Bounding Volume Hierarchy and Non-Uniform Rational B-Splines for Contact Enforcement in Large Deformation Finite Element Analysis of Sheet Metal Forming

Abstract

This dissertation presents a robust and efficient algorithm to enforce the unilateral contact condition, with friction, between a deformable body, modeled with a finite element mesh, and rigid surfaces modeled with a set of independent Non-Uniform Rational B-Splines (NURBS) patches. This algorithm is applied to the simulation of the stamping process.

The NURBS patches provide a surface that is accurate, smooth and easy to modify.The contact detection algorithm consists in building a series of surface envelopes called Axis Aligned Minimum Bounding Boxes (AAMBB) that cover the interior of the tools and are organized in hierarchical trees called Bounding Volume Hierarchy (BVH). Efficient tree traversal algorithms are used to build two contact detection tests. The first test inexpensively detects body points that do not penetrate the tools. If a point cannot be guaranteed to not penetrate the tool, its closest point projection on an individual NURBS patch is computed with a fast optimization algorithm that allows for solutions on the edges of a NURBS patch. The second test guarantees that the closest point projection is a global solution to the minimization of the distance between a body point and the entire surface of a tool. The impenetrability condition is enforced with a augmented Lagrangian method. Further, the enforcement of the Coulomb friction law guarantees that the friction force is in the direction opposite to the trajectory of the point and in the tangent plane of the surface. A large deformation elastoplastic constitutive law is used. The deformation gradient is decomposed multiplicatively into its elastic and plastic parts. The hyperelastic model and the plastic strain rate, based on the Lie derivative, guarantee the frame invariance of the constitutive behavior. The deformable body is modeled with solid finite elements with selective integration of the pressure component of the stress to avoid volumetric locking. The resulting system of nonlinear equations is solved with a dynamic relaxation method with adaptive parameters.

To validate the approach, numerical results are compared with analytical and experimental results. The proposed contact algorithm is shown to be four times faster than a brute force method on a numerical example.