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• XLME interpolants, a seamless bridge between XFEM and enriched meshless methods

  F. Amiri, C. Anitescu, M. Arroyo, S. Bordas, T. Rabezuk

  Computational Mechanics (2014). Vol. 53 (1), pp. 45-57 (Preprint)

  
  

  Abstract

  In this paper, we develop a method based on local maximum entropy shape functions together with enrichment functions used in partition of unity methods to discretize problems in linear elastic fracture mechanics. We obtain improved accuracy relative to the standard extended finite element method at a comparable computational cost. In addition, we keep the advantages of the LME shape functions, such as smoothness and non-negativity. We show numerically that optimal convergence (same as in FEM) for energy norm and stress intensity factors can be obtained through the use of geometric (fixed area) enrichment with no special treatment of the nodes near the crack such as blending or shifting.

  Abstract
  In this paper, we develop a method based on local maximum entropy shape functions together with enrichment functions used in partition of unity methods to discretize problems [...]

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• Shape control of active surfaces inspired by the movement of euglenids

  M. Arroyo, A. DeSimone

  Journal of the Mechanics and Physics of Solids (2014). Vol. 62, pp. 99-112 (Preprint)

  
  

  Abstract

  We examine a novel mechanism for active surface morphing inspired by the cell body deformations of euglenids. Actuation is accomplished through in-plane simple shear along prescribed slip lines decorating the surface. Under general non-uniform actuation, such local deformation produces Gaussian curvature, and therefore leads to shape changes. Geometrically, a deformation that realizes the prescribed local shear is an isometric embedding. We explore the possibilities and limitations of this bio- inspired shape morphing mechanism, by first characterizing isometric embeddings un- der axisymmetry, understanding the limits of embeddability, and studying in detail the accessibility of surfaces of zero and constant curvature. Modeling mechanically the active surface as a non-Euclidean plate (NEP), we further examine the mechanism beyond the geometric singularities arising from embeddability, where mechanics and buckling play a decisive role. We also propose a non-axisymmetric actuation strategy to accomplish large amplitude bending and twisting motions of elongated cylindrical surfaces. Besides helping understand how euglenids delicately control their shape, our results may provide the background to engineer soft machines

  Abstract
  We examine a novel mechanism for active surface morphing inspired by the cell body deformations of euglenids. Actuation is accomplished through in-plane simple shear along [...]

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• Nonsingular isogeometric boundary element method for stokes flows in 3D

  L. Heltai, M. Arroyo, A. DeSimone

  Comput. Methods Appl. Mech. Engrg., (2014). Vol. 268, pp. 514-539 (preprint)

  
  

  Abstract

  Isogeometric analysis (IGA) is emerging as a technology bridging computer aided geometric design (CAGD), most commonly based on Non-Uniform Rational B-Splines (NURBS) surfaces, and engineering analysis. In finite element and boundary element isogeometric methods (FE-IGA and IGA-BEM), the NURBS basis functions that describe the geometry define also the approximation spaces. In the FE-IGA approach, the surfaces generated by the CAGD tools need to be extended to volumetric descriptions, a major open problem in 3D. This additional passage can be avoided in principle when the partial differential equations to be solved admit a formulation in terms of boundary integral equations, leading to boundary element isogeometric analysis (IGA-BEM). The main advantages of such an approach are given by the dimensionality reduction of the problem (from volumetric-based to surface-based), by the fact that the interface with CAGD tools is direct, and by the possibility to treat exterior problems, where the computational domain is infinite. By contrast, these methods produce system matrices which are full, and require the integration of singular kernels. In this paper we address the second point and propose a nonsingular formulation of IGA-BEM for 3D Stokes flows, whose convergence is carefully tested numerically. Standard Gaussian quadrature rules suffice to integrate the boundary integral equations, and carefully chosen known exact solutions of the interior Stokes problem are used to correct the resulting matrices, extending the work by Klaseboer et al. (2012) [27] to IGA-BEM.

  Abstract
  Isogeometric analysis (IGA) is emerging as a technology bridging computer aided geometric design (CAGD), most commonly based on Non-Uniform Rational B-Splines (NURBS) surfaces, [...]

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• An adaptive meshfree method for phase-field models of biomembranes. Part II: A Lagrangian approach for membranes in viscous fluids

  C. Peco, A. Rosolen, M. Arroyo

  Journal of Computational Physics (2013). Vol. 249 (5), pp. 320-336 (Preprint)

  
  

  Abstract

  We present a Lagrangian phase-field method to study the low Reynolds number dynamics of vesicles embedded in a viscous fluid. In contrast to previous approaches, where the field variables are the phase-field and the fluid velocity, here we exploit the fact that the phasefield tracks a material interface to reformulate the problem in terms of the Lagrangian motion of a background medium, containing both the biomembrane and the fluid. We discretize the equations in space with maximum-entropy approximants, carefully shown to perform well in phase-field models of biomembranes in a companion paper. The proposed formulation is variational, lending itself to implicit time-stepping algorithms based on minimization of a time-incremental energy, which are automatically nonlinearly stable. The proposed method deals with two of the major challenges in the numerical treatment of coupled fluid/phase-field models of biomembranes, namely the adaptivity of the grid to resolve the sharp features of the phase-field, and the stiffness of the equations, leading to very small time-steps. In our method, local refinement follows the features of the phasefield as both are advected by the Lagrangian motion, and large time-steps can be robustly chosen in the variational time-stepping algorithm, which also lends itself to time adaptivity. The method is presented in the axisymmetric setting, but it can be directly extended to 3D. We present a Lagrangian phase-field method to study the low Reynolds number dynamics of vesicles embedded in a viscous fluid. In contrast to previous approaches, where the field variables are the phase-field and the fluid velocity, here we exploit the fact that the phase-field tracks a material interface to reformulate the problem in terms of the Lagrangian motion of a background medium, containing both the biomembrane and the fluid. We discretize the equations in space with maximum-entropy approximants, carefully shown to perform well in phase-field models of biomembranes in a companion paper. The proposed formulation is variational, lending itself to implicit time-stepping algorithms based on minimization of a time-incremental energy, which are automatically nonlinearly stable. The proposed method deals with two of the major challenges in the numerical treatment of coupled fluid/phase-field models of biomembranes, namely the adaptivity of the grid to resolve the sharp features of the phase-field, and the stiffness of the equations, leading to very small time-steps. In our method, local refinement follows the features of the phase-field as both are advected by the Lagrangian motion, and large time-steps can be robustly chosen in the variational time-stepping algorithm, which also lends itself to time adaptivity. The method is presented in the axisymmetric setting, but it can be directly extended to 3D.

  Abstract
  We present a Lagrangian phase-field method to study the low Reynolds number dynamics of vesicles embedded in a viscous fluid. In contrast to previous approaches, where the [...]

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• Blending isogeometric analysis and local maximum entropy meshfree approximants

  A. Rosolen, M. Arroyo

  Comput. Methods Appl. Mech. Engrg., (2013). Vol. 264, pp. 95-107 (preprint)

  
  

  Abstract

  We present a method to blend local maximum entropy (LME) meshfree approximants and isogeometric analysis. The coupling strategy exploits the optimization program behind LME approximation, treats isogeometric and LME basis functions on an equal footing in the reproducibility constraints, but views the former as data in the constrained minimization. The resulting scheme exploits the best features and overcomes the main drawbacks of each of these approximants. Indeed, it preserves the high fidelity boundary representation (exact CAD geometry) of isogeometric analysis, out of reach for bare meshfree methods, and easily handles volume discretization and unstructured grids with possibly local refinement, while maintaining the smoothness and non-negativity of the basis functions. We implement the method with B-Splines in two dimensions, but the procedure carries over to higher spatial dimensions or to other non-negative approximants such as NURBS or subdivision schemes. The performance of the method is illustrated with the heat equation, and linear and nonlinear elasticity. The ability of the proposed method to impose directly essential boundary conditions in non-convex domains, and to deal with unstructured grids and local refinement in domains of complex geometry and topology is highlighted by the numerical examples.

  Abstract
  We present a method to blend local maximum entropy (LME) meshfree approximants and isogeometric analysis. The coupling strategy exploits the optimization program behind LME [...]

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• An adaptive meshfree method for phase-field models of biomembranes. Part I: Approximation with maximum-entropy basis functions

  A. Rosolen, C. Peco, M. Arroyo

  Journal of Computational Physics (2013). Vol. 249 (15), pp. 303-319 (Preprint)

  
  

  Abstract

  We present an adaptive meshfree method to approximate phase-field models of biomembranes. In such models, the Helfrich curvature elastic energy, the surface area, and the enclosed volume of a vesicle are written as functionals of a continuous phase-field, which describes the interface in a smeared manner. Such functionals involve up to second-order spatial derivatives of the phase-field, leading to fourth-order Euler–Lagrange partial differential equations (PDE). The solutions develop sharp internal layers in the vicinity of the putative interface, and are nearly constant elsewhere. Thanks to the smoothness of the local maximum-entropy (max-ent) meshfree basis functions, we approximate numerically this high-order phase-field model with a direct Ritz–Galerkin method. The flexibility of the meshfree method allows us to easily adapt the grid to resolve the sharp features of the solutions. Thus, the proposed approach is more efficient than common tensor product methods (e.g. finite differences or spectral methods), and simpler than unstructured Cº finite element methods, applicable by reformulating the model as a system of second-order PDE. The proposed method, implemented here under the assumption of axisymmetry, allows us to show numerical evidence of convergence of the phase-field solutions to the sharp interface limit as the regularization parameter approaches zero. In a companion paper, we present a Lagrangian method based on the approximants analyzed here to study the dynamics of vesicles embedded in a viscous fluid. We present an adaptive meshfree method to approximate phase-field models of biomembranes. In such models, the Helfrich curvature elastic energy, the surface area, and the enclosed volume of a vesicle are written as functionals of a continuous phase-field, which describes the interface in a smeared manner. Such functionals involve up to second-order spatial derivatives of the phase-field, leading to fourth-order Euler–Lagrange partial differential equations (PDE). The solutions develop sharp internal layers in the vicinity of the putative interface, and are nearly constant elsewhere. Thanks to the smoothness of the local maximum-entropy (max-ent) meshfree basis functions, we approximate numerically this high-order phase-field model with a direct Ritz–Galerkin method. The flexibility of the meshfree method allows us to easily adapt the grid to resolve the sharp features of the solutions. Thus, the proposed approach is more efficient than common tensor product methods (e.g. finite differences or spectral methods), and simpler than unstructured C0C0 finite element methods, applicable by reformulating the model as a system of second-order PDE. The proposed method, implemented here under the assumption of axisymmetry, allows us to show numerical evidence of convergence of the phase-field solutions to the sharp interface limit as the regularization parameter approaches zero. In a companion paper, we present a Lagrangian method based on the approximants analyzed here to study the dynamics of vesicles embedded in a viscous fluid.

  Abstract
  We present an adaptive meshfree method to approximate phase-field models of biomembranes. In such models, the Helfrich curvature elastic energy, the surface area, and the [...]

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• Curved fluid membranes behave laterally as an effective viscoelastic medium

  M. Rahimi, A. DeSimone, M. Arroyo

  Soft Matter (2013). Vol. 9, pp. 11033-11045 (Preprint)

  
  

  Abstract

  The lateral mobility of membrane inclusions is essential in biological processes involving membrane-bound macromolecules, which often take place in highly curved geometries such as membrane tubes or small organelles. Probe mobility is assisted by the lateral fluidity, which is thought to be purely viscous for lipid bilayers and synthetic systems such as polymersomes. In previous theoretical studies, the hydrodynamical mobility is estimated assuming a fixed membrane geometry. However, fluid membranes are very flexible out-of-plane. By accounting for the deformability of the membrane and in the presence of curvature, we show that the lateral motion of an inclusion produces a normal force, which results in a nonuniform membrane deformation. Such a deformation mobilizes the bending elasticity, produces extra lateral viscous and elastic forces, and results in an effective lateral viscoelastic behavior. The coupling between lateral and out-of-plane mechanics is mediated by the interfacial hydrodynamics and curvature. We analyze the frequency and curvature dependent rheology of flexible fluid membranes, and interpret it with a simple four-element model, which provides a background for microrheological experiments. Two key technical aspects of the present work are a new formulation for the interfacial hydrodynamics, and the linearization of the governing equations around a cylindrical geometry.

  Abstract
  The lateral mobility of membrane inclusions is essential in biological processes involving membrane-bound macromolecules, which often take place in highly curved geometries [...]

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• Adhesion and friction control localized folding in supported graphene

  K. Zhang, M. Arroyo

  Journal of Applied Physics (2013). Vol. 113 (19), pp. 193501-1/193501-8 (Preprint)

  
  

  Abstract

  Graphene deposited on planar surfaces often exhibits sharp and localized folds delimiting seemingly planar regions, as a result of compressive stresses transmitted by the substrate. Such folds alter the electronic and chemical properties of graphene, and therefore, it is important to understand their emergence, to either suppress them or control their morphology. Here, we study the emergence of out-of-plane deformations in supported and laterally strained graphene with high-fidelity simulations and a simpler theoretical model. We characterize the onset of buckling and the nonlinear behavior after the instability in terms of the adhesion and frictional material parameters of the graphene-substrate interface. We find that localized folds evolve from a distributed wrinkling linear instability due to the nonlinearity in the van der Waals graphene-substrate interactions. We identify friction as a selection mechanism for the separation between folds, as the formation of far apart folds is penalized by the work of friction. Our systematic analysis is a first step towards strain engineering of supported graphene, and is applicable to other compressed thin elastic films weakly coupled to a substrate.

  Abstract
  Graphene deposited on planar surfaces often exhibits sharp and localized folds delimiting seemingly planar regions, as a result of compressive stresses transmitted by the [...]

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• Second order convex maximum entropy approximants with applications to high order PDE

  A. Rosolen, D. Millán, M. Arroyo

  Int. J. Numer. Meth. Engng. (2013). Vol. 94 (2), pp. 150-182 (Preprint)

  
  

  Abstract

  We present a new approach for second order maximum entropy (max-ent) meshfree approximants that produces positive and smooth basis functions of uniform aspect ratio even for non-uniform node sets, and prescribes robustly feasible constraints for the entropy maximization program defining the approximants. We examine the performance of the proposed approximation scheme in the numerical solution by a direct Galerkin method of a number of partial differential equations (PDEs), including structural vibrations, elliptic second order PDEs, and fourth order PDEs for Kirchhoff-Love thin shells and for a phase field model describing the mechanics of biomembranes. The examples highlight the ability of the method to deal with non-uniform node distributions, and the high accuracy of the solutions. Surprisingly, the first order meshfree max-ent approximants with large supports are competitive when compared to the proposed second order approach in all the tested examples, even in the higher order PDEs.

  Abstract
  We present a new approach for second order maximum entropy (max-ent) meshfree approximants that produces positive and smooth basis functions of uniform aspect ratio even for [...]

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• Nonlinear manifold learning for meshfree finite deformation thin shell analysis

  D. Millán, M. Arroyo, A. Rosolen

  Int. J. Numer. Meth. Engng. (2013). Vol. 93 (7), pp. 685-713 (Preprint)

  
  

  Abstract

  Calculations on general point-set surfaces are attractive because of their flexibility and simplicity in the preprocessing but present important challenges. The absence of a mesh makes it nontrivial to decide if two neighboring points in the three-dimensional embedding are nearby or rather far apart on the manifold. Furthermore, the topology of surfaces is generally not that of an open two-dimensional set, ruling out global parametrizations. We propose a general and simple numerical method analogous to the mathematical theory of manifolds, in which the point-set surface is described by a set of overlapping charts forming a complete atlas. We proceed in four steps: (1) partitioning of the node set into subregions of trivial topology; (2) automatic detection of the geometric structure of the surface patches by nonlinear dimensionality reduction methods; (3) parametrization of the surface using smooth meshfree (here maximum-entropy ) approximants; and (4) gluing together the patch representations by means of a partition of unity. Each patch may be viewed as a meshfree macro-element. We exemplify the generality, flexibility, and accuracy of the proposed approach by numerically approximating the geometrically nonlinear Kirchhoff–Love theory of thin-shells. We analyze standard benchmark tests as well as point-set surfaces of complex geometry and topology.

  Abstract
  Calculations on general point-set surfaces are attractive because of their flexibility and simplicity in the preprocessing but present important challenges. The absence of [...]

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