This work investigates a simplified approach to cope with the optimization of preliminary design of structures under local fatigue constraints along with a global enforcement on the overall compliance. The problem aims at the minimization of the weight of linear elastic structures under given loads and boundary conditions. The expected stiffness of the optimal structure is provided by the global constraint, whereas a set of local stress---based constraints ask for a structure to be fatigue resistant. A modified Goodman fatigue strength comparison is implemented through the same formalism to address pressure---dependent failure in materials as in Drucker---Prager strength criterion. As a simplification, the Sines approach is used to define the equivalent mean and alternating stresses to address the fatigue resistance for an infinite life time. Sines computation is based on the equivalent mean and alternate stress depending on the invariants of the stress tensor and its deviatoric part, respectively. The so---called singularity phenomenon is overcome by the implementation of a suitable qp-relaxation of the equivalent stress measures. Numerical examples are presented to illustrate the features of the achieved optimal layouts and of the proposed algorithm.

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Topology optimization for minimum weight with compliance and simplified nominal stress constraints for fatigue resistance

An iterative 2D finite-element-based optimization procedure has been developed which incorporates robust design philosophies. This has been used to determine precise free-form shapes for a hole in a plate example, with the aim of maximizing its fatigue-life when exposed to varying load orientations. Past methods have typically considered only a single nominal load orientation, with empirical approaches to deal with the orientation variability, thus resulting in suboptimal solutions. Here a robust stress method is developed that produces a notch shape that minimizes the peak stress and renders it constant for a range of load orientations. Furthermore, a more sophisticated robust fatigue-damage optimization method is then developed to minimize the peak fatigue damage for a given stochastic distribution of load orientations. Fatigue calculations for an example problem with significant load orientation variation show that the robust optimization methods provide fatigue-life extensions 2 to 8 times better than past methods. It is anticipated that the implementation of robust optimal shapes in metallic components would result in greater fatigue-life extension.

Robust shape optimization of notches for fatigue-life extension

In this paper a new method for an automated shape optimization of dynamically loaded components in mechanical systems is presented. The optimization is carried out by means of the results of a durability analysis based on finite elements. Load time histories, which are necessary for durability analyses, are derived from a multibody simulation. The whole optimization loop, which is an iterative procedure, incorporates all these gradual analysis steps and is implemented by the authors in a straightforward, batch-oriented manner using well-known standard software. Since the whole process involves several different analysis types, such as multibody system simulation and durability analysis, the resulting setup is rather complex. Furthermore, the reader may not be familiar with all the terms arising within the context of every single analysis domain. Therefore, some essential aspects of each of the stages involved in the process are explained to provide the reader with the necessary background. In the following, the required software setup as well as the implementation are described. Finally, an academic example is discussed to illustrate and clearly outline the potential of this method.

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Shape optimization of structural parts in dynamic mechanical systems based on fatigue calculations

An introduction and overview of the developments in shape optimization for fatigue problems is given, beginning with the main ideas in the field of static shape optimization. The use of numerical tools together with the incorporation of non-linear material properties in the continuum mechanics design has led to new possibilities for the analysis of fatigue behaviour in mechanical engineering structures. Based on publications by Lemaitre and Chaboche, the material behaviour is described by defining different partial differential equation systems on micro-scale and meso-scale. From this an optimal algorithm results for the shape optimization of mechanical engineering structures using continuum damage mechanics. Our numerical and experimental tests show a significant increase of lifetime in comparison to classical shapes.

Shape optimization under fatigue using continuum damage mechanics

Three-dimensional structural design optimisation based on fatigue implementing a genetic algorithm and a non-similitude crack growth law

Almost any machine part is dynamically loaded and fails in most cases by fatigue. Nevertheless many authors recommend just a static shape optimization based on the minimization of maximum stress to improve the lifetime. In this paper a more detailed description of fatigue within continuum damage mechanics (CDM) considering the loading history is introduced. On this basis, two new cost functions are determined for shape optimization of dynamically loaded machine parts. An optimization procedure is built up using the methods of mathematical programming (MP). The optimized parts show a remarkably increased lifetime in numerical results as well as in experiments.

J. Grunwald

Papers

A fatigue model for shape optimization.