COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
VCCM Rule-Based Meshing Algorithm for an Automatic 3D Analysis
of Crack Propagation of Mixed Mode
Kohei Murotani 1*, Kanda Yasuyoki 2, Hiroshi Okada 3, Toshimitsu Fujisawa 4, Genki Yagawa 1
1
2
3
4
Center for Computational Mechanics Research (CCMR), Toyo University Address: 2-36-5, Hakusan, Bunkyo-ku,
Tokyo, 112-0001 Japan
University of the Ryukyus, Japan
Department of Nano-Structure and Advanced Materials, Kagoshima University, Japan
Prometech Software, Inc., Tokyo, Japan
Email: [email protected]
Abstract An analysis of crack propagation using FEM need to perform re-meshing for an updated shape of
crack with each cycle. In an analysis of crack propagation of 3D mixed mode, the shape of crack seems to be
complex. In addition, we must effectively calculate parameters of fracture mechanics and exactly forecast a
direction of crack propagation. Because of this situation, we can analyze an automatic 3D analysis of crack
propagation of mixed mode, if we can perform meshing satisfied the two basic conditions of VCCM (Virtual
Crack Closure Method) for an arbitrary shape of crack. In this paper, we show an algorithm of a VCCM
rule-based meshing for the arbitrary shape of crack and calculation results.
Key words: FEM, crack propagation, VCCM
INTRODUCTION
Design and operation of structures that require a high level of safety like those related to nuclear power, chemical
plants and aerospace systems is extremely important. However, a lot of the simulations for assessment of structural
integrity depend on the experience and instinct of skilled technicians because a systematic, rapid, and precise method
of analysis has not been established. Major factors for this are “difficulty in constructing a computational mechanical
model for precise representation of a complex 3D shape of a large-scale structure”.
Our targets are an integration of excellent technologies, for example, large scale parallel analyses programs of the
finite element method (ADVENTURE and NEXST), Free Mesh Method (FMM), programs of parametric
computations in fracture mechanics analysis (three dimensional VCCM) and so on, under cooperation on CAE
software developers and practical applications of fundamental technologies in Universities.
Concretely speaking, mainly we develop a system for an assessment of integrity in large scale structures of nuclear
–related equipments. We develop a system with a speed and a reliability of a top class in the world for automatic crack
progress analyses and parametric analyses using PC clusters as calculation platforms. Characteristics of our system are
the followings.
(1) Practical operations: By using GUI based on Windows, our system can give simple operations to the various types
of analyses and by using promoting interfaces of commercial three dimensional CAD software, can give robust
performances to analyses of shapes for real machines. Additionally, as a result that a PC client control a PC cluster as
a calculation server on the block, our system can give user-friendly operational environments to designers and analysts
who can use latest computing machines.
(2) Large scale parallel parametric computations: Our system can automatically do a parametric analysis of more than
20 cases for a problem with sizes of one million DOF per 1PC in a day (24 hours). If a PC cluster with 20 PC can do a
parametric analysis of more than 400 cases in a day.
(3) Automatic crack progress analyses: Our system can automatically do an automatic crack progress analysis of more
than 20 steps for a problem with sizes of one million DOF per 1PC in a day (24 hours). If a PC cluster with 20 PC can
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do a parametrical analysis of more than 20 cases in a day.
AN AUTOMATIC 3D ANALYSIS SYSTEM OF CRACK PROPAGATION “FMM CRACK”
1. Previous version Our system automatically generates mesh, assigns boundary condition, performs elasticity
analysis by FEM, gets an amount of crack propagation from its result, and repeatedly performs remeshing and
elasticity analysis for 3D test pieces with elliptic type crack. We realize an automatic analysis of crack propagation of
3D mode by an automation of these processes. Our system consists of following five processes. Fig. 1 shows
Procedures for our system.
Process1. Meshing generation of surface patches for 3D test pieces with elliptic type crack
Process2. Meshing generation of volume meshes from the surface parches (using NEXST_FMM)
Process3. Assignment of boundary conditions
Process4. Elasticity analyses by FEM
Process5. Forecast of amount of crack propagations
Figure 1: Procedures for our system
Our system package “NEXST_FMM” and modules of processes Process1, Process3, Process4 and Process5 developed
by Prometech Software, Inc.. “NEXST_FMM” is a module of a tetrahedron mesh generator developed by “the Frontier
Simulation Software for Industrial Science project” of IT program sponsored by the Ministry of Education, Culture,
Sports, Science and Technology.
2. New version New version performs analyses for 3D crack of an arbitrary shape. Therefore, we must improve
Process1 from elliptic type crack to 3D crack of an arbitrary shape.
In previous version, we use “extrapolating displacement method” (direct method) in Process5. For we perform higher
precision forecast, we choose VCCM (Virtual Crack Closer-integral Method) [2]. From using VCCM, we require a
meshing generator of surface parches for VCCM.
We must require “VCCM rule-based meshing generator for an arbitrary shape of crack” for new version.
VCCM (VIRTUAL CRACK CLOSER-INTEGRAL METHOD)
VCCM is the method that evaluates the energy release rate and the stress intensity factor. VCCM has two merits. First
merit is that VCCM can split the energy release rate into its Mode Ⅰ, Ⅱ, Ⅲ contributions without any further
complications. Second merit is that a program for a post process of FEM can easily be constructed, because VCCM use
only nodal forces and relative nodal opening displacements. Fig. 2 briefly shows algorithm of VCCM.
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Figure 2: VCCM (VirtualCrack Closer-integral Method)
GENERATION OF SURFACE PARCHES FOR 3D TEST PIECES WITH CRACK OF AN
ARBITRARY SHAPE
We execute following three stages in a crack propagation problem as shown in Fig. 3, for we will develop an automatic
analysis of crack propagation of 3D mixed mode.
Stage 1. Modal I analysis of crack propagation for elliptic type crack
Stage 2. Modal I analysis of crack propagation for planar crack of an arbitrary shape
Stage 3. Modal mixed analysis of crack propagation for 3D crack of an arbitrary shape
Fig. 4 (a) is shape of elliptic type crack in Stage1. Fig. 4 (b) is shape of planar crack of an arbitrary shape in Stage2 and
we assume the direction of crack propagation is arbitrary direction in 2D plane. In Stage3, the shape of crack is
arbitrary and we assume the direction of crack propagation is arbitrary 3D direction out of plane.
Figure 3: 3D test pieces with crack
(a) Elliptic type crack
(b) Planar crack of an arbitrary shape
Figure 4: Crack models
A BASIC ELEMENT PARTITION RULES IN VCCM
Let front edge of crack be “crack front”, short of crack front be “crack” and one segment beyond crack front be
“ligament” as shown as Fig. 5. A basic element partition Rules in VCCM are following two.
Rule 1. Ligament is symmetrically allocated to mesh of crack on an extension of the crack front.
Rule 2. Widths of ligament and crack are length Δ or vary smoothly across crack front.
An example of an allocation of ligament, crack and crack front are shown in Fig. 5.
Amount of crack propagation is obtained from stress intensity factor by Paris law. In the case of the modal mixed
analysis, we require to forecast the direction of crack propagation out of planar. Therefore we use our extrapolation
method of mesh using GSSA (Generalized Singular Spectrum Analysis) [1] to forecast the direction of propagation.
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Figure 5: Allocation of ligament, crack and crack front in VCCM
VOLUME MESH GENERATION
The nearer to the crack front, the node density is the higher for analyses. Additionally, when crack front comes close to
back face, node density is given to not only the crack front but also the back face.
As stated above, we need to construct volume meshes for models with large density changes. For solving this problem,
we use advancing front around crack front and Delaunay at a point distance from crack front.
Fig. 6. Areas using Delaunay and advancing front
ANALYSES
We use the model as shown in Fig. 7 and perform two analyses. Angle φ is defined as shown in Fig. 8.
First analysis is comparison of stress intensity factor using VCCM and theoretical solutions at b/a = 0.5，b/a = 1.0，
b/a = 2.0.
Fig. 9 is the surface meshes with embedded elliptical crack at b/a = 0.5，b/a = 1.0，b/a = 2.0.
Fig. 10 shows variations of normalized stress intensity factor for the problems of embedded elliptical cracks computed
by VCCM and theoretical solutions. We can get the result that values by VCCM are almost similar to theoretical
solutions.
Second analysis is crack propagation for the case of b/a = 0.5. Let an amount of crack propagation be
da = cK m dt,
where K is stress intensity factor, c and m are material dependent parameters and dt is time step. In this case,
= 1.0×10−7, m = 2.0 and dt = 2.0×104.
c
Fig. 11 shows the relations of times and crack depths at angles 0° and 90° and Fig. 12 shows the relations of stress
intensity factors and crack depths at angles 0° and 90°. We can find that the depth at 90° approch the depth at 0° and
ellipse becomes circle, if time passes.
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Figure 7: Over view of the block, crack
configuration and bound conditions
Figure 8: Embedded crack
(a) Over view in the case of b/a = 0.5
(b) The view of plane of crack in the case of b/a = 0.5
(c) Over view in the case of b/a=1.0
(d) The view of plane of crack in the case of b/a=1.0
(e) Over view in the case of b/a=2.0
(f) The view of plane of crack in the case of b/a=2.0
Figure 9: The surface meshes with embedded elliptical crack
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Figure 10: Variations of normalized stress intensity factor for the problems of embedded
elliptical cracks computed by VCCM and theoretical solutions
Figure 11: The relations of times and crack depths at angles 0° and 90°
Figure 12: The relations of stress intensity factors and crack depths at angles 0° and 90°
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A PRESENT SITUATION AND FUTURE PROSPECTS
Now, we develop Stage1. For the future, we develop theoretical matter and programming matter in Stage2 and Stage3.
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