Airbag inflation simulations using coupled fluid-structure analysis
3. Explicit Time Integration and CBS method
An explicit time integrator based on the forward Euler scheme is used to solve the time dependent problem. This scheme has potential ability to reduce the time of the computation and especially for large-scale problems and for higher dimension. For the purpose of describing the explicit time integration scheme and the operator splitting of the CBS method the obtained discretized and stabilized system of equation can be written as
( ) ( ) ( ) ( )
( ) ( ) ( )
T
Φ Φ Φ
+ + + + + + = +
=
+ + + + =
&
&
d ? ?
e
d d
M M U A U B U U C C P DU F F
C U 0
M M F K U F K U F K F F
(6)
Where U, P and Φ are the vectors of unknowns of the (nodal values) velocities, pressures and the advection functions respectively. The construction of the matrices M, K, C, A, B, MT, Mδ, KT, Kδ, Cγ and the vectors F, Fγ are reviewed briefly in [5,6,7,8]. The forward Euler method of time integration, applied to equation (6) gives
1
+1 ( ) [ ( ) ( ) ( ) ]
n = n +dt + − + − n − n n − + n
U U M Mδ F Fγ A U B U U C Cγ P (7)
In order to use equation (7) to advance the velocity, the pressure at time tn has to be computed;
this is done by the CBS method, by combining the momentum equation with the time differentiated version of compressibility constraint in (6), (C UT & =0 since C UT =0for all time) to generate the consistent discretized Poisson equation for the pressure, evaluated at time tn,
1 1
(C MT L−C P) n =C MT L− [F+ Fγ −A U( n)−B U( n)Un −DU] (8) With Un that satisfies C UT =0 The sequence of steps of advancing the velocity and pressure from tn to tn+1 is thus
1[ ( ) ( ) ]
n L n n n
= − + − − −
G M F Fγ A U B U U DU (9)
(C M C PT -1L ) n =C GT n (10)
1
+1 [ ( ) ]
n = n +dt n − L− + n
U U G M C C γ P (11)
1
+1 ( ) [ ( ) ( ) ]
n = n +dt Φ + − Φ − n + +
F F M Mδ F K U F K δ U F KF ε (12)
The presence of the ML−1in above equation required the use of a lumped mass matrix.
4. Conclusion
The computation of the problem is very intensive, hence an efficient implementation is required in order to produce the desired accuracy and to minimize the computation effort. An object oriented programming using the C++ package of PETSC [10] is under processing. Furthermore we intend to improve the technique by combining the formulation with FE-adaptivity in the
vicinity of the interface. The formulation is more flexible in handling fixed mesh domain in order to track two-fluid interfaces.
REFERENCES
[1] J. A. Sethian, Level Set Methods, Evolving Interface in Geometry, Fluid Mechanics, Computer Vision, and Material Science, Cambridge University Press (1996).
[2] J. A. Sethian, Level Set Methods and Fast Marching Methods, Cambridge, Cambridge University Press (1999).
[3] M. Sussman, P. Smareka and S. Osher, “A Level Set Approach for Computing Incompressible Two-Phase Flows”, J. Comp. Phys., 114, 146-168, (1994).
[4] M. Sussman, E. Fatemi, P. Smareka and S. Osher, “A Improved Level Set Method for Incompressible Two-Phase Flows”, J. Computers and Fluids, Vol. 4, 212-223 (1996).¨
[5] N. H. Sharif and N.-E. Wiberg, "Free-Surface Flow Predictions by an Interface Capturing FE-Technique” International Journal for Computational Civil and Structural Engineering (IJCCSE), Vol. 1, Issue 2, 98-106, (2000).
[6] N. H. Sharif and N.-E. Wiberg, ”Free Surface Flow through Rock-Fill Dams Analyzed by FEM with Level Set Approach", to appear in Journal of Computer Modeling and Simulation in Engineering (C.M.S.E), Vol. 3, (2001).
[7] N. H. Sharif and N.-E. Wiberg, "Adaptive ICT-Procedure for Nonlinear Seepage Flows with Free Surface in Porous Media", accepted for publication in Communications in Numerical Methods in Engineering (CNME), June (2001).
[8] N. H. Sharif and N.-E. Wiberg, "Computing Steady Motion of Incompressible Fluid Interface in Porous Media Flow", on CD in the European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS Computational Fluid Dynamics Conference 2001, Swansea, Wales, UK, 4-7 September (2001).
[9] O.C. Zienkiewicz and R.L. Taylor, " The Finite Element Method" Volume 3, Fluid Dynamics, Fifth Edition. BH, (2001).
[10] PETSC WWW home page: http://www-fp.mcs.anl.gov/petsc/, 2000.
[11] S. Osher, and J. A., Sethian, "Fronts Propagating with Curvature Dependent Speed:
Algorithm Based on Hamilton-Jacobi Formulations", J. Comput. Phys. 79, 12-49 (1988) [12] S. Osher and R. P. Fedkiw, "Level Set Method: An overview and some Recent Results"
Recent UCLA Computational and Applied Mathematics Reports, Department of Mathematics, University of California, Los Angeles, California, February 2000
[13] T. J. Barth and J. A. Sethian, “Numerical Schemes for the Hamilton-Jacobi and Level Set Equations on Triangulated Domains”, J. Comp. Phys., 145, 1-40, (1998)
[14] T. Tezduyar, S. Aliabadi, M. Behr, “Enhanced-Discretization Interface-Capturing
Technique (EDICT) for Computation of Unsteady Flows with Interfaces”, Comp. Methods Appl. Mech. Eng., 155, pp. 235-248, (1998).
[15] Y. Saad and M. H. Schultz, “GMRES: A Generalized Minimal Residual Algorithm for Solving Non-symmetric Linear Systems”, J. Soc. Idust. Appl. Math. Vol. 7, No. 3, July (1986).
Interactive Finite Element Analysis by Java3D API
Ling Lu, Odd Tullberg Department of Structural Mechanics
Chalmers University of Technology, Gothenburg, Sweden ling@sm.chalmers.se, odd@sm.chalmers.se
Summary the paper presents a framework for performing interactive Finite Element Analysis (FEA) in Virtual reality (VR). The Java3d API has been chosen as the software tool to develop our system and perform interactive FEA in VR. A 3D world for interactive FEM by Java 3D API is set up where the 3D models for analysis are loaded from other CAD packages, such as 3DS and DXF.
1. INTRODUCTION
In recent years, there has been much excitement about Virtual Reality (VR). From its beginning in the field of scientific simulation, VR has gradually grown into a new phase and become a distinct field in the world of computing. The utility of VR have already been researched and used in car design, robot, medicine, chemistry, biology and education, as well as in building design and construction.
The Finite Element Method (FEM) and Finite Element Analysis (FEA) were created in the late 1940’s as a structural analysis tool that was to assist aerospace engineers design better aircraft structures [1]. Since then, aided by the rapid growth of computer power, the method has continually developed until it has become a sophisticated generic tool for accomplishing a wide array of engineering tasks. It has been used in civil, mechanical, geotechnique,, and environment engineering, as well as in many other areas. The software techniques for FEA are already very developed and mature.
The traditional way to express a design before the construction is finished is to use 2D or 3D drawings, which are drawn on 2D paper; Current Computer Aided Design (CAD) software, like AutoCAD or SolidWorks, can do semi-real-time visualisation and rendering of the 3D data on the computer display system. It is usually very difficult to imagine a complex building before it is constructed. VR systems present a new method to show the ideas in a realistic 3D world. In this world, the architects and final users can discuss how the building will look before it is constructed and reach agreement on design decisions in a more intuitive and efficient way. Engineers can see abstract data, such as deformation, stress or strain, which needed to be visualised in a “concrete and real” way. Constructors can choose a better way to finish construction of the building.
To perform interactive FEA in VR leads to a comprehensive understand of the analysis and the design in a 3D world. The basic ideas have already been around for some time. Interactive FEA in VR, as one useful application of VR is taking off and being actively developed [2].
By using virtual pointers (like 3D mouse) and sharing virtual workspaces, remote collaboration and accurate information sharing can be achieved. Apart from using VR in interactive FEA in the building and construction industry, it can also be used in the education of engineering students, to give students an intuitive understanding of the FEA technique.
2. FRAMEWORK OF INTERACTIVE FEM AND AIM OF THE RESEARCH
Most FEA package, such as ABAQUS and ANSYS, support importing geometry from other CAD packages, and have an interactive pre/post processor, which include modelling, managing, monitoring analysis jobs, and result animations. The aim of the research is to set up an interactive FEA system in VR. It will integrate CAD, FEA and VR to perform real time interactive FEA. The system consists of five components:
• Approximation module: Software to provide fast feedback and results to the user even at the expense of accuracy.
• FEA module: FEA code to perform analysis.
• VR module: Software to interact with the VR world.
• Visualisation module: Software to visualise the numerical results from the FEA module.
• Database module: Software to handle the input output and storage of results.
The “Glue” code is needed to make all the modules work together. The Layout of an interactive FEA framework is as Figure 1.
The commercial FEA package available will be explored for FEA module, and FEA in Java has been studied. This research will be concentrated on VR and Visualization modules.
The main aims of the research are:
• Set up a 3D world for interactive FEA in VR.
• Interactively modify input within VR.
• Interactively view of the results in VR.
• Implement FEA in view of the specific requirements of conducting FEA in VR (primary latency reduction)
Within this project, the main task will be setting up a 3D world and modifying the input to show some interesting results in VR.
Figure 1 Interactive FEA Framework
3. SET UP AN INTERACTIVE FEA 3D WORLD USING THE JAVA 3D API 3.1. Why Java and Java 3D API is chosen as the developing tool?
A language called “Oak” was created at Sun Microsystems, Inc. in 1991, but was renamed
“Java” in 1995. It is created with the initial motivation for the need of a platform-independent language. It can be used to create two types of programs: applications and applets. It has many properties, such as: simple, portable, secure, robust, multithreaded, distributed, high performance, Object-oriented. As an object oriented programming language, Java also has the three principles of OOP: Encapsulation, Inheritance, and Polymorphism.
The Java 3D API is an interface for writing programs to display and interact with 3D graphics. 3D geometric objects are created and manipulated reside in a virtual universe, which is then rendered. The Java 3D renderer is capable of rendering in parallel. The instances of Java 3D objects are put into a scene graph data structure. The scene graph is an arrangement of 3D objects in a tree structure that completely specifies the contents of the virtual universe, and how it should
be rendered. The virtual universe is referenced to Locale objects, and Locale objects serves as the root of multiple subgraphs of the scene graph. A BranchGroup (BG) object is the root of a subgraph, which has two categories: the view branch graph and the content branch graph. A content branch graph is assembled from objects to define the geometry, behaviours, sound, lights, location, appearance, etc. A view branch graph specifies viewing parameter. A TransformGroup (TG) is often used in the creation of scene graphics, which hold geometry and its transformation [3]. Figure 2 shows a simple scene graph.
Figure 2 A Simple Scene Graph
The objects in the Java 3D can be created either by code to specify the geometry or by loader classes, which are provided by Java 3D utility package. Loaders can create Java 3D visual objects from files created with 3D modelling software. Loaders today exist for VRML files, AutoCAD DXF files, 3D studio 3DS files, Solid Works SLD, Visual Toolkit VTK, and a variety of other 3D file formats.
To implement interaction, Behaviour objects can be defined in a Java 3D scene graph. It is used to change the scene graph or objects in scene graph, in respond to stimulus. Changes include adding objects, detaching objects, change attributes of the objects, or a combination of these behaviours.
3.2. Framework of the interactive FEA in Java 3D world
To implement interactive FEA in VR, The Java 3D API is used to set up a 3D world for the VR module. The scene graph is produced as figure 3.
BranchGroup BG1, BG2, BG3 are the main components for the 3D world. BG2 is mainly used for setting up the static scene by reading the model from 3DS files or from direct coding of geometry, and is usually used to set up the environment. BG1 is mainly used for dynamic reading in the model from the code of the geometry, which can be changed during the interactive procedure.
BG3 is used for setting up the view platform in the VR world. The interactive behaviour is mainly performed in BG1. BGN can be added if more components are needed in the scene graph.
4. CONCLUSIONS AND FUTURE WORK
Using VR in FEA gives a more comprehensive and intuitive way to understand the work.
The overview of the available VR system and graphic APIs, and the study of the related work show the feasibility of using VR in FEA.
A framework of performing interactive FEA has been set up. A 3D world for interactive FEM by Java 3D API has been set up where the 3D models for analysis are loaded from other CAD packages, such as 3DStudio.
The basic 3D world for interactive FEA has been set up; next we will further develop the interface, which include modelling the environment and the user, modelling interface for
interactive FEM and FEA, connecting with the FEA software to implement interactive finite element analysis in VR.
Figure 3 Interactive FEA in Java 3D World REFERENCE
[1] Zienkiewicz, O.C., The Finite Element Method, 4th edition, McGraw-Hill, London, 1994.
[2] Connell, M., Kettil, P., Tägnfors, H., Tullberg, O., Wibeerg, N., (2000, Greece) Integrating modelling, simulation and visualisation in immersive VR environments- A Tool in bridge design, computational methods for shell and spatial structures, IASS-IACM 2000, Athens, Greece.
[3] Java3d Homepage, accessed July 28, 2000, http://java.sun.com/products/java-media/3D/collateral/, Getting started with the Java 3D API, tutorial v1.5.1.