Willner, Kai, Prof. Dr.-Ing. habil.

Prof. Dr.-Ing. habil. Kai Willner

Department of Mechanical Engineering
Institute of Applied Mechanics (LTM, Prof. Steinmann)

Room: Raum 00.039
Egerlandstraße 5
91058 Erlangen
Deutschland

 

Modellierung und Simulation von Systemen mit unsicheren Parametern

Die meisten in den Ingenieurwissenschaften gebräulichen Rechenverfahren, wie z.B. die Finite Element Methode in der Mechanik, setzen genau bekannte Modellparameter als eine wesentliche Grundlage für den Erfolg ihrer Berechnungen voraus. Im Falle unsicherer Modellparameter muss so vor Beginn der Rechnung der “wahrscheinlichste” Wert für jeden dieser Parameter bestimmt und dem Rechenverfahren als “scharfer Wert” zur Verfügung gestellt werden. Die Ergebnisse des Verfahrens sind dann wiederum “scharfe Werte”, die dem Benutzer eine scheinbare Exaktheit vorspiegeln, die unter Berücksichtigung der unsicheren Voraussetzungen jedoch in der Realität kaum ihre Entsprechung findet.
Mit Hilfe der Fuzzy-Arithmetik ist es nun möglich, auch a priori unsichere Parameter mit ihren “unscharfen Werten” zu verarbeiten, wobei das anfängliche Mehr an Information im Rechenverfahren seine volle Berücksichtigung findet.
Es werden Konzepte, Verfahren und Programmpakete entwickelt, mit deren Hilfe verschiedenste ingenieurwissenschaftliche Problemstellungen unter Einbeziehung unsicherer Modellparameter zufriedenstellend gelöst werden können.

Kontaktmechanik

  • Entwicklung konstitutiver Kontaktgesetze
  • Effiziente Kontaktalgorithmen für die FEM

Methode der finiten Elemente

Die Finite-Elemente-Methode (FEM, englisch: finite element method) ist das am häufigsten eingesetzte Verfahren zur Berechnung komplexer Konstruktionen im Maschinenbau, im Apparatebau, in der Fahrzeugtechnik, in der Luft- und Raumfahrttechnik und im Bauwesen. Der Einsatz erfolgt dabei nicht nur für Standardprobleme der Festigkeitsberechnung und der Schwingungs- und Stabilitätsuntersuchung, sondern auch für Spezialaufgaben, wie z.B. für Aufgaben der Bruch- und Kontaktmechanik oder bei extrem großen Deformationen und plastischen Beanspruchungen, wie sie etwa bei Crash-Untersuchungen auftreten.
Alle genannten Beispiele entstammen der Strukturmechanik, jedoch ist die Methode der finiten Elemente nicht darauf beschränkt. Prinzipiell kann jedes andere Feldproblem, das durch partielle Differentialgleichungen beschrieben wird, mit Hilfe der FEM gelöst werden.
Ein typisches Beispiel ist die Wärmeleitung. Auch Probleme der Hydro- und Aerodynamik oder der Akustik lassen sich lösen. Hier sind jedoch andere Verfahren, wie die Randelemente-Methode (BEM, englisch: boundary element method), häufig besser geeignet, da es sich um unendliche oder halbunendliche Gebiete handeln kann, die durch eine Randformulierung sehr viel besser erfaßt werden können.
Die FEM ist von Vorteil, wenn es sich um ein klar begrenztes Gebiet handelt, wie zum Beispiel bei der Strömungsberechnung in einem Hafenbecken oder der Innenraumakustik eines Fahrzeugs. Ähnliches gilt bei der Untersuchung elektromagnetischer Felder.
Ein immer mehr in den Vordergrund tretender Aspekt ist die Behandlung gekoppelter Feldprobleme, wie zum Beispiel thermomechanische Aufgabenstellungen. Dies umfaßt die Berechnung von wärmeinduzierten Spannungen, aber auch die Berechnung von Formgedächtniselementen, die thermisch aktiviert werden. Die sich rapide ausbreitende Verwendung piezomechanischer, magnetostriktiver oder elektrorheologischer Materialien als Aktoren und Sensoren macht die gekoppelte Berechnung elektrischer bzw.magnetischer Felder mit mechanischen Größen nötig. Darüberhinaus treten gekoppelte Probleme als Interaktionsproblem zwischen Gebieten mit verschiedenen Feldgrößen auf. Ein typisches Beispiel ist hier die Fluid-Struktur-Kopplung bei akustischen Fragestellungen.
Einen Überblick über die FEM-Resourcen im Internet mit Zugang zu frei verfügbarer Software gibt Finite Element Analysis.

Aktuelle Forschungsprojekte:

  • Exploring Brain Mechanics (EBM): Understanding, engineering and exploiting mechanical properties and signals in central nervous system development, physiology and pathology

    (Third Party Funds Group – Overall project)

    Term: 1. January 2023 - 31. December 2026
    Funding source: DFG / Sonderforschungsbereich / Transregio (SFB / TRR)
    Abstract

    Thecentral nervous system (CNS) is our most complex organ system. Despite tremendousprogress in our understanding of the biochemical, electrical, and geneticregulation of CNS functioning and malfunctioning, many fundamental processesand diseases are still not fully understood. For example, axon growth patterns inthe developing brain can currently not be well-predicted based solely on thechemical landscape that neurons encounter, several CNS-related diseases cannotbe precisely diagnosed in living patients, and neuronal regeneration can stillnot be promoted after spinal cord injuries.

    Duringmany developmental and pathological processes, neurons and glial cells aremotile. Fundamentally, motion is drivenby forces. Hence, CNS cells mechanicallyinteract with their surrounding tissue. They adhere to neighbouring cells and extracellular matrix using celladhesion molecules, which provide friction, and generate forces usingcytoskeletal proteins.  These forces aretransmitted to the outside world not only to locomote but also to probe themechanical properties of the environment, which has a long overseen huge impacton cell function.

    Onlyrecently, groups of several project leaders in this consortium, and a few other groupsworldwide, have discovered an important contribution of mechanical signalsto regulating CNS cell function. For example, they showed that brain tissuemechanics instructs axon growth and pathfinding in vivo, that mechanicalforces play an important role for cortical folding in the developing humanbrain, that the lack of remyelination in the aged brain is due to an increasein brain stiffness in vivo, and that many neurodegenerative diseases areaccompanied by changes in brain and spinal cord mechanics. These first insights strongly suggest thatmechanics contributes to many other aspects of CNS functioning, and it islikely that chemical and mechanical signals intensely interact at the cellularand tissue levels to regulate many diverse cellular processes.

    The CRC 1540 EBM synergises the expertise of engineers, physicists,biologists, medical researchers, and clinicians in Erlangen to explore mechanicsas an important yet missing puzzle stone in our understanding of CNSdevelopment, homeostasis, and pathology. Our strongly multidisciplinary teamwith unique expertise in CNS mechanics integrates advanced invivo, in vitro, and in silico techniques across time(development, ageing, injury/disease) and length (cell, tissue, organ) scalesto uncover how mechanical forces and mechanical cell and tissue properties,such as stiffness and viscosity, affect CNS function. We especially focus on(A) cerebral, (B) spinal, and (C) cellular mechanics. Invivo and in vitro studies provide a basic understanding ofmechanics-regulated biological and biomedical processes in different regions ofthe CNS. In addition, they help identify key mechano-chemical factors forinclusion in in silico models and provide data for model calibration andvalidation. In silico models, in turn, allow us to test hypotheses without the need of excessive or even inaccessibleexperiments. In addition, they enable the transfer and comparison of mechanics data and findingsacross species and scales. They also empower us to optimise processparameters for the development of in vitro brain tissue-like matricesand in vivo manipulation of mechanical signals, and, eventually, pavethe way for personalised clinical predictions.

    Insummary, we exploit mechanics-based approaches to advance ourunderstanding of CNS function and to provide the foundation for futureimprovement of diagnosis and treatment of neurological disorders.

  • Modellbasierter Abgleich von ex vivo und in vivo Testdaten (X01)

    (Third Party Funds Group – Sub project)

    Overall project: SFB 1540: Erforschung der Mechanik des Gehirns (EBM): Verständnis, Engineering und Nutzung mechanischer Eigenschaften und Signale in der Entwicklung, Physiologie und Pathologie des zentralen Nervensystems
    Term: 1. January 2023 - 31. December 2026
    Funding source: DFG / Sonderforschungsbereich (SFB)
    Abstract

    X01 befasst sich mit dem Problem widersprüchlicher Ergebnisse mechanischer Eigenschaften von ultraweichen Materialien wie Hirngewebe, wenn unterschiedliche ex vivo und in vivo Testverfahren verwendet werden. Unsere Hypothese ist, dass es ein kontinuumsbasiertes Simulationsmodell ermöglichen wird, die verschiedenen experimentell beobachtbaren Regime in vivo und ex vivo zu vereinen. Damit können wir erstmals mechanische ex vivo Parameter verwenden, die aus verschiedenen Testmodalitäten gewonnen wurden, um das mechanische in vivo Verhalten des menschlichen Gehirns zu erklären.

  • Eine hybride Fuzzy-Stochastische-Finite-Element-Methode für polymorphe, mikrostrukturelle Unsicherheiten in heterogenen Materialien

    (Third Party Funds Single)

    Term: 1. December 2020 - 30. November 2023
    Funding source: DFG-Einzelförderung / Sachbeihilfe (EIN-SBH)
    Abstract
    Computational homogenization requires two separate finite element models: a model at the macroscale and a model of the materials’ underlying structure at the microscale. Computational homogenization involves two main ingredients: the transfer of the macroscopic loading to the microscale and averaging the corresponding response of the microstructure to obtain the effective macroscopic properties. A challenging aspect for computational homogenization is the proper modelling of material with uncertainty in the microstructure, as considered in this project. Uncertainties in the macroscopic response of heterogeneous materials result from various sources: the natural variability in the microstructure’s geometry and its constituent’s material properties and the lack of sufficient knowledge regarding the microstructure. The first type of uncertainty is denoted as aleatoric uncertainty and may be characterized by probabilistic approaches. The second type of uncertainty is denoted as epistemic uncertainty and may be described using fuzzy arithmetic. Models considering both sources of uncertainty are denoted polymorphic, requiring some combination of stochastic and fuzzy methods.In Phase I we developed methods for the accurate and efficient propagation of polymorphic uncertainty through the material’s microstructure and applied all proposed approaches to a benchmark problem. The objectives of the Phase II are further development of modelling techniques and their application to the engineering design of structures. The outcome of Phase II will be an accomplished methodology allowing the uncertainty propagation from the lowest level of a material microstructure through the macroscopic structure simulation to the engineering design and decision making. More precisely in Phase II the following challenges are considered:- We continue the development of advanced fuzzy-stochastic benchmark RVE for the microstructure of heterogeneous materials, resulting thus in a more realistic and precise description of polymorphic uncertainty in the material’s microstructure. - Modelling techniques for spectral non-deterministic finite element analysis will be enriched to non-deterministic eXtended Isogeometric Analysis.- The computational cost of full-order large scale simulations of systems in the presence of uncertainty is unacceptably high, in particular considering many-query or real-time applications. Thus, reduced order modeling is an essential tool which allows a speed up microscale simulations. - Reduced order models and metamodels provide a necessary bridge to the final stage of the project, in which a suitable metamodel will be used on the macroscale to run large size simulations of engineering structures. - Finally, the influence of uncertainty in the macrostructure on the static and the dynamic behavior of engineering structures under random loading will be analyzed.

  • A hybrid Fuzzy-Stochastic-Finite-Element-Method for polymorphic, microstructural uncertainties in heterogeneous materials

    (Third Party Funds Group – Sub project)

    Overall project: Polymorphic uncertainty modelling for the numerical design of structures
    Term: 1. December 2020 - 30. November 2023
    Funding source: DFG-Einzelförderung / Sachbeihilfe (EIN-SBH)
    Abstract
    Computational homogenization requires two separate finite element models: a model at the macroscale and a model of the materials’ underlying structure at the microscale. Computational homogenization involves two main ingredients: the transfer of the macroscopic loading to the microscale and averaging the corresponding response of the microstructure to obtain the effective macroscopic properties. A challenging aspect for computational homogenization is the proper modelling of material with uncertainty in the microstructure, as considered in this project. Uncertainties in the macroscopic response of heterogeneous materials result from various sources: the natural variability in the microstructure’s geometry and its constituent’s material properties and the lack of sufficient knowledge regarding the microstructure. The first type of uncertainty is denoted as aleatoric uncertainty and may be characterized by probabilistic approaches. The second type of uncertainty is denoted as epistemic uncertainty and may be described using fuzzy arithmetic. Models considering both sources of uncertainty are denoted polymorphic, requiring some combination of stochastic and fuzzy methods.In Phase I we developed methods for the accurate and efficient propagation of polymorphic uncertainty through the material’s microstructure and applied all proposed approaches to a benchmark problem. The objectives of the Phase II are further development of modelling techniques and their application to the engineering design of structures. The outcome of Phase II will be an accomplished methodology allowing the uncertainty propagation from the lowest level of a material microstructure through the macroscopic structure simulation to the engineering design and decision making. More precisely in Phase II the following challenges are considered:- We continue the development of advanced fuzzy-stochastic benchmark RVE for the microstructure of heterogeneous materials, resulting thus in a more realistic and precise description of polymorphic uncertainty in the material’s microstructure. - Modelling techniques for spectral non-deterministic finite element analysis will be enriched to non-deterministic eXtended Isogeometric Analysis.- The computational cost of full-order large scale simulations of systems in the presence of uncertainty is unacceptably high, in particular considering many-query or real-time applications. Thus, reduced order modeling is an essential tool which allows a speed up microscale simulations. - Reduced order models and metamodels provide a necessary bridge to the final stage of the project, in which a suitable metamodel will be used on the macroscale to run large size simulations of engineering structures. - Finally, the influence of uncertainty in the macrostructure on the static and the dynamic behavior of engineering structures under random loading will be analyzed.

  • FOR 2271: Prozessorientiertes Toleranzmanagement mit virtuellen Absicherungsmethoden

    (Third Party Funds Group – Overall project)

    Term: 1. June 2016 - 14. August 2023
    Funding source: DFG / Forschungsgruppe (FOR)
    URL: https://www.for2271.tf.fau.de/
    Abstract

    The comprehension of geometric part deviationsand their manufacturing and assembly related sources as well as the investigationof their effects on the function and quality of technical products builds theframework for the planned research group “process-oriented tolerance managementbased on virtual computer-aided engineering tools”. The aim of this researchgroup is the provision of holistic methods and efficient tools for thecomprehensive management of geometric deviations along the product originationprocess, which are to be validated in a model factory. In doing so, aparticular focus is set on the development of a procedure for the fruitfulcooperation of all departments involved in geometric variations management,from product development, to manufacturing, to assembly and to metrology, whichwill enable companies to quickly specify functional tolerances, which aremanufacturable and measurable, and consequently to save costs and to reduce thetime to market.

    In this regard, the vision of the researchgroup is to enable the close collaboration of product development,manufacturing, assembly and metrology in computer-aided tolerancing, i. e.the joint formulation of functional tolerances, which are manufacturable andmeasurable. By enabling this close collaboration, all manufacturing andassembly related sources of later problems regarding the product function andquality can be considered already during early phases of virtual product andprocess development. As a consequence, tolerances can be specified efficientlyand optimized inspection plans as well as robust manufacturing and operatingwindows can be identified, which allows the development of robust products tobe manufactured and measured at low costs.  

    Since geometric part deviations are inevitableand affect the function and quality of technical products, their managementalong the product origination process is essential for the development offunctioning products, which conform to the quality and usage requirements ofcustomers and are successful on international markets. As a consequence,tolerance management is a fundamental task in product development and reachesvarious fields of industry, from consumer to industrial goods. Due to steadilyincreasing requirements on quality and efficiency, it strongly gains importancenot only with large, but also small and medium-sized enterprises. In thiscontext, the industrial application of the scientific findings of the researchgroup will contribute to the success of the German economy.  

  • A hybrid Sampling-Stochastic-Finite-Element-Method for polymorphic, microstructural uncertainties in heterogeneous materials

    (Third Party Funds Group – Sub project)

    Overall project: SPP 1886: Polymorphic uncertainty modelling for the numerical design of structures
    Term: 1. April 2016 - 30. November 2020
    Funding source: DFG / Schwerpunktprogramm (SPP)
    Abstract

    The overarching goal of the proposed project at the methodological side is to establish a computationally tractable numerical method that is suited to capture polymorphic uncertainties in large-scale problems (as arising from the numerical analysis of heterogeneous materials microstructures). On the one hand the method will allow for fuzzy probability distributions of the random parameters (describing a microstructures geometry) and on the other hand the method will be based on only a few reduced basis modes. These ingredients will enable to capture epistemic uncertainties in addition to aleatoric uncertainties in a computationally accessible manner. The overarching goal of the proposed project at the application side is to establish a non-deterministic macroscopic material model. On the one hand the model accounts for the heterogeneity of the underlying material's microstructure by computational homogenization, and on the other hand it captures polymorphic uncertainties in the geometry description of the microstructure. The non-deterministic macroscopic material model then represents the necessary input for the mechanical design of macroscopic (engineering) structures under due consideration of polymorphic uncertainties in the heterogeneous materials microstructure.

  • Vibration reduction by energy transfer using shape adaption

    (Third Party Funds Group – Sub project)

    Overall project: SPP 1897: Calm, Smooth and Smart - Novel Approaches for Influencing Vibrations by Means of Deliberately Introduced Dissipation
    Term: 1. January 2016 - 31. December 2019
    Funding source: DFG / Schwerpunktprogramm (SPP)
    Abstract

    Lightweight design is one of the most important issues in engineering design. The objective is to reduce the mass of structural components for the purpose of saving costs, energy and resources in manufacturing and operation processes. However, the lighter the structure is, the more it is prone to unwanted vibrations. Such vibrations should be minimized in order to prevent the environment, products and human beings from being harmed and to maximize the lifetime of the products.Vibration reduction can be achieved by passive, semi-active or active measures, where passive means that no external energy is needed, while semi-active and active measures employ external energy to either control dissipation or directly counteract the vibrational motion, respectively. Since active measures usually do not rely on dissipation, they do not fall in the scope of the call for proposals and will not regarded in this project. In the realm of passive and semi-active measures, two general approaches can be used to reduce vibration in structures, namely that of damping, which is the dissipation of kinetic energy into another form of energy, or that of absorption, which is the transfer of kinetic energy from a critical mode into an uncritical mode.The envisioned approach will combine the concepts of damping and absorption in a novel way by integrating the functionality of a damped, tuned mass absorber into a shape adaptive structure. By dynamically adapting the stiffness of a slender, beam-like structure using shape adaption of the cross-section, kinetic energy will be transferred from the critical low-frequency bending modes into a specifically designed, higher frequency absorber mode, which can then be damped in an optimal way. Optimal design of the shape adaption mechanism and of the absorber mode will be pursued using compliant mechanisms. The dissipation will be optimized by a specifically designed friction damper.

  • Fuzzy-arithmetical modeling of processes with uncertain prarameters

    (Third Party Funds Group – Sub project)

    Overall project: FOR 2271: Prozessorientiertes Toleranzmanagement mit virtuellen Absicherungsmethoden
    Term: 1. January 2016 - 28. February 2019
    Funding source: DFG / Forschungsgruppe (FOR)

  • Structural dynamics of rotating systems

    (Own Funds)

    Term: 1. January 2015 - 31. May 2020

  • Reduced order modelling of non-linear gyroscopic systems in ALE formulation with frictional contact

    (Own Funds)

    Term: since 1. January 2015
    Abstract

    Rotating systems are subject to gyroscopic effects, which influence the structure’s dynamics. The Arbitrary-Lagrangian-Eulerian formulation in the finite element method offers an efficient way to include translational and rotatory guiding movement in the model in the course of decoupling this motion from the FE mesh. At the same time this approach aggravates the computation of frictional contact of the rotating body with other still-standing structures.
    This procedure stems from the field of rolling contact dynamics and is used in this project for the simulation of disc brakes. By means of these non-linear gyroscopic ALE-systems miscellaneous methods of reduced order modelling in structural dynamics are put to test and extended to meet the models peculiarities.

  • Material modelling of sheet-layered lamination stacks

    (Own Funds)

    Term: since 1. January 2015
    Abstract

    The numerical simulation of sheet-layered lamination stacks, which can be found in electric motors and transformers, is a challenging task in structural mechanics due to the layout of these components.  Depending on the manufacturing process, these sheets are either in frictional contact to each other or are linked together with the help of a bonding varnish. Especially the interlayer between individual sheets and their interaction have a strong influence on the structure and may be responsible for a nonlinear deformation behavior. In the context of performance and computational effort, it is desirable to avoid a full Finite-Element simulation incorporating every layer such that homogenization techniques are used in this project to derive a sophisticated surrogate material model, which takes the special micro-structure of these lamination stacks into account.

  • Investigation and reduction of nonlinear oscillation systems using modal approaches

    (Own Funds)

    Term: since 1. September 2012
    Abstract

    In this project nonlinear oscillating systems are investigated. The nonlinearity is caused by the effect of large deformations (geometrical nonlinearity) or by physical effects, like friction. A designated target is after a nonlinear modal analysis (for example on the basis of NNMs) a model reduction on the isolated nonlinear mode. Limitations for this approach are given by the nonlinear modal analysis.

  • Constitutive friction law for the description and optimization of tailored surfaces

    (Third Party Funds Group – Sub project)

    Overall project: TRR 73: Umformtechnische Herstellung von komplexen Funktionsbauteilen mit Nebenformelementen aus Feinblechen - Blechmassivumformung
    Term: 1. January 2009 - 31. March 2021
    Funding source: DFG / Sonderforschungsbereich / Transregio (SFB / TRR)
    URL: https://www.tr-73.de/
    Abstract

    A central challenge insheet-bulk metal forming is a partially uncontrolled material flow. Thisworsens the achievable geometrical accuracy of the parts. In this context, theobjective of the project is to control the material flow by local adjustmentsof the friction by modifying the workpiece or tool surface. Hereby, the diefilling of the functional elements is improved. Especially tool-sidedmodifications have a high potential, since they do not extend the processchain. However, for an efficient application, they must offer a high wearresistance, which is why the functional relationships between wear-inducedchanges in the tool topography and friction are being researched.

  • C3: Parameter and shape optimization in finite elastoplasticity

    (Third Party Funds Group – Sub project)

    Overall project: TRR 73: Umformtechnische Herstellung von komplexen Funktionsbauteilen mit Nebenformelementen aus Feinblechen - Blechmassivumformung
    Term: 1. January 2009 - 31. December 2016
    Funding source: DFG / Sonderforschungsbereich / Transregio (SFB / TRR)
    URL: http://www.tr-73.de

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