Timoshenko beam finite elements using trigonometric basis functions

AIAA Journal ◽  
1988 ◽  
Vol 26 (11) ◽  
pp. 1378-1386 ◽  
Author(s):  
G. R. Heppler ◽  
J. S. Hansen
Keyword(s):  
Finite Elements ◽  
Timoshenko Beam ◽  
Basis Functions ◽  
Trigonometric Basis Functions

scholarly journals Polyhedral Finite Elements Using Harmonic Basis Functions

2008 ◽  
Vol 27 (5) ◽  
pp. 1521-1529 ◽  
Author(s):  
Sebastian Martin ◽  
Peter Kaufmann ◽  
Mario Botsch ◽  
Martin Wicke ◽  
Markus Gross
Keyword(s):  
Finite Elements ◽  
Basis Functions

Fundamental Understanding of Wave Propagation in a Beam Using PZT Actuators and Sensors

2013 ◽  
Author(s):  
Vishnu Prasad Venugopal ◽  
Gang Wang
Keyword(s):  
Finite Element ◽  
Wave Propagation ◽  
Finite Elements ◽  
Timoshenko Beam ◽  
High Frequency ◽  
Beam Structure ◽  
Frequency Wave ◽  
Coupled Equations ◽  
High Frequency Wave ◽  
Spectral Finite Element

Embedded smart actuators/sensors, such as piezoelectric types, have been used to conduct wave transmission and reception, pulse-echo, pitch-catch, and phased array functions in order to achieve in-situ nondestructive evaluation for different structures. By comparing to baseline signatures, the damage location, amount, and type can be determined. Typically, this methodology does not require analytical structural models and interrogation algorithm is carefully designed with little wave propagation knowledge of the structure. However, the wave excitation frequency, waveform, and other signal characteristics must be comprehensively considered to effectively conduct diagnosis of incipient forms of damage. Accurate prediction of high frequency wave response requires a prohibitively large number of conventional finite elements in the structural model. A new high fidelity approach is needed to capture high frequency wave propagations in a structure. In this paper, a spectral finite element method (SFEM) is proposed to characterize wave propagations in a beam structure under piezoelectric material (i.e., PZT) actuation/sensing. Mathematical models are developed to account for both Uni-morph and bi-morph configurations, in which PZT layers are modeled as either an actuator or a sensor. The Timoshenko beam theory is adopted to accommodate high frequency wave propagations, i.e., 20–200 KHz. The PZT layer is modeled as a Timoshenko beam as well. Corresponding displacement compatibility conditions are applied at interfaces. Finally, a set of fully coupled governing equations and associated boundary conditions are obtained when applying the Hamilton’s principle. These electro-mechanical coupled equations are solved in the frequency domain. Then, analytical solutions are used to formulate the spectral finite element model. Very few spectral finite elements are required to accurately capture the wave propagation in the beam because the shape functions are duplicated from exact solutions. Both symmetric and antisymmetric mode of lamb waves can be generated using bimorph or uni-morph actuation. Comprehensive simulations are conducted to determine the beam wave propagation responses. It is shown that the PZT sensor can pick up the reflected waves from beam boundaries and damages. Parametric studies are conducted as well to determine the optimal actuation frequency and sensor sensitivity. Such information helps us to fundamentally understand wave propagations in a beam structure under PZT actuation and sensing. Our SFEM predictions are validated by the results in the literature.

Layer-adapted methods for a singularly perturbed singular problem

2011 ◽  
Vol 11 (2) ◽  
pp. 192-205
Author(s):  
Christian Grossmann ◽  
Lars Ludwig ◽  
Hans-Görg Roos
Keyword(s):  
Boundary Layer ◽  
Finite Elements ◽  
Boundary Value Problem ◽  
Bessel Functions ◽  
Basis Functions ◽  
Singularly Perturbed ◽  
Singular Problem ◽  
Dimensional Problem ◽  
Modified Bessel Functions ◽  
One Dimensional

Abstract In the present paper we analyze linear finite elements on a layer adapted mesh for a boundary value problem characterized by the overlapping of a boundary layer with a singularity. Moreover, we compare this approach numerically with the use of adapted basis functions, in our case modified Bessel functions. It turns out that as well adapted meshes as adapted basis functions are suitable where for our one-dimensional problem adapted bases work slightly better.

Isogeometric Stress, Vibration and Stability Analysis of In-Plane Laminated Composite Structures

2018 ◽  
Vol 18 (05) ◽  
pp. 1850070 ◽  
Author(s):  
S. Faroughi ◽  
E. Shafei ◽  
D. Schillinger
Keyword(s):  
Stability Analysis ◽  
Finite Elements ◽  
Isogeometric Analysis ◽  
Composite Structures ◽  
Degrees Of Freedom ◽  
Computational Study ◽  
Laminated Composite ◽  
Basis Functions ◽  
Energy Error ◽  
Laminated Composite Structures

We present a computational study that develops isogeometric analysis based on higher-order smooth NURBS basis functions for the analysis of in-plane laminated composites. Focusing on the stress, vibration and stability analysis of angle-ply and cross-ply 2D structures, we compare the convergence of the strain energy error and selected stress components, eigen-frequencies and buckling loads according to overkill solutions. Our results clearly demonstrate that for in-plane laminated composite structures, isogeometric analysis is able to provide the same accuracy at a significantly reduced number of degrees of freedom with respect to standard [Formula: see text] finite elements. In particular, we observe that the smoothness of spline basis functions enables high-quality stress solutions, which are superior to the ones obtained with conventional finite elements.

scholarly journals General polytopal H(div)-conformal finite elements and their discretisation spaces.

2020 ◽  
Author(s):  
Elise Le Meledo ◽  
Philipp Öffner ◽  
Remi Abgrall
Keyword(s):  
Finite Elements ◽  
Degrees Of Freedom ◽  
General Setting ◽  
Basis Functions ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Wide Range ◽  
Virtual Element ◽  
Set Up

We present a class of discretisation spaces and H(div) - conformal elements that can be built on any polytope. Bridging the flexibility of the Virtual Element spaces towards the element's shape with the divergence properties of the Raviart - Thomas elements on the boundaries, the designed frameworks offer a wide range of H(div) - conformal discretisations. As those elements are set up through degrees of freedom, their definitions are easily amenable to the properties the approximated quantities are wished to fulfil. Furthermore, we show that one straightforward restriction of this general setting share its properties with the classical Raviart - Thomas elements at each interface, for any order and any polytopial shape. Then, to close the introduction of those new elements by an example, we investigate the shape of the basis functions corresponding to particular elements in the two dimensional case.

Development of Timoshenko beam finite elements for vehicle/track vibration analysis

2003 ◽  
Vol 2003.12 (0) ◽  
pp. 345-348
Author(s):  
Kazuhiro KORO ◽  
Kazuhisa ABE ◽  
Makoto ISHIDA ◽  
Takahiro Suzuki
Keyword(s):  
Finite Elements ◽  
Timoshenko Beam ◽  
Vibration Analysis
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