A reliability-based non-probabilistic multiscale topology optimization (NRBMTO) method with stress constraints is proposed for thermo-mechanical continuous structures with uncontrollable stresses. The physical parameters, external loads and temperature values at the macro scale, are regarded as non-probabilistic uncertain parameters in the optimization of structural topologies with complex physical fields at multi-scale. The homogenization-based finite element method is employed to quantify thermo-mechanical structures with multi-scale uncertain parameters in the established multi-scale topology model. The ellipsoid model is applied to describe the uncertainty of non-probabilistic random variables, and the non-probabilistic reliability index is obtained by estimating the failure probability based on the first-order reliability method (FORM). The unit stresses are aggregated to the global maximum stresses with the normalized p-norm function, taking into account the mechanical and thermal stresses. The sensitivity information of the compliance and stress constraint to the macro- and micro-design variables and uncertain variables are derived simultaneously. The macro- and micro- design variables are solved by the method of moving asymptotes (MMA), respectively. Several numerical examples are given to verify the effectiveness and feasibility of the proposed NRBMTO method. The results demonstrate that the optimized structure based on the NRBMTO method provides better security with reliability index =3 and minimum compliance (244.39) while stress is controlled below 235 MPa compared to the classical deterministic multiscale topology optimization (DMTO) method.

中文翻译：

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具有应力约束的热力连续体结构的基于非概率可靠性的多尺度拓扑优化


针对具有不可控应力的热力连续结构，提出了一种基于可靠性的带应力约束的非概率多尺度拓扑优化（NRBMTO）方法。在多尺度复杂物理场结构拓扑优化中，宏观尺度的物理参数、外部载荷和温度值被视为非概率不确定参数。在建立的多尺度拓扑模型中，采用基于均质化的有限元方法对具有多尺度不确定参数的热力结构进行量化。应用椭球模型描述非概率随机变量的不确定性，基于一阶可靠性方法（FORM）估计失效概率得到非概率可靠性指标。考虑到机械应力和热应力，单位应力通过归一化 p 范数函数聚合为全局最大应力。同时导出柔度和应力约束对宏观和微观设计变量以及不确定变量的敏感性信息。宏观和微观设计变量分别通过移动渐近线（MMA）方法求解。给出了几个数值例子来验证所提出的NRBMTO方法的有效性和可行性。结果表明，与经典的确定性多尺度拓扑优化（DMTO）方法相比，基于NRBMTO方法的优化结构提供了更好的安全性，可靠性指数=3和最小合规性（244.39），同时应力控制在235 MPa以下。