On The Differentiability of the Gap Function
Please login to view abstract download link
The gap function is crucial in many engineering simulation problems such as contact mechanics. It is a constructed distance measure between two objects that can be used in quantities such as the impenetrability constraint. In finite element and other numerical methods, the discretized gap function can be formulated in different ways: pointwise or through the mortar method; through closest point projection or orthogonal normal projection (also called ‘ray tracing’). To solve the simulation problems that contain the gap functions, their first-order derivatives are often required. Unfortunately, the gap functions are not always differentiable. Despite the enormous success in applications, there has been limited research on the continuity and differentiability of the gap functions. In this talk, we discuss conditions under which mortar gap functions defined by orthogonal normal projections are smooth. We briefly discuss gap functions based on closest point projections. Further, we show that the mortar gap functions are not semi-smooth, making the adoption of existing nonsmooth optimization algorithms challenging. We propose several smoothing algorithms guided by the nonsmoothness analysis and illustrate their effectiveness via contact mechanics examples.
