Linear Analysis in StressCheck

The simplest type of solution to perform in StressCheck is a linear analysis. The linear analysis idealization is based on the following assumptions:

• Small-strain, small-deformation theory of elasticity (infinitesimal strains and rotations)
• Linear relation between stress and strain
• Equilibrium equations written in the original (undeformed) configuration

One of the powerful characteristics of StressCheck is that the linear analysis is just the first step in a hierarchy of analyses. The same model used to perform a linear analysis may be used as the starting point for a nonlinear analysis. It is also possible to use the result of a steady state heat transfer analysis to represent the temperature distribution for a thermal elasticity analysis.

These analysis extensions may be performed without changing the basic finite element model. All that is required is that additional attributes be defined, such as thermal coefficients or nonlinear material properties.

P-Extensions

There are three types of polynomial extensions (i.e. p-extensions) supported by the Linear solver: Upward-p, Downward-p and Adaptive-p. Polynomial levels (i.e. p-levels) for a Linear analysis may range from p=1 to p=8, and are specified via the “p-limits” fields. Note that the range of p-levels assigned to elements from the Linear analysis tab will affect only those elements which are considered to have a Variable p-Discretization attribute. This attribute may be assigned in the p-Discretization input class.

All elements are Variable by default. The p-Discretization attribute for each element may be assigned to be Variable, Uniform, Fixed, or Bounded.

Upward-p

The information required for a Linear analysis are the extension and the range of p-levels used to compute a sequence of solutions (p-limits). By “extension”, we mean the procedure by which the program repeats the analysis at different p-levels, or with different degrees of element refinement, in order to establish a convergence sequence from which an error estimate can be computed.

By default, StressCheck uses an “Upward-p” extension and assigns a constant p-level to all elements, increasing the p-level of all elements uniformly for each solution in the sequence. For example, if you specify upward extension, p=1 to 4, the program will automatically assign p=1 to all elements that do not have a specific p-Discretization assigned in the p-Discretization tab of the Input dialog, for the first solution in the sequence. During the second solution, the p-level for all elements will be set to p=2, and so on.

Downward-p

The Downward-p extension performs the highest p-level first and decreases the p-level until the lowest p-level of the specified range is reached. The most significant difference between Upward and Downward extensions is that during a downward extension the elemental stiffness matrices are saved on disk and reused during subsequent solutions. The matrices can be reused because the lower order terms are embedded in the higher order matrices and do not need to be computed again. This can save a significant amount of computation time.

However, there is a price to be paid for computational efficiency in terms of disk space to save the high order matrices. The stiffness matrix for a single 2D quadrilateral element in elasticity (p=8) requires 4465 terms or 32 thousand bytes. The stiffness matrix for a single 3D hexahedral element in elasticity requires 166,176 terms (p=8), or almost 1 megabyte.

Adaptive-p

Choosing “Adaptive-p” means that the polynomial order of the elements with a Variable or Uniform p-Discretization assignment will be determined based on the values of the error indicator for each run until convergence is obtained. The “Run Limit” input field refers to the maximum number of adaptive runs the program will perform if convergence is not obtained. The “p” input field is to provide the initial p-level for the adaptive p-extension.

Upward Model (Plate Only)

For plate bending analysis, you will also be given the opportunity to perform an analysis during which the hierarchic model is allowed to change, while the p-level is held constant.