亢战 白嵩

摘 要：该研究在不确定性结构鲁棒性优化和压电智能结构拓扑优化方面的研究结果。首先基于不确定性的非概率椭球凸模型描述，研究了桁架结构的鲁棒性优化设计问题。考虑桁架结构弹性模量的不确定，并用非概率椭球凸模型处理不确定参数。提出了一种量化的结构鲁棒性度量方法。基于该度量模型，提出了结构鲁棒性优化问题的数学模型，其目标是要达到在体积约束条件下，选出结构中鲁棒性最小的一个功能函数，使其鲁棒性最大化。数值算例验证了优化模型的正确性和算法的有效性。我们考虑连续体结构载荷幅度等参数的有界不确定性，利用非概率椭球凸模型进行不确定性参数的界限描述，研究连续体结构的鲁棒性拓扑优化设计的建模与数值方法。为提高求解效率，利用位移与载荷的线性关系，提出一种基于解析几何方法的鲁棒性度量方法，从而避免了求解双层优化问题。基于该度量方法，优化模型的目标是在体积分数约束条件下寻求最优拓扑形式以最大化结构的位移鲁棒性。数值算例验证了优化模型的正确性和算法的有效性。具有狭长形状的压电作动器有利于输出较大的位移，而采用周期拼装方式实现这类结构则具有制造成本相对较低的优点。我们提出了基于周期拼装的平面压电作动器结构拓扑优化设计的数学模型。其中，以位移输出点作功最大化为设计目标，考虑了材料体积和控制能耗约束，对结构基体材料和压电材料的分布以及控制电压的分布进行优化设计。该文给出了结构响应的设计灵敏度分析，并采用基于梯度的数学规划方法对优化问题进行求解。数值算例验证了该文提出的数学模型和算法的可用性与有效性。

关键词：结构优化 拓扑优化 不确定性 多场耦合 智能结构

Structural Optimization Considering Uncertainties and Multi-field Coupling

Kang Zhan Bai Song

（Dalian University of Technology）

Abstract：This report presents our recent progress on study of structural robust optimization under uncertainties and topology optimization of piezoelectric smart structures. Based on ellipsoid convex model description of uncertainties， we studied the robust design optimization problem of truss structures. In the study， the uncertainties of material properties of truss structures are considered and modeled by non-probabilistic ellipsoid convex model. A quantified measure of structural robustness was proposed， and based on this measure， the optimization formulation aims at choosing the robustness of the concerned structural behaviors with smallest robustness， and maximizing the chosen robustness under total volume constraint. Numerical example verified the validity of the proposed method. We studied the topology optimization formulation and numerical techniques of continuum structures with bounded loads on the basis of non-probabilistic ellipsoid convex model description of uncertainties. For the purpose of improving the numerical efficiency， by using the linear relationship between the displacements and the loads， a quantified measure of structural robustness is computed using analytical geometric method. Using this measure， the optimization formulation aims at finding the optimal topology layout to maximize the displacement robustness of structure under volume fraction constraint. The validity of the proposed method is verified by a numerical example. Piezoelectric actuators with large aspect ratio are suitable for deliver large displacements. Assembling this type of actuators by means of repetitive components has the advantage of low manufacturing cost. We present a mathematical model for topology optimization of planar piezoelectric actuators with repetitive components. The design objective is to maximum the work exported at the displacement output port. Constraints with regard to the control energy consumption and the material volume are imposed to the optimization problem. The distributions of the actuation voltage as well as the topologies of both host layers and piezoelectric layers are to be optimized. Numerical techniques for sensitivity analysis of structure response are presented and the proposed optimization problem is solved with a gradient-based mathematical programming approach. Illustrative examples are given to demonstrate the validity and applicability of the proposed approach.

Key Words：Structural optimization； Topology optimization； Uncertainty； Multi-field coupling； Smart structure

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