Introduction to MPC Equations and Rigid Elements Glenn Grassi, MSC Software May 2010
Agenda •
MSC.Nastran MSC.Nastran Set Definition Definitions s
•
What are MPC Equations
•
Forming and using MPC Equations
•
Rigid Elements in MSC.Nastran
•
Q&A Special thanks to Lanc La nce e Proc Procto torr Jim Jim Swa wan n Jack Jack Ca Cast stro ro for their contributions to this presentation.
MSC/MD Nastran Set Definitions •
•
Each degree of freedom of an MSC/MD Nastran analysis model is defined as being a member of a “user -set” These set notations, such a G-set, M-set, S-set, etc are identifications and classifications of how each degree of freedom (dof) in the analysis participates in the solution sequence
MSC/MD Nastran Set Definitions •
•
•
When a GRID entry is place in the bulk data section there are (6) dof’s added into the model Other entries such as SPOINT or EPOINT, for example, will add (1) dof in the model The collection of all dof’s that are entered into the model will initially be label as belonging to the G-set
MD Nastran Set Definitions •
•
The dof’s that are developed are initially placed into the G-set (Global set) in numerical order according to their ID number If the follow bulk data entries were defined the G-set would be defined as containing 13 dof’s – GRID,1,,0.,0.,0. – GRID,20,,1.,0.,0. – SPOINT,2
MD Nastran Set Definitions •
•
As the solution proceeds these dof’s may be relabeled and transferred to other sets. If dof’s are constrained using multipoint constraint (MPC) entries or rigid elements (RBAR, RBE2, RBE3, etc) then some dof’s are relabeled as now belonging to the M-set (Dependent Set)
MD Nastran Set Definitions •
•
The remaining dof’s that are not defined in the M-set would then be relabeled as belonging to the N-set (Independent Set)
Once a dof is labeled a belong to the M-set it cannot be relabeled again. This is know as a “mutually exclusive” set definition. In this example dof’s 7 and 9 belong to the M-set and cannot be redefined.
MD Nastran Set Definitions •
•
The dof’s in the N-set can be further reduced by adding single point constraints (SPC, AUTOSPC, PS) into the model.
Dof’s 3 and 8 now joint dof’s 7 and 9 as being “mutually exclusive”
MD Nastran Set Definitions •
If a dof was incorrectly specified in the M-set and the S-set then a Fatal Message (2101) would occur.
MD Nastran Set Definitions •
The dof’s that are not constrained will be relabeled as belonging to the F-set (Free).
MD Nastran Set Definitions As the analysis progresses, further set reductions and relabeling are possible
•
•
The original 13 dof problem has been trimmed down to a 5 dof problem
MD Nastran Set Definitions •
A complete description of user sets can be found in the MSC/MD Nastran Reference Manual and Dynamic User Guide.
MD Nastran Set Definitions •
A printout of the user set definitions can be obtained by including PARAM,USETPRT,2 in the analysis.
MPC Equations A MultiplePointConstraint Equation (MPC) is a linear relationship between two or more degrees of freedom that are expressed in the form
•
Σ j R j u j = 0
Where •
•
u j = any degree of freedom defined by a grid point or an spoint R j = user-defined scale factor
MPC Equations •
Multipoint constraints have many important practical applications and can be used to Tie GRIDs together
•
Determine relative motion between GRIDs
•
Maintain separation between GRIDs
•
Determine average motion between GRIDs
•
Model bell-crank or control system
•
Units conversion UY6 = UY7
MPC Equations Σ j R j u j = 0
+
+
M-set N-set
= 0
MPC Equations •
Simple example of an MPC Equation.
Y6 = Y7
MPC Equations •
Y6 = Y7
•
0 = Y7 - Y6
0 = - Y6 + Y7
or
Independent dof Dependent dof (first one listed) •
Add coefficients 0 = 1.0 * Y7 – 1.0 * Y6
MPC
1
7
2
1.0
6
2
-1.0
MPC Equations
MPC Equations
MPC Equations •
By including MPCFORCE=ALL the MPC forces acting on the grids can be printed.
MPC Equations
Angle ?
MPC Equations
SPOINT
100
SPOINT
200
300
SPOINT
401
THRU
430
•
SPOINT has an ID for a single dof
•
ID must be unique (cannot conflict with grids, epoints, etc)
•
There are no directions associated with the SPOINT – dof 0 is usually used in MPC equations – dof 1 can also be used
(cannot use 2 thru 6)
MPC Equations •
•
Calculate the relative angle in radians between GRID 6 and GRID 7 by introducing an SPOINT 100 Calculate the relative angle in degrees between GRID 6 and GRID 7 by introducing an SPOINT 200
•
MPC equation: SPOINT100 = RZ7 - RZ6 {SPOINT100 -
•
MPC equation: SPOINT200 = SPOINT100 x 57.2958
RZ7 + RZ6 = 0
}
{SPOINT200 - SPOINT100 x 57.2958 = 0 }
MPC + MPC
1 1
100
0
1.0
7
6
6 200
6 0
1.0 1.0
100
0
-1.0 -57.2958
MPC Equations
MPC Equations •
SPOINT 100 = 7.238917E-03 (radians)
•
SPOINT 200 = 4.147595E-01 (degrees)
How to enforce a 5.0 degree angle
MPC Equations MPC = 1 Begin Bulk $-------2-------3-------4-------5-------6-------7-------8-------9-------0------SPOINT
100
200
MPC
1
7
2
1.
6
2
-1.
MPC
1
100
1
1.0
7
6
-1.0
6
6
1.0
200
0
1.0
100
0
-57.2958
+ MPC
1
$-------2-------3-------4-------5-------6-------7-------8-------9-------0-------
MPC Equations MPC = 1 SPC = 1 Begin Bulk $-------2-------3-------4-------5-------6-------7-------8-------9-------0------SPOINT
100
200
MPC
1
7
2
1.
6
2
-1.
MPC
1
100
1
1.0
7
6
-1.0
6
6
1.0
200
0
1.0
100
0
-57.2958
+ MPC
1
$-------2-------3-------4-------5-------6-------7-------8-------9-------0------SPC
1
200
0
5.0
MPC Equations MPC = 1 SPC = 1 Begin Bulk $-------2-------3-------4-------5-------6-------7-------8-------9-------0------SPOINT
100
200
MPC
1
7
2
1.
6
2
-1.
MPC
1
100
1
1.0
7
6
-1.0
6
6
1.0
200
0
1.0
100
0
-57.2958
+ MPC
1
$-------2-------3-------4-------5-------6-------7-------8-------9-------0------SPC
1
200
0
5.0
MPC Equations MPC = 1 SPC = 1 Begin Bulk $-------2-------3-------4-------5-------6-------7-------8-------9-------0------SPOINT
100
200
MPC
1
7
2
1.
6
2
-1.
MPC
1
100
1
1.0
7
6
-1.0
6
6
1.0
200
0
1.0
100
0
-57.2958
+ MPC
1
$-------2-------3-------4-------5-------6-------7-------8-------9-------0------SPC
1
200
0
5.0
MPC Equations MPC = 1 SPC = 1 Begin Bulk $-------2-------3-------4-------5-------6-------7-------8-------9-------0------SPOINT
100
200
MPC
1
7
2
1.
6
2
-1.
MPC
1
100
1
1.0
7
6
-1.0
6
6
1.0
100
0
200
0
1.0
+ MPC
1
-57.2958
$-------2-------3-------4-------5-------6-------7-------8-------9-------0------SPC
1
200
0
5.0
MPC Equations MPC = 1 SPC = 1 Begin Bulk $-------2-------3-------4-------5-------6-------7-------8-------9-------0------SPOINT
100
200
MPC
1
7
2
1.
6
2
-1.
MPC
1
100
1
1.0
7
6
-1.0
6
6
1.0
100
0
200
0
1.0
+ MPC
1
-57.2958
$-------2-------3-------4-------5-------6-------7-------8-------9-------0------SPC
1
200
0
5.0
MPC Equations MPC = 1 SPC = 1 Begin Bulk $-------2-------3-------4-------5-------6-------7-------8-------9-------0------SPOINT
100
200
MPC
1
7
2
1.
6
2
-1.
MPC
1
100
1
1.0
7
6
-1.0
6
6
1.0
200
0
1.0
100
0
-57.2958
+ MPC
1
$-------2-------3-------4-------5-------6-------7-------8-------9-------0------SPC
1
200
0
5.0
MPC Equations MPC = 1 SPC = 1 Begin Bulk $-------2-------3-------4-------5-------6-------7-------8-------9-------0------SPOINT
100
200
MPC
1
7
2
1.
6
2
-1.
MPC
1
7
6
-1.0
100
0
1.0
6
6
1.0
200
0
1.0
100
0
-57.2958
+ MPC
1
$-------2-------3-------4-------5-------6-------7-------8-------9-------0------SPC
1
200
0
5.0
MPC Equations MPC = 1 SPC = 1 Begin Bulk $-------2-------3-------4-------5-------6-------7-------8-------9-------0------SPOINT
100
200
MPC
1
7
2
1.
6
2
-1.
MPC
1
7
6
-1.0
100
0
1.0
6
6
1.0
200
0
1.0
100
0
+ MPC
1
-57.2958
$-------2-------3-------4-------5-------6-------7-------8-------9-------0------SPC
1
200
0
5.0
MPC Equations MPC = 1 SPC = 1 Begin Bulk $-------2-------3-------4-------5-------6-------7-------8-------9-------0------SPOINT
100
200
MPC
1
7
2
1.
6
2
-1.
MPC
1
7
6
-1.0
100
0
1.0
6
6
1.0
100
0
-57.2958
200
0
1.0
+ MPC
1
$-------2-------3-------4-------5-------6-------7-------8-------9-------0------SPC
1
200
0
5.0
MPC Equations MPC = 1 SPC = 1 Begin Bulk $-------2-------3-------4-------5-------6-------7-------8-------9-------0------SPOINT
100
200
MPC
1
7
2
1.
6
2
-1.
MPC
1
7
6
-1.0
100
0
1.0
6
6
1.0
100
0
-57.2958
200
0
1.0
+ MPC
1
$-------2-------3-------4-------5-------6-------7-------8-------9-------0------SPC
1
200
0
5.0
MPC Equations
Enforced a 5.0 degree angle
MPC Equations MPC = 1 Begin Bulk $-------2-------3-------4-------5-------6-------7-------8-------9-------0------SPOINT
100
200
MPC
1
7
2
1.
6
2
-1.
MPC
1
100
1
1.0
7
6
-1.0
6
6
1.0
200
0
1.0
100
0
-57.2958
+ MPC
1
$-------2-------3-------4-------5-------6-------7-------8-------9-------0-------
MPC Equations MPC = 400 Begin Bulk $-------2-------3-------4-------5-------6-------7-------8-------9-------0------SPOINT
100
200
MPCADD
400
1
2
MPC
1
7
2
1.
6
2
-1.
MPC
2
100
1
1.0
7
6
-1.0
6
6
1.0
200
0
1.0
100
0
-57.2958
+ MPC
3
3
$-------2-------3-------4-------5-------6-------7-------8-------9-------0-------
MPC Equations Dependent DOF coefficient = -1.0 (pre-defined in MSC.Patran) 0 = 1.0 * Y7 - 1.0 * Y6 ( original ) 0 = -1.0 * Y7 + 1.0 * Y6 ( modified
Y6 = Y7
x -1.0)
MPC Equations MSC.Patran does not define SPOINT’s SPOINT’s – Use Create/Node/Edit
Constrain all dof’s except except the dof = 1
MPC Equations MPC = 1 Begin Bulk $-------2-------3-------4-------5-------6-------7-------8-------9-------0------SPOINT
100
200
MPC
1
7
2
1.
6
2
-1.
MPC
1
100
0
1.0
7
6
-1.0
6
6
1.0
200
0
1.0
100
0
-57.2958
+ MPC
1
$-------2-------3-------4-------5-------6-------7-------8-------9-------0------MPC = 1 Begin Bulk $-------2-------3-------4-------5-------6-------7-------8-------9-------0------GRID
100
23456
GRID
200
23456
MPC
1
7
2
1.
6
2
-1.
MPC
1
100
1
1.0
7
6
-1.0
6
6
1.0
200
1
1.0
100
1
-57.2958
+ MPC
1
$-------2-------3-------4-------5-------6-------7-------8-------9-------0-------
MPC Equations
MPC Equations 2
Use of MPC to tie GRIDs together 1
0 = -UX2 + UX1 0 = -UY2 + UY2 0 = -UZ2 + UZ2
MPC, 535, 2, 1, -1.0, 1, 1, +1.0 MPC, 535, 2, 2, -1.0, 1, 2, +1.0 MPC, 535, 2, 3, -1.0, 1, 3, +1.0
0 = -
X2
+
X1
MPC, 535, 2, 4, -1.0, 1, 4, +1.0
0 = -
Y2
+
Y1
MPC, 535, 2, 5, -1.0, 1, 5, +1.0
0 = -
Z2
+
Z1
MPC, 535, 2, 6, -1.0, 1, 6, +1.0
MPC Equations MPC used to Maintain Separation •
Enforce a separation between GRIDs – Similar to using a gap – Changes which DOF are dependent/independent
– Example:
1
– Initially 1” apart – Keep separation = 0.25”
1.0”
0.25”
2
MPC used to Maintain Separation 1 total shrink =
1.00
2.0 x -0.375 = -0.75
0.25 desired initial
Relative motion:
2
U1000 = U1 – U2
U1 = U2 + (desired – initial)
0 = -U1 + U2 + U1000
SPOINT,1000 MPC, 535, 1, 2, -1.0, +, , 1000, 1, +1.0 SPC, 2002, 1000, 1, -0.75
2, 2, +1.0
MPC Equations Use of MPCs for AVERAGE Motion •
Determine average motion of DOFs
U1000 = (U1+ U2 + U3 + U4 +U5 +U6)/6
4
5 3 6 2
1
0 = -6*U1000 + U1+ U2 + U3 + U4 +U5 +U6
MPC Equations MPCs as Bell-crank or Control System •
Output of 1 DOF scales another 1 5 6 . 1
U2 = U1 /1.65 0
2
= -1.65*U2 + U1
MPC
SID
G1
C1
A1
G2
C2
A2
MPC
535
2
1
-1.65
1
1
+1.0
Rigid Elements •
•
•
•
The multipoint constraint, or MPC entry, provides the capability to model rigid bodies and to represent other relationships which can be treated as rigid constraints. The MPC entry provides considerable generality but lacks user convenience since the user must supply all of the coefficients in the equations of constraint To enhance user convenience, nine rigid body elements (R-Type) are available in MSC.Nastran. These elements require only the specification of the degrees-of-freedom that are involved in the equations of constraint. All coefficients in these equations of constraint are calculated internally in MSC.Nastran.
Rigid Elements
Not Exactly Rigid - Averaging element
RBEs and MPCs •
Not necessarily “rigid” elements – Working Definition:
The motion of a DOF is dependent on the motion of at least one other DOF
Motion at one GRID drives another •
Simple Translation
X motion of Green Grid drives X motion of Red Grid
Motion at one GRID drives another •
Simple Rotation
Rotation of Green Grid drives X translation and Z rotation of Red Grid
Linear RBEs and MPCs
The motion of a DOF is dependent on the motion of at least one other DOF – Displacement, not elastic relationship – Not dictated by stiffness, mass, or force – Linear relationship – Small displacement theory – Dependent v. Independent DOFs – Stiffness/mass/loads at dependent DOF transferred to independent DOF(s)
Small Displacement Theory & Rotations •
Small displacement theory: sin( ) ≈ tan( ) ≈ cos( ) ≈ 1
•
For Rz @ A
TxB B
RzB = RzA= TxB = ( )*LAB TyB = 0 A
X
Y
Commonly used “Rigid” Elements in MSC.Nastran •
Geometry-based – RBAR – RBE2
•
}
Really- rigid “rigid” elements
Geometry- & User-input based – RBE3
•
User-input based – MPC
•
Less Common “Rigid” elements (not covered today) – RBAR1, RJOINT, RROD, RTRPLT, RTRPLT1, RBE1, RSSCON, RSPLINE
Common Geometry-Based Rigid Elements
•
RBAR – Rigid Bar with six DOF at each end
– RBE2
– Rigid body with independent DOF at one GRID, and dependent DOF at an arbitrary number of GRIDs.
The RBAR •
The RBAR is a rigid link between two GRID points – Proper rigid body motion is preserved
The RBAR B
– Most common to have all the dependent DOFs at one GRID, and all the independent DOFs at the other
A
– Can mix/match dependent DOF between the GRIDs, but this is rare – The independent DOFs must be capable of describing the rigid body motion of the element
RBAR
EID
GA
GB
CNA
RBAR
535
1
2
123456
CNB
CMA
CMB 123456
RBAR Example: Fastener •
Use of RBAR to “weld” two parts of a model together:
RBAR
EID
GA
GB
CNA
RBAR
535
1
2
123456
B
A
CNB
CMA
CMB 123456
RBAR Example: Pin-Joint •
Use of RBAR to form pin-jointed attachment
RBAR
EID
GA
GB
CNA
RBAR
535
1
2
123456
B
A
CNB
CMA
CMB 123
RBAR definition in Patran
The RBE2 •
One independent GRID (all 6 DOF)
•
Multiple dependent GRID/DOFs
RBE2 Example •
Rigidly “weld” multiple GRIDs to one other GRID:
RBE2
EID
RBE2
99
GN
CM
GM1 GM2 GM3 GM4 GM5
101 123456
1 1
3 4
101
2
2
3
4
RBE2 Example RBE2
EID
RBE2
99
•
GN
CM
GM1 GM2 GM3 GM4 GM5
101 123456
1
2
3
Note: No relative motion between GRIDs 1-4 ! – No deformation of element(s) between these GRIDs 1
3 4
101
2
4
Common RBE2/RBAR Uses •
RBE2 or RBAR between 2 GRIDs – “Weld” 2 different parts together 6DOF connection – “Bolt” 2 different parts together 3DOF connection •
•
•
RBE2 – “Spider” or “wagon wheel” connections – Large mass/base-drive connection
RBE2 definition in Patran
RBE3 Elements
• Motion at a dependent GRID is the weighted average of the motion(s) at a set of master (independent) GRIDs • NOT a “rigid” element • IS an interpolation element • Does not add stiffness to the structure (if used correctly)
RBE3 Description
RBE3 Description •
•
•
By default, the reference grid DOF will be the dependent DOF Number of dependent DOF is equal to the number of DOF on the REFC field Dependent DOF cannot be SPC’d, OMITted, SUPORTed or be dependent on other RBE/MPC elements – PARAM,AUTOMSET,YES can resolve conflicts
RBE3 Is Not Rigid! •
RBE3 vs. RBE2 – RBE3 allows warping and 3D effects – In this example, RBE2 enforces beam theory (plane sections remain planar)
RBE3
RBE2
RBE3: How it Works? – Applied Forces •
Forces/moments applied at reference grid are distributed to the master grids in same manner as classical bolt pattern analysis • Step 1: Applied loads are transferred to the CG of the weighted grid group using an equivalent Force/Moment
• Step 2: Applied loads at CG transferred to master grids according to each grid’s weighting factor
RBE3: How it Works? – Applied Forces •
Step 1: Transform force/moment at reference grid to equivalent force/moment at the weighted CG of master grids. FA
Reference Grid
MA
CG
e
FCG
CG
MCG
FCG=FA MCG=MA+FA*e
RBE3: How it Works? – Applied Forces •
Step 2: Move loads at CG to master grids according to their weighting values. • Force at CG divided amongst master grids according to weighting factors Wi • Moment at CG mapped as equivalent force couples on master grids according to weighting factors Wi
RBE3: How it Works? – Applied Forces •
Step 2: Continued… F1m
FCG
CG
MCG
F2m
Total force at each master node is sum of... Forces derived from force at CG: Fif = FCG{W / i Wi} Plus Forces derived from moment at CG: 2 2 2 Fim = {McgWir /(W i 1r1 +W2r2 +W3r3 )}
F3m
RBE3: How it Works? – Mass Distribution •
Masses smeared to the master grids similar to forces distribution • Mass is distributed to the master grids with weighting factors • Rotational inertia is transferred to master grids • Reference node inertial force is distributed in same manner as when static force is applied to the reference grid.
Example 1
•
RBE3 distribution of loads when force at reference grid at CG passes through CG of master grids
Example 1: Force Through CG •
•
Simply supported beam • 10 elements, 11 nodes numbered 1 through 11
100 LB. Force in negative Y on reference grid 99
Example 1: Force Through CG
•
Load through CG with uniform weighting factors results in uniform load distribution
Example 1: Force Through CG •
Comments… • RBE3 Require 6 RBMODES •
x rotation DOF is added to satisfy equilibrium
Example 2
•
Force does not pass thru CG of master grids
Example 2: Load not through CG •
The resulting force distribution is not intuitively obvious • Note forces in the opposite direction on the left side of the beam.
Upward loads on left side of beam result from moment caused by movement of applied load to the CG of master grids.
Example 3 •
Use of weighting factors to generate realistic load distribution: 100 LB. transverse load on 3D beam.
Example 3: Transverse Load on Beam •
If uniform weighting factors are used, the load is equally distributed to all grids.
Example 3: Transverse Load on Beam •
The uniform load distribution results in too much transverse load in flanges causing them to droop.
Displacement Contour
Example 3: Transverse Load on Beam •
•
•
Assume quadratic distribution of load in web Assume thin flanges carry zero transverse load Master DOF 1235. DOF 5 added to make RY rigid body motion determinate
Example 3: Transverse Load on Beam •
Displacements with quadratic weighting factors virtually equivalent to those from RBE2 (Beam Theory), but do not impose “plane sections remain planar” as does RBE2.
Example 3: Transverse Load on Beam
•
RBE3 Displacement Contour • Max Y disp=.00685
Example 3: Transverse Load on Beam
•
RBE2 Displacement contour • Max Y disp=.00685
Example 4 •
•
•
Use RBE3 to get “unconstrained” motion Cylinder under pressure Which Grid(s) do you pick to constrain out Rigid body motion, but still allow for free expansion due to pressure?
Example 4: Use RBE3 for Unconstrained Motion •
Solution: • Use RBE3 • Move dependent DOF from reference grid to selected master grids with UM option on RBE3 (otherwise, reference grid cannot be SPC’d) • Apply SPC to reference grid
Example 4: Use RBE3 for Unconstrained Motion •
Since reference grid has 6 DOF, we must assign 6 “UM” DOF to a set of master grids • Pick 3 points, forming a nice triangle for best numerical conditioning • Select a total of 6 DOF over the three UM grids to determine the 6 rigid body motions of the RBE3 • Note: “M” is the NASTRAN DOF set name for dependent DOF
How Do I create UM set in Patran?
Reassign Dependant terms Pick 3 nodes @ approx 120
UX, UY, UZ only Uy, Uz in cyl coord sys is determinate
Example 4: Use RBE3 for Unconstrained Motion •
For circular geometry, it’s convenient to use a cylindrical coordinate system for the master grids. • Put THETA and Z DOF in UM set for each of the three UM grids to determine RBE3 rigid body motion
What is the UM? •
2
1
UM fields can be used to move the dependent DOF away from the reference grid • For Example (in 1-D):
3 99
1
Equation (consider avg x disp of grid 99) Avg motion:
U99x = (U1x + U2x + U3x) / 3
Default MPC:
-3.*U99x + U1x + U2x + U3x =0
Rearrange UM:
U1 + U2 + U3 - 3 * U99 =0
2
3 99
First term in MPC equation is dependent; Same equation, different order
Example 4: Use RBE3 for Unconstrained Motion “UM” Grids
Example 4: Use RBE3 for Unconstrained Motion •
Result is free expansion due to internal pressure. (note: poisson effect causes shortening)
Example 4: Use RBE3 for Unconstrained Motion •
Resulting MPC Forces are numeric zeroes verifying that no stiffness has been added.
–PARAM,AUTOMSET,YES can also be used in many instances instead of UM
RBE3 – Non Uniform Distribution – CHEXA(8)
Coefficients 1.0, 0.5 and 0.25
Coefficients all 1.0
RBE3 – Non Uniform Distribution – CHEXA(8) Stress and Deflection
Coefficients 1.0, 0.5 and 0.25 Correct Stress = 2,500 Correct Disp = 2.5e-3
Max Stress 5,830 Max Disp = 2.86e-3
Coefficients all 1.0
RBE3 – Non Uniform Distribution – CHEXA(20)
Coefficients:
1.0
-2.0
0.5
-1.0
0.25
Coefficients all 1.0
RBE3 – Non Uniform Distribution – CHEXA(20) Stress and Deflection
Correct Stress = 2,500 Correct Disp = 2.5e-3
Coefficients all 1.0
Coefficients: 1.0 -2.0 0.5 -1.0 0.25
Max Stress 11,600 Max Disp = 3.21e-3
RBE3: Additional Reading •
Recommended TANs – – – – –
TAN#: TAN#: TAN#: TAN#: TAN#:
2402 3280 4155 4494 4497
RBE3 - The Interpolation Element. RBE3 ELEMENT CHANGES IN VERSION 70.5, improved diagnostics RBE3 ELEMENT CHANGES IN VERSION 70.7 Mathematical Specification of the Modern RBE3 Element AN ECONOMICAL METHOD TO EVALUATE RBE3 ELEMENTS IN LARGE-SIZE
MODELS
–Visit SimCompanion
http://simcompanion.mscsoftware.com)
