A space-time adaptive finite element method with exponential time integrator for the phase field model of pitting corrosion

Gao, Huadong; Ju, Lili; Li, Xiao; Duddu, Ravindra

doi:10.1016/j.jcp.2019.109191

Title: A space-time adaptive finite element method with exponential time integrator for the phase field model of pitting corrosion

Journal Article · 19 December 2019 · Journal of Computational Physics

DOI: https://doi.org/10.1016/j.jcp.2019.109191 · OSTI ID:1803035

Gao, Huadong ^[1]; Ju, Lili ^[2]; Li, Xiao ^[3];

^[4]

Huazhong Univ. of Science and Technology, Wuhan (China); Univ. of South Carolina, Columbia, SC (United States)
Univ. of South Carolina, Columbia, SC (United States)
Univ. of South Carolina, Columbia, SC (United States); The Hong Kong Polytechnic Univ. (Hong Kong)
Vanderbilt Univ., Nashville, TN (United States)

In this paper we propose a space-time adaptive finite element method for the phase field model of pitting corrosion, which is a parabolic partial differential equation system consisting of a phase variable and a concentration variable. A major challenge in solving this phase field model is that the problem is very stiff, which makes the time step size extremely small for standard temporal discretizations. Another difficulty is that a high spatial resolution is required to capture the steep gradients within the diffused interface, which results in very large number of degrees of freedom for uniform meshes. To overcome the stiffness of this model, we combine the Rosenbrock–Euler exponential integrator with Crank–Nicolson scheme for the temporal discretization. Moreover, by exploiting the fact that the speed of the corroding interface decreases with time, we derive an adaptive time stepping formula. For the spatial approximation, we propose a simple and efficient strategy to generate adaptive meshes that reduces the computational cost significantly. Thus, the proposed method utilizes local adaptivity and mesh refinement for efficient simulation of the corrosive dissolution over long times in heterogeneous media with complex microstructures. We also present an extensive set of numerical experiments in both two and three dimensional spaces to demonstrate efficiency and robustness of the proposed method.

View Accepted Manuscript (DOE)

Cite

Export

Save

Research Organization:: Univ. of South Carolina, Columbia, SC (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)

Grant/Contract Number:: SC0016540; SC0020270

OSTI ID:: 1803035

Journal Information:: Journal of Computational Physics, Journal Name: Journal of Computational Physics Vol. 406; ISSN 0021-9991

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

References (60)

Approximating the large sparse matrix exponential using incomplete orthogonalization and Krylov subspaces of variable dimension: Approximating the large sparse matrix exponential Vo, Huy D.; Sidje, Roger B. Numerical Linear Algebra with Applications, Vol. 24, Issue 3 https://doi.org/10.1002/nla.2090	journal	March 2017
Algorithms for Phase Field Computation of the Dendritic Operating State at Large Supercoolings Wang, Shun-Lien; Sekerka, Robert F. Journal of Computational Physics, Vol. 127, Issue 1 https://doi.org/10.1006/jcph.1996.0161	journal	August 1996
Adaptive Mesh Refinement Computation of Solidification Microstructures Using Dynamic Data Structures Provatas, Nikolas; Goldenfeld, Nigel; Dantzig, Jonathan Journal of Computational Physics, Vol. 148, Issue 1 https://doi.org/10.1006/jcph.1998.6122	journal	January 1999
Modeling Melt Convection in Phase-Field Simulations of Solidification Beckermann, C.; Diepers, H. -J; Steinbach, I. Journal of Computational Physics, Vol. 154, Issue 2 https://doi.org/10.1006/jcph.1999.6323	journal	September 1999
A method for exponential propagation of large systems of stiff nonlinear differential equations Friesner, Richard A.; Tuckerman, Laurette S.; Dornblaser, Bright C. Journal of Scientific Computing, Vol. 4, Issue 4 https://doi.org/10.1007/BF01060992	journal	December 1989
Numerical modeling of corrosion pit propagation using the combined extended finite element and level set method Duddu, Ravindra Computational Mechanics, Vol. 54, Issue 3 https://doi.org/10.1007/s00466-014-1010-8	journal	April 2014
A Posteriori Error Estimates and an Adaptive Finite Element Method for the Allen–Cahn Equation and the Mean Curvature Flow Feng, Xiaobing; Wu, Hai-jun Journal of Scientific Computing, Vol. 24, Issue 2 https://doi.org/10.1007/s10915-004-4610-1	journal	August 2005
Computation of dendrites using a phase field model Wheeler, A. A.; Murray, B. T.; Schaefer, R. J. Physica D: Nonlinear Phenomena, Vol. 66, Issue 1-2 https://doi.org/10.1016/0167-2789(93)90242-S	journal	June 1993
A phase field concept for multiphase systems Steinbach, I.; Pezzolla, F.; Nestler, B. Physica D: Nonlinear Phenomena, Vol. 94, Issue 3 https://doi.org/10.1016/0167-2789(95)00298-7	journal	July 1996
Prediction of dendritic growth and microsegregation patterns in a binary alloy using the phase-field method Warren, J. A.; Boettinger, W. J. Acta Metallurgica et Materialia, Vol. 43, Issue 2 https://doi.org/10.1016/0956-7151(94)00285-P	journal	February 1995
Pit growth studies in stainless steel foils. I. Introduction and pit growth kinetics Ernst, P.; Newman, R. C. Corrosion Science, Vol. 44, Issue 5 https://doi.org/10.1016/S0010-938X(01)00133-0	journal	May 2002
Electrochemical micromachining, polishing and surface structuring of metals: fundamental aspects and new developments Landolt, D.; Chauvy, P. -F.; Zinger, O. Electrochimica Acta, Vol. 48, Issue 20-22 https://doi.org/10.1016/S0013-4686(03)00368-2	journal	September 2003
A quantitative and thermodynamically consistent phase-field interpolation function for multi-phase systems Moelans, Nele Acta Materialia, Vol. 59, Issue 3 https://doi.org/10.1016/j.actamat.2010.10.038	journal	February 2011
Efficient simulation of geothermal processes in heterogeneous porous media based on the exponential Rosenbrock–Euler and Rosenbrock-type methods Tambue, A.; Berre, I.; Nordbotten, J. M. Advances in Water Resources, Vol. 53 https://doi.org/10.1016/j.advwatres.2012.12.004	journal	March 2013
An introduction to phase-field modeling of microstructure evolution Moelans, Nele; Blanpain, Bart; Wollants, Patrick Calphad, Vol. 32, Issue 2 https://doi.org/10.1016/j.calphad.2007.11.003	journal	June 2008
A simple moving mesh method for one- and two-dimensional phase-field equations Tan, Zhijun; Tang, Tao; Zhang, Zhengru Journal of Computational and Applied Mathematics, Vol. 190, Issue 1-2 https://doi.org/10.1016/j.cam.2005.01.042	journal	June 2006
An efficient second-order linear scheme for the phase field model of corrosive dissolution Gao, Huadong; Ju, Lili; Duddu, Ravindra Journal of Computational and Applied Mathematics, Vol. 367 https://doi.org/10.1016/j.cam.2019.112472	journal	March 2020
Isogeometric analysis of the Cahn–Hilliard phase-field model Gómez, Héctor; Calo, Victor M.; Bazilevs, Yuri Computer Methods in Applied Mechanics and Engineering, Vol. 197, Issue 49-50 https://doi.org/10.1016/j.cma.2008.05.003	journal	September 2008
Finite Volume model for diffusion- and activation-controlled pitting corrosion of stainless steel Scheiner, Stefan; Hellmich, Christian Computer Methods in Applied Mechanics and Engineering, Vol. 198, Issue 37-40 https://doi.org/10.1016/j.cma.2009.04.012	journal	August 2009
Computational simulation of multi-pit corrosion process in materials Pidaparti, Ramana M.; Fang, Long; Palakal, Mathew J. Computational Materials Science, Vol. 41, Issue 3 https://doi.org/10.1016/j.commatsci.2007.04.017	journal	January 2008
Stable pitting corrosion of stainless steel as diffusion-controlled dissolution process with a sharp moving electrode boundary Scheiner, S.; Hellmich, C. Corrosion Science, Vol. 49, Issue 2 https://doi.org/10.1016/j.corsci.2006.03.019	journal	February 2007
A numerical framework for the modeling of corrosive dissolution Sarkar, Swarnavo; Warner, James E.; Aquino, Wilkins Corrosion Science, Vol. 65 https://doi.org/10.1016/j.corsci.2012.08.059	journal	December 2012
A phase field model for simulating the pitting corrosion Mai, Weijie; Soghrati, Soheil; Buchheit, Rudolph G. Corrosion Science, Vol. 110 https://doi.org/10.1016/j.corsci.2016.04.001	journal	September 2016
A phase field model for simulating the stress corrosion cracking initiated from pits Mai, Weijie; Soghrati, Soheil Corrosion Science, Vol. 125 https://doi.org/10.1016/j.corsci.2017.06.006	journal	August 2017
New phase field model for simulating galvanic and pitting corrosion processes Mai, Weijie; Soghrati, Soheil Electrochimica Acta, Vol. 260 https://doi.org/10.1016/j.electacta.2017.12.086	journal	January 2018
A sequential non-iterative approach for modeling multi-ionic species reactive transport during localized corrosion Sun, Xiangming; Duddu, Ravindra Finite Elements in Analysis and Design, Vol. 166 https://doi.org/10.1016/j.finel.2019.103318	journal	November 2019
Efficient integration of large stiff systems of ODEs with exponential propagation iterative (EPI) methods Tokman, M. Journal of Computational Physics, Vol. 213, Issue 2 https://doi.org/10.1016/j.jcp.2005.08.032	journal	April 2006
Spectral implementation of an adaptive moving mesh method for phase-field equations Feng, W. M.; Yu, P.; Hu, S. Y. Journal of Computational Physics, Vol. 220, Issue 1 https://doi.org/10.1016/j.jcp.2006.07.013	journal	December 2006
An efficient fully linearized semi-implicit Galerkin-mixed FEM for the dynamical Ginzburg–Landau equations of superconductivity Gao, Huadong; Sun, Weiwei Journal of Computational Physics, Vol. 294 https://doi.org/10.1016/j.jcp.2015.03.057	journal	August 2015
An efficient exponential time integration method for the numerical solution of the shallow water equations on the sphere Gaudreault, Stéphane; Pudykiewicz, Janusz A. Journal of Computational Physics, Vol. 322 https://doi.org/10.1016/j.jcp.2016.07.012	journal	October 2016
An exponential time-integrator scheme for steady and unsteady inviscid flows Li, Shu-Jie; Luo, Li-Shi; Wang, Z. J. Journal of Computational Physics, Vol. 365 https://doi.org/10.1016/j.jcp.2018.03.020	journal	July 2018
An adaptive variational procedure for the conservative and positivity preserving Allen–Cahn phase-field model Joshi, Vaibhav; Jaiman, Rajeev K. Journal of Computational Physics, Vol. 366 https://doi.org/10.1016/j.jcp.2018.04.022	journal	August 2018
Phase-field simulations of solidification in binary and ternary systems using a finite element method Danilov, D.; Nestler, B. Journal of Crystal Growth, Vol. 275, Issue 1-2 https://doi.org/10.1016/j.jcrysgro.2004.10.139	journal	February 2005
Peridynamic modeling of pitting corrosion damage Chen, Ziguang; Bobaru, Florin Journal of the Mechanics and Physics of Solids, Vol. 78 https://doi.org/10.1016/j.jmps.2015.02.015	journal	May 2015
Implementation of a restarted Krylov subspace method for the evaluation of matrix functions Afanasjew, Martin; Eiermann, Michael; Ernst, Oliver G. Linear Algebra and its Applications, Vol. 429, Issue 10 https://doi.org/10.1016/j.laa.2008.06.029	journal	November 2008
Computing matrix functions Higham, Nicholas J.; Al-Mohy, Awad H. Acta Numerica, Vol. 19 https://doi.org/10.1017/S0962492910000036	journal	May 2010
Exponential integrators Hochbruck, Marlis; Ostermann, Alexander Acta Numerica, Vol. 19 https://doi.org/10.1017/S0962492910000048	journal	May 2010
Phase-field model of pitting corrosion kinetics in metallic materials Ansari, Talha Qasim; Xiao, Zhihua; Hu, Shenyang npj Computational Materials, Vol. 4, Issue 1 https://doi.org/10.1038/s41524-018-0089-4	journal	July 2018
Unitary quantum time evolution by iterative Lanczos reduction Park, Tae Jun; Light, J. C. The Journal of Chemical Physics, Vol. 85, Issue 10 https://doi.org/10.1063/1.451548	journal	November 1986
An extended finite-element model coupled with level set method for analysis of growth of corrosion pits in metallic structures Vagbharathi, A. S.; Gopalakrishnan, S. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 470, Issue 2168 https://doi.org/10.1098/rspa.2014.0001	journal	August 2014
Phase-field method for computationally efficient modeling of solidification with arbitrary interface kinetics Karma, Alain; Rappel, Wouter-Jan Physical Review E, Vol. 53, Issue 4 https://doi.org/10.1103/PhysRevE.53.R3017	journal	April 1996
Phase-field model for binary alloys Kim, Seong Gyoon; Kim, Won Tae; Suzuki, Toshio Physical Review E, Vol. 60, Issue 6 https://doi.org/10.1103/PhysRevE.60.7186	journal	December 1999
New Approach to Many-State Quantum Dynamics: The Recursive-Residue-Generation Method Nauts, André; Wyatt, Robert E. Physical Review Letters, Vol. 51, Issue 25 https://doi.org/10.1103/PhysRevLett.51.2238	journal	December 1983
An Extended Finite Element Method Based Approach for Modeling Crevice and Pitting Corrosion Duddu, Ravindra; Kota, Nithyanand; Qidwai, Siddiq M. Journal of Applied Mechanics, Vol. 83, Issue 8 https://doi.org/10.1115/1.4033379	journal	May 2016
Acceleration Techniques for Approximating the Matrix Exponential Operator Popolizio, M.; Simoncini, V. SIAM Journal on Matrix Analysis and Applications, Vol. 30, Issue 2 https://doi.org/10.1137/060672856	journal	January 2008
Stopping Criteria for Rational Matrix Functions of Hermitian and Symmetric Matrices Frommer, Andreas; Simoncini, Valeria SIAM Journal on Scientific Computing, Vol. 30, Issue 3 https://doi.org/10.1137/070684598	journal	January 2008
Exponential Rosenbrock-Type Methods Hochbruck, Marlis; Ostermann, Alexander; Schweitzer, Julia SIAM Journal on Numerical Analysis, Vol. 47, Issue 1 https://doi.org/10.1137/080717717	journal	January 2009
Numerical Studies of Discrete Approximations to the Allen–Cahn Equation in the Sharp Interface Limit Zhang, Jian; Du, Qiang SIAM Journal on Scientific Computing, Vol. 31, Issue 4 https://doi.org/10.1137/080738398	journal	January 2009
Efficient Solution of Parabolic Equations by Krylov Approximation Methods Gallopoulos, E.; Saad, Y. SIAM Journal on Scientific and Statistical Computing, Vol. 13, Issue 5 https://doi.org/10.1137/0913071	journal	September 1992
Computing the Action of the Matrix Exponential, with an Application to Exponential Integrators Al-Mohy, Awad H.; Higham, Nicholas J. SIAM Journal on Scientific Computing, Vol. 33, Issue 2 https://doi.org/10.1137/100788860	journal	January 2011
Analysis and Approximation of the Ginzburg–Landau Model of Superconductivity Du, Qiang; Gunzburger, Max D.; Peterson, Janet S. SIAM Review, Vol. 34, Issue 1 https://doi.org/10.1137/1034003	journal	March 1992
On Krylov Subspace Approximations to the Matrix Exponential Operator Hochbruck, Marlis; Lubich, Christian SIAM Journal on Numerical Analysis, Vol. 34, Issue 5 https://doi.org/10.1137/S0036142995280572	journal	October 1997
Exponential Integrators for Large Systems of Differential Equations Hochbruck, Marlis; Lubich, Christian; Selhofer, Hubert SIAM Journal on Scientific Computing, Vol. 19, Issue 5 https://doi.org/10.1137/S1064827595295337	journal	September 1998
Expokit: a software package for computing matrix exponentials Sidje, Roger B. ACM Transactions on Mathematical Software, Vol. 24, Issue 1 https://doi.org/10.1145/285861.285868	journal	March 1998
Phase-Field Simulation of Solidification Boettinger, W. J.; Warren, J. A.; Beckermann, C. Annual Review of Materials Research, Vol. 32, Issue 1 https://doi.org/10.1146/annurev.matsci.32.101901.155803	journal	August 2002
Phase-Field Models for Microstructure Evolution Chen, Long-Qing Annual Review of Materials Research, Vol. 32, Issue 1 https://doi.org/10.1146/annurev.matsci.32.112001.132041	journal	August 2002
Pitting Corrosion of Metals Frankel, G. S. Journal of The Electrochemical Society, Vol. 145, Issue 6 https://doi.org/10.1149/1.1838615	journal	January 1998
The Influence of Passive Film Damage on Pitting Corrosion Chen, Ziguang; Zhang, Guangfeng; Bobaru, Florin Journal of The Electrochemical Society, Vol. 163, Issue 2 https://doi.org/10.1149/2.0521602jes	journal	November 2015
Numerical Modeling of Localized Corrosion Using Phase-Field and Smoothed Boundary Methods Chadwick, Alexander F.; Stewart, James A.; Enrique, Raúl A. Journal of The Electrochemical Society, Vol. 165, Issue 10 https://doi.org/10.1149/2.0701810jes	journal	January 2018
Exponential Time Differencing Gauge Method for Incompressible Viscous Flows Ju, Lili; Wang, Zhu Communications in Computational Physics, Vol. 22, Issue 2 https://doi.org/10.4208/cicp.OA-2016-0234	journal	June 2017

Similar Records

Nonoverlapping Localized Exponential Time Differencing Methods for Diffusion Problems

Journal Article · 2020 · Journal of Scientific Computing · OSTI ID:1594011

Hoang, Thi-Thao-Phuong; Ju, Lili; Leng, Wei; +1 more

Linearly first- and second-order, unconditionally energy stable schemes for the phase field crystal model

Journal Article · 2017 · Journal of Computational Physics · OSTI ID:22622255

Yang, Xiaofeng; Han, Daozhi

High order explicit local time stepping methods for hyperbolic conservation laws

Journal Article · 2020 · Mathematics of Computation · OSTI ID:1594035

Hoang, Thi-Thao-Phuong; Ju, Lili; Leng, Wei; +1 more

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
adaptivity
exponential integrator
finite element method
phase field model
pitting corrosion
semi-implicit scheme

Title: A space-time adaptive finite element method with exponential time integrator for the phase field model of pitting corrosion

Citation Formats

References (60)

Similar Records

Related Subjects