a relaxation func-tion Ψ(τ) with an asymptotic time dependence of the form Ψ(τ) ∼ τ−α.Diffusion-advection equations have been widely used to describe the evolutionof carrier density in the materials, but it is known that in the case of disor-dered materials, integer order models do not describe accurately the process( [7]) and that is the reason why in the latest decades the use of fractionaltime derivatives have been proposed to improve those models. For example,in [3] the following model was considered:

Dαt y(x, t) = −v ∂y(x, t)

∂x+D

∂2y(x, t)

∂x2+ f(x, t), t ∈ (0, T ], x ∈ (0, L), (7.1)

with initial condition

y(x, 0) = g(x), x ∈ (0, L), (7.2)

and boundary conditions

y(0, t) = φ0(t), y(L, t) = φL(t), t ∈ (0, T ], (7.3)

where 0 < α < 1 and the fractional derivative is of the Caputo type which,for the considered values of α, is given by ( [2]):

Dαt y(t) =

1

Γ (1− α)

∫ t

0

(t− s)−αy′(s) ds. (7.4)

It was assumed that v > 0 is the average fluid velocity, D > 0 is the diffusioncoefficient and g, f , φ0 and φL are continuous functions in their respectivedomains.

Here we consider the general model (which obviously reduces to (7.1)-(7.3)in the case where λ = 0 with v(x) ≡ v > 0 and D(x) ≡ D > 0):

Title Suppressed Due to Excessive Length 53

Dα,λt u(x, t) =∂

∂x

(−v(x)u(x, t) +D(x)

∂u(x, t)

∂x

)+ f(x, t), in(0, T ]× (0, L),(7.5)

u(x, 0) = g(x), x ∈ (0, L), (7.6)

u(0, t) = φ0(t), u(L, t) = φL(t), t ∈ (0, T ], (7.7)

where Dα,λt u(x, t) is the tempered Caputo derivative with respect to the vari-able t of the function u(x, t) ( [1]):

Dα,λt (u(t)) = e−λtDαt

(eλtu(t)

)(7.8)

=e−λt

Γ (1− α)

∫ t

0

1

(t− s)αd(eλsu(s)

)ds

ds, 0 < α < 1, λ ≥ 0.

Note that if in the equation above, we consider λ = 0, the definition of theusual Caputo derivative (7.4) is recovered.

The total measured current I(t), produced by the extraction of carriersfrom the space between the electrodes, placed at x = 0 and x = L, is given( [4]) by the space average of the current density j(x, t), and since

j(x′, t) = − d

dt

∫ x′

0

qu(x, t)dx, (7.9)

where q is the carrier electrical charge, we obtain

I(t)

q= − d

dt

∫ L

0

(L− x)u(x, t)dx. (7.10)

7.2 Numerical method and results

In order to approximate the solution of (7.5)-(7.7), we first take (7.8) intoaccount, and note that (7.5) can be written as

Dαt

(eλtu(x, t)

)= −

∂(eλtv(x)u(x, t)

)∂x

+∂

∂x

(D(x)

∂(eλtu(x, t)

)∂x

)+eλtf(x, t).

Therefore, if we consider the function y(x, t) = eλtu(x, t), and we determinethe solution y(x, t) of problem:

Dαt y(x, t) = −∂ (v(x)y(x, t))

∂x+

∂

∂x

(D(x)

∂y(x, t)

∂x

)+ eλtf(x, t),(7.11)

y(x, 0) = g(x), x ∈ (0, L), (7.12)

y(0, t) = eλtφ0(t), y(L, t) = eλtφL(t), t ∈ (0, T ], (7.13)


then the solution of (7.5)-(7.7) is obtained through u(x, t) = e−λty(x, t).Therefore, we first approximate the solution of problem (7.11)-(7.13), byadapting the numerical scheme proposed in [3]. We now briefly describe thenumerical approach.We consider a uniform mesh in the interval [0, L], defined by the grid-pointsxi = ih, i = 0, 1, . . . ,K, where h = L

K , and we use the following second orderfinite difference approximations:

∂ (v(x)y(x, t))

∂x|x=xi

≈ v(xi+1)y(xi+1, t)− v(xi−1)y(xi−1, t)

2h, (7.14)

∂

∂x

(D(x)

∂y(x, t)

∂x

)|x=xi

≈ 1

h2

(D

(xi +

h

2

)y(xi+1, t)− (7.15)

−(D

(xi +

h

2

)+D

(xi −

h

2

))y(xi, t) +

+ D

(xi −

h

2

)y(xi−1, t)

), i = 1, . . . ,K − 1.

For the numerical approximation of the Caputo derivative of order α on theinterval [0, T ], we will use the non-uniform mesh

ti =

(i

n

)rT, (7.16)

where r ≥ 1 is the so-called grading exponent. The length of each one of theintervals defined with this partition is:

τi = ti+1 − ti =(i+ 1)r − ir

nrT, i = 0, 1, . . . , n− 1.

Note that if r = 1 we obtain a uniform mesh. We then use the followingapproximation for the Caputo derivative (see [3]):

Dαy(tk) ≈ 1

Γ (2− α)

k−1∑j=0

τ−αj aj,k (y(tj+1)− y(tj)) = Dαyk, (7.17)

where

aj,k =

(kr − jr

(j + 1)r − jr

)1−α

−(kr − (j + 1)r

(j + 1)r − jr

)1−α

, j = 0, 1, . . . , k−1, k = 1, . . . , n.

(7.18)Concerning the order of the approximation we have (see [6]):∣∣∣Dα

t y(tk)− Dαyk

∣∣∣ ≤ Ck−min2−α,rα,


which gives us an information about the proper choice of the grading exponent(see [6]).

Using (7.17), we obtain:

Dαt y(xi, tl) ≈

1

Γ (2− α)

l−1∑j=0

τ−αj aj,l (y(xi, tj+1)− y(xi, tj)) , (7.19)

i = 1, . . . ,K−1, l = 1, . . . , n, where the coefficients aj,l are defined in (7.18).Denoting by Y li ≈ y(xi, tl), f

li = f(xi, tl), D

(xi ± h

2

)= Di± 1

2, and substi-

tuting (7.14), (7.15) and (7.19) in (7.11), we obtain the following implicitnumerical scheme:

1

Γ (2− α)

l−1∑j=0

τ−αj aj,l

(Y j+1i − Y ji

)=Di+ 1

2Y li+1 −

(Di+ 1

2+Di− 1

2

)Y li +Di− 1

2Y li−1

h2

−vi+1Y

li+1 − vi−1Y li−1

2h+ eλtlf li , i = 1, . . . ,K − 1, l = 1, . . . , n, (7.20)

where, according to the initial and boundary conditions (7.12) and (7.13), wehave

Y 0i = g(xi), i = 1, . . . ,K − 1,

Y l0 = eλtlφ0(tl), Y lK = eλtlφL(tl), l = 1, . . . , n.

After having determined the unknowns Y li , i = 1, . . . ,K−1, l = 1, . . . , n, thesolution of (7.5)-(7.7) at the mesh-points will be given by u(xi, tl) ≈ U li =e−λtlY li .

Figures 7.1 to 7.3 present some numerical simulations for transient cur-rents, considering a narrow (gaussian profile) of photogenerated carriers, atthe position x = 0.2 of a layer of thickness L = 1. Both advection and diffu-sion coefficients are constants: v = 0.01 and D = 0, for Fig. 1 and v = 0.01and D = 0.01, for Fig. 2.

In Fig. 7.3 we show the current behavior when the carrier velocity hastwo different values along the the material layer, v = 0.001 in the first halfand v = 0.05 in the second half, in a diffusion-less situation, D = 0. Thesenumerical results are in agreement with the analytical results related to theTOF experiments in [8]. Since in most of the cases, the analytical solution isnot known, numerical methods are necessary and the one presented here isable to model many situations. In a forthcoming paper, we will prove that thisnumerical scheme is stable and convergent, namely with convergence ordersof (2 − α) and 2, time and space, respectively, with a proper choice of thegrading exponent.


Fig. 7.1: Transient current for α = 0.5(left) and α = 0.75(right), with λ = 0(tiny dash),

λ = 0.001(small dash), λ = 0.01(medium dash), λ = 0.1(large dash) and λ = 1(solid);v = 0.01 and D = 0.

Fig. 7.2: As in Fig. 7.1, with λ = 0(tiny dash), λ = 0.001(small dash), λ = 0.01(medium

dash) and λ = 0.1(large dash); v = 0.01 and D = 0.01.

Fig. 7.3: As in Fig. 7.1, for two layers material with λ = 0(tiny dash), λ = 0.001(smalldash) and λ = 0.01(medium dash); v = 0.001 in the first layer and v = 0.05 in the second;

D = 0 in both layers.

Acknowledgements The authors acknowledge the Fundacao para a Ciencia e a Tecnolo-

gia (Portuguese Foundation for Science and Technology) through projects UID/Multi/04621/2013and PEst-OE/EEI/LA0008/2013, respectively.


References

1. Baeumer, B., Meerschaert, M.M.: Tempered stable Levy motion and transient super-

diffusion. J. Comput. Appl. Math. 233, 2438–2448 (2010).2. Diethelm, K.: The analysis of fractional differential equations: An application-oriented

exposition using differential operators of Caputo type. Springer-Verlag, Berlin (2010).3. Morgado, L.F., Morgado, M.L.: Numerical modelling transient current in the time-

of-flight experiment with time-fractional advection-diffusion equations. J. of Math.

Chem. 53, 958–973 (2015).4. Philippa, B.W., White, R.D., Robson, R.E.: Analytic solution of the fractional

advection-diffusion equation for the time-of-flight experiment in a finite geometry.

Phys. Rev. E 84, 041138 (2011).5. Scher, H., Montroll, E.: Anomalous transit-time dispersion in amorphous solids. Phys.

Rev. B 12 6, 2455–2477 (1975).

6. Stynes, M., O’Riordan, E., Gracia, J.L.: Error analysis of a finite difference methodon graded meshes for a time-fractional diffusion equation. SIAM J. Numer. Anal., 55,

1057–1079 (2017).

7. Uchaikin,V.V., Sibatov, R.T.: Fractional Kinetics in Solids: Anomalous Charge Trans-port in Semiconductors, Dielectrics and Nanosystems. World Scientific (2012).

8. Sibatov, R.T., Morozova, E.: Tempered Fractional Model of Transient Current inOrganic Semiconductor Layers, in Theory and Applications of Non-integer Order

Systems, Springer International Publishing, 287-295 (2017).

Chapter 8

Damage detection in thin plates viatime-harmonic infrared thermography

Manuel Pena and Marıa-Luisa Rapun

8.1 Statement of the problem

Infrared thermography has become a powerful tool for non-destructive testingin a wide rank of applications, ranging from medical imaging, to buildingand material diagnosis. In this work we aim at finding small interior defectsinside metallic plates by processing time-harmonic thermograms, which areobtained after heating the plate to be inspected by a time-harmonic excitationfrom one lamp at the same side of the sample where the thermogram is taken,see Figure 8.1.

The plate R ⊂ Rd (where d = 2 or 3) is assumed to be surrounded by airat room temperature, Tair, with whom it exchanges heat by radiation andnatural convection. The convection coefficient h is assumed to be constantand the surface of the plate is modeled as a gray body with absorptance αand emissivity ε. The lamp is modeled as a point source which radiates ina time-harmonic manner with an amplitude I and frequency ω. The defectsconform a region D ⊂ R. For simplicity, we assume the thermal conductivityκ, the density ρ and the specific heat capacity c to be piecewise constantfunctions, i.e.:

κ (x) =

κe x ∈ R \ Dκi x ∈ D

, ρ (x) =

ρe x ∈ R \ Dρi x ∈ D

, c (x) =

ce x ∈ R \ Dci x ∈ D

.

(8.1)

Manuel Pena

Universidad Politecnica de Madrid, Pza. Cardenal Cisneros, 28040 Madrid, Spain, e-mail:[email protected]

Marıa-Luisa RapunUniversidad Politecnica de Madrid, Pza Cardenal Cisneros, 28040 Madrid, Spain, e-mail:

[email protected]

59

60 Manuel Pena and Marıa-Luisa Rapun

Fig. 8.1 Layout of the

experiments: The plate Ris defined as a bounded

box region of R2 or R3,

with one of its dimensionsmuch smaller than the

remaining ones. The lamp

and the thermal camera arelocated at the same side of

the plate. The illuminated

side is denoted as Γfront,and the opposite side is

Γback. The remaining sides,

much smaller in area, aredenoted as Γsides. The angle

between the incoming lightrays and the normal n is

denoted as θinc.

Then, the complex amplitude T (x) of the time-harmonic temperature distri-

bution T (x, t) = T (x) + Re (T(x)e−ıωt) (where T (x) is a steady mean value)satisfies the set of equations:

div (κe∇T ) + iωρeceT = 0 in R \ Ddiv (κi∇T ) + iωρiciT = 0 in DT+ − T− = 0 on ∂Dκe∂nT

+ − κi∂nT− = 0 on ∂Dκe∂nT = 0 on Γsides

κe∂nT +AT = −α qs on Γfront

κe∂nT +AT = 0 on Γback

(8.2)

where A = h+ 4εσT 3air is a constant that takes into account both the effects

of convective terms and a linearization around Tair of the radiative terms.The heat qs (x) coming from a lamp located at a point s is modeled by

qs (x) =I

2π

cos θinc|x− s|

orI

4π

cos θinc|x− s|2

(8.3)

for the two-dimensional and three-dimensional cases respectively.Given a thermogram Tfront, that is, a measurement of the temperature

distribution along Γfront for a given experiment, we would like to obtain thedomain Dapp such that TDapp

(x) = Tfront (x), for all x ∈ Γfront, where TDapp

stands for the solution of the set of equations (8.2) setting D = Dapp. Sinceexperimental errors of very different nature are expected, we will consider aless demanding constraint and seek for Dapp such that the functional

8 Damage detection in thin plates via time-harmonic infrared thermography 61

J(R \ Dapp

)=

∫Γfront

∣∣TDapp (x)− Tfront (x)∣∣2 d` (8.4)

attains a global minimum. This will be done by computing its topologicalderivative, defined in the next section.

8.2 Topological derivative

The topological derivative (TD in the sequel) of the shape functional J at apoint x measures the sensitivity of such functional to locating an infinitesimalball Bε(x) of radius ε > 0 at x, providing the asymptotic expansion (see [6]):

J(R \ Bε (x)

)= J (R) + DT (x) f (ε) + o (f (ε)) as ε→ 0+. (8.5)

where f (ε) is a positive increasing function chosen such that the expansion(8.5) holds. In our case, we can take f (ε) to be the measure of Bε(x). In viewof expansion (8.5), the points where DT attains large negative values are themost effective in minimizing the functional (8.4), and therefore, our guess ofDapp will be defined as [1, 5]:

Dapp :=

x ∈ R; DT (x) < λmin

y∈RDT(y)

, (8.6)

where 0 < λ < 1 is a parameter that can be tuned.Formula (8.5) is not practical from the numerical point of view. Adapting

the results in [1, 2, 4], we obtain a closed-form formula for the TD: for allx ∈ R,

DT (x) = Re

(dκe(κe − κi)

(d− 1)κe + κi∇T 0 (x)·∇V 0 (x)−iω (ρece − ρici)T 0 (x)V 0 (x)

),

(8.7)where T 0 is solution to the direct problem

div(κe∇T 0

)+ iωρeceT

0 = 0 in Rκe∂nT

0 = 0 on Γsides

κe∂nT0 +AT 0 = −α qs on Γfront

κe∂nT0 +AT 0 = 0 on Γback

(8.8)

and V 0 is solution to its associated adjoint problem

62 Manuel Pena and Marıa-Luisa Rapundiv(κe∇V 0

)− iωρeceV 0 = 0 in R

κe∂nV0 = 0 on Γsides

κe∂nV0 +AV 0 = Tfront − T 0 on Γfront

κe∂nV0 +AV 0 = 0 on Γback

. (8.9)

As can be seen, for the computation of the TD no a priori information isneeded about the number or size of defects, as both T0 and V0 are defined onthe plate without any defect. The adjoint problem compares the thermogramexpected at a healthy plate T 0 with the measured thermogram Tfront.

In general, we will have several experiments with the lamp at a numberNlamps of different positions si and a number Nfreq of different frequenciesωj . In that case, we replace the functional (8.4) by a functional of the form

J(R \ Dapp

)=

Nlamps∑i=1

Nfreq∑j=1

pij

∫Γfront

∣∣∣T (i,j)Dapp

(x)− T (i,j)front (x)

∣∣∣2 d`, (8.10)

where the superscripts stand for the different configurations (namely T (i,j)

corresponds to the temperature associated with the i-th position of the lampand the j-th frequency), and pij > 0 are weights that can be tuned. Bylinearity, the TD of (8.10) is nothing but the linear combination of eachindividual derivative. The weights pij are defined in terms of the inverse ofthe largest negative value of each individual TD, as done in [3].

8.3 Numerical experiments

In this section we present a couple of numerical experiments. The followingparameters model an aluminum plate with air defects:

• κe = 200 W/(m ·K), ρe = 2700 Kg/m3 and ce = 900 J/(Kg ·K)• κi = 0.025 W/(m ·K), ρi = 1 Kg/m3 and ci = 1000 J/(Kg ·K)• α = 0.4, ε = 0.08, Tair = 290 K and h = 15 W/(m2 ·K)

The lamp will have an amplitude I = 6000 W and will be located at a distance0.15 m from the plate.

Given that the thermograms are no actual measures but simulated data, agaussian random error is added to them to simulate noisy experimental data,see [5] for further details about the generation of such error.

First, we present a two dimensional example, where the plate is the box[0, `x] × [0, `y] with `x = 0.01 m and `y = 1 m, and contains three differentdefects: an elliptical hole located at (0.5`x, 0.25`y) and semi-axis of 0.1`xand 0.3`y, a circular hole located at (0.6`x, 0.6`y) with radius 0.125`x, and acircular hole located at (0.4`x, 0.8`y) with radius 0.1`x.


In Figure 8.2 we represent the TD for two different data sets. The truedefects have been superimposed in white. The plate is distorted in the non-zoomed drawing for a better visualization. The TD is normalized in such away that its largest negative value is equal to −1 for an easier comparisonbetween both experiments. It can be seen that the TD accurately pinpointthe position, size and number of defects (regions in blue) even for a relativehigh noise level. However it is unable to provide the correct depths, sincethe largest negative values are always attained in regions close to Γfront. Thebiggest region corresponds to the elliptical hole, which is bigger in size. Recon-structions correlate not only with size but also with depth. We observe thatthe smaller and less deep circular hole is better identified than the remain-ing one, which is bigger but is farther located from Γfront. When comparingboth figures, we see that for a fixed number of thermograms, reconstructionsare sharpener when thermograms correspond to both several locations of thelamp and several excitation frequencies.

Fig. 8.2: Left: TD for Nlamps = 48 different lamp positions marked as × at frequency of

1 Hz. Right: TD for Nlamps = 12 lamp positions at Nfreq = 4 linearly spaced frequenciesbetween 0.8 and 2 Hz. The level of noise in the thermograms for both experiments is 20%.

In Figure 8.3 we represent the TD at Γfront to better visualize the sharpnessof the minima. The sharper the minimum the less dependent is the recon-struction on the parameter λ in (8.6). In the same figure we also representthe (rotated) plate where the three true holes are in white and the recon-structed holes corresponding to different values of λ are shown in differentcolor regions.

To illustrate the performance of the method in the three dimensional case,we consider now the plate [− `x2 ,

`x2 ] × [− `y2 ,

`y2 ] × [− `z2 ,

`z2 ] with `x = 0.5 m,

`y = 0.01 m and `z = 1 m, which contains two different defects: a spheri-

64 Manuel Pena and Marıa-Luisa Rapun

0.0 0.2 0.4 0.6 0.8 1.0−1.0

−0.8

−0.6

−0.4

−0.2

0.0

0.2

0.4

DT(0, y)λ=0.25λ=0.5λ=0.75

0.0 0.2 0.4 0.6 0.8 1.0y(m)

0.00

0.01x(m)

0.0 0.2 0.4 0.6 0.8 1.0−1.0

−0.8

−0.6

−0.4

−0.2

0.0

0.2

0.4

DT(0, y)λ=0.25λ=0.5λ=0.75

0.0 0.2 0.4 0.6 0.8 1.0y(m)

0.00

0.01x(m)

Fig. 8.3: TD at Γfront and reconstructed defects for several values of λ .

cal hole located at (0, 0, 0) with radius of 0.3`y, and a box hole defined by

[`x−3`y

4 ,`x+3`y

4 ]× [− `y+3`y6 ,

−`y+3`y6 ]× [

`z−6`y4 ,

`z+6`y4 ]

The TD at Γfront for two different data sets is shown in Figure 8.4. In thefirst one, 4 positions and 6 frequencies are combined, and the thermogramscontain a 5% relative error. We identify the position of the rectangular box,however we can barely see the spherical defect. For the second experiment,we combine noisy thermograms with a 10% level of noise, corresponding to 9lamp positions and 6 frequencies. Although thermograms are more polluted,we can clearly identify the position and approximate size of the two defects.However, the method has again problems in detecting the correct depth.More sophisticated (and much more computational costly) iterative methodsusing the TD as a first step can be developed to try to overcome this difficulty.This will be done in future work. We have limited our study to time-harmonicexcitations. The extension to the full time-dependent heat equation could alsoovercome this problem, and will be considered in future.

Fig. 8.4: Left: TD for Nlamps = 4 lamp positions and Nfreq = 6 linearly spaced frequenciesbetween 0.8 and 2 Hz. Right: TD for Nlamps = 9 lamp positions and Nfreq = 6 linearly

spaced frequencies between 0.8 and 2 Hz.


Acknowledgments

The authors are supported by the Spanish Ministry of Economy and Compet-itiveness under the research projects MTM2014-56948-C2-1-P and TRA2016-75075-R.

References

1. Carpio A., Rapun M.L.: Solving inhomogeneous inverse problems by topological

derivative methods. Inverse Problems 24, art. 045014 (2008).2. Carpio A., Rapun M.L.: Hybrid topological derivative and gradient-based methods

for electrical impedance tomography. Inverse Problems 28, art. 095010 (2012).

3. Funes J.F., Perales J.M., Rapun M.-L., Vega J.M.: Defect detection from multi-frequency limited data via topological sensitivity. J. Math. Imaging Vis. 55, 19–35

(2016).

4. Guzina, B.B., Bonnet, M.: Small-inclusion asymptotic of misfit functionals for inverseproblems in acoustics. Inverse Problems, 22(5), 1761–1786 (2006).

5. Pena M., Rapun M.L.: Detecting damage in thin plates by processing infrared ther-

mographic data with topological derivatives. Advances in Mathematical Physics, 4,1–18 (2019).

6. Sokolowski, J., Zochowski, A.: On the topological derivative in shape optimization.

SIAM Journal on Control and Optimization 38, 1251–1272 (1999)

Chapter 9

Reverse Logistics Modelling of AssetsAcquisition in a Liquefied PetroleumGas Company

Cristina Lopes, Aldina Correia, Eliana Costa e Silva, Magda Monteiro, RuiBorges Lopes

Abstract In the business of liquefied petroleum gas (LPG), the LPG cylin-der is the main asset and a correct planning of its needs is critical. This workaddresses a challenge, proposed at an European Study Group with Industryby a Portuguese energy sector company, where the objective was to definean assets acquisition plan, i.e., to determine the amount of LPG cylinders toacquire, and when to acquire them, in order to optimize the investment. Theused approach to find the solution of this problem can be divided in threephases. First, it is necessary to forecast demand, sales and the return of LPGbottles. Subsequently, this data can be used in a model for inventory man-agement. Classical inventory models, such as the Wilson model, determinethe Economic Order Quantity (EOQ) as the batch size that minimizes thetotal cost of stock management. A drawback of this approach is that it doesnot take into account reverse logistics, which in this challenge (i.e. the returnof cylinders) plays a crucial role. At last, because it is necessary to considerthe return rate of LPG bottles, reverse logistic models and closed loop supplychain models are explored.

9.1 Problem Description

This work addresses an industrial challenge that consisted in planning theacquisition of liquefied petroleum gas (LPG) cylinders. The challenge was

Cristina Lopes

LEMA, CEOS.PP, ISCAP - Polytechnic of Porto, Portugal, e-mail: cristi-

[email protected]

Aldina Correia and Eliana Costa e SilvaCIICESI, ESTG - Polytechnic of Porto, Portugal e-mail: aic,[email protected]

Magda Monteiro and Rui Borges Lopes

CIDMA, University of Aveiro, Portugal

67

68 Lopes et al.

proposed at an European Study Group with Industry, by a Portuguese com-pany of the energy sector (named ALPHA for confidentiality reasons) thatstarted its activity in 2006 focusing in the production and distribution of bio-fuel and, since then, has extended its business areas to other fuels and energy.In this company, the LPG cylinder business started in 2012, and since thenit has experienced a continuous growth. ALPHA currently commercializespropane gas in two types of cylinders: type A with capacity 9kg, and type Bwith capacity 45kg.

In Portugal, companies selling LPG cylinders are also responsible for col-lecting the empty cylinders, regardless of the company from which the previ-ous cylinders were bought (direct replacement policy) [7]. The empty bottlesreturned to the company can be reinserted in the system, filled again andsold to the clients. As the acquisition of new bottles is expensive, reusing isthe key. Cylinders are assets owned by the companies: each competitor canonly refill its own cylinders. Companies experiencing growth have to purchaseadditional cylinders to meet demand. The cylinder is the main asset and acorrect planning of its needs is critical.

The industrial challenge was to find a model to forecast the demand andreturn rate of each type of cylinder, and to define an assets acquisition plan,i.e., to determine when to order to the external supplier new LPG bottles(Order Point) and how many should be bought (batch size), in order tooptimize the investment.

9.2 Literature Review

Classical inventory models, such as Wilson’s deterministic model [4,9], deter-mine the Economic Order Quantity (EOQ) as the batch size that minimizesthe total cost of stock management. The total cost is the sum of three com-ponents:

• CA Acquisition Costs (price of acquiring the assets)• CS Setup costs (fixed cost for every order, transportation, collect)• CH Holding costs (insurances, taxes, rent, electricity, salary, opportunity

costs)

Once the forecast of demand, sales and return of LPG cylinders is deter-mined, an EOQ model can be used for inventory management [2].

The EOQ model is an attempt to estimate the best order quantity bybalancing the conflicting costs of holding stock and of placing replenishmentorders. The effect of order quantity on stock-holding costs is that, the largerthe order quantity for a given item, the longer will be the average time instock and the greater will be the storage costs. On the other hand, the placingof a large number of small-quantity orders produces a low average stock, but

9 Reverse Logistics Modelling of Assets Acquisition in a LPG Company 69

a much higher cost in terms of the number of orders that need to be placedand the associated administrative and delivery costs.

Another classical approach is the Continuous Review Policy (s,Q), whichconsiders probabilistic demand.

A drawback of these approaches is that they do not take into accountreverse logistics, which in this challenge (i.e. the return of cylinders) playsa crucial role. The plan should take in account the empty bottles that arereturned to the company, which can be either reused or disposed of. Thereforewe started by applying to the data two inventory models with reverse flowsfound in literature, and then developed a deterministic model and stochasticmodel tailored for this case study.

9.2.1 Inventory models in literature with reverse flows

Richter [6] extended the EOQ model to allow the incorporation of used prod-ucts, which were repaired and incorporated in production. It assumes a sta-tionary demand in a model with two shops, where the first shop is producingnew products and repairing products used by the second shop.

Also considering deterministic demand and reverse logistics is the modelproposed by Teunter [8], differing in allowing to consider varying disposalrates and disaggregating holding costs. In this model (figure 9.1, M manu-facturing batches and R recovery batches succeed each other.

Fig. 9.1: Inventory stock model according to Teunter model [8] for case M=1, R=5

70 Lopes et al.

We implemented in an Excel file, for the company to use, all the formulasfrom Teunter model for computing the total cost per unit of time (case M=1),the optimal batch size for manufacturing Qm and for recovery Qr and thenumber of recovery batches.

Other developments on the EOQ model are by Alivoni et al. [1]. Theypropose a stochastic model where production or purchase of new items in-tegrates product reuse, in order to identify the need of placing a produc-tion/purchasing order to avoid stock-out situations.

9.3 Inventory models developed for the company basedon continuous replenishment

The models presented before do not contemplate all the specifications re-quired in this case study. In our case, the returned items arrive continuously,not in discrete moments, and can have three different destinations, as depictedin Figure 9.2. Most of the returned LPG bottles (98%) only need cleaning,and some of them (about 2%) need requalification. At the moment, becausethis business is relatively new for the company, there is no LPG bottles thatneed to be disposed of, but in the future this situation can also occur. Thecosts and time for each of these processes are different.

Fig. 9.2: Reverse flows and Inventory stock costs in our case study

The previous model considered that both the acquired and returned bot-tles arrive at discrete moments periodically in time, but actually that only


happens with the acquired bottles. The returned bottles arrive continuouslyto the warehouse, and are continuously cleaned and requalified and filled(with rate u+ d), as depicted in Figure 9.3. Therefore, a continuous replen-ishment model could be adapted to this case study. In this setting, two casescan happen:

Case λ > u+d : If demand exceeds the incoming flow, sometimes we haveto buy new cylinders from supplier. We derived a Deterministic Model Dwith continuous returns for this case.

Case λ ≤ u + d : If the returned bottles are enough to respond to thedemand, buying new bottles from the supplier is unnecessary. To addressthis case we present the Deterministic Model R without purchases.

9.3.1 Model D - Deterministic continuous returns

We developed a deterministic model D, based on EOQ (Wilson, 1934), whichconsiders deterministic continuous constant demand, deterministic discretereplenishment from supplier, and deterministic continuous constant replen-ishment from returned bottles, for the case when returns are not enough torespond to the demand and hence the company has to buy new bottles fromthe supplier (Figure 9.3).

Fig. 9.3: Deterministic Inventory stock model D developed for our case study

As in the classical EOQ formula, in this model the total costs consideredare the sum of the acquisition costs CA, setup costs CS and holding CostsCH .The acquisition costs in Equation 9.1 consider the cases where new bottlesare acquired from the supplier with a cost Cm, the bottles are reused with

72 Lopes et al.

just a cleaning cost Cu, or the case where the returned bottles have to berequalified with a cost Cd. In this three cases, a constant filling cost is alsoincluded. In the future, a disposal cost Cl could also be considered. At themoment, because this business is relatively new for the company, there is noLPG bottles that need to be disposed of. Hence, the rate of bottles returnedand disposed of (l) is zero. The acquisition costs are:

CA = Cm(1− r)(λ− I) + Cuu(λ− I) + Cd(r − u)(λ− I) (9.1)

where λ is the constant demand rate (units/units of time), I is the initialstock r is the return rate, u is the rate of bottles returned and cleaned andd = r− u is the rate of bottles returned and requalified. The setup costs are:

CS =Km(λ− I)(1− r)

Qm+Ku(λ− I)u

Qu+Kd(λ− I)(r − u)

Qd(9.2)

where Km is the production fixed setup costs, Ku is the reuse fixed setupcosts, Kd is the requalification fixed setup costs, Qm is the batch size forbuying new bottles, Qu is the batch size for reusing bottles, and Qd is thebatch size for requalifying bottles. The holding costs are:

CH = hm(1− r)Qm

2+ hu

uQu2

+ hd(r − u)Qd

2+ hi

I

2(9.3)

where hm is the holding cost per new item bought per year, hu is theholding cost per reused item per year, hd is the holding cost per requalifieditem per year, and hi is the holding cost per existent item in stock per year.

By deriving the total costs, it is possible to obtain the expression for theoptimal quantities Q∗m (batch size for buying new bottles), Q∗u (batch size forreuse) and Q∗d (batch size for requalification) that minimize the total costs.

Q∗m =

√2Km(λ− I)

hmQ∗u =

√2Ku(λ− I)

huQ∗d =

√2Kd(λ− I)

hd(9.4)

Fig. 9.4 Stock quantityacross time, Lead time, andOrder Point in Model D


From the triangle in Figure 9.4 we can find the Order Point: OPl = λ −

(u+ d)⇔ OP = (λ− (u+ d)) lThis model was also implemented in an Excel file for the company to use.

9.3.2 Model R - Deterministic without purchases

Our Deterministic Model R without purchases considers deterministic con-tinuous constant demand, unnecessary replenishment from supplier, and de-terministic continuous constant replenishment from returned bottles. In thissetting, there is a period T1 where there is simultaneously continuous replen-ishment of bottles (with rate u+d) and demand being satisfied (with rate λ),and a period T2 where replenishment is interrupted and there is only demandbeing satisfied.

Fig. 9.5: Deterministic Inventory stock model R for continuous returns without purchases

Therefore, from the slopes of the main triangles in Fig. 9.5, we have:

T1 =M

u+ d− λT2 =

M

λM = Q− λ · T1 = Q

(1− λ

u+ d

)(9.5)

where M is the maximum stock level, and the batch size corresponds to thetotal production during period T1, i.e., Q = (u + d)T1. The total costs aregiven by:

74 Lopes et al.

TC(Q) = Cuu(D−I)+Cdd(D−I)+ +(Ku +Kd)D

Q+ +Ch

Q

2

(1− λ

u+ d

)(9.6)

being D the demand for the planning horizon (year) and λ the daily de-mand. Deriving the total costs, the optimal quantity Q∗ that minimizes thetotal cost is:

Q∗ =

√2(Ku +Kd)D

Ch

√u+ d

u+ d− λ(9.7)

Fig. 9.6 Stock quantity

across time, Lead time, and

Order Point in Model R

If the lead time l is longer than the period of demand (l > T2) then fromthe slope in the blue triangle in Figure 9.6 we can derive the Order Point(formula 9.8).

M −OPl − T2

= u+ d− λ⇔ OP = M − (u+ d− λ)(l − T2) (9.8)

Replacing M and T2 using Equations (9.5), we can obtain the order pointOP as a function that depends only on the quantity of bottles Q, the demandrate and reutilization rate, and lead times:

OP = Q

(1 +

u+ d

λ

)+ l (λ− (u+ d)) (9.9)

9.3.3 Model S - Stochastic inventory model

At first we assumed a deterministic constant demand and return rate, but infact it is not constant nor deterministic. It shows seasonality and trend. Tocorrectly plan the acquisition of new cylinders from the supplier, we proceededto forecast not only the demand, but also the reverse logistic flows.


Forecasting of demand and returns was made using exponential smoothingand moving averages to compute seasonal coefficients and forecast demandand returns. Multiple regression models and Artificial neural networks werealso used to forecast [5]. Afterwords, we used a weighted linear combination ofthe probability density functions as in [3] for the final forecast. The forecastedmean and RMSE was used as input values for the stochastic inventory modelsdeveloped for the case study.

For this, we present a stochastic inventory model S, based on the continu-ous review policy (s,Q), which considers continuous stochastic demand, dis-crete replenishment from supplier, continuous stochastic replenishment fromreturned bottles, and constant lead times, as depicted in Figure 9.7.

Fig. 9.7: Stochastic Inventory stock model S

Assuming demand during lead time is dl ∼ N(µdl, σdl), then the OrderPoint is:

OP = µdl + zασdl (9.10)

where zα = Φ−1(1 − α) is the safety factor for a given Level of Service1 − α. The demand is replaced by a difference of normal random variablesλ−(u+d) where λ ∼ N(µλ, σλ), u ∼ N(µu, σu) and d ∼ N(µd, σd). Assumingindependence, we have:

µλ−(u+d) = µλ − µu − µd σλ−(u+d) =√σ2λ + σ2

u + σ2d (9.11)

Finally, the order point is given by

OP = (µλ − µu − µd)l + Zα

√l(σ2

λ + σ2u + σ2

d) (9.12)

76 Lopes et al.

9.4 Conclusions

A Portuguese company in the energy sector posed a challenge to define theacquisition plan of LPG bottles. To answer this industrial challenge, three in-ventory models with reverse flows were developed for the company. The modelD considers deterministic continuous constant replenishment from returnedLPG bottles and also discrete batches of new bottles that are bought fromthe supplier. The model R considers the future case of the company when thereturned bottles cover for the demand, and replenishment from the supplierwill be unnecessary. Finally, model S was developed to approach the anglethat demand and returns are not constant but continuous and stochastic,with discrete replenishment from the supplier.

Acknowledgements COST Action TD1409, Mathematics for Industry Network (MI-

NET), COST-European Cooperation in Science and Technology; CIDMA-Center for Re-search and Development in Mathematics and Applications; FCT-Portuguese Foundation

for Science and Technology, project UID/MAT/04106/2013. We would like to thank Ana

Sapata from University of Evora, and Claudio Henriques, Fabio Henriques e Mariana Pintofrom University of Aveiro for their contributions during the European Study Group.

References

1. A. Alinovi, E. Bottani, and R. Montanari. Reverse logistics: A stochastic EOQ-basedinventory control model for mixed manufacturing/remanufacturing systems with re-

turn policies. International Journal of Production Research, 50(5):1243–1264, 2012.

2. R.H. Ballou. Business Logistics Management. Prentice Hall, 4th edition, 2006.3. L. Cassettari, I. Bendato, M. Mosca, R. Mosca. A new stochastic multi source ap-

proach to improve the accuracy of the sales forecasts. foresight, 19(1):48–64, 2017.

4. F.W. Harris. Operations Cost Factory Management Series, Chicago: Shaw, 1915.5. I.C. Lopes, E. Costa e Silva, A. Correia, M. Monteiro, R. Borges Lopes. Combining

data analysis methods for forecasting liquefied petroleum gas cylinders demand. V

Workshop on Computational Data Analysis and Numerical Methods, ESTG, InstitutoPolitecnico do Porto, Portugal, 11-12 May 2018.

6. K. Richter. The extended EOQ repair and waste disposal model. International

Journal of Production Economics, 45(1-3):443–447, 1996.7. J. Sousa. Background of Portuguese domestic energy consumption at european level.

In: IT4Energy International Workshop on Information Technology for Energy Appli-cations, 2012.

8. R.H. Teunter. Economic ordering quantities for recoverable item inventory systems.

Naval Research Logistics (NRL), 48(6):484–495, 2001.9. R.H. Wilson. A Scientific Routine for Stock Control. Harvard Business Review. 13,

116–28, 1934.

Chapter 10

Stochastic Order Relations in aGambling-type Environment

Sandor Guzmics

Abstract In this work we examine some stochastic ordering relations,namely the increasing convex order and the Lorenz order, between randomvariables which arise from a simple lottery setting as well as the relation be-tween their natural continuous variants. We will provide stochastic orderingresults for the continuized random variables.

10.1 Introduction

The notion of stochastic dominance has been established originally for com-paring the riskiness of possible scenarios in financial and insurance math-ematics. Later these concepts have been also applied in other probabilis-tic environments, for instance in gamblings. We examine the structure of alottery type gambling by introducing advanced indicators that stem fromthe distribution of the random variables which are naturally associated withthe corresponding game. Our work is motivated by a standard 90/5 typelottery setting. The outcomes are described by five-tuples, and we considerthe ordered sample, and investigate the ordered differences between the el-ements of the ordered sample with respect to the increasing convex orderand the Lorenz order. We illustrate our computations by a data set ob-tained from Hungarian lottery history from 1957 to 2018. It consists of 3217five-tuples drawn from the set 1, . . . , 90, and it is available under the linkhttps://bet.szerencsejatek.hu/cmsfiles/otos.html . In addition we will exam-ine a na-tural continuization of the above setting, which possesses nicer math-ematical properties than the original discrete one. In particular, the ordereddifferences in the discrete setting are not ordererd in the Lorenz order, in

Sandor GuzmicsUniversity of Vienna, Department of Statistics and Operations Research,

1090 Vienna, Oskar Morgenstern Platz 1, Austria, e-mail: [email protected]

77

78 Sandor Guzmics

contrast to the continuous setting. Finally we present some ideas for possibleextensions.

10.2 The discrete setting

Let us consider a standard lottery setting, where five numbers are drawn formthe fundamental set H = 1, . . . , 90 without replacement. In accordancewith this, the players have to fill in a lottery coupon by crossing five numbersfrom the set H. It is well known, that the number of scores follows a hy-pergeometric distribution and it is also obvious, that if we denote the resultof one draw by the set-valued random variable X = X1, X2, X3, X4, X5 ,then X is uniformly distributed on the 5-element subsets of H, i.e., on the setH = h ⊆ H : |h| = 5 and P(X = h) = 1/

(905

)= 1/43949268 ' 2.2754 · 10−8

for all h ∈ H . (Since in the following it will play an important role that Hconsists of equally placed numbers, we will sometimes refer to such a discreteset as a grid.)

Let us introduce the usual notation X∗j for the ordered sample, i.e., in ourcase X∗j is j-th smallest out of the five drawn numbers (j = 1, . . . , 5). Due tothe current setting X∗1 < X∗2 < X∗3 < X∗4 < X∗5 holds with probability 1. Itis easy to see that the probability distribution function of X∗j ( j = 1, . . . , 5) is

P(X∗j = k) =

(k−1j−1)·(90−k5−j

)(905

) for k = j, . . . , 85 + j, (10.1)

and its expectation isE(X∗j ) ' 15.1667 · j . (10.2)

In order to obtain a better insight into the structure of this lottery, wefocus on the differences

Dj = X∗j+1 −X∗j (j = 1, . . . , 4) (10.3)

between the neighbouring elements in the ordered sample (X∗1 , X∗2 , X

∗3 , X

∗4 , X

∗5 ) ,

and we will investigate D∗j (j = 1, . . . , 4), i.e., the ordered sample of therandom variables Dj . Notice that each inequality of the general relationD∗1 ≤ D∗2 ≤ D∗3 ≤ D∗4 can also hold with equality (with positive probabil-ity). First we study the range of the sample, which coincides with the sum ofthe differences defined in (10.3).

The range of the sample.

Z := max1≤j≤5

Xj − min1≤j≤5

Xj = X∗5 −X∗1 =

4∑j=1

Dj =

4∑j=1

D∗j . (10.4)

10 Stochastic Order Relations in a Gambling-type Environment 79

It is easy to see that P(Z = k) = c · (90 − k) · (k − 1) · (k − 2) · (k − 3)for k = 4, 5, . . . , 89 , where c = 20

90·89·88·87·86 ' 3.7923 · 10−9. The expectedvalue is E(Z) ' 60.6667, while the mode of Z is 68 .

The distribution of D∗j (j = 1, . . . , 4) .

We provide the distribution of D∗j (j = 1, . . . , 4) via an implicit, combina-torial consideration, namely in terms of certain partitioning problems. Thederivation of an explicit form will only be possible for D∗1 . As an introductorystep, for each j = 1, . . . , 4 we give Nj , the largest possible value of D∗j , whichare respectively N1 = 22, N2 = 29, N3 = 43, N4 = 86.

Let us introduce the notion of a gap, the number of integers lying betweentwo neighbouring drawn numbers in some realization. As a distinction, wewill call initial gap the number of integers smaller than X∗1 . In notation

l0 := X∗1 − 1, lj := X∗j+1 −X∗j − 1 (j = 1, . . . , 4). (10.5)

It is clear that the sequence of gaps l0, l1, l2, l3, l4 uniquely determinesX∗1 , . . . , X

∗5 . Since a difference of size k between neighbouring drawn num-

bers corresponds to a gap of size k−1 , it is worth to introduce the followingindex sets:

I1 := 1 ≤ i ≤ 4 | li ≥ k − 1 , I2 := 1 ≤ i ≤ 4 | li > k − 1 . (10.6)

Using I1 and I2 , the distribution of D∗j can be written as

P(D∗j = k) =#(l0, l1, l2, l3, l4) | Condition 1., 2., 3. hold (

905

) for k = 1, . . . , Nj ,

(10.7)where

Condition 1. l0 + l1 + l2 + l3 + l4 ≤ 85,Condition 2. |I1| ≥ 5− j,Condition 3. |I2| ≤ 4− j.

Figure 10.1 shows the probability distributions of D∗1 , D∗2 , D

∗3 .D

∗4 along

with their realizations in the data that we have described in Section 10.1.Note that for visualizing purposes we display the probability distributionswith continuous curves, but meanwhile we have to keep their discreteness inmind.

Explicit formula for the probability distribution function of D∗1.

We do not attempt to solve the combinatorial problems given in (10.7), but

by another combinatorial consideration we get P(D∗1 ≥ k ) =(94−4k

5

)/(905

),

80 Sandor Guzmics

Fig. 10.1: The pdfs of the ordered differences along with their realizations in the data.

For the sake of convenience the pdfs are visualized by continuous curves.

which implies

P(D∗1 = k) =((

94−4k5

)−(90−4k

5

)) /(905

)(k = 1, . . . , 22). The expected value

is E(D∗1) ' 4.1844, and since P(D∗1 = k) is decreasing in k , the mode ofD∗1 is 1.

Numerical evaluations of the probability distribution function ofD∗

2, D∗3, D

∗4.

We succeeded in determining the pdfs P(D∗j = k), (j = 1, 2, 3) numerically(look at also Figure 10.1 ), and we computed the expected value and the modeof the distributions: E(D∗2) = 9.0528 and its mode is 7 , E(D∗3) ' 16.3838and its mode is 15 , E(D∗4) ' 31.0457 , and its mode is 28.

What would be a natural continuous analogue of the lottery setting de-scribed above? We define such a continuous analogy of the discrete setting,where the expectations of X∗1 , . . . , X

∗5 and D∗1 , . . . , D

∗4 nearly coincide with

those of the discrete setting. In order to obtain this, we suggest the followingcontinuous model.

10.3 The continuous setting

Let X1, X2, X3, X4, X5 be the sample drawn from the discrete grid1, . . . , 90 . Then


Yj := Xj + Uj j = 1, . . . , 5 ,

where Uj ∼ UNI[−0.5, 0.5] are independent of Xj and of each other. Itis obvious that Yj ∼ UNI[0.5, 90.5], furthermore the construction has thefavourable property that three or more Yi cannot fall very close to eachother, so an important feature of the discrete grid is preserved.

They also fulfill our previously described aim, that is, their expected val-ues nearly coincide with the expected values of the corresponding discretevariables, as the following table shows. For sake of simplicity we will use thenotation D∗1 ≤ . . . ≤ D∗4 for both the discrete and the continuous setting andwe will always make it clear which variant is actually meant.

dis

cr.

sett

ing1 E(X∗1 ) E(X∗2 ) E(X∗3 ) E(X∗4 ) E(X∗5 ) E(D∗1) E(D∗2) E(D∗3) E(D∗4)

15.1667 30.3333 45.5000 60.6667 75.8333 4.1844 9.0528 16.3838 31.0457

cont.

sett

ing2 E(Y ∗1 ) E(Y ∗2 ) E(Y ∗3 ) E(Y ∗4 ) E(Y ∗5 ) E(D∗1) E(D∗2) E(D∗3) E(D∗4)

15.1612 30.3281 45.4965 60.6652 75.8320 4.1606 9.0610 16.3918 31.0573

1 The values are exact and they are displayed up to four decimal place accu-racy.2 Values based on a sample of 10 Million drawn from the distribution(Y1, . . . , Y5).

10.4 Stochastic order relations in the Lorenz order andin the increasing convex order

It is worth to examine whether some stochastic order relation holds betweenthe ordered differences D∗1 , . . . , D

∗4 . Here we will consider the increasing

convex order and the Lorenz order. For their definitions we refer to Shakedand Shantikumar [5], [6], Denuit et al. [1], Scarsini [4], Lorenz [3], and Kampkeand Radermacher [2].

Investigations in the Lorenz order.

We found that the ordered differences in the discrete setting are not or-dered in the Lorenz order, while in the continuous case they are, i.e.,D∗1 L D∗2 L D∗3 L D∗4 .Figure 10.2 depicts the Lorenz curves. Looking at Figure 10.2a one mightconjecture an order relation for the discrete case, too, but by examining thelower tails of the distributions carefully (the left part of the Lorenz curves),the opposite can be concluded.Proposition 10.1. sg

(i) In the continuous setting D∗i L D∗j for 1 ≤ i < j ≤ 4 .(ii) In the discrete setting D∗i 6L D∗j for i 6= j .

82 Sandor Guzmics

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 10.2: Lorenz curves of D∗j (j = 1, . . . , 4) in the discrete (left) and continuous (right)

settings.

Sketch of the Proof. We have to examine the pointwise orderedness ofthe Lorenz curves. We omit the details but provide a graphical justification(Figure 10.2).

Investigations in the increasing convex order.

Proposition 10.2. s In both settings D∗i ICV X D∗j for 1 ≤ i < j ≤ 4 .

Sketch of the Proof. According to Denuit et al. [1] Proposition 3.4.6 wehave to examine whether E((D∗i − t)+) ≤ E((D∗j − t)+) holds for all t ∈ Rwhen i < j. In the discrete setting it is enough the examine this relation fort ∈ 1, . . . , 43. In the continuous setting Figure 10.3 confirms the statement.

Fig. 10.3

The expressions E(D∗i − t)+

(i = 1, . . . , 4) are plotted as

functions of t to illustrate

the stochastic order rela-tions in the increasing con-

vex order: D∗i ICV X D∗

jfor 1≤ i < j ≤ 4.

10.5 Summary

We have seen that the application of stochastic order relations can lead to abetter understanding in some settings which inherently possess a stochastic


nature, such as a lottery game. The perspective is wider, since some exten-sions of the discussed tools (e.g., the multivariate Lorenz dominance) mightenable us to investigate stochastic dominance in multivariate settings. Thatcan be the topic of possible future research.

References

1. Denuit, M., Dhaene, J., Goovaerts, M., Kaas, R.: Actuarial Theory for Dependent

Risks: Measures, Orders and Models. Hohn Wiley & Sons Ltd (2005)2. Kampke, T., Radermacher, F.J.: Lorenz Curves and Partial Orders. In: Income Mod-

eling and Balancing. Lecture Notes in Economics and Mathematical Systems, vol 679.

Springer, Cham (2015)3. Lorenz, M.O.: Methods of Measuring the Concentration of Wealth, Publications

of the American Statistical Association. Vol. 9 (New Series, No. 70) 209–219.doi:10.2307/2276207 (1905)

4. Scarsini, M.: Multivariate convex orderings, dependence, and stochastic equality. In:

Journal of Applied Probability, Vol. 35, pp. 93-103 (1998)5. Shaked, M., Shanthikumar, J.G.: Stochastic orders and their applications. Academic

Press, Boston (1994)

6. Shaked, M., Shanthikumar, J.G.: Stochastic orders. Springer-Verlag, New York (2007)

Index

dedication, v

foreword, vii

preface, ix

85

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