Development of geometric algorithms in architectural planning and architectural structures

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The results obtained are summarized as follows.1. We consider the problem of generating triangular meshes for planar regions and a class of curved surfaces using Steiner points that minimizes the maximum edge length ratio under the constraint that (1) the number of Steiner points are given or (2) the lower bound on minimum edge length is given. We have developed algorithms for the problem.2. Truss topology optimization problem for specified fundamental frequency has been shown to be formulated as SemiDefinite Programming (SDP) problem. An optimal topology with five-fold fundamental frequencies has been obtained without any difficulty. General forms of necessary and sufficient optimality conditions have been derived from the optimality conditions of the SDP problem, and the symmetry properties of optimal topologies have been investigated. It has also been shown that optimal solutions under linear buckling constraints can be found by successively applying SDP.3. Large-deformation analysis problem of cable networks has been formulated as Second-Order Cone Programming (SOCP) problem. It has been demonstrated that equilibrium shapes can be found without any even for an unstable equilibrium state.4. The design of room layout is determined by the adjacency of room, room size, shape, orientation and etc. We formulated the problem of optimal room layout as a mathematical programming problem from the viewpoint of geometric optimization. We have developed a new algorithm for the problem using orthogonal graph drawing algorithms.