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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
0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60
Adjoint
LCF
0.90
1.00
1.10
1.20
1.30
1.40
1.50
1.60
Finite Difference
LCF
Frozen Temp.
Variable Temp.
Baseline
Bisection
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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