Development of a Nonlinear, C1-Beam Finite-Element Code for Actuator Design of Slender Flexible Robots

2017 ◽  
Author(s):  
Hidenori Murakami ◽  
Oscar Rios ◽  
Takeyuki Ono
Keyword(s):  
Finite Element ◽  
Large Deformation ◽  
Cross Section ◽  
Beam Model ◽  
Finite Element Code ◽  
Shape Functions ◽  
Lagrange Equations ◽  
Element Discretization ◽  
Flexible Robots ◽  
Actuator Design

For actuator design and motion simulations of slender flexible robots, planar C1-beam elements are developed for Reissner’s large deformation, shear-deformable, curbed-beam model. Internal actuation is mechanically modeled by a rate-form of beam constitutive relation, where actuation curvature is prescribed at each time. Geometrically, a curbed beam is modeled as a frame bundle, whereby at each point on beam’s curve of centroids a moving orthonormal frame is attached to a cross section. After a finite element discretization, a curve of centroids is modeled as a C1-curve, employing cubic shape functions for both planar coordinates with an arc-parameter. The cubic shape functions have already been utilized in linear Euler-Bernoulli beams for the interpolation of transverse displacement. To define the rotation angle of each cross section or the attitude of the moving frame, quadratic shape functions are used introducing a middle node, resulting in three angular nodal displacements. As a result, each beam element has total eleven nodal coordinates. The implementation of a nonlinear finite element code is facilitated by the principle of virtual work, which yields Reissner’s large deformation curbed beam model as the Euler-Lagrange equations. For time integration, the Newmark method is utilized. Finally, as applications of the code, a few inchworm motions induced by different actuation curvature fields are presented.

scholarly journals A PETSc-Based Parallel Implementation of Finite Element Method for Elasticity Problems

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Jianfei Zhang
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Parallel Implementation ◽  
Good Choice ◽  
Finite Element Code ◽  
Finite Element Discretization ◽  
Element Analysis ◽  
Element Discretization ◽  
Elasticity Problems ◽  
Parallel Code

Starting a parallel code from scratch is not a good choice for parallel programming finite element analysis of elasticity problems because we cannot make full use of our existing serial code and the programming work is painful for developers. PETSc provides libraries for various numerical methods that can give us more flexibility in migrating our serial application code to a parallel implementation. We present the approach to parallelize the existing finite element code within the PETSc framework. Our approach permits users to easily implement the formation and solution of linear system arising from finite element discretization of elasticity problem. The main PETSc subroutines are given for the main parallelization step and the corresponding code fragments are listed. Cantilever examples are used to validate the code and test the performance.

Study on the collapse of tapered tubes subjected to oblique loads

2008 ◽  
Vol 222 (11) ◽  
pp. 2025-2039 ◽  
Author(s):  
P Hosseini-Tehrani ◽  
S Pirmohammad ◽  
M Golmohammadi
Keyword(s):  
Finite Element ◽  
Cross Section ◽  
Numerical Models ◽  
Point Of View ◽  
Finite Element Code ◽  
Energy Absorber ◽  
Explicit Finite Element ◽  
Bending Deformation ◽  
Oblique Loading

In this work, several antisymmetric tapered tubes with an inner stiffener under axial and oblique loading are studied and optimum dimensions of the tapered tube are derived from a crashworthiness point of view. The importance of detecting these dimensions is optimizing the weight while the crashworthiness of tube is not damaged. By using an internal stiffener, crashworthiness is improved against oblique loads, and the sensitivity of tubes with respect to oblique loads and bending deformation is diminished. The numerical models have been developed using the explicit finite element code LS-DYNA. The crashworthiness of the optimized tapered tube is compared with that of an octagonal-cross-section tube which is known as the best energy absorber model in the literature. It is shown that an optimized tapered tube has an average of 29.3 per cent less crushing displacement in comparison with octagonal-section tube when both tubes have the same weights and the same peaks of crushing load. Finally, the orientation of loading is changed and the best orientation is proposed.

Computational Wear and Contact Modeling for Fretting Analysis with Isogeometric Dual Mortar Methods

2016 ◽  
Vol 681 ◽  
pp. 1-18 ◽  
Author(s):  
Philipp Farah ◽  
Markus Gitterle ◽  
Wolfgang A. Wall ◽  
Alexander Popp
Keyword(s):  
Finite Element ◽  
Three Dimensions ◽  
Fretting Wear ◽  
Additional Contribution ◽  
Shape Functions ◽  
Element Discretization ◽  
B Splines ◽  
Mortar Methods ◽  
Deformation Regime ◽  
Incremental Scheme

A finite element framework based on dual mortar methods is presented for simulating fretting wear effects in the finite deformation regime. The mortar finite element discretization is realized with Lagrangean shape functions as well as isogeometric elements based on non-uniform rational B-splines (NURBS) in two and three dimensions. Fretting wear effects are modeled in an incremental scheme with the help of Archard’s law and the worn material is considered as additional contribution to the gap function. Numerical examples demonstrate the robustness and accuracy of the presented algorithm.

scholarly journals Consistent coupling of positions and rotations for embedding 1D Cosserat beams into 3D solid volumes

2021 ◽  
Author(s):  
Ivo Steinbrecher ◽  
Alexander Popp ◽  
Christoph Meier
Keyword(s):  
Finite Element ◽  
Cross Section ◽  
Lagrange Multiplier ◽  
Degrees Of Freedom ◽  
Element Discretization ◽  
Geometrically Exact Beam ◽  
Coupling Constraints ◽  
Rotational Degrees Of Freedom ◽  
3D Solid ◽  
Cosserat Continua

AbstractThe present article proposes a mortar-type finite element formulation for consistently embedding curved, slender beams into 3D solid volumes. Following the fundamental kinematic assumption of undeformable cross-section s, the beams are identified as 1D Cosserat continua with pointwise six (translational and rotational) degrees of freedom describing the cross-section (centroid) position and orientation. A consistent 1D-3D coupling scheme for this problem type is proposed, requiring to enforce both positional and rotational constraints. Since Boltzmann continua exhibit no inherent rotational degrees of freedom, suitable definitions of orthonormal triads are investigated that are representative for the orientation of material directions within the 3D solid. While the rotation tensor defined by the polar decomposition of the deformation gradient appears as a natural choice and will even be demonstrated to represent these material directions in a $$L_2$$ L 2 -optimal manner, several alternative triad definitions are investigated. Such alternatives potentially allow for a more efficient numerical evaluation. Moreover, objective (i.e. frame-invariant) rotational coupling constraints between beam and solid orientations are formulated and enforced in a variationally consistent manner based on either a penalty potential or a Lagrange multiplier potential. Eventually, finite element discretization of the solid domain, the embedded beams, which are modeled on basis of the geometrically exact beam theory, and the Lagrange multiplier field associated with the coupling constraints results in an embedded mortar-type formulation for rotational and translational constraint enforcement denoted as full beam-to-solid volume coupling (BTS-FULL) scheme. Based on elementary numerical test cases, it is demonstrated that a consistent spatial convergence behavior can be achieved and potential locking effects can be avoided, if the proposed BTS-FULL scheme is combined with a suitable solid triad definition. Eventually, real-life engineering applications are considered to illustrate the importance of consistently coupling both translational and rotational degrees of freedom as well as the upscaling potential of the proposed formulation. This allows the investigation of complex mechanical systems such as fiber-reinforced composite materials, containing a large number of curved, slender fibers with arbitrary orientation embedded in a matrix material.

General Beam Cross-Section Analysis Using a 3D Finite Element Slice

2014 ◽  
Author(s):  
Philippe Couturier ◽  
Steen Krenk
Keyword(s):  
Finite Element ◽  
Cross Section ◽  
Cross Sections ◽  
Degrees Of Freedom ◽  
Finite Thickness ◽  
Single Layer ◽  
Element Discretization ◽  
Cubic Polynomial ◽  
Hermitian Interpolation ◽  
Section Analysis

A formulation for analysis of general cross-section properties has been developed. This formulation is based on the stress-strain states in the classic six equilibrium modes of a beam by considering a finite thickness slice modelled by a single layer of 3D finite elements. The displacement variation in the lengthwise direction is in the form of a cubic polynomial, which is here represented by Hermitian interpolation, whereby the degrees of freedom are concentrated on the front and back faces of the slice. The theory is illustrated by application to a simple cross-section for which an analytical solution is available. The paper also shows an application to wind turbine blade cross-sections and discusses the effect of the finite element discretization on the cross-section properties such as stiffness parameters and the location of the elastic and shear centers.

scholarly journals Finite Element Analyses of the Modified Strain Gradient Theory Based Kirchhoff Microplates

Surfaces ◽  
2021 ◽  
Vol 4 (2) ◽  
pp. 115-157
Author(s):  
Murat Kandaz ◽  
Hüsnü Dal
Keyword(s):  
Finite Element ◽  
Variational Problem ◽  
Strain Gradient ◽  
Degrees Of Freedom ◽  
Weak Form ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Shape Functions ◽  
Lagrange Equations ◽  
Modified Strain Gradient Theory

In this contribution, the variational problem for the Kirchhoff plate based on the modified strain gradient theory (MSGT) is derived, and the Euler-Lagrange equations governing the equation of motion are obtained. The Galerkin-type weak form, upon which the finite element method is constructed, is derived from the variational problem. The shape functions which satisfy the governing homogeneous partial differential equation are derived as extensions of Adini-Clough-Melosh (ACM) and Bogner-Fox-Schmit (BFS) plate element formulations by introducing additional curvature degrees of freedom (DOF) on each node. Based on the proposed set of shape functions, 20-, 24-, 28- and 32- DOF modified strain gradient theory-based higher-order Kirchhoff microplate element are proposed. The performance of the elements are demonstrated in terms of various tests and representative boundary value problems. Length scale parameters for gold are also proposed based on experiments reported in literature.

Coupled Deformation Modes in the Large Deformation Finite-Element Analysis: Problem Definition

2006 ◽  
Vol 2 (2) ◽  
pp. 146-154 ◽  
Author(s):  
Bassam A. Hussein ◽  
Hiroyuki Sugiyama ◽  
Ahmed A. Shabana
Keyword(s):  
Large Deformation ◽  
Cross Section ◽  
Flexible Structures ◽  
Beam Theory ◽  
Problem Definition ◽  
Beam Model ◽  
Element Analysis ◽  
Elastic Line ◽  
The Cross ◽  
Deformation Modes

In the classical formulations of beam problems, the beam cross section is assumed to remain rigid when the beam deforms. In Euler–Bernoulli beam theory, the rigid cross section remains perpendicular to the beam centerline; while in the more general Timoshenko beam theory the rigid cross section is permitted to rotate due to the shear deformation, and as a result, the cross section can have an arbitrary rotation with respect to the beam centerline. In more general beam models as the ones based on the absolute nodal coordinate formulation (ANCF), the cross section is allowed to deform and it is no longer treated as a rigid surface. These more general models lead to new geometric terms that do not appear in the classical formulations of beams. Some of these geometric terms are the result of the coupling between the deformation of the cross section and other modes of deformations such as bending and they lead to a new set of modes referred to in this paper as the ANCF-coupled deformation modes. The effect of the ANCF-coupled deformation modes can be significant in the case of very flexible structures. In this investigation, three different large deformation dynamic beam models are discussed and compared in order to investigate the effect of the ANCF-coupled deformation modes. The three methods differ in the way the beam elastic forces are calculated. The first method is based on a general continuum mechanics approach that leads to a model that includes the ANCF-coupled deformation modes; while the second method is based on the elastic line approach that systematically eliminates these modes. The ANCF-coupled deformation modes eliminated in the elastic line approach are identified and the effect of such deformation modes on the efficiency and accuracy of the numerical solution is discussed. The third large deformation beam model discussed in this investigation is based on the Hellinger–Reissner principle that can be used to eliminate the shear locking encountered in some beam models. Numerical examples are presented in order to demonstrate the use and compare the results of the three different beam formulations. It is shown that while the effect of the ANCF-coupled deformation modes is not significant in very stiff and moderately stiff structures, the effect of these modes can not be neglected in the case of very flexible structures.

scholarly journals Coupled Bending-Bending-Torsion Vibration of a Rotating Pre-Twisted Beam with Aerofoil Cross-Section and Flexible Root by Finite Element Method

2004 ◽  
Vol 11 (5-6) ◽  
pp. 637-646 ◽  
Author(s):  
Bulent Yardimoglu ◽  
Daniel J. Inman
Keyword(s):  
Finite Element ◽  
Finite Element Model ◽  
Cross Section ◽  
Centrifugal Force ◽  
Axial Stress ◽  
Element Model ◽  
Torsional Stiffness ◽  
Extended Model ◽  
Beam Model ◽  
Cross Sectional

The purpose of this paper is to extend a previously published beam model of a turbine blade including the centrifugal force field and root flexibility effects on a finite element model and to demonstrate the performance, accuracy and efficiency of the extended model for computing the natural frequencies. Therefore, only the modifications due to rotation and elastic root are presented in great detail. Considering the shear center effect on the transverse displacements, the geometric stiffness matrix due to the centrifugal force is developed from the geometric strain energy expression based on the large deflections and the increase of torsional stiffness because of the axial stress. In this work, the root flexibility of the blade is idealized by a continuum model unlike the discrete model approach of a combination of translational and rotational elastic springs, as used by other researchers. The cross-section properties of the fir-tree root of the blade considered as an example are expressed by assigning proper order polynomial functions similar to cross-sectional properties of a tapered blade. The correctness of the present extended finite element model is confirmed by the experimental and calculated results available in the literature. Comparisons of the present model results with those in the literature indicate excellent agreement.

Finite Element Discretization of Piezothermoelastic Equations Using the Generalized Equation of Heat Conduction

2005 ◽  
Author(s):  
Zsolt Vi´zva´ry
Keyword(s):  
Finite Element ◽  
Heat Conduction ◽  
Generalized Equation ◽  
Finite Element Code ◽  
Spatial Discretization ◽  
Finite Element Discretization ◽  
Fourier’S Law ◽  
Galerkin’S Method ◽  
Element Discretization ◽  
Equation Of Heat Conduction

In this paper, the derivation of the finite element equations of piezothermoelasticity is presented using the classical and generalized Fourier’s law. Galerkin’s method is used to the spatial discretization of the equations. Based upon these equations, a finite element code is under development.

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