Sphere packing is an attractive way to generate high quality mesh. Several algorithms have been proposed in this topic, however these algorithms are not sufficiently fast for large scale problems. The paper presents an efficient sphere packing algorithm which is much faster and appears to be the most practical among all sphere packing methods presented so far for mesh generation. The algorithm packs spheres inside a domain using advancing front method. High efficiency has resulted from a concept of 4R measure, which localizes all the computations involved in the whole sphere packing process.
A general numerical approach was developed to simulate the mechanical properties and the failure of heterogeneous elasto-plastic materials using statistical distributions of the material properties. An appropriate elastic-plastic constitutive relation is used to describe the material behavior and failure in each element, with a two-parameter Weibull distribution used to produce the initial heterogeneous material property variations. An adaptive incremental load-step is applied so that only one or a few elements （or integration points） change their status （i.e., from elastic to plastic, or from plastic to strain failure） within one load step. A failed element is then assigned a very small modulus to simulate the failure rather than removing it from the model, which keeps the continuity of the geometric mesh. The numerical results show that the model is suitable for simulating the effective mechanical properties and failure of heterogeneous materials with local elasto-plastic constitutive relations.
