Uniqueness of the Geometric Representation in Large Rotation Finite Element Formulations

2010 ◽  
Vol 5 (4) ◽  
Author(s):  
Ahmed A. Shabana
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Computer Aided Design ◽  
Curvature Vector ◽  
Space Curve ◽  
Body Motion ◽  
Large Rotation ◽  
Space Curves ◽  
Curvature Continuity ◽  
Finite Element Formulations

Several finite element formulations used in the analysis of large rotation and large deformation problems employ independent interpolations for the displacement and rotation fields. As explained in this paper, three rotations defined as field variables can be sufficient to define a space curve that represents the element centerline. The frame defined by the rotations can differ from the Frenet frame of the space curve defined by the same rotation field and, therefore, such a rotation-based representation can provide measure of twist shear deformations and captures the rotation of the beam about its axis. However, the space curve defined using the rotation interpolation has a geometry that can significantly differ from the geometry defined by an independent displacement interpolation. Furthermore, the two different space curves defined by the two different interpolations can differ by a rigid body motion. Therefore, in these formulations, the uniqueness of the kinematic representation is an issue unless nonlinear algebraic constraint equations are used to establish relationships between the two independent displacement and rotation interpolations. Nonetheless, significant geometric and kinematic differences between two independent space curves cannot always be reduced by using restoring elastic forces. Because of the nonuniqueness of such a finite element representation, imposing continuity on higher derivatives such as the curvature vector is not straight forward as in the case of the absolute nodal coordinate formulation (ANCF) that defines unique displacement and rotation fields. ANCF finite elements allow for imposing curvature continuity without increasing the order of the interpolation or the number of nodal coordinates, as demonstrated in this paper. Furthermore, the relationship between ANCF finite elements and the B-spline representation used in computational geometry can be established, allowing for a straight forward integration of computer aided design and analysis.

An Element Independent Corotational Procedure for the Treatment of Large Rotations

1986 ◽  
Vol 108 (2) ◽  
pp. 165-174 ◽  
Author(s):  
C. C. Rankin ◽  
F. A. Brogan
Keyword(s):  
Finite Element ◽  
Rigid Body ◽  
Displacement Field ◽  
Rigid Body Motion ◽  
Rotation Vector ◽  
Body Motion ◽  
Large Rotation ◽  
Total Displacement ◽  
Large Rotations ◽  
Finite Element Formulations

A new corotational procedure is developed which enables existing finite element formulations to be used in problems that contain arbitrarily large rotations. Through the use of a nonsingular large rotation vector, the contribution of the rigid body motion of the element to the total displacement field is removed before element computations are performed, with the result that almost any element can be easily upgraded to handle large rotations. This paper contains a derivation of the theory, an outline of the implementation into the STAGS code, and a demonstration of performance for problems involving large rotations and moderate strains.

Performance of the Incremental and Non-Incremental Finite Element Formulations in Flexible Multibody Problems

1999 ◽  
Vol 122 (4) ◽  
pp. 498-507 ◽  
Author(s):  
Marcello Campanelli ◽  
Marcello Berzeri ◽  
Ahmed A. Shabana
Keyword(s):  
Finite Element ◽  
Multibody Systems ◽  
Absolute Nodal Coordinate Formulation ◽  
Flexible Multibody Systems ◽  
Large Displacement ◽  
Body Motion ◽  
Absolute Nodal Coordinate ◽  
Flexible Multibody ◽  
The Absolute ◽  
Finite Element Formulations

Many flexible multibody applications are characterized by high inertia forces and motion discontinuities. Because of these characteristics, problems can be encountered when large displacement finite element formulations are used in the simulation of flexible multibody systems. In this investigation, the performance of two different large displacement finite element formulations in the analysis of flexible multibody systems is investigated. These are the incremental corotational procedure proposed in an earlier article (Rankin, C. C., and Brogan, F. A., 1986, ASME J. Pressure Vessel Technol., 108, pp. 165–174) and the non-incremental absolute nodal coordinate formulation recently proposed (Shabana, A. A., 1998, Dynamics of Multibody Systems, 2nd ed., Cambridge University Press, Cambridge). It is demonstrated in this investigation that the limitation resulting from the use of the infinitesmal nodal rotations in the incremental corotational procedure can lead to simulation problems even when simple flexible multibody applications are considered. The absolute nodal coordinate formulation, on the other hand, does not employ infinitesimal or finite rotation coordinates and leads to a constant mass matrix. Despite the fact that the absolute nodal coordinate formulation leads to a non-linear expression for the elastic forces, the results presented in this study, surprisingly, demonstrate that such a formulation is efficient in static problems as compared to the incremental corotational procedure. The excellent performance of the absolute nodal coordinate formulation in static and dynamic problems can be attributed to the fact that such a formulation does not employ rotations and leads to exact representation of the rigid body motion of the finite element. [S1050-0472(00)00604-8]

Substructure analysis of complex systems using rigid body information of components

2002 ◽  
Vol 216 (10) ◽  
pp. 811-817 ◽  
Author(s):  
W S Hwang ◽  
D H Lee
Keyword(s):  
Finite Element ◽  
Complex Systems ◽  
Rigid Body ◽  
Computer Aided Design ◽  
Element Model ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Element Analysis ◽  
Design Data ◽  
Substructure Analysis

Frequency response function (FRF) based substructure analysis can predict the response of complex systems using the FRFs of substructures. It combines the FRFs of each substructure derived from finite element analysis or experiments depending on the situation. In general, the substructure with the excitation is separated from the others by rubber bushes to prevent the transmission of vibration from the source to the main structure. In this case, the substructure with the excitation shows rigid body motion up to the mid-frequency region. This paper presents a new FRF-based substructure analysis that uses the FRFs from the rigid body information not from the complex finite element model of the substructure with rigid body motion. The rigid body information including the mass, the moment of inertia and the coordinates of the mass centre comes from the computer-aided design data. Since the mechanism of this technique is very similar to the finite element formation, it can be applied to complex systems with ease. Through a simple example of a ladder structure and a practical example of the interior noise in a car, the accuracy and efficiency of this approach is proven.

Modal-Based Finite Elements for Efficient Wave Propagation Analysis

2013 ◽  
Author(s):  
German Capuano ◽  
Massimo Ruzzene ◽  
Julian J. Rimoli
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Wave Propagation ◽  
Finite Elements ◽  
Benchmark Problems ◽  
Fine Scale ◽  
Multi Scale ◽  
Natural Modes ◽  
Propagation Analysis ◽  
Finite Element Formulations

This paper presents an extension to the Geometric Multi-Scale Finite Element Method (GMsFEM) developed by Casadei et al. to predict the dynamic response of heterogeneous materials and structures. The proposed approach introduces elements enriched by the natural modes over their own domain. When heterogeneities are present, the auxiliary fine-scale mesh from GMsFEM is used to calculate the modes numerically. The enrichment scheme is also chosen in such a way that it automatically satisfies continuity across boundaries. The computational efficiency of the method is compared to that of traditional finite element formulations through selected benchmark problems.

Metal Forming and the Finite-Element Method

1989 ◽  
Author(s):  
Shiro Kobayashi ◽  
Soo-Ik Oh ◽  
Taylan Altan
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Metal Forming ◽  
Computer Aided Design ◽  
The Finite Element Method ◽  
Forming Technology ◽  
Computer Aided ◽  
Deformation Mechanics ◽  
Aided Design ◽  
Element Method

The application of computer-aided design and manufacturing techniques is becoming essential in modern metal-forming technology. Thus process modeling for the determination of deformation mechanics has been a major concern in research . In light of these developments, the finite element method--a technique by which an object is decomposed into pieces and treated as isolated, interacting sections--has steadily assumed increased importance. This volume addresses advances in modern metal-forming technology, computer-aided design and engineering, and the finite element method.

scholarly journals A New Mixed Functional-probabilistic Approach for Finite Element Accuracy

2020 ◽  
Vol 20 (4) ◽  
pp. 799-813
Author(s):  
Joël Chaskalovic ◽  
Franck Assous
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Asymptotic Relation ◽  
Probabilistic Approach ◽  
Random Variable ◽  
High Order ◽  
Relative Accuracy ◽  
The Difference ◽  
New Perspective

AbstractThe aim of this paper is to provide a new perspective on finite element accuracy. Starting from a geometrical reading of the Bramble–Hilbert lemma, we recall the two probabilistic laws we got in previous works that estimate the relative accuracy, considered as a random variable, between two finite elements {P_{k}} and {P_{m}} ({k<m}). Then we analyze the asymptotic relation between these two probabilistic laws when the difference {m-k} goes to infinity. New insights which qualify the relative accuracy in the case of high order finite elements are also obtained.

scholarly journals Development of a numerical compensation framework for geometrical deviations in bulk metal forming exploiting a surrogate model and computed compatible stresses

2021 ◽  
Author(s):  
Lorenzo Scandola ◽  
Christoph Büdenbender ◽  
Michael Till ◽  
Daniel Maier ◽  
Michael Ott ◽  
...  
Keyword(s):  
Finite Element ◽  
Surrogate Model ◽  
Metal Forming ◽  
Computer Aided Design ◽  
Transition Process ◽  
Reference Points ◽  
Compensation Method ◽  
Tool Geometry ◽  
Bulk Metal ◽  
Bulk Metal Forming

AbstractThe optimal design of the tools in bulk metal forming is a crucial task in the early design phase and greatly affects the final accuracy of the parts. The process of tool geometry assessment is resource- and time-consuming, as it consists of experience-based procedures. In this paper, a compensation method is developed with the aim to reduce geometrical deviations in hot forged parts. In order to simplify the transition process between the discrete finite-element (FE) mesh and the computer-aided-design (CAD) geometry, a strategy featuring an equivalent surrogate model is proposed. The deviations are evaluated on a reduced set of reference points on the nominal geometry and transferred to the FE nodes. The compensation approach represents a modification of the displacement-compatible spring-forward method (DC-SF), which consists of two elastic FE analyses. The compatible stress originating the deviations is estimated and subsequently applied to the original nominal geometry. After stress relaxation, an updated nominal geometry of the part is obtained, whose surfaces represent the compensated tools. The compensation method is verified by means of finite element simulations and the robustness of the algorithm is demonstrated with an additional test geometry. Finally, the compensation strategy is validated experimentally.

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