A Hilbert space framework for sensitivity filtering and vertex morphing

Abstract

AbstractA common problem in structural optimization is the formation of checkerboard patterns due to numerical instabilities. One way to mitigate it is to consider regularization techniques such as the application of a smoothing filter during the optimization procedure. This approach leads to what is now called sensitivity filtering, which has become a popular method in the engineering community due to easy implementation and applicability to large‐scale optimization problems. However, the method suffers from a lack of mathematical foundation and theoretical justification [1,2].In an attempt to merge sensitivity filtering into the standard optimization technology, a framework for sensitivity filtering in conjunction with mesh‐free methods and node‐based optimization was developed in [3]. The resulting method is referred to as vertex morphing. Intermediate design iterations of the method exhibit certain desirable properties, depending on the choice of filter radius, such as the attenuation of high‐frequency modes and the preservation of tiny details in the initial design. Since they can be seen as valid design choices, the method is usually stopped before convergence.In this contribution, we provide a mathematical foundation for the vertex morphing method and establish a connection with known regularization techniques. We consider design variables as fields defined on some domain D ⊆ ℝd that can be written as urn:x-wiley:16177061:media:PAMM202100258:pamm202100258-math-0001 where k ∈ L2 (D × D) is an integral kernel and p ∈ L2 (D) is the control variable. With this formulation, the so‐called filtering rules, which form the basis for the design update rules, can be derived rigorously. To analyze the behavior of the method, we regard the linear least squares problem and show that the intermediate results correspond to some regularized solution to the problem. A comparison between the vertex morphing method and other regularization techniques is made for the linear least squares problem and the compliance minimization of a thin shell structure, which is based on the assumed natural deviatoric strain (ANDES) formulation [4].