Recent Developments in Isogeometric Analysis for LS-DYNA ®

Isogeometric analysis (IGA) uses the spline functions from CAD (computer-aided design) for analysis with the objectives of 1) reducing the effort of moving from design to analysis, 2) using the actual geometry of the parts from CAD instead of approximating them with polynomials, and 3) obtaining higher order accuracy through the higher order basis functions of CAD. LS-DYNA is the first commercial code to support IGA through the implementation of generalized elements, and then through key word descriptions of patches of B-spline and NURBS elements. Three shell formulations and a solid element formulation are currently available. Additional recently added capabilities, including improved contact, anisotropic material modeling, spotweld models and support for unstructured spline capabilities through the specification of their Bezier extractions. These and other recent additions are described and demonstrated. In addition, some of the difficulties industry has encountered in moving to this new technology are discussed. Introduction to Isogeometric Analysis NURBS are the most commonly used functions in computer aided design (CAD) for representing the geometry of the boundaries of parts. They are popular because they can represent basic geometric shapes exactly with only a small amount of data, while also being able to define very general shapes that are not amendable to being described by simple analytical functions, e.g., car bodies. NURBS are defined on rectangular patches. To represent more general shapes, trimmed NURBS are defined by deleting parts of the patches specified by trimming curves. Isogeometric analysis [1] was introduced in 2005 by Thomas J. R. Hughes and his students, Austin Cottrell and Yuri Bazilevs. Its objective was succinctly summarized in the abstract: “Basis functions generated from NURBS (Non-Uniform Rational B-Splines) are employed to construct an exact geometric model. For purposes of analysis, the basis is refined and/or its order elevated without changing the geometry or its parameterization. ” Isogeometric analysis was originally tested in LS-DYNA through the generalized element formulation introduced and implemented in the code by D. Benson while on sabbatical at the University of Texas, Austin with Tom Hughes in 2007. During the next few years, they were added as part of the standard element library by the current development team of D. Benson, S. Hartmann, A. Nagy, and L. Li. This paper summarizes some of the recent additions made to the IGA capability in LS-DYNA that we believe will be of the greatest interest to the user community. The details of these additions are provided in separate papers. 15th International LS-DYNA® Users Conference Isogeometric Analysis June 10-12, 2018 2 The Importance of Geometry The primary motivation behind isogeometric analysis is the use of the exact geometry for the analysis. The two attributes of NURBS in CAD that provide the geometric flexibility are 1) using them to define the boundary representation and 2) trimming them to arbitrary shapes. Unfortunately, they introduce obstacles to the use of NURBS in analysis. The difficulty of using boundary representations (B-rep) for analysis is the absence of any basis functions on the interior of a solid object for representing the displacement field. This difficulty can be overcome by using boundary element methods (BEM), but that limits the solutions to linear problems. Trimmed NURBS surfaces have their own problems. Their arbitrary shape means that special integration rules need to be formulated for each trimmed element. Stability can also be a problem. By far, the biggest difficulty with them is joining the edges of trimmed boundaries together since their basis functions don’t line up like for untrimmed NURBS patches and standard finite element formulations. The CAD software packages introduce their own set of challenges. At the time that these packages were being developed, no thought was given to the possibility of using the NURBS patches in the future for analysis. Ideally, each part would be represented by a minimal number of patches to maximize the smoothness of the geometric description and the convergence rate of the analysis. However, CAD models are revised over and over again during their lifetime, with the result that patches proliferate and minute gaps in the geometry are filled in with patches that are little more than slivers. These issues not only have a negative effect on analysis, but also on other uses of CAD geometry such as 3-D printing and importing CAD models from one CAD program to another. As a result, there is an entire industry devoted to “healing” CAD models with products such as Elysium’s CADdoctor. The process of generating a mesh from a CAD model is referred to as “mesh generation” and in IGA, it is referred to as “creating an analysis suitable geometry” (ASG). Many of the basis functions used in the ASG are not commonly used in CAD. Since CAD defines solids in terms of two-dimensional boundary representations, even the simplest three-dimensional NURBS function is not found in CAD models. Some of these basis functions allow local refinement of the computational mesh and relatively unstructured meshes in comparison to NURBS. This is a very active area of research in the mathematics and engineering communities.