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12 Model order reduction and digital twins

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We are currently facing a substantial transformation of our industrial world and the way our economics are organized. This transformation, known as digitalization, is driven by the systemic integration of information technology in all kinds of devices, machines, and factories such that new smart networks are formed and new smart products have the ability to monitor, to forecast, and to control their behavior. One of the fundamental pillars of digitalization is simulation technology, since it enables the new intelligence layer in the form of digital twins which mirror the physical systems into the digital world - also named by Gartner Inc. as a top technology trend for 2017 and 2018. Creating such intelligence layers over several domains and life cycle phases requires, among other challenges, technologies for transforming and reducing complex simulation models. Exactly for this task a key technology is model order reduction (MOR). However, MOR is not only a key technology within emerging digital twins but also helps to reduce simulation times in the existing everyday business of simulation engineers. This is especially important when for a simulation model a large number of evaluations are needed. Within this chapter we present use cases where MOR is a key enabler for the realization of digital services and the reduction of simulation times. Furthermore we outline the potential of MOR in the context of realizing the digital twin vision.

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       

        

      

 We are currently facing a substantial transformation of our industrial world

and the way our economics are organized. This transformation, known as digitaliza-

tion, is driven by the systemic integration of information technology in all kinds of

devices, machines, and factories such that new smart networks are formed and new

smart products have the ability to monitor, to forecast, and to control their behavior.

One of the fundamental pillars of digitalization is simulation technology, since it en-

ables the new intelligence layer in the form of digital twins which mirror the physical

systems into the digital world – also named by Gartner Inc. as a top technology trend

for 2017 and 2018. Creating such intelligence layers over several domains and life cycle

phases requires, among other challenges, technologies for transforming and reducing

complex simulation models. Exactly for this task a key technology is model order re-

duction (MOR). However, MOR is not only a key technology within emerging digital

twins but also helps to reduce simulation times in the existing everyday business of

simulation engineers. This is especially important when for a simulation model a large

number of evaluations are needed. Within this chapter we present use cases where

MOR is a key enabler for the realization of digital services and the reduction of simu-

lation times. Furthermore we outline the potential of MOR in the context of realizing

the digital twin vision.

 digital twin, virtual sensors, control, predictive maintenance, circuit sim-

ulation

  35B30, 37M99, 41A05, 65K99, 93A15, 93C05

 

This chapter provides an overview of several projects which we worked on through-

out the last years. These projects were initiated from dierent directions and perspec-

tives since the authors of this chapter work in multiple departments across Siemens.

Nevertheless, all of our projects were either part of concrete business opportunities

             



         

     

       

           

                 

     



     

or of predevelopment activities to evaluate new business opportunities. This means

that the goal was always to improve products, to develop new products, or to evaluate

the potential lying in innovative business ideas. In this environment the application

of model order reduction (MOR) was not a goal in its own. Instead the application of

MOR was always triggered by the requirements coming from the project goals. In par-

ticular for predevelopment projects such a goal is typically to evaluate the commercial

benet lying in new technologies, which in our case was MOR.

In this chapter we start with outlining the underlying business visions of digital-

ization and digital twins and the role of MOR within this vision. This part is followed

by a report of our experience with productizing MOR algorithms. Finally, we report the

content, the challenges, and the results of some of our projects.

Throughout this chapter we try to give an insight into our work between the poles

of business models and technological challenges, which is sometimes even the great-

est challenge.

    

Complexity in today's industry is exploding. New production methods, miniaturiza-

tion of electronics, novel sensor technologies, and last but not least the Internet of

things have led to many disruptive developments implying more and more complex

products. On the one hand, this oers unique opportunities, e. g., in terms of eciency

or autonomy of components, products, and complex systems. On the other hand, it

challenges today's design, engineering, operation, and service paradigms mostly fo-

cusing on manual expert interaction, which can hardly, if at all, handle this enormous

complexity.

Digitalization changes everything everywhere. With the rise of new technology

trends, such as AI foundations, intelligent things, cloud to edge, or immersive expe-

riences [76], many of today's paradigms can be expected to be disrupted. Not only in

the consumer market, as we can clearly observe today, but also in the industrial and

medical sectors we see disruptions as proven by rst early adopters.

Digital twins will be one key answer to these challenges; see, e. g., [24, 35, 81] for

a broad overview from an engineering perspective. They are the next wave in simu-

lation technologies (Figure 12.1). Digital twins integrate all (electronic) information

and knowledge generated during the lifetime of a product, from the product deni-

tion and ideation to the end of its life. Examples of these data range from the initial

requirements which have led to the design of the product, the design and engineering

data, which have been generated during virtual design, to operation data such as sen-

sor values collected during operation. The data themselves are only a central asset, if

it can be used to make relevant predictions providing the right level of information at

        

                 

 

the right time. Ultimately, digital twins mirror products and systems from the real into

the digital world and vice versa.

From a high-level point of view, information included in digital twins can be

split in to two categories: (i) pure data values with only little additional structure

and knowledge associated, such as data gathered from sensors, and (ii) structured

executable model-based data, in particular simulation models. Thus from this point

of view digital twins bring together classical data-based schemes with model-based

approaches such as simulation and optimization (Figure 12.2).

Today, most model-based approaches, and in particular simulation, are domain-

specic and mostly used during design and engineering. The core concept of the dig-

    

   

   

   

 

     

ital twin is to extend their usage along the complete life cycle and to deliver new ser-

vices providing the right information at the right place in an ecient way, for exam-

ple, digital twins supporting early system conguration during the sales process or

optimization of operation and service concepts. This broad usage implies a number

of requirements to modeling and simulation which diverge from its classical use in

design and engineering:

–Interactivity – Speed and accuracy dene the value of simulation and digital

twins. Being very accurate, today's model and simulation approaches are ex-

tremely time-consuming. Speeding them up, while retaining the right level of

accuracy, is crucial for extending the use of digital twins.

–Reliability – Users of digital twins cannot be expected to be sophisticated experts,

like it can be expected during the use in design and engineering. Thus any predic-

tion by the digital twin must be fail-safe and/or provided along with condence

intervals such that no expertise is required to interpret the results or can be used

autonomously, e.g., by controls.

–Usability – Model-based and simulation tools are expert-centric today. Their re-

sources are limited and thus the use of corresponding tools today is limited by

the availability. Therefore, any digital twin solution must be accessible also for

nonexperts from a usability perspective.

–Security – Many business models based on the digital twin will require to ex-

change digital twins between dierent parties. Reverse engineering must be pre-

vented, such that no intellectual property is lost.

–Deployability – Digital twins will be used dierently from the place where they

have been created, e.g., on customer premises, in the cloud, on controls. Thus

deployment must be easy to reduce barriers and eorts.

The digital twin concept has been originally introduced in 2003 by Michael Grieves

[41] and rst put to public by NASA in 2012 [38]. Digital twins are considered so impor-

tant to business, that they were named one of Gartner's Top 10 Strategic Technology

Trends for 2017 [76]. They are becoming a business imperative, covering the entire life

cycle of an asset or process and forming the foundation for connected products and

services. Companies that fail to respond will be left behind. For example, it is predicted

that companies who invest in digital twin technology will see a 30% improvement in

cycle times of critical processes [77]. A potential market of 90 billion US dollar per year

associated to corresponding oerings is predicted [28].

To realize the vision of digital twins, MOR is a key technology. Other key technolo-

gies cover novel user interaction paradigms and devices (such as virtual, augmented,

or mixed reality), technologies for merging data and model-based approaches, or se-

mantic technologies to easier built-up systems of digital twins.

        

       

   

The digital twin vision extends the expert-centric focus of modeling, simulation, and

optimization technologies towards a digital assistance for everyone in day-to-day de-

cisions. This is supported by a double exponential growth of capability in simulation

technology. On the one hand, computational hardware is developing exponentially

according to Moore's law [94]. On the other hand, eciency of simulation algorithms

is subject to exponential growth as well [91]. With this growing capability, computer-

aided paradigms have become so powerful that they can provide novel simulation-

based assistance in many elds, for example, digital twins providing new services for

predicting failures, increasing operational eciency, or for service planning [76].

However, compared to computer-aided tools in engineering, computer-aided as-

sistance by means of digital twins is a niche application. The manual setup of corre-

sponding models is a tedious task requiring simulation experts. This limits the use of

model-corresponding concepts since corresponding eorts and costs are major road-

blockers for increased use [60]. Furthermore, the lack of rigorous concepts for quan-

tifying errors often implies very conservative safety margins, so that the full poten-

tial often cannot be exploited. Missing protection of intellectual property of models

and the lack of standards (the functional mock-up interface [FMI] is only adopted

slowly [10]) are hindering further. Thus today, digital twin-based approaches are only

adopted in applications of high value, e.g., heavy-duty vehicles [44]. MOR [5, 4, 95] is

a key technology to solve these challenges in the context of digital twins. By splitting

computations in an oine and an online phase, computational eort is shifted to an

oine phase allowing interactive simulation during the online phase. However, not

only does this imply a speedup of calculations, but due to their reduced information

set, reduced-order models (ROMs) protect intellectual property eciently. While ge-

ometries can be recovered from the meshes of three-dimensional simulations, this is

not the case for ROMs, in particular since the output generally focuses on the quantity

of interest, i. e., a temperature at a single location rather than a complete temperature

eld. This furthermore increases the usability, since only relevant information is ac-

cessible. In addition, ROMs can be eciently containerized using available standards

such as FMI [10], thus increasing usability. In particular in view of the challenges laid

down in Section 12.2, MOR is a key technology for digital twins.

A variety of concepts and approaches have been introduced in the last decades

mostly using projection-based approaches such as proper orthogonal decomposition

(e. g., [111]), balanced truncation (e. g., [42]), the reduced basis method (e. g., [80]),

or Krylov subspace methods (e. g., [7]). The key idea of most approaches is to reduce

the space of considered functions by means of an appropriate low-dimensional basis.

For (close-to-)linear models, MOR is state-of-the-art in computational engineering and

science. For nonlinear models it is a highly active eld of research (e.g., [8]).

     

In addition to classical MOR methods, machine learning oers an alternative ap-

proach. Many successful applications, such as the ecient operation of wind parks

[62], have been realized during the last years. Compared to model-based approaches,

machine learning concepts require comparably little manual eorts to be set up. How-

ever, being data-centric, machine learning is not applicable where only few data are

available. This is often the case in industrial applications, where relevant data cannot

be measured, cannot be shared (e. g., due to IP concerns), or is simply not available

(e. g., failure data for small lot products). On the one hand machine learning could be

used to speed up simulation models by means of learning the underlying simulation

data (e. g., [43]), but a combined approach with the ROM as the foundation and ma-

chine learning closing the accuracy gap seems to be a more promising approach [58].

However, such combined approaches have rarely been considered in the past and we

believe that it has a strong future potential.

Within the following sections we review the application of MOR in projects which

tackled concrete aspects of the digital twin vision described above. However, before

describing these projects we generally review the process of productizing algorithms

since the overall goal of every industrial R&D activity is to improve or deliver new

products or services.

       



 

The process of making an algorithm suitable for use in commercial (CAE) software,

also referred to as productizing, can be long and dicult to plan. Even if algorithms

are known in the literature to be generally robust, the applicability to commercial soft-

ware implementation is not always straightforward. In particular, it is challenging to

foresee the user's needs and desired application of a method so that the method's as-

sumptions do not lose validity. Moreover methods and software developers often face

strict boundary conditions regarding implementation variants that are dictated by,

e. g., the structure of the underlying physics engine or solver in which novel methods

are implemented. Furthermore, while algorithms are usually designed by experts, the

actual end-users are typically not experts in using those algorithms – they are experts

in their own domain. Hence successful productizing requires not only that algorithms

are robust with respect to applications, but also that their parameters can be (re)set

in an automatic and dynamic way: automatic to reduce the need for users to set pa-

rameters and dynamic because parameters may need to be adjusted not only at the

start of but also during the simulation. In this way the numerical methods become

transparent to the user while the freedom of the user to interact with the algorithm is

        

somehow restricted. A good balance between transparency and user freedom has to

be found. The situation becomes even more complicated if a working algorithm is not

available or if the problem at hand is not yet fully understood and analyzed.

In this section we describe the various phases from algorithms to products. We as-

sume that the problem to be solved is suciently well-dened (and constrained) and

end-user requirements are known, and hence we focus on the process of solving the

problem. As a concrete example, one could consider the typical MOR problem: given

a dynamical system, nd a reduced dynamical system that approximates the origi-

nal system with a controllable trade-o between error and speed, and preservation of

key properties like stability. We identify the following phases that will be discussed in

more detail in the next subsections:

– research: literature study and investigation of novel approaches;

– prototyping: implementation of stand-alone or integrated software to allow feasi-

bility studies;

– productizing: implementation in, or as, a product;

– customer feedback: closing the loop with new results and new requirements from

end-users.

These phases may overlap in practice and moreover the process might become iter-

ative: After customer feedback, but also during productizing, often new insights are

obtained which require further research and prototyping.

 

During the research phase, traditionally two activities are dominant: literature study

and design of novel approaches. Depending on the complexity and condentiality of

the problem, these activities are carried out by one or more researchers, e. g., a techni-

cal leader, a (team of) researcher(s), and a MSc/PhD student, or even outsourced to an

external party. For literature study, it is not only important to have the problem at hand

well-dened, one must also know which literature to study. In some cases the right

sources are naturally available because the researcher has experience on the topic. In

other cases the topic may be less or even not covered in existing literature, or not in the

context of the application at hand. Communication with colleagues (potentially in dif-

ferent divisions) and external parties like universities is then required to at least nd

a starting point. In several cases such contacts, for instance made during conferences

or European networks like EU-MORNET [30], may develop to long-lasting collabora-

tions with rewards such as scientic and commercial breakthroughs and stang op-

portunities. The circuit simulation-related MOR work described in Section 12.10, for

example, has been performed in collaboration with the TU Eindhoven, in the Euro-

pean project ASIVA14 [21], while the drivetrain dynamics simulation tools described

     

in Section 12.8 have been developed in cooperation with the KU Leuven and the Uni-

versity of Calabria within several years of research interactions and projects such as

the Marie Curie H2020 project DEMETRA [57].

Often the problem is not suciently covered in the literature: The context or ap-

plication may be dierent, the boundary conditions imposed by the main CAE solvers

could be a limiting factor to the implementation of original algorithms, or the problem

itself may simply be new. Even if the problem is well covered, one usually has to adapt

and tune the proposed methods to the problem at hand. This stage, which may vary

from simple changes of existing strategies to the design of novel approaches, typically

involves prototyping, which we discuss in more detail in the next subsection.

 

When a set of methods is dened to achieve a specic target it is time to develop the

rst prototype code in order to test if the assumptions made during the research stage

are valid and if the knowledge gained has application potential. In the prototyping

phase, usually, one or more method developers and/or software engineers start to de-

ne preliminary software architectures and begin the implementation of a prototype

code. Common choices for development environments are MATLAB [66] and Python

[79]. As a good practice, the developed mock-up code should be easy to extend, it

should be tested in a similar environment as compared to the target solver in which the

nal implementation is foreseen, and it should be exible enough to be tested in mul-

tiple scenarios and maintain a satisfactory level of user-friendliness. In this way new

extensions of the methods can be easily tested on multiple scenarios, the code can be

shared with colleague researchers and consultants for usage in bilateral projects, and

the risk of failure during the prototyping-to-product transition is reduced. Once the

set of algorithms is mature enough, it is important to perform stress tests in the largest

possible range of applications. Automatic testing is not mandatory but is surely an

added value.

Using the specic case of MOR-related algorithms, it can happen that a large

amount of parameters must be set by the user and that these are of dicult physical

and mathematical interpretation to nonexpert users. Moreover, automatic parameter

tuning algorithms are rarely available in the literature for the specic application fore-

seen for the implemented method. For this reason a big eort during the prototyping

phase is generally spent in making the numerical methods robust and the automatic

parameter setting transparent while still allowing advanced users to retain the de-

sired level of control on the numerical method. During the prototyping phase of the

method described in Section 12.8 the original number of parameters linked to the un-

derlying MOR strategy was drastically reduced thanks to automatic parameter setting

and the remaining parameters have been readapted to represent physical quantities

that are easy to understand from a user point of view. Similarly, for the MOR approach

        

described in Section 12.10, most of the low-level parameters have been combined into

macro-options that give the user (and developer) easy control over performance and

accuracy.

If this target is achieved, the prototype should be tested on real engineering cases

during, e. g., bilateral services projects and/or funded research projects. This step is

useful to conrm the potential of the method, nd out unforeseen usages, and detect

potential limitations.

Often, at the end of the prototyping stage, a preliminary user interface is created

to explore the usability of complex numerical solutions.

 

Once the set of algorithms has reached a satisfactory level of robustness and usabil-

ity the prototyping phase can be sided by the productizing phase. First the developed

methods should be assessed for their market value, general applicability, and strategic

importance. This stage is fundamental in order to assign a well-balanced amount of

development resources. After this assessment the correct number of resources – gen-

erally one or more developers and/or software engineers – is assigned the task of im-

plementation into the target commercial CAE solver. The goal is to translate customer

specications, design requirements, and prototype code into a professional and con-

sistent implementation. Especially during the implementation of novel methods, it is

of paramount importance that researchers and developers communicate on a regular

basis. In practice, the specic research knowledge and the application-oriented char-

acter of many methods makes it hard to make consistent and complete code design

specications. In this case, developers may face the challenge of interpreting proto-

type code and might implement nonintended behavior. It is advisable to initially allow

researchers and developers to spend time together and even promote pair-coding ac-

tivities. The more the algorithms are complex and havea dual theoretical-applied char-

acter, the more this practice should be promoted. During this period and in parallel

with the method implementation into commercial solvers, a team of developers might

also start to implement a user-friendly user interface. The more the numerical method

has been rened and made robust, the less the user interface creation process is chal-

lenging. During the creation of the MOR method applied to drivetrains described in

Section 12.8 a prototype user interface was also created in parallel with the research

and method prototyping. This and the strong cooperation between the research and

development units of Siemens allowed for a smooth transition of the prototype code

and prototype user interface into a commercially available solution for MOR applied to

drivetrain problems. One of the main challenges in the productizing of the methods in

Section 12.10 was the choice on which parameters to make available to the user. This

has been an iterative process itself, where researchers, developers, and application

engineers were involved.

     

  

No matter how sound the underlying theory is and no matter how many tests have

been done, the most useful feedback on the quality (performance, accuracy) of the

product is end-user feedback. The diculty, as mentioned before, is that the test cases

used by development teams typically do not cover completely the real cases used by

customers. Hence, there is always a risk involved with releasing improved or new func-

tionalities. The key is again communication to manage expectations, not only inter-

nally with sales and product engineering teams, but also with the customer (either

directly or via customer-based application engineers): Roughly speaking, one of the

rst things to do when a customer request (bug report or enhancement request) is

led is to analyze whether there is a real bug in the theory and/or implementation,

or whether the result is within accuracy tolerances but outside customer expecta-

tions. Ideally this rst analysis is done by application or test engineers, but depend-

ing on the complexity, development teams may need to be involved as well. When

the issue is identied as bug, apart from implementation errors, regularly one will

have to go back to the underlying theory, for instance to adjust initially made as-

sumptions or estimates, hence reiterating the phases described in the previous sub-

sections.

When the result is within accuracy tolerances but outside customer expectations,

the situation can become more complicated. Not only one has to be sure that the result

is indeed within tolerances, but one also has to explain this to the customer: Particu-

lar care has to be taken here to avoid breaking long-standing trust relations. Further-

more, it might also be an indication that certain settings and options in the software

are not clear for users, which may require software and/or documentation to be im-

proved.

During the circuit simulation-related MOR work described in Section 12.10 all of

the above-mentioned scenarios have happened. For example, a bug reporting a too

large dierence in signal delay was initially identied as a side eect of the way the

delay was computed during postprocessing of simulation data. A deeper analysis,

however, showed that while the actual delays were still within (user-settable) simu-

lation tolerances, the used error estimations in the code were in fact too optimistic,

and hence all phases above had to be reiterated in order to x the issue. After the re-

lease of the rst version of the drivetrain simulation tools described in Section 12.8, a

user signaled an extension request to improve the usability of the tool for large system-

level models that include multiple drivetrains. The user was contacted and asked for

feedback about the urgency of the required extension. It was then decided in agree-

ment within the party to take the time to develop a proper interface for the requested

extension and release it together with the ocial product release a few months af-

ter.

        

  

We conclude by repeating what was mentioned in the introduction: The phases de-

scribed in this section are typically visited in an iterative way. Moreover, they may in

fact be visited in any order, for instance when through the acquisition of software (or

company) one starts with an actual product that has to be integrated in a larger envi-

ronment.

     

 

The use of models to enhance or extend test-based engineering processes is one of the

key application elds of model-based system testing [27]. Test data exploitation can

be greatly enhanced by complementing sparse physical sensor measurements with

model-based virtual sensor data [107, 20]. Control system eciency can be increased

by providing optimal control inputs using quantities which cannot be measured di-

rectly and operating system performance can be tracked through monitoring internal

system states. Traditionally such control inputs or internal states of devices are mea-

sured during operation by hardware sensors [53]. However, due to cost restrictions or

extreme physical conditions it is not possible to place hardware sensors at any desired

position in any device. The goal of virtual sensors in all these applications is to provide

online information about internal conditions or system performance based on simu-

lation models instead of hardware sensors. These system models can be used oine

to expand data sets or may be running parallel to operation, permanently synchro-

nized with the current operation state, and report the desired internal states at the

usual rate of the hardware sensors. From a business perspective such virtual sensor

software modules may not only add value to the engineering process but can enable

new simulation-based products such as advanced condition monitoring for improved

availability or reduced downtimes. Furthermore, when virtual sensor algorithms and

existing controllers are integrated into one software architecture, novel model-based

controllers can be realized.

However, the systematic application of embedded simulation models for ex-

tended data analysis or parallel to operation is still a young eld of activity. On the

other hand, driven by the need to reduce development cycle times, simulation has be-

come a frequently used tool during the development of products [17]. To draw reliable

conclusions during the development process detailed three-dimensional simulation

models are needed and the evaluation of these simulation models typically involves

signicant simulation times. This makes their reusage inside virtual sensor software

and related state estimation a challenge.

     

In fact one of the central requirements for simulation models inside virtual sen-

sors is the capability for fast estimation or even real-time capability when the results

should be updated within the usual update frequency of hardware sensors. For this

reason, MOR [8, 5] is applied to, e.g., detailed three-dimensional simulation models

developed for design engineering purposes. This ensures reusage of the already avail-

able information and it allows to obtain fast or even real-time capable surrogate mod-

els which nevertheless operate within an acceptable accuracy.

  

In this section the required steps for a virtual temperature sensor are described.

For a virtual temperature sensor the starting point is the thermal energy equa-

tion which reads for heat conduction with Fourier's law q = −κ ∇T [67, 55, 105] for a

computational domain Ω as

𝜕t(Cp T )+ ∇ ⋅(−κ ∇T )=h in Ω,

q⋅ n=hf on Γ N ,(12.1)

q⋅ n=α( T −Tamb ) on Γ R .

Here, T is the temperature eld, T amb is the ambient temperature, Cp is the specic

heat capacity, κ is the heat conductivity, and αis the convection coecient [61, 56].

In a typical industrial setup, Dirichlet boundary conditions are not used. Instead, the

thermal losses are captured by the volume heat load h or the heat uxes hf at the

boundary. The most important boundary condition is the Robin boundary condition,

which is also known as Newton's law of cooling [56]. This boundary condition models

the thermal communication with the environment. Especially when a thermal model

contains only solid bodies which are surrounded by a coolant, the convective heat

transfer coming from the coolant ow can be modeled by a given distribution of con-

vection coecients. For example, this applies to thermal models of electric motors

which contain the solid parts of the stator, rotor, and housing, but not the ow do-

main of the cooling air ow.1

To start the MOR procedure, the thermal energy equation has to be written as a

state-space system in the form

Ed

dtx=A x +B u, (12.2)

y= C x.

1A full conjugate heat transfer model would lead to a dramatic increase in complexity and compu-

tational time, since the turbulent and thermal air ow in the rotor-stator gap and around the stator

cooling ns needs to be resolved [56].

        

Here x∈ℝ n is the system state, u∈ ℝm is the input which drives the system, and

y∈ℝ p is the measurable respectively observable system output. Furthermore, the

system matrices are of dimensions E, A∈ℝ n×n ,B∈ ℝn×m , and C∈ ℝp×n .

To obtain the thermal energy equation (12.1) as system (12.2), the following steps

need to be performed.

– The heat load h and heat ux hf are assumed to consist additively of contributions

which only vary in time, i. e., h= h 1 (t )+ ⋅⋅⋅ +hl ( t)and hf =hf,1 ( t) +⋅⋅⋅ +hf,k ( t ). This

assumption is fullled for a typical thermal simulation model in the industry since

the usual procedure in commercial three-dimensional simulation software is (a) to

mark the relevant model components on which the heat loads and the heat uxes

are applied and (b) to specify the total thermal losses which are produced by these

model components. In a subsequent step the commercial software distributes the

total thermal losses spatially homogeneous over the marked model component

[2]. This leads to m= l+ k inputs.

– A nite element method or nite volume discretization approach in space brings

the thermal energy equation almost into the desired state-space formulation.

Some minor changes are necessary since during the nite element method or -

nite volume assembly procedure a constant vector b 0occurs at the right-hand side

due to the Robin boundary condition [59]. This part is added to the input terms

by extending the input matrix B as B= (B , b 0 ) and the input vector u as u= (u , 1).

This leads in total to m= l+ k+ 1 inputs.

– Additionally the output matrix C has to build up according to the desired location

of the virtual sensors. This is done by marking for each virtual sensor its relevant

nodes or elements in the computational mesh. This determines for each virtual

sensor its corresponding row in the output matrix C.

The major technological challenge in this process is to access the assembled system

matrices from commercial CAE software. For Simcenter Thermal Flow [2, 99] this was

solved with a special subroutine and for NX Nastran [2, 71] this was solved with DMAP

[2, 25]. However, there are commercial CAE software packages which do not provide

any customization possibility to access the system matrices or some explicit solvers

even do not assemble global system matrices.

Once the state-space system corresponding to the thermal simulation model is

obtained, any MOR method can be applied which works on state-space systems of

type (12.2) [8, 5]. For thermal simulation models the matrices are huge in size but

sparse [59]. In our experience, typical industrial small-sized thermal models contain

up to 106degrees of freedom and typical medium-sized thermal models contain up

to 108degrees of freedom. For this reason the Krylov subspace MOR methods are a

good choice since Krylov subspaces have a long history in connection with linear it-

erative solvers for especially huge and sparse linear equation systems [39]. A detailed

review of Krylov subspace MOR methods can be found, e.g., in [5, 8, 7, 72]. Instead

of giving yet another introduction into Krylov subspace MOR methods we concentrate

     

on what is necessary to realize virtual sensors with these methods in an industrial

environment.

However, classical Krylov subspaces MOR methods such as [72] are feasible only

for linear and time-invariant systems (12.2). For products with temperature-dependent

material properties, the heat equation (12.1) becomes nonlinear due to κ= κ (T). In this

case nonlinear algorithms (e.g., [8, 114, 69]) need to be applied.

  

The rst goal was to establish a user-friendly work ow for generating ROMs from ex-

isting three-dimensional thermal simulation models in an industrial environment. For

this goal the determining factors are that (a) the simulation models are constructed in

a commercial CAE software and (b) the simulation engineers have profound knowl-

edge in their physical domain and the used CAE software but in general they are not

experts in MOR nor they are programmers; see Section 12.4 for more details. More pre-

cisely, since all commercial CAE software packages are used through graphical user

interfaces, simulation engineers are generally not used to run algorithms in command

line tools or software development environments.2 This starting position requires (a)

to interact with the commercial CAE software and (b) to hide the details of the MOR al-

gorithms from the user (Section 12.2). For this reason a MOR plug-in for Simcenter [26],

the agship product of Siemens in the CAE market, was developed. This MOR plug-in

adds a ribbon to the Simcenter Graphical User Interface (GUI) which guides the user

with buttons and following pop-up windows through the process of generating, apply-

ing, and exporting ROMs (Figure 12.3). This MOR plug-in was developed as Siemens

internal engineering tool and is in productive usage within dierent projects and de-

partments. To ensure that the resulting GUI matches the user expectations, several

     

2The main task of simulation engineers is to support or enable the product development process

based on simulative information. To accomplish this, simulation models with the relevant physical

information are built from CAD models. From the obtained simulation results conclusions are then

drawn, e. g., about the product design or the reliability, and this information is fed back in the devel-

opment process. This means that simulation engineers are focusing on product development and not

on algorithm development.

        

in-house simulation engineers were included in the process of designing the GUI and

the work ow of the plug-in (Section 12.4.5). With this plug-in the step to easily gener-

ate ROMs from existing three-dimensional thermal simulation models was solved.

For realizing virtual temperature sensors, the next step is to wrap the obtained

ROM inside a virtual sensor software module which is runnable on the target hardware

and software architecture. For the communication with the surrounding software ar-

chitecture, the virtual sensor software module must receive the current operating con-

ditions, transform these conditions into the required input for the ROM, call the ROM,

transform the ROM results into the required format, and feed the properly formatted

ROM results back into the surrounding software architecture. Furthermore, one com-

munication cycle of that kind must be done within an expected frequency.

A crucial point is the available information during operation. Typically the avail-

able information is not identical with the required input for the ROM. For example,

for electric motors the current is known during operation and can be fed into the vir-

tual sensor software module. However, the ROM obtained from the thermal simulation

model requires heat loads as inputs. Thus it must be part of the oine phase, i. e., the

creation phase of an ROM, to provide the required information for mapping the avail-

able inputs (e. g., current) to the required ones (e. g., heat loads). This task involves

detailed product-specic knowledge and is a central key for a vital and accurate vir-

tual sensor software module. In our projects this task was solved with detailed look-up

tables which were provided by the respective engineering departments.

Another important ingredient of a virtual sensor software module is to ensure that

the ROM is permanently synchronized with the actual operation condition of the prod-

uct. This requires that the virtual sensor software module receives and adequately pro-

cesses the relevant information about the current operation state to keep its internal

ROM synchronized. In our projects we solved this task with online ltering algorithms

[101, 100, 46, 52], such as Kalman lters, where the ltering was done based on the

available temperature hardware sensors and the corresponding temperatures coming

from the ROM for these locations.

The last step in the development procedure is torun system tests to improve the so-

lutions based on this feedback. Some of our projects have currently reached this stage,

whereas in our in-house hardware lab virtual temperature sensor software modules

are already running and tested.

   

The main challenge of our projects was that virtual temperature sensors based on

ROMs were realized for the rst time for the considered products. This means that not

only the software modules had to be developed but we also had to establish a work

ow of how to realize these virtual sensor software modules. While customized one-

time solutions are sucient for research projects and rst prototypes, they are not a

     

proper solution for new services or products. New products or services require a sus-

tainable work ow which is integrated into the existing development ecosystem of the

involved engineers (Section 12.4). The approach we put into practice started from ex-

isting three-dimensional thermal simulation models. These models were compressed

with MOR and the resulting ROMs were small and fast enough to be executed within

the usual hardware sensor update frequency, either in an embedded environment or in

a cloud environment which is connected to the product. In order to integrate this task

in the existing development ecosystem of simulation engineers,we developed a plug-

in for Simcenter, which is the standard CAE software within Siemens for simulation-

based engineering steps.

The following task of integrating the ROM into the target hardware and software

system was still realized as customized and manual solution for each product. A po-

tential future integration of this step into the existing development ecosystem of au-

tomation engineers are new state-observer blocks within the Totally Integrated Au-

tomation portal, which is the engineering platform from Siemens for all kinds of au-

tomation tasks [104]. During our projects prototypical blocks for such state-observers

based on ROMs were developed but a fully integrated solution is still pending. Nev-

ertheless, exactly the integration into existing automation engineering software tools

is the second important step in realizing virtual sensors in a standard way. Overall,

in a typical industrial development ecosystem, the simulation engineers create the

ROMs for virtual sensors and the automation engineers integrate the virtual sensors

into the software architecture of the products. Thus, to establish virtual sensors based

on ROMs there must be a fully integrated solution for both, the simulation and au-

tomation engineering ecosystems (Section 12.2).

     

Data-driven operation support has been a topic for about 10 years. The eciency of

methods such as condition-based monitoring or sensor-based fault detection depends

on the amount and the placement of sensors.

     

Very new is the demand of simulative operation support [29]. It allows monitoring ev-

ery position and physical size of a system at any time point. Due to this knowledge,

the system state may be predicted at any time. A simulation-based software program

runs in parallel to the operation and is synchronized by sensor values at every time

point. In many reports this is called the digital twin. The benets are summarized in

Figure 12.4. Among the most important benets are inspection and service planning,

        

      

           

lifetime prediction, advanced fault detection, and control and optimization during op-

eration. Selling not only the hardware of the system but also additional services can

be a huge advantage in countries with high salaries. There are existing rst clients

of Siemens who demand this kind of operation support. In a rst view some services

such as giving an availability guarantee may sound risky for a company. On the other

hand this is a unique selling point and selling the risk may bring good prot; see any

insurance company. A more accurate analysis leads to the conclusion that all partic-

ipants may benet from an availability guarantee (Figure 12.5). Giving an availability

guarantee for products cannot mean that there are no downtimes due to faults or in-

spections. Instead, the downtimes, especially the unexpected downtimes, should be

reduced. The main task is to detect faults at a very early stage and predict their degra-

dation. Thus, immediate downtimes are transferred to predictive downtimes. The base

for giving an availability guarantee is the early detection of faults. If a fault is detected,

then its degree of degradation is predicted. Depending on this prediction an inspec-

     

tion may be scheduled and/or the performance of the system is reduced in order to

achieve the inspection time. Often, the plant is located in very isolated regions. Thus,

the execution of maintenance and spare part supply must be planned very carefully.

The early knowledge of the cause of failure is of tremendous interest.

   

We consider a solid body Ω ⊂ ℝ 3 with boundary 𝜕 Ω= ΓD ∪ΓN , composed of a material

with Young's modulus E≥ 0 and Poisson ratio − 1≤ν≤ 0. 5. The body is subject to

volume forces f: Ω→ ℝ 3 and surface forces g :𝜕 Ω→ℝ 3 . Displacements d: Ω→ ℝ 3

from some appropriate function space ℰ (Ω, ℝ 3 )are determined by the equations of

linear elasticity (see, e. g., [49]):

−div(Ae(d )) =fin Ω,

(Ae(d ))⋅ n= g on ΓN ,

d=0 on ΓD , (12.3)

where the strain e (d ) is given by the symmetrized gradient of displacements,

e( d)=1

2∇d+ ∇dT ∈ ℝ3×3 , (12.4)

and the stress Ae( d) is given by

Ae( d)= 2μ e ( d)+λ trace e ( d ) I (12.5)

=2μe (d ) +λdiv(d)I.

Here, λ= νE

(1+ν )(1−2ν ) and μ= E

2(1+ν) are the Lame constants and I is the identity matrix.

Equation (12.3) is the strong formulation for linear static elasticity. A Galerkin dis-

cretization of the weak formulation (typically by nite elements) yields a linear system

Kd = f, K∈ ℝn×n , d, f∈ ℝn , (12.6)

where n is the dimension of the ansatz space, Kis the stiness matrix, and, by abuse

of notation, d is the vector of displacements and fthe vector of acting forces.

In the dynamic case, i. e., when d and f are time-dependent, equation (12.6) is

extended to [113]

M̈

d+ D(t ) ̇

d+ K(t) d= f(t ), (12.7)

where M∈ℝ n×n is the mass matrix and D∈ ℝn×n is the damping matrix. Note that

both D and Kmay be time-dependent. An important special case of this is the rotor

dynamic equation

M̈

d+D(ω )+ ω G  ̇

d+ K(ω) d= f(t , ω ), (12.8)

where ω denotes the angular velocity of the rotor and Gis the so-called gyroscopic

matrix [36].

        

   

In many real-world applications, the number nof degrees of freedom of the discretized

system (12.7) or (12.8) is large and its numerical integration is not possible in real-time.

MOR strategies introduce a reduced state q∈ ℝr with r≪ n , via d= Ψq ,Ψ∈ℝ n×r .

One way to obtain the reduction matrix Ψ for system (12.7) is to use modal reduc-

tion. Setting up the eigenvalue problem of equation (12.7)

ω2 Mθ = K θ (12.9)

and taking the rst r eigenvectors, the matrix Ψ may be dened by

Ψ={θ1 ,...,θr }. (12.10)

A preferable technique may be the Krylov subspace methods [7, 93]. The subspace is

dened by

Ψ=K −1

ωf,K −1

ωMK −1

ωf,...,K −1

ωM r−1K −1

ωf. (12.11)

The Krylov basis may be computed by the Arnoldi algorithm, which delivers an or-

thonormal basis of the subset.

Inserting this into (12.7) and multiplying by ΨT , one obtains the reduced equation

̂

M̈

q+̂

D(t ) ̇

q+̂

K(t) q=ΨT f(t ), (12.12)

where ̂

M=ΨT MΨ, (12.13)

̂

D=ΨT DΨ, (12.14)

̂

K=ΨT KΨ∈ℝr×r (12.15)

are the reduced matrices. In the case of rotor dynamics, D and K depend on ω . In

the ramp-up phase of an electric engine, where the rotation frequency increases, the

reduction operations (12.14) and (12.15) have to be performed in each time step. Inter-

polation schemes may reduce the computational eort. In the constant phase of the

engine, also the reduced matrices remain constant.

As ltering methods are generally applied to rst-order equations, we dene as

usual

x= q̇

q,u(t )= 0

̇

f(t ) , (12.16)

and, assuming that ̂

Mis invertible,

A(t )=0 I

̂

M−1 ̂

K(t ) ̂

M−1 ̂

D(t ) , B= 0

ΨT ,(12.17)

so that we obtain the equivalent rst-order system in state-space form

̇

x= A(t) x+ Bu(t ). (12.18)

     

        

         

Following [13], [64], and [63] we want to identify an unbalance of a rotor during op-

eration. The test conguration is an electric engine which drives directly a generator

(Figure 12.6). The two rotors are connected by a clutch. Starting point for the analysis is

the rotor dynamic model which was used in the design process of this particular driv-

etrain (Figure 12.7). According to the strategies described in Section 12.6.3, we reduced

the model in order to obtain real-time capability. In order to obtain realistic frequen-

cies for the model, also some nonlinearities in terms of the uid bearings have to be

considered. Therefore, a nested procedure was applied [86], which keeps the nonlin-

ear parts at the bearings and reduces the linear parts in term of the motor and the

generator. Both rotors from the considered drivetrain are equipped with four discrete

planes meant for balancing the rotor. The validation of the digital twin was done by

physically attaching a small test weight to one of the balancing planes. Four sensors

located at the bearings (red bars in Figure 12.7) provided the measurement data which

are compared to the simulation results.

The comparison or identication was performed by an augmented nonlinear

Kalman lter procedure. The unbalance itself enters the model in terms of an external

force (see equation (12.7)), where location, orientation, and magnitude are identi-

ed by the ltering algorithm. The result of the identication method is presented

in Figure 12.8. The blue peak in Figure 12.8 presents the location and the amount of

unbalance.

        

       

        

 

By combining MOR techniques and nonlinear identication methods, a digital twin

for detecting and localizing faults (in terms of unbalances) has been developed for

rotating systems. Further eorts are made in order to predict the increase of vibra-

tion during operation. The time a critical vibration is achieved denes the moment

for scheduling an inspection. Knowing this time at an early stage, an inspection and

spare part supply can be prepared.

     

   

The product race has become an innovation race, reconciling challenges of branding,

performance, time-to-market, and competitive pricing while complying with ecolog-

ical, safety, and legislation constraints. The answer lies in "smart" products of high

     

complexity, relying on heterogeneous technologies and involving active components.

The corresponding design and engineering process hence must take the integration of

control functions in the product explicitly into account. This adds an important addi-

tional complexity to the design engineering process where the interaction between the

control and the system requires these should be optimized concurrently. The current

industrial practice however still treats passive system design and controller design as

dierent and separate design loops with their own models and their own validation

and verication strategies. Suboptimal designs and unexpected integration problems

are the result. Not reusing the wealth of engineering models available from earlier

detailed system design stages furthermore leads to inconsistency problems and inef-

fective engineering processes. Closing this gap oers a signicant potential for opti-

mized designs, better product performance, and fewer and shorter design iteration

cycles [1, 106].

  

Two classes of challenges can be distinguished in relation to the integration of con-

trol functions in the product. The rst one targets the optimization of the controller

architectures, strategies, and settings for a controlled product, hereby using a sys-

tem or "plant" model in a virtual controller optimization process. The second chal-

lenge targets the design of an optimal control solution by including a model of the

controlled system into the controller itself, for example in a model predictive control

(MPC) approach [82]. For both cases, the used system models are typically developed

dedicated for the control application taking into account feasible complexities and

system simplications. One objective to do so is to allow fast virtual testing and opti-

mization cycles and to enable hardware validations in physical control environments.

The detailed (for example multiphysics) design engineering models from the system

design departments are typically not reused. The main reason for such a suboptimal

approach is that either detailed system models from the system engineering design

departments are not available in the control department or are of too high complexity

to be used together with control simulation.

Focusing on the rst challenge of optimizing the controller design, one can dis-

tinguish three phases:

Phase 1: The combination of the multiphysics simulation model with that of the

controller enables the design of the control logic and the performance engineering

of the intelligent system. This is referred to as "model-in-the-loop" (MIL). The simula-

tion is oine, i. e., there is no requirement for real-time performance of the simulation.

Basically, two interconnecting objectives can be distinguished: One is to perform sys-

tems engineering based on the multiphysics "plant" model, including the representa-

tion of (often simplied or idealized) control in the multiphysics model; the other is

to perform control engineering, using a model of the system to be controlled ("plant

        

model"). The rst objective, for example, serves the purpose of conguration design

(how many actuators and sensors, where to place them, etc.) or concept evaluation

studies or the optimization of the mechanical system design taking into account the

presence of control and certain control laws. The second objective is oriented towards

the development of the optimal control logic and the development and verication

of control hardware, control libraries, and embedded software up to the validation

and calibration of the control system on the electronic control unit (ECU) (Figure 12.9)

[106].

       

To couple the models, dierent approaches exist. One may embed state equations

with a description of the plant system (e. g., multibody simulation models or one-

dimensional ordinary dierential equation-based system simulation models) into

these of the control (or vice versa) to enable the use of one solver, or adopt a true

co-simulation approach where each system part runs its own solver [106, 40].

Alternatively, or in combination with the above approaches, a reduction of the

plant model (e. g., a nite element or complex, even nonlinear multibody simulation

model) into a description compatible with the controller model (e.g., state-space for-

mulation) can be used. The model reduction step mostly achieves its goals at the ex-

pense of the full observability and/or controllability of the physical phenomena, lead-

ing to a macroscopic "equivalence" but loosing direct insight in the microscopic ob-

servation domain. The challenge is to develop model compression methodologies that

allow maintaining a relation with the physical meaning of model parameters. Such

     

co-simulation and model reduction approaches are used both for MIL applications

for systems engineering and for control logic engineering.

Phase 2: The next step is the development and optimization of the "embedded"

control software. This needs also to be done in the context of the functioning of the

multiphysics system to be controlled. This is referred to as "software-in-the-loop."

While some of this can be done in oine simulation (provided software libraries of the

controller are available), the nal optimization needs to take into account the work-

ing of the software in real-time, requiring real-time capable multiphysics simulation

models.

Phase 3: The nal testing and calibration of the controller software and hardware

requires the controller to be connected to a multiphysics simulation model of the com-

ponents, subsystems or system, in a dedicated computing environment that is referred

to as "hardware-in-the-loop" [6]; of course, this requires real-time capable simulation

models.

The use of MOR is hence a key factor for enabling a true model-based engineering

approach where consistent engineering models can be used throughout the various

design phases. The applied MOR methods may depend on the reduction purpose and

the nature of the master models. For example, in mechatronics systems, these can be

nite element, multibody, or multiphysics models which can be linear or weakly or

strongly nonlinear. Two application cases will be briey discussed.

          



Active noise reduction (and sound shaping) is a widely studied research topic with

many potential industrial applications. Structural-acoustic solutions using smart ma-

terials as sensor and/or actuators are explored, enabling intelligent structures. Such

solutions are however typically developed as add-on systems which prevents optimiz-

ing their potential impact as part of the overall system. A model-based mechatronic

engineering approach was developed to enable an integrated solution [22, 23]. It was

applied to the active sound control of vehicle engine noise to the car interior. The

challenge consisted of relating the large three-dimensional, frequency-domain (nite

element- and boundary element-based) vibro-acoustic, and structural models for the

vehicle structure and structural components, interior vehicle cavities, and exterior

propagation eld, with models of smart material sensors and actuators and a time-

domain control model. A simplied vehicle structure was developed to allow experi-

mental validation. It consisted of a concrete car body, an engine compartment with an

articial source, and a exible rewall panel on which the structural-acoustic control

was to be applied with piezo-elements. The rigid walls of the concrete car structure can

be easily modeled and also treated with a dedicated damping surface (Figure 12.10).

        

    

MOR was a key element in the modeling approach, allowing to incorporate the re-

duced model as a plant model in the controller simulation. The component mode syn-

thesis (CMS) approach was used. Very large reduction factors were used, reducing the

large structural/vibro-acoustic nite element model (25,000 acoustic degrees of free-

dom but which can overall easily reach hundreds of thousands of degrees of freedom

when multiple exible panels are included) to a time-domain state-space model of

realistic size (200 degrees of freedom). The sensors and actuators were represented

by one-dimensional models for their functional performance, while their added mass

and stiness are accounted for in the three-dimensional nite element models. The

acoustic propagation was related to the structural outputs by means of an "acoustic

transfer vector" approach. The modeling approach included the following steps using

multiple software tools:

– generate structural mesh and apply material properties (nite element preproces-

sor);

– add actuator and sensor mechanical models (nite element preprocessor);

– run a modal analysis (nite element analysis);

– build the acoustic nite element model and perform modal analysis (nite ele-

ment analysis);

– import the structural model and couple it with the acoustic one (nite element

analysis);

– calculate actuator and sensor electromechanical coupling (extended nite ele-

ment analysis);

– reduce and convert the nite element model into a state-space model (MATLAB);

– implement and optimize the controller with the coupled state-space model (MAT-

LAB/Simulink).

The coupling between acoustic and structural models is shown in Figure 12.11.

After performing a coupled modal analysis, the desired degrees of freedom are

taken to derive the state-space model. In this case, the state-space model features two

inputs (one actuator on the rewall and a sound source in the engine compartment)

and four outputs (three pressures in the passenger compartment and one velocity on

the rewall). The state-space model derived from this coupled approach allows the

implementation of any controller involving the predened degrees of freedom, and

     

            

      

    

if the nite element approach involves the systematic representation of the sensors

and actuators, the resultant state-space model is, in fact, a representation of the fully

coupled electro-vibro-acoustic system, with any possible input-output relationships

allowed by the chosen degrees of freedom.

Using this model, an optimization procedure is performed. The costfunction takes

into account the sound pressure level at the drivers head, the actuator input energy,

and the weight of the solution. The rewall thickness and the velocity feedback con-

troller gain are the variables. The position of the collocated sensor/actuator pair (SAP)

can be considered xed or included in the optimization loop. Figures 12.12 shows the

cost function for each thickness as a function of the feedback gain, on the best SAP

position for each case.

        

The best SAP position and optimal feedback gain depend on the thickness, which

indicates that the global optimum can only be achieved in such a concurrent design

between the active and passive system characteristics, proving the eectiveness of an

integrated mechatronics simulation approach. The same model can be used to evalu-

ate dierent controller strategies such as combined feedforward/feedback, ltered-X

LMS, and NEX-LMS, and for dierent performance targets (noise level optimization

and/or sound quality control). A more extensive discussion of the various modeling

aspects and the detailed optimization procedures can be found in [22, 23].

      

  

The development of the control of mechatronic systems becomes more complex when

multiple actuators and sensors are interfacing with a highly dynamic multiphysical

system, often not having the sensors available to develop an optimal control. In the

automotive industry, controls have mostly been developed using rule-based meth-

ods, rst by directly writing code, later using a model-based approach. In the au-

tomotive industry, the complex balancing of multiple performances of the combus-

tion engine such as emissions, fuel economy, and acceleration performance have in-

creased the complexity of the engine actuators. To develop the controls for such com-

plex mechatronic systems, new methods are required. Optimal control such as MPC in

combination with methods to predict virtual controllable quantities using state esti-

mation technologies in combination with Kalman ltering are examples of such new

technologies that start to nd their entry in the automotive world. MPC and Kalman

ltering-based state estimation require, however, models that run fast while keeping

a certain level of accuracy. Often, ad hoc simplied models are (re)developed, neglect-

ing the availability of detailed engineering models. Reusing such models would not

only save modeling time but would also allow better consistency of the various design

engineering models over the dierent attributes and product versions and variants.

The simulation models for designing mechatronic systems are often created based

on a combination of detailed three-dimensional models and test data and have a high

level of accuracy but have a too slow calculation time to be used in MPC or state es-

timation methodologies in real-time on an ECU. To be able to convert such system

models into the context of optimal control in combination with virtual sensing, neu-

ral networks can be an ideal methodology to develop a control model directly from

the detailed plant model to be controlled [50, 65]. Figure 12.13 explains the dierent

steps in the process, showing the reduction of the detailed engine model to a neu-

ral network-based model for the virtual sensor as well as the optimal controller. The

neural network ROM allows real-time execution of the model for both the sensing and

the control action. Initial results using the controls model in closed loop with the de-

tailed plant model for tracking purposes indicate that the performance, for scenarios

     

          

              

for which the neural network is not trained, remains within good accuracy as long as

the important states of the model are kept observable. Further results in this eld will

bring more clarity in how broadly this technology can be used for engine controls or

other advanced vehicle controllers.

By applying multiple load cycles covering the full operating space of the system,

the neural networks can be trained to represent the relevant system behavior even in

the case of strongly nonlinear system characteristics.

 

Model-based approaches nd increasingly their way into the design of (optimal) con-

trol systems. In the majority of cases, however, the applied system models ("plant

models") are ad hoc developed low-complexity models that are not correlated to the

design engineering models developed in the mechanical design stages. This not only

leads to inecient processes redoing eorts that could be recovered, but also gives

rise to major issues related to consistency and traceability when design improvements,

        

versions, or variants are to be processed. MOR can oer an answer for both the control

design and control implementation and opens up new opportunities for concurrent

design of the mechanical and the control system. Major challenges are still presented

by the very large reductions factors to allow fast control optimization or even real-time

usage inside state estimators or MPC controllers. The use of a neural network-based

approach to reduce complex nonlinear models subject to an envelope of operating

conditions oers signicant potential that is to be further investigated.

     

     

The simulation of dynamical systems involving contacts between elastic bodies [112] is

a challenging and active research eld on its own. In particular high-frequency phe-

nomena, numerical stiness, high degree of nonlinearity, high dimensionality, and

multidisciplinary nature (mechanics, acoustics, uid dynamics, tribology) are among

the major challenges that researchers and software developers must address in or-

der to eciently solve these types of problems [11]. Despite its seemingly niche de-

scription, there is a wide range of applications in which these problems are found

and need to be solved. In particular, the simulation of geared transmissions or drive-

trains is practically ubiquitous if one has to deal with simulations of electromechan-

ical machines. Drivetrains contain a multitude of components, including bearings,

gears, clutches, and spline connections that are known to behave nonlinearly and

contain multiple contacts between exible objects. While several MOR methods have

been applied largely and successfully [32] in the eld of exible multibody simulations

[97] in both academic and industrial settings with a large growing body of literature,

MOR methods dedicated to the eld of contact mechanics have been only recently ex-

plored [12, 103]. Moreover the developed methods often target high-dynamic contact

mechanics simulations with fully exible bodies and dynamic interactions with exi-

ble eigenmodes of the structure [12]. These problems would indeed remain practically

intractable without the usage of MOR and/or computer clusters. On the other hand, a

large set of system-level related problems (such as large drivetrains or more complex

machines containing several drivetrains) might not need the level of delity and the

still relatively large computational times necessary to solve a fully nonlinear dynamic

problem. They might instead still benet from MOR solutions that speed up simulation

times and decrease memory usage and disk storage, while still retaining the required

level of delity on both system and component levels (Section 12.2).

The method discussed in this chapter exploits both an advanced MOR strategy

and physics-based considerations coming from the targeted application domain of

drivetrain simulation and combines them. The result is a numerical strategy that is

     

ecient and computes accurately most of the drivetrain dynamics-related scenarios

that are industrially relevant. The focus is on three-dimensional system-level simu-

lations including lightweight and internal gears and noise-vibration and harshness

(NVH) problems in a multibody simulation environment.

  

While the application challenge is relatively straightforward to summarize – eciently

and accurately solve three-dimensional multibody problems involving multiple gear con-

tacts for system-level and NVH purposes – the technical challenges connected to it are

multiple and have their root in the mathematical description of the equations of mo-

tion of a multibody system. The following set of equations is an index 3 dierential

algebraic equation describing the dynamic motion of a exible multibody problem:

M( x)̈

x+ Kx + GT ( x ) λ= f ext + fv , (12.19)

where M (x ) is the nonlinear mass matrix, Kis the linear stiness matrix, xis the vector

of generalized coordinates, Gis the Jacobian of the constraints, λis the vector of La-

grange multipliers, and f ext and fv are the vectors of external forces and the quadratic

velocity terms. Without entering in more details (which can be found in, e. g., [11]) we

mention that despite being a fully nonlinear problem, the equations describing the de-

formation of the exible bodies present in the system can be reduced by using linear

MOR methods for second-order systems in the following form:

M̈

u+ Ḋ

u+ Ku = f, (12.20)

where M , D ,K are the linear mass, damping, and stiness matrix of the underlying

nite element model, uis the vector of linear nodal deformations, and fis the vec-

tor of nodal forces. This system can be reduced thanks to Petrov–Galerkin projection

methods:

WT MV ̈

u+ WT DV ̇

u+ WT KVqu = WT f,(12.21)

where W and V are the left and right subspaces used to obtained the reduced system.

Galerkin methods can be used in exible multibody problems within the assumption

of large gross motion but small deformations within each of the body frames. This

assumption is amply satised in problems involving gear contacts.

While exible bodies that are not involved in contact interactions can be reduced

with very ecient techniques, such as balanced truncation [83, 42], Krylov [32], CMS

[19], etc., exible bodies that include contact interactions suer from the so-called in-

terface problem [102]. In practice, a very large and inecient reduction space needs

to be used for an accurate reduction. The size of the reduction space becomes propor-

tional to the amount of degrees of freedom potentially involved in the contact interac-

tions. This causes problems such as high memory usage, large precomputation time,

        

large storage requirements, and increased numerical stiness. Finally, the contact de-

tection phase is also computationally very costly and scales with the (large) number

of degrees of freedom that can be involved during contact.

In recent years several methods have been presented to maintain a high level of

accuracy – similar to nonlinear nite element full-order dynamic computations – but

drastically limit the impact of the above-mentioned issues. In particular, the following

works [11, 12] obtain very good results in terms of speedup and memory usage while

losing only a fraction of the accuracy obtained with nonlinear nite element problems.

The eld of hyperreduction [18, 31] is also exploited to tackle the contact detection

problem with very promising results. However, this methodology is relatively complex

to include in proprietary multipurpose multibody solvers and, moreover, the simula-

tion time and the user expertise needed to use these techniques do not match the re-

quirements of system-level three-dimensional multibody software. In order to develop

a novel method for gear contact simulation to be included in a commercial multibody

solver, the following decisions have been taken based on the available state of the art:

–Contact detection: Hyperreduction for contact detection is still in its infancy and

needs further development. For this reason the computational performances are

improved by using geometrical considerations that are related to the specic ap-

plication eld of drivetrain dynamics.

–Dynamic exibility: The majority of the applications that involve dynamic sim-

ulations must include the modal behavior of the full drivetrain but the eigen-

frequencies related to gears bodies and teeth themselves are outside of the fre-

quency range of interest for many applications. From this point of view, it was

decided to concentrate on a correct representation of the contact stiness to prop-

erly represent the quasi-static behavior of the gear contact and the overall three-

dimensional system-level dynamics. The accurate evaluation of the contact sti-

ness is of paramount importance in the denition of the dynamic modes of the

full drivetrain.

–Contact stiness formulation: The contact interactions of nite element meshes

require a very ne spatial discretization to properly capture the correct Hertzian

nonlinear behavior during contact. For this reason it was decided to focus the

attention on the development of a method that combines the advantages of both

general nite element formulations but exploits analytical formulas near the con-

tact regions.

While the rst point is important but out of the scope of this chapter, the second and

third bullets highlight how available techniques that can be found in the literature

[3] have been enhanced thanks to a cooperation between Siemens PLM Software and

the Mechanical Engineering Department of KU Leuven (Section 12.4.2). These devel-

opments lead to a method based on MOR [102, 12, 16] that allows to describe in an

accurate way the gear contact stiness, including eects such as gear body deforma-

     

        

tion, Hertzian nonlinear stiness, and teeth convective couplings, while remaining

extremely ecient. The reasons for the eciency and accuracy of the method are:

–Eciency: The problem is treated quasi-statically; thanks to the MOR technique,

very few degrees of freedom are retained to describe the teeth deformation. When

possible, potential symmetries in the gears geometry are also exploited for e-

ciency purposes.

–Accuracy: Despite the application of MOR, the contact interaction is statically

quasi-exact with respect to the full-order nite element model. Convective de-

formation terms that couple the deformation of dierent teeth are accurately

retained. Local dynamic eects such as teeth dynamic vibrations that are less

often of relevance during standard operations are instead discarded. While these

dynamic eects might be relevant for problems such as dynamic ring gears ex-

citations and high-speed applications, the method is implemented in a modular

way so that future extensions are ecient to implement.

  

The developed technology based on MOR achieves the objectives targeted at the be-

ginning of the development but a key component still needs to be addressed to al-

low a smooth user experience and limit the amount of expertise needed to use the

method: usability (Section 12.4). For this reason the parameters that control the level

of static completeness of the reduction space can be adjusted with a single parame-

ter that ranges between zero and one where the limit value of one represents exact

static completeness at the expense of some longer preprocessing time and slightly

slower simulations while lower values allow to obtain a trade-o between accuracy

and speed. Moreover the user is provided with a parametric mesher that automatically

creates the gears nite element meshes based on a few parameters (Figure 12.14) that

is used for the automatic generation of the reduction space while further minimizing

the user intervention.

The interfaces between the MOR method proposed and the multibody solver are

integrated into a user-friendly application-driven user interface – the Simcenter 3D

        

Transmission Builder (Figure 12.15) – that proposes also a simplied work ow for

the creation of complex drivetrains. Practically, thanks to the dedicated Simcenter 3D

Transmission Builder interface and application-specic choices related to the MOR

technique it was possible to obtain a seamless and user-friendly usage of advanced

MOR numerical techniques available to nonexpert users.

       

       



The described numerical method based on MOR is released as a product in Simcen-

ter 3D Motion under the name of Advanced–FE preprocessor gear contact. Before the

product release, given both the complexity of the method and the number of assump-

tions made during development, the methodology has been validated using multiple

numerical and experimental results. In this chapter we present a subset of the exper-

imental validation results to show the accuracy of the proposed method. For a larger

set of examples we refer to [85]. The validation processhas been carried out thanks to

the usage of an in-house precision gear test rig [74] jointly developed by Siemens PLM

Software, KU Leuven, and the University of Calabria. The test rig has been designed

and manufactured to assess typical gear-related physical quantities in static and dy-

namic conditions, under imposed conditions of misalignment and shaft compliances.

Particular attention is given to the measurement of gear pair transmission error (TE)

[75], which is a typical key performance indicator used for the assessment of drivetrain

NVH performances.

     

As an illustrative example we present validation results related to the complex

case of gear contact between two spur gears, including large friction, microgeome-

try modications, and dierent loading conditions. The measured TE is shown in Fig-

ure 12.16. It can be seen that despite the wide range of torques applied, the proposed

method is able to match the TE with a high degree of accuracy. This is particularly

striking since the eects of microgeometry, friction (as noticeable in the discontinu-

ous jumps in the TE), teeth exibility, and local contact nonlinearities are highly inter-

acting with each other. The experimental results and multiple numerical validations

carried out conrmed the good performances of the method in terms of both accuracy

and speed.

          

     

In this section, we consider lifetime analysis for railway axles as an example for a

digital twin in the design phase. The engineer would like to know already in the design

        

phase how lifetime depends on the inspection scheme, the inspection interval, the

type of load, and the fatigue crack growth. Stochastic MOR serves as a key element for

the realization of this digital twin.

     

An important part of lifetime analysis is the computation of small failure probabil-

ities, which is a challenge for practical problems with CPU time-intensive function

calls. As already pointed out in Chapter 10 of this volume of Model order reduction,

Section 10.2.3, failure probabilities are described by multidimensional probabilistic

integrals which may be discretized by a multivariate quadrature rule. In the case of

failure probabilities, the integrand of the probabilistic integral is discontinuous such

that common quadrature rules will not provide sucient accuracy. Here the key idea

is to reformulate the integral into an integral with a smooth integrand and then to ap-

ply the unscented Kalman lter (UKF) [51] for evaluating the integral. The reliability

of a product, system, or process is often indicated by a failure function:

g(x )= ≤ 0 unsafe,

>0 safe, (12.22)

where x is the vector of stochastic variables. The failure function g (x ) describes a dam-

age mechanism, e. g.,

1. the mechanical stress or the temperature exceeds a given threshold (nite element

analysis, computational uid dynamics);

2. the amplitude of oscillations exceeds a given threshold (linear and nonlinear

modal frequency analysis);

3. changes of the microstructure of the material as a prestage of cracks diers from

a given pattern (stochastic Voronoi techniques, nite element method);

4. the crack size exceeds a given length (nite element method +crack size analysis);

5. a chemical species exceeds a given concentration (computational uid dynamics,

ChemKin).

Using this failure function, the failure probability reads

P g(x )≤ 0= 

g(x)≤0

ρx (x) d x, (12.23)

where ρx (x ) is the stochastic density of x . We restrict our presentation to the case of

independent standard normally distributed variables x= (x 1 ,..., xn ) T . In the general

case of a failure function depending on nonnormally distributed variables ̃

xi , e. g.,

the Rosenblatt transformation may be applied [90], mapping ̃

xi to xi for i= 1,..., n. In

     

order to obtain an integral over ℝn , an indicator function is introduced,

P g(x )≤ 0= 

ℝn

Γg (x)ρx (x)d x , (12.24)

where

Γg (x )= 0g(x )> 0,

1g(x )≤ 0 . (12.25)

For practical applications, the Monte Carlo method is too expensive to evaluate the

integral in (12.24) because of the high computing time. So a stochastic MOR method

is required. A standard method for approximation of the integral in (12.24) is the rst-

order reliability method (FORM) [48, 47]. This method rst computes the so-called beta

point (the point of highest failure probability) and then constructs a linear approxima-

tion of the failure function in the beta point. For highly nonlinear failure functions g,

the failure probability of the FORM will not be accurate. An extension of the FORM,

the second-order reliability method [14], is more accurate but requires second-order

derivatives, which are often not available in practical applications. Because of the dis-

continuous integrand in (12.24), standard quadrature formulas will in general lead to

bad approximation properties. Our stochastic model order method now consists of a

reformulation of (12.24) in an integral with a continuous integrand and subsequent

application of a nonlinear lter method. The reformulation is possible if the failure

function g (x ) is continuously dierentiable in its coordinates and strictly monotone

in at least one coordinate (backmapping approach); see [110]. Without loss of gener-

ality, let g (x ) be monotone in xn . It follows that the critical ̄

xn dening the limit state

can be expressed as a function of x 1 ,..., xn−1 :

̄

xn = ζ (x1 ,...,xn−1 ), (12.26)

0=g(x 1 ,..., xn−1 , ̄

xn ).

Then the failure integral (12.24) reads

P g(x )≤ 0

=

ℝn−1

ρ1 (x1 )⋅⋅⋅ρn−1 (xn−1 ) ∞

 ̄

x

ρn (xn )dx1 ...dxn

=

ℝn−1

ρ1 (x1 )⋅⋅⋅ρn−1 (xn−1 )h(x1 ,...,xn−1 )dx1 ...dxn−1 , (12.27)

with

h( x1 ,..., x n−1 )= 1

2erfcζ(x 1 ,..., xn−1 )

2 .

        

The smoothness of h follows from the implicit function theorem. The failure proba-

bility P (g(x )≤ 0) can thus be interpreted as the mean of the smooth function h . For

evaluation of this mean a nonlinear lter method, called UKF [51], is applied. This l-

ter can be used to estimate the mean and covariance of a nonlinear stochastic process

f(w), where w∈ℝ n w is a normally distributed random vector with mean E(w ) and

covariance Pww ∈ℝ n w ×nw . So-called sigma points 𝒳 (i) , together with weights W mean

i

and W cov

i, are constructed and mapped to 𝒵 ( i)=f(𝒳 ( i))for i= 0,..., p . The unscented

lter then yields an approximation of the mean μ and covariance Pzz of the nonlinear

function by

E( z)≈ p



i=0

Wmean

i𝒵 ( i),

Pzz ≈ p



i=0

Wcov

i𝒵 ( i)−y𝒵 ( i)−E( z) T .

By Taylor expansion one can show second-order accuracy of mean and covariance

[51].

   

In stochastic crack growth, the crack depth ais a function of the stochastic parameter

vector x and load cycle N,

a= a(x , N ), (12.28)

and the failure function (12.22) is given by

g(x , N)= acrit −a(x , N), (12.29)

where a crit denotes a critical crack depth indicating failure of the component. We con-

sider elliptical surface cracks given by crack depth aand crack form b= a/ c . Both

the initial crack depth α= a 0and the initial crack form b 0 are stochastic. Further

stochastic parameters are given by the so-called POD curve and the probability of

crack initiation. In contrast to most other places in this handbook, here "POD" does

not mean "proper orthogonal decomposition," but "probability of detection." The POD

curve gives the probability of crack detection during inspection and so characterizes

the inspection scheme. The nal crack depth monotonically depends on the initial

crack size a 0such that the reformulation of the failure integral of Section 12.9.1 can be

applied with xn =a 0in (12.26). The goal is to compute the cumulative failure proba-

bility for a given number of equidistant inspection intervals, under consideration of:

– the replacement of a component if a crack is detected during inspection;

– the probability of crack initiation (input).

     

We use the following notation:

Ti i-th inspection time

[a min , a max ] domain of denition of crack depth a

αinitial crack depth

βminimum detectable crack size at time Tk

(according to POD curve)

[β min , β max ] domain of denition of β

̂

xvector with realizations of all stochastic variables

except for α and β

αn critical initial crack size leading to failure at time Tn ,

a( ̂

x, αn , Tn )= a crit

Pn

fprobability that the crack is not detected during

inspections and reaches the critical crack depth

at time Tn

Pn

dprobability that the crack does not exceed the critical

depth and is detected at time Tn

Pn

ccumulative failure probability at time Tn

under consideration of failure of replaced components

cn probability of crack initiation in interval [ Tn−1 , Tn ]

ρ̂

x,ρ α,ρ βstochastic densities of ̂

x, α, β

The probability of detection of a crack with depth a is given by

Iβ (a)= a



βmin

ρβdβ.

The probability of detection of a crack at time Tn is

In

β=I n

β(̂

x, α)= Iβ a( ̂

x, α, Tn ).

Probabilities Pn

dand P n

fare given by

Pn

d=

Ω

In

d(̂

x) ρ̂

xd̂

x, Pn

f=

Ω

In

f(̂

x) ρ̂

xd̂

x,(12.30)

with

In

d(̂

x)=









∫α 1

amin I 1

βρ αdα for n= 1,

∫α n

amin (1−I 1

β)⋅⋅⋅( 1−In−1

β)I n

βρ αdα for n> 1

and

In

d(̂

x)=









∫a max

α1 ρ α dα for n= 1,

∫α n−1

αn (1−I 1

β)⋅⋅⋅( 1−In−1

β)ρ α dα for n> 1.

        

The cumulative failure probability can then be computed by the following recursive

scheme:

P1

c=c 1 P 1

f,(12.31)

Pn+1

c=P 1

f(c 1+⋅⋅⋅+cn+ 1)+P 2

f(c 1+⋅⋅⋅+c n)+⋅⋅⋅ +P n+1

fc 1

+c1 P 1

dP n

c(12.32)

+c1 P 2

d+c 2 P 1

dP n−1

c

+⋅⋅⋅ +

+c1 P n

d+c 2 P n−1

d+⋅⋅⋅+c nP 1

dP 1

c.

The integrals in (12.30) are the mean values of In

d(̂

x)and In

f(̂

x)and are evaluated by UKF

as previously described. The critical initial crack depths αn−1 ,αn appearing as integral

limits in the denition of In

d(̂

x)and In

f(̂

x)are computed by a bisection algorithm. This

procedure of evaluating the failure integrals has been validated by Monte Carlo for a

model problem in [73].

    

For lifetime investigation of railway axles, we use the failure function in (12.29) with

acrit = 10 mm. The stochastic distributions of the initial depth a0 and initial form b0 of

the elliptical surface crack are given in Table 12.1. In this study two inspection schemes

are considered:

– ultrasound far end scan;

– ultrasound mechanized.

In the rst case the axle is scanned by sound from one end of the shaft to the other in

the longitudinal direction, in the second case in the radial direction. The POD curves of

these schemes are shown in Figure 12.17. The cumulative density function of crack ini-

tiation is given later together with the results of the results of lifetime analysis. Fracture

mechanics for railway axles are subject of current research [54, 108]. Here the fracture

mechanical simulations are accomplished by the simulation program ERWIN from the

      

   

b    [.,]        

b  = a/ c 

a   

 λ=    .

  

     

   

Fraunhofer Institute IWM in Freiburg, Germany [34]. It predicts the propagation of el-

liptical surface cracks, for dierent types of bendings. The stress intensity factors are

represented by so-called polynomial inuence factors [15]. Inputs of the crack sim-

ulation are internal and external stress proles (due to the external load spectrum),

da/ dN -curves, and the initial crack depth and form.

It should be noted that we call the crack simulation a black box. Inputs are the ini-

tial crack depth and length, and output is the nal crack depth, which is subsequently

used for evaluation of (12.29).

   

        

         

         

         

       

Four test cases are considered with dierent loads, inspection schemes, and inspec-

tion intervals (Table 12.2). The resulting lifetimes are shown in Figure 12.18. Figure 12.18

also shows the probability of crack initiation, which is input to the lifetime calcu-

lations. Lifetime is represented as a function of the deferred distance in kilometers,

where one kilometer corresponds to 354 load cycles. The results show how lifetime

depends on the probability of crack initiation, the type of load, and the inspection

scheme. As expected, the full load case with the worst inspection scheme and inspec-

tion interval 100,000 km (case 2) has the shortest lifetime. The second worst is the

partial load case (case 1) with the same inspection scheme and interval. Lifetime of

the full load case can be improved by either switching to shorter inspection intervals

        

           

 

(case 3) or to a better inspection scheme (case 4). Both cases lead to longer lifetimes

than cases 1 and 2, where shorter inspection intervals are more eective than a better

inspection scheme. One call of the crack simulation program takes approximately 15

CPU seconds on a 3.2 GHz processor. For each curve shown, 3,000 to 4,000 simulation

calls are required.

 

As an example of a digital twin in the design phase, lifetime analysis for railway axles

has been presented, for dierent inspection and load case scenarios. By using adapted

stochastic MOR methods, stochastic crack growth under consideration of inspections

can be computed in reasonable computational time, which would not be possible with

pure Monte Carlo methods. Key elements are the reformulation of the failure integrals

as mean values of continuous integrands so that a nonlinear lter like the UKF can be

applied with sucient accuracy.

     

     

MOR has been part of the standard techniques used in circuit simulation for a long

time, with publications dating back to at least 1990 [78]. The relation is bidirectional,

with the circuit simulation and semi-conductor industry providing several benchmark

cases [92, 88]. It is not only Moore's law [70], which states that circuit design complex-

ity roughly doubles every 2 years, that drives this relationship; it is also the growing

     

need of circuit designers to include more physical details in their simulations and at

the same time to have control over accuracy and performance. From a technological

point of view, MOR methods will have to be developed that can deal with the growing

complexity: The number of unknowns typically increases with the size of the design

and the advance of the technology node (decrease of transistor dimensions), while

the (Jacobian) density of the problem typically increases with the amount of detail in-

cluded (wire resistance, capacitive coupling, inductive coupling). The MOR challenge

lies hence not only in the problem's dimension, but also in the problem's complexity

in terms of coupling and detail. From a business point of view, it is clear that scalabil-

ity of simulation software is key (Section 12.2). What is less clear, however, is in which

part of the ow the scalability should apply. For instance, before actually simulating a

circuit, the dierential algebraic equations describing the behavior of the circuit rst

need to be constructed. This process is called extraction and the resulting description

of the circuit that can be translated into a system of dierential algebraic equations is

called netlist . Whether to apply MOR during this extraction phase and/or the simula-

tion phase is not always clear, not only for reasons of robustness and reliability, but

also for commercial reasons. In the remainder of this section we will focus mainly on

the technical challenges.

  

At rst sight, MOR problems arising in circuit simulation may seem easy as they fall

into the most elementary class of linear time-invariant dynamical systems. Electrical

circuits that include nonlinear elements such as CMOS transistors are described by

systems of dierential algebraic equations of the form

j(x)+d q(x)

dt = s(t),

with node voltages and currents x(t )∈ ℝn , (non)linear vector-valued q(t , x) , j(t , x) ∈

ℝn with the electrical branch contributions, and sources s(t )∈ ℝn . Typically, only a

linear subsystem is considered for reduction. This linear system models the behavior

of linear resistors (R) and capacitors (C) and is usually considered in the frequency

domain:3

Gv+ sCv= Bu,

with node voltages v∈ℝn , inputs u∈ ℝk , Laplace variable s , conductance and capac-

itance matrices G, C∈ℝ n×n , and input mapping B∈ ℝn×k . One distinguishes between

internal nodes and terminals (or ports): Internal nodes only have connections to other

3For simplicity we do not include inductors (L).

        

nodes via RLC elements, while terminals have also connections to non-RLC elements

like transistors. This means that internal nodes are candidates for elimination while

terminals need to be preserved, in general, because they connect the linear subsys-

tem to the rest of the system. In the circuit simulation community, MOR is also known

as netlist reduction or parasitic reduction, with the adjective parasitic referring to the

nonintentional nature of the RLC elements that model the wire resistance and capac-

itive and inductive coupling.

Methods from several well-known categories are used for reduction:

– Krylov subspace projection, among the rst of MOR methods to be applied to elec-

trical circuits [33, 72];

– balanced truncation, with a priori error bounds [84, 9];

– modal truncation, used for the construction of behavioral models [87];

– nodal elimination methods, which have as advantages the existence of error

bounds and ease of implementation [98, 96].

An advantage of nodal elimination methods (and to a lesser extent modal truncation),

especially in the context of circuit simulation software, is that the ROMs can naturally

be translated into a reduced circuit with meaningful RLC elements. For Krylov sub-

space and balanced truncation methods, the ROMs are typically dense with nonphys-

ical (negative) RLC elements, and integration requires interfaces to deal with matrix-

based circuit descriptions.

Despite the developments in the MOR domain, even the problem of reduction of

linear circuits is still not considered as solved. The following key challenges can be

identied for linear circuits:

– Linear solve costs: For subcircuits that contain only resistors or capacitors,

projection- and elimination-based MOR procedures are error-free [89, 98], but

the question of how to minimize the ll-in created by node elimination for the full

design system matrix factors is still open (and becomes more dicult for mixed

RLC circuits).

– Coupled problems: With decreasing feature sizes and increasing frequencies, ca-

pacitive and inductive coupling becomes stronger and denser. As a result, the orig-

inal system matrices become denser, the reduction procedure becomes more ex-

pensive, and the ROM may become even denser, rendering MOR less eective.

– Precise accuracy performance tuning: For users it is important to be able to trade

o between accuracy and performance. For instance, for top-level verication,

one can (and often has to) accept less accuracy in order to improve simulation

speed or to make simulation possible at all. The challenge is here twofold: (1) how

to estimate the eect on accuracy when integrating the reduced circuit into the full

design and (2) how to estimate the eect on the overall simulation time.

– Which method to apply when: There is not a single best method for MOR that ts

all problems. Hence there is a need to be able to select automatically and dynam-

ically, based on certain characteristics, which method to use.

     

– Variability-aware analysis: Uncertainty quantication of the impact of process

variability on design robustness requires ROMs that are valid for ranges of design

parameters (with as additional complication that there can be many parameters).

– With designs having a growing number of nonlinear devices like CMOS transis-

tors, also the need for robust and ecient MOR methods for nonlinear systems

increases.

     

SRAM memory designs [109] are of interest for MOR methods for several reasons.

SRAM designs typically have a relatively large memory block of 6- or 8-transistor

bitcells (depending on the memory size) and some smaller control blocks. Although

designers often replace the large bitcell matrix by a much smaller model (manually),

for top-level simulations an automatic, accuracy-preserving method is required. Fur-

thermore, the extraction (modeling) of the many and long wordlines and bitlines may

result in netlists with not only many resistors but also many coupling capacitors. Espe-

cially for so-called critical path simulations, where one wants to ensure that the delay

for read and write operations is within specications, reduction must be done with

care to guarantee that the delay error is within picoseconds or even less. Additionally,

to assess robustness of the design against process variations, one needs to run many

simulations and hence simulation time needs to be minimized. In short, for memory

designs, MOR has to deal with all the challenges mentioned in the previous section.

In Figure 12.19 we show the results for time-domain simulations with reduction

settings varying from conservative to aggressive. The main impact on accuracy (error

in delay) and performance (simulation time) is caused by how coupling capacitors are

reduced. It depends on the type of verication how much error is acceptable: This can

vary from tens of picoseconds to less than one picosecond. The results are produced

using the circuit simulator Eldo Premier [68].

  

Driven by rapidly increasing design sizes and complexity, MOR, also known as para-

sitic or netlist reduction, has become a standard and indispensable option in modern

circuit simulators. Although current MOR methods are suitable for robust accuracy

performance control of simulation of advanced CMOS designs, more advanced CMOS

nodes and verication requirements will require the development of new methods and

approaches. Not only accuracy and performance remain key priority, also methods

that can be used in the context of other applications, such as variability-aware de-

sign, are required. In particular parameterized MOR methods for linear and nonlinear

systems will need to be further developed in order to make them suitable for use in

future industrial software (Section 12.2).

        

       −         

             

           

             

               

                

         

   

Within this contribution, we have reviewed a number of industrial success stories

of MOR in the context of digital twins (Sections 12.5–12.10). Through MOR the corre-

sponding simulations could be accelerated and reduced in their memory footprint.

This enabled novel applications which would not have been possible without these

improvements. Therefore, MOR is a key enabler for a new generation of digital twins

(Sections 12.2 and 12.3). With respect to sustainable industrial applications and com-

mercial software packages containing MOR engines it is crucial to close the gap from

algorithms to products. Here, professional software development plays a crucial role,

which we have addressed in Section 12.4.

MOR allows to reduce computational execution time of models while controlling

accuracy. Application- and purpose-specic models with dierent requirements in

terms of speed and accuracy can be realized. At the same time, MOR liberates sim-

ulation models form their execution engines, i.e., their specic simulation tools and

numerical solvers. This allows a separation of the creator of a digital twin – typically

a simulation engineer – and the consumer – anyone downstream including machines

themselves – through appropriate application interfaces. Furthermore, this allows not

only to reuse the models during operation as highlighted by some of the use cases,

such as virtual sensors (Section 12.5), but also novel licenses and business models

[37], such as pay-per-execution time. In particular, realizing a pay-per-execution busi-

     

ness model during operation allows to scale with the number of products sold by a

company assuming that each product contains a digital twin. Typical business mod-

els in the context of simulation tools only scale with the number of engineers working

in a company, assuming that each engineer uses corresponding tools. Therefore, the

impact does not only lead to novel application areas but also to the way how industrial

value streams are organized.

Summarizing the current rapid advancementsin MOR, a novel generation of digi-

tal twins, so-called executable digital twins [45], is likely to emerge in the near future.

An executable digital twin is a specic encapsulated realization of a digital twin with

its execution engines. As such they enable the reuse of simulation models outside R&D

departments. In order to do so, the executable digital twin needs to be prepared suit-

ably for a specic application out of existing data and models. In particular, it must

have the right accuracy and speed. The executable digital twin can be instantiated on

edge devices, on premise servers, or in cloud environments and used autonomously

by a nonexpert or a machine through a limited set of specic application programming

interfaces (APIs).

In order to realize this vision, several key challenges remain open though many

of them are subject to active research eorts:

–How to prevent virtual reverse engineering? Thanks to fast execution times,

digital twins could be executed many times allowing to reverse engineer specic

features, e. g., optimal control logics.

–How to leverage MOR with black-box solvers? Many commercial simulation

tools do not provide APIs for systematic interaction with their kernels. However,

this is a central requirement for integrating novel MOR tools. At the same time the

development of such APIs will take signicant time due to the existing develop-

ment processes. That is, rst, such APIs must be ranked high enough in feature

backlogs for next software releases, and second, these features need to be vali-

dated and veried before they are available.

–How to provide certiable accuracy bounds for ROMs? The usage of MOR to

enhance operations, e. g., in the context of model predictive control, requires cer-

tiable models, e. g., ensuring conservation of important quantities.

–How to combine/integrate machine learning and MOR technologies better?

Machine learning technologies, e.g., neural networks, are rapidly expanding in

industrial applications addressing similar aspects as MOR. However, combined

concepts are still missing.

–How to package ROMs appropriately? Even though containerization technolo-

gies, e. g., Docker, have matured over the last years, it is not clear how to leverage

them in the context of MOR, e. g., appropriate interfaces and standards are miss-

ing.

        

Mathematical research in MOR as well as close collaboration with industrial software

providers and users will be key to address these challenges and ultimately realize the

vision of executable digital twins.



The digital twin concerning the predictive maintenance was implemented by Chris-

toph Ludwig; see again [13], [64], and [63].



             

         Mechatronics   

 

      Simulations with NX / Simcenter 3D: Kinematics, FEA, CFD, EM and

Data Management        

              

 J. Sound Vib.     

    Approximation of Large-Scale Dynamical Systems    

               

  Contemp. Math.   

       Decision and Control 2005 and 2005

European Control Conference, CDC-ECC'05, 44th IEEE Conference on   



           

 Appl. Numer. Math.    

                

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   

                

            

     Proceedings of the 8th International Modelica Conference;

March 20th–22nd; Technical University; Dresden; Germany, number 063  

    

   Model Reduction of Contact Problems in Flexible Multibody Dynamics with

Emphasis on Dynamic Gear Contact Problems    

              

      Int. J. Numer. Methods Eng.   



              

       15th Petroleum and Chemical Industry Conference

Europe   

    Asymptotic Approximations for Probability Integrals    

    

     

            

          

              

          

 Meccanica     

   Design Theory and Methods Using CAD/CAE: The Computer Aided Engineering

Design Series     

           

 SIAM J. Sci. Comput.     

            AIAA J.  

 

             

       Mech. Syst. Signal Process.  



        

                  

         J. Sound Vib.   

 

                   

           Mech. Syst. Signal

Process.    

       

  

  

 

  

 

               

            

   Structural Health Monitoring, Damage Detection & Mechatronics  

   

            

 

                

          

       PLM 14  

   

            

           

     Int. J. Numer. Methods Eng.   



             

    Multibody Syst. Dyn.    

               

  IEEE Trans. Comput.-Aided Des.   

         





        

               

   

      Rotordynamik   

        The st. Gallen business model navigator



                

  53rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials

Conference 20th AIAA/ASME/AHS Adaptive Structures Conference 14th AIAA   

         Matrix Computations      

     

               

          Multibody Syst.

Dyn.    

           White

paper 

                

  Int. J. Control     

              

Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and

Data Mining    

           Proc. Automotive

Powertrain Control Systems 

             

           Introduction to Mathematical Systems Theory: Linear

Systems, Identication and Control  

              

   Struct. Saf.    

           Struct. Saf.  

 

    Linear Static and Dynamic Finite Element Analysis     

           

          

               

         IEEE Trans. Autom. Control

   

             J. Basic Eng.   

 

             Sensor Technology Handbook

    

           

  ZEVrail    

      Modern Thermodynamics   

         Principles of Heat Transfer   

 

           "demetra" (design of

mechanical transmissions: Eciency, noise and durability optimization)    

  

     

                 

         

 

       The Finite Element Method: Theory, Implementation, and

Applications        

              Int. J. Control.

Autom. Syst.   

      The Classical Theory of Fields       

   

                

  Energy Conversion Congress and Exposition (ECCE), IEEE 2012  

 

            

        J. Vib. Control

 

             ISMA2018

Conference on Noise and Vibration Engineering   

                 

         ISMA2018 Conference on

Noise and Vibration Engineering   

  

       Non-Equilibrium Thermodynamics   



         



              

     Int. J. Numer. Model.    

          Electronics   



    

 

          

 IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst.    

            

   Proceedings of ASME Turbo Expo 2012   

 

                

        Advances in Condition Monitoring of

Machinery in Non-Stationary Operations    

               

          

         Mech. Syst. Signal Process.

  

          

 

         

 

             IEEE Trans.

Comput.-Aided Des. Integr. Circuits Syst.   

        

  

            

      J. Math. Ind.    

              

  

        Model predictive control: Theory and design  

            Math.

Comput. Model. Dyn. Syst.    

           

IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst.    

                 

            

 Proceedings of the 6th European Conference on Computational Mechanics (ECCM 6) 7th

European Conference on Computational Fluid Dynamics (ECFD 7) 

          J. Comput. Appl. Math.

  

   Methods for Eigenvalue Problems with Applications in Model Order Reduction

    

    

             IEEE Trans.

Comput.-Aided Des. Integr. Circuits Syst.    

        Ann. Math. Stat.   



                 

             

     

      

             

   Linear Algebra Appl.    

          IEEE Spectr.    

             Model Order Reduction: Theory,

Research Aspects and Applications    

          y− δ    

   ProRISC IEEE 14th Annual Workshop on Circuits, Systems and Signal

Processing   

     Dynamics of Multibody Systems     

       RC  IEEE Trans. Comput.-Aided Des. Integr.

Circuits Syst.    

    

 

   Optimal State Estimation: Kalman, H Innity, and NonlinearApproaches  



        System Identication      

      

              

        Comput. Methods Appl. Mech.

Eng.   

     

               

          International

Gear Conference   

  

 

                

 Encyclopedia of Physics     

                

      Eng. Comput.     

                

         Special Topics in

Structural Dynamics      

              

           Recent Trends in Fracture and

Damage Mechanics       

   Deep-Submicron CMOS ICs: From Basics to ASICs   

 

            

 Ind. Eng. Chem. Res.     

             

AIAA J.    

          Encyclopedia of

Computational Mechanics 

         Control of Surge in Centrifugal Compressors by Active

Magnetic Bearings: Theory and Implementation    

          Nonlinear heat transfer modelling and

reduction  

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In aerodynamics related design, analysis and optimization problems, flow fields are simulated using computational fluid dynamics (CFD) solvers. However, CFD simulation is usually a computationally expensive, memory demanding and time consuming iterative process. These drawbacks of CFD limit opportunities for design space exploration and forbid interactive design. We propose a general and flexible approximation model for real-time prediction of non-uniform steady laminar flow in a 2D or 3D domain based on convolutional neural networks (CNNs). We explored alternatives for the geometry representation and the network architecture of CNNs. We show that convolutional neural networks can estimate the velocity field two orders of magnitude faster than a GPU-accelerated CFD solver and four orders of magnitude faster than a CPU-based CFD solver at a cost of a low error rate. This approach can provide immediate feedback for real-time design iterations at the early stage of design. Compared with existing approximation models in the aerodynamics domain, CNNs enable an efficient estimation for the entire velocity field. Furthermore, designers and engineers can directly apply the CNN approximation model in their design space exploration algorithms without training extra lower-dimensional surrogate models.

The potential of the Augmented Kalman Filter algorithm is tested in this paper for joint state-input estimation in structural dynamics field. In view of inverse load identification, the filter is compared with the Transfer Path Analysis Matrix Inversion technique, commonly used for industrial applications. An existing Optimal Sensor Placement strategy for Kalman Filter is adopted and validated on real experimental data. The advantages of the proposed methods, through strain measurements information, are identified in the effort needed for data-acquisition and data-processing. The effectiveness of the filter and the quality of the results are demonstrated in this paper for an industrial test-case, such as a rear twistbeam suspension.

• Christoph Ludwig
• Oliver Junge
• U. Wever

Model based real-time parameter identification in oscillating systems is a topic of ongoing interest, especially in the context of fault diagnosis during the operation of the system. At the core is a sufficiently small model which is successively calibrated by measurement data. For smooth data such as temperatures, Kalman-based filtering works well. However, for highly oscillatory data from, for example, rotating systems which are often additionally disturbed by harmonic excitations, these methods are prone to failure. In this paper we present an identification method that is able to detect changes in the stiffness properties of the system characterized by a single fault parameter based on frequency data. Its superior performance is demonstrated by a mass–spring system as well as a rotating shaft.

• H. Brandtstaetter
• L. Huebner
• Artur Jungiewicz
• U. Wever

The potential of data driven operational support with respect to predictive analysis is limited. A new approach is the model based simulation of operational behavior. The simulation of specific physical effects allows monitoring of the system behavior even of data that cannot be measured directly. A simulation model that supports the plant monitoring is called digital twin. It provides additional information about the asset state. Better knowledge of the system behavior increases the availability of the plant and the possibility to predict potential faults during operation. This paper presents two examples of digital twins. The first one, which is realized for a 50MW electric drive train, is designed to identify the actual unbalance state of the rotor system. The second one is designed to optimize the run up routines for synchronous motors with DOL start. It calculates the current rotor temperature based on the transferred losses and predicts the temperature for switching-on scenarios. The mathematical methods to implement digital twins are explained in detail. The results of numerical simulations are compared to measurements on the real system. Finally, the benefits of the digital twin in terms of failure diagnosis and system state predictions are presented.

Gear Transmission Error (TE) is often considered as the main cause of gear whine. TE represents the difference between the perfectly kinematic transmission of motion and the one actually achieved. TE vibrations are extremely small and pose significant measurement challenges. This article demonstrates how low-cost digital encoders can be successfully used together with the Elapsed Time Method to simplify TE measurement with respect to the traditional Direct Method. A precision gear pair test rig is exploited to compare the two methods from a theoretical and an experimental point of view. Following the observations drawn from such comparison, a measuring chain is set up to validate the proposed procedure on a real case all-electric vehicle gearbox. It is shown how TE represents a useful gearbox NVH indicator and how it can be used to support gear microgeometry design.

In this work we present a novel method for the solution of gear contact problems in flexible multi-body. These problems are characterized by significant variation in the location and size of the contact area, typically requiring a high number of degrees of freedom to correctly capture deformation and stress fields. Therefore fully dynamic simulation is computationally prohibitive. To overcome these limitations, we exploit a combined analytic-numerical contact model within a parametric model order reduction (PMOR) scheme. The reduction space consists of a truncated set of eigenvectors augmented with a parameter dependent set of residual static shape vectors. Each static shape is computed by interpolating among a set of displacement modes of the interacting bodies, obtained from a series of precomputed static contact analyses. During the contact analyses, an analytic model based on the Hertz theory describes the teeth local deformation. We implement the proposed method in an in-house code and we apply it to spur and helical gears dynamic contact analyses. We compare the results with classical PMOR schemes highlighting how the combined use of the semi-analytic contact model allows to decrease further the model complexity as well as the computational burden, for both static and dynamic cases. Finally, we validate the methodology by means of a comparison with experimental data found in literature, showing that the numerical method is able to capture quantitatively the static transmission error measurements in case of both helical and spur geared transmission for different torque levels.

Design models can drastically improve the applicability of testing and allow measuring previously unmeasurable quantities and designing reduced test configurations. A common workflow is followed: a multiphysics system model provides a prediction of the system states which is corrected by the estimation algorithms using the measurement data. The model can then generate data of the non-measurable quantities (e.g. virtual sensors). A wide range of models can be used, including analytical, 1D lumped parameter and 3D distributed parameter models. Key is that they are easy to evaluate and have a small number of states, while capturing the dominant physics. Novel model order reduction techniques enable the use of more complex models. A wide range of state estimation approaches has been developed such as the (linear, extended, unscented, …) Kalman Filter and the Moving Horizon Estimator. All approaches require a trade-off between accuracy and computational load so that conventional estimators must be tailored to deal with high-fidelity nonlinear models of industrial complexity. The approach is illustrated with two cases: the estimation of hard-to-measure vehicle body forces using the extended Kalman filter and the application to an electro-mechanical drivetrain subject to unknown input forces. Methodological aspects are evaluated and different estimators are compared.

Current design rules for railway wheelsets do not directly address issues related to fatigue crack propagation. Nevertheless, the latter topic is a part of the revised safety concept for passenger trains recently adopted in German railway applications. Numerous research activities, including international cooperative projects, have been conducted in the past decade aiming at quantifying fatigue crack growth rates in railway axles and estimating their inspection intervals based on the fracture mechanics methodology. This paper summarizes some experience and findings obtained by the authors within several studies dealing with the assessment of fatigue crack propagation in railway steels. Particular aspects highlighted in the paper include material characterization, effects of the specimen geometry and crack tip constraint on fatigue crack growth rates, stress analyses of axles and wheelsets, the derivation of stress intensity factor solutions applicable to specific conditions achieved in railway axles, considerations of the variability and scatter of geometrical parameters and material data in fatigue crack growth calculations.

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Source: https://www.researchgate.net/publication/347907141_12_Model_order_reduction_and_digital_twins