Elastic Bending Energy Research Articles

Efficient linear schemes with unconditional energy stability for the phase field elastic bending energy model

In this paper, we study efficient numerical schemes of the classical phase field elastic bending energy model that has been widely used to describe the shape deformation of biological lipid vesicles, in which the free energy of the system consists of an elastic bending energy, a surface area constraint and a volume constraint. One major challenge in solving such model numerically is how to design appropriate temporal discretizations in order to preserve energy stability with large time step sizes at the semi-discrete level. We develop a first order and a second order time stepping scheme for this highly nonlinear and stiff parabolic PDE system based on the “Invariant Energy Quadratization” approach. In particular, the resulted semi-discretizations lead to linear systems in space with symmetric positive definite operators at each time step, thus can be efficiently solved. In addition, the proposed schemes are rigorously proved to be unconditionally energy stable. Various numerical experiments in 2D and 3D are presented to demonstrate the stability and accuracy of the proposed schemes.

Nucleation, Growth, and Bundling of GaN Nanowires in Molecular Beam Epitaxy: Disentangling the Origin of Nanowire Coalescence.

We investigate the nucleation, growth, and coalescence of spontaneously formed GaN nanowires in molecular beam epitaxy combining the statistical analysis of scanning electron micrographs with Monte Carlo growth models. We find that (i) the nanowire density is limited by the shadowing of the substrate from the impinging fluxes by already existing nanowires, (ii) shortly after the nucleation stage, nanowire radial growth becomes negligible, and (iii) coalescence is caused by bundling of nanowires. The latter phenomenon is driven by the gain of surface energy at the expense of the elastic energy of bending and becomes energetically favorable once the nanowires exceed a certain critical length.

Three-dimensional instabilities of pantographic sheets with parabolic lattices: numerical investigations

In this paper, we determine numerically a large class of equilibrium configurations of an elastic two-dimensional continuous pantographic sheet in three-dimensional deformation consisting of two families of fibers which are parabolic prior to deformation. The fibers are assumed (1) to be continuously distributed over the sample, (2) to be endowed of bending and torsional stiffnesses, and (3) tied together at their points of intersection to avoid relative slipping by means of internal (elastic) pivots. This last condition characterizes the system as a pantographic lattice (Alibert and Della Corte in Zeitschrift fur angewandte Mathematik und Physik 66(5):2855–2870, 2015; Alibert et al. in Math Mech Solids 8(1):51–73, 2003; dell’Isola et al. in Int J Non-Linear Mech 80:200–208, 2016; Int J Solids Struct 81:1–12, 2016). The model that we employ here, developed by Steigmann and dell’Isola (Acta Mech Sin 31(3):373–382, 2015) and first investigated in Giorgio et al. (Comptes rendus Mecanique 2016, doi: 10.1016/j.crme.2016.02.009 ), is applicable to fiber lattices in which three-dimensional bending, twisting, and stretching are significant as well as a resistance to shear distortion, i.e., to the angle change between the fibers. Some relevant numerical examples are exhibited in order to highlight the main features of the model adopted: In particular, buckling and post-buckling behaviors of pantographic parabolic lattices are investigated. The fabric of the metamaterial presented in this paper has been conceived to resist more effectively in the extensional bias tests by storing more elastic bending energy and less energy in the deformation of elastic pivots: A comparison with a fabric constituted by beams which are straight in the reference configuration shows that the proposed concept is promising.

Efficient and stable exponential time differencing Runge–Kutta methods for phase field elastic bending energy models

The Willmore flow formulated by phase field dynamics based on the elastic bending energy model has been widely used to describe the shape transformation of biological lipid vesicles. In this paper, we develop and investigate some efficient and stable numerical methods for simulating the unconstrained phase field Willmore dynamics and the phase field Willmore dynamics with fixed volume and surface area constraints. The proposed methods can be high-order accurate and are completely explicit in nature, by combining exponential time differencing Runge–Kutta approximations for time integration with spectral discretizations for spatial operators on regular meshes. We also incorporate novel linear operator splitting techniques into the numerical schemes to improve the discrete energy stability. In order to avoid extra numerical instability brought by use of large penalty parameters in solving the constrained phase field Willmore dynamics problem, a modified augmented Lagrange multiplier approach is proposed and adopted. Various numerical experiments are performed to demonstrate accuracy and stability of the proposed methods.

A two phase field model for tracking vesicle-vesicle adhesion.

A multi-phase-field model for simulating the adhesion between two vesicles is constructed. Two phase field functions are introduced to simulate each of the two vesicles. An energy model is defined which accounts for the elastic bending energy of each vesicle and the contact potential energy between the two vesicles; the vesicle volume and surface area constraints are imposed using a penalty method. Numerical results are provided to verify the efficacy of our model and to provide visual illustrations of the different types of contact. The method can be adjusted to solve endocytosis problems by modifying the bending rigidity coefficients of the two elastic bending energies. The method can also be extended to simulate multi-cell adhesions, one example of which is erythrocyte rouleaux. A comparison with laboratory observations demonstrates the effectiveness of the multi-phase field approach.

Shape optimization of self-avoiding curves

This paper presents a softened notion of proximity (or self-avoidance) for curves. We then derive a sensitivity result, based on shape differential calculus, for the proximity. This is combined with a gradient-based optimization approach to compute three-dimensional, parameterized curves that minimize the sum of an elastic (bending) energy and a proximity energy that maintains self-avoidance by a penalization technique. Minimizers are computed by a sequential-quadratic-programming (SQP) method where the bending energy and proximity energy are approximated by a finite element method. We then apply this method to two problems. First, we simulate adsorbed polymer strands that are constrained to be bound to a surface and be (locally) inextensible. This is a basic model of semi-flexible polymers adsorbed onto a surface (a current topic in material science). Several examples of minimizing curve shapes on a variety of surfaces are shown. An advantage of the method is that it can be much faster than using molecular dynamics for simulating polymer strands on surfaces. Second, we apply our proximity penalization to the computation of ideal knots. We present a heuristic scheme, utilizing the SQP method above, for minimizing rope-length and apply it in the case of the trefoil knot. Applications of this method could be for generating good initial guesses to a more accurate (but expensive) knot-tightening algorithm. • A new softened version of proximity (or self-contact) for 1-D curves is proposed. • Sensitivity of the proximity, with respect to shape differentiation, is given. • A method for finding minimizing curve shapes (that respect self-contact) is given. • The method is applied to simulating polymer strands adsorbed onto surfaces. • A heuristic method is given for approximating ideal knots.

Algorithm 952

The implementation of a library of basic functions for the construction and analysis of planar quintic Pythagorean-hodograph (PH) curves is presented using the complex representation. The special algebraic structure of PH curves permits exact algorithms for the computation of key properties, such as arc length, elastic bending energy, and offset (parallel) curves. Single planar PH quintic segments are constructed as interpolants to first-order Hermite data (end points and derivatives), and this construction is then extended to open or closed C 2 PH quintic spline curves interpolating a sequence of points in the plane. The nonlinear nature of PH curves incurs a multiplicity of formal solutions to such interpolation problems, and a key aspect of the algorithms is to efficiently single out the unique “good” interpolant among them.

Multiscale Stick-Slip Dynamics of Adhesive Tape Peeling.

Using a high-speed camera, we follow the propagation of the detachment front during the peeling of an adhesive tape from a flat surface. In a given range of peeling velocity, this front displays a multiscale unstable dynamics, entangling two well-separated spatiotemporal scales, which correspond to microscopic and macroscopic dynamical stick-slip instabilities. While the periodic release of the stretch energy of the whole peeled ribbon drives the classical macro-stick-slip, we show that the micro-stick-slip, due to the regular propagation of transverse dynamic fractures discovered by Thoroddsen et al. [Phys. Rev. E 82, 046107 (2010)], is related to a high-frequency periodic release of the elastic bending energy of the adhesive ribbon concentrated in the vicinity of the peeling front.

An adaptive finite element method for the modeling of the equilibrium of red blood cells

SummaryThis contribution is concerned with the numerical modeling of an isolated red blood cell (RBC), and more generally of phospholipid membranes. We propose an adaptive Eulerian finite element approximation, based on the level set method, of a shape optimization problem arising in the study of RBCs. We simulate the equilibrium shapes that minimize the elastic bending energy under prescribed constraints of fixed volume and surface area. An anisotropic mesh adaptation technique is used in the vicinity of the cell membrane to enhance the robustness of the method. Efficient time and spatial discretizations are considered and implemented. We address in detail the main features of the proposed method, and finally we report several numerical experiments in the two‐dimensional and the three‐dimensional axisymmetric cases. The effectiveness of the numerical method is further demonstrated through numerical comparisons with semi‐analytical solutions provided by a reduced order model. Copyright © 2015 John Wiley & Sons, Ltd.

A Model for Shaping Membrane Sheets by Protein Scaffolds

Membranes of peripheral endoplasmic reticulum form intricate morphologies consisting of tubules and sheets as basic elements. The physical mechanism of endoplasmic-reticulum shaping has been suggested to originate from the elastic behavior of the sheet edges formed by linear arrays of oligomeric protein scaffolds. The heart of this mechanism, lying in the relationships between the structure of the protein scaffolds and the effective intrinsic shapes and elastic properties of the sheets’ edges, has remained hypothetical. Here we provide a detailed computational analysis of these issues. By minimizing the elastic energy of membrane bending, we determine the effects of a rowlike array of semicircular arclike membrane scaffolds on generation of a membrane fold, which shapes the entire membrane surface into a flat double-membrane sheet. We show, quantitatively, that the sheet’s edge line tends to adopt a positive or negative curvature depending on the scaffold’s geometrical parameters. We compute the effective elastic properties of the sheet edge and analyze the dependence of the equilibrium distance between the scaffolds along the edge line on the scaffold geometry.

Confined disclinations: exterior versus material constraints in developable thin elastic sheets.

We examine the shape change of a thin disk with an inserted wedge of material when it is pushed against a plane, using analytical, numerical, and experimental methods. Such sheets occur in packaging, surgery, and nanotechnology. We approximate the sheet as having vanishing strain, so that it takes a conical form in which straight generators converge to a disclination singularity. Then, its shape is that which minimizes elastic bending energy alone. Real sheets are expected to approach this limiting shape as their thickness approaches zero. The planar constraint forces a sector of the sheet to buckle into the third dimension. We find that the unbuckled sector is precisely semicircular, independent of the angle δ of the inserted wedge. We generalize the analysis to include conical as well as planar constraints and thereby establish a law of corresponding states for shallow cones of slope ε and thin wedges. In this regime, the single parameter δ/ε^{2} determines the shape. We discuss the singular limit in which the cone becomes a plane, and the unexpected slow convergence to the semicircular buckling observed in real sheets.

Hydrophobic nanoparticles promote lamellar to inverted hexagonal transition in phospholipid mesophases.

This study focuses on how the mesophase transition behaviour of the phospholipid dioleoyl phosphatidylethanolamine (DOPE) is altered by the presence of 10 nm hydrophobic and 14 nm hydrophilic silica nanoparticles (NPs) at different concentrations. The lamellar to inverted hexagonal phase transition (Lα-HII) of phospholipids is energetically analogous to the membrane fusion process, therefore understanding the Lα-HII transition with nanoparticulate additives is relevant to how membrane fusion may be affected by these additives, in this case the silica NPs. The overriding observation is that the HII/Lα boundaries in the DOPE p-T phase diagram were shifted by the presence of NPs: the hydrophobic NPs enlarged the HII phase region and thus encouraged the inverted hexagonal (HII) phase to occur at lower temperatures, whilst hydrophilic NPs appeared to stabilise the Lα phase region. This effect was also NP-concentration dependent, with a more pronounced effect for higher concentration of the hydrophobic NPs, but the trend was less clear cut for the hydrophilic NPs. There was no evidence that the NPs were intercalated into the mesophases, and as such it was likely that they might have undergone microphase separation and resided at the mesophase domain boundaries. Whilst the loci and exact roles of the NPs invite further investigation, we tentatively discuss these results in terms of both the surface chemistry of the NPs and the effect of their curvature on the elastic bending energy considerations during the mesophase transition.

Conformation of a charged vesicle.

We report on the first systematic study of vesicle conformational change caused by Coulomb interaction between surface charges on a lipid vesicle. The equilibrium configuration of a charged vesicle is found, as the result of the competition between the local bending elastic energy and the long-range electrostatic interaction within the membrane where the counter-ion effects are neglected. Because of the Rayleigh instability, a charged vesicle undergoes conformational transitions as a function of the surface charge density.

Exact Solution of the Elastica with Transverse Shear Effects

A new derivation of Euler's Elastica with transverse shear effects included is presented. The elastic potential energy of bending and transverse shear is set up. The work of the axial compression force is determined. The equation of equilibrium is derived using the variation of the total potential. Using substitution of variables an exact solution is derived. The equation is transcendental and does not have a closed form solution. It is evaluated in a dimensionless form by using a numerical procedure. Finally, numerical examples of laminates made of composite material (fiber reinforced) and sandwich panels are provided.

Preface Issue 4-2014

The last few years have been a good time for solving long standing geometrical-topological conjectures. This issue reports on the solution of the Willmore conjecture –the “best” topological torus is a “real” torus with ratio of radii equal to √ 2– and one of Thurston’s conjectures –every hyperbolic 3-manifold can be fibered over a circle, up to passing to a finite cover. Starting in the 1960s, Thomas Willmore studied the integral of the squared mean curvature of surfaces in R3 as the simplest but most interesting frame invariant elastic bending energy. This energy had shown up already in the early 19th century – too early for a rigorous investigation. Willmore asked: What is the shape of a compact surface of fixed genus minimising the Willmore energy in this class? (In the 1990s, existence of minimisers was proved by Leon Simon, with a contribution by Matthias Bauer and Ernst Kuwert.) Willmore already knew that the genus-0minimiser is a sphere. Assuming that symmetric surfaces require less energy than asymmetric ones (which has not been proved, yet) he studied families of geometric tori with the smaller radius 1 fixed and found that the larger radius √ 2 would yield the minimum in this very special family. He conjectured that this particular torus would be the genus-1-minimiser. Almost 50 years later Fernando Marques and Andre Neves found and published a 100-page-proof. In the present survey article, disregarding technical details, they explain their approach and their fundamental ideas and provide also an excellent historical and mathematical background. After Thurston’s Geometrization Conjecture was finally solved about 10 years ago by Grigori Perelman, one of the most important questions in three-dimensional topology has been to understand the hyperbolic 3-manifolds. William Thurston’s 1982–paper “Three-dimensional manifolds, Kleinian groups and hyperbolic geometry” (see the “Classics Revisited”–contribution of this year’s first issue) lists 24 at that time open problems in 3-manifolds-theory. Problem No. 18 reads as follows: “Does every finite volume hyperbolic 3-manifold have a finite cover which fibers over the circle?” This conjecture was indeed proved 2012 by Ian Agol and Dani Wise. Stefan Friedl explains their work and the requisite background in his very comprehensible survey article “Thurstons Vision for 3-Manifolds and the Virtual Fibering Theorem.” As Stefan Friedl explains, after Agol & Wise, only “Thurston’s last challenge” concerning ratios of volumes of hyperbolic 3-manifolds remains to be solved. In this issue the “Classics Revisited”-contribution (in collaboration with the Zentralblatt fur Mathematik) is written by Jean Mawhin. It concerns the “Cours d’Analyse Infinitesimale” of Charles-Jean de La Vallee Poussin, who is still famous for his proof (independent of Hadamard’s) of the prime number theorem. These textbooks were used in Belgium for an exceptionally long time. Reviewers praise the originality, clearness and elegance of the exposition and its merits as a handbook in analysis. In the “Book Reviews”-section Gunter M. Ziegler gives us a critical view of Alexander Soifer’s “The Mathematical Coloring Book”. Johannes Huebschmann evaluates the book on “Poisson Structures” written by Camille Laurent-Gengoux, Anne Picherau, and Pol Vanhaecke.

Pattern formation by phase-field relaxation of bending energy with fixed surface area and volume.

We explore a wide variety of patterns of closed surfaces that minimize the elastic bending energy with fixed surface area and volume. To avoid complicated discretization and numerical instabilities for sharp surfaces, we reformulate the underlying constrained minimization problem by constructing phase-field functionals of bending energy with penalty terms for the constraints and develop stable numerical methods to relax these functionals. We report our extensive computational results with different initial surfaces. These results are discussed in terms of the reduced volume and are compared with the known results obtained using the sharp-interface approach. Finally, we discuss the implications of our numerical findings.

Simulating vesicle–substrate adhesion using two phase field functions

A phase field model for simulating the adhesion of a cell membrane to a substrate is constructed. The model features two phase field functions which are used to simulate the membrane and the substrate. An energy model is defined which accounts for the elastic bending energy and the contact potential energy as well as, through a penalty method, vesicle volume and surface area constraints. Numerical results are provided to verify our model and to provide visual illustrations of the interactions between a lipid vesicle and substrates having complex shapes. Examples are also provided for the adhesion process in the presence of gravitational and point pulling forces. A comparison with experimental results demonstrates the effectiveness of the two phase field approach.

Gamma-Limit of a Model for the Elastic Energy of an Inextensible Ribbon

A Γ-convergence result involving the elastic bending energy of a narrow inextensible ribbon is established. As a consequence of the result, the energy is reduced to a one-dimensional integral, over the centerline of the ribbon, in which the aspect ratio of the ribbon appears as a small parameter. That integral is observed to increase monotonically with the aspect ratio. The Γ-limit of the family of energies is taken in a Sobolev space of centerlines with nonvanishing curvature. In that space, it is shown that the Γ-limit is a functional first proposed by Sadowsky in the context of narrow ribbons that form Möbius bands. The results obtained here do not apply to such ribbons, since the centerline of a Möbius band must have at least one inflection point. As a first step toward dealing with such inflection points, a result concerning the lower semicontinuity of the Sadowsky functional with inflection points comprising a set of measure zero within the domain of an arclength parameterization is presented.

Formula omitted] Hermite interpolation with spatial Pythagorean-hodograph cubic biarcs

In this paper the C1 Hermite interpolation problem by spatial Pythagorean-hodograph cubic biarcs is presented and a general algorithm to construct such interpolants is described. Each PH cubic segment interpolates C1 data at one point and they are then joined together with a C1 continuity at some unknown common point sharing some unknown tangent vector. Biarcs are expressed in a closed form with three shape parameters. Two of them are selected based on asymptotic approximation order, while the remaining one can be computed by minimizing the length of the biarc or by minimizing the elastic bending energy. The final interpolating spline curve is globally C1 continuous, it can be constructed locally and it exists for arbitrary Hermite data configurations.

Error estimates for approximations of a gradient dynamics for phase field elastic bending energy of vesicle membrane deformation

In this paper, we study the numerical approximations of a gradient flow associated with a phase field bending elasticity model of a vesicle membrane with prescribed volume and surface area. A spatially semi-discrete scheme based on a mixed finite element formulation and a fully discrete in space and time scheme are analyzed. Optimal order error estimates are rigorously derived for these numerical schemes without any a priori assumption. Copyright © 2013 John Wiley & Sons, Ltd.