Acceleration of the spectral stochastic FEM using POD and element based discrete empirical approximation for a micromechanical model of heterogeneous materials with random geometry - Pivovarov, Dmytro; Steinmann, Paul; Willner, Kai (Computer Methods in Applied Mechanics and Engineering, 2019)
The spectral stochastic FEM with local basis functions in the stochastic domain (SL-FEM) is one of the most flexible and accurate stochastic methods, however, also the most computationally expensive. These expenses are traditionally associated with the extra large tangent stiffness matrix and a huge number of elements which need to be re-integrated in every iteration. In this work we incorporate the proper orthogonal decomposition (POD) into the SL-FEM, thus performing a drastic reduction of the stiffness matrix. In order to reduce the integration costs by hyperreduction, a novel element-based modification of the discrete empirical interpolation, the so-called element-based empirical approximation method (EDEAM), is developed and combined with the POD. Particular advantages of the SL-FEM for order reduction and hyperreduction compared to other stochastic techniques are discussed. The new reduced-order SL-FEM is applied to the computational homogenization of materials with random geometry of the microstructure, i.e. to a general class of problems exhibiting strongly nonlinear, non-smooth and sometimes discontinuous dependency of the solution on some random parameters. The reduced-order SL-FEM demonstrates a high accuracy and a high solution speed, whereby the solution time for the reduced-order SL-FEM is comparable to the solution time of only one single Monte-Carlo sample.
CRIS-Team Scopus Importer:2019-12-03
Keywords: Computational homogenization; Hyperreduction; POD; Random geometry; Reduced order modeling; Stochastic FEM

Challenges of order reduction techniques for problems involving polymorphic uncertainty - Pivovarov, Dmytro; Willner, Kai; Steinmann, Paul; Brumme, Stephan; Müller, Michael; Srisupattarawanit, Tarin; Ostermeyer, Georg Peter; Henning, Carla; Ricken, Tim; Kastian, Steffen; Reese, Stefanie; Moser, Dieter; et al. (GAMM-Mitteilungen, 2019)
Modeling of mechanical systems with uncertainties is extremely challenging and requires a careful analysis of a huge amount of data. Both, probabilistic modeling and nonprobabilistic modeling require either an extremely large ensemble of samples or the introduction of additional dimensions to the problem, thus, resulting also in an enormous computational cost growth. No matter whether the Monte-Carlo sampling or Smolyak's sparse grids are used, which may theoretically overcome the curse of dimensionality, the system evaluation must be performed at least hundreds of times. This becomes possible only by using reduced order modeling and surrogate modeling. Moreover, special approximation techniques are needed to analyze the input data and to produce a parametric model of the system's uncertainties. In this paper, we describe the main challenges of approximation of uncertain data, order reduction, and surrogate modeling specifically for problems involving polymorphic uncertainty. Thereby some examples are presented to illustrate the challenges and solution methods.
CRIS-Team Scopus Importer:2019-05-02
Keywords: approximation of uncertain data; model order reduction; sensitivity analysis; surrogate modeling; uncertainty quantification

Fuzzy dynamics of multibody systems with polymorphic uncertainty in the material microstructure - Pivovarov, Dmytro; Hahn, Verena; Steinmann, Paul; Willner, Kai; Leyendecker, Sigrid (Computational Mechanics, 2019)
This paper deals with the fuzzy dynamics of multibody systems with polymorphic uncertainty in the material microstructure. Macroscopic material properties are obtained using fuzzy-stochastic FEM based computational homogenization. In particular, the spectral stochastic local FEM is utilized to simulate a representative volume element of the microstructure. Forward dynamics of the macroscopic system is modeled using the Graph Follower algorithm. Thereby we propagate the uncertainty from the lowest level of material microstructure to the highest level of multibody dynamics. Differences in the propagation of epistemic and aleatoric uncertainties to the macroscale and their influence on the multibody dynamics are discussed. A particular example of a multibody system used in this paper is a multistory frame, whereby the considered heterogeneous material is a cement-based concrete.
CRIS-Team Scopus Importer:2019-07-09
Keywords: Computational homogenization; Epistemic uncertainty; Forward dynamics simulation; Fuzzy numbers; Stochastic FEM; α-cut-optimization

Fuzzy-Stochastic FEM-based homogenization framework for materials with polymorphic uncertainties in the microstructure - Pivovarov, Dmytro; Oberleiter, Thomas; Willner, Kai; Steinmann, Paul (International Journal for Numerical Methods in Engineering, 2018)

Modified SFEM for computational homogenization of heterogeneous materials with microstructural geometric uncertainties - Pivovarov, Dmytro; Steinmann, Paul (Computational Mechanics, 2016)
In the current work we examine the application of the Stochastic Finite Element Method (SFEM) to the modeling of Representative Volume Elements for heterogeneous materials. Uncertainties in the geometry of the microstructure result in the random nature of the solution fields thus requiring use of the stochastic version of the Finite Element Method. For considering large differences in the material properties of matrix and inclusions a standard SFEM approach proves not stable and results in high numerical errors compared to a brute-force Monte-Carlo evaluation. Therefore in order to stabilize the SFEM we propose an alternative Gauss integration rule as resulting from a truncation of the probability density function for the random variable. In addition we propose new basis functions substituting the common Polynomial Chaos Expansion, resulting in higher accuracy for the standard deviation in the homogenized stress at the macro scale.
Keywords: Stochastic FEM; Computational Homogenization; Random Geometry

Novel adaptive reduced order spectral non-deterministic FEM approach to problems with general interval-fuzzy-stochastic uncertainties - Pivovarov, Dmytro (Computational Mechanics, 2025)
The aim of the current work is the development of the fast, efficient, and accurate approach towards problems with uncertain parameters. This approach is initially based on the spectral local non-deterministic FEM that is able to deal with all types of uncertain parameters: interval-like, fuzzy, stochastic and mixed types. The on-fly order reduction via low-rank tensor decomposition is implemented into the algorithm in order to reduce computational costs and to achieve parsimonious spectral representation of the parameter dependencies. The non-intrusive spectral FEM formulation with sparse adaptive grid of samples is utilized. This allows to use any conventional FEM code as the black-box within the non-deterministic framework. The samples are selected on the basis of the semi-optimal greedy algorithm. The crucial aspect of the proposed novel approach is the use of the IsoGeometrical basis functions and adaptive automatic refinement in the non-deterministic domain. Comparison with the unmodified spectral FEM demonstrates persuasive advantages of the novel approach.
CRIS-Team Scopus Importer:2025-07-04
Keywords: Adaptive sampling; IGA-FEM; Low-rank tensor decomposition; Non-deterministic FEM; Order reduction

On periodic boundary conditions and ergodicity in computational homogenization of heterogeneous materials with random microstructure - Pivovarov, Dmytro; Zabihyan, Reza; Mergheim, Julia; Willner, Kai; Steinmann, Paul (Computer Methods in Applied Mechanics and Engineering, 2019)
Due to high computational costs associated with stochastic computational homogenization, a highly complex random material microstructure is often replaced by simplified, parametric, ergodic, and sometimes periodic models. This replacement is often criticized in the literature due to unclear error resulting from the periodicity and ergodicity assumptions. In the current contribution we perform a validation of both assumptions through various numerical examples. To this end we compare large-scale non-simplified and non-ergodic models with simplified, ergodic, and periodic solutions. In addition we analyze the Hill-Mandel condition for stochastic homogenization problems and demonstrate that for a stochastic problem there are more than three classical types of boundary conditions. As an example, we propose two novel stochastic periodic boundary conditions which possess a clear physical meaning. The effect of these novel periodic boundary conditions is also analyzed by comparing with non-ergodic simulation results. (C) 2019 Elsevier B.V. All rights reserved.
CRIS-Team WoS Importer:2019-10-25

On spectral fuzzy–stochastic FEM for problems involving polymorphic geometrical uncertainties - Pivovarov, Dmytro; Willner, Kai; Steinmann, Paul (Computer Methods in Applied Mechanics and Engineering, 2019)
In this work we review a unified formulation for spectral fuzzy, spectral stochastic, and spectral fuzzy–stochastic FEM. We propose some modifications for the fuzzy and fuzzy–stochastic FEM involving local bases in the non-deterministic dimensions and the incorporation of extended FEM into the non-deterministic FEM. These modifications were previously used for stochastic FEM only. We discuss advantages of the proposed techniques for problems with uncertainties in the geometry and demonstrate their application to computational homogenization of heterogeneous materials with geometrical uncertainties in the microstructure. We address also some theoretical aspects of fuzzy FEM, that have not been tackled in the literature, namely the inner product of fuzzy variables. We address also another important problem, which becomes often the target of criticism of fuzzy approach. The membership function of the fuzzy input has often some degree of arbitrariness: it is known only approximately or it is constructed based on some assumptions. Here we propose a fuzzy FEM representation, which requires only the modal value and support of the fuzzy input parameters and is completely independent of the membership function's shape. We also discuss the case of imprecise probabilities which allows for severe dimension reduction of the non-deterministic problem if the accurate spectral simulation technique is used. Application of dimension reduction is also demonstrated on the example of computational homogenization.
CRIS-Team Scopus Importer:2019-05-09
Keywords: eXtended FEM; Fuzzy FEM; Imprecise probability; Polymorphic uncertainties; Random geometry; Stochastic FEM

On stochastic FEM based computational homogenization of magneto-active heterogeneous materials with random microstructure - Pivovarov, Dmytro; Steinmann, Paul (Computational Mechanics, 2016)
In the current work we apply the stochastic version of the FEM to the homogenization of magneto-elastic heterogeneous materials with random microstructure. The main aim of this study is to capture accurately the discontinuities appearing at matrix-inclusion interfaces. We demonstrate and compare three different techniques proposed in the literature for the purely mechanical problem, i.e. global, local and enriched stochastic basis functions. Moreover, we demonstrate the implementation of the isoparametric concept in the enlarged physical-stochastic product space. The Gauss integration rule in this multidimensional space is discussed. In order to design a realistic stochastic Representative Volume Element we analyze actual scans obtained by electron microscopy and provide numerical studies of the micro particle distribution. The SFEM framework described in our previous work (Pivovarov and Steinmann in Comput Mech 57(1): 123–147, 2016) is extended to the case of the magneto-elastic materials. To this end, the magneto-elastic energy function is used, and the corresponding hyper-tensors of the magneto-elastic problem are introduced. In order to estimate the methods' accuracy we performed a set of simulations for elastic and magneto-elastic problems using three different SFEM modifications. All results are compared with ``brute-force'' Monte-Carlo simulations used as reference solution.
Keywords: Stochastic FEM; eXtended FEM; Computational homogenization; Random geometry; Nonlinear magneto-elasticity

Overview of eXtended IsoGeometrical FEM for non-deterministic problems - Pivovarov, Dmytro; Willner, Kai; Steinmann, Paul (Computational Mechanics, 2025)
The aim of this work is to review and analyze modern advanced finite element techniques, i.e. the IsoGeometrical analysis (IGA) and the eXtended IsoGeometrical analysis (XIGA), from the viewpoint of the general interval-fuzzy-stochastic problem setting. We demonstrate the incorporation of IGA and XIGA bases into the general n-dimensional non-deterministic spectral FEM and show their application for a few problems. The first is a computational homogenization of heterogeneous materials with uncertainties in the microstructure. The second is a simple tension experiment model with large uncertainty in samples geometry. We analyze also a few applications, where the IGA basis in the parametric space by far beats the traditional polynomial chaos and the piece-wise continuous finite element basis. In these applications, the arbitrary smoothness and oscillation-free behavior of the IGA basis allows to achieve better accuracy with less computational effort as compared to standard approaches.
CRIS-Team Scopus Importer:2025-10-16
Keywords: Fuzzy FEM; IGA-FEM; Spectral approach; Stochastic FEM; Uncertainty quantification; XIGA-FEM

Stochastic local FEM for computational homogenization of heterogeneous materials exhibiting large plastic deformations - Pivovarov, Dmytro; Mergheim, Julia; Willner, Kai; Steinmann, Paul (Computational Mechanics, 2021)
Computational homogenization is a powerful tool allowing to obtain homogenized properties of materials on the macroscale from simulations of the underlying microstructure. The response of the microstructure is, however, strongly affected by variations in the microstructure geometry. In particular, we consider heterogeneous materials with randomly distributed non-overlapping inclusions, which radii are also random. In this work we extend the earlier proposed non-deterministic computational homogenization framework to plastic materials, thereby increasing the model versatility and overall realism. We apply novel soft periodic boundary conditions and estimate their effect in case of non-periodic material microstructures. We study macroscopic plasticity signatures like the macroscopic von-Mises stress and make useful conclusions for further constitutive modeling. Simulations demonstrate the effect of the novel boundary conditions, which significantly differ from the standard periodic boundary conditions, and the large influence of parameter variations and hence the importance of the stochastic modeling.
CRIS-Team WoS Importer:2021-11-05

Two reduction methods for stochastic FEM based homogenization using global basis functions - Pivovarov, Dmytro; Steinmann, Paul; Willner, Kai (Computer Methods in Applied Mechanics and Engineering, 2018)
The computationally most expensive part of stochastic FEM based homogenization is the inversion of the stochastic stiffness matrix. Previously we studied different combinations of global and local basis functions in the stochastic domain while keeping the common FEM discretization in the physical domain. Simulation results showed that a local basis (SL-FEM) results in a sparse but bigger stiffness matrix compared to a global basis (SG-FEM). In this work we introduce a SFEM modification involving global basis functions in both the physical and the stochastic domain. We introduce two different strategies for the implementation of the global basis. The first is the application of a trigonometric basis within a global Galerkin framework. The second is the use of vibration modes of some simple deterministic auxiliary model as global basis functions. The second approach in particularly results in a basis reduction technique. However this approach differs markedly from stochastic spectral decomposition which uses the matrix of the original stochastic problem. We address two very important questions, namely the treatment of the Gibbs phenomenon in the case of physical-stochastic trigonometric basis functions and the evaluation of the eigenvectors for the stochastic problem which are suitable for further basis reduction. Convergence studies and detailed comparison with Monte-Carlo results are presented for both methods.
CRIS-Team Scopus Importer:2022-08-05
Keywords: Computational homogenization; eXtended FEM; Global basis; Random geometry; Reduced order modeling; Stochastic FEM