Fully implicit methodology for the dynamics of biomembranes and capillary interfaces by combining the level set and Newton methods

Laadhari, Aymen; Saramito, Pierre; Misbah, Chaouqi; Székely, Gábor

doi:10.1016/j.jcp.2017.04.019

Cited by 16 publications

(25 citation statements)

References 61 publications

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“…It features an affordable computational burden and maintains stability for significantly larger time steps. This is part of an ongoing work to model red blood cells [1,8,12,13]. We foresee the coupling with the bilayer bending force and the cytoskeleton elasticity model.…”

Section: Discussionmentioning

confidence: 99%

Exact Newton method with third-order convergence to model the dynamics of bubbles in incompressible flow

Laadhari

2017

Applied Mathematics Letters

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In this letter, we present a computational framework based on the use of the Newton and level set methods and tailored for the modeling of bubbles with surface tension in a surrounding Newtonian fluid. We describe a fully implicit and monolithic finite element method that maintains stability for significantly larger time steps compared to the usual explicit method and features substantial computational savings. A suitable transformation avoids the introduction of an additional mixed variable in the variational problem. An exact tangent problem is derived and the nonlinear problem is solved by a quadratically convergent Newton method. In addition, we consider a generalization to the multidimensional case of the Kou's and McDougall's methods, resulting in a faster convergence. The method is benchmarked against known results with the aim of illustrating its accuracy and robustness.

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Section: Discussionmentioning

confidence: 99%

Exact Newton method with third-order convergence to model the dynamics of bubbles in incompressible flow

Laadhari

2017

Applied Mathematics Letters

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“…In perspective, we aim to extend our theoretical work by exploring other types of fractional differential equations specifically designed for various physical systems, in addition to conducting appropriate numerical analyzes and simulations [27][28][29][30][31][32].…”

Section: Examplesmentioning

confidence: 99%

Nonexistence results for a sequential fractional differential problem

Kassim

Alqahtani

Tatar

et al. 2023

Math Methods in App Sciences

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“…The balance of surface forces and hydrodynamic stresses acting on the membrane allows the fluid-membrane coupling through the jump of the normal Cauchy stress (2.3f) [40,59]. Hence, the force F Γ can be written as a forcing term acting locally on the membrane in the right side of the momentum equation.…”

Section: Notations and Membrane Modelmentioning

confidence: 99%

“…As these are high-order derivatives of curvature, specific treatments are necessary if standard finite elements are used. To reduce the numerical oscillations and errors due to multiple projections and Lagrange interpolations (for lower order polynomials [30]) and non-physical adjustments of the level set function (for global mass conservation purposes [40,41]), while accurately assessing the membrane force, we use higher order polynomial approximations for finite element spatial discretization [33].…”

Section: Introductionmentioning

confidence: 99%

“…Although staggered strategies are widely used in the literature due to their attractiveness in terms of implementation and computational cost, they suffer from instabilities leading to strong constraints on the time step and are limited to very small regimes of inertia (in the Stokes limit) [42]. On the contrary, strongly coupled strategies (monolithic or iterative) exhibit more stability but are less used due to their complexity and difficulty in computing the Jacobian [35,40]. To solve the problem with a reasonable computational cost and allow a straightforward implementation using standard generalized Stokes solvers, we will relax the inextensibility constraint on velocity by a penalty approach, while being able to reproduce and draw conclusions related to cell behaviors in a non-Newtonian fluid.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Hydrodynamics simulation of red blood cells: Employing a penalty method with double jump composition of lower order time integrator

Laadhari,

Deeb,

Kaoui

2023

Math Methods in App Sciences

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We propose a numerical framework tailored for simulating the dynamics of vesicles with inextensible membranes, which mimic red blood cells, immersed in a non‐Newtonian fluid environment. A penalty method is proposed to handle the inextensibility constraint by relaxation, allowing a simple computer implementation and an affordable computational load compared to the full mixed formulation. To handle the high‐order derivatives in the stress jump across the membrane, which arise due to the high geometric order of the Helfrich functional, we employ higher degree finite elements for spatial discretization. The time integration scheme relies on the double composition of the Crank–Nicolson scheme to achieve faster fourth‐order convergence behavior. Additionally, an adaptive time‐stepping strategy based on a third‐order temporal integration error estimation is implemented. We address the main features of the proposed method, which is benchmarked against existing numerical and experimental results. Furthermore, we investigate the influence of non‐Newtonian rheology on the system dynamics.

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Fully implicit methodology for the dynamics of biomembranes and capillary interfaces by combining the level set and Newton methods

Cited by 16 publications

References 61 publications

Exact Newton method with third-order convergence to model the dynamics of bubbles in incompressible flow

Exact Newton method with third-order convergence to model the dynamics of bubbles in incompressible flow

Nonexistence results for a sequential fractional differential problem

Hydrodynamics simulation of red blood cells: Employing a penalty method with double jump composition of lower order time integrator

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