Basic Concept of the Beam Wave based Element for Mid and High
Frequency Analysis
SOUFIEN ESSAHBI
Department of Heavy Machinery Maintenance,
Higher Institute of Applied Sciences and Technology of Gafsa,
Sidi Ahmed Zarrouk University Campus,
2112 Gafsa,
TUNISIA
Abstract: - This paper describes a Hermite beam wave based element of the steady-state dynamic response of a
1D structure system. This study focuses on the development of beam wave based elements. Compared with
standard piecewise polynomial approximation, the wave basis is shown to give a considerable reduction in
computational degrees of freedom.
In practical terms, it is concluded that the degrees of freedom for which accurate results can be obtained, using
these new techniques, can be up to half of that of the conventional finite-element method.
Key-Words: - finite element, Trefftz method, beam wave based element, frequency depending.
Received: August 25, 2022. Revised: February 19, 2023. Accepted: March 18, 2023. Published: April 11, 2023.
1 Introduction
The finite element method (FEM), [1], [2], [3], is a
numerical technique that makes it possible to solve
approximately the differential equations or with
linear partial derivatives whatever the imposed
boundary conditions, especially for composite
structures, [4].
However, its implementation remains difficult
and costly in some cases. Indeed, the mesh must
obey certain rules, in particular, the elements must
not be crushed, to avoid the degeneration of the
associated Jacobian.
It is well known that the use of discrete
numerical methods (finite element method FEM) for
the solution of the dynamic structure equation is
limited to problems in which the wavelength under
consideration is not small in comparison with the
domain size. The limitation arises because
conventional elements, based on polynomial shape
functions, can reliably capture only a limited portion
of the sinusoidal waveform. In fact, an accurate
description of the problem needs the use of about
eight to ten degrees of freedom per full wavelength
[5], [6]. To overcome these problems, we developed
a beam wave based element, this method is based on
the indirect Trefftz method, [1], [7], [8], [9], [10].
In this paper, we describe the basic concept of
the beam wave based element. The idea is the
enrichment of the conventional shape functions by
the solution of the homogeneous equation. This
technique makes the formation of matrices more
complicated. To illustrate this technique two
examples are presented. The numerical validation of
this element is made by calculating the percentage
of an error on the whole structure.
2 Problem Formulation
Consider an elastic thin beam of length ,
thickness, density , Poisson's coefficient and
elasticity modulus .
The beam makes an angle from the horizontal, the
Fig.1 below shows the problem geometry.
The problem to study is governed by the dynamic
equation of the structure and the boundary
conditions given by the following equation:
Dynamic equation
4 ' '
4'
4' ,
b
d w x q x
k w x
d x D
in
s
(1)
With
2
4s
b
t
kD
: Structural bending (1)
3
2
12 1
Et
D
: Bending stiffness. (2)
Boundary Conditions
- Clamped - clamped beam
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'
'
00
0
00
0
w
wL
dw
dx
dw L
dx
(3)
- simply supported beam
2
2'
2
2'
00
0
00
0
w
wL
dw
dx
dwL
dx
(4)
Either the virtual displacement
vM
, arbitrary and
regular in the domain
s
, the weighting of the
structure dynamic equation by
vM
leads after
integrations to:
4 ' '
4 ' '
4' 0
s
b
d w x q x
k w x v x d
d x D
(6)
v
kinematically admissible
The mathematical transformation by two
integrations by parts we arrive at the following weak
form:
2 ' 2 ' '
4 ' ' '
2 ' 2 '
3 3 2 2
3 ' 3 ' ' 2 ' ' 2 '
0 0 0 0 0
s s s
b
d w x d v x q x
d k w x v x d v x d
d x d x D
d w d w dv d w dv d w
v L L v L L
d x d x dx d x dx d x
(5)
In the case of a beam simply supported on both
sides the displacement and the bending moment are
zero where the second derivatives of
w
are zero
and we can take
vM
equal to zero at
'0x
and
'xL
.
'
o
M
'
x
L
0
x
0
y
x
y
o
Fig.1: Elastic thin beam.
3 Finite Element Approximations
The FEM, [11], [12], is a well-known simulation
technique to model the steady-state dynamic
behaviour of complex structures. The technique
determines an approximate solution to the problem
described by the beam dynamic equation (1) and the
imposed structural boundary conditions (4) and (5).
The finite element used in this study is the beam
linear finite element with two degrees of freedom
per node. Fig.2 shows the geometry of the beam
element and these degrees of freedom.
Fig. 2: Finite beam element.
The FEM approximates the exact solution for each
of the structural deformation fields by a weighted
sum of simple (polynomial) shape functions.
The displacement of the structure is approximated
on a finite element by:
1
1,
1 2 3 4
2
2,
, , , x
s s s s s n
x
w
w
w N N N N N u
w
w
(6)
With
; 1,2,3,4
m
s
Nm
are the shape functions of
high precision of Hermite type given by: (9)
22
22
11 2 1 1 1 2 1 1
4 2 2
sll
N
Applying a Galerkin weighted residual formulation,
[10], these functions are expressed as a linear
1
1,x
w
w
2
2,x
w
w
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combination of the same basis functions as used in
the deformation approximations (8)
The equation (5) is written in the following matrix
form: (10)
4
,,
00
3
3'
2
2'
,,
3
3'
2
2'
0
0
00
LL
s xx s xx b s s n s
xx
v N N k N N dx u v N f x dx
dw
dx
dw
dx
v v v L v L dw L
dx
dwL
dx
4 Enriched Finite Element
The idea is to enrich the basis of the standard finite
elements with a base derived from the homogeneous
solution of the dynamic equation, [13], of the
structure.
The solutions of the homogeneous equation are
given by:
'nb
j k x
n n n
w w e
(7)
With
We enriched the shape functions
Σφάλμα! Το αρχείο προέλευσης της αναφοράς δεν βρέθηκε.
on the basis of the structure mode, using some of
the propagating modes
12
,
bb
ik x ik x
ee
.
Therefore the new shape functions are given by:
(12)
With and is the element length.
The displacement of the structure will be
approximated on an element by:
(8)
With are the structure waves
amplitudes.
The test function is chosen equal to the conjugated
shape function.
Fig. 3 shows the geometry of the beam-enriched
element and these degrees of freedom.
Fig. 3: Enriched finite beam element.
5 Numerical Results
A comparison between the numerical results
obtained by the enriched finite element and the
standard finite element is made. The example of a
simply supported beam is presented, and two cases
are studied. The first case is the case of a load
distributed over the beam and the second is the case
of a concentrated force applied in the middle of the
beam.
The percentage of error between numerical values
and analytical ones in the middle of the beam is
calculated.
The error according to the number of degrees of
freedom is presented.
The percentage of relative error, [14], is given by:
With
: The analytic displacement of the beam,
: The numerical displacement of the
beam.
In this study we use an aluminum beam whose
characteristics are the following:
: Density,
: elasticity modulus,
: Poisson's coefficient,
: Moment of inertia.
The displacement of the beam can be decomposed
on its modal base as follows:
(9)
With
: Mode ,
: Modal component of displacement corresponds
to the mode .
In the case of a simply supported beam the modes of
the beam can be written as [15], [16]:
(10)
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And the modal component of the displacement is
given by:
(11)
With
(12)
5.1 Loading Distributed on the Beam
Fig.4 shows a simply supported beam excited by a
distributed load .
'
()qx
Fig. 4: Simply supported beam with a distributed
loading.
The analytic displacement of the beam writes:
′
′
Subsequently, we present in Fig.5, Fig.6, Fig.7,
Fig.8, Fig.9, Fig.10 the error according to the
number of degrees of freedom for different
frequencies of excitations.
010 20 30 40 50 60 70
10-8
10-6
10-4
10-2
100
102
ddl
Error(%)
Error = f(ddl) ; freq = 10
Hermite
Hermite enrichi
Fig. 5: Error according to degrees of freedom for
.
0100 200 300 400 500 600
10-8
10-6
10-4
10-2
100
102
ddl
Error(%)
Error = f(ddl) ; freq = 2 kHz
Hermite
Hermite enrichi
Fig. 6: Error according to degrees of freedom for
.
0100 200 300 400 500 600
10-6
10-4
10-2
100
102
ddl
Error (%)
Error = f(ddl) ; freq = 4 kHz
Hermite
Hermite enrichi
Fig. 7: Error according to degrees of freedom for
.
0500 1000 1500 2000 2500
10-6
10-4
10-2
100
102
ddl
Error (%)
Error = f(ddl) ; freq = 8 kHz
Hermite
Hermite enrichi
Fig. 8: Error according to degrees of freedom for
.
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500 1000 1500 2000 2500 3000
10-6
10-4
10-2
100
102
104Error = f(ddl) ; freq = 40 kHz
ddl
Error (%)
Hermite
Hermite enrichi
Fig. 9: Error according to degrees of freedom for
.
500 1000 1500 2000 2500 3000 3500
10-6
10-4
10-2
100
102
104
ddl
Error (%)
Error = f(ddl) ; freq = 100 KHz
Hermite
Hermite enrichi
Fig. 10: Error according to degrees of freedom for
.
According to these results, it is noted that to have
the same error for the two elements, it is necessary
to use more than the double degrees of freedom for
the not enriched Hermite element. And we note that
the enriched element converges faster than the non-
enriched Hermite element.
5.2 Concentrated Force Applied in the
Middle of the Beam
Fig. 11 shows a simply supported beam submitted to
a concentrated force in the middle.
c
F
Fig. 11: Concentrated force.
The analytical displacement of the beam writes:
′
′
′
Subsequently, we present in Fig.12, Fig.13, Fig.14,
Fig.15, Fig.16, and Fig.17 the error according to the
number of degrees of freedom for different
frequencies of excitations.
0100 200 300 400 500 600 700
10-6
10-4
10-2
100
102
ddl
Error (%)
Error = f(ddl) ; freq = 1 kHz
Hermite
Hermite enrichi
Fig. 12: Error according to degrees of freedom for
, concentrated force case.
0100 200 300 400 500 600 700 800 900
10-4
10-2
100
102
ddl
Error(%)
Error = f(ddl) ; freq = 4 kHz
Hermite enrichi
Hermite
Fig. 13: Error according to degrees of freedom for
, concentrated force case.
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0200 400 600 800 1000 1200 1400
10-4
10-2
100
102
104
ddl
Error (%)
Error = f(ddl) ; freq = 8 kHz
Hermite
Hermite enrichi
Fig. 14: Error according to degrees of freedom for
, concentrated force case.
0200 400 600 800 1000 1200
10-4
10-2
100
102
104
ddl
Error (%)
Error = f(ddl) ; freq = 16 kHz
Hermite enrichi
Hermite
Fig. 15: Error according to degrees of freedom for
, concentrated force case.
0500 1000 1500 2000 2500 3000 3500
10-8
10-6
10-4
10-2
100
102
ddl
Error (%)
Error =f(dll) ; freq = 40 kHz
Hermite
Hermite enrichi
Fig. 16: Error according to degrees of freedom for
, concentrated force case.
500 1000 1500 2000 2500 3000 3500
10-6
10-4
10-2
100
102
104Error = f(ddl) ; freq = 100 kHz
ddl
Error (%)
Hermite
Hermite enrichi
Fig. 17: Error according to degrees of freedom for
, concentrated force case.
These results show the efficiency of the Hermite-
enriched element developed in low, medium, and
high frequencies.
According to these results, we note that the enriched
element converges faster than the non-enriched
Hermite element, in addition, the use of this element
allows us to reduce the number of degrees of
freedoms necessary to half.
6 Conclusion
This article describes the beam plane wave element.
This paper aimed to study this enriched element
according to the frequency so the comparison with
the standard finite element.
This element is of Hermite type enriched by a
base deduced from the homogeneous solution of the
dynamic equation of the structure. The validation
was done by treating two examples of a simply
supported beam. The first was the case of a
distributed constant loading and the second was the
case of a concentrated force. The results found
showed the effectiveness of the developed element
at low, medium, and high frequencies. Thus, these
results showed that the developed enriched Hermite
elements converged faster than those of the non-
enriched Hermite type.
The obtained results show that, while increasing
the frequency of excitation, the necessary number of
degrees of freedom for the solution of problems
with a given level of error decreases. So the results
prove that the enriched element converges more
quickly than the Hermite non-enriched element.
From the perspective of the continuity of this work
and to broaden its field of application, it would be
interesting to develop the extension of the method to
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cases of problems associated with 2D, 3D, and
composite material.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
Soufien ESSAHBI contributed to this research at all
stages, from the formulation of the problem to the
final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflict of interest to declare
that is relevant to the content of this article.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
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International Journal on Applied Physics and Engineering
DOI: 10.37394/232030.2023.2.3
Soufien Essahbi
E-ISSN: 2945-0489
21
Volume 2, 2023
