How Cell Shape Affects the Stresses in a Cell Sheet

2000 ◽  
Author(s):  
G. W. Brodland ◽  
Jim H. Veldhuis ◽  
Daniel I-Li Chen
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Cell Shape ◽  
Cell Sheet ◽  
Element Formulation ◽  
Analytical Mechanics ◽  
Cell Sheets ◽  
Average Cell ◽  
Real Cell ◽  
A Cell

Abstract Computer simulations and analytical mechanics are used to investigate the mechanics of cell sheets. The simulations are based on a recent finite element formulation [1] in which each cell is modeled using multiple finite elements, and cells can rearrange. Sheet stresses calculated using an analytical expression based on average cell shape are found to agree well with those calculated in the finite element simulations. This is an important step towards the development of constitutive equations to describe real cell sheets.

Cell-Level Finite Element Studies of Viscous Cells in Planar Aggregates

2000 ◽  
Vol 122 (4) ◽  
pp. 394-401 ◽  
Author(s):  
Helen H. Chen ◽  
G. Wayne Brodland
Keyword(s):  
Experimental Data ◽  
Wound Healing ◽  
Finite Element ◽  
Cell Shape ◽  
Cell Sheet ◽  
Element Formulation ◽  
Cell Level ◽  
Cell Sheets ◽  
Real Cell ◽  
Sheet Stretching

A new cell-level finite element formulation is presented and used to investigate how epithelia and other planar collections of viscous cells might deform during events such as embryo morphogenesis and wound healing. Forces arising from cytoskeletal components, cytoplasm viscosity, and cell-cell adhesions are included. Individual cells are modeled using multiple finite elements, and cell rearrangements can occur. Simulations of cell-sheet stretching indicate that the initial stages of sheet stretching are characterized by changes in cell shape, while subsequent stages are governed by cell rearrangement. Inferences can be made from the simulations about the forces that act in real cell sheets when suitable experimental data are available. [S0148-0731(00)01404-7]

scholarly journals Polygonal finite elements for three-dimensional Voronoi-cell-based discretisations

2012 ◽  
Author(s):  
Kaliappan Jayabal ◽  
Andreas Menzel
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Three Dimensional ◽  
Specific Property ◽  
Voronoi Cell ◽  
Polycrystalline Materials ◽  
Element Formulation ◽  
Hybrid Finite Element ◽  
Grain Structures ◽  
Polygonal Elements

Hybrid finite element formulations in combination with Voronoi-cell-based discretisation methods can efficiently be used to model the behaviour of polycrystalline materials. Randomly generated three-dimensional Voronoi polygonal elements with varying numbers of surfaces and corners in general better approximate the geometry of polycrystalline microor rather grain-structures than the standard tetrahedral and hexahedral finite elements. In this work, the application of a polygonal finite element formulation to three-dimensional elastomechanical problems is elaborated with special emphasis on the numerical implementation of the method and the construction of the element stiffness matrix. A specific property of Voronoi-based discretisations in combination with a hybrid finite element approach is investigated. The applicability of the framework established is demonstrated by means of representative numerical examples.

scholarly journals International Workshop on Finite Elements for Microwave Engineering

2016 ◽  
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Dominant Role ◽  
Piecewise Linear ◽  
International Workshop ◽  
American Mathematical Society ◽  
Element Formulation ◽  
Microwave Engineering ◽  
History Of ◽  
Element Activity

When Courant prepared the text of his 1942 address to the American Mathematical Society for publication, he added a two-page Appendix to illustrate how the variational methods first described by Lord Rayleigh could be put to wider use in potential theory. Choosing piecewise-linear approximants on a set of triangles which he called elements, he dashed off a couple of two-dimensional examples and the finite element method was born. … Finite element activity in electrical engineering began in earnest about 1968-1969. A paper on waveguide analysis was published in Alta Frequenza in early 1969, giving the details of a finite element formulation of the classical hollow waveguide problem. It was followed by a rapid succession of papers on magnetic fields in saturable materials, dielectric loaded waveguides, and other well-known boundary value problems of electromagnetics. … In the decade of the eighties, finite element methods spread quickly. In several technical areas, they assumed a dominant role in field problems. P.P. Silvester, San Miniato (PI), Italy, 1992 Early in the nineties the International Workshop on Finite Elements for Microwave Engineering started. This volume contains the history of the Workshop and the Proceedings of the 13th edition, Florence (Italy), 2016 . The 14th Workshop will be in Cartagena (Colombia), 2018.

Modelling of rotors with three-dimensional solid finite elements

2001 ◽  
Vol 36 (4) ◽  
pp. 359-371 ◽  
Author(s):  
A Nandi ◽  
S Neogy
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Cross Sections ◽  
Three Dimensional ◽  
Element Model ◽  
Finite Element Formulation ◽  
Element Formulation ◽  
Two Dimensional ◽  
Inertia Effects ◽  
Solid Finite Elements

A shaft is modelled using three-dimensional solid finite elements. The shear-deformation and rotary inertia effects are automatically included through the three-dimensional elasticity formulation. The formulation allows warping of plane cross-sections and takes care of gyroscopic effect. Unlike a beam element model, the present model allows the actual rotor geometry to be modelled. Shafts with complicated geometry can be modelled provided that the shaft cross-section has two axes of symmetry with equal or unequal second moment of areas. The acceleration of a point on the shaft is determined in inertial and rotating frames. It is found that the finite element formulation becomes much simpler in a rotating frame of reference that rotates about the centre-line of the bearings with an angular velocity equal to the shafts spin speed. The finite element formulation in the above frame is ideally suited to non-circular shafts with solid or hollow, prismatic or tapered sections and continuous or abrupt change in cross-sections. The shaft and the disc can be modelled using the same types of element and this makes it possible to take into account the flexibility of the disc. The formulation also allows edge cracks to be modelled. A two-dimensional model of shaft disc systems executing synchronous whirl on isotropic bearings is presented. The application of the two-dimensional formulation is limited but it reduces the number of degrees of freedom. The three-dimensional solid and two-dimensional plane stress finite element models are extensively validated using standard available results.

Efficient Residual and Matrix-Free Jacobian Evaluation for Three-Dimensional Tri-Quadratic Hexahedral Finite Elements With Nearly-Incompressible Neo-Hookean Hyperelasticity Applied to Soft Materials on Unstructured Meshes in Parallel, With PETSc and libCEED

2020 ◽  
Author(s):  
Arash Mehraban ◽  
Jed Brown ◽  
Valeria Barra ◽  
Henry Tufo ◽  
Jeremy Thompson ◽  
...  
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Optimization Design ◽  
Solid Mechanics ◽  
Soft Materials ◽  
Biological Tissues ◽  
Higher Order ◽  
Mixed Formulation ◽  
Element Formulation ◽  
Matrix Free

Abstract Soft materials such as rubber, elastomers, and soft biological tissues mechanically deform at large strain isochorically for all time, or during their initial transient (when a pore fluid, typically incompressible such as water, does not have time to flow out of the deforming polymer or soft tissue porous skeleton). Simulating these large isochoric deformations computationally, such as with the Finite Element Method (FEM), requires higher order (typically quadratic) interpolation functions and/or enhancements through hybrid/mixed methods to maintain stability. Lower order (linear) finite elements with hybrid/mixed formulation may not perform stably for all mechanical loading scenarios involving large isochoric deformations, whereas quadratic finite elements with or without hybrid/mixed formulation typically perform stably, especially when large bending or folding deformations are being simulated. For topology-optimization design of soft robotics, for instance, the FEM solid mechanics solver must run efficiently and stably. Stability is ensured by the higher order finite element formulation (with possible enhancement), but efficiency for higher order FEM remains a challenge. Thus, this paper addresses efficiency from the perspective of computer science algorithms and programming. The proposed efficient algorithm utilizes the Portable, Extensible Toolkit for Scientific Computation (PETSc), along with the libCEED library for efficient compiler optimized tensor-product-basis computation to demonstrate an efficient nonlinear solution algorithm. For preconditioning, a scalable p-multigrid method is presented whereby a hierarchy of levels is constructed. In contrast to classical geometric multigrid, also known as h-multigrid, each level in p-multigrid is related to a different approximation polynomial order, p, instead of the element size, h. A Chebyshev polynomial smoother is used on each multigrid level. Algebraic MultiGrid (AMG) is then applied to the assembled Q1 (linear) coarse mesh on the nodes of the quadratic Q2 (quadratic) mesh. This allows low storage that can be efficiently used to accelerate the convergence to solution. For a Neo-Hookean hyperelastic problem, we examine a residual and matrix-free Jacobian formulation of a tri-quadratic hexahedral finite element with enhancement. Efficiency estimates on AVX-2 architecture based on CPU time are provided as a comparison to similar simulation (and mesh) of isochoric large deformation hyperelasticity as applied to soft materials conducted with the commercially-available FEM software program ABAQUS. The particular problem in consideration is the simulation of an assistive device in the form of finger-bending in 3D.

Stability of Non-Axisymmetric Rotor and Bearing Systems Modeled With Three-Dimensional-Solid Finite Elements

2019 ◽  
Vol 142 (1) ◽  
Author(s):  
Joseph Oh ◽  
Alan Palazzolo ◽  
Lingnan Hu
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Computational Efficiency ◽  
Degrees Of Freedom ◽  
Monodromy Matrix ◽  
Three Dimensional ◽  
Element Formulation ◽  
System Matrix ◽  
Rotor Bearing ◽  
Solid Finite Elements

Abstract Although rotors are simplified to be axisymmetric in rotordynamic models, many rotors in the industry are actually non-axisymmetric. Several authors have proposed methods using 3D finite element, rotordynamic models, but more efficient approaches for handling a large number of degrees-of-freedom (DOF) are needed. This task becomes particularly acute when considering parametric excitation that results from asymmetry in the rotating frame. This paper presents an efficient rotordynamic stability approach for non-axisymmetric rotor-bearing systems with complex shapes using three-dimensional solid finite elements. The 10-node quadratic tetrahedron element is used for the finite element formulation of the rotor. A rotor-bearing system, matrix differential equation is derived in the rotor-fixed coordinate system. The system matrices are reduced by using Guyan reduction. The current study utilizes the Floquet theory to determine the stability of solutions for parametrically excited rotor-bearing systems. Computational efficiency is improved by discretization and parallelization, taking advantage of the discretized monodromy matrix of Hsu's method. The method is verified by an analytical model with the Routh–Hurwitz stability criteria, and by direct time-transient, numerical integration for large order models. The proposed and Hill's methods are compared with respect to accuracy and computational efficiency, and the results indicate the limitations of Hill's method when applied to 3D solid rotor-bearing systems. A parametric investigation is performed for an asymmetric Root's blower type shaft, varying bearing asymmetry and bearing damping.

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