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.

Topology Optimization Using Node-Based Smoothed Finite Element Method

2015 ◽  
Vol 07 (06) ◽  
pp. 1550085 ◽  
Author(s):  
Z. C. He ◽  
G. Y. Zhang ◽  
L. Deng ◽  
Eric Li ◽  
G. R. Liu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Topology Optimization ◽  
Optimization Design ◽  
Solid Mechanics ◽  
Optimality Criteria ◽  
Topology Optimization Problem ◽  
Design Variables ◽  
Smoothed Finite Element Method ◽  
Element Method

The node-based smoothed finite element method (NS-FEM) proposed recently has shown very good properties in solid mechanics, such as providing much better gradient solutions. In this paper, the topology optimization design of the continuum structures under static load is formulated on the basis of NS-FEM. As the node-based smoothing domain is the sub-unit of assembling stiffness matrix in the NS-FEM, the relative density of node-based smoothing domains serves as design variables. In this formulation, the compliance minimization is considered as an objective function, and the topology optimization model is developed using the solid isotropic material with penalization (SIMP) interpolation scheme. The topology optimization problem is then solved by the optimality criteria (OC) method. Finally, the feasibility and efficiency of the proposed method are illustrated with both 2D and 3D examples that are widely used in the topology optimization design.

Eulerian FEA With Volume of Solid (VOS) Application in Metal Forming

2010 ◽  
Author(s):  
K. S. Al-Athel ◽  
M. S. Gadala
Keyword(s):  
Finite Element ◽  
Free Surface ◽  
Process Improvement ◽  
Metal Forming ◽  
Material Flow ◽  
Solid Mechanics ◽  
Uniform Mesh ◽  
Element Formulation ◽  
Volume Of Fluid Method ◽  
Whole Process

The adaptation of the volume of fluid method (VOF) to solid mechanics (VOS) is presented in this work with the focus on metal forming applications. The method is discussed for a general non-uniform mesh with Eulerian finite element formulation. The implementation of the VOS method in metal forming applications is presented by focusing on topics such as the contact between the tool and the workpiece, tracking of the free surface of the material flow and the connectivity of the free surface during the whole process. Improvement on the connectivity of the free surface and the representation of curves is achieved by considering the mechanics of different metal forming processes. Different applications are simulated and discussed to highlight the capability of the VOS method.

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.

Modeling Beam–Solid Finite Element Interfaces: A Stabilized Formulation for Contact and Coupled Systems

2018 ◽  
Vol 10 (09) ◽  
pp. 1850094 ◽  
Author(s):  
Jorge A. Montero ◽  
Ghadir Haikal
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Computational Models ◽  
Solid Mechanics ◽  
Pressure Field ◽  
Coupled Systems ◽  
Weak Continuity ◽  
Finite Element Discretizations ◽  
Geometric Compatibility ◽  
Element Discretizations

A number of engineering applications involve contact with bodies modeled using specialized theories of solid mechanics like beams or shells. While computational models for contact in 2D and 3D solid mechanics have been extensively developed in the literature, problems involving contact with beams or shells have received less attention. When modeling contact between a solid body represented with beam or shell theory and a domain discretized with solid finite elements, the contact model faces the typical challenges of enforcing geometric compatibility and the transfer of a complete pressure field along the contact interface, with the added complications stemming from the different underlying mathematical formulations and finite element discretizations in the connecting domains. Resultant-based beam and shell theories do not provide direct estimates of surface tractions, therefore rendering the issue of pressure transfer on beam–solid and shell–solid interfaces more problematic. In the absence of specialized contact formulations for solid–beam and solid–shell interfaces, contact models have relied almost exclusively on the Node-To-Surface (NTS) geometric compatibility approach. This formulation suffers from well-known drawbacks, including instability, surface locking and incomplete pressure fields on the interface. The NTS approach, however, remains the method most readily applicable to contact with beam or shell elements among the vast variety of available methods for computational contact modeling using finite elements. The goal of this paper is to bridge the gap in the literature on coupling domains with beam and solid finite element discretizations. We propose an interface formulation for beam–solid interfaces that ensures the transfer of a complete pressure field while enforcing geometric compatibility using standard NTS constraints. The formulation uses a stabilization approach, based on a special form of the Discontinuous Galerkin method, to enforce weak continuity between the stress fields on the solid side of the interface, and the moment and shear resultants in the contacting beam. We show that the proposed formulation is a robust approach for satisfying compatibility constraints while ensuring the transfer of a complete pressure field on beam–solid finite element interfaces that can be used with bilinear and quadratic interpolations in the solid, and Euler or Timoshenko formulations for the beam.

scholarly journals Finite Element Implementation of Mechanochemical Phenomena in Neutral Deformable Porous Media Under Finite Deformation

2011 ◽  
Vol 133 (8) ◽  
Author(s):  
Gerard A. Ateshian ◽  
Michael B. Albro ◽  
Steve Maas ◽  
Jeffrey A. Weiss
Keyword(s):  
Finite Element ◽  
Porous Media ◽  
Finite Deformation ◽  
Biological Tissues ◽  
Solid Matrix ◽  
Element Formulation ◽  
Momentum Exchange ◽  
Deformable Porous Media ◽  
Chemical Effects ◽  
Highly Nonlinear

Biological soft tissues and cells may be subjected to mechanical as well as chemical (osmotic) loading under their natural physiological environment or various experimental conditions. The interaction of mechanical and chemical effects may be very significant under some of these conditions, yet the highly nonlinear nature of the set of governing equations describing these mechanisms poses a challenge for the modeling of such phenomena. This study formulated and implemented a finite element algorithm for analyzing mechanochemical events in neutral deformable porous media under finite deformation. The algorithm employed the framework of mixture theory to model the porous permeable solid matrix and interstitial fluid, where the fluid consists of a mixture of solvent and solute. A special emphasis was placed on solute-solid matrix interactions, such as solute exclusion from a fraction of the matrix pore space (solubility) and frictional momentum exchange that produces solute hindrance and pumping under certain dynamic loading conditions. The finite element formulation implemented full coupling of mechanical and chemical effects, providing a framework where material properties and response functions may depend on solid matrix strain as well as solute concentration. The implementation was validated using selected canonical problems for which analytical or alternative numerical solutions exist. This finite element code includes a number of unique features that enhance the modeling of mechanochemical phenomena in biological tissues. The code is available in the public domain, open source finite element program FEBio (http://mrl.sci.utah.edu/software).

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