The P-Finite Element Integrated Thermal-Structural Program (PITS 2D) solves the equations for the combined effects of thermal expansion and elasticity in a two-dimensional or an axisymmetric three-dimensional structure. The name of this program reflects the use of p-version finite elements, in contradiction to the traditional use of h-version finite elements. In comparison with older finite-element programs that utilize the h-version, PITS 2D predicts physical response more accurately in the presence of large gradients.

A p-version finite element in PITS 2D (see figure) can be either a quadrilateral or a triangle with concave and/or convex curved sides. A quadrilateral element is defined by eight nodes: four of them define the corners, and each of the remaining four nodes defines the approximate midpoint of a curved side. Similarly, a triangular element is defined by six nodes, three lying at the corners and each of the three remaining nodes lying at about the midpoint of a curved side.

In an h-version finite element, the displacements, stresses, strains, temperature, and other physical quantities are represented by trial first-order functions of spatial coordinates. The solution that one seeks is embodied in the coefficients of the functions. In a p-version finite element, the physical quantities are represented by polynomials that can be of orders higher than first. Thus, the polynomials of the p-version can fit physical data points more smoothly and accurately than can the linear functions of the h-version. In a given case, the error in the solution can be made smaller by using a polynomial of higher order. In PITS 2D, the polynomials can be of any order from first to eighth.

Examples of phenomena that can be analyzed by use of the current version of PITS 2D include stresses and/or temperatures in two-dimensional or axisymmetric three-dimensional structures made of isotropic or orthotropic materials, bending and buckling in thick plates made of isotropic or orthotropic materials, torsion in beams, and potential flow of fluid around a body. Nonlinear effects of thermal radiation (without stress) can be analyzed in cases of isotropy or orthotropy; nonlinear effects of thermal radiation (with stress) can be analyzed in cases of isotropy. Boundary conditions on displacement, stress, temperature, thermal, flux, and convection can include quadratic spatial variations.

PITS 2D is based largely on an older finite-element program called "BUCKY." It was developed and tested on computers running the Linux, HP-UX, and SunOS operating systems. A standard Unix Fortran 77 compiler and linker are necessary for execution.

This work was done by James P. Smith of Johnson Space Center. MSC-22832

##### NASA Tech Briefs Magazine

This article first appeared in the November, 1999 issue of NASA Tech Briefs Magazine.

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