Understanding R-type Elements and Degrees-of-Freedom
7.2 Rigid Element Processing Options
Rigid elements are represented mathematically as a system of linear constraint equations that can be represented in matrix form as:
[RG]{x} = {0}
A constraint equation exists for each dependent degree of freedom (DOF).
When NX Nastran processes RBAR, RBE1, RBE2, RBE3, RROD and RTRPLT rigid elements, the software uses either the linear elimination method or the Lagrange multiplier method. You can select the rigid element processing method that you want by using the RIGID case control command.
• Use RIGID = LINEAR to select the linear elimination method. This is the default method.
• Use RIGID = LAGRAN to select the Lagrange multiplier method.
When determining which rigid element processing method to select, note that:
• Only the Lagrange multiplier method allows for thermal expansion in rigid elements.
• The Lagrange multiplier method may yield a performance improvement for dynamic solutions even though it adds DOF to the problem. This occurs because the Lagrange multiplier method more efficiently processes the very densely populated mass matrices that rigid elements can produce.
For other cases, the linear elimination method is preferred because the artificial stiffness that is added to the model when using the Lagrange multiplier method can produce either ill-conditioned stiffness matrices or overly stiff models.
Linear elimination method
The linear elimination method partitions the system of linear constraint equations into dependent (M-set) and independent (N-set) DOF groupings.
The dependent DOF are then solved for in terms of the independent DOF as follows:
The equation of motion before elimination of dependent DOF is:
The linear elimination method reduces the equation of motion to just the independent DOF as follows:
Note that the mass, stiffness and loads on the dependent DOF (M-set) get distributed onto the independent DOF (N-set). This can result in very densely populated matrices, which is a disadvantage of the elimination method. For example, a single lumped mass distributed onto a large number of grids using an RBE3 element will generate a fully populated mass matrix on the associated DOF. For these cases, the Lagrange multiplier method may improve performance.
Lagrange multiplier method
An alternate approach is the Lagrange multiplier method. Unlike the linear elimination method, NX Nastran does not differentiate between dependent and independent DOF when using the Lagrange multiplier method. Instead, NX Nastran defines a Lagrange multiplier for each constraint equation and then treats the Lagrange multipliers as additional DOF. The resulting equation of motion is:
Although the Lagrange multiplier method adds DOF to the problem rather than removing them, the mass matrix remains very sparse and the stiffness matrix remains relatively sparse.
The problem is that the augmented stiffness matrix is often singular. Because the augmented stiffness matrix is often singular, NX Nastran optionally adds artificial stiffness terms that connect the DOF in each constraint. The artificial stiffness can be thought of as beam elements that span the same DOF as the constraint equations.
With the artificial stiffness added, singularity is eliminated and the equation of motion becomes:
If the constraints are for a rigid element like an RBE2, the artificial stiffness terms do not alter the problem because the relationship between the constrained DOF represents rigidity already.
However, if the constraints are for an RBE3 element, the artificial stiffness terms can produce:
• A numerically ill-conditioned stiffness matrix if the artificial stiffness is too little.
• An overly stiff stiffness matrix if the artificial stiffness is too large.
To control artificial stiffness when RBE3 elements are present, two scale factors, cland cK, are available. The equation of motion with the scale factors included is:
The clscale factor multiplies the constraint equation matrix, [RG], and can be set to help avoid a numerically ill-conditioned stiffness matrix. The cKscale factor multiplies the artificial stiffness terms and can be set to help avoid over stiffening the stiffness matrix.
The values for the scale factors are problem dependent.
• Set the clscale factor so that the terms in the constraint equations are the same order of magnitude as the terms in the stiffness matrix. The clscale factor is set directly by specifying the LMFACT parameter. The LMFACT parameter is applicable to all rigid elements. The default value is 1.0 x 106, which is an appropriate value for most models.
• Set the cKscale factor indirectly using the new LMSTAT and LMDYN parameters. The LMSTAT and LMDYN parameters are applicable to RBE3 elements only.
For a statics solution, specify the LMSTAT parameter, where cKand the LMSTAT setting are related by:
The default value for LMSTAT is 6, which results in cK= 1.0 x 10-6. To completely remove artificial stiffness from the problem, set LMSTAT to -1. When LMSTAT is set to -1, cKis set to zero.
For a dynamics solution, specify the LMDYN parameter, where cKand the LMDYM setting are related by:
The default value for LMDYN is -1. When LMSTAT is set to -1, cKis set to zero and artificial stiffness is completely removed from the problem.
For modal equations, there is no requirement that the stiffness matrix be non-singular. Thus, cK can always be set to zero. However for static solutions, some non-zero value is normally required because a DOF in the M-set which has zero stiffness will be restrained by the AUTOSPC
operation.
Thermal expansion of rigid elements using Lagrange multiplier method
Thermal expansion on the RBAR, RBE1, RBE2, RBE3, RROD and RTRPLT elements is calculated when:
• ALPHA is defined on the rigid element definition.
• The case control RIGID = LAGRAN is defined.
• The grid point initial and load temperatures are defined using the TEMPERATURE case control command on the grids defining the rigid element connectivity.
When thermal strain is included, the right hand side of the equation of motion includes the change in displacement, uT, that results from the thermal loading.
The change in temperature for the thermal strain is the difference between the grid point temperatures selected with the TEMPERATURE(LOAD) and TEMPERATURE(INIT) case control commands. The load and initial temperature values used on RBAR, RROD, and RTRPLT elements is an average calculated from the grid point values. On RBE1, RBE2 and RBE3 elements, an average load and initial temperature value is calculated for each leg of the element using the values on the independent/dependent grid pairs such that each leg can have a different thermal strain if the temperatures vary at the grids.
The rigid element thermal strains are calculated from:
thermal= a(AVGTEMP(LOAD) – AVGTEMP(INIT))
If load or initial temperatures are undefined, they are assumed to be zero.
Rigid element processing example
In this three DOF example, x1and x2are independent, and x3is dependent.
The equation of motion before applying constraints is:
The constraint equation is:
Solving the constraint equation for x3 gives:
Using the elimination method, x3is removed from the equation of motion as follows:
This equation can be easily solved, though it is clear that the mass and stiffness for x3 has been redistributed onto x1and x2.
Now consider the Lagrange multiplier method. The equation of motion now includes an extra term as follows:
Although the Lagrange multiplier method has added DOF to the equation of motion, the sparsity of the matrices is retained.