A preconditioner for the Ohta-Kawasaki equation

We propose a new preconditioner for the Ohta-Kawasaki equation, a nonlocal Cahn-Hilliard equation that describes the evolution of diblock copolymer melts. We devise a computable approximation to the inverse of the Schur complement of the coupled second-order formulation via a matching strategy. The preconditioner achieves mesh independence: as the mesh is refined, the number of Krylov iterations required for its solution remains approximately constant. In addition, the preconditioner is robust with respect to the interfacial thickness parameter if a timestep criterion is satisfied. This enables the highly resolved finite element simulation of three-dimensional diblock copolymer melts with over one billion degrees of freedom

Matching Schur complement approximations for certain saddle-point systems

The solution of many practical problems described by mathematical models requires approximation methods that give rise to linear(ized) systems of equations, solving which will determine the desired approximation. This short contribution describes a particularly effective solution approach for a certain class of so-called saddle-point linear systems which arises in different contexts

Sharp-interface problem of the Ohta-Kawasaki model for symmetric diblock copolymers

The Ohta-Kawasaki model for diblock-copolymers is well known to the scientific community of diffuse-interface methods. To accurately capture the long-time evolution of the moving interfaces, we present a derivation of the corresponding sharp-interface limit using matched asymptotic expansions, and show that the limiting process leads to a Hele-Shaw type moving interface problem. The numerical treatment of the sharp-interface limit is more complicated due to the stiffness of the equations. To address this problem, we present a boundary integral formulation corresponding to a sharp interface limit of the Ohta-Kawasaki model. Starting with the governing equations defined on separate phase domains, we develop boundary integral equations valid for multi-connected domains in a 2D plane. For numerical simplicity we assume our problem is driven by a uniform Dirichlet condition on a circular far-field boundary. The integral formulation of the problem involves both double- and single-layer potentials due to the modified boundary condition. In particular, our formulation allows one to compute the nonlinear dynamics of a non-equilibrium system and pattern formation of an equilibrating system. Numerical tests on an evolving slightly perturbed circular interface (separating the two phases) are in excellent agreement with the linear analysis, demonstrating that the method is stable, efficient and spectrally accurate in space.Comment: 34 pages, 10 figure

Optimal design of chemoepitaxial guideposts for directed self-assembly of block copolymer systems using an inexact-Newton algorithm

Directed self-assembly (DSA) of block-copolymers (BCPs) is one of the most promising developments in the cost-effective production of nanoscale devices. The process makes use of the natural tendency for BCP mixtures to form nanoscale structures upon phase separation. The phase separation can be directed through the use of chemically patterned substrates to promote the formation of morphologies that are essential to the production of semiconductor devices. Moreover, the design of substrate pattern can formulated as an optimization problem for which we seek optimal substrate designs that effectively produce given target morphologies. In this paper, we adopt a phase field model given by a nonlocal Cahn--Hilliard partial differential equation (PDE) based on the minimization of the Ohta--Kawasaki free energy, and present an efficient PDE-constrained optimization framework for the optimal design problem. The design variables are the locations of circular- or strip-shaped guiding posts that are used to model the substrate chemical pattern. To solve the ensuing optimization problem, we propose a variant of an inexact Newton conjugate gradient algorithm tailored to this problem. We demonstrate the effectiveness of our computational strategy on numerical examples that span a range of target morphologies. Owing to our second-order optimizer and fast state solver, the numerical results demonstrate five orders of magnitude reduction in computational cost over previous work. The efficiency of our framework and the fast convergence of our optimization algorithm enable us to rapidly solve the optimal design problem in not only two, but also three spatial dimensions.Comment: 35 Pages, 17 Figure

Sharp-interface problem of the Ohta-Kawasaki model 2 for symmetric diblock copolymers

Abstract The Ohta-Kawasaki model for diblock-copolymers is well known to the scientific community of diffuse-interface methods. To accurately capture the long-time evolution of the moving interfaces, we present a derivation of the corresponding sharp-interface limit using matched asymptotic expansions, and show that the limiting process leads to a Hele-Shaw type moving interface problem. The numerical treatment of the sharp-interface limit is more complicated due to the stiffness of the equations. To address this problem, we present a boundary integral formulation corresponding to a sharp interface limit of the Ohta-Kawasaki model. Starting with the governing equations defined on separate phase domains, we develop boundary integral equations valid for multi-connected domains in a 2D plane. For numerical simplicity we assume our problem is driven by a uniform Dirichlet condition on a circular far-field boundary. The integral formulation of the problem involves both double- and single-layer potentials due to the modified boundary condition. In particular, our formulation allows one to compute the nonlinear dynamics of a non-equilibrium system and pattern formation of an equilibrating system. Numerical tests on an evolving slightly perturbed circular interface (separating the two phases) are in excellent agreement with the linear analysis, demonstrating that the method is stable, efficient and spectrally accurate in space

Emerging Developments in Interfaces and Free Boundaries

The field of the mathematical and numerical analysis of systems of nonlinear partial differential equations involving interfaces and free boundaries is a well established and flourishing area of research. This workshop focused on recent developments and emerging new themes. By bringing together experts in these fields we achieved progress in open questions and developed novel research directions in mathematics related to interfaces and free boundaries. This interdisciplinary workshop brought together researchers from distinct mathematical fields such as analysis, computation, optimisation and modelling to discuss emerging challenges

Productive and efficient computational science through domain-specific abstractions

In an ideal world, scientific applications are computationally efficient, maintainable and composable and allow scientists to work very productively. We argue that these goals are achievable for a specific application field by choosing suitable domain-specific abstractions that encapsulate domain knowledge with a high degree of expressiveness. This thesis demonstrates the design and composition of domain-specific abstractions by abstracting the stages a scientist goes through in formulating a problem of numerically solving a partial differential equation. Domain knowledge is used to transform this problem into a different, lower level representation and decompose it into parts which can be solved using existing tools. A system for the portable solution of partial differential equations using the finite element method on unstructured meshes is formulated, in which contributions from different scientific communities are composed to solve sophisticated problems. The concrete implementations of these domain-specific abstractions are Firedrake and PyOP2. Firedrake allows scientists to describe variational forms and discretisations for linear and non-linear finite element problems symbolically, in a notation very close to their mathematical models. PyOP2 abstracts the performance-portable parallel execution of local computations over the mesh on a range of hardware architectures, targeting multi-core CPUs, GPUs and accelerators. Thereby, a separation of concerns is achieved, in which Firedrake encapsulates domain knowledge about the finite element method separately from its efficient parallel execution in PyOP2, which in turn is completely agnostic to the higher abstraction layer. As a consequence of the composability of those abstractions, optimised implementations for different hardware architectures can be automatically generated without any changes to a single high-level source. Performance matches or exceeds what is realistically attainable by hand-written code. Firedrake and PyOP2 are combined to form a tool chain that is demonstrated to be competitive with or faster than available alternatives on a wide range of different finite element problems.Open Acces