SHOE (Sandia Higher-Order Elements) is a research program to investigate the visualization of higher-order finite-element simulation results. Finite-element simulations typically use low-order (linear or occasionally quadratic) polynomials to approximate solutions of differential equations describing interesting situations.
One example is solid mechanics, which approximates the deformation of a mechanical part in response to imposed forces. These finite-element simulations require the domain (i.e., the mechanical part) to be decomposed into many small regions in order for a low-order polynomial to approximate the true deformation of the part.
By using higher-order polynomials, fewer elements may be needed. For certain types of simulations, reducing the number of high-order finite elements in a polynomial approximation can speed up a computer's calculation time by orders of magnitude.
Since the polynomial approximation is simply a large set of numbers—the coefficients of the polynomials —understanding the results requires visualization. Unfortunately, not many visualization techniques can be applied to higher-order finite elements because existing techniques make assumptions about polynomials that do not hold for higher-order polynomials. This complicates the development and use of simulations that provide higher-order finite elements.
SHOE has been able to advance the state of the art in two ways:
Sandia has also extended the programming interface to perform a more robust and correct sampling of higher-order finite elements when isosurfacing. To their detriment, current visualization techniques create a low-order approximation of the higher-order approximation of the differential equation. In contrast, SHOE simply divides the higher-order finite elements further into regions where the mathematical properties required for proper visualization still hold.
