Initial-interval driven biasing of nodal concentration along a fixed curve

Element size transitioning in the construction of spatial meshes for finite element models is often controlled by biasing the concentration of nodes, towards one end or the other, along each of a set of curves in the model. A simple, common and efficient scheme to implement such nodal concentration biasing along a given curve is to require that the nodal spacings δ_i be (sequence) terms bⁱδ₀ of a geometric series. Current practice takes the parameter value b, or its equivalent, as an independent input, so that the initial nodal spacing δ₀ must be a computed output. This is the most straightforward approach, but the lack of direct control over the value δ₀ is a significant shortcoming. In an element size transitioning scenario, δ₀ is often a parameter for which the model builder/analyst has independent quantitative information. It may represent the a priori known thickness of a thin bond or weld, for example. A more rational choice for these cases, proposed by this paper, is a scheme for which δ₀ is an independent input parameter instead of b. The parameter b is computed by a convergence-guaranteed algorithm for which the existence of b as a single-valued function of its input is proven.