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(a, φ ≈ 0.8): Low insertion speed, "spiral" morphology. Implicit integration converges even at large constant ∆t. (b, φ ≈ 0.3): High insertion speed, constant ∆t. Implicit integration convergence failure sets in at very few self-contacts. (c, φ ≈ 0.6): High insertion speed, adaptive time stepping. Convergence is retained throughout the simulation. The color encodes the wire-wire contact energy from zero (blue) to high (red). For better visibility, the three energy scales are distinct.
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A finite element program is presented to simulate the process of packing and
coiling elastic wires in two- and three-dimensional confining cavities. The
wire is represented by third order beam elements and embedded into a
corotational formulation to capture the geometric nonlinearity resulting from
large rotations and deformations. The hyperbolic e...
Contexts in source publication
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... path through energy valleys getting narrower and narrower, there is an increasing chance of starting far enough from the sought energy minimum that no path is found any more. In such situations, adaptive time step control is indispensable for avoiding divergence dead ends and to ensure the found local minimum corresponds to the physical solution. Fig. 4 visualizes the implications in practice. A two-dimensional circular cavity with effective size R/r = 50 is packed with a frictionless elastic wire, at two different insertion velocities. The "spiral" morphology ( Stoop et al., 2008) evolving from a slow insertion can be simulated to large packing densities even implicitly at relatively ...
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... in practice. A two-dimensional circular cavity with effective size R/r = 50 is packed with a frictionless elastic wire, at two different insertion velocities. The "spiral" morphology ( Stoop et al., 2008) evolving from a slow insertion can be simulated to large packing densities even implicitly at relatively large constant time steps ∆t (Fig. 4a). Using the same step size but an insertion speed high Leung and Wong (2000) 58.51 40.46 22.23 51.92 48.69 18.53 46.82 53.6 15.76 Rhim and Lee (1998) 58.58 40.31 22.16 - 47.07 53.46 15.59 Lo (1992) 58.8 40.1 22.3 52.3 48.4 18.6 47.2 53.4 15.8 Crisfield (1990) 58.53 40.53 22.16 51.93 48.79 18.43 46.84 53.71 15.61 Sandhu et al. (1990) 58 ...
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... 47.2 53.4 15.8 Crisfield (1990) 58.53 40.53 22.16 51.93 48.79 18.43 46.84 53.71 15.61 Sandhu et al. (1990) 58 enough to introduce dynamic effects from the insertion, the rough valley structure emerging in the local energy landscape inhibits the line search solver from converging as soon as wire-wire contacts reach a certain degree of complexity (Fig. 4b). Convergence here means meeting one of the convergence criteria within 1000 nonlinear iterations and at line search step sizes λ > λ min = 10 −12 . The used convergence criteria include r 2 < 10 −50 , r 2 < 10 −10 r 0 2 , and ∆u 2 < 10 −10 , where r 0 is the initial residual vector, and ∆u is the proposed NR increment. Simulating the ...
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... include r 2 < 10 −50 , r 2 < 10 −10 r 0 2 , and ∆u 2 < 10 −10 , where r 0 is the initial residual vector, and ∆u is the proposed NR increment. Simulating the "classic" morphology ( Stoop et al., 2008) to large densities in reasonable CPU time is within the realm of possibility for the fully updated implicit solver only with adaptive time stepping (Fig. 4c). Given the steep energy landscape inherently arising from strongly confined wire packings, it may seem tempting to artificially smoothen it by treating contact forces constant during a single NR solver sweep. That is, not updating f ext during solving, and hence setting J ext = 0, as indicated at the end of Section 3.2. While this ...
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Citations
... The manner by which filaments pack inside a given shape is an important problem in many fields, from the ordering of microtubules inside plant cells [1][2][3], via packing of genetic material inside viral capsids [4][5][6][7][8][9][10], through spooling of filaments and biopolymers [11][12][13][14][15][16][17]. Most approaches to such problems are numerical and regard highly symmetric volumes, as these are hard problems that involve elasticity, entropy, self-avoidance, and global (shape) constraints. ...
... The results also support effects seen in other works regarding the configurations of flexible filaments in nontrivial containers, such as those in [4,[7][8][9][10][11][12][13]44,[63][64][65][66][67], and serve to both strengthen the importance of the container geometry, and expand those works by adding analytical insight to the complex phase space that results from considering both geometry and self-avoidance. ...
... Finally, this work does not consider asymmetric spheroids or other shapes of containers [4,[7][8][9][10][11][12][13]44,[63][64][65][66][67], charged filaments [34,[37][38][39][40][41][42][43][44], filaments that exhibit torsion [68,69] or FIG. 9. σ as function of δ for u = 0 and = 10, 000, approaching the limit of pure cold, non-self-avoiding filaments. That σ has a local minimum at small δ's suggests that a transition similar to the one described in Ref. [23] occurs. ...
Using a statistical-mechanics approach, we study the effects of geometry and self-avoidance on the ordering of slender filaments inside nonisotropic containers, considering cortical microtubules in plant cells, and packing of genetic material inside viral capsids as concrete examples. Within a mean-field approximation, we show analytically how the shape of the container, together with self-avoidance, affects the ordering of the stiff rods. We find that the strength of the self-avoiding interaction plays a significant role in the preferred packing orientation, leading to a first-order transition for oblate cells, where the preferred orientation changes from azimuthal, along the equator, to a polar one, when self-avoidance is strong enough. While for prolate spheroids the ground state is always a polar-like order, strong self-avoidance results with a deep metastable state along the equator. We compute the critical surface describing the transition between azimuthal and polar ordering in the three-dimensional parameter space (persistence length, eccentricity, and self-avoidance) and show that the critical behavior of this system is in fact related to the butterfly catastrophe model. We calculate the pressure and shear stress applied by the filament on the surface, and the injection force needed to be applied on the filament in order to insert it into the volume. We compare these results to the pure mechanical study where self-avoidance is ignored, and discuss similarities and differences.
... Notice that in Eq. 19, we assume inextensibility of the polygon edges in the bending forces for simplicity, as proposed in [53]. Moreover, to avoid the discontinuity in the coordinate of the closest point of approach between interacting hoop edges [55], we set ⃗ ∇iξ = 0, such that the contact forces are applied in normal direction, which considerably simplifies the interaction expression: ...
... PolyHoop thus enables simulations of the packing of elastic rods (previously performed on linear rods [55,62]) with circular hoops. ...
We present PolyHoop, a lightweight standalone C++ implementation of a mechanical model to simulate the dynamics of soft particles and cellular tissues in two dimensions. With only few geometrical and physical parameters, PolyHoop is capable of simulating a wide range of particulate soft matter systems: from biological cells and tissues to vesicles, bubbles, foams, emulsions, and other amorphous materials. The soft particles or cells are represented by continuously remodeling, non-convex, high-resolution polygons that can undergo growth, division, fusion, aggregation, and separation. With PolyHoop, a tissue or foam consisting of a million cells with high spatial resolution can be simulated on conventional laptop computers.
... The problem of packing a thin flexible object in a closed cavity has been extensively studied in the literature [110]. Based on that, numerical approaches have been derived to simulate also the coiling procedure. ...
... Neuroepithelial cells at the NP midline have a lower apicobasal height than lateral neuroepithelial cells due to SHH signaling from the notochord (22,23,38). Bending energy scales with the tissue thickness cubed (39), such that tissues subject to transverse forces tend to bend at the thinnest sites. Midline tapering was present during folding in both our mouse and human data ( Fig. 2 G and H ). It may thus facilitate MHP formation and tube folding. ...
... The NNE apicobasal height was measured at 7.2 ± 1.2 μm (SD) in human embryos, which is ∼6 times shorter compared to the NP (Fig. 3E). According to linear elastic beam theory (39), this makes NNE about 6 3 = 216 times less rigid in bending than the NP. After incorporating this information into model I, midline tapering (with h NP /h MHP = 2, w HP = 2h NP ) and midline intrinsic curvature (κ mid = κ max , w HP = 2h NP ) were no longer sufficient to create a tube shape (Fig. 3F ). ...
... We employed third-order beam elements containing an additional energy term for transverse shear that depends on Poisson's ratio ν. A detailed description can be found in ref. 39. A predictor-corrector method from the Newmark family was used to integrate Newton's equations of motion in time, using a uniform homogeneous mass density ρ in the tissue. ...
Significance
Spinal neural tube folding shows a transition from medial to dorsolateral neural plate bending sites (hinge points) along the embryo’s body axis. While hinge points are considered an important driving force in neural tube closure, their biomechanical origin is poorly understood. Here we use a computational model based on mouse and human embryo imaging data to show that the median hinge point emerges in simulations combining mesoderm expansion, nonneural ectoderm expansion, and neural plate adhesion to the notochord. Furthermore, dorsolateral hinge points emerge through a medial pulling force exerted by dorsal nonneural ectoderm (zippering). This indicates that spinal neural tube folding could be primarily driven by mesoderm expansion and zippering, with hinge points passively emerging in response to extrinsic forces.
... The manner by which filaments pack inside a given shape is an important problem in may fields. From the ordering of micro-tubules inside plant cells [2][3][4], via packing of genetic material inside viral capsids [5][6][7][8], through spooling of filaments and biopolymers [9][10][11][12][13][14][15]. Most approaches to such problems are numerical, and regard highly symmetric volumes, as these are hard problems that involve elasticity, entropy, self-avoidance, and global (shape) constraints. ...
Using a statistical - mechanics approach, we study the effects of geometry and entropy on the ordering of slender filaments inside non-isotropic containers, considering cortical microtubules in plant cells, and packing of genetic material inside viral capsids as concrete examples. Working in a mean - field approximation, we show analytically how the shape of container, together with self avoidance, affects the ordering of stiff rods. We find that the strength of the self-avoiding interaction plays a significant role in the preferred packing orientation, leading to a first-order transition for oblate cells, where the preferred orientation changes from azimuthal, along the equator, to a polar one, when self avoidance is strong enough. While for prolate spheroids the ground state is always a polar like order, strong self avoidance result with a deep meta-stable state along the equator. We compute the critical surface describing the transition between azimuthal and polar ordering in the three dimensional parameter space (persistence length, container shape, and self avoidance) and show that the critical behavior of this systems, in fact, relates to the butterfly catastrophe model. We compare these results to the pure mechanical study in arXiv:1910.08171, and discuss similarities and differences. We calculate the pressure and shear stress applied by the filament on the surface, and the injection force needed to be applied on the filament in order to insert it into the volume
... This is because merging colonies and the accumulated stress trigger multi-layer formation and limit the size of monolayer region around the edge (Figure 4figure supplement 1). We investigated whether these multi-layered structures affect the radial alignment during inward growth by performing three-dimensional (3D) FEM simulations based on recently developed algorithms (Yaman et al., 2019;Vetter et al., 2015;Vetter et al., 2013). Our previous computational tool (GRO) cannot simulate bacterial growth in 3D. ...
... For the 3D computer simulations, we employed an open-source (https://libmesh.github.io/) parallel finite element library written in C++ (Vetter et al., 2013). Sample files and scripts used for analysis are available on GitHub (https://github.com/mustafa-basaran/Large_Scale_Orientation_Bacteria, ...
During colony growth, complex interactions regulate the bacterial orientation, leading to the formation of large-scale ordered structures, including topological defects, microdomains, and branches. These structures may benefit bacterial strains, providing invasive advantages during colonization. Active matter dynamics of growing colonies drives the emergence of these ordered structures. However, additional biomechanical factors also play a significant role during this process. Here we show that the velocity profile of growing colonies creates strong radial orientation during inward growth when crowded populations invade a closed area. During this process, growth geometry sets virtual confinement and dictates the velocity profile. Herein, flow-induced alignment and torque balance on the rod-shaped bacteria result in a new stable orientational equilibrium in the radial direction. Our analysis revealed that the dynamics of these radially oriented structures also known as aster defects, depend on bacterial length and can promote the survival of the longest bacteria around localized nutritional hot spots. The present results indicate a new mechanism underlying structural order and provide mechanistic insights into the dynamics of bacterial growth on complex surfaces.
... Neuroepithelial cells at the NP midline have a lower apicobasal height than lateral neuroepithelial cells, due to SHH signaling from the notochord [18,19,31]. Bending energy scales with the tissue thickness cubed [32], such that tissues subject to transverse forces tend to bend at the thinnest sites. Thus, midline tapering may facilitate MHP formation and tube folding. ...
... The NNE apicobasal height was measured at 7.2 ± 1.2 µm (SD) in human embryos, which is approximately 6 times shorter compared to the NP (Fig. 3E). According to linear elastic beam theory [32], this makes NNE 6 3 ≈ 216 times less rigid in bending than the NP. After incorporating this information into Model I, midline tapering (with hNP/hMHP = 2, wHP = 2hNP) and midline intrinsic curvature (κ mid = κmax, wHP = 2hNP) were no longer sufficient to create a tube shape (Fig. 3F). ...
... To understand these observed differences along the rostrocaudal axis, we studied which factors determine MHP curvature. As elastic structures tend to deform at their thinnest point, where least force is required [32], we considered midline tapering. To see how midline tapering changes over developmental time between Mode 1 and Mode 3 folding, we analyzed high-resolution plastic sections and HREM data of the mouse PNP between Modes 1-3 (SS11-SS31). ...
Neurulation is the process in early vertebrate embryonic development during which the neural plate folds to form the neural tube. Spinal neural tube folding in the posterior neuropore changes over time, first showing a median hingepoint, then both the median hingepoint and dorsolateral hingepoints, followed by dorsolateral hingepoints only. The biomechanical mechanism of hingepoint formation in the mammalian neural tube is poorly understood. Here, we employ a mechanical finite element model to study neural tube formation. The computational model mimics the mammalian neural tube using microscopy data from mouse and human embryos. While intrinsic curvature at the neural plate midline has been hypothesized to drive neural tube folding, intrinsic curvature was not sufficient for tube closure in our simulations. We achieved neural tube closure with an alternative model combining mesoderm expansion, non-neural ectoderm expansion and neural plate adhesion to the notochord. Dorsolateral hingepoints emerged in simulations with low mesoderm expansion and zippering. We propose that zippering provides the biomechanical force for dorsolateral hingepoint formation in settings where the neural plate lateral sides extend above the mesoderm. Together, these results provide a new perspective on the biomechanical and molecular mechanism of mammalian spinal neurulation.
... As tube collapse is the same along the tube length up to minor boundary effects for a cylindrical geometry, we exploited the symmetry by considering only a two-dimensional cross-section perpendicular to the tube axis to simulate epithelial tube collapse (Fig. 5). A full technical description of the custom finite element simulation framework that we employed to simulate epithelial tube collapse can be found in Vetter et al. (2013). Originally developed for more general linearly (visco-)elastic thin structures, it allows for efficient transient simulation of thickness-preserving epithelial tissues in 2D. ...
... Here, M denotes the finite element mass matrix, D the viscous damping matrix, x the vector holding the translational and rotational degrees of freedom and f the total generalized force vector including all forces from elasticity (Àr x U ), contact, applied pressure and the volume constraint. Elastic contact between tube segments and between the epithelium and the rigid clamps was modelled using Hertzian contact mechanics where applicable, as also detailed in Vetter et al. (2013). In the volume-controlled collapse scenario, the volume enclosed by the epithelium was imposed with a Lagrange multiplier and reduced linearly over time. ...
During lung development, epithelial branches expand preferentially in a longitudinal direction. This bias in outgrowth has been linked to a bias in cell shape and in the cell division plane. How this bias arises is unknown. Here, we show that biased epithelial outgrowth occurs independent of the surrounding mesenchyme, of preferential turnover of the extracellular matrix at the bud tips and of FGF signalling. There is also no evidence for actin-rich filopodia at the bud tips. Rather, we find epithelial tubes to be collapsed during early lung and kidney development, and we observe fluid flow in the narrow tubes. By simulating the measured fluid flow inside segmented narrow epithelial tubes, we show that the shear stress levels on the apical surface are sufficient to explain the reported bias in cell shape and outgrowth. We use a cell-based vertex model to confirm that apical shear forces, unlike constricting forces, can give rise to both the observed bias in cell shapes and tube elongation. We conclude that shear stress may be a more general driver of biased tube elongation beyond its established role in angiogenesis.
This article has an associated ‘The people behind the papers’ interview.
... For the 3D computer simulations, we employed a parallel finite element program written in C++ [36]. Analogous to [15], the bacteria were modeled as an isotropic, linearly elastic continuum whose intial stress-free shapes were spherocylinders. ...
During colony growth, complex biomechanical interactions regulate the bacterial orientation, leading to the formation of large-scale ordered structures including topological defects. These structures may benefit bacterial strains, providing invasive advantages during colonization. Thus far, the emergence of 1/2 topological defects has been observed and extensively assessed. Here, we experimentally and numerically investigate aster formation in bacterial colonies during inward growth when crowded populations invade a closed area. Herein, the velocity field and torque balance on the rod-shaped bacteria significantly differed, resulting in new stable orientational equilibrium in the radial direction and to the formation of +1 topological defects. The dynamics of these defects depend on bacterial length and can promote the survival of the longest bacteria around localized nutritional hot spots. The present results indicate a new mechanism underlying defect formation and provide mechanistic insights into the dynamics of bacterial growth on complex surfaces.
... To address this issue, a computational approach is a reasonable alternative to illustrate specific effects of mechanical parameters and conditions on the coil structure and thereby speculate on these underlying mechanisms. Because the problem of dense packing of thin flexible objects is encounterd in various fields of natural science, this general issue has attracted much attention in the physics literature [13][14][15][16][17][18] , and computational approaches have been applied to construct structures of densely packed objects by using discrete or finite element methods [13,19] . Existing studies have demonstrated that the structure of the densely packed object can be classified into several patterns according to its mechanical properties such as axial torsion [13] , friction [17] , and material plasticity [18] . ...
... Its three-dimensional behavior, including large deflection, is specifically addressed by the idea of corotational beam element formulation, in which the term "corotational " refers to the provision of a single element frame that continually rotates with the beam element [20] . In this approach, large deflection of the beam can be treated as separately divided into small deformation on the rotating frame and finite rotation of the frame independently; thus, the constitutive law of the beam can be considered to be linear [19] . This design allows us to use apparent axial, flexural, and torsional rigidities of the coil determined by the single helical shape of the coil in computation. ...
... A dot shows derivative with respect to time. We simplified the mass matrix as a diagonal matrix at each coil node p , given by M = diag [ , , , , , ] , where m p is the mass calculated by linear interpolation in each beam element using the apparent density of the coil (i.e., coil mass/beam volume) and J p is the inertia moment simplified as the values of isotropic sphere based on [19] , since the inertia effect can be insufficient in the coil deployment process. The damping matrix C is represented by Rayleigh ansatz C = M . ...
In this paper we develop a computational model of endovascular coiling for cerebral aneurysms to interpret the mechanical behavior of the coil during the deployment process into the aneurysm. Three-dimensional coil behavior, including large deflection, is expressed by corotational beam element formulation, and transitions of the coil configuration during deployment are obtained by solving the equation of motion while considering friction. Numerical examples of the straight-shape coil deployment to an idealized aneurysm via a catheter demonstrate that the coil forms an ordered loop structure at the initial stage of coil deployment, while axial buckling is found at the part of the coil just released from the catheter during deployment. Although the cause of the axial buckling can be well explained by frictional resistances exerted on the coil placed in the aneurysm, the post-buckling structure of the coil shows two patterns depending on the frictional coefficient. At low friction, the whole coil placed in the aneurysm slides along the aneurysm surface and maintains its ordered structure. At high friction, however, large deflection occurs where the coil has just been released from the catheter and after pushing itself into place in the aneurysm, the ordered coil structure collapses and formation of a disordered structure with uniform distribution ensues. The present model is valuable for interpreting the mechanical behavior of endovascular coils and for estimating the extent of spatial uniformity achieved by a coil in an aneurysm.
