Discrete geometric modeling of slender flexible structures for interactive assembly simulation in automotive industry

In industry, there is an increasing demand for using fast (i.e. real time) realistic simulation of slender flexible structures in software tools for CAD, digital mock-up and virtual assembly, which nowadays can only handle rigid geometries. Typical examples of such structures from automotive industry are cables, tubes and hoses. For slender one-dimensional deformable objects, the theory of Cosserat rods provides a suitable framework for physically correct simulation of deformations like stretching bending and twisting. The static equilibrium equations of such Cosserat models are given as a set of nonlinear differential equations, which are usually discretized with the Finite Element Method in computational mechanics. This discretization technique provides very accurate simulation results, but is, used in the standard way, computationally far too demanding for fast simulations with an interactive modification of the boundary conditions. The kinematics of Cosserat rods are closely related to the differential geometry of framed curves, whereby the strain measures of the rods corresponds to the differential invariants of the curves, like arc length or curvature. Therefore, ideas from the discrete differential geometry of framed curves are utilized to construct discrete Cosserat rods. This approach leads to qualitatively correct results, even for very coarse discretizations. Hence, only a small number of degrees of freedom are needed to generate moderately accurate results. This leads to fast computational performance, which makes the discrete Cosserat rod model suitable for interactive simulations. In our contribution, we present such a geometry based discretization approach for flexible one-dimensional structures, together with some application examples of assembly simulation of cables and hoses in automotive industry.