Zhiqiang Luo
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WCSA研究进行中Research in progress

Efficient Worst-Case
Structural Analysis for Robust Shape Optimization

不预设载荷，找到形状的薄弱之处。再用小幅几何变化，降低最坏情况应力。Find where a shape is vulnerable, without prescribing load cases. Then make small geometric changes to reduce worst-case stress.

Zhiqiang LuoEtienne VougaQixing Huang

查看优化结果Explore the results理论与方法Theory & method
A character mesh before and after stress-driven shape optimization, alongside overlaid cross-sections of the two shapes
01 /从薄弱区域到几何改进。论文中的应力场与截面对比；颜色展示空间分布，定量下降幅度见下方独立实验。From weak regions to geometric improvement. Stress fields and overlaid cross-sections from the manuscript; colors show spatial patterns, with quantitative reductions from separate experiments below.

我的工作重点My contribution

理论推导与形状优化结果：将表面载荷下的 von Mises 最坏情况分析转化为可计算的谱问题，并研究临界载荷如何驱动受约束的几何优化。Theoretical derivation and shape optimization results: formulating surface-load von Mises WCSA as a tractable spectral problem, and studying how its critical loads drive constrained geometric optimization.

01 / 形状优化Shape optimization

改变局部形状，缓解应力集中。Small shape changes. Lower stress concentrations.

WCSA 不仅告诉我们哪里薄弱，还给出导致薄弱区域高应力的表面载荷。选取空间位置和载荷模式各异的临界载荷后，用平滑的径向基函数调整形状，并约束体积变化与位移幅度。WCSA returns both a weak-region map and the surface loads that expose those weaknesses. A diverse set of critical loads guides a smooth radial-basis deformation, constrained by volume change and displacement bounds.

以下为论文草稿中的选定设置：固定载荷、30 次优化迭代。数字为报告的应力下降比例，并非所有设置中的最大值。Selected configurations reported in the manuscript: fixed loads, 30 optimization iterations. These are reported stress reductions, not maxima over all settings.

猫Cat28.10%

应力下降stress reduction

1 个临界载荷1 critical load
蝎子Scorpion26.61%

应力下降stress reduction

3 个临界载荷3 critical loads
树Tree19.91%

应力下降stress reduction

5 个临界载荷5 critical loads

探索实验Explore the experiment

需要多少个临界载荷？How many critical loads are enough?

应力下降 (%) · 越高越好Stress reduction (%) · higher is better论文选用设置Selected configuration

所选临界载荷数量Number of selected critical loads

一个载荷已获得大部分改善；增加载荷的收益较小。One critical load captures most of the improvement; additional loads offer only a small gain.

查看精确数据View exact values
猫Cat · 固定载荷，30 次迭代Fixed loads, 30 iterations
载荷数Loads123456789101112
应力下降Reduction28.10%28.47%28.20%28.51%28.51%28.54%28.61%28.61%28.57%28.58%28.47%28.46%

02 / 理论推导Theoretical formulation

把未知载荷，转化为一个小型谱问题。From unknown loads to a small spectral problem.

给定形状与支撑条件，我们寻找固定表面质量范数预算内、使某个应力采样点的 von Mises 应力最大的载荷。核心观察是：在最后取 von Mises 范数之前，载荷到局部应力的映射是线性的。Given a shape and its supports, we seek the surface load that maximizes von Mises stress at a stress sample, under a fixed surface-mass norm budget. The key observation: the load-to-stress map is linear before the final von Mises norm.

  1. 01

    定义载荷空间Define the load space

    用表面质量矩阵衡量载荷密度，统一不同表面离散下的载荷预算。Measure surface-load density using the surface mass matrix, giving a consistent load budget over the discretized boundary.

    fᵀ Ms f ≤ C²
  2. 02

    构建表面响应模态Build a surface response basis

    对表面 Schur 补刚度求解广义特征问题，保留前 k 个非零模态，构成紧凑的应力响应矩阵。Solve a generalized eigenproblem for the surface Schur-complement stiffness. Retain k nonzero modes to build a compact stress response matrix.

    K̃s φj = μj Ms φj
  3. 03

    求最坏应力，恢复临界载荷Find the stress, recover the load

    最大奇异值给出截断空间内的最坏应力；对应的奇异向量恢复引发该应力的表面载荷。The largest singular value gives the worst stress within the truncated space; its singular vector recovers the critical surface load.

    Φk = C σmax(Rk)
展开：响应矩阵与截断误差A closer look: the response matrix and truncation error

在固定应力采样点，令 Pᵥₘ = LLᵀ 为 von Mises 矩阵分解，G 为位移到应力的映射，Φⱼ 为体积位移模态。第 j 列响应为：At a fixed stress sample, let Pᵥₘ = LLᵀ factor the von Mises matrix, G map displacement to stress, and Φⱼ denote the volumetric displacement mode. The j-th response column is:

rj = μj−1 Lᵀ G Φj,   Rk = [r₁ … rk]

因此只需对 RₖRₖᵀ 求最大特征值，其维数由 von Mises 因子空间决定，与表面自由度数无关。截断应力是所保留子空间中的最优值；若完整的被省略响应能量已知，还可得到逐点误差界：The largest eigenvalue of RₖRₖᵀ solves the problem in a space whose dimension is independent of the surface degrees of freedom. Truncated stress is optimal within the retained subspace. If the complete omitted response energy is known, it also gives a pointwise error bound:

0 ≤ Φall − Φk ≤ C (Σj>k ‖rj‖²)½

这是有限元离散问题上的逐点关系。仅观察有限个模态的衰减，并不等于已获得完整尾项或连续体最大应力的认证。This is a pointwise relation for the finite-element discretization. Observed decay over a finite set of modes alone does not certify the complete tail or the continuum maximum stress.

Side-by-side exact and truncated worst-case stress maps on the manuscript's example meshes
02 /全空间参考解与截断求解器的薄弱区域对比。这里展示空间分布的一致性，不将近似热点等同于唯一热点的严格认证。Weak-region maps from the full-space reference and truncated solver. The comparison illustrates spatial agreement, rather than a certificate of a unique hotspot.

03 / 更多结果More results

保留整体形状，调整关键区域。Preserve the form. Refine the vulnerable regions.

茶壶、草莓与猫模型的结果展示了局部几何调整。截面叠加与局部放大图呈现优化实际改变的位置；外轮廓的变化与应力场一起评估。The teapot, strawberry, and cat examples show localized geometric changes. Overlaid cross-sections and close-ups reveal where the optimization acts, allowing shape changes to be assessed alongside stress fields.

Teapot, strawberry, and cat models with before and after stress fields, overlaid cross-sections, and zoomed geometric changes
03 /论文原始结果图。由左至右：优化前、优化后、截面变化与局部放大。点击图片查看大图。Original manuscript results. Left to right: before, after, cross-section change, and a close-up. Select the figure to view it at full size.

关于这个项目About this project

从力学推导，走向可用的设计工具。Connecting mechanics to practical design tools.

这项研究连接计算机图形学、有限元分析、谱方法与受约束优化。我的工作重点是理论推导和形状优化结果。This research connects computer graphics, finite-element analysis, spectral methods, and constrained optimization. My focus is the theoretical derivation and shape optimization results.

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