Emmanuel Witrant

Nicolas Marchand

Modeling and Feedback Control for Air Flow Regulation in Deep Pits

We consider the problem of regulating the air quality in underground extraction rooms for mining industry. This is a challenging control problem where the flow dynamics, the interconnections between subsystems and the time-varying topology have to be taken into account along with real-time computation constraints. Our work is focused on the deep pit part of the ventilation system, which brings fresh air at a specified pressure to the extraction levels. The flow interactions and main automation elements are first presented, with a real-time engineering model of the complete mine ventilation system. A novel control-oriented model focused on the pressure dynamics is then introduced, as a convective-resistive partial differential equation (PDE) with multiple inputs where the time-varying transport coefficients are estimated based on the distributed measurements. A fast predictive controller (FPC) is finally proposed to compensate the pressure losses due to friction and multiple flow exhausts thanks to the ventilation pit input pressure regulation. Simulation results illustrate the efficiency of the modeling and control algorithms.

Introduction

Mining ventilation is an interesting example of a large scale system with high environmental impact. Indeed, one of the first objectives of modern mining industry is to fulfil ecological ⋆ E-mail: Emmanuel.Witrant, Nicolas.Marchand @gipsa-lab.grenoble-inp.fr specifications during the ore extraction and ore crushing, by optimizing the energy consumption or the production of polluting agents. This motivates the development of new control strategies for large scale aerodynamic processes based on appropriate automation and the consideration of the global system. The approach presented in this paper is focused on the mining ventilation process, as 50 % of the energy consumed by ore extraction goes into ventilation (including heating the air). It is clear that investigating automatic control solutions and minimizing the amount of pumped air to save energy consumption (proportional to the cube of airflow quantity) is of great environmental and industrial interest. The mine ventilation topology is depicted in Figure 1. It is achieved by a turbine and a heater connected on the surface to a deep pit (vertical shaft) that conducts the airflow to the extraction levels. The heater is introduced (in winter time at least) to avoid freezing in the upper part of the shaft and the air is cooled down at high depths (more than 1000 meters) because of the geothermal heating effect. From the deep pit, fans located at each extraction level pump fresh air to the extraction rooms via tarpaulin tubes. Bad quality air is naturally driven by the pressure gradient and flows from the extraction rooms back to the exhaust ventilation shaft (similar but separate from the inflow ventilation shaft).

From a control point of view, we can divide the regulation problem in two parts:

1. pressure regulation in the deep pit thanks to the ground turbine control and distributed sensors within the shaft; 2. air quality regulation in the extraction rooms thanks to the fans and chemical sensors located in the rooms.

Both problems are connected by the flow properties in the deep pit at the extraction rooms level and we consider the pressure as the interconnection variable. From a modeling point of view, the first problem typically has a clear geometry while the second one is strongly varying in geometry (rooms are blasted every day), characteristics (tarpaulin tube length and shape) and disturbances (trucks) even within the same mine. Computational Fluid Dynamics (CFD) models, such as the one presented in [START_REF] Witrant | HynX team. Air flow modelling in deep wells: application to mining ventilation[END_REF], can then be envisioned for the deep pit while greybox identification or global models focused on the main dynamics should be preferred for the second problem. The mine automation, communication network, historical background on real-time control and closed-loop control strategies for the second problem are detailed in [START_REF] Witrant | Wireless ventilation control for large-scale systems: the mining industrial case[END_REF]. The efficiency of these control strategies strongly depends on the available pressure in the vertical shaft, which is the topic of the present work. We then focus on the boundary control (turbine operation) of a deep pit flow (vertical shaft) subject to distributed losses (friction and fans exhausts).

Starting from the fundamental equation describing the flow dynamics (Navier-Stokes), we first present the simplifying hypotheses that allows the design for a real-time simulator of the complete mine aerology. This simulator is subsequently used as a reference to investigate the performance and limitations of the proposed regulation approach. Defining the pressure distribution as the regulated variable, we then propose a novel control-oriented model for large Poiseuille flows that integrates the distributed pressure measurements thanks to an online parameter identification method. The importance of a diffusive term to model the considered flow is discussed, which motivates a regulation approach focused on the convective and resistive phenomena. Based on this analysis, we finally propose a dedicated fast predictive control approach to ensure the real-time constraints and performance objectives.

This chapter is organized as follows. The physical properties and a real-time simulator of the ventilation system are presented in Section 2, along with the main physical hypotheses. A control-oriented model and an estimation method that provides for online tracking of the flow transport parameters are derived in Section 3. Section 4 is focused on a FPC feedback approach that allows for real-time pressure regulation at the bottom of the pit.

Ventilation in a deep pit

The aim of this section is to describe key issues that are necessary to analyse and model the airflow behavior in a deep pit. We first mention the physical principles, which provide fundamental dynamics for the mine ventilation model. This process model includes flow interactions, automation devices and can be used in real-time thanks to a 0-D modeling approach. Focusing on the pressure dynamics, we then propose a 1-D model that specifically describes the deep pit part and will be used in the next sections to set the regulation law.

Physical model

The dynamics of the flow is obtained from the conservation of mass (density ρ), momentum M = ρV (V being the flow speed) and energy (per unit mass) E, along with the perfect gas equation of state p = ρRT , where p is the pressure, R the specific gas constant and T the temperature. Choosing a conservative form (in the numerical sense) of Navier-Stokes equations, we have that [START_REF] Hirsch | Numerical Computation of Internal & External Flows: the Fundamentals of Computational Fluid Dynamics[END_REF]:

∂ ∂t   ρ ρV ρE   + ∇ •   ρV ρV T ⊗ V + pI -ς ρVH -ς • V -k∇T   =   0 0 q  
where ⊗ is the tensor product of two vectors, ς the shear stress vector, k the thermal conductivity, H the total enthalpy and q(x, y, t) the rate of heat addition (see [START_REF] Brown | Response of wall heat transfer to flows along a cylindrical cavity and to seepage flows in the surrounding medium[END_REF] for a precise description). Neglecting the viscous effects (which is reasonable for air and the targeted simplified model), we obtain Euler equation:

∂ ∂t   ρ ρV ρE   + ∇ •   ρV ρV T ⊗ V + pI ρVH   =   0 0 q   (1)
A direct use of this equation to model the complete mining ventilation process involve complex CFD considerations (the airflow in the vertical shaft is mainly turbulent), typically out of reach for industrial applications and not suitable for real-time control objectives. An additional step toward model simplification and real-time computation analysis is then needed, as presented in the next subsection. QmX SF _ 2

Bond graph model and real-time simulator

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Step 2

Step 1

Step This section briefly summarizes the main results presented in [START_REF] Witrant | HynX team. Air flow modelling in deep wells: application to mining ventilation[END_REF], where a non-dimensional model was proposed thanks to the bond graph approach and a real-time simulator was designed. Non-dimensional modeling has an increasing use in the design, validation and tuning of control laws, as it allows for the integration of as much physical properties as possible (avoid data mapping) and a reduced computation time (close to real-time, approximately 10 times slower in the worst cases).

Signal 1 F 1 F Q SV F 1 F 2 Q SV F 2 F 1 Q SV F 2 F 1 Q SV F 2 F 1 Q SV F 2 F 1 Q SV F 1 F 2 Q SV F 1 F 2 Q SV F 1 F 2 Q SV F 2 F 1 Q SV F 2 F 1 Q SV F 1 F 2 Q SV F 1 F 2 Q SV F 1 F 2 Q SV F 1 F 2 Q SV F 1 F 2 Q SV F 1 F 2 Q SV F 1 F 2 Q SV F 1 F 2 Q SV F 2 F 1 Q SV F 2 F 1 Q SV F 2 F 1 Q SV F 2 F 1 Q SV F 2 F 1 Q SV F 1 F 2 Q SV F 2 F 1 Q SV F 2 F 1 Q SV F 1 F 2 Q SV F 1 F 2 Q SV F 1 F 2 Q SV F 1 F 2 Q SV F1 F2 Q SV F 1 F 2 Q SV F 2 F 1 Q SV F 2 F 1 Q SV F 1 F 2 Q SV F 2 F 1 Q SV F 2 F 1 Q SV F 1 F 2 Q SV F 1 F 2 Q SV F 1 F 2 Q SV F 1 F 2 Q SV F 1 F 2 Q SV F 2 F 1 Q SV F 2 F 1 Q SV F 2 F 1 Q SV F 2 F 1 Q SV F 2 F 1 Q SV F 2 F 1 Q SV F 2 F 1 Q SV F 2 F 1 Q SV F 1 F 2 Q SV F 1 F 2 Q SV F 1 F 2 Q SV F 2 F 1 Q SV F 1 F 2 Q SV F 1 F 2 Q SV F 2 F 1 Q SV F 1 F 2 Q SV F 1 F 2 Q SV F 1 F Q SV F 1 F 2 Q SV F 1 F 2 Q SV F 1 F Q SV F 1 F 2 Q SV F 1 F 2 Q SV F 2 F 1 Q SV F 1 F 2 Q SV F 1 F 2 Q SV F 1 F 2 Q SV [ Pps 3 ] [Ppe 3 ] [ Pps 2 ] [Ppe 2 ] [ Pps 1 ] [Ppe 1 ] [Pps 3 ] [ Ppe 1 ] [Pps 1 ] [Pps 2 ] Q_in Q_pollu Q_out
The deep pit is described by interconnected cells containing height-averaged flow values, as depicted in Figure 2. Note that this is equivalent to a control volume discretization of the flow, where a single volume is used at each height. The equivalent bond graph description is then obtained for each cell by deriving a flow/effort model from Euler equation (1) under the following hypotheses: H1. the impulsive term is negligible compared to the pressure: ρv 2 << p in the momentum equation, which is approximated with an algebraic relationship (such as Saint-Venant or Darcy-Weisbach equation) that writes as M ≈ a(M, ρ) ∇p, where a(•) is a continuously differentiable function and ∇p is the pressure gradient exerted on the control volume considered; H2. only the static pressure is considered, implying that the kinetic energy term in the energy conservation equation is omitted:

H = E + p/ρ; H3. the gas is calorically perfect E = c v T , where c v = R/(γ -1) is the specific heat at constant volume and γ = 1.4.
The mine ventilation simulator is constructed based on this flow description and the fans models proposed in [START_REF] Talon | Engine control model based design with achille library[END_REF]. The deep pits (inflow and exhaust) are both discretized with 28 control volumes and we consider three extraction levels. The turbine and fans regulation is done by setting their rotational speed, and flow speed, pressure and temperature can be measured in each control volume. The resulting computation time is 34 times faster than real-time in an Intel Centrino 1.83 GHz PC. This simulator is used in the next sections to illustrate the estimation and control strategies. It may also be considered in the futur to develop virtual sensing capabilities in real-time operation schemes.

Distributed pressure dynamics

For real-time control purposes, we are specifically interested in the pressure dynamics, which provides for the regulated variable. In order to achieve model-based feedback control, the distributed measurements can be used to reduce the model complexity thanks to on-line parameter estimation, as described in the next section. The appropriate simplified model is obtained from H1-H3 by expressing the energy equation in terms of pressure (perfect gas equation) as:

∂ρE ∂t = - ∂ ∂x M • E + p ρ + q ⇔ ∂p ∂t = - ∂ ∂x M ρ • 1 + R c v p + R c v q (2)
Note that the momentum can be obtained by implicit resolution of the Darcy-Weisbach equation, which then provides the dynamics for the density evolution.

Distributed measurements and online parameters estimation

As mentioned previously, we suppose that distributed measurements (i.e. obtained thanks to a wireless sensor network) are available to set the control law. One of the main advantages is the possibility to constrain the pressure model according to the measured behavior of the flow. In this section, we derive a generic model architecture from (2) that depicts convective-resistive dynamics. An online parameter estimation method is then introduced to set the transport coefficients according to distributed measurements.

Control oriented model

To propose an efficient model-based control of the flow dynamics, we need to capture the main tendencies of the pressure evolution. The momentum and density impact on the pressure dynamics are considered with their volume-averaged values:

M (t) = 1 V V M (v, t)dv, ρ(t) = 1 V V ρ(v, t)dv
where V corresponds to the volume of the pit. The energy losses are generated by pressure losses at the exhausts locations and friction on the walls, which implies that:

R c v q(x, t) = s(t)̺ x (x)f f an (∆p f an (t), η(t)) + r(t)p(x, t)
where s quantifies the impact of the exhausts on the main flow, ̺ x is set by the mine topology (i.e. ̺ x = 1 at the exhausts location and 0 otherwise), f f an depends on the fan model, the vector ∆p f an contains the pressure gradients across each fan, η is the blades rotational speed and r denotes the resistance on the pit sides. ̺ x , ∆p f an and η are considered as known engineering parameters. The physical model ( 2) is then approximated with the averaged controloriented model:

pt = c(t)p x + r(t)p + s(t)p ext (x, t) (3)
where c(t) . = -M (t)/ ρ(t) • (1 + R/c v ), p ext (x, t) . = ̺ x (x)f f an (∆p f an (t), η(t)) and p y . = ∂p/∂y. The boundary conditions are set by the inflow pressure p(0, t) = p in (t) (regulated thanks to a local PID feedback on the turbine rotation speed) and the dead end at the bottom of the shaft px (L, t) = 0, L being the shaft length.

Online parameter estimation

A classical estimation problem is to find the set of parameters ϑ that minimizes the difference between the measured and estimated data, given a specific model architecture. This is done in this section by deriving a variation law for ϑ such that the estimation error is exponentially decreasing. We consider the general class of systems that involve transport phenomena described as:

p t = d(t)p xx + c(t)p x + r(t)p + s(t)p ext (x, t) (4)
where ϑ(t) = {d(t), c(t), r(t), s(t)} denote the diffusive, convective, resistive and source coefficients, respectively, and p ext (x, t) is a distributed source term. The boundary conditions are given by p(O, t) = p 0 (t) and p x (L, t) = 0, for x ∈ [0, L]. The variation of ϑ is provided by the following theorem.

       pt = A(p, px , pxx , u, θ) θ + γ(p -p) a 1 px (0, t) + a 2 p(0, t) = a 3 a 4 px (L, t) + a 5 p(L, t) = a 6 θ = A(p, px , pxx , u, θ) † [p t + λ(p -p)] (6)
where A † is the Moore-Penrose inverse of A.

Proof. Considering the estimation error ǫ(x, t) . = p(x, t) -p(x, t), we have from (6) that:

ǫ t = p t -[p t + λ(p -p) + γ(p -p)] = -(γ + λ)ǫ
with the boundary conditions ǫ(0, t) = 0 and ǫ x (L, t) = 0. Then, to investigate the stability in the L 2 sense, we introduce the Lyapunov function:

L(t) . = 1 2 L 0 ǫ 2 dx
It follows that:

dL dt = -(γ + λ) L 0 ǫ 2 dx = -2(γ + λ)L
which directly implies that ||ǫ(x, t)|| 2 2 = e -2(γ+λ)t ||ǫ(x, 0)|| 2 2 and concludes the proof.

Remark 1. The parameter λ is introduced to moderate the variations of θ and γ ensures the convergence of the estimated dynamics. The use of two parameters instead of one, which would be sufficient from a stability point of view, is motivated by the need to adjust the estimator performances according to robustness constraints and non-smooth signal variations.

Example 1. Consider the model ( 4) with the inputs depicted in Figure 3, a Dirichlet boundary condition at the surface and a Neumann one at x = L. Such a model writes as [START_REF] Witrant | HynX team. Air flow modelling in deep wells: application to mining ventilation[END_REF] with: and the estimated values of the transport coefficients θ are directly obtained according to (6).

       A(p, p x , p xx , u, ϑ) = [p xx p x p p ext ] ϑ = [d c r s] T a 1 = 0, a 2 = 1, a 3 = p in a 4 =
The corresponding algorithm is designed by discretizing the diffusivity using central difference and the conductivity thanks to a Lax-Wendroff scheme (the model thus remains stable for d(t) = 0, provided that the conditions on the associated Courant number are satisfied).

The reference model algorithm is set by introducing P ∈ R N as the vector of discretized pressures and with the variation law:

P k+1 1 = P k 1 +
D(P i-1 , P i , P i+1 ) . = P i-1 -2P i + P i+1 C(P i-1 , P i , P i+1 , α) . = α -1 2 P i-1 -αP i + α + 1 2 P i+1
where P i , i = 2 . . . N -1 denotes the i th component of P , k the time index, t s the sampling time, -α = -ct s /∆x the CFL number and ∆x the spatial step. The estimated pressure vector P is computed using a similar algorithm and the parameter estimation is achieved with: A =

        1 ∆x 2 D(p 0 , P1 , P2 ) 1 ∆x C(p 0 , P1 , P2 , α) P1 P ext,1 . . . . . . . . . . . . 1 ∆x 2 D( Pi-1 , Pi , Pi+1 ) 1 ∆x C( Pi-1 , Pi , Pi+1 , α) Pi P ext,i . . . . . . . . . . . . 1 ∆x 2 D( PN-1 , PN , PN ) 1 ∆x C( PN-1 , PN , PN , α) PN P ext,N        
The resulting model pressure and estimation error are depicted in Figure 4 while the exact and estimated values of the transport coefficients are presented in Figure 5. We can see that the proposed estimator achieves a very precise tracking of the transport parameters and quickly converges to the exact values. The large perturbations induced by the steps in the boundary and distributed input signals are reflected as peaks in the estimated parameters (at 30, 50 and 60 s) but are quickly damped.

Application to the mining ventilation case

The test case is defined as follows, for three extraction levels in a L = 1100 m deep mine. The 1 st level fan (at L/4) is not operated (a small flow is naturally driven by the pressure gradient), the 2 nd level fan (at L/2) goes from 0 to 150 rpm at t = 2000 s and the 3 rd level fan (at 3L/4) is operated at 200 rpm. A step is set on the turbine rotation speed at t = 1000 s. Pressure measurements at the turbine output and at the bottom of the pit are available for feedback control and presented in Figure 7 (top). We consider two different estimated models, with:

1. four transport parameters: diffusion, convection, resistance and source; 2. three parameters, excluding the diffusion according to (3).

The resulting estimation error and parameters values can be compared in Figures 6 and7, respectively. We can see that the estimation error quickly decreases to a negligible value in both cases, which validates the proposed estimation strategy. Furthermore, the slight estimation improvement obtained thanks to the additional diffusive term, noticeable but not crucial, validates the volume-averaging of the momentum and density as a possible model simplification for feedback control.

Fast model predictive control

We consider that the heater is operated based on the atmospheric conditions (i.e. it acts as an external input) and that the regulated input is the turbine downflow pressure (a local control loop is set on the turbine to adjust its rotational speed according to a desired pressure). The control problem is then to ensure a minimum pressure within the shaft (at each extraction level) based on the turbine actuation. As the pressure can only decrease as we go deeper, the control objective can be reduced to ensuring a minimum pressure at the bottom of the pit. We consider that distributed pressure measurements are available to set the state feedback control law. The aim of this section is to propose a predictive control of the mine process. Predictive control is widely used in process industry because of the usual robustness of the method and of the slow dynamics of the processes. Classical predictive control requires the optimization of a criterium whose optimum gives the control to apply over the next time instant. Unfortunately, predictive control relies on finding the extremum of an optimization problem that can hardly be guaranteed, especially for systems described by PDE like the present system. Recently, fast predictive control approaches were developed that enable to have an analytical solution to the predictive control problem without any optimization by using the structure of the system. This is what we propose to apply here.

We start from the control oriented model and assume that the time varying parameters c(t), r(t) and d(t) and external disturbance s(t)p ext (x, t) are slowly varying parameters and can therefore be considered as constants: c(t) = c, r(t) = r, d(t) = d and s(t)p ext (x, t) = sp ext (x). By slowly varying, it is only necessary to assume that the perturbation related to the opening and closing of the extraction rooms levels p ext (x) is piecewise constant. The diffusive term d is assumed to be small and is neglected: d ≈ 0. Note that these assumptions are only made to design the control law that is tested and applied to the global system without any simplifications. Therefore, we focus on the following equation:

p t = cp x + rp + sp ext (x) (7)
Predictive control consists in finding an open-loop control profile τ → p(0, τ ) at each instant t such that the solution of the dynamical system (here the solution of (7)) has "some" properties on a time horizon [t, t + w] where w is the prediction horizon (may be infinite). These properties can be very various and have to ensure the stability of the closed loop scheme. The most classical property is optimality in optimal predictive control. This openloop control profile is applied at its first instant p(0, 0) and the scheme is repeated at the next time instant t + dt.

The aim of the control law is to give the inflow pressure control profile p(0, τ ) in order to ensure a given down pressure p(L, ∞) in closed loop. Once the steady state has been achieved, a corresponding constant inflow pressure p(0, ∞) will ensure the desired down pressure p(L, ∞). At time t, we assume that the system starts with some constant initial inflow p(0, t), an established pressure profile in the ventilation pit and p(L, t) at the bottom. This established steady state field of pressure p st (x, p 0 ) results from the application of the constant inflow pressure p 0 and is simply given by (7) assuming no time derivatives, that is:

p st (x, p 0 ) = p 0 e -r c x - s c x 0 e -r c (x-z) p ext (z)dz
Note that according to this notation, one has p(L, ∞) = p st (L, p(0, ∞)) and, as soon as the system is in a steady state regime at time t, p(L, t) = p st (L, p(0, t)).

We propose to approximate the pressure open-loop profile p(x, t + τ ), τ ∈ [0, +∞], by: p(x, t + τ ) = pe -αx e -βτ + p st (x, p(0, ∞))

When τ increases, p(x, t + τ ) then converges to the steady state solution that ensures the desired down pressure p(L, ∞). At time t (τ = 0), it is natural to take:

p = e αL [p(L, t) -p st (L, p(0, ∞))]
in order to ensure the initial condition at least at x = L. Finally, to ensure that p is a solution of (7), we impose that β = αc -r.

In this scheme, α is a free parameter. However:

• α > r/c is necessary to ensure the convergence of the open-loop trajectory to p(L, ∞);

• the convergence speed is inversely proportional to the difference between α and r/c;

• the closer α is to r/c, the closer p(x, t) is to p(x, t), that is to the exact solution of ( 7).

The open-loop control law profile, defined at time t for any τ > 0, is therefore given by:

p(0, t + τ ) = (p(L, t) -p(L, ∞)) e -α(x-L) e -βτ + p(0, ∞)
where p(L, t) is the current down pressure, p(L, ∞) is the desired down pressure and:

p(0, ∞) = e r c L p(L, ∞) + s c L 0 e -r c (L-z) p ext (z)dz
To prove the efficiency of the proposed approach, two scenarii are compared:

1. the inflow pressure is set at some value that ensures a down pressure of 1 hP a for the initial ventilation topology; 2. the inflow pressure is automatically computed and adapted online according to the above control law using the down pressure measurement.

In both cases, we consider the test case described in Subsection 3.3 with the 2 nd level fan operated at t = 1000 s instead of t = 2000 s. The obtained control is actually not directly set on the system but acts as a reference to the turbine outflow pressure. Indeed, the shaft inflow pressure is set using the corresponding turbine speed by means of a local control law (proportional-integral control) on the turbine.

On Figure 8, one can see the desired and real inflow pressure as well as its corresponding down pressure. The control law clearly enables 1) a fastest convergence to the desired down pressure and 2) to maintain the down pressure even in case of fan operation at different extraction levels.

Conclusions

In this work, we considered the problem of airflow modeling and control in deep pits, such as the ones encountered in mines. Starting from the fundamental physical equations of the flow and engineering models of the automation components, we first proposed a real-time model of the complete mine aerology based on a bond graph approach. Specific hypotheses related to the flow properties were then expressed to obtain a 1-D model of the pressure dynamics.

It was then shown that, considering the averaged impact of the momentum and density, the pressure dynamics could be reduced to a transport equation with time-varying coefficients. A specific estimation method was derived to infer these coefficients from the distributed pressure measurements and the interest for a diffusive term in the flow model was discussed. A model-based feedback control approach based on fast predictive control was proposed. Thanks to this approach, a reference down pressure can be ensured even in case of pressure perturbation due to air usage for underground extraction rooms.

The overall results illustrate the validity of simplified models and the interest for dedicated model-based control methodologies to regulate large airflows, thus allowing for an increased security and a reduced energy consumption. The proposed theoretical results on flow model simplification, transport parameters estimation and FPC for boundary regulation can be applied more widely on large Poiseuille flows with distributed outputs.

Fig. 1

 1 Fig. 1 Airflows in an underground mine.

Fig. 2 2

 2 Fig. 2 2-D (left) and control-volume (middle) description of the deep pit, simulator of the complete mine aerology with three ore extraction levels (right).

1 , a 5 =Fig. 3

 153 Fig. 3 Boundary condition and distributed outputs.

Fig. 4

 4 Fig. 4 Model pressure P (t) and squared estimation error ||ǫ(x, t)|| 2 2 .

Fig. 5 Fig. 6

 56 Fig. 5 Exact ('-') and estimated ('-• -') transport coefficients values.

Fig. 7

 7 Fig. 7 Parameter estimation for the mine ventilation shaft (three '-' and four '-• -' parameters).

Fig. 8

 8 Fig. 8 Down pressure regulation in a mine with (right) and without (left) fast predictive control

  where p is the state, u a known exogenous input, ϑ ∈ R M denotes a set of time-varying parameters, A(p, p x , p xx , u, ϑ) ∈ R 1×M sets the input-to-state relationship and a i are scalar real coefficients. The estimated state p(x, t) converges exponentially to p(x, t) in the L 2 sense

	Theorem 1. Consider the general class of systems described by:
	 	p t = A(p, p x , p xx , u, ϑ)ϑ
		a 1 p x (0, t) + a 2 p(0, t) = a 3	(5)
		a 4 p x (L, t) + a 5 p(L, t) = a 6
	and:
	||p(x, t) -p(x, t)|| 2 2 = e -2(γ+λ)t ||p(x, 0) -p(x, 0)|| 2

2

where || • || 2 denotes the L 2 norm and γ, λ are positive scalar parameters, if:


Jørgen Vitting Andersen
email: jorgen-vitting.andersen@univ_paris1.fr

Ioannis Vrontos

Petros Dellaportas

Serge Galam

Communication impacting financial markets

Background: Since the attribution of the Nobel prize in 2002 to Kahneman for prospect theory, behavioral finance has become an increasingly important subfield of finance. However the main parts of behavioral finance, prospect theory included, understand financial markets through individual investment behavior. Behavioral finance thereby ignores any interaction between participants.

Methodology:

We introduce a socio-financial model that studies the impact of communication on the pricing in financial markets. Considering the simplest possible case where each market participant has either a positive (bullish) or negative (bearish) sentiment with respect to the market, we model the evolution of the sentiment in the population due to communication in subgroups of different sizes. Nonlinear feedback effects between the market performance and changes in sentiments are taking into account by assuming that the market performance is dependent on changes in sentiments (e.g. a large sudden positive change in bullishness would lead to more buying). The market performance in turn has an impact on the sentiment through the transition probabilities to change an opinion in a group of a given size. The idea is that if for example the market has observed a recent downturn, it will be easier for even a bearish minority to convince a bullish majority to change opinion compared to the case where the meeting takes place in a bullish upturn of the market.

Conclusions: Within the framework of our proposed model, financial markets stylized facts such as volatility clustering and extreme events may be perceived as arising due to abrupt sentiment changes via ongoing communication of the market participants. The model introduces a new volatility measure which is apt of capturing volatility clustering and from maximum likelihood analysis we are able to apply the model to real data and give additional long term insight into where a market is heading.

INTRODUCTION

The quote "In the short run, the market is a voting machine, but in the long run it is a weighing machine" is attributed to Benjamin Graham [START_REF] Graham | was an influencial American econmist mentor to several investors of whom the most known is probably Warren Buffet[END_REF]. Graham himself used the quote to argue that investors should use the so-called fundamental value investment approach and concentrate on analyzing accurately the worth of a given financial asset. That is, he suggested ignoring the short run "voting machine" part, and instead concentrate on the long run, where somehow the "weighing machine"of the market would ensure to end up with a pricing of the true worth of an asset. The interesting part of Graham's quote is the allusion to the decision making of people and its impact on the markets. Graham however does not refer to how the decision making process actually takes places, and this will be our focus in the following.

In situations of uncertainty people often consult others to get more information and thereby a better understanding of the situation. This is in particular true with respect to financial markets, where market participants consult the media or other colleagues to get an idea of the origin behind price movements, or to assess which impact a given information could have on the markets. For example, copying the behavior of others is one of the predominant mechanisms in decisions to purchase [START_REF] Bikhchandani | Learning from the behavior of others: conformity, fads, and informational cascades[END_REF]. Social households, i.e. those that interact with their neighbors, have also been shown to have an impact in the level of stock market investments. In [START_REF] Hong | Social interaction and stock-market participation[END_REF] it was shown how the stock-market participation rates were higher in places that had a higher sociability. However the sole knowledge of the role of imitation is not sufficient to describe in an operative frame the price dynamics of financial markets since indeed the rules governing opinion dynamics and group decisions still needs to be understood and identified in more detail.

In [START_REF] Shiller | Survey evidence on diffusion of interest and information among investors[END_REF] interpersonal communication were found to be extremely important in investors decisions. Questionnaire surveys of institutional and individual investors were found to reveal a strong influence by word-of-mouth communications. In [START_REF] Arnswald | Investment behaviour of German equity fundmanagers an exploratory analysis of survey data[END_REF] it was shown that among fund managers in Germany information exchange with other financial and industry experts was the second most important factor influencing their investment decisions, complemented by conversations with their colleagues and reports from media. More recently impact of large scale human network structes and its flow of information were shown to have an impact on the asset pricing [START_REF] Ozsoylev | Asset pricing in large information networks[END_REF]. Communication and opinion formation among market participants therefore seems to be an important ingredient in the price formation of any financial market, the question however is how to quantify a general framework of such a scenario at the micro level? In the following we will introduce a socio-financial [START_REF] Vitting Andersen | An Introduction to Socio-Finance[END_REF] model that in a quantitative manner does exactly this.

ANALYSIS

Insert Figure 1 Consider a population of market participants, shown schematically as circles in Fig. 1A. We will proceed as in the so-called Galam model of opinion formation [START_REF] Galam | Local dynamics vs. social mechanisms: a unifying frame[END_REF][START_REF] Galam | Sociophysics: A Physicists Modeling of Psycho-political Phenomena[END_REF][START_REF] Biondi | Formation of share market prices under heterogeneous beliefs and common knowledge[END_REF] and for simplicity imagine that people have just two different opinions on the market, which we can characterize as either "'bullish"' (black circles) or "bearish" (white circles). Letting B(t) denote the proportion of bullishness in a population at time t, the proportion of bearishness is then 1 -B(t).

Figure 1A represents the opinions of the participants at the beginning of a given day. During the day people meet in random subgroups of different sizes, as illustrated by the different boxes in Fig. 1B, to update their view of the market. Take, for example, the leftmost box in Fig. 1B with six persons, two bullish, four bearish, who we can imagine are sitting around a table, or having a conference call, discussing the latest market developments. The outcome of the discussions for the different groups are illustrated in Fig. 1C. For simplicity we have illustrated the case where a majority opinion in a given subgroup manages to polarize the opinion of the group by changing the opinion of those who had an opinion belonging to the minority. If we take the afore mentioned group of six persons we can see that after discussing, because of the majority polarizing rule, they have all become bearish.

More realistically, we will in the following instead assume that is a certain probability for a majority opinion to prevail, and that even under certain conditions a minority could persuade a part of the majority to change their opinion.

For a given group of size k with j agents having a bullish opinion and k -j a bearish opinion, we let m k,j denote the transition probability for all (k) members to adopt the bullish opinion as a result of their meeting. After one update taking into account communications in all groups of size k with j bullish agents, the new probability of finding an agent with a bullish view in the population can therefore be written:

B(t + 1) = m k,j (t)C k j B(t) j [1 -B(t)] k-j (1)
where

C k j ≡ k! j!(k -j)! (2)
are the binomial coefficients. Notice that the transition probabilities m k,j depend on time, since we assume that they change as the market performance changes (this point will be explained further below).

Taking the sum over different groups of different sizes and different composition of bullishness within each group(see Fig. 1B) one obtains a general term, B(t + 1), for the bullishness in a population at time t + 1 due to the Documents de Travail du Centre d'Economie de la Sorbonne -2014.29 outcome of meetings of groups with different sizes and different composition of bullishness:

B(t + 1) = L ∑ k=1 a k k ∑ j=0 m k,j (t)C k j B(t) j [1 -B(t)] k-j (3) L ∑ k=1 a k = 1 ; a k ≡ 1 L (4)
With L denoting the size of the largest group and a k denoting the weight of the group of size k. The link between communication and its impact on the markets can then be taken into account by assuming that the price return r(t)

changes whenever there is a change in the bullishness. The idea is that the bullishness itself is not the relevant factor determining how prices will change. Those feeling bullish would naturally already hold long positions on the market. Rather, when people change their opinion, say becoming more negative about the market, or less bullish, this will increase their tendency to sell. Assuming the return to be proportional to the percentage change in bullishness, RB(t), as well as economic news, η(t), the return r(t) is given by:

r(t) = RB(t) λ + η(t), λ > 0 (5)
with

RB(t) = B(t)-B(t-1) B(t-1)
the change or "return" of the bullishness. The variable η(t) = r(t) -RB(t) λ is assumed to be Gaussian distributed with a mean µ ≡ 0 and a standard deviation that varies as a function of time depending on changes in sentiment. We will assume that the market will react to fundamental economic news represented by η but that the amplitude of the reaction depends on changes in the sentiment RB(t):

σ(t) = σ 0 exp ( |RB(t)| β ), σ 0 > 0, β > 0 (6)
The influence of the financial market on decision-making can now be included in a natural way by letting the strength of persuasion depend on how the market has performed since the last meeting of the market participants.

The idea is that, if for example the market had a dramatic downturn at the close yesterday, then in meetings the next morning, those with a bearish view will be more likely to convince even a bullish majority of their point of view. In the formal description below, this is taken into account by letting the transition probabilities for a change of opinion, i.e., the probabilities of transitions like Fig. 1.B → Fig. 1.C, depend on the market return over the last period: where α defines the scale for which a given return r(t) impacts the transition probabilities. The condition m k,j (t = 0) ≡ j/k describes the initially unbiased case where in average no market participant changes opinion.

m k,j (t) = m k,j (t -1) exp ( r(t) α ); m k,j (t = 0) ≡ j/k, α > 0 (7

RESULTS

As a first demonstration of the properties of the model, Eqs. (3-7), Figure 2a shows an illustration of the link between the bullishness of the market participants obtained through communication (thin solid line) and market prices (thick solid line). One observes a clear almost continuous decrease in bullishness, whereas the market itself first decreases but then regains what it lost at the end of the time period, essentially ending up unchanged compared to its initial level. This illustrates the competition between persistence through memory effects in the bullishness (1-7) and the randomness of the return via the η(t)-term in [START_REF] Shiller | Survey evidence on diffusion of interest and information among investors[END_REF]. The decaying rate of the return of the bullishness over the last part of the time period and the simultaneous rise of the prices illustrates indeed the complex and highly nonlinear relationship between returns and sentiments. The model is able to reproduce the most important of the so-called stylized facts seen in real financial market data [START_REF] Cont | Empirical properties of asset return s: stylized facts and statistical issues[END_REF]. Figure 2b illustrates clustering of the volatility of the return of the time series in Fig. 2a. Furthermore the returns show fat-tailed behavior in the returns (Fig. 2c), with fat tail exponents similar to what is found in real markets [START_REF] Plerou | Scaling of the distribution of fluctuations of financial market indicies[END_REF]. Finally let us also mention that there are no arbitrage possibilities within the framework of the model seen by the zero autocorrelations of returns (thin line, Fig. 2d), whereas long memory effects are seen in the volatility, seen via the slowly decaying autocorrelations of volatility (thick line, Fig. 2d).

Insert Figure 2 about here

Having verified that the model is indeed able to describe the main characteristics of real market data, the next question concerns how to find the optimal parameters describing real market data. Using inference based on maximum likelihood (see appendix) we have first conducted a series of simulations in order to retrieve the parameter values used to generate the simulated data (Figure 2), namely λ = 1.1, σ 0 = 0.01 , β = 0.001 and a = 400.

In order to avoid the positivity restrictions for the parameters, λ > 0, σ 0 > 0, β > 0 and α > 0 we use the logarithmic transformation, so that λ * = ln (λ), σ * 0 = ln Different starting values of the model parameters were used to ensure that the maximization/minimization algorithm converges to the true simulated parameters. We present one example of optimization using T = 5000 data points (see Table 1) noting that other values of T gave similar results. The first column of Table 1 shows the 'true' Insert Table 2 about here

( σ 2
Insert Figure 4 about here

To estimate the model parameters we apply the maximum likelihood method as mentioned above. Table 2 presents the estimates θ * of the transformed parameters θ * and their standard errors (columns 1 and 2, respectively), the corresponding estimated parameter values θ (column 3), and the value of the gradient of the log-likelihood function evaluated at the parameter estimates θ * (last column). Different starting values for the model parameters been used in the maximum likelihood estimation procedure (see Panel A-C). These results provide evidence of convergence of the maximization algorithm since the estimates found by using different starting values are very similar, and the value of the gradient of the log-likelihood evaluated at the parameter estimates is near zero.

Next, we estimate the proportion of bullishness across time and the time-varying conditional volatilities. In Figure 4, we present the FTSE-20 prices P (t), the corresponding returns r(t), the estimated bullishness proportions

B(t)

DISCUSSION

The main parts of traditional finance as well as behavioral finance, prospect theory [START_REF] Kahneman | Prospect theory: an analysis of decision-making under risk[END_REF] included, understand pricing in financial markets through individual investment behavior, thereby ignoring any interaction between market participants. We have on the contrary proposed a socio-financial model that focus on interaction between market participants via communication. We have shown how communication can impact the pricing in financial Consider the probability distribution of r(1), the first observation in the sample. Since η(t) is assumed Gaussian, the density of the first observation, conditional on B(0) and m k,j (0) = j/k (corresponding to the neutral case), takes the form

f [r(1)|B(0), m k,j (0), θ] = 1 √ 2πσ(1) exp { - 1 2σ(1) 2 [ r(1) - 1 λ RB (1) ] 2 }
, where θ = (λ, σ 0 , β, α) ′ denotes the parameter vector to be estimated. Next, conditioning on r(1), the density of the second observation r(2) is

f [r(2)|r(1), B(0), m k,j (0), θ] = 1 √ 2πσ(2) exp { - 1 2σ(2) 2 [ r(2) - 1 λ RB (2) ] 2 }
.

Proceeding in this fashion, the conditional density of the t -th observation can be calculated as

f [r(t)|r(t -1), ..., r(1), B(0), m k,j (0), θ] = 1 √ 2πσ(t) exp { - 1 2σ(t) 2 [ r(t) - 1 λ RB(t) ] 2 } .
Therefore, the likelihood of the complete sample can be written as

f [r|B(0), m k,j (0), θ] = (2π) -T 2 T ∏ t=1 σ(t) -1 • exp { - 1 2 T ∑ t=1 1 σ(t) 2 [ r(t) - 1 λ RB(t) ] 2 }
.

The log-likelihood function, denoted L T (r|θ), can be written as

L T (r|θ) = - T 2 ln (2π) - T ∑ t=1 ln σ(t) - 1 2 T ∑ t=1 1 σ(t) 2 [ r(t) - 1 λ RB(t) ] 2 .
Documents de Travail du Centre d'Economie de la Sorbonne -2014.29

Clearly, the value of θ that maximizes the conditional likelihood is identical to the value that maximizes the conditional log-likelihood.

A)

Communication in groups of different sizes leads to a majority concensus in each group

  about here Documents de Travail du Centre d'Economie de la Sorbonne -2014.29

)

  Documents de Travail du Centre d'Economie de la Sorbonne -2014.29

0 )

 0 , β * = ln (β) and α * = ln (α). Thus the parameter vector to estimate is θ * = (λ * , σ * 0 , β * , α * ) ′ . The technique of reparameterizing the model parameters is very useful in order to ensure that a numerical optimization algorithm always provides parameter values within certain specified boundaries. An attractive feature of the method of maximum likelihood estimation is its invariance to one-to-one transformations of the parameters of the log-likelihood. That is, the maximum likelihood solution is invariant under transformation of parameters. Finally, due to the highly non-linear nature of the proposed model, maximization of the conditional log-likelihood with respect to the model parameters is achieved by using numerical optimization algorithms. Documents de Travail du Centre d'Economie de la Sorbonne -2014.29

  parameter values θ = (λ, σ 0 , β, α) ′ used to simulate the data, while column 2 presents the 'true' transformed parameter values θ * = (λ * , σ * 0 , β * , α * ) ′ . The maximum likelihood estimates θ * of the transformed parameters θ * and their standard errors are presented in column 3, while column 4 shows the corresponding estimated parameter values θ. Finally, in the last column we present the value of the gradient of the log-likelihood function evaluated at the parameter estimates θ * . Different starting values for the model parameters have been used in the maximum likelihood estimation procedure (see Panel A-C). For example, the starting values used in Panel C are the 'true'parameter values used in the simulation experiments. The results presented in Table1indicate that the maximum likelihood inferential procedure provide estimates that are close to the 'true' parameter values, taking into account the estimates and the corresponding standard errors. The result provide evidence of convergence of the maximization/minimization algorithm since the estimates found by using different starting values are very similar, and the value of the gradient of the log-likelihood evaluated at the parameter estimates is near zero.Insert Figure3about here Next, we examine whether the proposed algorithm can estimate adequately the proportion of bullishness across time and the time-varying conditional volatilities. To this end, in Figure3, we present the simulated prices P (t), and the corresponding returns r(t), as well as the simulated bullishness proportions B(t) across time, for T = 5000 data points. We also present the estimated bullishness proportions B(t) and the estimated conditional volatilities σ(t) which are based on the parameter estimates of the model. Looking at Figure 3(b) and Figure 3(d) which illustrate the 'true' and the estimated bullishness proportions, respectively, we can observe that these proportions are almost identical. Thus, the proposed optimization procedure manages to correctly identify and detect the proportions of a bullish view on the simulated prices. Finally, looking at Figure 3(c) and Figure 3(e), which illustrate the simulated returns and the estimated conditional volatilities, respectively, we can see that periods of lower or higher deviation in the return series can be detected by the models volatility estimates. Therefore, the proposed model is able to capture the volatility clustering phenomenon of the return series. Application to the FTSE-20 Athens stock index: We present an empirical application of the proposed sociofinancial model to the FTSE-20 Athens stock exchange index. The idea is to consider a very volatile market to study abrupt and large changes in market performance and map the corresponding evolution in sentiments. The data consists of 1247 daily prices over the 7/11/2008-6/11/2013 period. We compute and analyse the returns of the FTSE-20 index. Documents de Travail du Centre d'Economie de la Sorbonne -2014.29

Figure 4

 4 Figure4c, one observes a steady decline in the bullishness proportions even after a relatively long (two years: from mid-2011 to mid-2013) and stable price evolution. We consequently suggest to consider this as indicating a 'bad signal" for the following FTSE-20 price levels. Finally, comparing the observed volatility of the FTSE-20, Figure4(b), and the estimated volatility from the model, Figure4(d), one notices that periods of high volatility of the market is indeed detected by volatility estimates of the socio-financial model. Therefore our proposed model can be seen as introducing an alternative measure to capture volatility clustering phenomena observed in return series of financial market data.

  markets. Opinion formation of the market participants is modeled by communication that takes place in subgroups of different sizes. Nonlinear feedback effects were introduced to describe how market performance can change sentiments and vice versa. Our socio-financial model was shown to be able to reproduce the main stylized facts of financial market data. By estimating the model parameters via maximum likelihood method we are able to study extreme behavior of markets swings and sentiments illustrated by a case of the FTSE-20 Athens stock index Documents de Travail du Centre d'Economie de la Sorbonne -2014.29 over the last five years. Recent price behavior of the FTSE-20 index seems to indicate a reversal of the bearish trend whereas the sentiment measure obtained from our method seems to indicate that the market still has room for further downturns. Finally our model introduces a new volatility measure apt of capturing volatility clustering seen in the real financial market data APPENDIX In the following we present an inferential method adopted to estimate the parameters of the model (3-7) based on the maximization of the conditional likelihood. Below we describe the calculation of the likelihood function for the model for a sample of T observations r = (r(1), r(2), ..., r(T )) under the assumption of a Normal distribution for the error process η(t).

FIG. 1 :FIG. 2 : 5 .FIG. 3 :FIG. 4 :

 12534 FIG.1:Changing the "bullishness" in a population via communications in subgroups. a): At the beginning of a given day t a certain percentage B(t) of bullishness. b): During the day communication takes place in random subgroups of different sizes. c): illustrates the extreme case of complete polarization m k,j = ±1 created by a majority rule in opinion. In general m k,j ≃ j/k corresponds to the neutral case where in average the opinion remains unchanged within a subgroup of size k. d): due to the communication in different subgroups the "bullishness" at the end of the day is different from the beginning of the day.

λ 1 . 1 λ- 9 .

 119 * 0.095 λ * 0.1738 0.8100 λ 1.1898 -0.0007 σ 0 0.01 σ * 0 -9.210 σ * 0 1891 0.0277 σ 0 0.0101 -0.0208 β 0.001 β * -6.908 β * -6.8884 0.0417 β 0.0010 -0.0219 α 400 α * 5.991 α * 5.9746 0.0382 α 393.32 0.0883 True initial: denotes the 'True' simulated parameter values θ, True trans.: denotes the 'True' transformed simulated parameter values θ * , MLEs trans: denote the maximum likelihood estimates θ * of the transformed parameters θ * , StdErr: denotes the standard errors of the transformed parameters θ * , MLEs: denote the maximum likelihood estimates θ of the parameters θ, Gradient: denotes the value of the gradient of the log-likelihood evaluated at the parameter estimates. Documents de Travail du Centre d'Economie de la Sorbonne -2014.29

Table 1 :

 1 Maximum likelihood estimation results of the simulation experiments (T = 5000).

	Panel A: Starting values λ * =-0.6931, σ * 0 =-6.4378, β * =-3.5066, α * =5.7038
	True initial True trans.	MLEs trans. StdErr MLEs	Gradient
	λ	1.1 λ * 0.095 λ *	0.1747 0.8095 λ 1.1909	0.0004
	σ 0 0.01 σ * 0 -9.210 σ * 0	-9.1891 0.0277 σ 0 0.0101	0.0087
	β 0.001 β * -6.908 β * -6.8884 0.0417 β 0.0010	-0.0167
	α 400 α * 5.991 α *	5.9746 0.0382 α 393.32	0.8311
	Panel B: Starting values λ * =-2.9957, σ * 0 =-13.8155, β * =-6.9078, α * =6.2146
	True initial True trans.	MLEs trans. StdErr MLEs	Gradient
	λ	1.1 λ * 0.095 λ *	0.1737 0.8063 λ 1.1897	-0.0048
	σ 0 0.01 σ * 0 -9.210 σ * 0	-9.1891 0.0277 σ 0 0.0101	0.0777
	β 0.001 β

* -6.908 β * -6.8883 0.0402 β 0.0010 -0.0461 α 400 α * 5.991 α * 5.9746 0.0348 α 393.29 -0.3053 Panel C: Starting values λ * =0.0953, σ * 0 =-9.2103, β * =-6.9078, α * =5.9915 True initial True trans. MLEs trans. StdErr MLEs Gradient

Table 2 :

 2 Maximum likelihood estimation results of the FTSE-20 stock index return series. Panel A: Starting values λ * =-0.6931, σ * 0 =-6.4378, β * =-3.5066, α * =5.7038 Starting values λ * =-0.6931, σ * 0 =-9.2103, β * =-4.6052, α * =6.6846 Starting values λ * =0.0953, σ * 0 =-9.2103, β * =-6.9078, α * =5.9915 MLEs trans: denote the maximum likelihood estimates θ * of the transformed parameters θ * , StdErr: denotes the standard errors of the transformed parameters θ * , MLEs: denote the maximum likelihood estimates θ of the parameters θ, Gradient: denotes the value of the gradient of the log-likelihood evaluated at the parameter estimates.

		MLEs trans. StdErr	MLEs	Gradient
	λ *	1.2825 3.7466 λ 3.6055	-0.0001
	σ * 0	-7.4537 0.0702 σ 0 0.0241	0.0001
	β * -4.5640 2.9009 β 0.0104	-0.0000
	α *	6.7058 2.9388 α 817.16	-0.0004
	Panel B: MLEs trans. StdErr	MLEs	Gradient
	λ *	1.2869 3.7417 λ 3.6214	0.0002
	σ * 0	-7.4537 0.0702 σ 0 0.0241	-0.0001
	β * -4.5615 2.8968 β 0.0104	-0.0001
	α *	6.7033 2.9345 α 815.12	-0.0001
	Panel C: MLEs trans. StdErr	MLEs	Gradient
	λ *	1.2832 3.7284 λ 3.6082	-0.0001
	σ * 0	-7.4537 0.0702 σ 0 0.0241	-0.0007
	β * -4.5641 2.8921 β 0.0104	0.0005
	α *	6.7059 2.9298 α 817.23	0.0003

Documents de Travail du Centre d'Economie de la Sorbonne -2014.29

ACKNOWLEDGMENTS

The research leading to these results has received funding from the European Union Seventh Framework Programme (FP7-SSH/2007-2013) under grant agreement n 320270 SYRTO.


Jean-Matthieu Etancelin
email: jean-matthieu.etancelin@imag.fr

Georges-Henri Cottet

Franck Pérignon

Christophe Picard

P Érignon

Multi-CPU and multi-GPU

Keywords: Multi-scale method, Hybrid computing, Remeshed particle method

published or not. The documents may come

The transport of a scalar in a turbulent flow is a central issue in many fields such as combustion, environmental flows or multiphase flows. The problem to solve consists in the coupling of the incompressible Navier-Stokes equations with an advection-diffusion equation. The ratio of the flow viscosity to the scalar diffusivity, called the Schmidt number, characterizes the dynamics of this model. For Schmidt numbers higher than one, the smallest turbulence length scale of the flow motion is larger than the one of the scalar fluctuations [START_REF] Lagaert | Hybrid spectral-particle method for the turbulent transport of a passive scalar[END_REF]. In these situations, the use of a multi-scale hybrid method is mandatory to avoid solving the flow at the smallest scale.

In our calculations, both the flow and scalar solvers rely on remeshed particle methods. These semi-Lagrangian methods were originally designed for advection dominated problems. Particle remeshing enables to preserve a regular particle distribution and to control the accuracy of the method [START_REF] Koumoutsakos | High resolution simulation of the flow around an impulsively started cylinder using vortex methods[END_REF]. High order remeshing kernels combined with dimensional splitting allows to derive methods which combine stability, even for large time-steps, high order of accuracy and affordable cost [START_REF] Cottet | High order semi-lagrangian particle methods for transport equations: numerical analysis and implementations issues[END_REF].

In practice, the two parts of the problem (flow and scalar solvers) are computed simultaneously on different processes. The hybrid computing strategy consists in defining a subset of the processes that only solves the flow using the parallel CPU advection solver from [START_REF] Lagaert | Hybrid spectral-particle method for the turbulent transport of a passive scalar[END_REF] in addition to finite differences and FFT-based Poisson solver. Meanwhile, the other processes compute the scalar transport on several GPUs by extending the library introduced in [START_REF] Cottet | High order semi-lagrangian particle methods for transport equations: numerical analysis and implementations issues[END_REF]. The proposed implementation involves three levels of parallelism: tasks split among all processes, multi-CPU solvers within the tasks, and multi-core computations using GPUs. These levels are implemented using the MPI programming model and the OpenCL framework on the GPU devices.

The benchmark shown on Figure 1 concerns the transport of a passive scalar in a turbulent plane jet in a periodic unit box. Velocity and scalar fields are initialized with hyperbolic tangent profiles complemented by a small random perturbations (see [START_REF] Magni | Accurate, non-oscillatory, remeshing schemes for particle methods[END_REF] for details). The Schmidt number is equal to 10 and the Reynolds number, based on the jet width, is equal to to 10 3 . The flow and scalar meshes use 128 3 and 512 3 grid points, respectively.

Jean-Matthieu Etancelin, Georges-Henri Cottet, Franck Pérignon and Christophe Picard The computational times per iteration are given on Table 1 for the above simulation. The overall time to solution is driven by the flow solver when using a single GPU. In the multi-GPU case, we observe on preliminary results a large difference between the total time and the computational parts. This is explained by the amount of communications, from and to GPU devices and between hosts, which are not yet fully optimized. However, in each part, we obtain a speedup close to two when doubling the amount of resources. The optimization of the necessary communications between the GPUs for the scalar transport is the subject of ongoing work.

Figure 1 :

 1 Figure 1: Side and top views of the contours of a passive scalar in a turbulent plane jet in a hybrid CPU/GPU computations with 128 3 and 512 3 grid points for the velocity and the scalar, respectively.

Table 1 :

 1 Computational times (in seconds per iteration) for a plane jet simulation on a system with two hexacore Xeon E5-2640 and two Tesla K20m.

	Configuration	Total time Flow Passive Scalar
	5 CPU + 1 GPU	2.56	2.49	0.87
	9 CPU + 1 GPU	1.43	1.36	0.86


Lex Hendriks
email: a.hendriks@uva.nl

Akin Kazakci

Imagining Knowledge, a Formal Account of Design

Design, as in designing artifacts like cars or computer programs, is one of those aspects of rational agency hardly even mentioned in traditional logical theory. As an engineering discipline, design obviously involves reasoning but seems to depend much more on a mix of factual knowledge, experimenting and imagination. We will present a formal framework for the dynamic interplay between knowledge and imagination inspired by C-K theory Hatchuel and Weil (2003a) and discuss the possible directions for further development of a 'logic of design'.

Introduction

While there are extensive studies on modeling and theorizing about design 1 , most engineers still perceive design as practice or even art. Considering the large variety of design practices, it is arguably difficult to determine a fixed universal set of core characteristics and phenomena defining design science.

Indeed, what is scientific about design? One possible answer is the specific mode of reasoning design requires. Design is a reasoning process by which new descriptions for new classes of objects are generated. Thus, the properties of this creative process constitute a core phenomenon for design research.

Yet, traditional formal theories and models of design usually treat creativity as external phenomena, for example, due to imagination, as e.g. in [START_REF] Simon | The Sciences of the Artificial[END_REF] and [START_REF] Simon | Problem forming, problem finding and problem solving in desing[END_REF]. A first category of models aim at structuring the process, G. Pahl and Beitz (1984)), and hence, they provide models for the organization of design activities.

A second category describe design as the progressive refinement of the object being designed, [START_REF] Gero | Design prototypes:a knowledge representation schema for design[END_REF] or [START_REF] Braha | Topological structures for modelling engineering design processes[END_REF], and hence, propose a model of the evolution of design artifacts. In both cases, the creative mechanisms of design reasoning are not deemed as central (i.e. the models do not seek to explain where do refinements come from or what sort of organizing allows better creativity).

One notable exception to these works is the concept-knowledge (C-K) theory of design [START_REF] Hatchuel | C-K theory: Notions and applications of a unified design theory[END_REF]2003a) and [START_REF] Hatchuel | C-K design theory: an advanced formulation[END_REF]. C-K theory places the creative generation of new definitions and objects at the heart of design through a notion of conceptual expansion. It describes design as a reasoning process where an object's identity is revised by changing its definition through the introduction of new properties. If successful, a new class of objects has been designed.

Inspired by C-K theory and building on our previous work [START_REF] Kazakci | A formalization of C-K design theory based on intuitionistic logic[END_REF], [START_REF] Hendriks | A formal account of the dual expansion of concepts and knowledge in C-K theory[END_REF], the current work presents a framework demonstrating how C-K type design reasoning can be formalized within logic.

Our contribution in the current paper is twofold. First, we introduce formal Design Operators replacing the underspecified operators of C-K theory, and next we sketch how Design Scenario's can be build with such operators to fully explain the phenomena described in C-K theory, such as the conceptual expansion mentioned before.

This opens up perspectives of automating, at least partially, C-K type design reasoning, along with many research questions. Computer aided design support has been one of the main inspirations for the application of formal methods to design theory since [START_REF] Simon | The Sciences of the Artificial[END_REF]. How precisely the use of constructive reasoning in design, e.g. intuitionistic logic, can yield a recipe for an artifact based on a proof of its possible existence, is one such a research question. The use of Design Scenarios on the other hand points toward another possible connection with logic, via dynamic logic.

The plan of the paper is as follows. Section 2 gives a summary of C-K theory and previous work on its formal foundations. Section 3 introduces the basic logical framework we use. Design stages and design concepts and body are defined and some of their properties are summarized. Section 4 introduces design operators and then extends these to Design Scenarios. Some concluding remarks can be found in section 5.

A dynamic perspective on Design Reasoning

C-K theory describes design reasoning as the dynamic interplay of knowledge and concepts. Concepts are (partial) descriptions of a new object which existence (or the impossibility of such existence) cannot be decided based on the current knowledge. Each stage in the design process is defined by its current knowledge and its current concept. The design space can be considered as the product of the knowledge space and the concept space.

• Knowledge Space The elements in the Knowledge Space are sets of knowledge, representing all the knowledge available to a designer (or to a group of designers) at a given time.

• Concept Space The elements of the Concept Space are (partial) descriptions of unknown objects that may or may not be possible to exist.

Whereas knowledge consists of true propositions, concepts are propositions whose status is unknown (based on the given knowledge). Such propositions can neither be stated as true, nor false by the designer at the moment of their creation (e.g., 'some tires are made of dust').

According to C-K theory, creative design begins by adding a new and unusual property to an existing concept C to form a new concept C ′ (e.g. 'tires for life'). The elaboration of concept can then be continued either by further expansions (tires for life are made of silicon) or by restrictions (that is by adding usual properties of the initial concept, e.g. tires for life are round). Such conceptual expansions or restrictions are called partitioning in C-K theory.

Note that, according to C-K theory, new concepts are formed by combinations of concepts occurring in the propositions of existing knowledge. The designer will use his or her body of knowledge K either to partition further the concepts, or to attempt a validation of a given concept. This last type of operation (K-validation) corresponds to the evaluation of the feasibility of a design description (e.g. could it exist).

Often the validation of a concept will not be readily possible. In order to validate concept C, new knowledge warranting the existence conditions of such an object should be acquired. In terms of C-K theory, knowledge should be expanded (K-expansion). Such new knowledge may bring new concepts into the game, allowing for new expansions and restrictions of the design concept C.

The central proposition of C-K theory is thus "design is the interaction and dual expansions concepts and knowledge" Hatchuel and Weil (2003a;[START_REF] Kazakci | A formalization of C-K design theory based on intuitionistic logic[END_REF].

Previous work on formal aspects of C-K theory

Despite the mathematical references and metaphors used in the presentation of the theory, a full mathematical presentation of the theory has not been provided to date.

Nevertheless, some steps for formalizing C-K theory have been taken in some recent work. Hatchuel and Weil (2003b) argues that there are significant similarities between the type of reasoning described by C-K theory and Forcing, a technique used in Set Theory for constructing alternative set theoretic models with desired properties. It is claimed that the parallel between Forcing and C-K theory is an important step for design theory in general but this issue needs more formal investigation.

In a complementary approach, [START_REF] Kazakci | A model of ck design theory based on based on a term logic: a formal background for a class of design assistants[END_REF]shows that C-K type reasoning can be implemented with much more simple formalisms. They use propositional term logic to model the basic ideas of C-K theory. They suggest a notion of "models of K space" to emphasize that different structures (or formalism) used to model knowledge will yield different conceptive power and degrees of flexibility in reasoning.

In [START_REF] Kazakci | A formalization of C-K design theory based on intuitionistic logic[END_REF] a first-order logical formal account of C-K theory's core notions is presented. To emphasize the constructive aspects of a design process, intuitionistic logic is used to study the interaction and expansion of concepts and knowledge, based on the definitions of the basic notions. Building on this work, [START_REF] Hendriks | A formal account of the dual expansion of concepts and knowledge in C-K theory[END_REF] complements this approach in that we consider the core proposition of the theory, the dual expansion of concepts and knowledge, and investigates the logical implications of such a principle.

Producing concepts from knowledge

One of the intriguing idea from C-K theory is the emerging of new concepts from (new) knowledge. Can we generate concepts using logic? Recall that in C-K theory for C to be a concept it has to be 'unknown', in as far as our body of knowledge allows us, whether there are instances of the concept or not. From the examples in C-K theory literature we can reconstruct a simple mechanisms at work here, using the language L K in which our body of knowledge K is represented. If we assume L K is a first order logic language enriched with a set of constants for specific individuals and predicates (the signature of L K ), it is only natural that extending K may also extend the signature. The new part of the body of knowledge may introduce new constants (Planck's h, the star Vega-β, President Obama) and predicates (being married to, prime, being the president of).

Concepts can now be generated from knowledge by recombination of expressions used in the body of knowledge. If 'Beatrix is the queen of The Netherlands' then 'x is the queen of y' is a phrase in L K . Which allows us to form an expression like: 'Planck is the queen of Vega-β'.

If we model phrases as formulas with one free variable x (a restriction we may lift later on) this amounts to forming conjunction of existing phrases, like in Boat(x) ∧ Flies(x).

Such a new phrase (if K does not imply that ∀x.Boat(x) ∧ Flies(x) and neither that ¬∀x.Boat(x) ∧ Flies(x), could be used as a concept C and starting point for design.

Combined with knowledge extension this simple mechanism will supply the design process with a wealth of new concepts without any appeal to imagination or hidden creative powers.

The mechanisms above can easily be extended further, e.g. by lifting the restriction on the type of phrases used. Like in the example 'Planck is the queen of Vega-β', where we could start the design turning this into the question 'Wouldn't it be nice if Planck is the queen of Vega-β?' This could 'branch' (in a series of steps) into 'absorbing very bright light to produce both energy and comfortable back ground illumination'.

Generating knowledge from concepts

Knowledge may guide the designer in avoiding branches that are known to be undesirable (like in case C contains some constraints on the amount of money to spend on producing a Car(x), combined with the knowledge that Gold(x) will raise the cost of production considerably).

One simple mechanism of knowledge expansion occurs when the designer (or the design team) is aware of the body of knowledge K, say there knowledge is a part of K called K 0 . Extending this K 0 could be done be Googling the web, searching Wikipedia, asking experts etc.

A second mechanism could involve further research in the field K. The experts in the field might be unable to answer the questions of the design team. Further research and experimentation could be necessary. For example we might want to use carbon-epi-hexa-fluor-plexitude for the heat shield of our Vega-β-surveyor, but it is unknown what will happen if carbon-epi-hexa-fluorplexitude is heated above 5,000 degrees Celsius.

A third mechanism may occur when we try to combine parts of theories, say the nanotechnology with the neurobiology of humain brain, say, in order to use nanotechnological devices to record firing patterns of neurons. Whether such thing is possible may be a complete new subject for scientific research.

A logical framework

In this section, we will describe the design process as generating design stages K; C , where K is some body of knowledge and C is a concept. In C-K theory one is especially interested in design stages where C is totally new for K. Here we will allow degenerated stages that can be discarded once they are seen as inconsistent or in fact not new at all (hence C turns out to be feasible already based on K).

The design process may extend a design stage in principle in infinitely many ways. One could try to imagine all 'existing' possible 'bodies of knowledge' or all 'possible concepts' and try to describe these as (a special sort of) sets, such that the operations in the design process which transform a stage K; C into a new K ′ ; C ′ can be defined as a special kind of (extended) 'search operations', not unlike known search algorithms (e.g. on databases or in linear programming).

Not only Ockham's Razor makes the 'existence' of such 'Knowledge Spaces' or 'Concept Spaces' suspect, simply from a pragmatic point of view, some sort of constructive reasoning in the design process seems to be attractive. Operations in the design process defined using 'mental images' of infinite collections containing 'lawless' sequences are certainly not constructive [START_REF] Kazakci | A formalization of C-K design theory based on intuitionistic logic[END_REF]. Note that according to C-K theory there is no 'algorithm' or construction that will determine the next step in a design path. Such a path can be seen to behave 'lawless' in Intuitionist sense [START_REF] Kazakci | A formalization of C-K design theory based on intuitionistic logic[END_REF]. Therefore, at least from a pragmatic point of view, it seems reasonable to assume that at each point in the design process the body of knowledge has a finite representation.

We introduce a basic logical framework that will allow us to represent in logical terms some of the ideas in C-K theory. We will be a little vague on the precise logical language L and the rules of the logic used. This is because we want to extend the framework later on. This could be done in different directions and we don't want to assume too much about either the expressivity of the language or the strength of the logical rules.

Without any problem the reader may assume L is the language of predicate logic and the logic is the usual classical logic (although all our results will be valid in intuitionist logic as well: we will simply not use the axiom A ∨ A or the rule A ⊢ A of classical logic). Note that, in a constructivist type of logic, like intuitionist logic, a proof of C from K is a construction, one that under a certain conditions can be used as a recipe to construct an instance of C(x) based on the constructions that exist according to K -which is a suitable characteristic for modeling design endeavor.

In our notation T ⊢ A means a formula A is provable from the set of formulas T. T * will be the set of all formulas (in L) derivable from T (T * = {A ∈ L | T ⊢ A}). So T * ⊆ S * and T ⊢ A implies S ⊢ A.

Design stages and design space

Let us model the body of knowledge (as a first approximation) as a finite set of formula, in the language of first order logic (predicate logic). Such a finite theory K will always be 'partial knowledge' and hence extendable. Again in a first approximation, we could model the 'concept' C as a formula (in predicate logic).

Our definition of design reasoning is based on the one in C-K theory, where design is defined as a reasoning activity, starting with a proposition of the design concept and proceeding by operators adding knowledge or expanding the design concept. The definition below is slightly more abstract, which would seem to allow more design stages and design steps than described in C-K theory itself. Definition 3.1. A design stage is a pair s = K; C , where K a finite set of sentences, called the body-of-knowledge of s and C a sentence, called the design concept of s.

A design stage K; C is called consistent if K ¬C. A design stage K; C is called open if K C ∨ ¬C.
A design step is a pair (s 0 , s 1 ) where s 0 and s 1 are design stages. We will often use the notation s 0 ⇒s 1 for the design step (s 0 , s 1 ). Usually we will also assume that

s 0 = K 0 ; C 0 , s 1 = K 1 ; C 1 etc. A design step s 0 ⇒s 1 is called sound, written as s 0 s ⇒s 1 , if s 1 is consistent and for all A ∈ K 0 it is true that K 1 ⊢ A (i.e. K 1 ⊢ K 0 ) and K 1 , C 1 ⊢ C 0 . Design step s 0 implies s 1 if K 0 → C 0 ⊢ K 1 → C 1 .
Design step s 0 is equivalent with s 1 if s 0 implies s 1 and s 1 implies s 0 .

We will use the notation s 0 ⊢ s 1 if s 0 implies s 1 and s 0 ≡ s 1 when they are equivalent.

Note that in the definition above there are no constraints on the type of sentence used as the design concept. Our C does not necessarily have the form ∃x.P 0 (x) ∧ . . . ∧ P n (x), where the P i are the desired properties for the object that we wish to come out of the design process. It is even not required that C is an existential formula. One can imagine for example that the 'thing' we try to design is a way of transforming all x with property A into some y with relationship R between the x and the y. So C = ∀x(A(x) → ∃yR(x, y) would be a conceivable design concept.

That we assume the body-of-knowledge K to be finite is not a real restriction in practice (at any moment of time each finite group of people could only be aware of a finite number of facts). We could allow for infinite K in principle, but this would slightly complicate our formulas like in the definition of s 0 ⊢ s 1 , where for an infinite K the conjunction K formally is not defined.

The following facts follow directly from our definitions.

Observation 1. Let s 0 = K 0 ; C 0 , s 1 = K 1 ; C 1 and s 2 = K 2 ; C 2 be design stages. 1. If s 0 is consistent then K 0 is consistent (i.e. K 0 ⊥). 2. If s 0 ⇒s 1 is sound and K 0 ⊆ K 1 then s 0 is consistent.
3. If s 0 ≡ s 1 and s 1 (or s 0 ) is consistent then s 0 ⇒s 1 is sound. 4. If s 0 ⇒s 1 and s 1 ⇒s 2 are sound, so is s 0 ⇒s 2 .

Our definition of sound design steps is one way of formalizing the intuitive notion of one design stage 'implying' another, found in most descriptions of design (i.e. in [START_REF] Hatchuel | C-K design theory: an advanced formulation[END_REF] and [START_REF] Braha | Topological structures for modelling engineering design processes[END_REF]). The informal notion is sometimes used in a loose sense of 'having some reason to go from s 0 to s 1 '. On other occasions the 'implication chain' is a series of stages where apparently more logic is involved. Our notion of soundness is a weak form of such a logical connection, whereas the defined implication (s 0 ⊢ s 1 ) is rather strong2 Two special cases of sound design steps may clarify the often observed difference in direction of the 'implication' between 'refining' the specifications and 'refining' the (structural) knowledge. Note that if K 0 = K 1 and s 0 ⇒s 1 is sound, then K 0 , C 1 ⊢ C 0 and hence s 1 ⊢ s 0 . On the other hand, if C 0 = C 1 and s 0 ⇒s 1 is sound, K 1 ⊢ K 0 and hence s 0 ⊢ s 1 . As long as we confine ourselves to sound design steps that only change the design concept or only the body-of-knowledge (which theoretically we could obviously always do by splitting up steps if necessary, see figure 3.1), we could use the implication of one stage by another as a basis3 for describing the design process (taking care each time using the right direction of the 'implication' between the states). In a more realistic model of design steps, where both parts of the state can change in one single step, the implication between states (e.g. in the way defined above) becomes awkward to deal with.

t t t ✒ ✲ ✻ s 0 s 1 s 2 s 0 s ⇒s 2 s 0 ⊢ s 1 s 1 ⊣ s 2 K 0 ; C 0 K 1 ; C 0 K 1 ; C 1
Simple examples of sound steps are:

• (Adding knowledge) K; C ⇒ K, A; C • (Adding properties) K; ∃xC(x) ⇒ K; ∃x(C(x) ∧ P(x)
• (Introducing a definition) K; C ∧ D ⇒ K, P↔C ∧ D; P An example of s 0 ⊢ s 1 where s 0 ⇒s 1 is not sound would be s 0 = K; C and

s 1 = K; C ∨ D . If K D → C then K together with C ∨ D does not imply C.
As can be concluded from the examples above, our formalism does not require the design concept to be of the form ∃xC(x) as perhaps expected from the informal description of the design concept as a description of something we seek to exist. Allowing the full generality of a first-order sentence provides us with a lot more flexibility as we will show in the sequel.

As promised in the introduction we would prove the conjecture that using sound design steps leading to a closed design stage will provide a proof for the original design concept.

Theorem 1. Let s n = K n ; C n be a closed design stage reached after n sound design steps from s

0 = K 0 ; C 0 (so K n ⊢ C n ) then K n ⊢ C 0 .
Proof. From the fact 1.3 we can conclude that s 0 s ⇒s n . Hence, by definition, K n , C n ⊢ C 0 . As K n ⊢ C n by the Cut-rule it follows that K n ⊢ C 0 . ⊣

Design scenarios

In C-K theory design operations are made within or between 'K-space' and 'C-space'. Translated into our formalism we get:

Create K -→ K; C Forming the first concept from properties of K. Refine K; C -→ K; C ′ Adding a property from K to C. Enhance K; C -→ K ′ ; C Using properties from C to find additional knowledge. Expand K; C -→ K ′ ; C ′ Combining Refine and Enhance. Validate K ′ ; C -→ K; C
Adding knowledge about the existence of C to K This may or may not add C or ¬C to K. Design concepts often are considered as (partial) definitions of an artifact, formalized as formulas φ(x) with exactly one free variable x. In the publications on C-K theory the design concept is introduced as a set of properties. The idea behind this (e.g. in [START_REF] Kazakci | A formalization of C-K design theory based on intuitionistic logic[END_REF] is that the concept C is a sentence of the form ∃x.P 1 (x) ∧ . . . P n (x) and {P 1 , . . . , P n } is the set of properties that make up the concept C.

To treat the design concept as a set of properties within our formal framework forces us to introduce properties, sets of such properties and some operator ∃ to bridge the gap between sets of properties and the propositions C introduced before. Definition 4.1. Let S be a set of properties. ∃S will be the proposition claiming the existence of an object with all the properties in S. For the design stage K; ∃S we will also use (K; S).

A further analysis of the design operations listed above shows that aparently also other basic extra-logical operations on (sets) of properties and propositions, like Query, Test and operators to extract properties from formulas, will play a role in describing the dynamic interplay between knowledge and imagination. Definition 4.2. Let A be a formula and P a predicate occurring in A, than both P( x) and ¬P( x) are properties of A. Let A be a formula and S a set, then the basic design operations are defined as: ¬ For property P, ¬P is property: the complement of P. .Prop A.Prop is the set of properties of A. ! !S is an element of S. ! ⊆ ! ⊆ S is a subset of S. ?

If S is a set of properties, ?S is a sentence (obtained by a query). test

If A is a formula test(A) is a formula.

In our extended language we will allow the use of set operations, relations (e.g. ∪, ∩, ∈, ⊆, =, ) and the contstant ∅ (for the empty set).

How exactly selection, e.g. of a subset of properties is made, or how the query ? or the test are performed we keep for now formally undefined. Several options may be investigated within (and by expanding) our formal framework. Our definition of the complement of a property does not rule out that ¬¬P is the same property as P, nor does it requires so. Usually one surely would expect ¬¬¬P to equal ¬P, but for the moment we will not fix the rules of the formalism at this level of detail.

For selection one may thinks of a random choice. The query may be a Google-search with a the English names of the properties, out of the result of which somehow some piece of knowledge is selected and translated in the language L K ′ , where K ′ may be an extension of K containing some new properties.

Scenarios

We are now ready explain what we consider the core of C-K theory, the dual expansion of concept and knowledge, in terms of our basic operations. Consider the following scenario: expand (K; S) := { P := !S; K P := ?{P}; A := !K P ; Q := !A.Prop; K P := ?{Q}; R := !{Q, ¬Q}; return (K ∪ K Q ∪ K P ; S ∪ {R}) } This scenario can be read as: Based on a property of S we expand our knowledge to K P . This new knowledge, resulting from a query, may introduce some new properties, e.g. the property Q. A query fed by Q could lead to new knowledge K Q . The set of properties for the design concept is now extended with either Q or its complement, whereas the body of knowledge meanwhile is extended with both K P and K Q .

Assume we started with K ∃S, it now might be the case that K, K P , K Q ⊢ ∃S, or even K, K P , K Q ⊢ ∃(S ∪ {R}).

The scenario above kept close to the description of expansion in C-K theory, as in [START_REF] Hatchuel | C-K theory: Notions and applications of a unified design theory[END_REF].

Conclusions

Our approach in this paper has been a mix of formal reasoning and informal analysis of engineering design, especially as described by C-K theory. The result is a formal framework with some 'dynamic 'features, without a well defined semantics.

Figure 1 :

 1 Figure 1: Splitting design steps in a K-and a C-component

Another notion of (strict) implication would be s 0 ∼ s 1 defined asK 0 ⊢ C 0 ⇒ K 1 ⊢ C 1 .Implication of states implies strict implication, but not the other way around.

This would result in a more restrictive relationship than soundness.