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

Element distortion refers to the difference between the shape of a standard element (a perfect cube in the case of a hexahedral element for example) and the shape of the element in the mesh. The transformation between the standard element and the element in the mesh is called mapping. Mapping functions (Q) establish a relationship between the global coordinates of the element in the mesh (x,y,z) and the local coordinates of the standard element (ξ,η,ζ).

For example, the image below shows how a tetrahedral element in the mesh (right) is mapped from its standard shape (left): In traditional FEA packages, where low order (e.g. p=1 or 2) approximations of the displacement field are used, the effect of distorted elements (or more specifically, distorted tetrahedral elements) is discouraged due to the negative impact of distortion on the quality of the finite element solution. Lower-order implementations are very susceptible to element distortions; as a result, meshes are typically dense.

The implementation of the finite element method in StressCheck is less susceptible to element distortion because the mapping functions and the functions used for the approximation of the displacements are independent. In practice, the elements implemented in StressCheck can be reasonably distorted (e.g. vertex angles in the range 5 ≤ θ ≤ 175, where θ is the solid angle at each corner of the element) and still be acceptable for detailed analysis. Thus, observation of element distortion should not automatically prompt the user to refine the mesh.

However, if the element is too distorted, such as a tetrahedral element with a vanishing angle, the mapping function Q becomes ill-conditioned and its negative impact on solution quality is only partially compensated by the higher polynomial order of approximation of the displacement functions. Since it is not possible to know a-priori the effect of distortion on the quality of the approximation, StressCheck provides for the extraction of every engineering quantity of interest as a function of the number of degrees of freedom (DOF) to check for convergence. Please refer to this article for the recommended quality control procedures in StressCheck.

The following examples demonstrate how auto-meshed tetrahedral elements with reasonable distortion may be used in StressCheck to perform 3D high-quality detailed stress analyses:

• StressCheck Demo: Aircraft Keel Beam Stress Analysis
• StressCheck Demo: 3D Tie Rod Stress Concentration Factor Study

Note: In 3D the use of hexahedral or pentahedral elements is always preferred over tetrahedral elements, as they are more computationally efficient. A user may hand-mesh any part (or automatically mesh constant thickness parts by extrusion) with geometrically mapped hexahedral or pentahedral elements if auto-mesh generated tetrahedral elements are too distorted and convergence checks following p-extension indicate that the errors are larger than what is considered acceptable. For reference, many 3D examples in the StressCheck Handbook folders (Edit > Handbook, click Activate Open File Dialogue to browse the Handbook library) are hand-meshed with hexahedral and/or pentahedral elements.

For complex geometry where the use of automatic mesh generation is the only practical option, local mesh refinement must be used to eliminate highly distorted elements in the regions of primary interest.

Threading and Processing

Much of the current StressCheck Professional code base is single-threaded, though over time ESRD is working to improve as much of the code as possible to be multi-threaded. Therefore, the current release of StressCheck is more multi-threaded than previous releases, and less multi-threaded than it will be in future releases. Additionally, the current release of StressCheck does not yet support parallel/multi-core processing.

CPU Core Allocation

Regarding the number of cores, the Windows OS handles all the core allocation for StressCheck processes. It is possible to set processor affinity for specific applications such as StressCheck: https://www.tekrevue.com/tip/restrict-apps-cpu-cores-processor-affinity/