SIAM Journal on Scientific Computing, Volume 42, Issue 4, Page B1092-B1114, January 2020.
Structural optimization searches for the optimal shape and topology of components such that specific physical quantities are optimized, for instance, the stability of the resulting structure. Problems involving multiple scales, i.e., structures on a microscopic and a macroscopic level, can be efficiently solved by homogenization-based two-scale approaches. In each optimization iteration, many computationally expensive tensors $E$ describing the macroscopic behavior of a given microstructure have to be calculated, implying that the solution of one optimization problem can take weeks. The computational complexity can be greatly reduced with surrogates $\tilde{E}$ that are constructed in advance in an offline phase via interpolation and that can be reused for different scenarios. Three main issues arise in this context: First, the curse of dimensionality renders conventional interpolation schemes infeasible even for moderate dimensionalities $> 4$. Therefore, we use sparse grid interpolation combined with a novel problem-tailored boundary treatment to drastically reduce the necessary grid size with only slightly higher approximation errors. Second, common sparse grid bases are not continuously differentiable. Hierarchical B-splines achieve lower approximation errors and supply exact continuous gradients of $\tilde{E}$, which enables gradient-based optimization without approximating gradients of $E$. Third, the interpolated tensors are usually required to be positive definite, which is not fulfilled by common interpolation methods. We are able to preserve positive definiteness of the interpolated tensors by interpolating Cholesky factors instead. Combining these three contributions allows computing optimized structures for two- and three-dimensional optimization scenarios with speedups of up to 86 when compared to non-surrogate-based solutions.

中文翻译：

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稀疏网格上具有B样条的基于梯度的两尺度拓扑优化

SIAM科学计算杂志，第42卷，第4期，第B1092-B1114页，2020年1月。
结构优化会搜索组件的最佳形状和拓扑，以便优化特定的物理量，例如，最终结构的稳定性。涉及多个尺度的问题，即微观和宏观层面的结构，可以通过基于均化的两尺度方法有效解决。在每个优化迭代中，必须计算许多描述给定微结构的宏观行为的计算量大的张量$ E $，这意味着一个优化问题的解决可能需要数周时间。代用$ \ tilde {E} $可以大大降低计算复杂度，代用$ \ tilde {E} $可以通过插值在离线阶段预先构建，并且可以在不同的场景中重复使用。在这种情况下出现了三个主要问题：首先，维度的诅咒使得即使对于中等维度$> 4 $，常规的插值方案也不可行。因此，我们将稀疏网格插值与新颖的问题量身定做的边界处理结合使用，以大幅度减少必要的网格大小，而仅产生稍高的逼近误差。第二，常见的稀疏网格基础无法连续区分。分层B样条曲线可实现较低的逼近误差，并提供$ \ tilde {E} $的精确连续梯度，从而可以在不逼近$ E $的情况下进行基于梯度的优化。第三，通常要求插值张量是正定的，这是普通插值方法无法满足的。我们可以通过内插Cholesky因子来保持内插张量的正定性。