Adjoint Method to Calculate Shape Gradients of Failure Probabilaties for Turbomachinery Components

TL;DR

This paper introduces an adjoint method for efficiently calculating shape sensitivities of failure probabilities in turbomachinery components, enabling probabilistic design optimization for complex geometries.

Contribution

It presents a novel finite element-based adjoint approach for shape gradient computation of failure probabilities under probabilistic low cycle fatigue criteria.

Findings

High accuracy compared to finite difference methods

Significant computational efficiency improvements

Effective for complex geometries like blisks

Abstract

In the optimization of turbomachinery components, shape sensitivities for fluid dynamical objective functions have been used for a long time. As peak stress is not a differential func- tional of the shape, such highly efficient procedures so far have been missing for objective functionals that stem from mechan- ical integrity. This changes, if deterministic lifing criteria are replaced by probabilistic criteria, which have been introduced recently to the field of low cycle fatigue (LCF). Here we present a finite element (FEA) based first discretize, then adjoin approach to the calculation of shape gradients (sen- sitivities) for the failure probability with regard to probabilistic LCF and apply it to simple and complex geometries, as e.g. a blisk geometry. We review the computation of failure probabilities with a FEA postprocessor and sketch the computation of the relevant quantities for the adjoint method. We demonstrate high accuracy and computational efficiency of the adjoint method compared to finite difference schemes. We discuss implementation details for rotating components with cyclic boundary conditions. Finally, we shortly comment on future development steps and on poten- tial applications in multi criteria optimization.