A stabilized finite element formulation for liquid shells and its application to lipid bilayers

Sauer, Roger A.; Duong, Thang X.; Mandadapu, Kranthi K.; Steigmann, David J.

doi:10.1016/j.jcp.2016.11.004

Cited by 62 publications

(83 citation statements)

References 66 publications

(196 reference statements)

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“…On the one hand, for a time-evolution problem such as the last example, the matrices can be computed at the beginning of the simulation and used during the whole dynamicsOn the other hand, in all problems considered here, we have found that, for relatively large meshes, the computational time required for solving the system of equations (either with an iterative solver such as GMRES with domain decomposition or with a parallel direct solver such as MUMPS) is always larger than the time required to assemble the matrix. The LMP method can be used to discretize the models in [8,9,10,12] involving vector-and tensor-valued PDEs coupled to surface evolution laws. On a time-evolving surface, Γ t , an extension of the method should provide an algorithm to evolve the local parametrization around each node.…”

Section: Discussionmentioning

confidence: 99%

Approximation of tensor fields on surfaces of arbitrary topology based on local Monge parametrizations

Torres-Sánchez

Santos-Oliván

Arroyo

2020

Journal of Computational Physics

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We introduce a new method, the Local Monge Parametrizations (LMP) method, to approximate tensor fields on general surfaces given by a collection of local parametrizations, e.g. as in finite element or NURBS surface representations. Our goal is to use this method to solve numerically tensor-valued partial differential equations (PDE) on surfaces. Previous methods use scalar potentials to numerically describe vector fields on surfaces, at the expense of requiring higher-order derivatives of the approximated fields and limited to simply connected surfaces, or represent tangential tensor fields as tensor fields in 3D subjected to constraints, thus increasing the essential number of degrees of freedom. In contrast, the LMP method uses an optimal number of degrees of freedom to represent a tensor, is general with regards to the topology of the surface, and does not increase the order of the PDEs governing the tensor fields. The main idea is to construct maps between the element parametrizations and a local Monge parametrization around each node. We test the LMP method by approximating in a least-squares sense different vector and tensor fields on simply connected and genus-1 surfaces. Furthermore, we apply the LMP method to two physical models on surfaces, involving a tension-driven flow (vector-valued PDE) and nematic ordering (tensor-valued PDE). The LMP method thus solves the long-standing problem of the interpolation of tensors on general surfaces with an optimal number of degrees of freedom.

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Section: Discussionmentioning

confidence: 99%

Approximation of tensor fields on surfaces of arbitrary topology based on local Monge parametrizations

Torres-Sánchez

Santos-Oliván

Arroyo

2020

Journal of Computational Physics

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“…with κ B being the bending modulus, H denoting the local mean curvature, H 0 the spontaneous curvature, κ K the Gaussian modulus and K the Gaussian curvature. On the one hand, from a given energy functional, in-plane surface stresses and moments can be deduced by functional derivatives with respect to the strain tensor [74,87] and curvature tensor, respectively [78,91,92]. On the other hand, in numerical algorithms the explicit introduction of stresses is typically bypassed and forces on the nodes discretizing the membrane are often computed directly by deriving a discretized version of the energy functional, here equations (7) and (8), with respect to the node positions [50,78,93].…”

Section: Physical Modelmentioning

confidence: 99%

Computational modeling of active deformable membranes embedded in three-dimensional flows

Bächer

Gekle

2019

Phys. Rev. E

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Active gel theory has recently been very successful in describing biologically active materials such as actin filaments or moving bacteria in temporally fixed and simple geometries such as cubes or spheres. Here we develop a computational algorithm to compute the dynamic evolution of an arbitrarily shaped, deformable thin membrane of active material embedded in a 3D flowing liquid. For this, our algorithm combines active gel theory with the classical theory of thin elastic shells. To compute the actual forces resulting from active stresses, we apply a parabolic fitting procedure to the triangulated membrane surface. Active forces are then dynamically coupled via an Immersed-Boundary method to the surrounding fluid whose dynamics can be solved by any standard, e.g. Lattice-Boltzmann, flow solver. We validate our algorithm using the Green's functions of [Berthoumieux et al. New J. Phys. 16, 065005 (2014)] for an active cylindrical membrane subjected (i) to a locally increased active stress and (ii) to a homogeneous active stress. For the latter scenario, we predict in addition a so far unobserved non-axisymmetric instability. We highlight the versatility of our method by analyzing the flow field inside an actively deforming cell embedded in external shear flow. Further applications may be cytoplasmic streaming or active membranes in blood flows. Published as: Phys. Rev. E 99(6), 062418 (2019)

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“…The theory and its application example presented here provide a rigorous basis for the development of numerical methods, for example in the framework of finite elements, that are applicable to both solidand fluid-like material behavior. This is currently being considered in the framework of C 1 -continuous surface discretizations [20,18].…”

Section: Unconstrained Liquid Shellmentioning

confidence: 99%

On the theoretical foundations of thin solid and liquid shells

Sauer

Duong

2016

Mathematics and Mechanics of Solids

129

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This paper gives a concise summary of the general theoretical framework suitable to describe shells with solid-like and liquid-like behavior. Thin-shell kinematics are considered and used to derive the equilibrium equations from linear-and angular-momentum balance. Based on the mechanical power balance and the mechanical dissipation inequality, the constitutive equations for the hyperelastic material behavior of constrained shells are derived and their material stability is examined. Various constitutive examples are considered and assessed for their stability. The governing weak form of the formulation is derived and decomposed into in-plane and out-of-plane components. The presented work provides a very general framework for a unified description of solid and liquid shells and illustrates what leads to their loss of material stability. This framework serves as a basis for developing computational shell formulations based on rotation-free shell discretizations. Therefore the full linearization of the formulation is also presented here.

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A stabilized finite element formulation for liquid shells and its application to lipid bilayers

Cited by 62 publications

References 66 publications

Approximation of tensor fields on surfaces of arbitrary topology based on local Monge parametrizations

Approximation of tensor fields on surfaces of arbitrary topology based on local Monge parametrizations

Computational modeling of active deformable membranes embedded in three-dimensional flows

On the theoretical foundations of thin solid and liquid shells

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