论文标题
用于计算高阶P谐波下降方向及其形状优化的极限
Towards computing high-order p-harmonic descent directions and their limits in shape optimization
论文作者
论文摘要
我们向矢量值$ p $ -laplacian提出了用于经典标量$ p $ -laplace dirichlet问题的算法的扩展名，具有混合边界条件，以便使用$ p $ harmonic方法在形状优化中解决形状的问题。提出的方法的主要优点是，不需要迭代，因此需要$ p $，因此可以有效地计算更高订单的解决方案。我们表明，所需数量的牛顿迭代相对于网格点的数量仍然多项式，并通过考虑形状变形的数值实验来验证结果。此外，我们讨论从分析和数值的角度考虑这些问题的限制时会引起的挑战，尤其是在源术语中的符号变化方面。
We present an extension of an algorithm for the classical scalar $p$-Laplace Dirichlet problem to the vector-valued $p$-Laplacian with mixed boundary conditions in order to solve problems occurring in shape optimization using a $p$-harmonic approach. The main advantage of the proposed method is that no iteration over the order $p$ is required and thus allow the efficient computation of solutions for higher orders. We show that the required number of Newton iterations remains polynomial with respect to the number of grid points and validate the results by numerical experiments considering the deformation of shapes. Further, we discuss challenges arising when considering the limit of these problems from an analytical and numerical perspective, especially with respect to a change of sign in the source term.
