Organizer: Michael Kaliske (TU Dresden)
DFG-PP 7 : Polymorphic Uncertainty Modeling for the Numerical De-sign of Structures
Thursday 14:00 - 16:00 Coudraystr. 9A, Lecture hall 6
Polymorphic Uncertainty Modeling for the Numerical Design of Structures (SPP 1886)
M. Kaliske (TU Dresden) 14:00
Advanced engineering solutions are characterized by inherent robustness and flexibility as essential features for a faultless life of structures and systems under uncertain and changing conditions. An implementation of these features in a structure or system requires a comprehensive consideration of uncertainty in the model parameters and loads
as well as other types of intrinsic and epistemic uncertainties.
Numerical design of structures/systems should be robust with respect to uncertainties inherently present in resistance of materials, boundary conditions e.g. environmental and man imposed loads, physical and numerical models. This requires in turn the avail-ability of a reliable numerical analysis, assessment and prediction of the lifecycle of a structure/system taking explicitly into account the effect of the unavoidable uncertain-ties.
Challenges in this context involve, for example, limited information, human factors, subjectivity and experience, linguistic assessments, imprecise measurements, dubious information, unclear physics etc. Due to the polymorphic nature and characteristic of the available information both probabilistic and set-theoretical approaches are relevant for solutions.
Comparison of nested collocation and projection algorithms for mixed aleatory and epistemic uncertaintites using a probability-box approach
A. Fau (Leibniz Universität Hannover), M. Dannert (Leibniz Universität Hannover), M. Broggi (Leibniz Universität Hannover), U. Nackenhorst (Leibniz Universität Hannover), M. Beer (Leibniz Universität Hannover)
14:20
Most numerical models for engineering applications include some uncertainties caused for example, by load, geometry or material properties. Two types of uncertainties may be distinguished, aleatory uncertainties due to random phenomena and epistemic un-certainties due to a lack of knowledge or data [1]. Input variables have generally both, aleatory and epistemic uncertainties. To model such mixed uncertain variables, proba-bility bounds analysis has shown to be useful. The probaproba-bility of an event is here not described by a unique function [2] but within an upper and a lower bound, which form a so-called probability-box (p-box) distribution function. One key issue in the applications is numerical effort [3]. Therefore, collocation and projection methods are proposed to improve numerical efficiency.
In this work, a stochastic finite element analysis for an elasto-plastic problem involv-ing uncertain constitutive parameters is implemented. The underlyinvolv-ing random field is modelled by a p-box. A nested collocation scheme, where samples are chosen within a certain lattice construction [4], and a projection method [5] are examined in a com-parative study. The algorithms are tested with one- and two-dimensional example and verified by Monte Carlo simulation. A critical review of the two methods for mixed uncertainties description will be proposed in this contribution.
[1] A. Der Kiureghian and O. Ditlevsen. “Aleatory or epistemic? Does it matter?”
Structural Safety, 31(2), pp. 105-112, 2009.
[2] S. Ferson and J. Hajagos.“Arithmetic with uncertain numbers: rigorous and (often) best possible answers”, Reliability Engineering and System Safety, 85(1-3), pp. 135-152, 2004.
[3] H. Zhang et al. Structural reliability analysis on the basis of small samples: An
interval quasi-Monte Carlo method, Mechanical Systems and Signal Processing, 37(1-2), pp. 137-151, 2013.
[4] D. Xiu and J. S. Hesthaven. High-Order Collocation Methods for Differential Equa-tions with Random Inputs. SIAM J. Sci. Comput., 27(3), pp. 1118-1139, 2005.
[5] O.P. Le Maître and O.M. Knio. Spectral Methods for Uncertainty Quantification, Chap. Non-intrusive Methods. Springer, 2010.
On uncertainty in friction measurements
G. Ostermeyer (Technische Universität Braunschweig), M. Mueller (Technische Universität Braunschweig), T. Srisupattarawanit, A. Völpel
14:40
In the fields of mechanical engineering, friction is ubiquitous. It is a fundamental cause of energy loss and wear. Another concern is the occurrence of comfort-relevant friction induced vibrations. Prominent examples of this are NVH phenomena such as brake squeal, which is being investigated at great expense by the German automotive industry and academia, to determine its causes and potential influencing factors.
For this purpose, numerous specialized measurements are performed, and models of varying complexity are used. All of these measurements and models have the common trait that the coefficient of friction, defined as the ratio of the tangential force and normal force, has a decisive influence on the systems’ stability.
To parameterize the friction coefficient in macroscopic models, measurements must be performed. In this case, often an average over time and over various loading procedures is used. As the measurements reveal, the coefficient of friction is in reality not constant, but is subject to a high degree of dynamics on various time scales, caused by complex processes in the boundary layer. A treatment of the coefficient of friction as a steady-state parameter, or even as a constant, is thus a major reduction.
The large variability of the coefficient of friction causes a corresponding variance in the stability limits of the models considered. This phenomenon is observed in the real world, where squealing seems to have a non-deterministic behavior. This suggests uncertainties in the modeling of the friction coefficient. Due to the various types of uncertainty (vari-ability, incompleteness and inaccuracy), the entire problem is a matter of polymorphic uncertainty.
This paper focuses on the modeling of the friction coefficient, taking into account the various causes of uncertainty. Selections of raw data obtained at the authors institute throughout many years of research on friction phenomena in brake systems will be eval-uated and classified with respect to its uncertainty properties.
Polymorphic Uncertainty Modeling of Heterogeneous Thermo - Hydro - Me-chanical Coupled Systems under Vague Assumptions on Parameter Correla-tions
T. Lahmer (Bauhaus-Universität Weimar), C. Könke
(Bauhaus-Universitaet Weimar), L. Nguyen-Tuan (Bauhaus-Universität Weimar), A. Schmidt
15:00
New hybrid materials and constructions with heterogeneous material properties gain
in-creasing importance due to the development of new light-weight building concepts. Fur-thermore, many natural and technical materials applied show a heterogeneous material distribution in the existing engineering constructions. Examples are typical geoscience materials or aggregate-matrix materials w.r.t their meso- and microscopic treatment in a multi-scale approach. The modeling of the material behavior during forecast simulations can either be done by the estimation of upper and lower bounds or by the application of multi-dimensional random fields. In multi-field situations, e.g. coupled thermal-hydro-mechanical systems like dams, dikes or subsoil deposits, there are a series of sensitive material properties which can be modeled via random fields. Often, these fields exhibit a certain correlation, e.g. in regions of deteriorated material the hydraulic permeability might be increased while the mechanical stiffness and strength is reduced. The question arising is, if this has also effects on other material properties and how the interdepen-dency can be taken effectively into consideration in this case. In the given research, a methodology is derived which allows, based on a polymorphic uncertainty model, the generation of random fields for multi-physic applications, where however, the degree of interdependence of the different fields and their model parameters is not fully known.
Therefore, the talk comprises first steps in the the development, analysis and applica-tion of a polymorphic uncertainty model, which captures the random variability of the material properties with vague information concerning their correlations and correlation lengths. First simulations for simplified THM models will be presented. The talk will also give some conclusions about the assessed structural reliability while using the newly proposed method.
On Using Fuzzy Arithmetic in Optimization Problems with Uncertain Con-straints
M. Mäck (University of Stuttgart), M. Hanss (University of Stuttgart) 15:20
In engineering problems it is often required to not only find an optimal solution, but also to ensure that the solution is robust with respect to potential uncertainties. Existing algorithms for optimization deliver accurate solutions for a multitude of problems and a broad field of applications. A proper functioning of these algorithms, however, requires the provision of exactly known parameters, and thus, the availability of a detailed knowl-edge of the considered system as well as its potential restrictions and constraints. In reality, however, the model parameters might be uncertain and the constraints might be vaguely defined or even unknown, making the determination of an optimal and prefer-ably robust solution very challenging. This deficiency of knowledge about the system is often referred to as uncertainty of epistemic type and it cannot be handled satisfac-torily by methods of traditional probability theory, as used for uncertainty of aleatoric type. In contrast, possibilistic approaches, such as fuzzy arithmetic, prove well suited to represent uncertainty caused by incomplete information or deficient knowledge and to quantitatively analyze systems with epistemic uncertainties.
In this contribution, a general approach is presented where fuzzy arithmetic is used to tackle optimization problems with uncertain constraints. Based on the fuzzy arithmetical formulation, a special robustness criterion can be defined which allows for the
determina-tion of an optimal soludetermina-tion in due consideradetermina-tion of uncertain, fuzzy-valued constraints.
Additionally, uncertain model parameters can be considered, leading to fuzzy-valued parameters also in the objective function, and thus, extending the methodology to a generalized approach to optimization under miscellaneous uncertain conditions. The potentials and drawbacks of the approach are discussed and illustrated on the basis of a meaningful example.
Solution Space approach to address polymorphic uncertainties in early phase structural design
F. Duddeck (Technische Universität München) 15:40
The main uncertainties encountered in early phase structural design are epistemic, i.e.
they relate to lack of knowledge w.r.t. design decisions made later in the development process. Assessments of structural concepts are required based a partial knowledge of structural layouts of the relevant components. The solution space approach enables now a decoupled development of components assuring the overall fulfillment of design criteria. This is achieved by (i) establishing a simplified model for complete structural performances, (ii) evaluations using this simplified model to derive requirements for components, and (iii) optimizing these component requirements for maximal flexibility.
This flexibility offers a lack-of-knowledge approach for epistemic uncertainties where single components can be developed allowing modifications of components later in the development process without endangering the overall concept behavior. The optimiza-tion w.r.t. flexibility is then followed by structural optimizaoptimiza-tions (e.g. size, shape and topology) where further uncertainties (aleatoric) can be considered. A polymorphic un-certainty approach can be realized combining different robustness methods for epistemic and aleatoric uncertainties. The approach was originally introduced and further devel-oped for crashworthiness and then extended to driving dynamics. New developments, mainly for crashworthiness, will be presented in this paper addressing approaches for reduction of complexity related to multi-load case optimization, optimized communal-ity of several vehicles and computational efficiency. The approach is based on Solution Spaces using decoupled corridors (upper and lower limit) for force-displacement curves for each component. Structural shape and topology optimization is then used in an exemplary manner to illustrate the method with particular focus on load case, material, and geometry uncertainties. An outlook will be given concerning generalization of the approach to other disciplines.
DFG-PP 7 : Polymorphic Uncertainty Modeling for the Numerical De-sign of Structures
Thursday 16:30 - 18:30 Coudraystr. 9A, Lecture hall 6
Polymorphic uncertainty quantification for stability analysis of fluid satu-rated soil and earth structures
C. Henning, T. Ricken (TU Dortmund University) 16:30
Nowadays, numerical simulations enable the description of mechanical problems in many application fields, e.g. in soil or solid mechanics. During the process of physical and computational modeling, a lot of theoretical model approaches and geometrical approx-imations are sources of errors. These can be distinguished into aleatoric (e.g. model parameters) and epistemic (e. g. numerical approximation) uncertainties. In order to get access to a risk assessment, these uncertainties and errors must be captured and quantified. For this aim a new priority program SPP 1886 has been installed by the DFG which focuses on the so called polymorphic uncertainty quantification. In this sub-project, which is part of the SPP 1886 (sp12), the focus is driven on quantification and assessment of polymorphic uncertainties in computational simulations of earth struc-tures, especially for fluid-saturated soils. To describe the strongly coupled solid-fluid response behavior, the theory of porous media (TPM) will be used and prepared within the framework of the finite element method (FEM) for the numerical solution of initial and boundary value problems [1, 2]. To capture the impacts of different uncertainties on computational results, two promising approaches of analytical and stochastic sensitivity analysis will enhance the deterministic structural analysis [3, 4, 5]. A simple consolida-tion problem already provided a high sensitivity in the computaconsolida-tional results towards variation of material parameters and initial values. The variational and probabilistic sensitivity analyses enable to quantify these sensitivities. The variational sensitivities are used as a tool for optimization procedures and capture the impact of different pa-rameters as continuous functions. An advantage is the accurate approximation of the solution space and the efficient computation time, a disadvantage lies in the analyti-cal derivation and algorithmic implementation. In the probabilistic sensitivity analysis from the field of statistics, the expense only increases proportionally to the problems dimension. Instead of a constant value, the model parameters are defined as probability distribution, which provides random values. Thus, a set of solution data is built up by several cycles of the simulation. Different approaches of the Bayes statistics will enable to receive accurate information with just a few simulations. The overall objective is the development of more efficient methods and tools for the sizing of earth structures in the long-run.
[1] R. De Boer: Theory of Porous Media, Springer, New York, 2008.
[2] W. Ehlers and J. Bluhm: Foundations of multiphasic and porous materials, Porous
Media: Theory, Experiments and Numerical Applications, pages 3-86., Springer-Verlag, 2002.
[3] F.-J. Barthold: Theorie und Numerik zur Berechnung und Optimierung von Struk-turen aus isotropen, hyperelastischen Materialien Technische Universität Braun-schweig, Forschungs- und Seminararbeit aus dem Bereich Mechanik der Universität Hannover, Bericht-Nr. F 93/2, 1993.
[4] J. Stieghan: Variationelle Sensitivitätsanalyse in der Theorie poröser Medien Tech-nische Universität Carolo-Wilhelmina zu Braunschweig, 2008.
[5] J. E. Oakley and A. O’Hagan: Probabilistic sensitivity analysis of complex models:
a Bayesian approach, J. R. Statist. Soc. B, 66(3):751-769, 2004.
Multivariate stochastic finite element method with correlated material pa-rameters
E. Penner (Paderborn University), I. Caylak (Paderborn University), R.
Mahnken (Paderborn University)
16:50
It is well known that material properties are uncertain due to the manufacturing process or the heterogeneity. In addition, there are measurement errors and incomplete infor-mation on material properties, geometry and boundary conditions. These uncertainties also lead to uncertainty in the mechanical response. Therefore continuum modeling, where input parameters of the stochastic model are uncertain, should be designed by a stochastic partial differential equation (SPDE) instead of a deterministic PDE. A pos-sible numerical method for solving the SPDEs is the stochastic finite element method (SFEM), where the Monte Carlo (MC) method or polynomial chaos expansion (PCE) as presented in [1] for small deformation problems, are commonly used.
The key idea of our contribution is to consider the uncertainty in material parameters by modeling them as multivariate stochastic variables. Furthermore, a statistical analysis of the random material parameters including their correlations is performed. Usually, the material parameters are considered as stochastically independent. However, in our work we consider the dependency including the correlation obtained from experimental data.
The multivariate stochastic variables, in our case the material parameters, are formulated in the multivariate PCE. In order to determine the corresponding PC coefficients for correlated stochastic material parameters, we use the Cholesky decomposition.
In a numerical example we consider the static problem for uniaxial tension of a rectan-gular plate. This problem is investigated under a plane stress state in order to represent exactly the experimental setup conditions. Based on experimental data, statistics are generated for material parameters. Then, PC coefficients are calculated and a multi-variate stochastic finite element analysis is performed. Finally, the probability density functions of the system response of simulation and experiment are compared with each other.
[1] R. G. Ghanem and P. D. Spanos, Stochastic Finite Elements: A Spectral Approach, Springer-Verlag, New York, 1991.
A possibilistic evaluation using Fuzzy finite element method based on sparse experimental data
A. Dridger (Paderborn University), I. Caylak (Paderborn University), R.
Mahnken (Paderborn University)
17:10
In many engineering applications the uncertainty quantification has become important in order to obtain authentic results using, e.g. the stochastic finite element method (SFEM). Generally, the corresponding partial differential equations (PDEs) are sub-jected with different kinds of uncertainties, e.g. in material parameters, in boundary conditions or in the geometry. The uncertainties may arise from linguistic, imprecise, in-complete or statistical properties and are categorized into aleatoric and epistemic types.
The aleatoric uncertainties may be described by stochastic methods characterized by randomness, whereas epistemic uncertainties may be described by the fuzzy approach using fuzzy sets and/or membership functions [1], respectively.
In this work we investigate the combination of aleatoric and epistemic uncertainties called “imprecise probabilities” [2], for linear elastic continua. For this reason, a mem-bership function is characterized as a “possibility distribution” which encode a family of probability distributions. The associated probability density functions are transformed into possibility distributions using the probability-possibility transformation [3]. The main objective is the numerical computation of the solution considering the interaction of corresponding fuzzy parameters. Possibilistic evaluation of the fuzzy finite element method (FFEM) is presented. The α-level discretization technique [1] is applied in order to reduce the fuzzy arithmetic based FEM to an interval arithmetic based FEM. In this context, our method is applied in a numerical example for a plate with a ring hole.
[1] B. Moeller and M. Beer, Fuzzy Randomness - Uncertainty in Civil Engineering and Computational Mechanics, Springer, 2004.
[2] D. Dubois and H. Prade, Possibility Theory and Its Applications: Where Do We Stand? Springer, pp. 31-60, 2015.
[3] M. Hamed and M. Serrurier, Representing uncertainty by possibility distributions encoding confidence bands, tolerance and prediction intervals, Springer, pp. 233-246, 2012.
Tuesday and Wednesday 11:00 - 11:30
Weimar hall, Foyer Organizers: Dominik Kern (TU Chemnitz)
Melanie Todt (TU Wien, Institute of Lightweight Design and Structural Biomechanics)
Variational Integrators for Standard and Non-standard Heat Transfer D. Kern (TU Chemnitz), J. Blomberg Ghini (Conseil Européen pour la
Recherche Nucléaire (CERN))
Variational Integrators, sometimes referred to as symplectic integrators, were originally developed for conservative mechanical systems. Their advantages motivated further extensions, i.a. for simulations of thermomechanical systems.
The standard heat transfer by Fourier’s law is not conservative. Therefore it needs to be included by D’Alembert terms. This kind of heat transfer leads to the classical, parabolic heat equation.
Whereas non-standard heat transfer of Green & Naghdi type II fits well into the vari-ational framework, as it directly enters the varivari-ational formulation via the free energy.
The resulting partial differential equation is hyperbolic.
In this contribution we focus on the nonstationary thermal problem and derive a vari-ational integrator that covers both, Fourier’s law as well as Green & Naghdi type II.
A one-dimensional continuum, a bar, serves as model problem for both kinds of heat transfer.
High performance optimization algorithms for interface identification prob-lems
M. Siebenborn (Trier University)
This poster presents optimization approaches for large scale interface identification prob-lems prepared for supercomputers. In many applications, which are modeled by partial differential equations, there is a small number of spatially distributed materials or pa-rameters separated by interfaces. Often these interfaces form complex contours forcing high resolutions in the discretization schemes. The challenge is thus to combine HPC techniques with shape optimization in order to come up with scalable algorithms even for very large problems. This can be achieved by a combination of multigrid strategies and quasi Newton methods. It is also shown how different shape metrics affect the quality of finite element meshes and which are the most suitable ones.
Shell-based ply-level models of layered composites
M. Todt (TU Wien, Institute of Lightweight Design and Structural
M. Todt (TU Wien, Institute of Lightweight Design and Structural