考慮不確定性與多場耦合的結構優化

亢戰 白嵩

摘 要：該研究在不確定性結構魯棒性優化和壓電智能結構拓撲優化方面的研究結果。首先基于不確定性的非概率橢球凸模型描述，研究了桁架結構的魯棒性優化設計問題。考慮桁架結構彈性模量的不確定，并用非概率橢球凸模型處理不確定參數。提出了一種量化的結構魯棒性度量方法。基于該度量模型，提出了結構魯棒性優化問題的數學模型，其目標是要達到在體積約束條件下，選出結構中魯棒性最小的一個功能函數，使其魯棒性最大化。數值算例驗證了優化模型的正確性和算法的有效性。我們考慮連續體結構載荷幅度等參數的有界不確定性，利用非概率橢球凸模型進行不確定性參數的界限描述，研究連續體結構的魯棒性拓撲優化設計的建模與數值方法。為提高求解效率，利用位移與載荷的線性關系，提出一種基于解析幾何方法的魯棒性度量方法，從而避免了求解雙層優化問題。基于該度量方法，優化模型的目標是在體積分數約束條件下尋求最優拓撲形式以最大化結構的位移魯棒性。數值算例驗證了優化模型的正確性和算法的有效性。具有狹長形狀的壓電作動器有利于輸出較大的位移，而采用周期拼裝方式實現這類結構則具有制造成本相對較低的優點。我們提出了基于周期拼裝的平面壓電作動器結構拓撲優化設計的數學模型。其中，以位移輸出點作功最大化為設計目標，考慮了材料體積和控制能耗約束，對結構基體材料和壓電材料的分布以及控制電壓的分布進行優化設計。該文給出了結構響應的設計靈敏度分析，并采用基于梯度的數學規劃方法對優化問題進行求解。數值算例驗證了該文提出的數學模型和算法的可用性與有效性。

關鍵詞：結構優化 拓撲優化 不確定性 多場耦合 智能結構

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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