March 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

* Corresponding author

Received  February 2018 Revised  June 2018 Published  October 2018

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.

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
References:
[1]

D. Aregba-DriolletF. 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. BangerthR. 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. ClarelliA. 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. GiavariniM. L. SantarelliR. 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. NataliniC. NitschG. 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.

[17]

BS EN ISO 4287:2000, Geometrical product specification (GPS). Surface texture. Profile method. Terms, definitions and surface texture parameters.

show all references

References:
[1]

D. Aregba-DriolletF. 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. BangerthR. 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. ClarelliA. 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. GiavariniM. L. SantarelliR. 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. NataliniC. NitschG. 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.

[17]

BS EN ISO 4287:2000, Geometrical product specification (GPS). Surface texture. Profile method. Terms, definitions and surface texture parameters.

Figure 1.  Evolution of $c$ along the left vertical bounder for $r$ piecewise and $\nu (r)$ a) linear and b) parabolic
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
[1]

Tong Yang, Fahuai Yi. Global existence and uniqueness for a hyperbolic system with free boundary. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 763-780. doi: 10.3934/dcds.2001.7.763

[2]

François James, Nicolas Vauchelet. One-dimensional aggregation equation after blow up: Existence, uniqueness and numerical simulation. Networks & Heterogeneous Media, 2016, 11 (1) : 163-180. doi: 10.3934/nhm.2016.11.163

[3]

Emil Minchev. Existence and uniqueness of solutions of a system of nonlinear PDE for phase transitions with vector order parameter. Conference Publications, 2005, 2005 (Special) : 652-661. doi: 10.3934/proc.2005.2005.652

[4]

Robert Jensen, Andrzej Świech. Uniqueness and existence of maximal and minimal solutions of fully nonlinear elliptic PDE. Communications on Pure & Applied Analysis, 2005, 4 (1) : 199-207. doi: 10.3934/cpaa.2005.4.187

[5]

Chunxiao Guo, Fan Cui, Yongqian Han. Global existence and uniqueness of the solution for the fractional Schrödinger-KdV-Burgers system. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1687-1699. doi: 10.3934/dcdss.2016070

[6]

Dominique Blanchard, Nicolas Bruyère, Olivier Guibé. Existence and uniqueness of the solution of a Boussinesq system with nonlinear dissipation. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2213-2227. doi: 10.3934/cpaa.2013.12.2213

[7]

Thi-Bich-Ngoc Mac. Existence of solution for a system of repulsion and alignment: Comparison between theory and simulation. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3013-3027. doi: 10.3934/dcdsb.2015.20.3013

[8]

Hany A. Hosham, Eman D Abou Elela. Discontinuous phenomena in bioreactor system. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-15. doi: 10.3934/dcdsb.2018294

[9]

Shuyin Wu, Joachim Escher, Zhaoyang Yin. Global existence and blow-up phenomena for a weakly dissipative Degasperis-Procesi equation. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 633-645. doi: 10.3934/dcdsb.2009.12.633

[10]

Caixia Chen, Shu Wen. Wave breaking phenomena and global solutions for a generalized periodic two-component Camassa-Holm system. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3459-3484. doi: 10.3934/dcds.2012.32.3459

[11]

Shihui Zhu. Existence and uniqueness of global weak solutions of the Camassa-Holm equation with a forcing. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 5201-5221. doi: 10.3934/dcds.2016026

[12]

Hiroshi Matano, Yoichiro Mori. Global existence and uniqueness of a three-dimensional model of cellular electrophysiology. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1573-1636. doi: 10.3934/dcds.2011.29.1573

[13]

Gong Chen, Peter J. Olver. Numerical simulation of nonlinear dispersive quantization. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 991-1008. doi: 10.3934/dcds.2014.34.991

[14]

Nicolas Vauchelet. Numerical simulation of a kinetic model for chemotaxis. Kinetic & Related Models, 2010, 3 (3) : 501-528. doi: 10.3934/krm.2010.3.501

[15]

Petr Bauer, Michal Beneš, Radek Fučík, Hung Hoang Dieu, Vladimír Klement, Radek Máca, Jan Mach, Tomáš Oberhuber, Pavel Strachota, Vítězslav Žabka, Vladimír Havlena. Numerical simulation of flow in fluidized beds. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 833-846. doi: 10.3934/dcdss.2015.8.833

[16]

Fangfang Jiang, Junping Shi, Qing-guo Wang, Jitao Sun. On the existence and uniqueness of a limit cycle for a Liénard system with a discontinuity line. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2509-2526. doi: 10.3934/cpaa.2016047

[17]

Kota Ikeda. The existence and uniqueness of unstable eigenvalues for stripe patterns in the Gierer-Meinhardt system. Networks & Heterogeneous Media, 2013, 8 (1) : 291-325. doi: 10.3934/nhm.2013.8.291

[18]

Shouming Zhou, Chunlai Mu, Liangchen Wang. Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 843-867. doi: 10.3934/dcds.2014.34.843

[19]

Christian Heinemann, Christiane Kraus. Existence of weak solutions for a PDE system describing phase separation and damage processes including inertial effects. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2565-2590. doi: 10.3934/dcds.2015.35.2565

[20]

Hassan Khassehkhan, Messoud A. Efendiev, Hermann J. Eberl. A degenerate diffusion-reaction model of an amensalistic biofilm control system: Existence and simulation of solutions. Discrete & Continuous Dynamical Systems - B, 2009, 12 (2) : 371-388. doi: 10.3934/dcdsb.2009.12.371

2017 Impact Factor: 0.884

Metrics

  • PDF downloads (22)
  • HTML views (111)
  • Cited by (0)

[Back to Top]