Elements
Zebulon makes a clear distinction between what is an element, what
is a material, and what is the element geometry. Element geometries contain
only information about the number of nodes, dimension, thickness, degree
of interpolation, degree of integration, and so on. Zebulon of course
has a complete set of element geometries for 1D, axisymmetric, planar 2D,
axisymmetric shells, beams, 3D shells, and solids.
Materials are attributes of any element, and are therefore independently
applied. There is no concept therefore of a linear elastic element, or a
plastic one.
Most importantly here is the definition of the element itself -
only variational statement or formulation of the internal reaction.
Such a distinct factorization of tasks ensures compatibility and
robustness of implementation. Some examples are given below.
Mechanical elements
- Small deformation elements are available using either
the initial configuration or in an updated form which integrates
over the current time step volume (at the end of an increment).
This element is available for all geometries, and all material behaviors
with the "primal" variable of eto (symmetric small strain tensor),
and "dual" variable sig (Cauchy stress).
- Plane stress Plane stress elements have an additional degree
of freedom at each Gauss point for the eto33 strain component, which
will be solved such that sig33=0. This increases the number of
solution variables, but does not enlarge the front size appreciably
so it is still computationally efficient. For single element solutions
the cost is approximately 1.2, and diminishes as the problem size increases.
the eps33 variable can also be set to a uniform value throughout a mesh
to give generalized plane strain. This method furthermore allows very complex
material laws to run in plane stress without modification.
- 2.5 D
The 2.5 dimension elements in Zebulon have 6 degrees of freedom per node.
These include 3 rotations. The material law is evaluated using 6 components
of stress and strain.
- Total Lagrangian
are available in a classical formulation.
- Total Lagrangian Incompressible
elements are available with either penalty or mixed formulations,
and have reduced interpolation for the volumetric dof.
- Updated Lagrangian
Updated Lagrangian formulations are available. Material behaviors
must be "modified" with a transformation layer which implements
the transformation from a F-sig primal-duel variable to a E-S
local material primal-dual couple. Integrated rotation (Jaumann) or
polar decomposition based (Green-Naghdi) corotational formulations are
available.
- Mindlin shell
formulations in Zebulon are designed especially in the interest of
being able to handle arbitrary material behaviors. For anisotropic
behavior such as the single crystal, full 3D tensors (6 component)
are required because even simple tension creates 3D strain (the graphic at
right for example). The element
employs a similar philosophy as the plane stress element to enforce the
shell normal stress equal to zero. The shell also has a generalized
section property, where any number of "elements" with any number of
integration points can be assigned through the thickness.
- Springs
Several spring formulations are available, with various "spring behaviors."
One node springs are available which can apply a force against any displacement.
with very low modulus, these "springs" are useful to stabilize
free bodies before contact occurs.
- Periodic elements
Periodic elements are available to apply mean states of
mixed uniform strain or stress through a periodic geometry.
- Cosserat continuum
The cosserat continuum is a higher order continuum where "micropolar"
rotations exist and are correlated to alignment of material microstructures.
Because the stress and strain tensors are not symmetric, special material
behaviors are required. Currently elastic and classical plasticity is
available, while a much broader class of materials are under development.
- Representative volume elements
are special element formulations with the full strain tensor as
degrees of freedom. The element is used in 1-element meshes as a means
to simulate material behavior under mixed stress/strain state loading.
Its functionality is similar to the simulation module, except it uses
iterations to solve the system.
Other elements
- Thermal continuum
Thermal elements are available with either stationary or transient
solutions. The thermal heat flux and capacity term are determined
from the thermal behavior selected, which can include anisotropic
terms, phase change (latent heat), etc.
- Diffusion continuum
is a classical diffusion element.
- Diffusion multi-variable with phase change
is an element specially formulated for problems with phase change, and
multiple diffusing variables. Diffusion DOFs are divided by solubility
to allow discontinuities in C for phase boundaries.
- Viscous fluid (under development)
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