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SO2 aggression to calcium carbonate stones MONUM, 05-09-08 Modelisation and simulation of sulphur dioxide aggression to calcium carbonate stones D. Aregba–Driollet IMB, Bordeaux 1 university Joint work with R. Natalini and F. Diele IAC-CNR IMB, Bordeaux 1 university D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08 Aggression of buildings by pollutants: industry, transportation, heating. Here we study a model of deterioration of calcium carbonate stones CaCO 3 by sulphur dioxide SO 2 . Formation of gypsum CaSO 4 · 2 H 2 O . Simplified one-step reaction: CaCO 3 + SO 2 + 1 20 2 → H 2 O CaSO 4 · 2 H 2 O + CO 2 IMB, Bordeaux 1 university D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08 The model � � A ∂ t ( ϕ ( c ) s ) = − ϕ ( c ) sc + d ∇ · ( ϕ ( c ) ∇ s ) m c � � A ∂ t c = − ϕ ( c ) sc m s c : density of CaCO 3 . s : porose concentration of SO 2 . ϕ : porosity. ϕ = intermediate limit of volume of void total volume ϕ ( c ) = α c + β, α, β > 0 . IMB, Bordeaux 1 university D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08 A , d : positive constants. m c , m s , m g : molecular mass of calcite, SO 2 and gypsum. The amount of gypsum g is then given by c + m c g = c 0 + m c g 0 m g m g IMB, Bordeaux 1 university D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08 Analytical results F. Guarguaglini and R. Natalini 1. Local existence is classical 2. Global existence is more di ffi cult to degenerate parabolic problem: • no a priori H¨ older estimates • no a priori H s estimates • Nonlinear term in the GRADIENT ∇ s · ∇ c : not only a L ∞ estimate • no coupling conditions (Kawashima-Shizuta) IMB, Bordeaux 1 university D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08  ∂ t ( ϕ ( c ) s ) = div( ϕ ( c ) ∇ s ) − ϕ ( c ) cs ,      (1)      ∂ t c = − ϕ ( c ) cs ,   for ( x , t ) ∈ [ Ω × [0 , T ] ( T > 0 , Ω ⊂ R N ). ϕ ( c ) = α c + β > 0 in [0 , � c 0 � ∞ ] min { β, α � c 0 � ∞ } + β } ≥ ϕ m > 0 Initial conditions s ( x , 0) = s 0 ( x ) , c ( x , 0) = c 0 ( x ) bdy conditions for s s ( x , t ) = ψ ( x , t ) for ( x , t ) ∈ ∂ Ω × (0 , T ] IMB, Bordeaux 1 university D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08 Let P = N if N > 2, P > 2 if N = 2, P = 2 if N = 1. The data c 0 , s 0 , ψ are nonnegative functions such that s 0 ∈ W 2 , P 2 ( Ω ) ∩ L ∞ ( Ω ) , c 0 ∈ W 1 , P ( Ω ) ∩ L ∞ ( Ω ) ; (2) ψ ∈ C ([0 , T ]; W 2 , P P 2 ( Ω )) ∩ W 2 , P 2 ( Ω )) ∩ C 1 ([0 , T ]; L 2 ( Q T ) for all T > 0 ; (3) the trace of the function ψ verifies ψ ∈ L ∞ ( ∂ Ω × (0 , T )) for all T > 0 . (4) IMB, Bordeaux 1 university D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08 W 2 , q ( Ω ) , ψ C ([0 , T ]; W 2 , q ( Ω )) ∩ Theorem Let s 0 , c 0 , ψ 0 , s 0 , c 0 ≥ ∈ ∈ C 1 ([0 , T ]; L q ( Ω ))) ∩ W 2 , q 2 ( Q T ) for all T > 0 and q > P . Then there ex- ists a nonnegative bounded global weak solution to problem (1), with ( s , c ) ∈ ( C ([0 , T ]; W 2 , q ( Ω )) ∩ C 1 ([0 , T ]; L q ( Ω ))) 2 . IMB, Bordeaux 1 university D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08 Numerical study of the 1D scaled model � ϕ ( c ) ∂ x s � = − ϕ ( c ) sc ∂ t ( ϕ ( c ) s ) − ∂ x ∂ t c = − ϕ ( c ) sc x ∈ Ω = ]0 , 1[ or ]0 , + ∞ [, t > 0. Initial conditions: s ( x , 0) = 0, c ( x , 0) = c 0 > 0. Boundary conditions: s (0 , t ) = s 0 ( t ), eventual Neumann condition for x = 1. IMB, Bordeaux 1 university D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08 Two approaches: finite di ff erences and finite elements. 1. Finite di ff erences: • Simple, no linear system to solve, easy to modify the time integration scheme. • Problems: – Meshing di ffi culties for 2D or 3D extensions. – Di ffi cult to increase the accuracy. 2. Finite elements: • Meshing, accuracy: great flexibility. • There is a linear system to solve. In both cases: semi-implicit treatment of the nonlinear source term No nonlinear algebraic system to solve. IMB, Bordeaux 1 university D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08 Finite di ff erences Main unknowns: ρ s = ϕ ( c ) s concentration of SO 2 , c . Approximation of of ∂ x ( a ( x ) ∂ x r ): ∆ m ( a , r ) : = ( a m + a m + 1 )( r m + 1 − r m ) − ( a m − 1 + a m )( r m − r m − 1 ) . 2 ∆ x 2 Notation: S ( ρ s , c ) = − ρ s c IMB, Bordeaux 1 university D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08 The scheme ( θ ∈ [0 , 1]):  m e − ∆ t ρ n c n + 1 = c n s , m  m    ρ n + 1 s , m − ρ n ϕ n , ρ n  � � s , m s = S ((1 − θ ) ρ n s , m + θρ n + 1 s , m , c n + 1  − ∆ m m ) .   ϕ n  ∆ t  This is a semi-implicit first order scheme. As S is linear with respect to ρ s it can be written explicitly. The di ff erential term is not implicited: this would lead to a nonsymmetric linear system. IMB, Bordeaux 1 university D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08   ρ s   Denoting u =   :        c  u n + 1 − u n = F ( u n , ∆ x ) . lim ∆ t ∆ t → 0 Higher order in time by discretization of u ′ ( t ) = F ( u n , ∆ x ) . IMB, Bordeaux 1 university D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08 Proposition For all n ≥ 0, all m = 1 , . . . , N : ρ n c n s , m ≥ 0 , m ∈ [0 , c 0 ] under the time step restriction: β ∆ x 2 ∆ t ≤ ϕ 0 + β (1 + ∆ x 2 c 0 (1 − θ )) where ϕ 0 = ϕ ( c 0 ). IMB, Bordeaux 1 university D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08 Finite elements P1 approximation for s P0 approximation for c . Denote σ ( x , t ) = s ( x , t ) − s (0 , t ): ∂ t ( ϕ ( c ) σ ) − ∂ x ( ϕ ( c ) ∂ x σ ) = F ( σ, c , t ) Functionnal space: H = { u ∈ H 1 (]0 , 1[) , u (0) = 0 } IMB, Bordeaux 1 university D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08 Variational formulation Find ( σ, c ) ∈ C 1 ([0 , + ∞ [ , H × L 2 (]0 , 1[)) such that for all ( p , q ) ∈ H × L 2 (]0 , 1[): � � �  ∂ t ϕσ pdx + ϕ∂ x σ∂ x pdx = pFdx ,     ]0 , 1[ ]0 , 1[ ]0 , 1[        � �    ∂ t ϕ ( σ + s (0 , . )) cqdx . cqdx = −     ]0 , 1[ ]0 , 1[ IMB, Bordeaux 1 university D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08 Discrete problem Regular mesh: [0 , 1] = ∪ 1 ≤ i ≤ N [ x i , x i + 1 ] , x i = ( i − 1) ∆ x , ∆ x = 1 / N . p i , i = 1 , . . . , N + 1 : classical P1 basis functions. q i , i = 1 , . . . , N :characteristic function of [ x i , x i + 1 [. The solution ( s , c ) is approximated by N + 1 N � � s h ( x , t ) = ξ i ( t ) p i ( x ) , c h ( x , t ) = η k ( t ) q k ( x ) . i = 1 k = 1 IMB, Bordeaux 1 university D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08 The discrete variational problem (without boundary condition):  ∂ t ( M ( η ) ξ ) + K ( η ) ξ = 0 ,          ∂ t η k = − ( ξ k + ξ k + 1 )   η k ( αη k + β ) = − γ k η k ( αη k + β ) , k = 1 , . . . , N .    2 M ( η ) and K ( η ) are tridiagonal symmetric matrices. � x k + 1 N � M ij ( η ) = ϕ ( η k ) p i p j dx , x k k = 1 � x k + 1 N � � � p ′ i p ′ K ij ( η ) = ϕ ( η k ) j + η k p i p j dx x k k = 1 IMB, Bordeaux 1 university D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08 Resolution 1. Fix ξ n and solve exactly on [ t n , t n + 1 ] the equation for η k , k = 1 , . . . , N : one obtains η n + 1 . 2. Fix η = η n + 1 and solve the system for ξ by the θ method: one obtains ξ n + 1 . Order two in time: by Heun scheme. One has to solve a linear system. IMB, Bordeaux 1 university D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08 Proposition Suppose that θ ∈ ]1 / 3 , 1] and ∆ x 2 < 3(3 θ − 1) . If the time c 0 step satisfies the condition ∆ x 2 ∆ x 2 θ (6 − c 0 ∆ x 2 ) < ∆ t ≤ (1 − θ )(3 + c 0 ∆ x 2 ) then for all x ∈ [0 , 1], c h ( x , . ) is a non increasing function of t and for all t ≥ 0: ρ s , h ( x , t ) ≥ 0 , c h ( x , t ) ∈ ]0 , c 0 ] . Moreover the condition is not empty. No upper bound on ∆ t for θ = 1. Uniform bound also for ρ s with additional conditions. IMB, Bordeaux 1 university D. Aregba-Driollet

SO2 aggression to calcium carbonate stones MONUM, 05-09-08 Numerical results 1. Finite di ff erence and finite element methods give comparable results 2. For the finite element method with θ = 1: good results with ∆ t = ∆ x 3. For θ = 1 / 2: the numerical order of accuracy is γ = 2. (Heun scheme for FE) 4. Order 3 in time does not improve the numerical order of accuracy for FD IMB, Bordeaux 1 university D. Aregba-Driollet