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Solving Nonlinear, High-Order Partial
Differential Equations Using a High-Performance
Isogeometric Analysis Framework
Adriano M.A. Cortes 1 , 2 , Philippe Vignal 1 , 3 ,AdelSarmiento 1 , 4 ,
Daniel Garcıa 1 , 5 , Nathan Collier 1 , 6 , Lisandro Dalcin 1 , 7 ,andVictorM.Calo 1 , 2 , 4
1 Center for Numerical Porous Media,
King Abdullah University of Science and Technology, Thuwal, Saudi Arabia
2 Earth Sciences & Engineering,
King Abdullah University of Science and Technology, Thuwal, Saudi Arabia
3 Material Science & Engineering,
King Abdullah University of Science and Technology, Thuwal, Saudi Arabia
4 Applied Mathematics & Computational Science, Earth Science & Engineering,
King Abdullah University of Science and Technology, Thuwal, Saudi Arabia
5 Mechanical Engineering,
King Abdullah University of Science and Technology, Thuwal, Saudi Arabia
6 Oak Ridge National Laboratory, Oak Ridge, Tennessee, USA
7 Consejo Nacional de Investigaciones Cientıficas y Tecnicas and Universidad
Nacional del Litoral, Santa Fe, Argentina
Abstract. In this paper we present PetIGA, a high-performance imple-
mentation of Isogeometric Analysis built on top of PETSc. We show its
use in solving nonlinear and time-dependent problems, such as phase-
field models, by taking advantage of the high-continuity of the basis
functions granted by the isogeometric framework. In this work, we focus
on the Cahn-Hilliard equation and the phase-field crystal equation.
Keywords: Isogeometric analysis, high-performance computing, high-
order partial differential equations, finite elements, phase-field modeling.
1 Introduction
The recent interest in Isogeometric Analysis (IGA) [1], a spline-based finite ele-
ment method, motivated an ecient implementation of this numerical method.
PetIGA [2] pursues this goal. It is built on top of PETSc [3,4], an ecient and
parallel library tailored for the solution of partial differential equations.
To highlight some of the features we have access to through the use of PetIGA,
we decided to focus on two phase-fields models of great interest in the material
science community, namely, the Cahn-Hilliard equation, and the phase-field crys-
tal equation. These are high-order and nonlinear partial differential equations.
Their discretization in space can be be simplified through the use of IGA when
compared to traditional finite elements, as the higher-continuous basis functions
required can be trivially generated within this setting.
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