Cahn–Hilliard-type EquationsRobust Discretization and Efficient Implementation
Cahn–Hilliard-type Equations
Robust Discretization and Efficient Implementation
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DissertationDate of Exam
09.04.2009Date of Publication
11.05.2009Advisor
Otto, FelixCo-Referee
Rumpf, MartinMetadata
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Abstract
In this work, a robust and efficient numerical method to simulate Cahn–Hilliard-type equations is presented. The considered equations are of fourth order and contain two nonlinearities: one is the concentration-dependent mobility and the other one is the potential.
Using the fact that Cahn–Hilliard-type equations are a gradient flow, we can derive symmetric time-discrete equations for the flux, even though the mobility is concentration dependent. As time-step scheme, we use the TR-BDF2 scheme.
For spatial discretization, we use H(div)-conformal finite elements, more precisely Raviart–Thomas elements. The introduced discrete gradient leads to the appearance of the inverse of the mass matrix. Since there is no mass-lumping method for Raviart–Thomas elements, this inverse is dense. To circumvent this difficulty, we adapt a technique from the area of mixed methods to our situation. It is based on the introduction of inter-element multipliers to guarantee conformality and yields a block-diagonal mass matrix. This approach is up to fifty times faster than more naïve ones like using a matrix-free method.
The discretization was efficiently implemented in a space- and time-adaptive finite element software, which was written from scratch in C++. We use the software to simulate Cahn–Hilliard-type equations arising in two different physical contexts.
The first one is the decomposition of a binary mixture both for the case of a constant mobility and a double-well potential (shallow quench) and the case of a degenerate mobility and a concave potential (deep quench). The latter is approximated with a variable-quench Ansatz and low temperature. We are able to compute large domains and long times. Theoretical results on the coarsening rate were reproduced accurately.
The second application is thin-film crystal growth. We use a diffuse-interface approximation to the standard BCF model which includes the Ehrlich–Schwoebel (ES) barrier. With this approximation we can handle topological changes. As an example, we consider the pinch-off caused by an instability due to the ES barrier. Thanks to the discretization via the flux, we can easily simulate step trains using skew-periodic boundary conditions.
Using the fact that Cahn–Hilliard-type equations are a gradient flow, we can derive symmetric time-discrete equations for the flux, even though the mobility is concentration dependent. As time-step scheme, we use the TR-BDF2 scheme.
For spatial discretization, we use H(div)-conformal finite elements, more precisely Raviart–Thomas elements. The introduced discrete gradient leads to the appearance of the inverse of the mass matrix. Since there is no mass-lumping method for Raviart–Thomas elements, this inverse is dense. To circumvent this difficulty, we adapt a technique from the area of mixed methods to our situation. It is based on the introduction of inter-element multipliers to guarantee conformality and yields a block-diagonal mass matrix. This approach is up to fifty times faster than more naïve ones like using a matrix-free method.
The discretization was efficiently implemented in a space- and time-adaptive finite element software, which was written from scratch in C++. We use the software to simulate Cahn–Hilliard-type equations arising in two different physical contexts.
The first one is the decomposition of a binary mixture both for the case of a constant mobility and a double-well potential (shallow quench) and the case of a degenerate mobility and a concave potential (deep quench). The latter is approximated with a variable-quench Ansatz and low temperature. We are able to compute large domains and long times. Theoretical results on the coarsening rate were reproduced accurately.
The second application is thin-film crystal growth. We use a diffuse-interface approximation to the standard BCF model which includes the Ehrlich–Schwoebel (ES) barrier. With this approximation we can handle topological changes. As an example, we consider the pinch-off caused by an instability due to the ES barrier. Thanks to the discretization via the flux, we can easily simulate step trains using skew-periodic boundary conditions.
Subjects
Cahn-Hilliard, finite elements, concentration dependent mobility, phase field equation, free boundary problems, TR-BDF2, Raviart-Thomas, static condensation, coarsening, molecular beam epitaxy, epitaxial growth, diffuse interface, Ehrlich-Schwoebel, pinch-off
Classification (DDC)
510 MathematikPenzler, Patrick: Cahn–Hilliard-type Equations : Robust Discretization and Efficient Implementation. - Bonn, 2009. - Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn.
Online-Ausgabe in bonndoc: https://nbn-resolving.org/urn:nbn:de:hbz:5N-17445
Online-Ausgabe in bonndoc: https://nbn-resolving.org/urn:nbn:de:hbz:5N-17445
@phdthesis{handle:20.500.11811/4071,
urn: https://nbn-resolving.org/urn:nbn:de:hbz:5N-17445,
author = {{Patrick Penzler}},
title = {Cahn–Hilliard-type Equations : Robust Discretization and Efficient Implementation},
school = {Rheinische Friedrich-Wilhelms-Universität Bonn},
year = 2009,
month = may,
note = {In this work, a robust and efficient numerical method to simulate Cahn–Hilliard-type equations is presented. The considered equations are of fourth order and contain two nonlinearities: one is the concentration-dependent mobility and the other one is the potential.
Using the fact that Cahn–Hilliard-type equations are a gradient flow, we can derive symmetric time-discrete equations for the flux, even though the mobility is concentration dependent. As time-step scheme, we use the TR-BDF2 scheme.
For spatial discretization, we use H(div)-conformal finite elements, more precisely Raviart–Thomas elements. The introduced discrete gradient leads to the appearance of the inverse of the mass matrix. Since there is no mass-lumping method for Raviart–Thomas elements, this inverse is dense. To circumvent this difficulty, we adapt a technique from the area of mixed methods to our situation. It is based on the introduction of inter-element multipliers to guarantee conformality and yields a block-diagonal mass matrix. This approach is up to fifty times faster than more naïve ones like using a matrix-free method.
The discretization was efficiently implemented in a space- and time-adaptive finite element software, which was written from scratch in C++. We use the software to simulate Cahn–Hilliard-type equations arising in two different physical contexts.
The first one is the decomposition of a binary mixture both for the case of a constant mobility and a double-well potential (shallow quench) and the case of a degenerate mobility and a concave potential (deep quench). The latter is approximated with a variable-quench Ansatz and low temperature. We are able to compute large domains and long times. Theoretical results on the coarsening rate were reproduced accurately.
The second application is thin-film crystal growth. We use a diffuse-interface approximation to the standard BCF model which includes the Ehrlich–Schwoebel (ES) barrier. With this approximation we can handle topological changes. As an example, we consider the pinch-off caused by an instability due to the ES barrier. Thanks to the discretization via the flux, we can easily simulate step trains using skew-periodic boundary conditions.},
url = {https://hdl.handle.net/20.500.11811/4071}
}
urn: https://nbn-resolving.org/urn:nbn:de:hbz:5N-17445,
author = {{Patrick Penzler}},
title = {Cahn–Hilliard-type Equations : Robust Discretization and Efficient Implementation},
school = {Rheinische Friedrich-Wilhelms-Universität Bonn},
year = 2009,
month = may,
note = {In this work, a robust and efficient numerical method to simulate Cahn–Hilliard-type equations is presented. The considered equations are of fourth order and contain two nonlinearities: one is the concentration-dependent mobility and the other one is the potential.
Using the fact that Cahn–Hilliard-type equations are a gradient flow, we can derive symmetric time-discrete equations for the flux, even though the mobility is concentration dependent. As time-step scheme, we use the TR-BDF2 scheme.
For spatial discretization, we use H(div)-conformal finite elements, more precisely Raviart–Thomas elements. The introduced discrete gradient leads to the appearance of the inverse of the mass matrix. Since there is no mass-lumping method for Raviart–Thomas elements, this inverse is dense. To circumvent this difficulty, we adapt a technique from the area of mixed methods to our situation. It is based on the introduction of inter-element multipliers to guarantee conformality and yields a block-diagonal mass matrix. This approach is up to fifty times faster than more naïve ones like using a matrix-free method.
The discretization was efficiently implemented in a space- and time-adaptive finite element software, which was written from scratch in C++. We use the software to simulate Cahn–Hilliard-type equations arising in two different physical contexts.
The first one is the decomposition of a binary mixture both for the case of a constant mobility and a double-well potential (shallow quench) and the case of a degenerate mobility and a concave potential (deep quench). The latter is approximated with a variable-quench Ansatz and low temperature. We are able to compute large domains and long times. Theoretical results on the coarsening rate were reproduced accurately.
The second application is thin-film crystal growth. We use a diffuse-interface approximation to the standard BCF model which includes the Ehrlich–Schwoebel (ES) barrier. With this approximation we can handle topological changes. As an example, we consider the pinch-off caused by an instability due to the ES barrier. Thanks to the discretization via the flux, we can easily simulate step trains using skew-periodic boundary conditions.},
url = {https://hdl.handle.net/20.500.11811/4071}
}
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