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PAT328, Section 3, March 2001 S1-1 MAR120, Lecture 4, March 2001 MAR120, Section 1, December 2001 SECTION 1 OVERVIEW OF MSC.MARC AND MSC.PATRAN ~ PART 1
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Page 1: PAT328, Section 3, March 2001S1-1MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001 SECTION 1 OVERVIEW OF MSC.MARC AND MSC.PATRAN ~ PART 1.

PAT328, Section 3, March 2001 S1-1MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001

SECTION 1

OVERVIEW OF MSC.MARC AND MSC.PATRAN ~ PART 1

Page 2: PAT328, Section 3, March 2001S1-1MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001 SECTION 1 OVERVIEW OF MSC.MARC AND MSC.PATRAN ~ PART 1.

PAT328, Section 3, March 2001 S1-2MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001

TABLE OF CONTENTS

Section Page

1.0 Overview of MSC.Marc and MSC.Patran ~ Part 1Company Overview ……………………………………………………………………………………………….. 1-3Features and Benefits of MSC.Marc…….………………………………………………………………………. 1-7Stretch Forming using MSC.Marc………………………………………………………………………………… 1-8Superplastic Forming using MSC.Marc………………………………………………………………………….. 1-9Analysis of a Rubber Boot Using MSC.Marc……………………………………………………………………. 1-10Profile Rolling Using MSC.Marc………………………………………………………………………………….. 1-12Finger Biomedical Pump Simulation Using MSC.Marc………………………………………………………… 1-14Ink Jet Printing Simulation Using MSC.Marc……………………………………………………………………. 1-15What is MSC.Marc……….………………………………………………………………………………………… 1-16MSC.Marc Documentation………………………………………………………………………………………… 1-17Overview What is MSC.Marc……………………………………………………………………………………… 1-18What is MSC.Patran……………………………………………………………………………………………..… 1-19MSC.Patran useful features example:Problem with Trimmed Surfaces……………………………………… 1-20MSC.Patran useful features example:Solution with Composite Surfaces…………………………………… 1-21MSC.Patran useful features example:Solution with Tessellated Surfaces…………………………………... 1-22MSC.Patran useful features Tools Aiding Creation of Tessellated Surfaces………………………………... 1-23MSC.Patran useful features example:Surface Decompose…………………………………………………… 1-24MSC.Patran Useful Features Verifying Mesh Topology……………………………………………………….. 1-25MSC.Patran Useful Features Parameters Smoothing Meshes………………………………………………. 1-26MSC.Patran Useful Features Parameters Smoothing Meshes………………………………………………. 1-27MSC.Patran Useful Features Hard Curves Controlling Meshing……………………………………………… 1-28MSC.Patran Useful Features Hard Points Controlling Meshing Fem Creation Tools………………………. 1-29Summary Of MSC.Marc Structural Solution Procedures And MSC.Patran Marc Preference Support…… 1-30Summary Of MSC.Marc Non-structural Solution Procedures And MSC.Patran Marc Preference Support. 1-31Example Of Highly Nonlinear Problem: Deep Drawing Of A Sheet To A Box………………………………. 1-32 General Solution Features ……………………………………………………………………………………….. 1-33Rigid-deformable And Deformable Deformable Contact………………………………………………………. 1-34Elastic, Plastic, Hyperelastic, Creep And Viscoelastic Material Models……………………………………… 1-36Large Element Library……………………………………………………………………………………………... 1-37Advanced Solution And Modeling Features…………………………………………………………………….. 1-38Local Adaptive Remeshing……………………………………………………………………………………….. 1-39Global Adaptive Remeshing……………………………………………………………………………………… 1-40Where On MSC.Patran?…………………………………………………………………………………………... 1-41

Page 3: PAT328, Section 3, March 2001S1-1MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001 SECTION 1 OVERVIEW OF MSC.MARC AND MSC.PATRAN ~ PART 1.

PAT328, Section 3, March 2001 S1-3MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001

COMPANY OVERVIEW

The MSC.Software Corporation (formerly MacNeal-Schwendler Corporation) has been supplying sophisticated computer-aided engineering (CAE) tools since 1963

MSC.Software is the developer, distributor, and supporter of the most complete and widely-used structural analysis program in the world, MSC.Nastran as well as the first commercial nonlinear analysis program in the world, MSC.Marc.

MSC.Nastran MSC.Marc MSC.Dytran MSC.Patran MSC.Marc Mentat

MSC.MVision MSC.Fatigue MSC.Laminate Modeler MSC.SuperForm MSC.SuperForge

…and more

Page 4: PAT328, Section 3, March 2001S1-1MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001 SECTION 1 OVERVIEW OF MSC.MARC AND MSC.PATRAN ~ PART 1.

PAT328, Section 3, March 2001 S1-4MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001

COMPANY OVERVIEW (CONT.) MSC.Nastran

Linear Analysis Vibration Classic Dynamics

MSC.Marc Spring back Welding Superplastic Forming

MSC.Dytran Sheet Metal Forming Plastic Container Forming Crash Analysis

MSC.SuperForm General Purpose Bulk

Forming MSC.SuperForge

3D Forging

Manufacturing Solver Carriers

Page 5: PAT328, Section 3, March 2001S1-1MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001 SECTION 1 OVERVIEW OF MSC.MARC AND MSC.PATRAN ~ PART 1.

PAT328, Section 3, March 2001 S1-5MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001

COMPANY OVERVIEW (CONT.)

1971 The MARC Analysis Research Corporation was founded.

1972 MSC releases proprietary version of NASTRAN, called MSC.Nastran.

1972 MARC Corp releases the first proprietary version of MARC, the first commercial Nonlinear finite element analysis program.

1994 MSC merged with PDA Engineering (Developer of PATRAN) to become the largest single provider of finite element analysis (FEA) software to the CAE market.

1999 MSC.Software merged with MARC Analysis Research to lead both the linear and the nonlinear analysis worldwide CAE market.

MSC.Software Milestones

1963 Company founded by Dr. Richard MacNeal and Mr. Robert Schwendler. Developed first program called SADSAM (for Structural Analysis by Digital Simulation of Analog Methods.) This was the forerunner of MSC’s flagship program, MSC.Nastran.

1965 MSC participates in NASA-sponsored project to develop a unified approach to computerized structural analysis. The program became known as NASTRAN (NASA Structural Analysis Program)

1965 A team of researchers at Brown University initiated the development of the technology leading to the MARC program.

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PAT328, Section 3, March 2001 S1-6MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001

With corporate headquarters in Santa Ana, California, MSC.Software maintains regional sales and support offices worldwide.

MSC Technical Support Hotline 1-800-732-7284 (USA/Canada). Staffed Monday through Friday,

7:00 a.m. to 3:00 p.m. Pacific Standard Time. (10:00 a.m. to 6:00 p.m. Eastern Standard Time.)

E-mail support (USA/Canada) at

[email protected] MSC.Marc, MSC.Marc Mentat, MSC.Patran Marc Preference support

[email protected] –other than Marc Preference- support)

Support (USA/Canada) Fax 714-979-2900 Internet support http://www.mscsoftware.com

MSC CLIENT SUPPORT

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PAT328, Section 3, March 2001 S1-7MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001

FEATURES AND BENEFITS OF MSC.MARC

MSC.Marc has helped analyze and influence final design decisions on

Automotive parts Nuclear reactor housings Biomedical equipment Offshore platform components Coated fiberglass fabric roof structures Rocket motor casings Ship hulls Elastomeric motor mounts Space vehicles Electronic components Steam-piping systems Engine pistons Tires Jet engine rotors Welding, casting, and quenching processes Large strain metal extrusions

Page 8: PAT328, Section 3, March 2001S1-1MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001 SECTION 1 OVERVIEW OF MSC.MARC AND MSC.PATRAN ~ PART 1.

PAT328, Section 3, March 2001 S1-8MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001

This series of images show a simulation of the stretch forming of skin panels for a major commercial aircraft. In stretch forming, the panels are stretched over a mandrel and then released, whereupon they spring back. As a result of these simulations, the panels can be formed properly the first time and their final shape matches the design without reworking.

STRETCH FORMING USING MSC.MARC

For example, Boeing is using MSC.Marc for simulating the forming of aircraft panels including springback

Page 9: PAT328, Section 3, March 2001S1-1MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001 SECTION 1 OVERVIEW OF MSC.MARC AND MSC.PATRAN ~ PART 1.

PAT328, Section 3, March 2001 S1-9MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001

This benchmark is a simulation of a manufacturing process that is taking a greater importance: Super Plastic Forming, or SPF; this is a process that permits extremely complicated shapes to be formed with a highly uniform thickness through the application of high temperature and controlled pressure to certain kinds of material. In this example the metal changes from its initial flat shape as catenaries are formed and then drapes over a protrusion to assume the final desired shape. The problem involves automated 3-D contact and large plastic deformation.

SUPERPLASTIC FORMING USING MSC.MARC

MSC.Marc 2001 added direct control loading so that strain rates are adhered to; this avoids writing a user subroutine.

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PAT328, Section 3, March 2001 S1-10MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001

The constant velocity boot is modeled using either the Mooney or Ogden model for nonlinear incompressible materials. Viscoelastic behavior for both small and large problems may be included. These material models are simulated with MSC.Marc’s Hermann elements specially formulated for incompressible materials. Very large strains result because of self contact between bellows of the boot. As the shaft rotates, one side of the boot goes into compression, where local buckling may occur; the opposite side goes into tension, which may result in the eventual failure of the material because of fatigue. Damage models are available to model the degradation of elastomeric materials.

ANALYSIS OF A RUBBER BOOT USING MSC.MARC

See detailed image in next page.

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PAT328, Section 3, March 2001 S1-11MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001

ANALYSIS OF A RUBBER BOOT USING

MSC.MARC (CONT.)

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PAT328, Section 3, March 2001 S1-12MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001

This series of images shows a simulation of the profile rolling process. Here, a set of rollers deforms a flat sheet into a channel shape. This simulation shows half of the environment which, in the end, is used to form a U-shaped channel.

PROFILE ROLLING USING MSC.MARC

See detailed image in next page.

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PAT328, Section 3, March 2001 S1-13MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001

PROFILE ROLLING USING MSC.MARC (CONT.)

Metal Forming

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PAT328, Section 3, March 2001 S1-14MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001

This 13-finger biomedical pump was easily modeled using MSC.Marc. The tube is a simple cylinder made of a polymer material, while the milking motion was simulated by defining the position versus time of each of the fingers. Because of the flexibility of MSC.Marc’s automated contact analysis capabilities, the user was able to optimize the pump’s performance simply by charging the position versus time tables.

FINGER BIOMEDICAL PUMP SIMULATION

USING MSC.MARC

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PAT328, Section 3, March 2001 S1-15MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001

Virtual Manufacturing includes the simulation of processes involved in the operation and performance of a product. This example shows an analysis of the paper motion through an ink jet printer. The paper is pushed and pulled by rollers through the preheating stage on the right and then, under the print head at the top. The first analysis (first two images) the paper curls away from the pre-heater. Uniform heating of the paper is an important quality issue, since large gradients will cause distortion in the text. The second analysis (last two images) resolves the problem by adjusting the push/pull force of the rollers, resulting in uniform temperature for a variety of paper thicknesses at the designed operating speed.

INK JET PRINTING SIMULATION USING MSC.MARC

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PAT328, Section 3, March 2001 S1-16MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001

MSC.Marc includes a wide array of element types and material models

WHAT IS MSC.MARC

X

Y

Z

Large Displacements

Contact Resolution

Collapse

Non-Linear Material(Hyperelastic rubber

in this example)

MSC.Marc is a full featured, comprehensive software package for linear and nonlinear finite element structural and thermal analyses.

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PAT328, Section 3, March 2001 S1-17MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001

PDF files

MSC.MARC DOCUMENTATION Volume A: User Information and Theory Volume B: Element Library Volume C: Program Input Volume D: User Subroutines

see ~marc2001/user (all Fortran templates)

Volume E: Demonstration Problems

see ~marc2001/demo (over 400 demos) Volume F: Background Papers see ~marc2001/primer

Page 18: PAT328, Section 3, March 2001S1-1MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001 SECTION 1 OVERVIEW OF MSC.MARC AND MSC.PATRAN ~ PART 1.

PAT328, Section 3, March 2001 S1-18MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001

MSC.Marc Mentat MAR101 (other) course

MSC.Patran Marc MAR120 (this) course

OVERVIEW WHAT IS MSC.MARC?

Analysis Solution of Problem Types:

Structural Thermal Coupled Thermal-Structural Fluid Mechanics Electromagnetics

Friendly Engineered GUIs: MSC.Patran Marc Preference MSC.Marc Mentat

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PAT328, Section 3, March 2001 S1-19MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001

WHAT IS MSC.PATRAN

MSC.Patran is a finite element pre- and post-processor, which has been integrated with several nonlinear analysis solvers including MSC.Marc, MSC.Nastran, and Abaqus/Standard for implicit solutions; and MSC.Dytran and LS-Dyna3D for explicit solutions.

Together, MSC.PATRAN and MSC.MARC are particularly useful for modeling complex non-linear problems.

MSC.AFEA combines MSC.Patran and MSC.Marc (The analysis may only be launched from within the graphics user interface.)

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PAT328, Section 3, March 2001 S1-20MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001

Trimmed SurfacesPave mesh of trimmed surfaces. Note mesh paths do not follow edge

Surface constraints lead to a poor mesh Vertices, hard points, hard curves, adjoining meshed edges

MSC.PATRAN USEFUL FEATURES EXAMPLE: PROBLEM WITH TRIMMED SURFACES

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PAT328, Section 3, March 2001 S1-21MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001

Composite Surface

Trimmed Surface Mesh

Create a Composite Surface to eliminate restrictions Vertices, Meshed Edges

Composite Surface Mesh

MSC.PATRAN USEFUL FEATURES EXAMPLE: SOLUTION WITH COMPOSITE SURFACES

Page 22: PAT328, Section 3, March 2001S1-1MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001 SECTION 1 OVERVIEW OF MSC.MARC AND MSC.PATRAN ~ PART 1.

PAT328, Section 3, March 2001 S1-22MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001

Original Geometry

Tri Mesh

Tessellated SurfaceShell Mesh from Tessellated Surface

MSC.PATRAN USEFUL FEATURES EXAMPLE: SOLUTION WITH TESSELLATED SURFACES

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PAT328, Section 3, March 2001 S1-23MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001

Tessellated Surfaces use the node positions to build a surface.

Mesh quality is not of concern except where severe curvature must be captured.

FEM - Modify/Mesh/Sew tool sews gaps on a tri meshed region for the purpose of building a tessellated surface.

MSC.PATRAN USEFUL FEATURES TOOLS AIDING CREATION OF TESSELLATED

SURFACES

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PAT328, Section 3, March 2001 S1-24MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001

New mesh after Decompose

Trimmed Surface Mesh

New Decomposed Surfaces

Decompose trimmed surfaces to allow for better meshing control

Select vertices at edge or internal positions or existing points.

MSC.PATRAN USEFUL FEATURES EXAMPLE: SURFACE DECOMPOSE

Page 25: PAT328, Section 3, March 2001S1-1MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001 SECTION 1 OVERVIEW OF MSC.MARC AND MSC.PATRAN ~ PART 1.

PAT328, Section 3, March 2001 S1-25MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001

Geometry - Verify / Surface / Boundary

Incongruent Topology Congruent Topology

Topology defines adjacency relationships. Shared edges of a surface are termed topologically congruent if they occupy the same model space and use the same vertices.

Topological congruency determines whether coincident nodes are created on common boundaries.

Verify/Surface/Boundaries checks for free and manifold edges. If free edges exist internal to the model then these entities are topologically incongruent.

MSC.PATRAN USEFUL FEATURES VERIFYING MESH TOPOLOGY

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PAT328, Section 3, March 2001 S1-26MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001

Tri Pattern for 90o

cornered Surfaces

IsoMesh Parameters Define mesh smoothing

parameters and mesh patterns.

Paver Parameter Allows for a tri element if element count on

the boundary is odd numbered Curvature check allows for refinement of

elements on highly curved boundaries. Control for internal element size. Default

range is set to largest and smallest element on the boundary.

MSC.PATRAN USEFUL FEATURES PARAMETERS SMOTHING MESHES

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PAT328, Section 3, March 2001 S1-27MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001

Curves defining core region

Tessellated Surface from the

Course Grid

Hard Points

Hard Curves

1

2

3

4

Mesh follows Core Curves

Existing Course grid model of a panel Hard Geometry

Detail of Panel is a laminated panel

Create points at the edge to define nodal positions.

New Fine Grid FEM Model

MSC.PATRAN USEFUL FEATURES HARD CURVES CONTROLLING MESHING

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PAT328, Section 3, March 2001 S1-28MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001

Noncongruent mesh withouthard points at T-points

Congruent mesh withhard points at T-points

MSC.PATRAN USEFUL FEATURES HARD POINTS CONTROLLING MESHING

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PAT328, Section 3, March 2001 S1-29MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001

Glide Bar elements

along curve 62

Guide Bar elements

along curve 61

Example: Sweep Glide-Guide sweeps elements along glide curve and maintains orientation by the guide curve.

Can be Used to mesh ducting

FEM CREATION TOOLS

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PAT328, Section 3, March 2001 S1-30MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001

Linear and Nonlinear solutions

Static and Transient Analysis

Buckling and Post-Buckling

Time and Frequency based (Classical) Dynamics

Frequency (and Modal) Extraction

Euler Buckling

Direct (Transient)

Modal (Transient)

Frequency Response (Steady State)

Spectrum Response (Steady State)

Supported by MSC.Patran 2001

SUMMARY OF MSC.MARC STRUCTURAL SOLUTION PROCEDURES AND MSC.PATRAN

MARC PREFERENCE SUPPORT

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PAT328, Section 3, March 2001 S1-31MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001

Fully Coupled Thermal/Structural AnalysisSteady StateTransientCreep

Hydrodynamic Bearing Electrostatic Analysis Magnetostatic Analysis Electromagnetic Analysis Acoustic Analysis Fluid Mechanics

Fluid behavior onlyFluid-thermal coupled behaviorFluid-solid coupled behaviorFluid-thermal-solid coupled behaviorFluid-Soil (pore pressure)

Heat Transfer Steady State Analysis Transient Analysis

Not Supported by MSC.Patran 2001

Supported by MSC.Patran 2001

SUMMARY OF MSC.MARC NON-STRUCTURAL SOLUTION PROCEDURES

AND MSC.PATRAN MARC PREFERENCE SUPPORT

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PAT328, Section 3, March 2001 S1-32MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001

MSC.Marc provides a large collection of examples. These are documented in Volume E. This example is given in section 8.38 and data files are from case e8x38a to case e8x38d which use four different techniques.

This example demonstrates the deep drawing of a box modeled with shell elements.

The punch and holder are modeled in MSC.Marc with Nurbs using the CONTACT option. It can be set up in the MSC.Patran Marc Preference in the Loads/BCs form.

EXAMPLE OF HIGHLY NONLINEAR PROBLEM: DEEP DRAWING OF A SHEET TO A BOX

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PAT328, Section 3, March 2001 S1-33MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001

Newton-Raphson and Arc-Length Methods

Manual or Automated Load Incrementation Procedures

GENERAL SOLUTION FEATURES

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PAT328, Section 3, March 2001 S1-34MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001

Click in a cell to toggle T, G, or nil

Analytic or Discrete Rigid Contact Surfaces with Velocity, Force/Moment, or Displacement Control

Glued, Stick-Slip or Continuous Friction Models

RIGID-DEFORMABLE AND DEFORMABLE DEFORMABLE CONTACT

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PAT328, Section 3, March 2001 S1-35MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001

When two bodies come into contact, MSC.Marc finds out which is the area of contact and calculates the contact stresses (Normal Stress or contact “pressure” and Shear Stress or contact “friction stress”)

The user may also request the calculation of Normal and Friction (nodal) Forces.

RIGID-DEFORMABLE AND DEFORMABLE DEFORMABLE CONTACT (CONT.)

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PAT328, Section 3, March 2001 S1-36MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001

Ample Library of Built in Material Models

Includes Composite, Damping and Failure Materials

ELASTIC, PLASTIC, HYPERELASTIC, CREEP AND VISCOELASTIC MATERIAL MODELS

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PAT328, Section 3, March 2001 S1-37MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001

0D (Point) Elements1D (Bar) Elements2D Solid (Continuum) Elements2D Shell Elements3D Solid (Continuum) Elements

0-D, 1-D, 2-D and 3-D Elements may be CombinedUser Control on Integration Methods

Example: First modal shape of beam model combining element types

LARGE ELEMENT LIBRARY

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PAT328, Section 3, March 2001 S1-38MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001

Fracture Mechanics

Design Sensitivity and Optimization

Parallel Processing using Automatic Domain Decomposition

Not Supported by MSC.Patran 2001

Automatic Global and Local Adaptive Remeshing

User Subroutines

Restart capabilities

Parallel Processing using Manual DomainDecomposition

ADVANCED SOLUTION AND MODELING FEATURES

Supported by MSC.Patran 2001

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PAT328, Section 3, March 2001 S1-39MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001

Automatic MPC

LOCAL ADAPTIVE REMESHING

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PAT328, Section 3, March 2001 S1-40MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001

GLOBAL ADAPTIVE REMESHINGAvailable for 2D only

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PAT328, Section 3, March 2001 S1-41MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001

In many cases all model definition, analysis submittal and results evaluation can be

done thru MSC.PATRAN and driven via the graphical user interface.

MSC.PATRAN on-line help facility includes

documentation for all GUI forms and topics.

WHERE ON MSC.PATRAN?

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PAT328, Section 3, March 2001 S1-42MAR120, Lecture 4, March 2001MAR120, Section 1, December 2001


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