Computationally solving the equations of elasticity is a key component in many materials science and mechanics simulations. Phenomena such as deformation-induced microstructure evolution, microfracture, and microvoid nucleation are examples of applications for which accurate stress and strain fields are required. A characteristic feature of these simulations is that the problem domain is simple (typically a rectilinear representative volume element (RVE)), but the evolution of internal topological features is extremely complex. Traditionally, the finite element method (FEM) is used for elasticity calculations; FEM is nearly ubiquituous due to (1) its ability to handle meshes of complex geometry using isoparametric elements, and (2) the weak formulation which eschews the need for computation of second derivatives. However, variable topology problems (e.g. microstructure evolution) require either remeshing, or adaptive mesh refinement (AMR) - both of which can cause extensive overhead and limited scaling. Block-structured AMR (BSAMR) is a method for adaptive mesh refinement that exhibits good scaling and is well-suited for many problems in materials science. Here, it is shown that the equations of elasticity can be efficiently solved using BSAMR using the finite difference method. The boundary operator method is used to treat different types of boundary conditions, and the "reflux-free" method is introduced to efficiently and easily treat the coarse-fine boundaries that arise in BSAMR. Examples are presented that demonstrate the use of this method in a variety of cases relevant to materials science: Eshelby inclusions, fracture, and microstructure evolution. Reasonable scaling is demonstrated up to $\sim$4000 processors with tens of millions of grid points, and good AMR efficiency is observed.
翻译:计算弹性等方程式是许多材料科学和机械模拟中的一个关键组成部分。 畸形导致的微结构进化、微裂变和微无核核化是需要精确压力和压力字段的应用实例。 这些模拟的一个特征是问题领域简单( 典型的直线代表性体积元素( RVE) ), 但内部地形特征的演化是极其复杂的。 传统上, 有限元素法( FEM) 用于弹性计算; FEM 几乎无所不在, 原因是:(1) 它有能力使用等分数元素处理复杂几何的模层, 以及(2) 微弱的配方, 需要精确的压力和压力字段。 然而, 不同的表层问题( 例如, 微结构演化) 需要重新显示, 或者适应缩缩缩法( AMR) —— 两者都能导致广泛的上流和缩放。 结构化的ARM( BAM( BAM) 是一个调整法的精细化方法, 显示, 将精细化法的精细化法的精细的精细的精细的精细化过程, 使用BML 方法可以显示, 方法的精细的精细的精细的精细的精细的比 。