Efficient and Accurate Separable Models for Discretized Material Optimization: A Continuous Perspective Based on Topological Derivatives

Multi-material design optimization problems can, after discretization, be solved by the iterative solution of simpler sub-problems which approximate the original problem at an expansion point to first order. In particular, models constructed from convex separable first order approximations have a lo...

Full description

Saved in:
Bibliographic Details
Published in:The Journal of geometric analysis 2024, Vol.34 (7), p.206, Article 206
Main Authors: Gangl, Peter, Nees, Nico, Stingl, Michael
Format: Article
Language:English
Subjects:
Abstract Harmonic Analysis
Accuracy
Approximation
Asymptotes
Continuity (mathematics)
Convex and Discrete Geometry
Design optimization
Differential Geometric PDE Control
Differential Geometry
Discretization
Dynamical Systems and Ergodic Theory
Fourier Analysis
Global Analysis and Analysis on Manifolds
Iterative solution
Mathematical analysis
Mathematics
Mathematics and Statistics
Shape Optimization and Applications
Topology
Citations: Items that this one cites
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:Multi-material design optimization problems can, after discretization, be solved by the iterative solution of simpler sub-problems which approximate the original problem at an expansion point to first order. In particular, models constructed from convex separable first order approximations have a long and successful tradition in the design optimization community and have led to powerful optimization tools like the prominently used method of moving asymptotes (MMA). In this paper, we introduce several new separable approximations to a model problem and examine them in terms of accuracy and fast evaluation. The models can, in general, be nonconvex and are based on the Sherman–Morrison–Woodbury matrix identity on the one hand, and on the mathematical concept of topological derivatives on the other hand. We show a surprising relation between two models originating from these two—at a first sight—very different concepts. Numerical experiments show a high level of accuracy for two of our proposed models while also their evaluation can be performed efficiently once enough data has been precomputed in an offline stage. Additionally it is demonstrated that suboptimal decisions can be avoided using our most accurate models.
ISSN:1050-6926
1559-002X
1559-002X
DOI:10.1007/s12220-024-01663-0