Accurate and fast algorithms for contour plotting in 2D and 3D domains for finite element analysis (FEA) data

Abstract

Contour plotting is one of the basic operations performed in many engineering analyses where visual inspection of results is to be carried out. Such analyses produce a large amount of data which is cumbersome to interpret in its numerical form. The graphical representation of the numerical data in the form of contour lines in 2D and contour surfaces in 3D makes the analyses more informative and faster. Finite Element Analysis (FEA) is one of such analyses which produces voluminous data and whose graphical representation becomes a necessity. Contour plotting is one method which is used for graphical representation. It substitutes a large amount of numerical data into graphical patterns, which helps to perceive the physical consequences of a calculation. To users, not familiar with the details of FEA, it is possibly one of the best ways to perceive the analysis results. So, once the ‘solver’ determines the element resultants, a post-processor in the form of contour plots is used to graphically display the domain response to the applied loads and boundary conditions. For a user, the accuracy of the analysis results depends on the accuracy of the graphical patterns which are displayed as contour lines or surfaces on the screen. Generally, higher order elements are used for accurate analysis, but are degenerated into linear elements for contour plotting to avoid complexity. This leads to loss of information in the graphical plotting. In some cases, to visualize the solutions effectively, approximations on finer meshes are required. In the present work, an attempt is made to develop accurate and fast algorithms for different meshes used for analysis in FEA. All the algorithms depend upon the contour equations developed using ‘Shape Functions’ for these meshes. The simplest way to plot contours on 2D domain is to use linear interpolation over triangular elements. This method works well only if the accuracy of higher order is not required. Higher order triangular elements are generally degenerated into linear triangular elements for fast contour plotting but on expense of accuracy. Higher accuracy may be desirable in regions where the variation in physical quantity is large or in applications such as imaging. In the present work, an attempt is made to use higher order interpolation using a six-node triangular element. The algorithm traces contours as accurately as the analysis results obtained in FEA. The quadratic interpolation function representing contours is degenerated into various conic sections and the special characteristics of these conic sections are exploited to accurately interpolate the function. After contour segments are traced, they are joined together to form contour lines for fast display on graphical display devices. Quadrilateral element is the second type of 2D element taken for contour plotting. The contour equation is developed over linear and quadratic quadrilateral elements. An attempt is made to enhance the speed of contour generation without compensating with accuracy. In four-node quadrilateral element, the developed contour equation represents a rectangular hyperbola. The contours are joined based on location of asymptotes of this rectangular hyperbola, thus making the algorithm fast. Further, the accuracy of contour is improved by using elementary calculus. Similarly, in eight-node quadrilateral element, an attempt is made to develop an algorithm to join the contours accurately and quickly. In 3D, the problems are mainly analyzed using tetrahedral and hexahedral elements. As discussed in case of 2D, to simplify the 3D contour surface generation, a higher order element is splited into linear elements and linear interpolation of the physical quantity is performed, thus resulting in the linear patches of contour surfaces. Even a hexahedral element is decomposed into 6 four-node tetrahedral elements for contour surface generation, thus leading to the linear interpolation of the physical quantity. In the present work, an attempt is made to develop accurate and fast contour surface plotting algorithm by first deriving the boundary curves and then locating its interior points. The method provides the contour surface as a group of bounding curves over an element. Towards the end, the contour surfaces over a quadratic tetrahedral element are generated. The contour equation is developed using shape functions which represents the various quadrics. The detailed numerical experimentations after implementing the different algorithms are given in respective chapters.