Development of geometric algorithms in architectural planning and architectural structures

主要研究结果如下：1.考虑在给定Steiner点个数或给定最小边长下界的约束下，用Steiner点生成平面区域和一类曲面的三角网格，使最大边长比最小.我们已经开发了算法的问题。2.针对给定基频的桁架拓扑优化问题，已被证明是半定规划（SDP）问题。一个最佳的拓扑结构与五倍的基本频率已获得没有任何困难。从SDP问题的最优性条件出发，导出了最优性必要条件和充分条件的一般形式，并研究了最优拓扑的对称性。它也已被证明，线性屈曲约束下的最佳解决方案，可以通过连续应用SDP。索网的大变形分析问题已转化为二阶锥规划问题。已经证明，即使对于不稳定的平衡态，也可以在没有任何平衡形状的情况下找到平衡形状.房间布局的设计是由房间的相邻性、房间的大小、形状、朝向等因素决定的。本文从几何优化的角度将房间布局优化问题转化为一个数学规划问题。我们已经开发了一个新的算法，使用正交图绘制算法的问题。

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.