TY - GEN
T1 - Isogeometric analysis of phase-field models
T2 - International ECCOMAS Multidisciplinary Jubilee Symposium - New Computational Challenges in Materials, Structures, and Fluids, EMJS 2008
AU - Gomez, H.
AU - Calo, V. M.
AU - Hughes, T. J.R.
N1 - Publisher Copyright:
© Springer Science + Business Media B.V. 2009.
PY - 2009
Y1 - 2009
N2 - The Cahn-Hilliard equation involves fourth-order spatial derivatives. Finite element solutions to the Cahn-Hilliard equation are not common because primal variational formulations of fourth-order operators are only well defined and inte-grable if the finite element basis functions are piecewise smooth and globally l1 -continuous. There are a very limited number of two-dimensional finite elements possessing l1 -continuity applicable to complex geometries, but none in three-dimensions. We propose isogeometric analysis as a technology that possesses a unique combination of attributes for complex problems involving higher-order differential operators, namely, higher-order accuracy, robustness, two- and three-dimensional geometric flexibility, compact support, and, most importantly, the possibility of l1 and higher-order continuity. A NURBS-based variational formulation for the Cahn-Hilliard equation was tested on two- and three-dimensional problems. We present steady state solutions in two-dimensions and, for the first time, in three-dimensions. To achieve these results an adaptive time-stepping method is introduced. We also present a technique for desensitizing calculations to dependence on mesh refinement. This enables the calculation of topologically correct solutions on coarse meshes, opening the way to practical engineering applications of phase-field methodology.
AB - The Cahn-Hilliard equation involves fourth-order spatial derivatives. Finite element solutions to the Cahn-Hilliard equation are not common because primal variational formulations of fourth-order operators are only well defined and inte-grable if the finite element basis functions are piecewise smooth and globally l1 -continuous. There are a very limited number of two-dimensional finite elements possessing l1 -continuity applicable to complex geometries, but none in three-dimensions. We propose isogeometric analysis as a technology that possesses a unique combination of attributes for complex problems involving higher-order differential operators, namely, higher-order accuracy, robustness, two- and three-dimensional geometric flexibility, compact support, and, most importantly, the possibility of l1 and higher-order continuity. A NURBS-based variational formulation for the Cahn-Hilliard equation was tested on two- and three-dimensional problems. We present steady state solutions in two-dimensions and, for the first time, in three-dimensions. To achieve these results an adaptive time-stepping method is introduced. We also present a technique for desensitizing calculations to dependence on mesh refinement. This enables the calculation of topologically correct solutions on coarse meshes, opening the way to practical engineering applications of phase-field methodology.
KW - Cahn-Hilliard
KW - Isogeometric analysis
KW - NURBS
KW - Phase-field
UR - http://www.scopus.com/inward/record.url?scp=84962888201&partnerID=8YFLogxK
U2 - 10.1007/978-1-4020-9231-2_1
DO - 10.1007/978-1-4020-9231-2_1
M3 - Conference contribution
AN - SCOPUS:84962888201
SN - 9781402092305
T3 - Computational Methods in Applied Sciences
SP - 1
EP - 16
BT - ECCOMAS Multidisciplinary Jubilee Symposium
A2 - Eberhardsteiner, Josef
A2 - Hellmich, Christian
A2 - Mang, Herbert A.
A2 - Périaux, Jacques
PB - Springer
Y2 - 18 February 2008 through 20 February 2008
ER -