Figure 1.
Evolution of $c$ along the left vertical bounder for $r$ piecewise and $\nu (r)$ a) linear and b) parabolic
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2019, 18(2): 977-998.
doi: 10.3934/cpaa.2019048
A nonlinear model for marble sulphation including surface rugosity: Theoretical and numerical results
| 1. | Dipartimento di Matematica "F. Enriques", Università degli Studi di Milano, Via C. Saldini 50, 20133 Milano, Italy |
| 2. | Istituto di Matematica Applicata e Tecnologie Informatiche "Enrico Magenes", CNR, Via Ferrata 1, 27100 Pavia, Italy |
| 3. | Dipartimento di Ingegneria e Architettura, Università degli Studi di Parma, Parco Area delle Scienze 181/A, 43124 Parma, Italy |
| 4. | Dipartimento di Matematica, Politecnico di Milano, Via E. Bonardi 9, 20133 Milano, Italy |
| 5. | Istituto per le Applicazioni del Calcolo "M. Picone", CNR, Via dei Taurini 19, 00185 Roma, Italy |
We consider an evolution system describing the phenomenon of marble sulphation of a monument, accounting of the surface rugosity. We first prove a local in time well posedness result. Then, stronger assumptions on the data allow us to establish the existence of a global in time solution. Finally, we perform some numerical simulations that illustrate the main feature of the proposed model.
Keywords:
Global existence and uniqueness,
bulk-surface PDE system,
sulphation phenomena,
numerical simulation.
Mathematics Subject Classification:
Primary: 35M33; Secondary: 65M60, 74N25.
Citation:
Elena Bonetti, Cecilia Cavaterra, Francesco Freddi, Maurizio Grasselli, Roberto Natalini. A nonlinear model for marble sulphation including surface rugosity: Theoretical and numerical results. Communications on Pure & Applied Analysis,
2019, 18
(2)
: 977-998.
doi: 10.3934/cpaa.2019048
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References:
| [1] |
D. Aregba-Driollet, F. Diele and R. Natalini,
A mathematical model for the sulphur dioxide aggression to calcium carbonate stones: numerical approximation and asymptotic analysis, SIAM J. Appl. Math., 64 (2004), 1636-1667.
doi: 10.1137/S003613990342829X. |
| [2] |
C. Baiocchi,
Sulle equazioni differenziali astratte lineari del primo e del secondo ordine negli spazi di Hilbert, Ann. Mat. Pura Appl., 76 (1967), 233-304.
doi: 10.1007/BF02412236. |
| [3] |
W. Bangerth, R. Hartmann and G. Kanschat,
deal.II - A general-purpose object-oriented finite element library, ACM Trans. Math. Softw., 33 (2007), 1-27.
doi: 10.1145/1268776.1268779. |
| [4] |
E. Bonetti and M. Frémond,
Analytical results on a model for damaging in domains and interfaces, ESAIM Control Optim. Calc. Var., 17 (2011), 955-974.
doi: 10.1051/cocv/2010033. |
| [5] |
F. Clarelli, A. Fasano and R. Natalini,
Mathematics and monument conservation: free boundary models of marble sulfation, SIAM J. Appl. Math., 69 (2008), 149-168.
doi: 10.1137/070695125. |
| [6] |
F. Freddi and G. Royer-Carfagni,
Regularized variational theories of fracture: A unified approach, Journal of the Mechanics and Physics of Solids, 58 (2010), 1154-1174.
doi: 10.1016/j.jmps.2010.02.010. |
| [7] |
M. Frémond, Non-smooth Thermomechanics, Springer-Verlag, Berlin, 2002.
doi: 10.1007/978-3-662-04800-9. |
| [8] |
C. Giavarini, M. L. Santarelli, R. Natalini and F. Freddi,
A non-linear model of sulphation of porous stones: Numerical simulations and preliminary laboratory assessments, Journal of Cultural Heritage, 9 (2008), 14-22.
doi: 10.1016/j.culher.2007.12.001. |
| [9] |
F. R. Guarguaglini and R. Natalini,
Global existence of solutions to a nonlinear model of sulphation phenomena in calcium carbonate stones, Nonlinear Anal., 6 (2005), 477-494.
doi: 10.1016/j.nonrwa.2004.09.007. |
| [10] |
F. R. Guarguaglini and R. Natalini,
Nonlinear transmission problems for quasilinear diffusion systems, Netw. Heterog. Media, 2 (2007), 359-381.
doi: 10.3934/nhm.2007.2.359. |
| [11] |
F. R. Guarguaglini and R. Natalini,
Fast reaction limit and large time behavior of solutions to a nonlinear model for sulphation phenomena, Comm. Partial Differential Equations, 32 (2007), 163-189.
doi: 10.1080/03605300500361438. |
| [12] |
A. Mielke,
A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems, Nonlinearity, 24 (2011), 1329-1346.
doi: 10.1088/0951-7715/24/4/016. |
| [13] |
R. Natalini, C. Nitsch, G. Pontrelli and S. Sbaraglia,
A numerical study of a nonlocal model of damage propagation under chemical aggression, European J. Appl. Math., 14 (2003), 447-464.
doi: 10.1017/S0956792503005205. |
| [14] |
C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.
|
| [15] |
M. Taylor,
Partial Differential Equations I. Basic Theory. Applied Mathematical Sciences,
115, Springer, New York, 2011.
doi: 10.1007/978-1-4419-7055-8. |
| [16] |
D. Whitehouse, Surfaces and Their Measurement, Butterworth-Heinemann, Boston, 2012.Google Scholar |
| [17] |
BS EN ISO 4287:2000, Geometrical product specification (GPS). Surface texture. Profile method. Terms, definitions and surface texture parameters.Google Scholar |
show all references
References:
| [1] |
D. Aregba-Driollet, F. Diele and R. Natalini,
A mathematical model for the sulphur dioxide aggression to calcium carbonate stones: numerical approximation and asymptotic analysis, SIAM J. Appl. Math., 64 (2004), 1636-1667.
doi: 10.1137/S003613990342829X. |
| [2] |
C. Baiocchi,
Sulle equazioni differenziali astratte lineari del primo e del secondo ordine negli spazi di Hilbert, Ann. Mat. Pura Appl., 76 (1967), 233-304.
doi: 10.1007/BF02412236. |
| [3] |
W. Bangerth, R. Hartmann and G. Kanschat,
deal.II - A general-purpose object-oriented finite element library, ACM Trans. Math. Softw., 33 (2007), 1-27.
doi: 10.1145/1268776.1268779. |
| [4] |
E. Bonetti and M. Frémond,
Analytical results on a model for damaging in domains and interfaces, ESAIM Control Optim. Calc. Var., 17 (2011), 955-974.
doi: 10.1051/cocv/2010033. |
| [5] |
F. Clarelli, A. Fasano and R. Natalini,
Mathematics and monument conservation: free boundary models of marble sulfation, SIAM J. Appl. Math., 69 (2008), 149-168.
doi: 10.1137/070695125. |
| [6] |
F. Freddi and G. Royer-Carfagni,
Regularized variational theories of fracture: A unified approach, Journal of the Mechanics and Physics of Solids, 58 (2010), 1154-1174.
doi: 10.1016/j.jmps.2010.02.010. |
| [7] |
M. Frémond, Non-smooth Thermomechanics, Springer-Verlag, Berlin, 2002.
doi: 10.1007/978-3-662-04800-9. |
| [8] |
C. Giavarini, M. L. Santarelli, R. Natalini and F. Freddi,
A non-linear model of sulphation of porous stones: Numerical simulations and preliminary laboratory assessments, Journal of Cultural Heritage, 9 (2008), 14-22.
doi: 10.1016/j.culher.2007.12.001. |
| [9] |
F. R. Guarguaglini and R. Natalini,
Global existence of solutions to a nonlinear model of sulphation phenomena in calcium carbonate stones, Nonlinear Anal., 6 (2005), 477-494.
doi: 10.1016/j.nonrwa.2004.09.007. |
| [10] |
F. R. Guarguaglini and R. Natalini,
Nonlinear transmission problems for quasilinear diffusion systems, Netw. Heterog. Media, 2 (2007), 359-381.
doi: 10.3934/nhm.2007.2.359. |
| [11] |
F. R. Guarguaglini and R. Natalini,
Fast reaction limit and large time behavior of solutions to a nonlinear model for sulphation phenomena, Comm. Partial Differential Equations, 32 (2007), 163-189.
doi: 10.1080/03605300500361438. |
| [12] |
A. Mielke,
A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems, Nonlinearity, 24 (2011), 1329-1346.
doi: 10.1088/0951-7715/24/4/016. |
| [13] |
R. Natalini, C. Nitsch, G. Pontrelli and S. Sbaraglia,
A numerical study of a nonlocal model of damage propagation under chemical aggression, European J. Appl. Math., 14 (2003), 447-464.
doi: 10.1017/S0956792503005205. |
| [14] |
C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.
|
| [15] |
M. Taylor,
Partial Differential Equations I. Basic Theory. Applied Mathematical Sciences,
115, Springer, New York, 2011.
doi: 10.1007/978-1-4419-7055-8. |
| [16] |
D. Whitehouse, Surfaces and Their Measurement, Butterworth-Heinemann, Boston, 2012.Google Scholar |
| [17] |
BS EN ISO 4287:2000, Geometrical product specification (GPS). Surface texture. Profile method. Terms, definitions and surface texture parameters.Google Scholar |
Figure 2.
Evolution of $s$ along the left vertical bounder for $r$ piecewise and $\nu (r)$ a) linear and b) parabolic
Figure 3.
Evolution of $r$ along the left vertical bounder for $r$ piecewise and $\nu (r)$ a) linear and b) parabolic
Figure 4.
Evolution of $c$ along a horizontal line within the solid located at a) $x_2 = 0.25$ and b) $x_2 = 0.75$ assuming parabolic relationship for $\nu (r)$
Figure 5.
Evolution of $s$ along a horizontal line within the solid located at a) $x_2 = 0.25$ and b) $x_2 = 0.75$ assuming parabolic relationship for $\nu (r)$
Figure 6.
Concentration of $SO_2$ within the solid at different time step a) $n = 5$ and b) $n = 15$ assuming parabolic relationship for $\nu (r)$
Figure 9.
Evolution of $r$ along the left vertical bounder for $r$ random and $\nu (r)$ a) linear and b) parabolic
Figure 7.
Evolution of $c$ along the left vertical bounder for $r$ random and $\nu (r)$ a) linear and b) parabolic
Figure 8.
Evolution of $s$ along the left vertical bounder for $r$ random and $\nu (r)$ a) linear and b) parabolic
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