Subdivision surfaces provide an elegant isogeometric analysis framework for
geometric design and analysis of partial differential equations defined on
surfaces. They are already a standard in high-end computer animation and
graphics and are becoming available in a number of geometric modelling systems
for engineering design. The subdivision refinement rules are usually adapted
from knot insertion rules for splines. The quadrilateral Catmull-Clark scheme
considered in this work is equivalent to cubic B-splines away from
extraordinary, or irregular, vertices with other than four adjacent elements.
Around extraordinary vertices the surface consists of a nested sequence of
smooth spline patches which join C1 continuously at the point itself. As
known from geometric design literature, the subdivision weights can be
optimised so that the surface quality is improved by minimising
short-wavelength surface oscillations around extraordinary vertices. We use the
related techniques to determine weights that minimise finite element
discretisation errors as measured in the thin-shell energy norm. The
optimisation problem is formulated over a characteristic domain and the errors
in approximating cup- and saddle-like quadratic shapes obtained from
eigenanalysis of the subdivision matrix are minimised. In finite element
analysis the optimised subdivision weights for either cup- or saddle-like
shapes are chosen depending on the shape of the solution field around an
extraordinary vertex. As our computations confirm, the optimised subdivision
weights yield a reduction of 50% and more in discretisation errors in the
energy and L2​ norms. Although, as to be expected, the convergence rates are
the same as for the classical Catmull-Clark weights, the convergence constants
are improved.Partial support through Trimble Inc and Cambridge Trust is gratefully acknowledged