COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
3D Animation For Free Mesh Method
S. Nagaoka1*, M. Inaba2, G. Yagawa1
1
2
Center for Computational Mechanic Research(CCMR) Toyo University, 2-36-5,Hakusan,Bunkyo-ku,
Tokyo,112-8611,Japan.
Techno Star, Japan
Email: gn0500200@toyonet.toyo.ac.jp
Abstract The Free Mesh Method is one of the meshless methods. It becomes possible to perform large-scale analysis
with high precision by use of the free mesh method, and development of a computer. However, an analysis result will
become complicated if large-scale destructive analysis is performed. In order to cope with this problem, it is thought
important to animate an analysis result. This paper describes the 3D animation system for the free mesh method which
is intended to be a post-processor for dynamic phenomena.
Key words: Free Mesh Method, 3D Animation;
PREFACE
In recent years, computer performance has been improving by leaps and bounds. Simultaneously, numerical
computation technology is also developing greatly. The finite element method is broadly used in the engineering field
as a leading tool for this. On the other hand, candidates for analysis are becoming increasingly large-scale and
complicated. Various problems in analysis have arisen with this increase in scale and complexity. For example,
problems that have arisen include how the numerical analysis portion which solves the dispersed partial differential
equation by approximation can be processed at high speed; and how the labor necessary for carrying out element
division can be mitigated. As a method of dealing with these problems, a new approach called the meshless method is
performed widely. The technique called the meshless method eliminates the necessity of element-node connectivity
information that is required for conventional finite-element-method analysis programs, and evaluation of a stiffness
matrix is performed in each node unit. By utilizing improvements in computer processing performance, and the free
mesh method, it has become possible to perform large-scale analysis with high precision.
IMPORTANCE OF INCLUDING ANIMATION
Improvement in computer processing performance and the introduction of FMM allows large-scale analysis to now be
performed with high precision. However, large-scale analysis sometimes computes a very complicated result and it is
often very difficult to understand a complicated analysis result with only numerical data. However, if an analysis result
can be obtained in visual form, it is much easier to understand a complicated analysis result. Moreover, by generating
the animation simultaneously with the analysis itself, reduction of the labor necessary to generate the animation after
the analysis is completed is also possible. So this research describes the elastic analysis using the free mesh method
including real-time animation.
THE FREE MESH METHOD
Fundamentally, at each node, a triangle element is temporarily made from other nodes (satellite nodes) and the node of
the neighborhood (central node). The elemental rigidity matrices derived from these temporary triangle elements
determine the contribution of each main node and this information is combined to create an overall stiffness matrix.
⎯ 812 ⎯
Figure 1: Concept of Free Mesh Method
Specifically, the process can be considered in the following manner. As seen in Fig. 2, the satellite nodes (m, n, o, p…)
surrounding a central node, l, are collected. These satellite nodes are arranged in a clockwise/counterclockwise
rotation about the central node, l, and two points are taken in order, to temporarily create triangle elements (lmn, lno,
lop, lpq -- ..) with the central node. For each triangle elements (for example, triangle lmn), a local stiffness matrix
[k]lmn is created in the usual manner, each line concerning the central node l of [k]lmn is added as a component of a
global stiffness matrix [K] concerning the node l. The work which solves simultaneous linear equations based on the
obtained global stiffness matrix can be processed with various kinds of direct methods and iteration methods like the
conventional finite element method. As a storage form of an overall stiffness matrix, , this is the same process as that of
the conventional finite element method. That is, the skyline method, a non-zero component, or something which can be
evaluated each time without storing the information is employable.
The features of the Free Mesh Method are summarized as follows.
(1) Easy to generate a large-scale mesh automatically and speedily
(2) Applicable without conscious mesh generation
(3) The result being equivalent to that of the finite element method
Figure 2: Local radial mesh around central node
DETERMINING OBJECT CONTACT
Moving in various ways, an object in contact with the ground or other objects receives a certain power. At this time,
through contact, various things rotate and rebound. However, it is difficult to calculate exactly the power produced on
an object at the time of contact. So, in this research, the node base penalty method was introduced to determine contact.
The concept of the penalty method is as follows. For example, in case the object faces the wall like in Fig. 3, it is
assumed that an object sinks into a wall as in Fig 3. The quantity that caved in is designated ⊿x. The power committed
on an object from a wall should multiply this ⊿x by a proportionality constant k. This is the view of the penalty
method. It is as follows when expressed with a formula.
⎯ 813 ⎯
F = −k×⊿x
(1)
Where, F is the force committed on an object from a wall.
With actual objects, many may move along with the rotation and according to the results of analysis, some objects
collide with others during rotation, while some objects begin rotation having collided with others so the patterns of
motion and contact are many and varied. In order to express these actions, penalty power for every node is calculated
in this research. Fig. 4 shows the force produced on an object when it falls and collides with the ground. As shown in
the figure, the node which collided with the ground receives power from the ground. However, other nodes do not
receive the power from the ground. It is possible to calculate the moment of the whole object by calculating the power
produced in all nodes in consideration of the above thing. Angle quantity of motion and angular velocity are calculated
from the moment N of the whole object. It is possible to express the action after collision of the object with the ground
or with other objects by introducing the penalty method for every node as mentioned above including the rotation in
three dimensions.
⊿x
Figure 3: Virtual displacement to calculate the penalty force
GROUND
Figure 4: Nodal forces when the body contacts the ground
LOAD DISTRIBUTION
When a rigid body collides with other elastic bodies, the power produced in each node is not necessarily equal. it is
possible to determine the load produced when a certain rigid body collides with an elastic body in the following
manner. As shown in Fig. 5, it is supposed that a rigid body collides with an elastic body. At this time, the rigid body
G is presupposed to collide with the elastic body from the left-hand side of Fig. 5. The load that the rigid body G will
⎯ 814 ⎯
apply to the elastic body at collision is set to 100 [Pa]. The method in which the node that receives load is determined
in the following manner. First, load will be gapplied to the element which includes the coordinates of the contact point
in the domain. Load will be applied to the node that generates Element A in the example of Fig. 5. Furthermore, in this
example, the rigid body G should have moved and collided from the left-hand side of Fig. 5. Therefore, load was
applied to Node a and Node c among the nodes a, b, and c which generate Element A. Moreover, the distribution of
load applied to the node of an elastic body in the case of a collision is determined by calculating the distance between
the contact point and each node. In the example of Fig. 5, the appropriate coordinates are (100,200) for Node a and
(100,100) for Node c. Furthermore, the coordinates of the contact point e of the collision with rigid body G are
(110,160). Then, the distances between the two points and the contact point which gives load are calculated. The
distance between the y coordinates of Node a and the contact point e is 40. The distance between the y coordinates of
Node c and the contact point e is 60. That is, the rate of the distance between each node and the contact point is set to
6:4. The whole load 100 [Pa] shall be distributed and given to each node according to this rate. Since the whole load is
100 [Pa], the load is divided into 40 [Pa] and 60 [Pa]. The point nearer to the contact point receives greater load.
Therefore, in this example, the load of 60 [Pa] is given to Node a, and the load of 40 [Pa] is given to Node c. The load
distribution at the time of contact is performed as mentioned above.
a
b
a:(x,y)=(100,200)
G
e
[A]
b:(x,y)=(200,200)
c:(x,y)=(100,100)
[B]
d:(x,y)=(200,100)
e:(x,y)=(110,160)
c
d
Figure 5: Nodal load distribution at the time of contact
SOME REMARKS
The conventional finite element method assumes displacement, distortion, and the stress point are in the same domain.
This is also assumed when performing structural analysis using the free mesh method. With the conventional method,
there is a fault that accuracy of strain and stress will be insufficient compared with that of displacement.
So Yagawa et al proposed a variant Free Mesh Method based on the Hellinger-Reissner principle. This method is
called the Enriched Free Mesh Method (EFMM). Displacement, strain, and the stress are defined differently domains
so it is thought that, with the introduction of EFMM, analysis of higher precision is now possible.
REFERENCES
1. Yagawa G, Yamada T. Free mesh method: A new meshless finite element method, Comp. Mesh., 1996; 18:
383-386.
2. Zienkiewicz OC, Taylor RL. The Finite Element Method, Volume 1. Fourth Edition, Japanese Edition,
CADTechs Inc., 1996.
3. Timoshenko S. Strength of Materials Part 1. Third Edition, Japanese Edition, D. Van Nostrand Company, Inc.,
1955.
⎯ 815 ⎯