Reza Zabihyan

Bio: Reza Zabihyan is an academic researcher from University of Erlangen-Nuremberg. The author has contributed to research in topics: Boundary value problem & Homogenization (chemistry). The author has an hindex of 4, co-authored 6 publications receiving 62 citations.

A numerical study of different projection-based model reduction techniques applied to computational homogenisation

Abstract: Computing the macroscopic material response of a continuum body commonly involves the formulation of a phenomenological constitutive model. However, the response is mainly influenced by the heterogeneous microstructure. Computational homogenisation can be used to determine the constitutive behaviour on the macro-scale by solving a boundary value problem at the micro-scale for every so-called macroscopic material point within a nested solution scheme. Hence, this procedure requires the repeated solution of similar microscopic boundary value problems. To reduce the computational cost, model order reduction techniques can be applied. An important aspect thereby is the robustness of the obtained reduced model. Within this study reduced-order modelling (ROM) for the geometrically nonlinear case using hyperelastic materials is applied for the boundary value problem on the micro-scale. This involves the Proper Orthogonal Decomposition (POD) for the primary unknown and hyper-reduction methods for the arising nonlinearity. Therein three methods for hyper-reduction, differing in how the nonlinearity is approximated and the subsequent projection, are compared in terms of accuracy and robustness. Introducing interpolation or Gappy-POD based approximations may not preserve the symmetry of the system tangent, rendering the widely used Galerkin projection sub-optimal. Hence, a different projection related to a Gauss-Newton scheme (Gauss-Newton with Approximated Tensors- GNAT) is favoured to obtain an optimal projection and a robust reduced model.

Aspects of computational homogenization in magneto-mechanics: Boundary conditions, RVE size and microstructure composition

Abstract: In the present work, the behavior of heterogeneous magnetorheological composites subjected to large deformations and external magnetic fields is studied. Computational homogenization is used to derive the macroscopic material response from the averaged response of the underlying microstructure. The microstructure consists of two materials and is far smaller than the characteristic length of the macroscopic problem. Different types of boundary conditions based on the primary variables of the magneto-elastic enthalpy and internal energy functionals are applied to solve the problem at the micro-scale. The overall responses of the RVEs with different sizes and particle distributions are studied under different loads and magnetic fields. The results indicate that the application of each set of boundary conditions presents different macroscopic responses. However, increasing the size of the RVE, solutions from different boundary conditions get closer to each other and converge to the response obtained from periodic boundary conditions.

FE2 simulations of magnetorheological elastomers: influence of microscopic boundary conditions, microstructures and free space on the macroscopic responses of MREs

Abstract: In the current work, the response of heterogeneous magnetorheological elastomers (MREs) which are loaded by external magnetic fields in the absence and also in the presence of free space is studied. A fully-coupled two-scale finite element computational homogenization procedure is used to derive the material response at the macro-scale from the averaged response of the underlying micro-scale problem. Different combinations of boundary conditions, that satisfy the Hill-Mandel condition and are based on the primary variables of the magneto-elastic enthalpy and energy functionals are applied to solve the micro-scale boundary value problem. Furthermore, the influences of various microstructures on the macroscopic response of the MREs are investigated. The results indicate that the choice of microscopic boundary conditions and microstructure types can significantly affect the macroscopic responses of MREs.

On periodic boundary conditions and ergodicity in computational homogenization of heterogeneous materials with random microstructure

Abstract: 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.

Nonlinear Finite Elements For Continua And Structures

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A recurrent neural network-accelerated multi-scale model for elasto-plastic heterogeneous materials subjected to random cyclic and non-proportional loading paths

Abstract: An artificial Neural Network (NNW) is designed to serve as a surrogate model of micro-scale simulations in the context of multi-scale analyses in solid mechanics. The design and training methodologies of the NNW are developed in order to allow accounting for history-dependent material behaviors. On the one hand, a Recurrent Neural Network (RNN) using a Gated Recurrent Unit (GRU) is constructed, which allows mimicking the internal variables required to account for history-dependent behaviors since the RNN is self-equipped with hidden variables that have the ability of tracking loading history. On the other hand, in order to achieve accuracy under multi-dimensional non-proportional loading conditions, training of the RNN is achieved using sequential data. In particular the sequential training data are collected from finite element simulations on an elasto-plastic composite RVE subjected to random loading paths. The random loading paths are generated in a way similar to a random walking in stochastic process and allow generating data for a wide range of strain-stress states and state evolution. The accuracy and efficiency of the RNN-based surrogate model is tested on the structural analysis of an open-hole sample subjected to several loading/unloading cycles. It is shown that a similar accuracy as with a FE 2 multi-scale simulation can be reached with the RNN-based surrogate model as long as the local strain state remains in the training range, while the computational time is reduced by four orders of magnitude.

Recent advances in hard-magnetic soft composites: Synthesis, characterisation, computational modelling, and applications

Abstract: Hard-magnetic soft composites consist of magneto-active polymers (MAPs) where the fillers are composed of hard-magnetic (magnetically polarised) particles. These novel multifunctional materials are experiencing a great advance from the last few years. This rise has been motivated by the possibility of controlling ferromagnetic patterns during the manufacturing process. Thus, structures with programmable functionalities can be conceptualised and implemented, opening new routes into the design of smart components with great opportunities in the biomedical engineering and soft robotics fields. In this work, we provide an overview of the state of the art of such MAPs, providing the key fundamentals and reference works. To this end, we present the current synthesis and experimental characterisation methods, the different computational modelling approaches across scales, and a detailed presentation of their current potential applications. Finally, we provide an overall discussion on future perspectives.

Two-stage data-driven homogenization for nonlinear solids using a reduced order model

Abstract: The nonlinear behavior of materials with three-dimensional microstructure is investigated using a data-driven approach. The key innovation is the combination of two hierarchies of precomputations with sensibly chosen sampling sites and adapted interpolation functions : First, finite element (FE) simulations are performed on the microstructural level. A sophisticated sampling strategy is developed in order to keep the number of costly FE computations low. Second, the generated simulation data serves as input for a reduced order model (ROM). The ROM allows for considerable speed-ups on the order of 10–100. Still, its performance is below the demands for actual twoscale simulations. In order to attain the needed speed-ups, in a third step, the use of radial numerically explicit potentials (RNEXP) is proposed. The latter combine uni-directional cubic interpolation functions with radial basis functions operating on geodesic distances. The evaluation of the RNEXP approximation is realized almost in real-time. It benefits from the computational efficiency of the ROM since a higher number of sampling points can be realized than if direct FE simulations were used. By virtue of the dedicated sampling strategy less samples and, thus, precomputations (both FE and ROM) are needed than in competing techniques from literature. These measures render the offline cost of the RNEXP manageable on workstation computers. Additionally, the chosen sampling directions show favorable for the employed kernel interpolation. Numerical examples for highly nonlinear hyperelastic (pseudo-plastic) composite materials with isotropic and anisotropic microstructure are investigated. Twoscale simulations involving more than 10 6 DOF on the structural level are solved using the RNEXP and the influence of the microstructure on the structural behavior is quantified.

Recent advances in hard-magnetic soft composites: Synthesis, characterisation, computational modelling, and applications

Abstract: Hard-magnetic soft composites consist of magneto-active polymers (MAPs) where the fillers are composed of hard-magnetic (magnetically polarised) particles. These novel multifunctional materials are experiencing a great advance from the last few years. This rise has been motivated by the possibility of controlling ferromagnetic patterns during the manufacturing process. Thus, structures with programmable functionalities can be conceptualised and implemented, opening new routes into the design of smart components with great opportunities in the biomedical engineering and soft robotics fields. In this work, we provide an overview of the state of the art of such MAPs, providing the key fundamentals and reference works. To this end, we present the current synthesis and experimental characterisation methods, the different computational modelling approaches across scales, and a detailed presentation of their current potential applications. Finally, we provide an overall discussion on future perspectives.