COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Parallel Computing for Enriched Free Mesh Method (EFMM)
Y. Kobayashi *, G. Yagawa
Centor for Computational Mechanics Research (CCMR), Toyo University, 2-36-5, Hakosan, Bunkyo-ku, Tokyo,
112-8611, Japan
Email: gn0500492@toyonet.toyo.ac.jp
Abstract In this paper a structural analysis is performed using the Parallel Free Mesh Method, a kind of mesh-free
method of analysis. In order to improve the accuracy of a solution using the Free Mesh Method, we have used the
Enriched Free Mesh Method, where the displacement field and the strain field are independently assumed. The
unknown parameters of these fields are determined by employing the Hellinger-Reissner principle. An analysis result
is demonstrated by real-time visualization processing. Some parallel performance study is also processed with a
cluster computer.
Key words: Free Mesh Method, Enriched Free Mesh Method, Parallel Computing
INTRODUCTION
Now, complex super-structures (such as highrise buildings, transportation networks, and communications facilities)
extend in all directions. On the other hand, the problem of damage resulting from natural disasters (such as
earthquakes) or human-inflicted disasters has also surfaced. Therefore, it can be said that the rise of these
super-structures and super-systems has led to an increase in the phenomena of potentially fatal catastrophes.
From this perspective, in order to maintain safety standards, it is necessary to raise the accuracy of soundness
evaluation. Moreover, in order to analyze a model which exists in reality, it is necessary to deal with a huge and
complicatedsubject and thus there is a strong requirement for the analysis model to be large-scale. This study is
performing Parallel Computing with a high precision analysis technique in order to cope with this demand.
We chose the Free Mesh Method (hereinafter referred to as “FMM”)[1] as the base technique of analysis. It is known
that this technique has good affinity with Parallel Computing in comparison[2] with the Finite Element Method
(hereinafter referred to as “FEM”). On the other hand, there were various problems in raising the level of accuracy of
analysis. However, Enriched FMM (hereinafter referred to as “EFMM”), an original technique is developed by
Yagawa et al conquered these problems. The purpose of this research is to perform Parallel Computing on EFMM to
further extend its potential.
FREE MESH METHOD
FFM is an analysis technique based on FEM. In usual FEM analysis, node information and element information are
needed as input data. Also, integration by domain is performed for every element. In contrast, the input data of FFM
requires only the information on a node. Clustered local elements shown in Fig. 1 are temporarily created around the
Central Node to replace the need for element information.
More precisely, the difference between with usual FEM and FMM is that the rigidity matrix is created with
information not only from every element but from every node. Thereby, since this is a system that temporarily
generates and calculates a local element domain locally, the whole element information is not needed. Therefore, the
intense labor involved in mesh generation becomes unnecessary.
That is, FFM can obtain a rigidity matrix and a solution equivalent to FEM by smoothing the procedure of obtaining
a stiffness matrix, without needing any element information. Moreover, affinity with Parallel Computing is improved
with this change of procedure.
⎯ 808 ⎯
Central Node
Satellite Nodes
Other Nodes
Figure 1: Clustered local elements
HIGHLY PRECISE FREE MESH METHOD
Usually, in the case of structual analysis performed by FMM, as with FEM, the displacement field and strain field are
proposed to be in the same domain.
Therefore, accuracy of strain analysis is insufficient the 1st order of displacement. Usage of high accuracy elements
shown in Fig. 2 can be an effective method of raising accuracy. However, in local processing like FMM, the use of
these is difficult. For example, adjustments that memorize new neighborhood information are required to implement
the usual algorithm. Moreover, since the number of nodes increases, the rigidity matrix will grow large. On the other
hand, the technique of increasing node flexibility[3] is also feasible. However this also leads directly to an over-sized
rigidity matrix.
Figure 2: Intermediate nodes
ENRICHED FMM
EFMM is an analysis technique which has simultaneously possesses an independent displacement field proposed for
every local element in addition to the independent strain field assumed for every cluster of local elements that is
characteristic of FMM. The extension local elements shown in Fig. 3 propose an arbitrary strain field at the local
element cluster level, and proposes the same displacement field as FEM at the local element level. Thus, a solution
with high accuracy can be obtained by proposing displacement and strain in independent domains.
Node-wise
strain field
Element-wise
displacement fields
Mixed
Figure 3: Enriched clustered local elements
⎯ 809 ⎯
HELLINGER-REISSNER PRINCIPLE
The technique based on Hellinger-Reissner principle, which is a variant method of associating the displacement field
of these independent places, and the strain field mentioned above is proposed by Yagawa et al. The Hellinger-Reissner
principle which makes displacement {u} and strain {ε} independent variables is expressed with the following
formula.
∏ HR = ∫ ε T DSu dΩ
Ω
(1)
1 T
~
ε DεdΩ − ∫ u T b dΩ − ∫ u T t dS
Ω 2
Ω
Sσ
− ∫
In this formula, {b} is the body force, t is the surface force working at the boundary area and omega is an analysis
domain. Additionally, suppose that there is a relationship in the upper formula as follows:
Su = B u , ε = N ε ε
(2)
Here, the displacement field is independently proposed with each element as with FEM or FFM. Accordingly, the
strain field is independently proposed by an arbitrary polynomial expression based on the local element clusters.
Then, an approximation equation is obtained from the conditional stationary of the functional[4]. The approximation
equation obtained is condensed at whole local element clusters. Since the rigidity matrix associated with this strain
field is proposed independently for every local element cluster, accuracy is increased.
Thus, this technique differs from regular mixed forms of FEM formulization.
THE EXAMPLE OF ANALYSIS
Figure 4: Mesh model
In order to verify the accuracy of displacement-strain analysis by EFMM in the case of a mesh model as shown in Fig.
4, a Cantilever beam model as shown in Fig. 5 has been chosen as an analysis model.
The confirmation of analysis accuracy is performed using the maximum displacement which is normalized with the
explicit solution technique[5]. I am going to perform the report during the presentation.
ν= 0.２ E = 1.94E4
P=100[Pa]
Figure 5: Cantilever beam model
Figure 6: The visualization of an output
⎯ 810 ⎯
PARALLEL COMPUTING FOR EFMM
With a single processor, there is a limit on throughput but for large-scale analysis improved speed and efficiency of
calculation processing is necessary. Therefore, Parallel Computing which uses many processors simultaneously is
indispensable to large-scale analysis.
The technology related to Parallel Computing, such as the parallel computer, the parallel language processor, the
Parallel Computing library,is progressing rapidly and constantly. In this research, Parallel Computing is performed by
PC cluster using the Parallel library MPI.
The advantage of using Parallel Computing for EFMM analysis is the analysis technique of the node base, as it also is
with FMM. The Parallel Computing is thereby based on the node numbers. Moreover, since an equation is constituted
node by node, Parallel Computing becomes easy.
Therefore, the Parallel Computing method at this point relies on domain division of nodes.
In domain division of nodes, in order to make processing uniform across processors, the number of nodes is made
uniform. Moreover, the domain boundary area for commincation control is minimized..
After applying division as shown in Fig. 7, a local matrix is generated for every node in each processor domain.
PE #4
PE #3
PE #2
PE #１
Figure 7: Processor domain division of nodes
SUMMARY AND REMARKS
We are creating a structural analysis application using EFMM based on Hellinger-Reissner principle. We are going to
report the check of verification of the accuracy of EFMM by announcement.
From now on, it is necessary for us to perform Parallel Computing as a step towards expanding the scale of analysis.
Therefore, We are performing EFMM using Parallel Computing with a cluster computer and hence apply EFMM to
large-scale analysis.
REFERENCES
1. Yagawa G, Yamada T. Free Mesh Method: A New Meshless Finite Element Method. Computational. Mechanics,
1996; 18: 383-386.
2. Yagawa G, Yamada T. Performance of Parallel Computing of Free Mesh Method. Japan NCTAM, 1996, pp.
51-52.
3. Kanto Y. Accurate Free Mesh Method by using Mixed Element. Transactions of JSCES, Paper No. 20000036,
2000.
4. Zienkiewicz OC, Taylor RL. The Finite Element Method Volume 1. Fourth Edition, Japanese Edition,
CADTechs, Inc.,: 1996.
5. Timoshenko S. Strength of Materials Part 1. Third Edition, Japanese Edition, D. Van Nostrand Company, Inc.,
1955.
⎯ 811 ⎯