Nonlinear finite element analyses confront users with many choices. An understanding of the fundamental concepts of
nonlinear finite element analysis is necessary if you do not want to use the finite element program as a black box.
The purpose of this manual is to describe the numerical methods included in Radioss.
Kinematic constraints are boundary conditions that are placed on nodal velocities. They are mutually exclusive for each degree
of freedom (DOF), and there can only be one constraint per DOF.
The stability of solution concerns the evolution of a process subjected to small perturbations. A process is considered
to be stable if small perturbations of initial data result in small changes in the solution. The theory of stability
can be applied to a variety of computational problems.
A large variety of materials is used in the structural components and must be modeled in stress analysis problems.
For any kind of these materials a range of constitutive laws is available to describe by a mathematical approach the
behavior of the material.
Explicit scheme is generally used for time integration in Radioss, in which velocities and displacements are obtained by direct integration of nodal accelerations.
The performance criterion in the computation was always an essential point in the architectural conception of Radioss. At first, the program has been largely optimized for the vectored super-calculators like CRAY. Then, a first parallel
version SMP made possible the exploration of shared memory on processors.
Kinematic constraints are boundary conditions that are placed on nodal velocities. They are mutually exclusive for each degree
of freedom (DOF), and there can only be one constraint per DOF.
A rigid body is defined by a master node and its associated slave nodes. Mass and inertia may be
added to the initial master node location. The master node is then moved to the center of mass,
taking into account the master node and all slave node masses. Figure 1 shows an idealized rigid body.
Rigid Body Mass
The mass of the rigid body is calculated by:(1)
The rigid body's center of mass is defined by:(2)
(3)
(4)
Where,
Master node mass
Slave node masses
, ,
Coordinates of the mass center
Rigid Body Inertia
The six components of inertia of a rigid body are computed by:(5)
(6)
(7)
(8)
(9)
(10)
Where,
Moment of rotational inertia in the
direction
Master node added inertia
Rigid Body Force And Moment
Computation
The forces and moments acting on the rigid body are calculated by:(11)
(12)
Where,
Force vector at the master node
Force vector at the slave nodes
Moment vector at the master node
Moment vector at the slave nodes
Vector from slave node to the center of mass
Resolving these into orthogonal components, the linear and rotational acceleration may be computed as:
Linear Acceleration(13)
Rotational Acceleration(14)
(15)
(16)
Where,
Principal moments of inertia of the rigid body
Rotational accelerations in the principal inertia frame (reference frame)
Rotational velocity in the principal inertia frame (reference frame)
Moments in the principal inertia frame (reference frame)
Time Integration
Time integration is performed to find velocities of the rigid body at the master
node:(17)
(18)
Where, is the linear velocity vector. Rotational velocities are computed
in the local reference frame.
The velocities of slave nodes are computed by:(19)
(20)
Boundary Conditions
The boundary conditions given to slave nodes are ignored. The rigid body has the boundary conditions given to the master node only.
A kinematic condition is applied on each slave node, for all directions. A slave node is not allowed to have any other kinematic conditions.
No kinematic condition is applied on the master node. However, the rotational velocities are computed in a local reference frame. This reference frame is not compatible with all options imposing rotation such as imposed velocity, rotational, rigid link.
The only exception concerns the rotational boundary conditions for which a special treatment is carried out. Connecting shell, beam or spring with rotation stiffness to the master node, is not yet allowed either.