  Block copolymers provide a wonderful platform in studying the soft condensed
matter systems. Many fascinating ordered structures have been discovered in
bulk and confined systems. Among various theories, the self-consistent field
theory (SCFT) has been proven to be a powerful tool for studying the
equilibrium ordered structures. Many numerical methods have been developed to
solve the SCFT model. However, most of these focus on the bulk systems, and
little work on the confined systems, especially on general curved surfaces. In
this work, we developed a linear surface finite element method, which has a
rigorous mathematical theory to guarantee numerical precsion, to study the
self-assembled phases of block copolymers on general curved surfaces based on
the SCFT. Furthermore, to capture the consistent surface for a given
self-assembled pattern, an adaptive approach to optimize the size of the
general curved surface has been proposed. To demonstrate the power of this
approach, we investigate the self-assembled patterns of diblock copolymers on
several distinct curved surfaces, including five closed surfaces and an
unclosed surface. Numerical results illustrate the efficiency of the proposed
method. The obtained ordered structures are consistent with the previous
results on standard surfaces, such as sphere and torus. Certainly, the proposed
numerical framework has the capability of studying the phase behaviors on
general surfaces precisely.
