In engineering applications one often has to trade-off among several objectives as, for example, the mechanical stability of a component, its efficiency, its weight and its cost. We consider a biobjective shape optimization problem maximizing the mechanical stability of a ceramic component under tensile load while minimizing its volume. Stability is thereby modeled using a Weibull-type formulation of the probability of failure under external loads. The PDE formulation of the mechanical state equation is discretized by a finite element method on a regular grid. To solve the discretized biobjective shape optimization problem we suggest a hypervolume scalarization, with which also unsupported efficient solutions can be determined without adding constraints to the problem formulation. FurthIn this section, general properties of the hypervolumeermore, maximizing the dominated hypervolume supports the decision maker in identifying compromise solutions. We investigate the relation of the hypervolume scalarization to the weighted sum scalarization and to direct multiobjective descent methods. Since gradient information can be efficiently obtained by solving the adjoint equation, the scalarized problem can be solved by a gradient ascent algorithm. We evaluate our approach on a 2 D test case representing a straight joint under tensile load.

在工程应用中，通常必须在多个目标之间进行权衡，例如，组件的机械稳定性，效率，重量和成本。我们考虑了一个双目标形状优化问题，即最大化陶瓷组件在拉伸载荷下的机械稳定性，同时将其体积最小化。因此，使用外部载荷下失效概率的威布尔型公式对稳定性进行建模。机械状态方程的PDE公式通过有限元方法在规则网格上离散化。为了解决离散化的双目标形状优化问题，我们建议进行超量标量，使用该标量也可以确定不受支持的有效解决方案，而无需对问题公式添加约束。在本节中，超级卷的一般属性更多，最大程度地控制主导的超容量可帮助决策者确定折衷解决方案。我们研究了超量标量与加权和标量的关系，并指导多目标下降方法。由于可以通过求解伴随方程来有效地获得梯度信息，因此可以通过梯度上升算法来解决标量问题。我们在一个2D测试用例上评估了我们的方法，该测试用例表示在拉伸载荷下的直缝。标量问题可以通过梯度上升算法解决。我们在一个2D测试用例上评估了我们的方法，该测试用例表示在拉伸载荷下的直缝。标量问题可以通过梯度上升算法解决。我们在一个2D测试用例上评估了我们的方法，该测试用例表示在拉伸载荷下的直缝。