scholarly journals Optimal control theory and advanced optimality conditions for a diffuse interface model of tumor growth

2020 ◽  
Vol 26 ◽  
pp. 71 ◽  
Author(s):  
Matthias Ebenbeck ◽  
Patrik Knopf
Keyword(s):  
Optimal Control ◽  
Tumor Growth ◽  
Phase Field ◽  
Chemical Potential ◽  
Type Equation ◽  
Field Variable ◽  
Reaction Diffusion Equation ◽  
Diffuse Interface ◽  
Interface Model ◽  
Diffuse Interface Model

We investigate a distributed optimal control problem for a diffuse interface model for tumor growth. The model consists of a Cahn–Hilliard type equation for the phase field variable, a reaction diffusion equation for the nutrient concentration and a Brinkman type equation for the velocity field. These PDEs are endowed with homogeneous Neumann boundary conditions for the phase field variable, the chemical potential and the nutrient as well as a “no-friction” boundary condition for the velocity. The control represents a medication by cytotoxic drugs and enters the phase field equation. The aim is to minimize a cost functional of standard tracking type that is designed to track the phase field variable during the time evolution and at some fixed final time. We show that our model satisfies the basics for calculus of variations and we present first-order and second-order conditions for local optimality. Moreover, we present a globality condition for critical controls and we show that the optimal control is unique on small time intervals.

scholarly journals Quasi-Brittle Fracture Modeling of Preflawed Bitumen Using a Diffuse Interface Model

2016 ◽  
Vol 2016 ◽  
pp. 1-7 ◽  
Author(s):  
Yue Hou ◽  
Fengyan Sun ◽  
Wenjuan Sun ◽  
Meng Guo ◽  
Chao Xing ◽  
...  
Keyword(s):  
Brittle Fracture ◽  
Phase Field ◽  
Field Method ◽  
Field Variable ◽  
Diffuse Interface ◽  
Phase Field Method ◽  
Interface Model ◽  
Diffuse Interface Model ◽  
Bituminous Materials ◽  
Fracture Modeling

Fundamental understandings on the bitumen fracture mechanism are vital to improve the mixture design of asphalt concrete. In this paper, a diffuse interface model, namely, phase-field method is used for modeling the quasi-brittle fracture in bitumen. This method describes the microstructure using a phase-field variable which assumes one in the intact solid and negative one in the crack region. Only the elastic energy will directly contribute to cracking. To account for the growth of cracks, a nonconserved Allen-Cahn equation is adopted to evolve the phase-field variable. Numerical simulations of fracture are performed in bituminous materials with the consideration of quasi-brittle properties. It is found that the simulation results agree well with classic fracture mechanics.

scholarly journals Optimal control of time-discrete two-phase flow driven by a diffuse-interface model

2019 ◽  
Vol 25 ◽  
pp. 13 ◽  
Author(s):  
Harald Garcke ◽  
Michael Hinze ◽  
Christian Kahle
Keyword(s):  
Optimal Control ◽  
Phase Field ◽  
Control Variable ◽  
Necessary Optimality Conditions ◽  
Time Discretization ◽  
Diffuse Interface ◽  
Interface Model ◽  
Diffuse Interface Model ◽  
Two Phase ◽  
Fully Discrete

We propose a general control framework for two-phase flows with variable densities in the diffuse interface formulation, where the distribution of the fluid components is described by a phase field. The flow is governed by the diffuse interface model proposed in Abelset al.[M3AS22(2012) 1150013]. On the basis of the stable time discretization proposed in Garckeet al.[Appl. Numer. Math.99(2016) 151] we derive necessary optimality conditions for the time-discrete and the fully discrete optimal control problem. We present numerical examples with distributed and boundary controls, and also consider the case, where the initial value of the phase field serves as control variable.

scholarly journals Optimal control of stochastic phase-field models related to tumor growth

2020 ◽  
Vol 26 ◽  
pp. 104
Author(s):  
Carlo Orrieri ◽  
Elisabetta Rocca ◽  
Luca Scarpa
Keyword(s):  
Optimal Control ◽  
Tumor Growth ◽  
Phase Field ◽  
Phase Field Model ◽  
Growth Dynamics ◽  
Necessary Optimality Conditions ◽  
Reaction Diffusion ◽  
Adjoint System ◽  
Reaction Diffusion Equation ◽  
Cahn Hilliard Equation

We study a stochastic phase-field model for tumor growth dynamics coupling a stochastic Cahn-Hilliard equation for the tumor phase parameter with a stochastic reaction-diffusion equation governing the nutrient proportion. We prove strong well-posedness of the system in a general framework through monotonicity and stochastic compactness arguments. We introduce then suitable controls representing the concentration of cytotoxic drugs administered in medical treatment and we analyze a related optimal control problem. We derive existence of an optimal strategy and deduce first-order necessary optimality conditions by studying the corresponding linearized system and the backward adjoint system.

scholarly journals Optimal distributed control of a diffuse interface model of tumor growth

Nonlinearity ◽  
2017 ◽  
Vol 30 (6) ◽  
pp. 2518-2546 ◽  
Author(s):  
Pierluigi Colli ◽  
Gianni Gilardi ◽  
Elisabetta Rocca ◽  
Jürgen Sprekels
Keyword(s):  
Tumor Growth ◽  
Distributed Control ◽  
Diffuse Interface ◽  
Interface Model ◽  
Diffuse Interface Model ◽  
Optimal Distributed Control

On micro–macro-models for two-phase flow with dilute polymeric solutions — modeling and analysis

2016 ◽  
Vol 26 (05) ◽  
pp. 823-866 ◽  
Author(s):  
G. Grün ◽  
S. Metzger
Keyword(s):  
Two Phase Flow ◽  
Type Equation ◽  
Momentum Equation ◽  
Diffuse Interface ◽  
Phase Flow ◽  
Interface Model ◽  
Diffuse Interface Model ◽  
Two Phase ◽  
Energy Functionals ◽  
Spring Force

By methods from nonequilibrium thermodynamics, we derive a diffuse-interface model for two-phase flow of incompressible fluids with dissolved noninteracting polymers. The polymers are modeled by dumbbells subjected to general elastic spring-force potentials including in particular Hookean and finitely extensible, nonlinear elastic (FENE) potentials. Their density and orientation are described by a Fokker–Planck-type equation which is coupled to a Cahn–Hilliard and a momentum equation for phase-field and gross velocity/pressure. Henry-type energy functionals are used to describe different solubility properties of the polymers in the different phases or at the liquid–liquid interface. Taking advantage of the underlying energetic/entropic structure of the system, we prove existence of a weak solution globally in time for the case of FENE-potentials. As a by-product in the case of Hookean spring potentials, we derive a macroscopic diffuse-interface model for two-phase flow of Oldroyd-B-type liquids.

scholarly journals Formal asymptotic limit of a diffuse-interface tumor-growth model

2015 ◽  
Vol 25 (06) ◽  
pp. 1011-1043 ◽  
Author(s):  
Danielle Hilhorst ◽  
Johannes Kampmann ◽  
Thanh Nam Nguyen ◽  
Kristoffer George Van Der Zee
Keyword(s):  
Tumor Growth ◽  
Growth Model ◽  
Singular Limit ◽  
Diffuse Interface ◽  
Interface Model ◽  
Diffuse Interface Model ◽  
Sharp Interface Limit ◽  
Field System ◽  
Tumor Growth Model ◽  
Phase Field System

We consider a diffuse-interface tumor-growth model which has the form of a phase-field system. We characterize the singular limit of this problem. More precisely, we formally prove that as the coefficient of the reaction term tends to infinity, the solution converges to the solution of a novel free boundary problem. We present numerical simulations which illustrate the convergence of the diffuse-interface model to the identified sharp-interface limit.

scholarly journals Optimal control of time discrete two-phase flow governed by a diffuse interface model

PAMM ◽  
2016 ◽  
Vol 16 (1) ◽  
pp. 785-786 ◽  
Author(s):  
Harald Garcke ◽  
Michael Hinze ◽  
Christian Kahle
Keyword(s):  
Optimal Control ◽  
Two Phase Flow ◽  
Diffuse Interface ◽  
Phase Flow ◽  
Interface Model ◽  
Diffuse Interface Model ◽  
Two Phase
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