H Ghazale
email: hasan.ghazale@promes.cnrs.fr

N Mazet

P Neveu

M Perier-Muzet

Experimental analysis of a hybrid thermochemical cycle for simultaneous cold & work productions driven by low grade heat sources

Keywords: Hybrid thermochemical cycle, Heat waste recovery, Cogeneration, Cold and work production

An original tri-thermal hybrid thermochemical multifunctional cycle is proposed in this study: a sorption cycle based on a reversible endothermic/exothermic solid-gas reaction, whose originality lies in the integration of an expander on the gas line to provide mechanical work. This discontinuous cycle is able to recover medium grade waste heat between 150 and 250 °C during a charging phase, store energy in its intermediate phase, produce cold by the evaporation of the gas working fluid, and produce mechanical power by its expansion during its flow in the production phase. For this purpose, an experimental prototype with MnCl2(6/2) NH3 (solid-gas reactants) and a 1 kWe scroll expander, is developed at the laboratory. Primary results of this experimental prototype showed a validation of the hybrid thermochemical cycle concept, where cold and mechanical production are achieved continuously during the production phase. The dynamic behaviours of the components and their couplings are analysed in several thermodynamic conditions (thermodynamic equilibrium deviations (T-P) at the reactor, expander rotational speed and volumetric ratio, mechanical and cold productions…), and such experimental results lead to the validation of a numerical dynamic model to go further in the energetic, exergetic, and global performance optimization studies of the cycle (internal heat recovery of the exothermic reaction, cold temperature at the expander's outlet, control on the coupling between the expander and the electrical generator…).

Introduction

Energy policies include important issues such as the use of low-temperature heat sources (renewable, industrial waste heat), the management of variabilities in both sources and demands, and a particular attention to the growing demand of electricity and cooling. In order to respond to this, thermochemical processes called 'hybrid' combining energy multigeneration and storage functions are being developed [START_REF] Wang | A resorption cycle for the cogeneration of electricity and refrigeration[END_REF][START_REF] Thiele | Efficiency of the Lamm-Honigmann thermochemical energy storage[END_REF]. As with any thermochemical cycle, it's based on a reversible chemical reaction where one of its phases is endothermic and the second is exothermic:

𝑆 ′ + ∆ℎ 𝑟 ⇄ 𝑆 + 𝐺. In the forward direction of the reaction, known as the charging or decomposition phase, heat is supplied to the solid-gas compound S' to decompose it into another poor solid-gas compound S. The desorbed gas, G, is condensed and stored at ambient conditions. In the backward direction of the reaction, the discharging or synthesis phase, the condensed reactive fluid evaporates to flow, react with S, and reform again S' releasing the heat of the synthesis reaction. The endo or exothermic effects generated in these processes can be used to produce useful thermal effects: storage, cold and/or heat productions. The hybridization of such a cycle with a vapour power cycle by the integration of an expander between the reactor and the condenser or the evaporator of the thermochemical cycle allows adding to the thermal effects a mechanical production effect (fig. 1). Thus, this hybrid thermochemical cycle is a multifunctional cycle that valorises, beside the thermal energy, the mass flow of the reactive gas between the components.

Figure 1-Simultaneous hybrid thermochemical cycle scheme

The thermochemical solid/gas reaction as well as the liquid/vapour phase change process are mono-variant thermodynamic equilibria described by equilibrium lines in the Clausius Clapeyron diagram. In the diagrams, figures 2-4, the high temperature equilibrium (HTE) refers to the solid/gas reaction equilibrium, while the low temperature equilibrium (LTE) refers to the liquid/vapor equilibrium of the reactive fluid. Hybrid thermochemical cycles can be adapted to a wide range of heat sources temperatures, depending on the chosen reactants and their equilibria with respect to the operating conditions [START_REF] Godefroy | Hybrid thermochemical cycles for lowgrade heat storage and conversion into cold and/or power[END_REF]. Several cycle modes that differ in the phase of production of useful effects are defined for this process:

• The separate mode (fig. 2), where mechanical production occurs during the charging phase and cold effect is produced during the discharging phase. In this mode, there is no storage of the mechanical work production. • The simultaneous mode (fig. 3), where mechanical and cold productions occur simultaneously during the discharging phase after being stored. • The combined mode (fig. 4) combines the two previous modes and thus mechanical power is produced in both phases of the cycle. The application of the concept of hybridization to the solid/gas sorption cycle has only been in use since the 2010s. Despite being a relatively new area of study, the bibliography on these cycles is limited. An analysis of the available information reveals the following key points:

• Previous studies on this topic have primarily focused on static thermodynamic analyses.

They have shown the feasibility of using these cycles to recover heat at low temperatures, below 200°C [START_REF] Wang | A resorption cycle for the cogeneration of electricity and refrigeration[END_REF][START_REF] Bao | Chemisorption power generation driven by low grade heat -Theoretical analysis and comparison with pumpless ORC[END_REF]. • The dynamic behaviour has been investigated in only one study. It highlighted the significant impact of the coupling between the expander and the reactor [START_REF] Bao | Modelling a chemisorption refrigeration & power cogeneration system[END_REF].

• The proposed cycle architectures integrate two thermochemical reactors. Such processes avoid the management of liquid ammonia, but they imply delicate control of the process [START_REF] Jiang | Exploration of ammonia resorption cycle for power generation by using novel composite sorbent[END_REF]. • The functionality of these systems in regards to storage has only been addressed in one study, which investigated the use of phase change material in a heat storage system combined with a thermochemical hybrid process [START_REF] Jiang | Performance prediction on a resorption cogeneration cycle for power and refrigeration with energy storage[END_REF].

• Only one experimental study has been conducted, but the hybridization concept was not achieved. The failure was attributed to an underestimate of the impact of heat transfer resistances, resulting in an undersized reactor [START_REF] Jiang | Experimental study on a resorption system for power and refrigeration cogeneration[END_REF]. The main objectives of the research work on these hybrid cycles conducted at CNRS-PROMES since 2016 are to numerically and experimentally analyse the behaviour and performance of single-reactor hybrid thermochemical processes, which seem simpler to control. This paper provides a summary of the work conducted on the simultaneous mode. The main results of a thermodynamic analysis conducted in a static regime are presented, which was aimed at determining the potential of these cycles. Then, it summarizes the numerical dynamic analysis of the coupling between the expansion device and the thermochemical reactor. Finally, the paper describes the experimental prototype that has been designed to demonstrate the proof of concept, and presents the first results of its experimentation.

Numerical analysis

1-Steady state thermodynamic analysis

The aim of this thermodynamic study, based on equilibrium conditions and energy calculations, is to explore the potential of hybrid thermochemical cycles with different reactive salts and to retrieve the most promising ones. This study is based on usual assumptions (steady-state process, heat losses and pressure drop inside components are neglected), detailed in a previous work [START_REF] Godefroy | Hybrid thermochemical cycles for lowgrade heat storage and conversion into cold and/or power[END_REF]. Several temperature pinches are set for heat exchanges and for the deviation from chemical reaction equilibrium in the reactors. The cold source and ambient sink temperatures are set at Tcold = 0 °C, Tamb = 20 °C respectively, and a maximal hot source temperature is set at Thot,max = 250 °C, while the required heat source temperature Thot depends on the reactive salt. A review of 103 reactive ammoniated salts from a database by Touzain and from CNRS-PROMES expertise has been carried out. For given operating conditions (Tamb and Tcold), the salts selected for the analysis are those that can be decomposed with a hot source temperature < Thot,max and that can be synthesized at a pressure lower than the evaporation pressure of the reactive fluid. For each of these salts, the required heat source temperature, the energy and exergy efficiencies of the cycle, and the mechanical work production ratio are calculated as follows:

▪ energy efficiency:

𝜂 𝐼 = |𝑊 𝑚𝑒𝑐 | + 𝑄 𝑐𝑜𝑙𝑑 𝑄 ℎ𝑜𝑡
where 𝑊 𝑚𝑒𝑐 is the produced mechanical work, 𝑄 𝑐𝑜𝑙𝑑 is the cold production and 𝑄 ℎ𝑜𝑡 is the heat supplied by the hot source. ▪ exergy efficiency:

𝜂 𝑒𝑥 = |𝑊 𝑚𝑒𝑐 | + 𝐸𝑥 𝑐𝑜𝑙𝑑 𝐸𝑥 ℎ𝑜𝑡
where

𝐸𝑥 𝑐𝑜𝑙𝑑 = 𝑄 𝑐𝑜𝑙𝑑 . ( 𝑇 𝑎𝑚𝑏 𝑇 𝑐𝑜𝑙𝑑 -1)
and

𝐸𝑥 ℎ𝑜𝑡 = 𝑄 ℎ𝑜𝑡 . (1 - 𝑇 𝑎𝑚𝑏 𝑇 ℎ𝑜𝑡 )
▪ ratio of mechanical power production:

𝜏 𝑤 = |𝑊 𝑚𝑒𝑐 | |𝑊 𝑚𝑒𝑐 | + 𝑄 𝑐𝑜𝑙𝑑 Figure 5
presents the results for the most common salts. It shows that the required hot source temperature is always higher than 100 °C and that the energetic and exergetic efficiencies of the cycle are weakly dependent on the reactive salts, 𝜂 𝐼 between 40 and 50 % and 𝜂 𝑒𝑥 between 20 and 30%. However, the ratio of mechanical production (w) is strongly dependent on the salt: this ratio increases with the requested hot source temperature of the cycle. This increase in the ratio of mechanical production is directly related to the equilibrium of the salt: each salt imposes its synthesis reaction pressure at the outlet of the expander leading to variable mechanical productions and thus variable mechanical production ratios.

Figure 5-Energetic and exergetic analysis of the cycle as function of several salts and temperatures

Among this set, two salts, CaCl2 and MnCl2, are particularly interesting for the first experimental investigations. Indeed, they are well known (used in commercial thermochemical processes) and stable; they have interesting thermodynamic performances; and their decomposition temperatures are reasonable for experimental developments in a research laboratory.

2-Analysis of the coupling between the expander and the reactor

In order to analyse the dynamic behaviour of hybrid thermochemical processes, numerical models were developed for each component (reactor, condenser, evaporator, and expander). In each component, internal variables (thermodynamic state of the reactive fluid, wall temperature, reaction advancement …) were considered uniform (nodal approach). The model of the reactor, the condenser and the evaporator are based on the 1 st law of thermodynamics and mass balances, solid/gas reaction or liquid/vapour phase change kinetics, and the ideal gas assumption. As a first approach, the expander is supposed to operate in steady-state and its global performance is described by two parameters: the isentropic efficiency ηis and the volumetric efficiency ηv. Moreover, its exhaust pressure is fixed to simulate user control. The reactant pair of CaCl2.8/4NH3 is chosen and standard heat transfer coefficients have been employed, with 100 W.m -2 .K -1 for the heat exchange between the reactive bed and the wall, and 1000 W.m -2 .K -1 for the heat exchange between the wall and heat transfer fluid. The hybrid cycle, composed of these models, was coded and simulated using the Python programming language (in an object-oriented method). The coupling between the expander and the reactor involves antagonistic behaviours of these components, and it is a key issue in the operation of thermochemical hybrid cycles [START_REF] Godefroy | Analyse thermodynamique et performances dynamiques de cycles hybrides impliquant des procedes a sorption[END_REF]. In the discharging phase of the simultaneous mode, the generation of mechanical power by the expander (𝑊 ̇𝑚𝑒𝑐 ) is dependent on the pressure levels of admission (𝑃 𝑒𝑥 𝑖𝑛 ) and exhaust (𝑃 𝑒𝑥 𝑜𝑢𝑡 ), as well as the mass flow rate of ammonia (𝑚̇𝑔), expressed as:

𝑊 ̇𝑚𝑒𝑐 = 𝑚̇𝑔. 𝑔(𝑃 𝑒𝑥 𝑖𝑛 , 𝑷 𝒆𝒙 𝒐𝒖𝒕 )
In the reactor, the advancement of the synthesis reaction (X) is a function of the exhaust pressure of the expander, and expressed as:

𝑑𝑋(𝑡) 𝑑𝑡 = 𝐾 𝑠 . (1 -𝑋(𝑡)). 𝑓(𝑇 𝑅 -𝑇 𝑒𝑞 (𝑷 𝒆𝒙 𝒐𝒖𝒕 (𝑡)))
where, 𝑋(𝑡) is the advancement of the chemical reaction defined as the ratio of the number of moles of the synthetized salt (S') to the total mole number of the salt at an instant t, expressed as:

𝑋(𝑡) = 𝑛 𝑠 ′ (𝑡) 𝑛 𝑠 (𝑡) + 𝑛 𝑠 ′ (𝑡)
𝐾 𝑠 is the kinetics constant coefficient of the synthesis reaction, TR is the reactive solid temperature and 𝑇 𝑒𝑞 is the equilibrium temperature of the reactant at the pressure of the synthesis reaction i.e. the exhaust pressure of the expander. The numerical analysis of the discharging phase, for a fixed cold source temperature at 0 °C (evaporation pressure 𝑃 𝑒𝑣𝑎𝑝 = 4.26 bar), shows a variation of the mean mechanical power as the outlet pressure at the expander varies (fig. 6). While having low pressure differences at the expander's extremities, the mechanical work production is unfavourable. With varying the outlet pressure at the expander in this dynamic study between 0.5 and 3 bars, the mean mechanical power increases as the outlet pressure at the expander decreases below 3 bars until reaching an optimum of production at 1.5 bar. On the other side, while the pressure decreases more below 1.5 bars, the mean mechanical power decreases due to low mass flow rates of ammonia gas caused by a slow chemical reaction rate while getting closer to the equilibrium of the reaction (the pressure at the outlet of the expander approaches the equilibrium pressure of the synthesis reaction at the imposed temperature). From an energy point of view, as the exhaust pressure decreases, the mechanical energy increases even though the reaction's advancement slows down. This antagonistic behaviour between the expander and the reactor, leads to the definition of the optimum operating conditions for the discharging phase. A prototype of the simultaneous mode has been developed. This prototype schematized in figure 7 includes five main parts: the reactor, the expander coupled to an electric generator, an evaporator, a condenser and a throttling valve. A thermo-regulator with a heat power of 14 kW operating between 20 and 180 °C is used to heat the reactor during decomposition phase and to cool it during the synthesis one. The reversible thermochemical reaction that takes place in the reactor tubes is:

𝑀𝑛𝐶𝑙 2 . 6𝑁𝐻 3 + ∆ℎ 𝑟 ⇄ 𝑀𝑛𝐶𝑙 2 . 2𝑁𝐻 3 + 4𝑁𝐻 3
The expander is a scroll type with a nominal mechanical power of 1 kW, volumetric expansion ratio of 3.5, a displacement of 14.5 cm 3 /Rev, maximum working pressure of 13.8 bars and a nominal rotating speed of 3600 rpm bought from Airsquared. The expander is coupled through a magnetic coupling to a DC electrical generator which is electrically coupled to a rheostat (0 ≤ Relec ≤ 100 Ohm) to vary the electrical load of the electrical circuit.

The evaporator is a stainless-steel plates heat exchanger with a total heat transfer area of 0.414 m². The evaporation heat is supplied by glycolic water loop, between 0 °C and 25 °C. An electrical flow heater of 0.6 kW is connected to the outlet of the evaporator to control the superheating of the NH3 flow.

The condenser is also a stainless-steel plates heat exchanger with a total heat transfer area of 0.276 m². The condensation heat is evacuated by a pressurized water circuit circulating between 10 °C and 30 °C. The throttling valve is an electronic one, coupled to a regulator in order to control the NH3 flow rate and maintain the superheating condition at the outlet of the evaporator for a fixed value.

After a tuning phase (tightness control at high pressure and vacuum, evacuation of noncondensable gases...), classical thermochemical cycles (bypassing the expander) are carried out to test and validate the thermochemical part of the prototype before integrating the expander. The hybrid cycle is then exploited. A first discharging phase was done in an open circuit electrical condition (at the generator), to validate the concept of the hybrid thermochemical cycle. Once validated, experiments with different electrical charges were done. The purpose of such electrical variations is to study the effect of the variation of the electrical charge on the behaviour of the expander and thus the whole performance of the cycle: cold production, mechanical power, and reaction kinetics.

2-Charging phase: reproducibility and kinetics

The first decomposition was done under the following operating conditions: the inlet temperature of the reactor heat transfer fluid was set at 180 °C and that of the condenser heat transfer fluid at 12 °C imposing a condensation pressure of 7 bars. This experiment was repeated twice under the same conditions for reproducibility tests. Then, the decomposition operating conditions were changed (by decreasing the reactor's inlet temperature) to analyse the evolution of the reaction kinetics at different deviations from the thermodynamic equilibrium conditions. Figure 8 presents the advancement of the reaction for these 6 decomposition phases. It is noteworthy that X equals 1 in the event of complete synthesis of the salt (MnCl2.6NH3), and X equals 0 in the event of total decomposition of the salt (MnCl2.2NH3).

Figure 8-Evolution of the advancement of the decomposition reaction for different deviations from equilibrium

The first three decomposition reactions, carried out in the same conditions, leading to a temperature deviation from the equilibrium of the salt of 43 °C (∆𝑇 𝑒𝑞 = 𝑇 ℎ𝑡𝑓,𝑅,𝑖𝑛 -𝑇 𝑒𝑞 (𝑃 𝑅 )), have identical evolution of the advancement demonstrating the reproducibility of the decomposition phase. For the three other decomposition reactions where the inlet temperature of the condenser heat transfer fluid remains at 12 °C and the deviation from the salt's equilibrium decreases to ∆Teq,4 = 33 °C, ∆Teq,5 = 23 °C and ∆Teq,6 = 13 °C: as expected, the reaction's rate becomes slower when the deviation from equilibrium decreases. For instance, at the time when the first three decompositions were finished (X→ 0 i.e., at 350 min), the reaction's advancement was still at X = 0.1, X = 0.3 and X = 0.43 for the 4 th , 5 th and 6 th reactions, respectively.

3-Discharging phase with an open circuit electrical generator

The heat transfer fluid inlet temperature at the evaporator is fixed at 10 °C, the reactor's temperature setpoint is 35 °C and the expander freely rotates with no electrical load. The reaction's advancement and cold production profiles of this experiment are shown in figure 9, and the expander's rotational speed, pressure ratio, and mechanical production are shown in figure 10.

Before the start of the synthesis phase, the solid reactant is kept at low temperature (35 °C) and therefore the pressure in the reactor is low. At the beginning of the synthesis phase (when the reactor is connected to the evaporator through the expander) the reactor sorbs a high flow of ammonia by maintaining a high-pressure difference with the evaporator which results in a peak of cold and mechanical power. Due to the exothermic synthesis reaction, the solid reactant rises in temperature and pressure, reducing the pressure difference across the expander and the sorbed ammonia flow rate is modulated so that the heat of reaction can be dissipated by the coolant. A change in the slope of the advancement and an increase in the cold production are noted after 50 minutes. The cold power production increased from 1.5 kW to 2.5 kW (as average values) and the saturation temperature of ammonia decreased to -5 °C. During the whole discharging phase, the expander is rotating and performing mechanical work (figure 10).

In the first 50 mins, the expander rotates freely at 450 rpm producing around 4 W of mechanical work. After, the expander accelerates to around 800 rpm and produces around 10 W of mechanical power. This acceleration in the reaction that favours the production at the second period of the synthesis reaction is referred, as a first hypothesis, to a variable thermal exchange at the level of the reactor where the heat transfer fluid evacuates weaklybut increasinglythe heat of the synthesis reaction. where, 𝑊 ̇𝑚𝑒𝑐 is the mechanical power of the expander in W, 𝟂 is the rotational speed in tr/min, 𝝩 is the torque force in N.m, 𝜂 𝑒𝑥𝑝 is the efficiency of the expander, 𝑚̇𝑔 is the mass flowrate of ammonia in kg/s, ℎ 𝑒𝑥 𝑖𝑛 and ℎ 𝑒𝑥 𝑜𝑢𝑡,𝑖𝑠 are the inlet enthalpy and outlet isentropic enthalpy at the extremities of the expander respectively in J/kg. During these experiments, the effects of the electrical resistances are shown on the behaviour of the expander. These effects on the operating conditions of the expander are summarized in table 1. The analysis of these results shows a concave evolution of the maximum mechanical power with the evolution of the electrical resistance. An optimum is obtained for these experiments with a load between 57 Ohm and the blocked expander.

Conclusions

A hybrid thermochemical concept to valorise low or mid-grade heat in cold and mechanical power is analysed numerically and developed experimentally. A prototype involving MnCl2NH3(6/2) ammoniated salts and a 1 kWe scroll expander was mounted. First, the reproducibility and kinetics of the cycles were checked for several operating conditions of the reactor. In a second step, the hybrid process with the expander was exploited. A first cycle was done with no electrical charges at the generator's level to validate the concept of the integration of an expander in a thermochemical process. Other experiments were followed where the evaporating conditions were fixed and the electrical resistance at the generator was varied. These experiments showed the effect of the electrical charge on the performance of the expander: by the increase of the electrical charge (electrical potential difference at the generator), the expander rotates faster with lower coupled torque forces applied by the generator. These variations in the behaviour of the expander with the generator affected the mechanical power production and the efficiency of the expander. A presence of an important internal leakage in the expander is observed, reducing the performance of that latter. Furthermore, the synthesis reaction has shown an acceleration, which could be attributed to a possible hypothesis that suggests a weak and gradually improving heat evacuation performance at the reactor level.

More experiments are in progress to better understand the coupling between the elements of the hybrid thermochemical cycle. Several experimental protocols are to be done, such as fixing the electrical resistance at the generator and varying the evaporating pressure to analyse the behaviour of the expander in such conditions. The numerical dynamic model of the cycle is set to be validated with the experimental results to head finally towards sensibility studies, energetic and exergetic studies and analysis of the components irreversibilities, along with a global performance optimization of the cycle.

Figure 2 -Figure 3 -Figure 4 -

 234 Figure 2-Separate mode of the cycle described in Clausius Clapeyron diagram Figure 3-Simultaneous mode of the cycle described in Clausius Clapeyron diagram

Figure 6 -

 6 Figure 6-Evolution of the mean mechanical power as function of the exhaust expander's pressure for fixed evaporating conditions Experimental analysis 1-Prototype description

Figure 7 -

 7 Figure 7-Schematic description of the thermochemical hybrid prototype7 reactor tubes connected together and placed inside a cylindrical shell with baffle plates consist the whole thermochemical reactor. As noted in the static thermodynamic analysis, MnCl2 salt was selected for this prototype because it shows high thermodynamic performance, and is a well-known salt with good stability. Each reactor tube contains the reactive material composed of 0.69 kg of Expanded Natural Graphite (to enhance conductive transfer), 4.15 kg of MnCl2 charged with 3.35 kg of NH3. Mineral oil is used as a heat transfer fluid in the reactor shell. A thermo-regulator with a heat power of 14 kW operating between 20 and 180 °C is used to heat the reactor during decomposition phase and to cool it during the synthesis one. The reversible thermochemical reaction that takes place in the reactor tubes is:

Figure 9 -

 9 Figure 9-Cold power, cold temperature and advancement of the chemical reaction over time during the discharging phase

Figure 11 -

 11 Figure 11-Advancement of the reactions Figure 12-Cold production during the discharging phase

Figure 13 -

 13 Figure 13 -Ammonia volumetric flowrate at the expander's inlet

Figure 15 -

 15 Figure 15-Expander rotational speed Figure 16-Torque force applied by the generator on the expander

  3rd to 6th of September 2023 The University of Edinburgh, Scotland

Table 1 -

 1 Maximum rotational speed, torques and mechanical power of the expander for different electrical resistances

	Experiment	Electrical	𝟂max	𝝩max (N.m)	𝑾 ̇𝒎𝒂𝒙 (W)
		resistance (Ω)	(rev/min)
	1	-	0	1.2	0
	2	7	350	0.6	22
	3	57	500	0.3	16
	4	100	600	0.25	15
	5	∞	800	0.1	10

Acknowledgments

This work has been financially supported by the French National Research Agency (ANR) for the purpose of the ThermHyVal project under contract ANR-20-CE05-0036 (https://anr.fr/Projet-ANR-20-CE05-0036) and by a prematuration program of the French National Centre for Scientific Research (CNRS).


Mohamed Aziz Boukraa

Franck Delvare

Fading regularization method and DKQ plate elements for the Cauchy problem associated with the biharmonic equation

mohamed.boukraa@uca.fr Motivation: Inverse problems in thin plate bending Thin plates find applications in several fields of science and engineering. We recall here the case of plate elements for a bridge deck. In some situations, it may be mandatory to ensure the embedding and support of these plates for the correct installation of this structure. However, direct measurements may be impossible or difficult to obtain during construction or during maintenance. An inverse problem then arises! Objectives : The aim of this study is to solve the inverse Cauchy-type problem for the bilaplacian operator, which governs the bending of thin plates, such as boundary conditions are available only on part of the boundary. We propose to apply the fading regularization method and to implement it numerically using Discrete Kirchhoff (DK) plate elements. The constitutive equation for the bending plate problem, under Kirchhoff-Love theory, is :

canambridges.com

Ω (L∇) t D(L∇)w D∆ 2 w δw dxdy = Ω q(x, y)δω dxdy + Γ M n ∂δw ∂n -V n δw ds + i δw i R i .
It is characterized by the biharmonic operator to which boundary conditions, related to the transverse displacement w, the normal rotation ∂w ∂n , the moment M n and the shear force V n , are added at any point of the boundary.

M n = -D ∆w + (1 -ν) 2n x n y ∂ 2 w ∂x∂y -n 2 y ∂ 2 w ∂x 2 -n 2 x ∂ 2 w ∂y 2 V n = -D ∂∆w ∂n + (1 -ν) ∂ ∂s n x n y ( ∂ 2 w ∂y 2 - ∂ 2 w ∂x 2 ) + (n 2 x -n 2 y ) ∂ 2 w

∂x∂y

These boundary conditions are not always accessible to measurements. Hence, in the case of an inaccessible part Γ i of the boundary and an over-specified part of the boundary Γ d , the resulting Cauchy problem is ill-posed in the sense of Hadamard. Neglecting the shear force by applying discrete Kirchhoff's theory (at corner nodes and element edges)

             ∆ 2 w = 0 ∀x ∈ Ω w = ϕ d ∀x ∈ Γ d w ,n = ψ d ∀x ∈ Γ d M n = M d ∀x ∈ Γ d V n = V d ∀x ∈ Γ d q ∂Ω = Γ d ∪ Γ i et Γ d ∩ Γ i = ∅.
1 4 3 2 w 1 θ x 1 θ y 1 ∆ s 5 ∆ s 6 ∆ s 7 ∆ s 8
3 degrees of freedom per node :

w = N d, d =   d 1 d 2 . . .   , d i =   w i θ x i θ y i   =   w i w ,y i -w ,x i  
The stiffness matrix of the DKQ element is defined as

K = Ω (L∇w) t D (L∇w) dΩ where L∇w =         ∂θ y ∂x ∂θ x ∂y ∂θ x ∂y + ∂θ y ∂x         =          - ∂ 2 w ∂x 2 ∂ 2 w ∂y 2 2 ∂ 2 w ∂x∂y         
The corresponding finite element formulation is :

E(V) := Kd -F b = 0, such as b = (M n , V n , R).
The fading regularization method

• Idea: Find the solution of the equilibrium equation that best fits the boundary data while maintaining independence of the regularization parameter and stability towards noisy data

V k+1 = Argmin V∈H(Γ) V -Φ d 2 H(Γ d ) + c V -V k 2 H(Γ)
• The fading regularization algorithm:

V k+1 = Argmin V∈R 5N J k+1 c (V) with V = (d, M n , V n ) = (W, θ ,x , θ ,y , M n , V n ) under the equality constraints E(V) = 0
where

J k+1 c (V) = W |Γ d -φ d 2 L 2 (Γ d ) + n y θ ,x + n x θ ,y |Γ d -µ d 2 L 2 (Γ d ) + M n|Γ d -M d 2 L 2 (Γ d ) + V n|Γ d -V d 2 L 2 (Γ d ) + c W -W k 2 L 2 (Γ) + θ ,x -θ ,x k 2 L 2 (Γ) + θ ,y -θ ,y k 2 L 2 (Γ) + M n -M k n 2 L 2 (Γ) + V n -V k n 2 L 2 (Γ)

Numerical simulations

We consider the biharmonic analytical solution:

u an (x, y) = 1 2 x(sin x cosh y -cos x sinh y), (x, y) ∈ Ω,
where Ω is a two-dimensional domain such as Cauchy data are given only on a part Γ d of its boundary Γ.

Two example of domains are considered :

Γ i Γ d (a) Γ d Γ d Γ i Γ i (b)

Figure 1 :

 1 Figure 1: Plate elements for a bridge deck

  No boundary condition is given on the part Γ i ! Discrete Kirchhoff plate element

Figure 2 :Figure 3 :Figure 4 :

 234 Figure 3: Reconstructions of the boundary conditions on the inner boundary of an annular domain (Fig. 2a) such as data are given on the outer boundary.