Why is the p-version FEM important?
The finite element method (FEM) provides an approximate solution. In engineering practice it is important to know not only the information one wishes to compute but also to have an indication about the size of error of FEM approximation. The p-version FEM makes it convenient and efficient to obtain error estimates in terms of the data of interest.
When was the p-version FEM developed?
Research on the p-version dates back to the late 1960's. Many important advances occurred in the 1970's. The theoretical basis was established in 1981 and optimal meshing strategies appropriate for the p-version were developed during the period 1984-1985. For details, refer to Szabo and Babuska, Finite Element Analysis, John Wiley & Sons, Inc. (1991). Beginning in 1985, these developments were made available for use in professional practice. The p-version FEM is a more recent technology than the h-version, the development of which began in the late 1950's.
Are error estimation procedures available in h-version codes?
Most h-version software programs offer some form of adaptive capability. The theory of adaptive mesh construction was developed in the 1970's by Babuska and Rheinboldt. The objective of an h-adaptive process is to obtain a sequence of finite element meshes in such a way that the error measured in energy norm is minimal, or nearly minimal, for each mesh. Subsequently, Zienkiewicz and Zhou proposed an adaptive scheme, variants of which have been implemented in h-version programs. In general, h-version programs do not provide convenient and reliable means for making an assessment of the quality of computed information.
Does the p-version have clear advantages over the h-version?
Yes. For typical design problems in mechanical and civil engineering practice the errors of approximation are reduced at an exponential rate when the number of degrees of freedom are increased, provided that the finite element mesh is properly constructed. The h-version can provide algebraic convergence rates only. This makes error control much more effective in the p-version. Furthermore, a converging sequence of solutions is much more naturally and conveniently obtained with the p-version than with the h-version. This makes it feasible to employ quality control procedures in the setting of practical engineering decision-making processes.
Are there classes of problems which can be handled by the p-version but not the h-version?
In principle, any problem which can be solved by the h-version can be solved by the p-version, and vice-versa. The h-version (p fixed at 1 or 2) is a subset of the more general p-version for which shape functions are approximated by higher order polynomials (up to p=8). The use of high-order p elements makes it possible to model and solve with high fidelity, classes of problems comprised of very thin domain. For example, the p-version has clear and substantial advantages for laminated composites and adhesively bonded joints, both of which require elements with very large aspect ratios. Other classes of problems include fracture mechanics and structural plates and shells. READ MORE...
What are the advantages of StressCheck over other FEA programs?
There are several important advantages. A key advantage relates to the quality of the computed information. StressCheck is the only FEA software in existance today which was designed for controlling both the errors of discretization and idealization. The errors of discretization are the errors controllable by the finite element mesh and the polynomial degree of the element shape functions. The errors of idealization are the errors associated with the restrictions incorporated in mathematical model (e.g. assumptions about material properties, boundary conditions, etc.). StressCheck provides valuable feedback that helps us answer the questions:
- Am I solving the equations right? Control errors associated with the numerical method (FEA), and
- Am I solving the right equations? Control errors associated with modeling (idealization).
Without control of both types of errors, it is impossible to validate the model.
Another competitive advantage is StressCheck's unique parametric handbook framework which enables the use of sophisticated FEA technology by designers and non-FEA experts. Any attribute of the model (geometry, material properties, and boundary conditions) can be cast in parametric form and rules are built in to ensure the integrity and validity of the model. This framework allows experts to develop parametric finite element models to which the solution and extraction procedures are predefined for the specific problem. Once validated, these models are entered into a handbook library and deployed to the design team.