Projects in the research field of “Numerical Methods and Uncertainity Quantification”
Modeling of Aleatoric and Epistemic Uncertainties in the Simulation of Multiphase Steel Structures
DFG (Deutsche Forschungsgemeinschaft) project BA 2823/121 within
Priority Program SPP1886
"Polymorphic uncertainty modelling for the numerical design of structures"
In this project we aim to develop methods for the computational analysis of multiphase steel structures taking into account aleatoric and epistemic uncertainties, which are associated with the material's microstructure and mechanical properties. The main goal is to predict the probability of failure (PoF) of an engineering structure in a certified sense by numerically identifying optimal bounds on the PoF, which take into account incomplete or imprecise statistical data. The approach is based on minimizing/maximizing a finitedimensional parameterization of the PoF in terms of Dirac masses to obtain the sharpest bounds possible. The available statistical data associated with macroscopic material properties enter the optimization problem as constraints. This statistical data will be partially obtained by numerical calculations performed at the microscale and homogenization of the microscopic response. Therein, incomplete data regarding the microstructure is incorporated in terms of bounds on statistical descriptors of higher order that describe the microstructure morphology. For this purpose, statistically similar representative volume elements are considered allowing for an efficient representation of the real microstructure and enabling access to targeted variations of microstructure morphology, which may not necessarily follow explicit probability distributions.
 N. Miska and D. Balzani [2019], "Quantification of uncertain macroscopic material properties resulting from variations of microstructure morphology based on statistically similar volume elements: application to dualphase steel microstructures", Computational Mechanics, Vol. 64, pp. 16211637
 N. Miska and D. Balzani [2020], "Computation of the Sharpest Bounds on Probabilities under the Influence of Polymorphic Uncertainties", Proceedings in Applied Mathematics and Mechanics, Published online.
 N. Miska and D. Balzani [2019], "Optimal Bounds for the Probability of Failure of Sheet Metal Forming Processes of DP Steel", Proceedings in Applied Mathematics and Mechanics, Published online.
 N. Miska and D. Balzani [2018], "A Method to Quantify Material Parameter Uncertainties Resulting from Microstructure Variation based on Artificial Microstructures", Proceedings in Applied Mathematics and Mechanics, Published online.
 K. Kremer, P. Edler, N. Miska, F. Leichsenring, D. Balzani, S. Freitag, W. Graf, M. Kaliske and G. Meschke [2019], "Modeling of structures with polymorphic uncertainties at different length scales", Surveys for Applied Mathematics and Mechanics (GAMMMitteilungen), Published online.
Robust and Efficient Finite Element Discretizations for Higher Order Gradient Formulations
DFG (Deutsche Forschungsgemeinschaft) project within the
SPP (German Priority Program) 1748
"Reliable Simulation Techniques in Solid Mechanics. Development of Nonstandard Discretization Methods, Mechanical and Mathematical Analysis"
Cooperation with:
Mira Schedensack (Universität Leipzig)
Abstact:
In solid mechanics, the development of advanced numerical models that can capture phenomena such as lengthscale dependent constitutive behavior or material softening due to microdamage are a field of ongoing research. More and more, nonlocal approaches including higherorder derivatives are considered, because not only are they able to model these phenomena but also to regularize geometrical singularities that lead to spurious mesh sensitivities when using standard local models. However, the finite element implementation of these nonlocal gradient models is a challenging task. So far, state of the art approaches are either incompatible with classical approximation schemes because of higher continuity requirements or relatively cost intensive due to a high number of independent variables. Within this project, a new approach is investigated: With the help of the rotationfree condition of gradient fields, the displacement gradient is taken as the only independent variable. This way, on the one hand the high continuity requirement is relaxed enabling incorporation in existing FEsoftware packages and on the other hand enhanced efficiency towards existing mixed formulations is expected.
 J.Riesselmann, J.Ketteler, M.Schedensack and D.Balzani [2019], "Threefield mixed finite element formulations for gradient elasticity at finite strains", GAMMMitteilungen, Published online, DOI: 10.1002/gamm.202000002
 J.Riesselmann, J.Ketteler, M.Schedensack and D.Balzani [2019], "A New C0Continuous FEFormulation for Finite Gradient Elasticity", PAMM Proc. Appl. Math. Mech., Vol. 19, No. 1
 J.Riesselmann, J.Ketteler, M.Schedensack and D.Balzani [2019], "C0 continuous finite elements for gradient elasticity at finite strains", Proc. of 8th GACM Colloq. on Comp. Mech., pp. 2730
 J.Riesselmann, J.Ketteler, M.Schedensack and D.Balzani [2021], "Rotfree mixed finite elements for gradient elasticity at finite strains", International J Numer Methods in Eng.,
 J.Riesselmann, J.Ketteler, M.Schedensack and D.Balzani [2021], "Rotfree finite elements for gradientenhanced formulations at finite strains", Proceedings in Applied Mathematics and Mechanics (PAMM), 20:1,
Risk Assessment of Regeneration Paths for Supporting Simultaneous Decisions
Shorash Miro in cooperation with M. Beer (Leibniz Universität Hannover)
This research is part of the Collaborative Research Center SFB 871 "Regeneration of Complex Capital Goods" at Leibniz Universität Hannover within subproject D5 "Risk assessment of regeneration paths". The aim of the project is the development of riskbased decision metrics for the regeneration path using advanced system reliability techniques based on the survival signature approach. The efficient system modeling in this project encompasses both aleatoric and epistemic uncertainties in terms of stochastic variability of system component's characteristics and vagueness in their subjective assessment. The risks associated due deteriorated parts of the complex mechanical system are quantified in terms of timedependent system reliability (survivability) of the capital good. Subsequently, relative importance measures are used to capture the most critical parts (components) to the functionality of the regarded system. Consequently, a decisionsupport paradigm is established to enable an optimal selection and adjustment of the regeneration paths of the capital good. The application and validation example of this research is reliabilitybased decisionsupport for regenerating a highpressure axial compressor of a turbojet engine due to roughness effects of its stator and rotor blades. In this example, an illustrative functionbased system model is developed based on a onedimensional aerodynamic flow model of the compressor. After that, the survival signature approach along with the relative importance measures are used to estimate the system reliability and prioritize the regeneration activities.
Calculation of Optimal Bounds for the Probability of Failure in Soft Biological Tissues
Cooperation with:
M. Ortiz (California Institute of Technology CALTECH, Pasadena, USA)
Abstract:
The computational simulation of soft biological tissues is of high interest to medical doctors since these calculations can provide additional information leading to an improvement of diagnosis and treatment. In particular for surgical intervention as e.g. during balloonangioplasty the quantification of probabilities of failure is important. However, due to a general lack of experimental data, classical uncertainty quantification methods cannot be applied because the full probability distribution of the random input data (as e.g. stiffness of the material) can not be assumed to be given. Then the calculation of bounds on the probability of failure may be a suitable approach which at least incorporates the information that may be assumed to be known, as e.g. the mean of the probability distribution. In this project we focus on the calculation of optimal bounds which are obtained by solving minimization/maximization problems in the probability space.
Novel finite element technologies for anisotropic media
DFG (Deutsche Forschungsgemeinschaft) project within the
SPP (German Priority Program) 1748
"Reliable Simulation Techniques in Solid Mechanics. Development of Nonstandard Discretization Methods, Mechanical and Mathematical Analysis"
Cooperation with:
J. Schröder (Universität DuisburgEssen),
P. Wriggers (Leibniz Universität Hannover)
Abstract:
Considering solid mechanics, a majority of the problems can be solved using the standard Galerkin method. Although this method is used as a standard tool for predicting the behavior of a variety of engineering structures, certain problems limit the applicability. In general, incompressible and/or anisotropic materials could lead to not wellposed formulations. Finiteelement formulations, which are available today using a purely volumetricisochoric split, are not sufficient for anisotropic materials. Therefore, in this research project, the primary goal is to develop new finiteelement formulations as a suitable basis for the stable calculation of complex materials in nonlinear applications. In order to achieve this goal new ideas have to be pursued since there is no obvious approach available at the moment to overcome these difficulties. Therefore, we follow three main strategies:

Different approximations of the kinematic quantities entering the isotropic and anisotropic parts of the free energy function provide the possibility to relax the constraints arising from anisotropy. In this approach the structure of the polyconvex energy function is preserved.

Special approximations of the minors of the deformation gradient lead to mixed formulations suitable for more general polyconvex strain energy functions. Thereby the ansatz spaces for the mixed variables approximating the minors are balanced.

The VFEM, which was formulated so far only for small strain problems, is extended to large strains based on isotropic/anisotropic polyconvex strain energy functions. Thereby, the advantage to discretize nonconvex regions using VFEM is exploited for the application to highly distorted deformed meshes.
 J. Schröder, N. Viebahn, D. Balzani and P. Wriggers [2016], "A novel mixed finite element for finite anisotropic elasticity; the skaelement  simplified kinematics for anisotropy", Computer Methods in Applied Mechanics and Engineering, Vol. 310, pp. 475494.
 J. Schröder, P. Wriggers and D. Balzani [2011], "A new Mixed Finite Element based on Different Approximations of the Minors of Deformation Tensors", Computer Methods in Applied Mechanics and Engineering, Vol. 200, pp. 3583–3600.
Robust Numerical Approximation Schemes for TensorValued Derivatives of Higher Order
Cooperation with:
M. Tanaka (Toyota R&D Laboratories, Inc., Yokomichi, Japan)
Abstract:
For the numerical simulation of nonlinear problems in the context of finite elements the stresses and their derivatives, the tangent moduli, are required for the calculation of element residual vectors and stiffness matrices. For complex materials these are typically timeconsuming to be derived and implemented. For certain numerical methods as e.g. the FE²scheme an analytical tangent does not even exist, since the stresses are numerically calculated. One approach to reduce development time here is to use numerical approximations of these derivatives. The main disadvantage of the classical approach where forward differences are considered is that they suffer from approximation errors and roundoff errors. Contrary to this, the methods developed here are based on complexstep derivative approximations and hyper dual numbers. Thereby, robust approximation schemes are developed that lead to computer accuracy although being insensitive with respect to perturbation values. The resulting approximation schemes are successfully applied to hyperelastic, inelastic and thermoplastic problems at finite strains.
 M. Tanaka, M. Fujikawa, D. Balzani, J. Schröder [2014], "Robust numerical calculation of tangent moduli at finite strains based on complexstep derivative approximation and its application to localization analysis, Computer Methods in Applied Mechanics and Engineering, Volume 269, Pages 454470,
 M. Tanaka, T. Sasagawa, R. Omote, M. Fujikawa, D. Balzani, J. Schröder [2015], "A highly accurate 1st and 2ndorder differentiation scheme for hyperelastic material models based on hyperdual numbers", Computer Methods in Applied Mechanics and Engineering, Volume 283, Pages 2245.