Gear Forging Simulation Using Cyclic Symmetry

Gangcheng Huang

MSC Software Corp. 2300 Traverwood Drive, Ann Arbor, MI 48105, USA

Abstract. The forging processes of gears, such as, helical and spiral bevel gears, are not symmetric but cyclic symmetric. Metal forming simulation software can utilize this cyclic symmetry to save CPU time and improve analysis. This paper presents such applications with MSC.SuperForm.

INTRODUCTION

Making gears with forging helps material grain flow to follow the configuration of the teeth. It increases the load bearing capacity of a gear without having to increase the tooth size. It also minimizes the effort of machining to produce the finished gear.

Metal forming simulation can be used in the process and tool design to check material flow, ensure the die filling and reduce material waste. In many cases, only a symmetric portion of the gear needs to be analyzed. However when forging gears, such as, helical, spiral bevel gear and sometimes direct gear, the gear geometry is not symmetric but cyclic symmetric.

FIGURE 1. Examples of Cyclic Symmetry Gears

Cyclic symmetry is observed when die, workpiece geometry, forces or boundary conditions are changed in a cyclic manner. In this situation, the material flow or deformation also behaves cyclically. Utilizing this feature in the metal forming simulation allows us to focus analysis on a small portion of the gear, thus save computational effort and improve accuracy of the analysis.

THE FINITE ELEMENT METHOD

The finite element method has been used in metal forming simulation for the last 30 years. The “flow formulation” or “rigid plastic” approach was studied by Zienkiewicz [1] and Kobayashi [2] in the early 1970s. The “solid” approach which considers the elastic effect in the material was published by McMeeking [3]. The two approaches, though different in methodologies and approximations, conclude with the solving of the following equations:

FuK =ˆ (1)

uu =ˆ (2) where K is the stiffness matrix, u the nodal incremental displacement or velocity vector and F the force vector. Equation (2) indicates typical displacement boundary constraints.

In the hot forging simulation, thermal mechanical coupled analysis is required. The heat transfer analysis is carried out in a stagger manner with the mechanical analysis as described by Rebelo and Kobayashi [4]. The following transient heat transfer equation is solved:

QTHTC =+ ˆ&̂ (3)

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admineditedbyS.Ghosh,J.C.Castro,andJ.K.Lee

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where T is the nodal temperature vector and Q the heat source vector. A single-step time updating scheme can be employed to compute the rate of the nodal temperature as follows

( )

−∆−

−+

∆=+ nTtnTnT

tnT ˆ1ˆ1ˆ11ˆ && β

β (4)

where n denotes nth time step and ∆t the time step length.

Cyclic Symmetry Constraints

For the symmetric boundary constraints, Equation (2) can simply be defined as

0ˆ =nu (5)

where n denotes the surface normal direction. If the transformation matrix R is known, Equation (5) can be described in the normal direction in the global coordinate system,

0ˆ =uRn (6)

FIGURE 2. Cyclic Symmetry Constraints

With the cyclic symmetric boundaries, the constraints are enforced on both sides as shown in Figure 2, where

AB uu ˆˆ =′ (7)

In Figure 2, Point B equals Point A after rotating certain degree in Z direction. For every point on one side of the cyclic symmetric boundaries, one can always find a corresponding point on the other side. This means if we know the cyclic symmetry angle and its rotation axis, we can define any two sides of the cyclic symmetric boundaries. These two sides of the geometry are identical with a cyclic symmetry angle between them. If R is the transformation matrix with respect to the cyclic symmetry angle and its rotation axis, Equation (7) can be further expressed as

AB uRu ˆˆ = (8)

Equation (8) together with Equation (1) can then be solved. The rigid rotation mode can be suppressed by adding constraints with respect to the rotation axis.

If the finite element mesh does not provide the nodes exactly matched on both sides of the cyclic symmetric boundaries, a tying constraint will be added to the equations. In Figure 2, if point B does not match point A but some point C between point A1 and A2, then Equation (8) can be rewritten as

iACB uRNuRu ˆˆˆ == (9)

where N is an interpolation function and i=1,2. The same constraints are used for the heat transfer analysis where the nodal temperatures on the cyclic symmetry surface are constrained as

iAB TNT ˆˆ = (10)

Re-meshing Strategy

For most of the metal forming simulations, re-meshing is required when Lagrange based finite element mesh gets too distorted. In gear forging, re-meshing is important to ensure the successful simulation. In order to maintain correct cyclic symmetry after the re-meshing procedure, the following approaches can be adopted:

• Duplicate the new mesh nodes on the cyclic symmetry boundary surfaces

• Add constraints to enforce new mesh nodes to be generated within the cyclic symmetry surfaces

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In the first approach, new mesh nodes are created firstly on one of the cyclic symmetric surfaces. The mesh on the second cyclic symmetric surface is then generated with rotation of the mesh from the first cyclic symmetric surface. The first approach guarantees that nodes on cyclic symmetry surfaces are exactly matched. This is possible in 2-D applications but can be difficult in 3-D applications. The second approach, which is used in this paper, projects the new mesh nodes to the cyclic symmetric surfaces defined based on the previous mesh. In the second approach, nodes that are not matched exactly on the other side of the symmetric surface are to be constrained as described in Equation (9) and (10). After every re-meshing, the cyclic symmetric constraints will be checked and redefined based on the new mesh nodes.

EXAMPLES

The following examples demonstrate the gear forging simulation with the use of the cyclic symmetry constraints. The method is implemented in MSC.SuperForm, a commercial software product from MSC Software Corporation for metal forming simulation.

Direct Gear Forging – 2-D Application

Based on the plane strain assumption, a direct gear forging with tapered punch and cyclic symmetric die profile is simulated (See Figure 4). In the forging process, a tapered punch pushes the material to flow sideways and form the gear teeth.

The cyclic symmetric angle is 30 degrees. Only one of the 12 teeth is analyzed. Figure 4 shows the simulation steps and the expansion to the full view. One can see that the two curved cyclic symmetric boundaries match well with each other.

Comparisons are made between 30-degree symmetry, 90-degree symmetry and full analysis in Figure 5, Figure 6 and Table 1. Figure 5 shows similar effective plastic strain contours between these different analysis runs. The flow lines are used to display the material flow. In Figure 6, the flow lines are used in the full model to compare the material flow with the 30-degree cyclic symmetry model. One can see that the cyclic symmetric boundaries shown in Figure 4 match well with the flow lines from the full model analysis. In Table 1, one can see that the CPU time saving is very significant. The analysis was performed on a P4 Dell computer with 3.0GHz clock speed. A

speed up of 70 times is recorded from a 30-degree symmetry model.

FIGURE 3. Direct Gear Forging

FIGURE 4. Direct Gear Forging Simulation

FIGURE 5. 90-degree symmetry and full model

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FIGURE 6. Flow Lines fiom the Full model

TABLE 1. Comparisons in Simulation

Max Plastic Strain

Symmetry

CPU time (sec)

1.55 1.56 1.66

Helical Gear Forging - 3-D Application

The total CPU time in simulation is about 1.5 hours on the same machine as in the previous example.

The CPU time saving can be even greater when 3- D gear forging simulation is performed using cyclic symmetry. Helical gear forging in this example (Figure 7) would require a considerable amount of computational effort without using the cyclic symmetry.

There are 10 teeth in this helical gear simulation. A 36-degree cyclic symmetric angle is used. Tetrahedral elements and re-meshing are adopted in the simulation.

Figure 7 shows the results of the effective plastic strain. The portion of the analyzed gear and the full- expanded gear views are shown.

FIGURE 7. Helical Gear Forging

Y' I

I

FIGURE 8. Helical Gear Forging Simulation

Bevel Gear Forging - 3-D Application

Producing this spiral bevel gear needs a hot forging process which in simulation requires a thermal- mechanical coupled analysis. Just like in mechanical analysis, cyclic symmetry can also be applied to the thermal analysis. The nodal temperatures will be constrained in the same way as for the nodal displacement in Equation (10).

There are 40 teeth in this bevel gear. Simulation is performed on one single tooth (see Figure 9). Temperature distribution is shown in Figure 10 together with its expanded view at the final step.

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The total CPU time used in the simulation is 0.6 hours with about 15000 tetrahedral elements. This performance could not have been achieved without using the cyclic symmetry.

7

FIGURE 9. Bevel Gear Forging

1

FIGURE 10. Bevel Gear Forging Simulation

REMARKS

forming simulation, especially the gear forging simulation as presented in this paper.

ACKNOWLEDGMENTS

The author wishes to thank his colleague for their contributions and support to the work of this paper.

REFERENCES

1 . Zienkiewicz,O.C, and Godbole, P.N.: “Flow of Plastic and Viscoplastic Solids with Special Reference to Extrusion and Forming Processes”, Znt.J.Num.Meth. in Eng., 8, pp.3-16 (1974)

2. Lee, C.H., and Kobayashi, S.: “New Solutions to Rigid Plastic Deformation Problems using a Matrix Method”, J.Engfor.Znd, Trans., ASME, 95, pp.865-873(1973)

3. McMeeking, R.M., and Rice, J.R.: “Finite-Element Formulation for Problems of Large Elastic-plastic Deformation”, Znt.J.SoZids Structures, 11, p.601 (1975)

4. Rebelo, N. and Kobayashi, S.: “ Acoupled Analysis of Viscoplastic Deformation and Heat Transfer -1I”, Znt. J.Mech.Sci., V01.22, pp.707-718 (1980)

Cyclic symmetry is a special feature that can be utilized to greatly improve the efficiency of the metal

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MachiningSimulations of High-Speed Machining Using a Multi-Material Finite Element Formulation (N015)Investigation of the Effect of Tool Edge Geometry Upon Cutting Variables, Tool Wear and Burr Formation Using Finite Element Simulation — A Progress Report (N019)Application of Three Dimensional Finite Element Modeling for the Simulation of Machining Processes (N028)Advances in Machining Process Modeling (N178)Thermo-Elasto-Visco-Plastic Constitutive Equations Fully Coupled with Ductile Damage. Application to Metal Cutting by Chip Formation (N179)A New Approach for Fem Simulation of NC Machining Processes (N188)Experimental and Numerical Investigation of Ductile Regime Machining of Silicon Nitride (N217)Adaptation of an Asperity Ploughing Model to Measured Roll Topographies (N278)A Finite-Element-Analysis of Orthogonal Metal Cutting Processes (N279)Experimental Investigation of Workpiece/Fixture Contact and Its Implementation in Numerical Simulation of Machining Processes (N309)Influence of Cutting Edge Radius of Coated Tool in Orthogonal Cutting of Alloy Steel (N369)Material Flow Stress Sensitivity Analysis in Numerical Cutting Modeling (N404)Prediction of Surface Roughness in the Machining of Carbon Steels by Cutting Tools (N465)

Non-Conventional Materials ProcessingNumerical Simulation of Laser/IR Assisted Micro-Embossing (N314)Chopper Gun Trajectory Optimization for Spray Forming in Automotive Manufacturing (N025)Analysis of Magnetic Abrasive Finishing with Slotted Magnetic Pole (N058)Laser Deposition Process Design Via Thermal Analysis-Thermal Model Development (N101)Finite Element Analysis of Laser Engineered Net Shape (LENS™) Tungsten Clad Squeeze Pins (N114)Influence of Process Conditions on Bowing Phenomenon in Successive Biaxial Stretching with Tenter (N116)Warpage Analysis of Silicon Wafer in Ingot Slicing by Wire-Saw Machine (N139)Finite Element Analysis for Development of Next-Generation Heater in Heat Treatment Process for Silicon Wafer (N140)Micro/Nano Patterning Characteristics in Hot Embossing Process (N198)Design and Strength Evaluation of Structural Joint Made by Electro-Magnetic Forming(EMF) (N199)Quasi-3D Strain Analysis in Equal Channel Angular Pressing (N214)Special Steel Production on Common Carbon Steel Production Line (N222)Forming and Bending of Metal Foams (N308)3D Finite Element Analysis of Orbital Forming and Inverse Analysis for Determination of Flow Stress of the Workpiece (N353)Thermo-Mechanical Analysis with Phase Transformations (N391)Simulation of Steady Laser Hardening by an Arbitrary Lagrangian Eulerian Method (N392)Finite Element Modeling of Multi-Pass Equal Channel Angular Extrusion (N402)Simulation of Double-Seaming in a Two-Piece Aluminum Can (N488)Three-Dimensional Heat Transfer Modeling of a Moving Plate in Forming Process Applications (N492)Development of Advanced Coatings for Laser Modifications Through Process and Materials Simulation (N128)

Modeling Materials at Micro and Nano-ScalesA Study of Micro Injection Molding for High-Aspect-Ratio Optical Fiber Ferrules (N335)Large Deformation Simulations of Nanocrystalline Materials (N559)Deformation Mechanisms during Hot Working of Titanium (Abstract only) (N543)Molecular Dynamics Modeling of Carbon Nanotubes and Their Composites (N512)Peridynamic 3D Models of Nanofiber Networks and Carbon Nanotube-Reinforced Composites (N514)Linking Atomistic and Continuum Mechanics Using Multiscale Models (N515)Studies on Deformation Mechanism and Punch Taper Effects on Nanoimprint Processes by Molecular Dynamics (N165)A First Principles Study of Alumina Nanoparticles (N298)A General Thermo-Mechanical Shape Memory Alloy Model: Formulation and Applications (N312)Two-Scale Characterization of Deformation-Induced Anisotropy of Polycrystalline Metals (N395)Activation Energy for Irreversible Deformation Processes in Spatially Extended Crystalline Systems (N403)Computational Homogenization Method for Atom-to-Continuum Modeling (N077)Development of Triple Scale Finite Element Analyses Based on Crystallographic Homogenization Methods (N417)Effect of Dispersion State on the Rheology of Multi-Walled Carbon Nanotube Suspensions in Shear Flow (N429)Crystal Level Simulations Using Eulerian Finite Element Methods (N491)Influence of Statistical Cell Dispersion on the Local Strain and Overall Properties of Cellular Materials (N502)

Material Characterization and Constitutive Modeling at Different ScalesStrengthening Mechanisms in Ti-6Al-4V/Tic Composites (N018)An Advanced Constitutive Model in the Sheet Metal Forming Simulation: The Teodosiu Microstructural Model and the Cazacu Barlat Yield Criterion (N024)A New Constitutive Model for Prediction of Springback in Sheet Metal Forming (N299)Development of Dynamic Explicit Crystallographic Homogenization Finite Element Analysis Code to Assess Sheet Metal Formability (N419)Design of Plane Strain Bending Based on Ideal Flow Theory (N518)Constitutive Behaviour of the Metastable Stainless Steel: Sandvik Nanoflex™ (N144)Hot Deformation Behavior of Bearing Steels (N174)Superplastic Forming Behaviors and Microstructure Characters of Magnesium Alloy Sheet AZ31B (N154)Instantiation of Polycrystal Plasticity Models to Predict Heterogeneous Straining in Aluminum Alloys (N561)Influence of Deformation Mechanisms on the Mechanical Behavior of Metals and Alloys: Experiments, Constitutive Modeling, and Validation (Abstract only) (N523)The Potential Advantages of Microstructure Modeling of Titanium to the Aerospace Industry (N526)Beta Grain Growth Kinetics in Ti-6Al-4V (N535)Predicting the Microstructure Dependent Mechanical Performance of Materials for Early-Stage Design (N542)Breakthrough Technologies in Titanium Refinement Methods (Abstract only) (N544)Microstructural Characterization Using 3-D Orientation Data Collected by an Automated FIB-EBSD System (N546)Modeling Cyclic Deformation of HSLA Steels Using Crystal Plasticity (N508)Multiple Time Scale Modeling for Cyclic Deformation with Crystal Plasticity (N510)The Parameter Identification of Thermal Visco-Plastic Model Considering Dynamic Recrystallization (N153)FEM Analysis of Forming and Microstructure and Prediction of Properties for Wayshaft (N196)Including Dislocation Flux in a Continuum Crystal Plasticity Model to Produce Size Scale Effects (N197)Models for Forming Simulations of Metastable Austenitic Stainless Steel (N215)Decreasing Computation Time in Finite Element Simulations Coupled with Polycrystalline Plasticity (N260)A Method for Determining Property Variance in Polycrystalline Materials (N289)Identification of Plane and Spatial Clustered Distributions of Particulate Inclusions (N206)Viscoplastic Selfconsistent Modelling of the Anisotropic Behavior of Voided Polycrystals (N321)A Macro Modeling of Phase Transition in Austenitic Stainless Steels and TRIP Steels (N330)Modeling the Anomalous Flow Behavior of Ni3Al Intermetallic Single Crystals (Abstract only) (N352)Multiscale Modeling of Large Deformations in 3-D Polycrystals (N365)Modeling the Mechanical Behavior of Polycrystalline Interconnect Lines (N366)A Crystal-Plasticity Model for the Flow Behavior of Two-Phase Alloy Systems (N388)On the Time-Dependent Inelastic Deformation of Metals (N405)Analysis of Isotropic Elastoplastic Models at Finite Strains Used in Numerical Modeling (N437)A Formulation for the Pseudo-Saturation Behavior Observed during Variable Amplitude Multiaxial Cyclic Plasticity (N487)A Comparative Study of Constitutive Algorithms for 3D Rate-Dependent Single Crystals (N460)Lattice Based Microstructure Evolution Model for Monte Carlo Finite Element Analysis of Polycrystalline Materials (N507)

Localization and Damage Modeling in Materials and ProcessingNumerical Study of Damage Evolution and Failure in an Electromagnetic Corner Fill Operation (N022)An Efficient Algorithm of Plastic Integration for Damage Modelling in Sheet Forming Process (N045)Forming Limit Prediction of Powder Forging Process by the Energy-Based Elastoplastic Damage Model (N043)Investigation of Damage Features in Hot Metal Forming (N055)Simulation of Ductile Damage in Metal Matrix Composites (N064)Numerical Simulation of Wheel Forming Process Including Damage and Thermal Effects (N180)Determination of Ductile Damage Parameters by Notched Round Bar Tension Test Using Image Analysis (N195)Analysis of Hydro Formed Complex Shape Parts Using a Ductile Fracture Criterion (N432)Prediction of Thermal Fatigue in Tooling for Die-Casting Copper Via Finite Element Analysis (N113)Simulation of Material and Structural Instability Phenomena during the Flaring Process of Cylindrical Shells (N119)A Voronoi Cell Finite Element Model for Ductile Damage in MMCs (N555)Evaluation of Material JR and Fracture Toughness Transition Curves Using Micro-Mechanical Modeling (N553)Simulation of Microdamage in Ceramics Deformed under High Confinement (N552)Multi-Scale Model for Damage Analysis in Fiber-Reinforced Composites with Debonding (N524)Finite Element Simulation of Surface Defects in the Automobile Door Outer Panel (N265)Finite Element Analysis for the Milli-Forming of Crystalline Materials with Damage: Application to Milli-Rolling (N268)Simulation of Fracture Initiation in Sliding Contact of Brittle Coatings on Elastoplastic Substrates (N287)Gradient Damage Models in Metal Forming Problems (N294)The Workability Criteria for Adiabatic Shear Band Phenomena in the Dual-Phase Steel Cold Heading Process (N189)Using the Finite Element Method and Artificial Neural Networks to Predict Ductile Fracture in Cold Forming Processes (N470)Micromechanism Study on the Cracking Tendency of Deformation of Anisotropic Plasticity Sheets (N493)The Role of Fatigue Variability in Life Prediction of an alpha+betaTitanium Alloy (N541)Analysis of Crack Development, both Growth and Closure, in Steel Oxide Scale under Hot Compression (N548)

Process Design and OptimizationModeling of Optimization Strategies in the Incremental CNC Sheet Metal Forming Process (N071)An Interactive and Flexible Approach to Stamping Design and Optimization (N240)Optimum Design of Addendum Surfaces in Sheet Metal Forming Process (N253)Computer Aided Process Planning for Non-Axisymmetric Deep Drawing Products (N455)Response Surface Methodology for the Design of Sheet Metal Forming Parameters to Control Springback Effects Using the Inverse Approach (N478)Optimization of the Blankholder Force with Application to the Numisheet'02 Deep Drawing Benchmark Test B1 (N494)Design of Optimal Temperature Distribution Using FEA for Warm Forming of Lightweight Materials (N135)Sequential Optimization and Reliability Assessment Method for Metal Forming Processes (N136)Multi-Length Scale Design of Deformation Processes for Control of Orientation (Texture) Dependent Properties (N138)FEM Optimization of Spin Forming Using a Fuzzy Control Algorithm (N166)Sensitivity Analysis and Optimization Algorithms for 3D Forging Process Design (N167)Optimum Design of Forging Process Parameters and Preform Shape under Uncertainties (N168)3-D Preform Shape Optimization in Metal Forming (N173)Flexible Aluminum Tubes and a Least Square Multi-Objective Non-Linear Optimization Scheme (N121)Study on Carbide Anvil of Hinged Cubic Press for Synthetic Diamond by FEM (N090)Using Material Processing Simulation Software to Predict a Part “In Use” Properties (N040)Modified Upper Bound Elemental Technique (MUBET) for Preform Design in Closed Die Forging (N126)Three-Dimensional Simulation and Design Sensitivity Analysis of the Injection Molding Process (N223)A Gradient Optimization Method for Efficient Design of Three-Dimensional Deformation Processes (N191)Parallel Optimization of Forging Processes for Optimal Material Properties (N238)Shape Optimization of Flow Guide in Three-Dimensional Profile of an H-Section Extrusion (N242)Optimization of the Shape of a Lip Die in Extrusion of a Plate Wider than the Diameter of a Round Billet Using a Lip Die (N243)About Distributed Simulation-Based Optimization of Forming Processes Using a Grid Architecture (N257)New Trends in Computer Simulation as Integrated Tool for Automotive Components Development (N306)Finite Element Method Applied to 2D and 3D Forging Design Optimization (N307)Application of Numerical Methods in Metal Forming Tribology (N318)An Optimization Strategy for the Determination of Material and Process Parameters to Avoid Segregation Defects during Metal Injection Powder (N331)Minimization of Anisotropic Effect during Thin Cup Free Bulging Process Using the Artificial Neural Networks Optimization Technique (N406)Application of Modified Inverse Method to Determine Flow Stress Function of AlMgSiNAO Alloy under Hot Forming Conditions (N420)Optimal Topology Design of Products Using Genetic Algorithm (N440)Optimization of Forming Processes in Microstructure Sensitive Design (N444)Development of an Inverse Analysis Method Coupled with A 3D Finite Element Model (N454)Automated Topology Design for Electromagnetic Devices (N477)Design and Optimization of Extruder Flow Channel with Flow Resistance Concept (N481)A Crisp Solution in Multi-Criteria Optimization Using a Taguchi Approach: Application to a Cold Heading Process (N496)Simulation of Coupled Processes and Product Properties (N501)

Advances in Numerical MethodsA New Hexahedral Solid Element for 3D FEM Simulation of Sheet Metal Forming (N050)Contact Algorithm for Eulerian Finite Element Method (N272)A Second Order Approach for Finite Deformation Implicit Contact Analysis, Including Friction, in Deformable Bodies (N283)Meshless Local Petrov-Galerkin Analysis for Thermoelastic Deformations of Functionally Graded Beams (N296)A Finite Element Solver for the Advection Equation Applied to Interface Tracking (N303)Tet-To-Hex Conversion for Finite Element Analysis (N027)A Novel Arbitrary Reference Configuration NARCO Lagrangian Formulation and Contrast to Total/Updated Lagrangian Formulation in Finite Strain Applications (N531)Enhanced Rotation-Free Basic Shell Triangle for Sheet Stamping Problems (N516)Development of a One Point Quadrature EAS Solid-Shell Element (N159)Enhanced Assumed Strain Shell and Solid-Shell Elements: Application in Sheet Metal Forming Processes (N172)A Quasi-Symmetric Contact Formulation for 3D Problems. Application to Prediction of Tool Deformations in Forging (N176)A Posteriori Error Estimation and Three-Dimensional Adaptive Remeshing: Application to Error Control of Non-Steady Metal Forming Simulations (N177)A General Purpose One Point Quadrature Shell Element Based on Resultant-Stress (N315)A Diffusion-Split Method to Deal with Thermal Shocks Using Standard Linear Tetrahedral Finite Elements (N347)A Finite Element Study of Capstan Friction Test (N357)Parallel Processing of 3-D Rigid-Plastic Finite Element Method Using Diagonal Matrix (N374)A Mixed Continuous Formulation for Solving Thermomechanical Equilibrium during Hot Forming Process Simulation (N411)Application of Space-Time Finite Elements to Elastodynamics Problems (N438)

APPENDIX ON CD-ROM ONLYPAM-RTM (Resin Transfer Molding) Simulation-Advances in Composite Manufacturing (N397)PAM-STAMP 2G Version 2003 Release News (N394)SYSWELD Welding Simulation Tool: Version 2003 Product News (N396)

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