Spectral decomposition for graded multi-scale topology optimization
Spectral decomposition for graded multi-scale topology optimization
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DOI：
10.1016/j.cma.2021.113670
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发表时间：
2021-04
影响因子：
7.2
通讯作者：
T. Kumar;S. Sridhara;B. Prabhune;K. Suresh
中科院分区：
文献类型：
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作者：
T. Kumar;S. Sridhara;B. Prabhune;K. Suresh
多尺度拓扑优化（MTO）今天被利用在需要具有大表面积与体积比的设计的应用中。此外，随着增材制造的出现，MTO获得了显著的突出地位。然而，MTO的一个主要缺点是它在计算上是昂贵的。作为替代方案，已经提出了分级MTO，其中较小尺度的设计特征是单一微结构的分级变化。这导致了显着降低计算成本，同时保留了许多的优点MTO.Graded MTO基本上取决于弹性矩阵的插值。不幸的是，今天使用的直接插值法不能保证所得矩阵的正定性。因此，在分级MTO算法的应变能可能会成为负的和nonphysical.In本文中，我们提出了一个简单而有效的基于谱分解的方法，保证正定弹性矩阵。所提出的方法依赖于弹性矩阵的实例的谱（本征）分解，然后通过本征值的回归和本征向量方向的插值。由此产生的弹性矩阵，然后可以用于稳定的优化。通过数值实验，比较了直接分解法和谱分解法的鲁棒性、准确性和速度。
Multi-scale topology optimization (MTO) is exploited today in applications that require designs with large surface-to-volume ratio. Further, with the advent of additive manufacturing, MTO has gained significant prominence. However, a major drawback of MTO is that it is computationally expensive. As an alternate, graded MTO has been proposed where the design features at the smaller scale are graded variations of a single microstructure. This leads to significant reduction in computational cost, while retaining many of the benefits of MTO.Graded MTO fundamentally rests on interpolation of elasticity matrices. Thedirectmethod of interpolation used today unfortunately does not guarantee positive-definiteness of the resulting matrices. Consequently, during the graded MTO algorithm the strain energy may become negative and non-physical.In this paper, we propose a simple but effective spectral decomposition-based approach which guarantees positive-definite elasticity matrices. The proposed method relies on a spectral (eigen) decomposition of instances of the elasticity matrices, followed by regression of eigenvalues and interpolation of eigenvector orientations. The resulting elasticity matrix can then be used for stable optimization. The direct and spectral decomposition methods are compared here for robustness, accuracy and speed, through several numerical experiments.