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A

M

CM

Bergische Universit¨

at Wuppertal

Fachbereich Mathematik und Naturwissenschaften

Institute of Mathematical Modelling, Analysis and Computational

Mathematics (IMACM)

Preprint BUW-IMACM 20/02

Jan Backhaus, Matthias Bolten, Onur Tanil Doganay, Matthias

Ehrhardt, Benedikt Engel, Christian Frey, Hanno Gottschalk,

Michael G¨

unther, Camilla Hahn, Jens J¨

aschke, Peter Jaksch, Kathrin

Klamroth, Alexander Liefke, Daniel Luft, Lucas M¨

ade, Vincent

Marciniak, Marco Reese, Johanna Schultes, Volker Schulz, Sebastian

Schmitz, Johannes Steiner and Michael Stiglmayr

GivEn – Shape Optimization for Gas Turbines in

Volatile Energy Networks

February 2020

http://www.math.uni-wuppertal.de

GivEn – Shape Optimization for Gas Turbines in

Volatile Energy Networks

Jan Backhaus, Matthias Bolten, Onur Tanil Doganay, Matthias Ehrhardt, Benedikt

Engel, Christian Frey, Hanno Gottschalk, Michael G¨

unther, Camilla Hahn, Jens

J¨

aschke, Peter Jaksch, Kathrin Klamroth, Alexander Liefke, Daniel Luft, Lucas

M¨

ade, Vincent Marciniak, Marco Reese, Johanna Schultes, Volker Schulz,

Sebastian Schmitz, Johannes Steiner, and Michael Stiglmayr

Abstract This paper describes the project GivEn that develops a novel multicrite-

ria optimization process for gas turbine blades and vanes using modern ”adjoint”

shape optimization algorithms. Given the many start and shut-down processes of

gas power plants in volatile energy grids, besides optimizing gas turbine geometries

for efﬁciency, the durability understood as minimization of the probability of fail-

ure is a design objective of increasing importance. We also describe the underlying

coupling structure of the multiphysical simulations and use modern, gradient based

Jan Backhaus, Christian Frey

Institute of Propulsion Technology, German Aerospace Center (DLR), 51147 K ¨

oln, Germany,

e-mail: {jan.backhaus,christian.frey}@dlr.de

Matthias Bolten, Onur Tanil Doganay, Matthias Ehrhardt, Hanno Gottschalk, Michael G ¨

unther,

Camilla Hahn, Jens J¨

aschke, Kathrin Klamroth, Marco Reese, Johanna Schultes and Michael

Stiglmayr

Bergische Universit¨

at Wuppertal, Fakult¨

at f¨

ur Mathematik und Naturwissenschaften, IMACM,

Gaußstrasse 20, 42119 Wuppertal, Germany, e-mail: {bolten, doganay,ehrhardt,

guenther,hgotsch, chahn,jaeschke, klamroth,reese, jschultes,

stiglmayr}@uni-wuppertal.de

Vincent Marciniak, Alexander Liefke and Peter Jaksch

Siemens AG, Power and Gas, Common Technical Tools Mellinghoffer Str. 55, 45473 M¨

ulheim

an der Ruhr, Germany, e-mail: {vincent.marciniak,alexander.liefke,peter.

jaksch}@siemens.com

Daniel Luft and Volker Schulz

Universit¨

at Trier, Fachbereich IV, Research Group on PDE-Constrained Optimization, 54296

Trier, Germany, e-mail: {luft,volker.schulz}@uni-trier.de

Lucas M¨

ade, Johannes Steiner and Sebastian Schmitz

Siemens Gas and Power GmbH & Co. KG, Probabilistic Design, GP PGO TI TEC PRD, Huttenstr.

12, 10553 Berlin, Germany, e-mail: {lucas.maede,johannes.steiner,schmitz.

sebastian}@siemens.com

Benedikt Engel

University of Nottingham, Gasturbine and Transmission Research Center (G2TRC), NG72RD

Nottingham, United Kingdom, e-mail: engel.benedikt@nottingham.ac.uk

1

2 The GivEn Consortium

multicriteria optimization procedures to enhance the exploration of Pareto-optimal

solutions.

1 Introduction

The diverse applications of gas turbines in the context of the energy system trans-

formation, such as backup power plants or hydrogen turbines, go hand in hand

with speciﬁc design requirements, in particular with regard to the efﬁciency of en-

ergy conversion and the reliability and ﬂexibility of operation. These different re-

quirements are intensively related to the coupled ﬂuid dynamic simulation and the

structural mechanical fatigue calculation. The use of integrated, multi-physical tool

chains and optimization software therefore plays an important role in gas turbine

design. This joint project links six different simulations – ﬂuid dynamics, laminar

convective heat transfer, 1D ﬂux networks and turbulent convective heat transfer,

heat conduction, structural mechanics, probabilistic modelling of material fatigue

– which are computed on a complex turbo geometry. These simulations are cou-

pled in the multi-objective shape optimization process. See Fig. 1 for a schematic

illustration of the multi-physical simulation/optimization cycle.

heat con-

duction

cooling

channels

CFD-model

gradient-based

multiobjective

optimization

thermo-

mechanical

equation

temperature

distribution

boundary

conditions

aerodynamic

objective

LCF

objective

geometry

update

of the

component

Fig. 1 Information ﬂow and dependencies between project parts.

GivEn – Shape Optimization for Gas Turbines in Volatile Energy Networks 3

The challenge for the GivEn project thereby is to adjoin a highly multi-physical

simulation chain with continuous coupling, to determine form gradients and form

Hessians with respect to different objectives, and to make it usable in the multicri-

teria optimization for the turbine design process.

Coupled multi-physics simulation is an ongoing topic in turbo-machinery. For

recent surveys of these ﬂuid dynamics and heat transfer topics, see e.g. [101, 103].

The important topic of turbo-machinery life calculation is often treated separately

and from a materials science point of view, see e.g. [15, 19, 80]. In contrast to

the traditional separation of the mechanical and the ﬂuid dynamics properties, the

approach we follow in GivEn preserves a holistic viewpoint.

The algorithmic optimization of turbo-machinery components by now has a long

history. While in the beginning genetic algorithms were used predominantly, in re-

cent times data driven methods like Gaussian processes or (deep) neural networks

predominate [1, 21, 98]. The strength of such procedures lie in global search ap-

proaches. As an alternative, gradient based optimization using adjoint equations are

seen as a highly effective local search method [34, 38], see also [59] for a recent

review including a comparison of the methods and [6] for bringing the data driven

and the adjoint world together using gradient enhanced Gaussian processes [1, 33].

When combining the challenge of multi-physics and multi-criteria optimiza-

tion, it would be desirable to treat mechanical and ﬂuid dynamic aspects of turbo-

machinery design on the same footing. A necessary prerequisite for this is the prob-

abilistic modelling of the mechanisms of material damage, as this enables the ap-

plication of the adjoint method [14, 39, 41, 42, 43, 44, 91, 68]. This is not possible

with a deterministic calculation of the lifetime of the weakest point, as taking the

minimum over all points on the component is a non differential operation.

The GivEn consortium exploits these new opportunities for multi-criteria and

multiphysics optimization. It brings together a leading original equipment manufac-

turer (Siemens Power and Gas), technology developing institutions (German Aero

Space Center (DLR) and Siemens CT) as well as researchers from academia (Uni-

versities of Trier and Wuppertal). Since 2017 this consortium addresses the chal-

lenges described in a joint research effort funded by the BMBF under the funding

scheme ”mathematics for innovation”. With the present article, we review the re-

search done so far and give an outlook on future research efforts.

This paper is organized as follows. In Section 2 we describe our research work on

the different physical domains including the usage of adjoint equations, improved

shape gradients and gradient based multi-criteria optimization. Following the de-

sign scheme outlined in Figure 1, we start with aerodynamic shape optimization in

Section 2.1 using modern mesh morphing based on the Steklov-Poincar´

e deﬁnition

of shape gradients [92, 93, 94], then proceed to heat transfer and the thermal loop

in Section 2.2. Section 2.3 includes related probabilistic failure mechanisms. The

model range from empirical models based on Weibull analysis and point processes

to elaborate multi scale models. Section 2.4 presents shape optimization methods

that are based on the probability of failure and develops a highly efﬁcient compu-

tational framework based on conformal ﬁnite elements. Section 2.5 presents novel

fundamental results on the existence of Pareto fronts in shape optimization along

4 The GivEn Consortium

with algorithmic developments in multi-criteria gradient based shape optimization

including scalarization, bi-criteria gradient descent and gradient enhanced Gaussian

processes.

In Section 3 we describe the industrial perspective from the standpoint of the

DLR and Siemens energy. While in Section 3.1 the DLR gives a description of the

interfaces with and the possible impact to the DLR’s own R&D roadmap, Siemens

Power & Gas in Section 3.2 relates adjoint based multi-criteria optimization with

adjoint multi-criteria tolerance design and presents an application on real world geo-

metries of 102 casted and scanned turbine vane geometries.

Let us note that this work is based on the papers [13, 20, 26, 41, 42, 60, 61, 62,

66, 68] that have been published with (partial) funding by the GivEn consortium so

far. As this report is written after about half of the funding period of the project, we

also give comments on future research plans within GivEn and beyond.

2 Areas of Mathematical Research and Algorithmic

Development

The project GivEn researches the multiobjective free-form optimization of turbo ge-

ometries. For this purpose, the thermal and mechanical stress of the turbine blades

and their aerodynamic behavior must be modelled, simulated and optimised. In the

following we describe the components of the multiphysical simulation and opti-

mization, namely aerodynamic shape optimization, heat transfer and thermal loop,

probabilistic objective functionals for cyclic fatigue, shape optimization for proba-

bilistic structure mechanics, multiobjective optimization, and probabilistic material

science.

2.1 Aerodynamic Shape Optimization

Shape optimization is an active research ﬁeld in mathematics. Very general basic

work on shape calculus can be found in [99, 50, 27]. Aerodynamic investigations

can be found in [87, 89]. New approaches understand shape optimization as the

optimization on shape manifolds [92, 105] and thus enable a theoretical framework

that can be put to good practical use, while at the same time leading to mathematical

challenges, as no natural vector space structure is given. Otherwise, applications

usually use ﬁnite dimensional parameterizations of the form, which severely limits

the space of allowed shapes.

In the shape space setting, the use of volume formulations has been shown in

combination with form metrics of the Steklov-Poincar´

e type [94, 93] were shown

to be numerically very advantageous, since the volume formulation in comparison

to the formally equivalent boundary formulation for canonical discretizations have

better approximation properties and also weaker smoothness requirements of the

GivEn – Shape Optimization for Gas Turbines in Volatile Energy Networks 5

functions involved. Additionally the Steklov-Poincar´

e type metrics require a free

combination of volume and boundary formulations together with an inherently good

approximation of the Shape-Hessian operators.

In order to exploit these theoretical advances for industrial applicability meet-

ing high-end standards, the TRASOR (TRACE Shape Optimization Routine) soft-

ware package for non-parametric shape optimization routines has been created. This

software package is built on several solver bundles connected by an interface in

Python 2.7 and 3.5. One major package bundle provided by the DLR and incor-

porated in TRASOR is TRACE 9.2, which is an interior ﬂow simulator (cf. [10]).

The TRASOR software incorporates shape gradient representations using Steklov-

Poincar´

e-metrics (cf. [94, 93]) based on shape sensitivities derived by automatic

differentiation provided by adjointTRACE [81, 6].

TRASOR also interfaces with FEniCS 2017.2.0 [5, 63], which is a Python based

ﬁnite element software utilizing several sub-modules, such as the Uniﬁed Form Lan-

guage (UFL [4]), Automated Finite Element Computing (DOLFIN [64, 65]) and

PETSc [7] as a linear algebra backend, in order to solve differential equations based

weak formulations. Various solver options, including CG, GMRES, PETCs’s built

in LU solver and preconditioning using incomplete LU and Cholesky, SOR or alge-

braic multigrid methods are available in FEniCS and thus applicable in TRASOR.

FEniCS/PETSc also offers the possibility to parallelize ﬁnite element solving, mak-

ing the Steklov-Poincar´

e gradient calculation scalable in processor number.

Features of the software package TRASOR include

–automatic ﬁle generation and management for TRACE and adjointTRACE

–interface between TRACE and FEniCS, including automatic FEniCS mesh gene-

ration from .cgns ﬁles

–steepest descent optimization using TRACE intern gradients

–steepest descent optimization using Steklov-Poincar´

e gradients calculated in

FEniCS

–target parameter selection for various parameters found in TRACE, including all

parameters listed in [37]

–generation of .pvd and .vtu ﬁles of gradients, sensitivities, meshes and ﬂow simu-

lation data for visual post processing

TRASOR features are tested on the low-pressure turbine cascade T106A de-

signed by MTU Aero Engines (cf. [53]). The algorithm using Steklov-Poincar´

e gra-

dients is outlined in Algorithm 1.

In order to exploit FEniCS it is necessary to create an unstructured computational

mesh with vertices prescribed by TRACE. As FEniCS 2017.2.0 is not fully capa-

ble of supporting hexahedral and quadrilateral elements (this should be available

with FEniCS 2020), hexahedral and quadrilateral elements used in TRACE are par-

titioned to conforming tetrahedral and triangular elements respecting the structured

TRACE mesh. The conversion process including the data formats for TRACE to

FEniCS mesh conversion are depicted in Fig. 2 (cf. [86, 65])

6 The GivEn Consortium

TRACE.cgns TRACE.dat FEniCS.msh FEniCS.xdmf/-

.h5 FEniCS mesh

POST TRASOR meshio DOLFIN

Fig. 2 TRACE to FEniCS Pipeline

For representing the TRACE generated mesh sensitivities Dad J(Ωext,k)as a

Steklov-Poincar´

e gradient a sufﬁcient metric has to be chosen. According to [94],

we implemented the following linear elasticity model

ZΩext,k

σ∇StP J(Ωext,k):ε(V)dx =Dad J(Ωext,k)[V]∀V∈H1

0(Ωext,k,Rd)

∇StP J(Ωext,k) = 0 on Γ

Inlet/Outlet

σ(V) = λTrε(V)I+2µε(V)

ε(V) = 1

2(∇V+∇V>),

(1)

where λ∈R,µ∈R+are the so called Lam´

e parameters. If Dis the entire duct

including the shape of the turbine blade Ωkat iteration kof the shape optimization

procedure, Ωext,k=D\Ωkis the external computational domain where the ﬂuid

dynamics takes place. Continuous Galerkin type elements of order one are used

for target and test spaces in the FEniCS subroutine conducting the shape gradient

calculation.

An exemplary comparison of a Steklov-Poincar´

e gradient calculated by solving

the linear elasticity system (1) with Lam´

e parameters λ≡0 and constant µ>0, and

a TRACE gradient, which is generated by solving a linear elasticity mesh smooth-

ing system with Dirichlet boundaries being the lattice sensitivities DadJ(Ωext,k), for

the isentropic total pressure loss coefﬁcient in relative frame of reference based on

dynamic pressure is portrayed in Fig. 3. We can see additional gain of regularity in

the gradient through Steklov-Poincar´

e representation, in particular the pronounced

rise in sensitivity at the trailing edge is handled by redistributing sensitivities at the

pressure side in a smooth manner, thus guaranteeing better stability of the mesh

morphing routine.

An in-depth comparison of shape optimization routines involving both types of

gradient representation will be subject of a follow-up study. Further, a Steklov-

Poincar´

e gradient representation using different bilinear forms matching the shape

Hessian of the RANS ﬂow and the target at hand are object of further studies, which

might open new possibilities with superior convergence and mesh stability behavior.

The following Algorithm 1 is a prototype for a shape optimization problem, in-

cluding the Steklov-Poincar´

e gradient representation.

GivEn – Shape Optimization for Gas Turbines in Volatile Energy Networks 7

(a) Steklov-Poincar´

e gradient (b) TRACE Mesh Smoothing gradient

(c) FEniCS computational mesh of the T106A

Fig. 3 Comparison of Steklov-Poincar´

e gradient (upper left) with TRACE Mesh Smoothing gra-

dient (upper right) on a FEniCS computational mesh of the T106A (lower center)

1Set ﬂow parameters in TRACE.cgns, optimization parameters and targets in TRASOR.py

2Build TRASOR ﬁle architecture

3Assemble and load FEniCS data from TRACE.cgns

4while k∇StP J(Ωext,kk>εshape do

5Flow simulation and (AD) checkpoint creation using TRACE

6Calculate mesh sensitivities by automatic differentiation using adjointTRACE

7Pass mesh sensitivities to FEniCS setup

8Generate Steklov-Poincar´

e gradient in FEniCS:

9Calculate Lam´

e-Parameters

10 Solve linear elasticity problem (1)

11 Extract target and ﬂow values to update/ create protocols and .pvd/.vtu ﬁles

12 Deform FEniCS mesh using FEniCS Steklov-Poincar´

e gradient and ALE (Arbitrary

Lagrangian-Eulerian)

13 Create TRACE deformation.dat ﬁles from FEniCS Steklov-Poincar´

e gradient

14 Deform TRACE mesh using PREP

15 end

Algorithm 1: TRASOR algorithm using Steklov-Poincar´

e gradients

8 The GivEn Consortium

2.2 Heat Transfer and the Thermal Loop

The numerical simulation of coupled differential equation systems is a challenging

topic. The difﬁculty lies in the fact that the (P)DEs involved may differ in type and

also in order, and thus require different types and quantities of boundary conditions.

[8] The question of the correct coupling is closely related to the construction of so-

called transparent boundary conditions, which are based on the coupling of interior

and exterior solutions.

The numerical simulation of coupled differential equation systems by means of

co-simulation has the innate advantage that one can choose the optimal solver for

each sub-system, for example by employing pre-existing simulation software. Most

of the work done in this ﬁeld concerns transient, i.e. time dependent, problems. In

our case, however, we are interested in steady state systems which rarely get special

attention in current research.

Our model problem arises from the heat ﬂow in a gas turbine blade. Since higher

combustion temperatures result in better efﬁciency [84], engineering always strives

for means to achieve these. However, this is limited by the material properties of the

turbine blade, especially its melting point. One way to mitigate this, is by cooling

the blade from the inside. This is done by blowing air through small cooling ducts.

These ducts have a complex geometry to increase turbulence of the airﬂow and

maximize heat transfer from the blade to the relatively cool air. For an overview,

see for example [49]. Due to the small length-scales and high turbulence, regular

ﬂuid dynamics simulation techniques are infeasible for the simulation of the airﬂow

within the ducts. Instead, they are modeled as a one-dimensional ﬂow with paramet-

ric models for friction and heat transfer, similar to the work in [72] and [101].

w∂v

∂x=A∂p

∂x+A

2Dh

fρv2+Aρω2r∂r

∂x

∂(vT )

∂x=S

ρ=w

vA

p=ρRsT

(2)

Here, wis the mass ﬂow through the channel which is assumed constant (i.e. only

one inlet and outlet), vis the ﬂuid velocity, ρis the ﬂuid’s density and pand Tdenote

the pressure and Temperature of the ﬂuid, respectively. Ais the cross-sectional area

and Dhthe hydraulic diameter of the channel, with fas the Fanning friction factor.

ωand rare only relevant in the rotating case and denote the angular velocity and

distance from the axis of rotation. Rsis the speciﬁc gas constant of the ﬂuid and S

denotes the heat source term from the heat ﬂux through the channel walls.

The system (2) takes the form of a DAE, but can be transformed into a system of

ODEs by some simple variable substitutions. Since the physical motivation behind

the terms is easier to understand in the DAE form, this is omitted here.

GivEn – Shape Optimization for Gas Turbines in Volatile Energy Networks 9

The heat conduction within the blade material is modeled by a PDE. In the tran-

sient case, this would be a heat equation. In the stationary case, it is given by a

Laplace equation. The heat transfer across the boundary is given by Robin bound-

ary conditions, that prescribe a heat ﬂux across the boundary depending on the tem-

perature difference between “inside” and “outside”. Temperature in (3) is denoted

by Uto signify that it is mathematically a different entity than the temperature in

the cooling channel, denoted by Tin (2). kis the thermal conductivity of the blade

metal, while hint and hext are the heat transfer coefﬁcients of the internal and external

boundary.

k∇2U=0 on Ω

−k∂U

∂n=hint(U−Uint)on ∂ Ωint

−k∂U

∂n=hext(U−Uext)on ∂ Ωext

(3)

The coupling between equations (2) and (3) is realized via the boundary condi-

tion, more speciﬁcally the internal boundary temperature Uint as a function of T, on

the conduction side and the source term Sin the cooling duct equations, which is a

function of the values of ∂U/∂non the cooling duct boundary.

The coupled system is discretized using a ﬁnite elements scheme for the con-

duction part (3). This is done, because it ensures we can choose the mesh for the

conduction part in a way that it is identical with the mesh used for the structural me-

chanics simulation described in Section 2.4, that uses the calculated temperatures as

an input. For the cooling duct part (2), we use a ﬁnite volume scheme, as that makes

it easier to have energy conservation across the boundary and provides a clear map-

ping of the PDE boundary to cooling channel elements. The resulting discretized

system is then solved by solving each subsystem and updating the boundary condi-

tion respectively the right hand side of the other system, alternating between the two

subsystems until the solutions of two consecutive iterations differ by a sufﬁciently

small margin. This back-and-forth iteration is reminiscent of a Gauß-Seidel iteration

scheme, or more general, a ﬁxed-point iteration.

Numerical tests have shown that this iterative solution indeed exhibits linear con-

vergence as seen in Fig. 4, with the solution behaving like a dampened oscillation

approaching the ”correct” solution. These numerical tests also indicated that the

convergence is not unconditional, but depends on the parameter values chosen for

the system, especially the thermal conductivity kand the heat transfer coefﬁcients

h. High values of hlead to divergence and turn the aforementioned dampened oscil-

lation into one with an exponentially increasing amplitude as seen in Fig. 5.

10 The GivEn Consortium

Fig. 4 Behavior of the cooling duct outlet temperature (left) and error-estimate (right) for a con-

verging set of parameters (hint =hext =5000,k=25)

Fig. 5 Behavior of the cooling duct outlet temperature (left) and error-estimate (right) for a diverg-

ing set of parameters (hint =hext =5000,k=40)

2.3 Probabilistic Objective Functionals for Material Failure

Since the pioneering work of Weibull [104], the probabilistic modelling of material

failure has been an established ﬁeld of material science, see about [9]. Applications

to the Low Cycle Fatigue (LCF) damage mechanism can be found in [76, 29, 100]. In

these studies, crack formation is modelled by percolation of intra-granular cracks or

by kinetic theory for the combination of cracks. The mathematical literature mainly

contains generic volume or surface target functions without direct material refer-

ence. In numerical studies, global compliance is usually chosen as the objective

functional, which also does not establish a direct relationship to material failure, see

e.g. [18].

The objective functional used in GivEn for the probability of failure originates

[43, 44, 68] see also [39] for multi-scale modeling. A connection between proba-

bilistic functional objectives of materials science and the mathematical discipline of

shape optimization is produced for the ﬁrst time in [43], see also [12, 14, 13].

GivEn – Shape Optimization for Gas Turbines in Volatile Energy Networks 11

The aim in this sub-area is the probabilistic modelling of material damage mech-

anisms and the calculation of form derivatives and form Hessian operators for the

failure probabilities of thermal and mechanically highly stressed turbine blades.

The physical cause of the LCF mechanism in the foreground is the sliding of

crystal dislocations along lattice planes with maximum shear stress and is therefore

dependent on the random crystal orientation. For this reason, so-called intrusions

and extrusions occur at the material surface, which eventually lead to crack forma-

tion [15, 79]. In an effective approach, this scattering of material properties can be

empirically investigated within the framework of reliability statistics.

In the deterministic approach prevalent in mechanical engineering, life expectancy

curves are used to determines the service life at each point of the blade surface. The

shortest of these times is to failure over all points is then, under consideration of

safety discounts, converted to the permitted safe operating time of the gas turbine.

The minimum formation inherent in this process means that the target functions

cannot be differentiated. Probabilistic target functions, on the other hand, can be

deﬁned according to the form and continuously adjusted.

In particular, the stability of discretization schemes must be examined in both

with regard to geometric approximation of the forms as well as the solutions. The

background is that H1solutions are insufﬁcient for a ﬁnite probabilistic target func-

tional, especially if notch support is also considered [67]. Sufﬁce of this must be

used a Wk,psolution and approximation theory [17].

Next, the calculation of the shape gradients and shape Hesse operators of the

functionals essentially follows [99], with open questions about the existence and

properties of shape gradients for the surface- and stress-driven damage mechanism

LCF still to be clariﬁed. This program has been started within the GivEn research

initiative, cf. [11]. Analogous to [43], the solution strategy is based on a uniform

regularity theory for systems of elliptic PDEs, cf. [2, 3, 17]. In particular, the math-

ematical status of the continuously-adjusted equation deserves further attention, as

this has a high regularity loss for surface-driven LCF.

In the following we present a hierarchy of probabilistic failure models that give

rise to objective functionals related to reliability. We start with the simple Weibull

model, proceed with a probabilistic model for LCF proposed by [91, 68] and then

give an outlook on the multi-scale modeling of the scatter in probabilistic LCF, see

[26].

2.3.1 The Weibull Model via Poisson Point Processes

Technical ceramic has multiple properties such as heat or wear resistance that make

them a widely used industrial material. Different to other industrial material, the

physical properties of ceramic materials highly depend on the manufacturing pro-

cess. What determines the failure properties the most, are small inclusions that stem

from the sintering process. These make ceramic a brittle material, leading to a some-

what high possibility of failure of the component under tensile load often before the

ultimate tensile strength is reached [15].

12 The GivEn Consortium

When applying tensile load, these inclusions may become the initial point of a

crack, developing into a rupture if a certain length of the radius of the crack is ex-

ceeded at a given level of tensile stress. Therefore, the probability of failure under

a given tensile load is the probability that a crack of critical length occurs. Or to

phrase it differently, the survival probability in this case is the probability that ex-

actly zero of these critical cracks occur. Thus, for a given domain Ω⊂Rd,d=2,3

with a suitable counting measure N[54], we can express the failure probability in

the following way,

PoF(Ω) = 1−PN(Ac(Ω)) = 0,(4)

where Ac(Ω)) is the set of critical cracks. The probability, that one of the inclusions

grow into a critical crack, mainly depends on the local stress tensor σn(u), which

itself is determined by a displacement ﬁeld u∈H1(Ω,Rd), that is the solution of

a linear elasticity equation. As there is no other indication, it is feasible to assume

that the location, size and orientation of the initial inclusions are independent of

each other and uniformly randomly distributed. Under these assumptions, it follows

that the counting measure N(Ω)is a Poisson point process (PPP). Taking further

material laws into account it follows that [14]

PoF(Ω|u) = 1−P(N(Ac(Ω,u)) = 0) = 1−exp{−ν(Ac(Ω,u))},(5)

with the intensity measure of the PPP

ν(Ac(Ω,u)) = Γ(d

2)

2πd

2Z

ΩZ

Sd−1

∞

Z

ac

dνa(a)dndx.(6)

With some reformulations we ﬁnd our objective functional of Weibull type

J1(Ω,u):=ν(Ac(Ω,u)) = Γ(d

2)

2πd

2Z

ΩZ

Sd−1σn

σ0m

dndx.(7)

This functional (7) will be one of the objective functionals in the following (multi-

objective) gradient based shape optimization.

2.3.2 Probabilistic Models for LCF

Material parameters relevant for fatigue design, like the HCF fatigue resistance

were considered as a random variable for a long time [97, 75, 79] and distribu-

tions and their sensitivities were even recorded in general design practice standards

[74, 52, 30, 31]. The existence of ﬂaws, such as crystal dislocations, non-metallic

inclusions or voids in every material has early lead to the discovery of the statistical

size effect [78, 70, 71, 51]. Within the last decade, a local probabilistic model for

LCF based on the Poisson point process was developed by Schmitz et al. [91, 90]

GivEn – Shape Optimization for Gas Turbines in Volatile Energy Networks 13

for predicting the statistical size effect in any structural mechanics FEA model. It

approximates the material LCF life statistics with a Weibull distribution which al-

lows developing a closed form integral solution for the distribution scale η(see

equation (10)). Recently, M¨

ade et al. [68] presented a validation study of the com-

bined size and stress gradient effect modeling approach within the framework of

Schmitz et al. [91]. If stress gradients are present in components, they can have an

increased LCF life. The beneﬁt is proportional to the stress gradient but also mate-

rial dependent [28, 95, 96, 106]. A stress gradient support factor nχ=nχ(χ|ϑ),

a functional of the normalized stress gradient χand material-speciﬁc parameters ϑ

[96], is introduced to quantify the effect. While the size effect as described by the

surface integral (10) causes an actual delay or acceleration in fatigue crack initia-

tion, researchers share the interpretation that stress gradient support effects in LCF

root back to retarded propagation of meso-scale cracks in the decreasing stress ﬁeld

[79, 58, 69, 56, 22, 82]. In order to compare the stress gradient effect for different

materials, a common detectable, “technical” crack size must be deﬁned. Since the

stress gradient χ(x)is, like the stress ﬁeld, a local property, it was integrated into the

calculation of the local deterministic life Ndet(x)with the Cofﬁn-Manson-Basquin

model:

εa(x)

nχ(χ(x)|ϑ)=σ0

f

E·(2Ndet(x))b+ε0

f·(2Ndet(x))c.(8)

Here, the stress is computed with the aid of the linear elasticity equation, which this

time is not a technical tool for smoothing gradients as in (13), but represents the

physical state, namely

∇·σ(u) + f=0 in Ω

σ(u) = λ(∇·u)I+µ(∇u+∇u>)in Ω

u=0 on ∂ ΩD

σ(u)·n=gon ∂ ΩN.

Here, Ωrepresents the component, λ>0 and µ>0 are Lam´

e coefﬁcients and u:

Ω→R3is the displacement ﬁeld on Ωobtained as a reaction to the volume forces

fand the surface loads g. We connect the topic of optimal probabilistic reliability

to shape optimization elasticity PDE as state equation and classify Poisson point

process models according to their singularity [11]. Following [91, 68], we obtain

for the probability of failure at a number of use cycles n

PoF(Ω,n) = 1−e−nmJR(Ω,u)(9)

The functional J(Ω,u)that is arising out of this framework is given by:

JR(Ω,uΩ):=Z∂Ω 1

Ndet(∇uΩ(x),∇2uΩ(x)) m

dA.(10)

14 The GivEn Consortium

low high

0,3

1

size eﬀect shift

χ-eﬀect

shift

Ni(Cycle,log)

εa(%,log)

Fit, smooth specimen

Fit, notch specimen

Pred., Cool-Hole Spec.-7

Pred., Cool-Hole Spec.-5

η=R∂Ω

1

Nm

det(χ(x),εa(x)) dA−1/m

Combined size- and stress gradient eﬀect modeling

Fig. 6 Strain W ¨

ohler plot of LCF test data, calibrated (dashed) and predicted (solid) median curves

for smooth (), notch (+) and cooling hole specimens (♦,×). All W¨

ohler curves are interpolated

median values of the Weibull LCF distributions exemplary indicated with the thin density function

plot.

Ndet denotes the deterministic numbers of life cycles at each point of the surface of

the component and mis the Weibull shape parameter.

M¨

ade et al. have calibrated the material parameters ϑ,E,σ0

f,ε0

f,b,cas well as

the Weibull shape parameter mwith the Maximum-Likelihood method simultane-

ously using smooth and notch specimen data simultaneously [67, 68]. The resulting

model was able to predict the LCF life distribution for certain component-similar

specimens (see Fig. 61).

In the following, we apply this model as cost functional in order to optimize the

component Ωw.r.t. reliability.

2.3.3 Multi-Scale Modeling of Probabilistic LCF

While the Weibull-based approach from the previous subsection allows a closed-

form solution and therefore fast risk assessment computation times, the microstruc-

tural mechanisms of LCF suggest a different distribution shape [73]. Since this is

not yet assessable by LCF experiments in a satisfying way, Engel et al. have used

numerical simulations of probabilistic Schmid factors to create an LCF model con-

sidering the grain orientation distribution and material stiffness anisotropy in cylin-

drical Ni-base superalloy specimens [26].

Polycrystalline FEA models were developed to investigate the inﬂuence of local

multiaxial stress states a result of as grain interaction on the resulting shear stress in

the slip systems. Besides isotropic orientation distributions also the case of a pref-

1Reprinted from Comp. Mat. Sci., 142, (2018) pp. 377–388, M¨

ade et al., Combined notch and size

effect modeling in a local probabilistic approach for LCF, Copyright (2017), with permission from

Elsevier

GivEn – Shape Optimization for Gas Turbines in Volatile Energy Networks 15

erential orientation distribution was analysed. From the FE analyses, a new Schmid

factor distribution, deﬁned by the quotient of max(τrss)maximum resolved shear

stress at the slip systems and von Mises stress σvM, was derived as a probabilistic

damage parameter at each node of the model. Qualitatively as well as quantitatively,

they differ largely from the single grain Schmid factor distribution of Moch [73]

and also from the maximum Schmid factor distribution of grain ensembles pre-

sented by Gottschalk et al. [45]. Experimental LCF data of two different batches

presented by Engel [25, 24] showed different LCF resistances and microstructural

analyses revealed a preferential grain orientation in the specimens which withstood

more cycles (see Fig. 7). By combining the Schmid factor based LCF life model of

Moch [73] and the Schmid factor distribution generated by FEA, Engel et al. were

able to predict just that LCF life difference [26]. Ultimately, it was found that the

microstructure-based liﬁng model is able to predict LCF lives with higher accuracy

than the Weibull approach by considering the grain orientation and their impact on

the distributions of Young’s moduli and maximum resolved shear stresses. How-

ever, the application is computationally demanding and its extension to arbitrary

components still has to be validated.

Fig. 7 Strain W ¨

ohler plot of LCF test data, calibrated (dashed) and predicted (solid) median curves

for specimens with isotropic and preferential grain orientation distribution (coarse and ﬁne grain)

by Engel et al. [26]. Calibration and prediction was carried out using the Schmid factor based LCF

life distribution. The underlying Schmid factor distribution was derived from polycrystalline FEA

simulations which considered the lattice anisotropy and orientation distribution in both specimen

types.

2.4 Shape Optimization for Probabilistic Structure Mechanics

This sub-area deals with two kinds of failure mechanisms, failure of brittle material

under tensile load and low cycle fatigue (LCF). As explained in the preceding sec-

tion, the probability of failure for both failure mechanisms can be expressed as local

16 The GivEn Consortium

Fig. 8 Visualization of adaption of the grid2

integral of the volume (ceramics) or surface (LCF) over a non-linear function that

contains derivatives of the state uΩsubject to the elasticity equation (2.3.2).

Since problems in shape optimization generally do not result in a closed solution,

the numerical solution plays a major role, e.g. in [50]. Typically, the PDEs occurring

as a constraint are discretized using the ﬁnite element method. Here, a stable and

accurate mesh representation of Ωis needed for stable numerical results. Deciding

for a mesh always means balancing the need for accuracy of the representation on

the one hand and on the other hand the time to solution. Especially in applications

such as shape optimization, where usually hundreds to thousands of iterations and

thus changes in the geometry and mesh are needed to ﬁnd a converged solution, the

meshing in each iteration often becomes a bottle neck in terms of computational

cost. Recent research therefore aims to ﬁnd methods to move the grid points of

a given representation in a stability preserving way, rather than to perform a re-

meshing. These approaches result in unstructured grids. To exploit the means of

high performance computing however, structured grids are way more desirable. This

let us to consider an approach ﬁrst developed in [46, 47, 48]. For demonstration

purposes, the technique is described in two dimensions but easily extends to three

dimensions.

We consider a rectangular domain ˜

Ωwhich is discretized by a regular triangu-

lar grid. We assume that all admissible shapes that occur during the optimization

process lie in this domain. The regular grid on ˜

Ωis denoted by ˜

T, the number of

elements by Nel and the number of nodes by Nno. In a second step, the boundary

δ Ω0of the shape to be optimized Ω0is superimposed onto the grid, see Fig. 8a.

The regular grid is then adapted to the boundary by moving the closest nodes to the

intersections of the grid and the boundary, see Fig. 8b. The adapted grid is denoted

by ˜

T0. As the nodes are moved only, the connectivity of ˜

T0is the same as before.

During the adaption process, the cells of ˜

T0are assigned a status as cells lying in-

side or outside of the component. The computations are only performed on those

cells that are inside the component. When updating the grid according to the new

shape of the domain Ω1and so forth, the adaption process starts with ˜

Ωagain, while

taking into account the information about the previous domain, the step length and

2Reprinted from Progress in Industrial Mathematics at ECMI 2018, pp. 515-520, Bolten et al.,

Using Composite Finite Elements for Shape Optimization with a Stochastic Objective Functional,

Copyright: Springer Nature Switzerland AG 2019

GivEn – Shape Optimization for Gas Turbines in Volatile Energy Networks 17

(a) Standard gradient (b) Smoothed gradient

Fig. 9 Gradient in standard scalar product and smoothed gradient; Nx=64,Ny=32

search direction that led to the current domain. By this, only the nodes that lie in

a certain neighborhood of the previous boundary have to be checked for adaption.

This leads to a speed up in meshing compared to actual re-meshing techniques.

Additionally, as the grid is otherwise regular and the connectivity is kept, the gov-

erning PDE only has to be changed in entries representing nodes on elements that

have been changed, hence no full assembly is needed, which is an advantage to both

re-meshing and mesh morphing techniques.

The objective functional (7) is discretized via ﬁnite elements. For reduction of

the computational cost, the adjoint approach than leads to the derivative

dJ1X,U(X)

dX =∂J1(X,U)

∂X+Λ∂F(X)

∂X−∂B(X)

∂XU(11)

B>(X)Λ=∂J1(X,U)

∂U(12)

B(X)U=F(X),(13)

with (12) being the adjoint equation giving the adjoint state and (13) is the dis-

cretized linear elasticity equation, giving the discrete displacement U.Xrepresents

the discretized domain Ω.

With (13) and (12) the derivative (11) is calculated on the structured mesh as

visualized in Fig. 9a. For the optimization, more closely described in the follow-

ing section 2.5, the gradient is smoothed using a Dirchlet-to-Neumann map [88]

(see Fig. 9b). This provides the shape gradients needed for further gradient based

optimization steps in the following section.

18 The GivEn Consortium

2.5 Multiobjective Optimization

The engineering design of complex systems like gas turbines often requires the con-

sideration of multiple aspects and goals. Indeed, the optimization of the reliability

of a structure usually comes at the cost of a higher volume and, hence, a higher

production cost. Other relevant optimization criteria are, for example, the minimal

buckling load of a structure or its minimal natural frequency [50]. In this section,

we consider both the mechanical integrity and the cost of a ceramic component in

a biobjective PDE constrained shape optimization problem. Further objective func-

tions can (and should) be added to the model depending on the application at hand.

Towards this end, we model the mechanical integrity J1(Ω,u)of a component Ω

as described in Section 2.4, see (7), while the cost J2(Ω)is assumed to be directly

proportional to the volume of the component, i.e., J2(Ω) = RΩdx.

Multiobjective shape optimization including mechanical integrity as one objec-

tive is widely considered, see, e.g., [16] for a recent example. Most of these works

neither consider probabilistic effects nor use gradient information. The formulation

introduced in Section 2.4 overcomes these shortcomings. It was ﬁrst integrated in

a biobjective model in [20], where two alternative gradient-based optimization ap-

proaches are presented. We review this approach and present new numerical results

based on structured grids and advanced regularization.

2.5.1 Pareto Optimality

Multiobjective optimization asks for the simultaneous minimization of pconﬂicting

objective functions J1,...,Jp, with p≥2. We denote by J(Ω) = (J1(Ω),...,Jp(Ω))

the outcome vector of a feasible solution Ω∈Oad (i.e., an admissible shape). Since

in general the optimal solutions of the objectives J1,...,Jpdo not coincide, a mul-

tidimensional concept of optimality is required. The so-called Pareto optimality is

based on the component-wise order [23]: A solution Ω∈Oad is Pareto optimal

or efﬁcient, if there is no other solution Ω0∈Oad such that J(Ω0)6J(Ω), i.e.,

Ji(Ω0)≤Ji(Ω)for i=1,...,pand J(Ω0)6=J(Ω). In other words, a solution is efﬁ-

cient if it can not be improved in one objective Jiwithout deterioration in one other

objective function Jk.

2.5.2 Foundations for Multi-Physics Multi-Criteria Shape Optimization

The existence of Pareto fronts for the multi-criteria case are considered in a simpli-

ﬁed analytical model replacing the RANS equations by potential theory with bound-

ary layer losses. Pareto fronts can be replaced by scalarizations using the techniques

from [43, 12] or [35, 14]. Their continuous course is investigated by variation of the

scalarization and the associated optimality conditions. The convergence of the dis-

cretized Pareto optimum solutions against the continuous Pareto optimum solution

shall be studied according to the approach of [50].

GivEn – Shape Optimization for Gas Turbines in Volatile Energy Networks 19

Optimizing the design of some component in terms of reliability, efﬁciency and

pure performance includes the consideration of various physical systems that in-

teract with each other. This leads naturally to a multicriteria shape optimization

problem over a shape space Owith requirements represented by cost functionals

J= (J1,...,Jl):

(Find Ω∗∈Osuch that

(Ω∗,vΩ∗)is Pareto optimal w.r.t. J.

The class of cost functional we use to model the requirements on the compo-

nent are connected with physical state equations and arise from the probabilistic

framework. They are described by

Jvol(Ω,v):=ZΩ

Fvol (x,v,∇v,...,∇kv)dx,

Jsur(Ω,v):=Z∂ Ω

Fsur (x,v,∇v,...,∇kv)dA.

Due to the fact that the event of failure, e.g. the crack initiation process takes place

on the surface of the component, the physical systems need to fulﬁll regularity con-

ditions in order to be includable in this setting. We describe the possible designs of

the component in this shape optimization problem by H¨

older-continuous functions

which give us the possibility to freely morph the shapes in various designs while

remaining the premised regularity conditions. In this situation uniform regularity

estimates for the solutions of the physical system are needed in order to ensure the

existence of a solution to this design problem in terms of Pareto optimality. The aim

of this subproject is to translate a multi physical shape optimal design problem into

the context of a well-posed multicriteria optimization problem.

We couple internal and external PDEs in order to describe the various forces

that are inﬂected on the component. In this framework, using techniques based on

pre-compactness of embedding between H¨

older spaces of different index like in

[43, 12], we are able to show [40] the existence of Pareto optimal shapes in terms

of subsection 2.5.1 which form a Pareto front, see also [20] for a related result. We

also prove the completeness of the Pareto front in the sense that the Pareto front

coincides with the Pareto front of the closure of the feasible set (which is equivalent

to the fact that every non-Pareto admissible shape is dominated by a Pareto optimal

shape).

Further we investigated scalarization techniques which transform the multi-

objective optimization problem into a uni-variate problem. In particular we con-

sidered the so-called achievement function and ε-constraint methods which depend,

besides on the cost functional, on an additional scalarization parameter that repre-

sents the different weightings of the optimization targets, as e.g. reliability or efﬁ-

ciency. Hence, the shape space on which the optimization process takes place can

also depend on this parameter and with it the corresponding space of optimal shapes

20 The GivEn Consortium

as well. Under suitable assumptions on the contuinuous dependency of the scalar-

ization method on the scalarization parameter, we are able to show a continuous

dependency of the optimal shapes spaces on the parameter as well. For details we

refer to the forthcoming work [40].

2.5.3 Multiobjective Optimization Methods

Algorithmic approaches for multiobjective optimization problems can be associated

with two common paradigms: scalarization methods and non-scalarization methods.

In [20], two algorithmic approaches are described: the weighted sum method as an

example for a scalarization method, see, e.g., [23], and a multiobjective descent

algorithm as an example for a non-scalarization method, see [32]. Gradient descent

strategies were implemented for both methods to search for Pareto critical points,

i.e., points for which no common descent direction for all objectives exists. In this

section, we focus on weighted sum scalarizations and present new numerical results

for a biobjective test case.

The weighted sum method replaces the multiobjective function Jby the weighted

sum of the objectives Jω(Ω) = ∑p

i=1ωiJi(Ω). Here, ω>0 is a weighting vector that

represents the relative importance of the individual objective functions. We assume

without loss of generality that ∑p

i=1ωi=1. The resulting scalar-valued objective

function Jωcan then be optimized by (single-objective) gradient descent algorithms,

see e.g. [36]. If a global minimum of the weighted sum scalarization Jωis obtained,

then this solution is a Pareto optimal solution of the corresponding multiobjective

optimization problem, see, e.g., [23]. The converse is not true in general, i.e., not

every Pareto optimal solution can be obtained by the weighted sum method. Indeed,

the weighted sum method can not be used to explore non-convex parts of the Pareto

front. Nevertheless, an approximation of the Pareto front can be obtained by appro-

priately varying the weights.

2.5.4 Case Study and Numerical Implementation

n

g

ˆ

Ω

Ω

∂ ΩNﬁxed

∂ ΩNfree

∂ ΩD

Fig. 10 Case study: general setup and starting solution

GivEn – Shape Optimization for Gas Turbines in Volatile Energy Networks 21

In our case study we focus on objectives J1and J2as introduced above (i.e., me-

chanical integrity and cost), and consider a ceramic component Ω⊂R2made from

beryllium oxide (BeO) (with material parameter setting equal to [20], in particular

Weibull’s modulus m=5). The volume force fis set to 1000 Pa. See Fig. 10 for an

illustration of a possible (non-optimal) shape. This shape is used as starting solution

for the numerical tests described below. The component is of length 0.6 m and is

assumed to have a thickness of 0.1m. It is ﬁxed on the left boundary ΩDand the

tensile load is acting on the right boundary ΩNﬁxed. The parts ΩDand ΩNﬁxed are

ﬁxed, while the part ΩNfree can be modiﬁed during the optimization process. The

biobjective shape optimization problem is then given by

min

Ω∈Oad J(Ω):= (J1(Ω,u),J2(Ω))

s.t. u∈H1(Ω,R2)is the solution of a linear elasticity equation.

(14)

The component is discretized using a regular 45×25 grid, see Section 2.4 and Fig. 9.

We use a gradient descent method to minimize the weighted sum objective func-

tion Jωfor different weight vectors ω>0. This is implemented using the negative

gradient as search direction and the Armijo-rule to determine a step-size, see e.g.

[36]. During the iterations, the component is modiﬁed by free form deformations

using the method developed in Section 2.4. Since we have a regular mesh inside the

component, only the grid points close to the boundary have to be adapted. When the

modiﬁcation during one iteration is too large, a complete remeshing is performed,

still using the approach described in Section 2.4. To avoid oscillating boundaries and

overﬁtting, we apply a regularization approach based on [94]. Numerical results for

three choices of the weight vector ωare shown in Fig. 11, and an approximation

of the Pareto front is given in Fig. 12. In these cases no remeshing step had to be

performed, because the step length was restricted to the mesh size.

Fig. 11 Near Pareto critical solutions obtained by the weighted sum method

2.5.5 Gradient Enhanced Kriging for Efﬁcient Objective Function

Approximation

To cut computational time of the optimization process one can apply surrogate mod-

els to estimate expensive to compute objective functions. Optimization on the sur-

rogate model is relatively cheap and yields new points which then in a next step are

evaluated with the expensive original objective function. In the biobjective model

presented above, the mechanical integrity J1is expensive, while the volume J2can

22 The GivEn Consortium

20 40 60 80 100

4.5

5

5.5

6

·10−2

J1

J2

Fig. 12 Approximated Pareto front obtained by the weighted sum method

be easily evaluated. We thus suggest to replace only the expensive objective J1by a

model function.

Let {Ω1,...,ΩM} ⊂ Oad be sampled shapes with responses {y1,...,yM}:=

{J1(Ω1),...,J1(ΩM)}.Kriging is a type of surrogate model that assumes that the re-

sponses {y1,...,yM}are realizations of Gaussian random variables {Y1,...,YM}:=

{Y(Ω1),...,Y(ΩM)}from a Gaussian random ﬁeld {Y(ˆ

Ω)}ˆ

Ω∈Oad . For an unknown

shape Ω0the Kriging model then predicts

ˆy(Ω0) = E[Y(Ω0)|Y(Ω1) = y1,...,Y(ΩM) = yM],

i.e., the estimated objective value of Ω0is the conditional expectation of Y(Ω0)un-

der the condition that the random ﬁeld is equal to the responses at the sampled

shapes, or in other words the predictor is an interpolator. An advantage of this

method is, that the model also provides information about the uncertainty of the

prediction, denoted as ˆs(Ω0), see [55] for more details.

If, as in our case, gradient information is available, one can incorporate this into

the Kriging model which is then called gradient enhanced Kriging. One follows

the same idea: the gradients {¯y1,..., ¯y¯

M}:={∇J1(Ω1),...,∇J1(Ω¯

M)}are assumed

to be realizations of the Gaussian random variables/vectors {¯

Y1,..., ¯

Y¯

M}. Adding

these random variables to the ones w.r.t. the objective values enables one to predict

objective values and gradients at unknown shapes Ω0, see also [55] for more details.

In the optimization choosing the predictor ˆy(Ω0)as the objective to acquire new

points to evaluate with the original function may yield poor results. Since then one

GivEn – Shape Optimization for Gas Turbines in Volatile Energy Networks 23

assumes that the prediction has no uncertainty, i.e. ˆs(Ω0) = 0, and areas, for which

the predictor has bad values and a high uncertainty while the original function has

better values than the best value at the moment, may be overlooked. Hence, one has

to choose an acquisition function that incorporates ˆy(Ω0)and ˆs(Ω0)to balance the

exploitation and exploration in the optimization.

We note that this gradient enhanced Kriging approach is a direction of ongo-

ing development for the in house optimization process AuoOpti at the German

Aerospace Center (DLR), see e.g. [85, 57] for design studies using the AutoOpti

framework. In our future work, we therefore intend to benchmark the EGO-based

use of gradient information with the multi criteria descent algorithms and identify

their respective advantages for gas turbine design.

3 Applications

In this section we present the industrial implications of the GivEn consortium’s re-

search. The German Aerospace Center (DLR), Institute for Propulsion Technology,

here is an important partner with an own tool development that involves an in-house

adjoint computational ﬂuid dynamics solver TRACE as well as a multi-criteria op-

timization toolbox AutoOpti. As TRACE and AutoOpti are widely used in the Ger-

man turbo-machinery industry, a spill-over of GivEn’s method to the DLR assures

an optimal and sustainable distribution of the research results.

In a second contribution, Siemens Energy shows that results developed in the

GivEn project can also be directly used in an industrial context, taking multi-criteria

tolerance design as an example.

3.1 German Aerospace Center (DLR)

Industrial turbomachinery research at DLR includes several activities that beneﬁts

from the insights gained in this project. These activities primarily pursue two goals:

(i) Assessment of technology potential for future innovations in the gas turbine

industry.

(ii) Development of efﬁcient design and optimization tools that can be used, for

instance, to perform (i).

3.1.1 Challenges of Industrial Turbomachinery Optimization

The gain in aerodynamic performance of both stationary gas turbines and aircraft

engines that has been achieved over the last decades, leaves little room for im-

provement if solely aerodynamics is considered. More precisely, aerodynamic per-

formance enhancements that neglect the issues of manufacturing, structural dynam-

24 The GivEn Consortium

ics or thermal loads, will typically not ﬁnd their way into application. One of the

reasons for this is the fact that real engines currently designed already have small

”safety” margins. Summarizing, one can conclude that aerodynamic performance,

structural integrity as well as manufacturing and maintenance costs have become

competing design goals. Therefore, the design of industrial turbomachinery has be-

come a multi-disciplinary multi-criteria optimization problem. Accordingly, DLR is

highly interested in advances concerning both simulation tools for coupled problems

and multi-disciplinary optimization (MDO) techniques.

Multi-criteria optimizations based on high-ﬁdelity simulations are ﬁrmly es-

tablished in the design of turbomachinery components in research and industry

[102, 59]. As explained above, current developments increasingly demand the

tighter coupling of simulations from multiple disciplines. Reliable evaluations of

such effects require the simultaneous consideration of aerodynamics, aeroelasticity

and aerothermodynamics. Moreover, optimization should account for the inﬂuence

of results from these disciplines on component life-times.

Gradient-free optimization methods, typically assisted by surrogate modeling,

prevail in today’s practical design processes [77, 83]. A tendency towards optimiz-

ing with higher level of detail and optimizing multiple stages simultaneously leads

to higher dimensional design spaces, making gradient-free methods increasingly ex-

pensive and gradient-based optimization the better suited approach.

3.1.2 Expected Impact of GivEn Results

The methods described in the preceding sections describe how derivatives for a fully

coupled aerothermal design evaluation process can be computed efﬁciently and how

a gradient based optimization procedure can be constituted for the design criteria

of efﬁciency and component life-time. The exemplary process, developed in the

frame of this project, serves as a research tool and a base to adopt the methods

for other applications resulting in different levels of simulation ﬁdelity, different

sets of disciplines as well as different objectives. The partners from DLR consider

the goals of this project as an important milestone that could enable researchers to

tackle, among others, the following problems:

–Concurrent optimization of turbine aerodynamics and cycle performance with

the goal of reducing cooling air mass ﬂows. Such optimization should take into

account the redistribution and mixing of hot and cold streaks to be able to predict

aerodynamic loads in downstream stages.

–Assessment of the technology potential of operating the burner at partial admis-

sion (or even partial shutdown) conditions in order to achieve good partial load

performance while avoiding a signiﬁcant increase in emissions.

–Assessment of potentials of thermal clocking. The idea is to be able to reduce

cooling air if the relatively cold wake behind cooled vanes is used in downstream

stages. Such optimizations will be based on unsteady ﬂow predictions.

GivEn – Shape Optimization for Gas Turbines in Volatile Energy Networks 25

The necessary changes to create adjoints for existing evaluation processes stretches

from parametrization to the simulation and post-processing codes for the different

disciplines involved. Moreover, DLR expects to beneﬁt from the coupling strategies

developed here. These should be both sufﬁciently accurate and apt for an appropriate

strategy to deﬁne coupled adjoint solvers. A particularly important milestone that

this project is to achieve is the establishment of a life-time prediction that is suitable

for gradient-based optimization.

DLR will not only apply these advances in aerothermal design problem but

also hopes that general conclusions can be drawn that carry over to other multi-

disciplinary design problems that involve coupled simulations of ﬂows and struc-

tures.

Theoretical concepts are shared in joint seminars in the early stages of the project

to lay the foundation of common implementation of prototypes in later phases of

the project. The role of these prototypes is to explore the implementation of the

methods and simultaneously are used to communicate about changes to existing

design evaluation chains and requirements for practical optimizations.

3.2 Siemens Energy

In accordance with the ﬁnal aim of the GivEn project, the main usage of adjoint

codes in the turbomachinery industry concerns the development of optimization

process tool chain for design purpose. Recently the use of adjoint codes to consider

the effects of manufacturing variations has been proposed. This new application can

have a major impact in the very competitive market of gas turbine for power gener-

ation at relatively low cost and time horizon. The following sections describe some

aspects of the development of such a method using the example of a turbine vane.

3.2.1 Effects of Manufacturing Variations

The vanes and blades of a gas turbine are designed to deﬂect and expand the ﬂow

leaving the combustion chamber and generate torque at the rotor shaft. It implies that

these components must operate at very high temperature while maintaining good

aerodynamics and mechanical properties. Therefore, the material chosen for such

hot gas section parts usually belongs to the Ni-base superalloy family and the man-

ufacturing process involves precision casting which is a very complex and expensive

technique. During the manufacturing process, the shape accuracy of each casting is

assessed by geometrical measurements. If all of the measured coordinates lie within

a speciﬁed tolerance band, the part is declared compliant and otherwise scrapped.

In order to reduce uneconomic scrapping, the re-evaluation of acceptable tolerances

is a constantly present challenge. In the past, point-based measurements and subse-

quent combination with expert judgement were conducted. Nowadays, advanced 3D

scanners enable the acquisition of highly detailed geometric views of each part in a

26 The GivEn Consortium

short amount of time which are better suited to characterize the analyze geometrical

deviations. They furthermore allow to focus on the effects of geometric variations

on the component’s functions rather than on the geometric variations themselves.

Nonetheless, evaluating the characteristics of each component produced using tra-

ditional computational methods - CFD and FEM - would not be industrially feasible

due to the computational resource required. In this context, the capability of adjoint

codes to take into account very efﬁciently the effects of a great number of param-

eters on one objective function is a crucial asset. It allows to beneﬁt from the high

resolution scans and model the manufacturing variations on the surface directly as

the displacement of each point on the surface.

However, since adjoint equations are linearized with respect to an objective func-

tion, the accuracy of the predictions provided with the help of an adjoint code will

be limited by the magnitude of the deformations. In other words, the deformations

must be small enough such that a linear approximation of the effects on the objec-

tive function is appropriate. Therefore the usefulness of adjoint codes in an industrial

context must be investigated using real manufacturing deviations. To this aim, 102

scans of heavy-duty turbine vanes have been used for the validation of the adjoint

codes. Since the role of a turbine vane consists in fulﬁlling aerodynamic and me-

chanical functions, the objectives chosen for the investigation must be chosen within

these two families.

3.2.2 Validation of the Tools

In a context of a more volatile energy production induced by renewable sources,

gas turbines must be started or stopped very often and produce energy the most

efﬁciently possible.

For turbine vanes, starting or stopping a machine more often implies that impor-

tant temperature gradients will be experienced also more often leading potentially

to LCF problems. The occurrence of LCF issues can be numerically estimated by

computing the maximal number of start/stop cycles above which a crack appears on

the vane surface. For this assessment, the probabilistic Weibull-based approach by

Schmitz et al. has been chosen. It was combined with the primal and adjoint equa-

tions and has been implemented into a Siemens in-house software. In Fig. 13, left,

a comparison of the gradients of the Weibull scale parameter ηLCF computed with

the adjoint code and the ﬁnite difference for each of the 102 vanes considered is

presented. Actually two sets of data are present: one assuming that the temperature

ﬁeld on the vane is independent of the actual geometry - the so-called frozen temper-

ature hypothesis - while the other considers on the contrary a variable temperature

ﬁeld. In both cases, all points are very close to the ﬁrst bisectional curve, meaning

that for each of the 102 vanes, the manufacturing deviations have a linear impact on

the Weibull scale parameter. In other words, the magnitude of the real manufactur-

ing deviations are small enough so their effects on LCF can be successfully taken

into account by an adjoint code. A more detailed investigation can be found in the

publication of Liefke et al. [60].

GivEn – Shape Optimization for Gas Turbines in Volatile Energy Networks 27

Fig. 13 Weibull scale parameter (ηLCF , right) and isentropic efﬁciency (η, left) obtained by adjoint

codes versus ﬁnite differences.

The efﬁciency of a gas turbine depends of course on the aerodynamic characteris-

tic of the vanes. However the behaviour of each single vane makes only sense if they

are integrated into a row together. When considering manufacturing variations, an

additional difﬁculty appears namely the interaction of the manufacturing deviations

of adjacent blades with each other. In this case, it would be interesting to compare

the gradients obtained with the adjoint code versus the ﬁnite difference, not only

for a purely axis-symmetrical situation (e.g. one passage) but also for several differ-

ent arrangements (e.g. more than one passage). Fig. 13, right, presents the gradients

of the stage isentropic efﬁciency for such a conﬁguration using DLR’s CFD suite

TRACE [6]. It can be seen that independently of the arrangement considered - 2,4

or 8 passages - there is a very good agreement between the gradients predicted by

the adjoint code and those obtained with the help of the ﬁnite difference. The work

of Liefke et al. [61] summarizes the complete investigation of this case.

3.2.3 Industrial perspectives

The previous section demonstrated that adjoint codes can be successfully used in

an industrial context to quantify the impact of manufacturing variations on low-

cycle fatigue and isentropic efﬁciency. It could possible to extend this approach to

additional physical phenomena such as high-cycle fatigue or creep, given that the

equations modelling these phenomena can be differentiated and of course that the

impact of the manufacturing variation remains linear.

In addition to the direct and short-term beneﬁts for the manufacturing of tur-

bomachine components, the results presented in this section also demonstrate that

adjoint codes can be successfully deployed and used in an industrial context. The

conﬁdence and experience gained will pave the way for other new usage within

Siemens Energy. Especially, the tools and concepts developed within the GiVen

28 The GivEn Consortium

project will greatly contribute to the creation of more rapid, efﬁcient and robust mul-

tidisciplinary design optimization of compressors and turbines either based fully on

adjoint codes or in combination with surrogate models.

Acknowledgements The authors were partially supported by the BMBF collaborative research

project GivEn under the grant no. 05M18PXA.

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