Phase Field Models Versus Parametric Front Tracking Methods: Are They Accurate and Computationally Efficient?

Barrett, John W.; Garcke, Harald; Nürnberg, Robert

doi:10.4208/cicp.190313.010813a

Cited by 10 publications

(6 citation statements)

References 73 publications

(97 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [33] the reader can found more details relative to a more extensive class of problems on the type those treated in this paper (reaction-diffusion equation), as well as different types for the nonlinear term F (ϕ). For numerical methods we direct the readers to consult the works [2,3,[6][7][8][9]11,15,16,18,21,23,28,29], and references therein.…”

Section: Introductionmentioning

confidence: 99%

Analysis of an iterative scheme of fractional steps type associated to the reaction–diffusion equation endowed with a general nonlinearity and Cauchy–Neumann boundary conditions

Moroşanu

Croitoru

2015

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

Keywords:Boundary value problems for nonlinear parabolic PDE Stability and convergence of numerical method Thermodynamics Heat transferThe paper studies the existence, uniqueness, regularity and the approximation of solutions to the reaction-diffusion equation endowed with a general nonlinear regular potential and Cauchy-Neumann boundary conditions. The convergence and error estimate results for an iterative scheme of fractional steps type, associated to the nonlinear parabolic equation, are also established. The advantage of such method consists in simplifying the numerical computation necessary to be done in order to approximate the solution of nonlinear parabolic equation.

show abstract

Section: Introductionmentioning

confidence: 99%

Analysis of an iterative scheme of fractional steps type associated to the reaction–diffusion equation endowed with a general nonlinearity and Cauchy–Neumann boundary conditions

Moroşanu

Croitoru

2015

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

show abstract

“…But accounting for fluid flow can be expected to be a (if not the most) significant contribution to the computational cost. Therefore, comparative studies on two-phase flow [43,44] give some insight and suggest that the effort of develop tracking methods pays off. But we also note that parallel test and benchmark computations of 3D problems are missing, and capturing methods involving only bulk differential equations seem more straightforward to parallelise and load-balance.…”

Section: Interface Capturingmentioning

confidence: 99%

Mathematical modelling in cell migration: tackling biochemistry in changing geometries

Stinner

Bretschneider

2020

Biochemical Society Transactions

View full text Add to dashboard Cite

Directed cell migration poses a rich set of theoretical challenges. Broadly, these are concerned with (1) how cells sense external signal gradients and adapt; (2) how actin polymerisation is localised to drive the leading cell edge and Myosin-II molecular motors retract the cell rear; and (3) how the combined action of cellular forces and cell adhesion results in cell shape changes and net migration. Reaction–diffusion models for biological pattern formation going back to Turing have long been used to explain generic principles of gradient sensing and cell polarisation in simple, static geometries like a circle. In this minireview, we focus on recent research which aims at coupling the biochemistry with cellular mechanics and modelling cell shape changes. In particular, we want to contrast two principal modelling approaches: (1) interface tracking where the cell membrane, interfacing cell interior and exterior, is explicitly represented by a set of moving points in 2D or 3D space and (2) interface capturing. In interface capturing, the membrane is implicitly modelled analogously to a level line in a hilly landscape whose topology changes according to forces acting on the membrane. With the increased availability of high-quality 3D microscopy data of complex cell shapes, such methods will become increasingly important in data-driven, image-based modelling to better understand the mechanochemistry underpinning cell motion.

show abstract

“…Trade-off curves for (b) and (c) are obtained through (19) and (20) respectively. For (a) we use the Matlab toolbox Jitterbug [22].…”

Section: A Integrator Examplementioning

confidence: 99%

“…For the unstable example we use the following parameters: Note that (c) needs to be simulated in this example, since the average sampling period h avg cannot be determined directly from V and J. Case (a) is still evaluated using Jitterbug, and (b) is evaluated using (20).…”

Section: B Unstable Examplementioning

confidence: 99%

See 1 more Smart Citation

On event-based sampling for LQG-optimal control

Andrén

Bernhardsson

Cervin

et al. 2017

2017 IEEE 56th Annual Conference on Decision and Control (CDC)

View full text Add to dashboard Cite

We consider the problem of finding an event-based sampling scheme that optimizes the trade-off between average sampling rate and control performance in a linear-quadratic-Gaussian (LQG) control problem setting with output feedback. Our analysis is based on a recently presented sampled-data controller structure, which remains LQG-optimal for any choice of sampling scheme. We show that optimization of the sampling scheme is related to an elliptic convection-diffusion type partial differential equation over a domain with free boundary, a so called Stefan problem. A numerical method is presented to solve this problem for second order systems, and thus obtain an optimal sampling scheme. The method also directly generalizes to higher order systems, although with a higher computational cost. For the special case of multidimensional integrator systems, we present the optimal sampling scheme on closed form, and prove that it will always outperform its periodic counterpart. Tight bounds on the improvement are presented. The improved performance is also demonstrated in numerical examples, both for an integrator system and a more general case.

show abstract

Phase Field Models Versus Parametric Front Tracking Methods: Are They Accurate and Computationally Efficient?

Cited by 10 publications

References 73 publications

Analysis of an iterative scheme of fractional steps type associated to the reaction–diffusion equation endowed with a general nonlinearity and Cauchy–Neumann boundary conditions

Analysis of an iterative scheme of fractional steps type associated to the reaction–diffusion equation endowed with a general nonlinearity and Cauchy–Neumann boundary conditions

Mathematical modelling in cell migration: tackling biochemistry in changing geometries

On event-based sampling for LQG-optimal control

Contact Info

Product

Resources

About