  Fine scale elastic structures are widespread in nature, for instances in
plants or bones, whenever stiffness and low weight are required. These patterns
frequently refine towards a Dirichlet boundary to ensure an effective load
transfer. The paper discusses the optimization of such supporting structures in
a specific class of domain patterns in 2D, which composes of periodic and
branching period transitions on subdomain facets. These investigations can be
considered as a case study to display examples of optimal branching domain
patterns.
  In explicit, a rectangular domain is decomposed into rectangular subdomains,
which share facets with neighbouring subdomains or with facets which split on
one side into equally sized facets of two different subdomains. On each
subdomain one considers an elastic material phase with stiff elasticity
coefficients and an approximate void phase with orders of magnitude softer
material. For given load on the outer domain boundary, which is distributed on
a prescribed fine scale pattern representing the contact area of the shape, the
interior elastic phase is optimized with respect to the compliance cost. The
elastic stress is supposed to be continuous on the domain and a stress based
finite volume discretization is used for the optimization. If in one direction
equally sized subdomains with equal adjacent subdomain topology line up, these
subdomains are consider as equal copies including the enforced boundary
conditions for the stress and form a locally periodic substructure.
  An alternating descent algorithm is employed for a discrete characteristic
function describing the stiff elastic subset on the subdomains and the solution
of the elastic state equation. Numerical experiments are shown for compression
and shear load on the boundary of a quadratic domain.
