NURBS-based solutions to inverse boundary problems in droplet shape prediction

Abstract

Inverse boundary problems are common in many areas of engineering. They arise in fluidstatics when liquid droplets are constrained between arbitrarily shaped solid surfaces, as when a molten solder droplet is used to assemble microelectronic devices. In this paper, we present a new method based on the Non-Uniform Rational B-Spline (NURBS) representation for solving the inverse problems arising in three-dimensional (3D) droplet shape prediction. The developed algorithm very naturally handles the contact (wetting) between the solid and liquid surfaces, without the need for additional contact detection schemes, by embedding a single parameter curve within the two parameter solid surface. An adaptive Legendre–Gauss integration method is used on the NURBS surfaces to calculate the potential energy of the system, thereby eliminating the need for mesh generation. Three problems motivated by microelectronics are solved to demonstrate the developed methodology: solder joints with circular and square pad shapes (with and without misalignment), and one where the droplet is constrained between a circular pad and a cylindrical surface. The meshless procedure developed here is expected to have wider applications to the iterative optimal design of engineering systems where mesh generation inhibits the automation of the design process.