Stabilized second-order convex splitting schemes for Cahn-Hilliard models with application to diffuse-interface tumor-growth models

Enhancement of damaged-image prediction through Cahn-Hilliard image inpainting

We assess the benefit of including an image inpainting filter before passing damaged images into a classification neural network. We employ an appropriately modified Cahn-Hilliard equation as an image inpainting filter which is solved numerically with a finite-volume scheme exhibiting reduced computational cost and the properties of energy stability and boundedness. The benchmark dataset employed is Modified National Institute of Standards and Technology (MNIST) dataset, which consists of binary images of handwritten digits and is a standard dataset to validate image-processing methodologies. We train a neural network based on dense layers with MNIST, and subsequently we contaminate the test set with damages of different types and intensities. We then compare the prediction accuracy of the neural network with and without applying the Cahn-Hilliard filter to the damaged images test. Our results quantify the significant improvement of damaged-image prediction by applying the Cahn-Hilliard filter, which for specific damages can increase up to 50% and is advantageous for low to moderate damage.

Parallel splitting solvers for the isogeometric analysis of the Cahn-Hilliard equation

Modeling tumor growth in biological systems is a challenging problem with important consequences for diagnosis and treatment of various forms of cancer. This growth process requires large simulation complexity due to evolving biological and chemical processes in living tissue and interactions of cellular and vascular constituents in living organisms. Herein, we describe with a phase-field model, namely the Cahn-Hilliard equation the intricate interactions between the tumors and their host tissue. The spatial discretization uses highly-continuous isogeometric elements. For fast simulation of the time-dependent Cahn-Hilliard equation, we employ an alternating directions implicit methodology. Thus, we reduce the original problems to Kronecker products of 1 D matrices that can be factorized in a linear computational cost. The implementation enables parallel multi-core simulations and shows good scalability on shared-memory multi-core machines. Combined with the high accuracy of isogeometric elements, our method shows high efficiency in solving the Cahn-Hilliard equation on tensor-product meshes.

A Distributed Control Problem for a Fractional Tumor Growth Model

In this paper, we study the distributed optimal control of a system of three evolutionary equations involving fractional powers of three self-adjoint, monotone, unbounded linear operators having compact resolvents. The system is a generalization of a Cahn–Hilliard type phase field system modeling tumor growth that has been proposed by Hawkins–Daarud, van der Zee and Oden. The aim of the control process, which could be realized by either administering a drug or monitoring the nutrition, is to keep the tumor cell fraction under control while avoiding possible harm for the patient. In contrast to previous studies, in which the occurring unbounded operators governing the diffusional regimes were all given by the Laplacian with zero Neumann boundary conditions, the operators may in our case be different; more generally, we consider systems with fractional powers of the type that were studied in a recent work by the present authors. In our analysis, we show the Fréchet differentiability of the associated control-to-state operator, establish the existence of solutions to the associated adjoint system, and derive the first-order necessary conditions of optimality for a cost functional of tracking type.

A framework for studying dynamics and stability of diffusive-reactive interfaces with application to Cu6Sn5 intermetallic compound growth

Often during phase growth, the rate of accretion, on the one hand, is determined by a competition between bulk diffusion and surface reaction rate. The morphology of the phase interface, on the other hand, is determined by an interplay between surface diffusivity and surface reaction rate. In this study, a framework to predict the growth and the morphology of an interface by modelling the interplay between bulk diffusion, surface reaction rate and surface diffusion is developed. The framework is demonstrated using the example of Cu-Sn intermetallic compound growth that is of significance to modern microelectronic assemblies. In particular, the dynamics and stability of the interface created when Cu and Sn react to form the compound Cu6Sn5 is explored. Prior experimental observations of the Cu6Sn5-Sn interface have shown it to possess either a scalloped, flat or needle-shaped morphology. Diffuse interface simulations are carried out to elucidate the mechanism behind the interface formation. The computational model accounts for the bulk diffusion of Cu through the intermetallic compound, reaction at the interface to form Cu6Sn5, surface diffusion of Cu6Sn5 along the interface and the influence of the electric current density in accelerating the bulk diffusion of Cu. A stability analysis is performed to identify the conditions under which the interface evolves into a flat, scalloped or needle-shaped structure.