What is a P-Extension?

The term “p-extension” refers to the process of systematically and hierarchically increasing the polynomial order of the element shape functions on a fixed mesh, thereby increasing simulation degrees of freedom (DOF) and representing more complex displacement fields for the mesh. Conversely, traditional FEA implementations require successive mesh refinements on elements of fixed polynomials, or h-extensions, to demonstrate convergence.

While both p-extensions and h-extensions are capable of converging to the exact solution of a mathematical problem solved by the finite element method, p-extensions will achieve convergence with less computational burden. Via the Wikipedia article on p-FEM:

The theoretical foundations of the p-version were established in a paper published Babuška, Szabó and Katz in 1981^[2] where it was shown that for a large class of problems the asymptotic rate of convergence of the p-version in energy norm is at least twice that of the h-version, assuming that quasi-uniform meshes are used. Additional computational results and evidence of faster convergence of the p-version were presented by Babuška and Szabó in 1982.^[3]

StressCheck is capable of automatically performing p-extensions during the linear solution process, as shown in the below Linear solver tab:

P-extension from p=2 to 8

Note: it is not always necessary to solve to the maximum polynomial level, p=8. Convergence may be assessed after three (3) hierarchically increasing polynomial levels to determine if continuing the p-extension is necessary to achieve the desired solution quality.

This feature makes it very easy to verify convergence of any data of interest, as all runs of increasing DOF are stored for live dynamic extractions of results:

Key Quality Check #4: Peak Stress Convergence (courtesy StressCheck Professional)

For more information and examples of p-extensions in practice:

• Linear Analysis Overview
• What Are the Key Quality Checks for FEA Solution Verification?
• Watch StressCheck Demos of Digital Engineering.com FEA Case Studies
• StressCheck Tutorial: Defining Solution and Extraction Settings for Parametric Models
• StressCheck Tutorial: Setting Up Solution Convergence Criteria
• Our Simulation Technology
• Simulation Technology References