Yield and Buckling Stress Limits in Topology Optimization of Multiscale Structures

TL;DR

The paper addresses integrating yield stress constraints with multiscale buckling criteria in topology optimization. It introduces a two-parameter microstructure and a homogenization-based workflow to compute stiffness, yield, and buckling, coupled via a stress-based Yield Load Factor that can act as an objective or constraint. A de-homogenization pathway maps homogenized designs to realizable microstructures, enabling body-fitted mesh evaluation and validation against homogenized predictions. The L-beam example demonstrates material-dependent design strategies, where yield-dominated materials favor simpler topologies and high-buckling materials drive multiscale infill, yielding practical, safer multiscale structures with good cross-scale agreement.

Abstract

This study presents an extension of multiscale topology optimization by integrating both yield stress and local/global buckling considerations into the design process. Building upon established multiscale methodologies, we develop a new framework incorporating yield stress limits either as constraints or objectives alongside previously established local and global buckling constraints. This approach significantly refines the optimization process, ensuring that the resulting designs meet mechanical performance criteria and adhere to critical material yield constraints. First, we establish local density-dependent von Mises yield surfaces based on local yield estimates from homogenization-based analysis to predict the local yield limits of the homogenized materials. Then, these local Yield-based Load Factors (YLFs) are combined with local and global buckling criteria to obtain topology optimized designs that consider yield and buckling failure on all levels. This integration is crucial for the practical application of optimized structures in real-world scenarios, where material yield and stability behavior critically influence structural integrity and durability. Numerical examples demonstrate how optimized designs depend on the stiffness to yield ratio of the considered building material. Despite the foundational assumption of separation of scales, the de-homogenized structures, even at relatively coarse length scales, exhibit a high degree of agreement with the corresponding homogenized predictions.

Figures (20)

• Figure 1: Illustration of the separation of scales of an isotropic multiscale structure for topology optimization: The macrostructure in the global frame $\bm{x}$ and the microstructure in the local frame $\bm{y}$ which is described by a repetition of the unit cell.
• Figure 2: Illustration of the multifield method used to create the physical design field $\bm{\rho}^m$ from the density field $\bm{x}$ and the void indicator field $\bm{s}$. The illustration is made on an arbitrary design field.
• Figure 3: Illustration of the two-parameter microstructure defined by $f_{geom}$ and $f_{sharp}$ with two values of $\alpha \in [0,0.1]$ and three values of $\bar{\eta} \in[0,0.4,0.8]$.
• Figure 4: Illustration of the yield strength analyses for the two-parameter microstructure. (a) Illustration of the zone of evaluated macroscopic stress states. (b) Microstructural yield strength for all principal stress states and cell orientations of the microstructure geometry in the center. The dashed line indicates the inscribed von Mises yield surface.
• Figure 5: Results of the parameter sweep over $\alpha$ and $\bar{\eta}$. (a) Volume fraction with isocontours indicating constant volume paths. (b) Effective relative Young's modulus with volume isocontours projected onto the stiffness surface. Peach colored dots mark the maximum Young's modulus and blue dots the maximum yield strength along each isocontour. (c) Relative yield strength with volume isocontours projected onto the surface. (d) Parameters $\alpha$ and $\bar{\eta}$ relative to the volume fraction for the yield strength optimal microstructures.