- arXiv:1905.07566
**Gradient Based Biobjective Shape Optimization to Improve Reliability and Cost of Ceramic Components****Authors:**Onur T. Doganay, Camilla Hahn, Hanno Gottschalk, Kathrin Klamroth, Johanna Schultes, Michael Stiglmayr

**Abstract:**We consider the simultaneous optimization of the reliability and the cost of a ceramic component in a biobjective PDE constrained shape optimization problem. A probabilistic Weibull-type model is used to assess the probability of failure of the component under tensile load, while the cost is assumed to be proportional to the volume of the component. Two different gradient-based optimization methods are suggested and compared at 2D test cases. read moreThe numerical implementation is based on a first discretize then optimize strategy and benefits from efficient gradient computations using adjoint equations. The resulting approximations of the Pareto front nicely exhibit the trade-off between reliability and cost and give rise to innovative shapes that compromise between these conflicting objectives. - arXiv:1904.08650
**Efficient Techniques for Shape Optimization with Variational Inequalities using Adjoints****Authors:**Daniel Luft, Volker H. Schulz, Kathrin Welker

**Abstract:**In general, standard necessary optimality conditions cannot be formulated in a straightforward manner for semi-smooth shape optimization problems. In this paper, we consider shape optimization problems constrained by variational inequalities of the first kind, so-called obstacle-type problems. Under appropriate assumptions, we prove existence of adjoints for regularized problems and convergence…read moreto adjoints of the unregularized problem. Moreover, we derive shape derivatives for the regularized problem and prove convergence to a limit object. Based on this analysis, an efficient optimization algorithm is devised and tested numerically. - arXiv:1705.05776
**Shape optimization to decrease failure probability****Authors:**Matthias Bolten, Hanno Gottschalk, Camilla Hahn, Mohamed Saadi

**Abstract:**Ceramic is a material frequently used in industry because of its favorable properties. Common approaches in shape optimization for ceramic structures aim to minimize the tensile stress acting on the component, as it is the main driver for failure. In contrast to this, we follow a more natural approach by minimizing the component’s probability of failure under a given tensile load. Since the…read morefundamental work of Weibull, the probabilistic description of the strength of ceramics is standard and has been widely applied. Here, for the first time, the resulting failure probabilities are used as objective functions in PDE constrained shape optimization. To minimize the probability of failure, we choose a gradient based method combined with a first discretize then optimize approach. For discretization finite elements are used. Using the Lagrangian formalism, the shape gradient via the adjoint equation is calculated at low computational cost. The implementation is verified by comparison of it with a finite difference method applied to a minimal 2d example. Furthermore, we construct shape flows towards an optimal / improved shape in the case of a simple beam and a bended joint. - arXiv:1806.04389
**Shape gradients for the failure probability of a mechanical component under cyclical loading****Authors:**Hanno Gottschalk, Mohamed Saadi

**Abstract:**This work provides a numerical calculation of shape gradients of failure probabilities for mechanical components using a first discretize, then adjoint approach. While deterministic life prediction models for failure mechanisms are not (shape) differentiable, this changes in the case of probabilistic life prediction. The probabilistic, or reliability based, approach thus opens the way for…read moreefficient adjoint methods in the design for mechanical integrity. In this work we propose, implement and validate a method for the numerical calculation of the shape gradients of failure probabilities for the failure mechanism low cycle fatigue (LCF), which applies to polycrystalline metal. Numerical examples range from a bended rod to a complex geometry from a turbo charger in 3D. - arXiv:1803.01216
**Adjoint Method to Calculate Shape Gradients of Failure Probabilaties for Turbomachinery Components****Authors:**Hanno Gottschalk, Mohamed Saadi, Onur Tanil Doganay, Kathrin Klamroth, Sebastian Schmitz

**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 functional of the shape, such highly efficient procedures so far have been missing for objective functionals that stem from mechanical integrity. This changes, if deterministic lifing criteria are replaced by…read moreprobabilistic 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 potential applications in multi criteria optimization. - arXiv:1702.05759
**Probabilistic LCF Risk Evaluation of a Turbine Vane by Combined Size Effect and Notch Support Modeling****Authors:**Lucas Mäde, Sebastian Schmitz, Hanno Gottschalk, Tilman Beck

**Abstract:**A probabilistic risk assessment for low cycle fatigue (LCF) based on the so-called size effect has been applied on gas-turbine design in recent years. In contrast, notch support modeling for LCF which intends to consider the change in stress below the surface of critical LCF regions is known and applied for decades. Turbomachinery components often show sharp stress gradients and very localized…read morecritical regions for LCF crack initiations so that a life prediction should also consider notch and size effects. The basic concept of a combined probabilistic model that includes both, size effect and notch support, is presented. In many cases it can improve LCF life predictions significantly, in particular compared to curve predictions of standard specimens where no notch support and size effect is considered. Here, an application of such a combined model is shown for a turbine vane. - Conference: ECCM-ECFD 2018, At Glasgow
**Using Adjoint CFD to Quantify the Impact of Manufacturing Variations on a Heavy Duty Turbine Vane****Authors:**Alexander Liefke, Vincent Marciniak, Uwe Janoske, Hanno Gottschalk

**Abstract:**Turbine efficiency is one of the main design criteria for heavy duty gas turbines. After the design, margin adaption factors are applied on the baseline to predict the impact of manufacturing variations (MV). These margins are normally based on testbed experience. A more detailed knowledge of the impact of MV, prior to testing, would therefore improve the margin prediction accuracy and could benefit in product cost and global efficiency. …read moreFor turbomachines the impact of MV can be quantified with a Monte Carlo (MC) simulation in combination with steady non-linear CFD calculations e.g. RANS. The drawback of this approach is the large number of RANS computations needed to quantify the impact of MV, which is prohibitive for a daily use in an industrial context. Assuming that the MV are small enough, the adjoint CFD method, which linearizes the governing equations, can be an alternative to the RANS evaluations. This kind of approach has been successfully used for compressors and turbines.

The first part of this paper presents a systematic approach to evaluate a hand-derived and an algorithmic-derived version of the discrete adjoint CFD solver TRACE. To do so, the ERCOFTAC axial flow turbine known as Aachen Turbine has been selected. For the adjoint version comparison a NACA-like parametrization is applied to compare and validate the adjoint-generated with finite difference gradients.

In the second part the adjoint-based method is applied to an industrial turbine vane to quantify the impact of MV. For this case real MV have been measured using 102 optical blade scans. The scans are used to generate the corresponding deformed geometries for which an adjoint and a RANS simulation are computed. The comparison between each computation demonstrates that the impact of realistic MV can be handled by the adjoint approach.