Conference Agenda

Ferroelectric as well as ferromagnetic materials are widely used in smart structures and devices as actuators, sensors etc. Regarding their nonlinear behavior, a variety of models has been established in the past decades. Investigating hysteresis loops or electromechanical/magnetoelectric coupling effects, only simple boundary value problems (BVP) are considered. In [1] a new scale–bridging approach is introduced to investigate the polycrystalline ferroelectric behavior at a macroscopic material point (MMP) without any kind of discretization scheme, the so–called Condensed Method (CM). Besides classical ferroelectrics, other fields of application of the CM have been exploited, e.g. [2, 3, 4]. Since just the behavior at a MMP is represented by the CM, the method itself is unable to solve complex BVP, which is technically disadvantageous if a structure with e.g. notches or cracks shall be investigated.

In this paper, a concept is presented, which integrates the CM into a Finite Element (FE) environment. Considering the constitutive equations of a homogenized MMP in the weak formulation, the FE framework represents the polycrystalline behavior of the whole discretized structure, which finally enables the CM to handle arbitrary BVP. A more sophisticated approach completely decouples the constitutive evolution from the FE discretization, by introducing an independent material grid. Furthermore, energetic consistencies of scale transitions from grain to MMP and MMP to macroscale are investigated. Numerical examples are finally presented in order to verify the approach.

References

[1] Lange, S. and Ricoeur, A., A condensed microelectromechanical approach for modeling tetragonal ferroelectrics, International Journal of Solids and Structures 54, 2015, pp. 100 – 110.

[2] Lange, S. and Ricoeur, A., High cycle fatigue damage and life time prediction for tetragonal ferroelectrics under electromechanical loading, International Journal of Solids and Structures 80, 2016, pp. 181 – 192.

[3] Ricoeur, A. and Lange, S., Constitutive modeling of polycrystalline and multiphase ferroic materials based on a condensed approach, Archive of Applied Mechanics 89, 2019, pp. 973 – 994.

[4] Warkentin, A. and Ricoeur, A., A semi-analytical scale bridging approach towards polycrystalline ferroelectrics with mutual nonlinear caloric–electromechanical couplings, International Journal of Solids and Structures 200 – 201, 2020, pp. 286 – 296.

Components used in the aerospace or automotive industries are often exposed to multi-physical loading conditions and thus may simultaneously be subjected to high stresses and strains as well as temperature changes. Therefore, high-strength and high-temperature resistant materials such as metals are commonly used for applications in this field. Since the overall material behavior is directly influenced by the distribution, size and morphology of the individual grains of the underlying polycrystalline microstructure, detailed knowledge of this microstructural behavior is required in order to accurately predict the macroscopic material response. Hence, multi-scale simulation approaches have been developed. Considering a two-scale finite element (FE) and fast Fourier transform (FFT)-based simulation approach [1, 2], the macroscopic and microscopic boundary value problems are first solved individually by assuming scale separation. In this context, the homogeneous macroscale is subdivided into a discrete number of finite elements. The microscopic boundary value problem is attached to each macroscopic integration point and solved using the FFT-based simulation approach. The scale transition is then performed by defining the macroscopic quantities as the average value over the corresponding local fields. This simulation approach is an efficient alternative to the classical FE² method for the simulation of periodic unit cells [3]. To illustrate the applicability of our model, we will present several numerical examples.

[1] J. Spahn, H. Andrä, M. Kabel, and R. Müller. A multiscale approach for modeling progressive damage of composite materials using fast Fourier transforms. Computer Methods in Applied Mechanics and Engineering, 268, 871–883, 2014

[2] J. Kochmann, S. Wulfinghoff, S. Reese, J. R. Mianroodi, and B. Svendsen. Two-scale FE–FFT- and phase-field-based computational modeling of bulk microstructural evolution and macroscopic material behavior. Computer Methods in Applied Mechanics and Engineering, 305, 89–110, 2016

[3] C. Gierden, J. Kochmann, J. Waimann, B. Svendsen, and S. Reese. A review of FE-FFT-based two-scale methods for computational modeling of microstructure evolution and macroscopic material behavior. Archives of Computational Methods in Engineering, 29(6), 4115-4135, 2022.

Numerical simulation of complex geometries and microstructures can be costly and time consuming, in particular due to the long process of preparing the geometry for meshing and the meshing process itself [1]. Several methods were proposed to overcome this issue, such as the extended finite element, meshless, Fourier transform and immersed boundary methods. Immersed boundary methods rely on embedding the physical domain into a Cartesian grid of finite elements and resolving the geometry only by adaptive numerical integration schemes. For instance, the isogeometric finite cell method (FCM) exploits the accuracy of higher-order, smooth B-Spline basis functions for the discretization and employs an octree scheme in order to refine the quadrature rule in trimmed elements. FCM has been applied successfully to various problems in solid mechanics, including linear and nonlinear elasticity, elasto-plasticity, and thermo-elasticity [2]. However, FCM typically requires several levels of refinement of the quadrature rule in order to deliver accurate results, which may lead to high computation times, especially for nonlinear, internal variable, and coupled multiphysics problems.

In this work, we adopt a novel algorithm for boundary-conformal quadrature based on a high-order reparameterization of trimmed elements [3] to solve small and large deformation thermo-elastic problems using spline-based immersed isogeometric analysis (IGA) without the need for a body conformal finite element mesh. In particular, the Gauss points on trimmed elements are obtained by a NURBS reparameterization of the physical subdomains of the cut elements of the Cartesian grid. This ensures an accurate integration with a minimal number of quadrature points. Furthermore, using periodic B-Spline discretizations, periodic boundary conditions for homogenization can be automatically fulfilled. Several numerical examples are presented to show the accuracy and efficacy of the boundary-conformal quadrature algorithm.

REFERENCES

[1] T. Hughes, J. Cottrell, and Y. Bazilevs. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering, 194:4135– 4195, 2005.

[2] Schillinger, D. and Ruess, M., 2015. The Finite Cell Method: A review in the context of higher-order structural analysis of CAD and image-based geometric models. Archives of Computational Methods in Engineering, 22(3), pp.391-455.

[3] Wei, X., Marussig, B., Antolin, P. and Buffa, A., 2021. Immersed boundary-conformal isogeometric method for linear elliptic problems. Computational Mechanics, 68(6), pp.1385-1405.

Metamaterials are attracting growing attention in industry and academia due to their unique mechanical behaviour. However, when the scale separation does not hold, they show size-effects. Generalized continua can model such materials as a homogeneous continuum with capturing the size-effects.

The relaxed micromorphic model [1] describes the kinematics of each material point via a displacement vector and a second-order micro-distortion field. It has demonstrated many advantages over other higher-order continua such as using fewer material parameters and the drastically simplified strain energy compared to the classical micromorphic theory. Moreover, the relaxed micromorphic model operates between two bounds; linear elasticity with the micro and macro elasticity tensors. The strain energy function in the relaxed micromorphic model employs the Curl of the micro-distortion field and therefore H(Curl)-conforming FEM implementation is necessary [2-3].

In our talk, we will present our recent results in identifying the material parameters and boundary conditions in the relaxed micromorphic model [4].

REFERENCES

[1] P. Neff, I.D. Ghiba, A. Madeo, L. Placidi and G. Rosi. A unifying perspective: the relaxed linear micromorphic continuum. Continuum Mechanics and Thermodynamics 26,639-681(2014).

[2] J. Schröder, M. Sarhil, L. Scheunemann and P. Neff. Lagrange and H(curl,B) based Finite Element formulations for the relaxed micromorphic model, Computational Mechanics 70, pages 1309–1333 (2022).

[3] A. Sky, M. Neunteufel, I. Muench, J. Schöberl, and P. Neff. Primal and mixed finite element formulationsfor the relaxed micromorphic model. Computer Methods in Applied Mechanics and Engineering 399, p. 115298 (2022).

[4] M. Sarhil, L. Scheunemann, J. Schröder, P. Neff. Size-effects of metamaterial beams subjected to pure bending: on boundary conditions and parameter identification in the relaxed micromorphic model. https://arxiv.org/abs/2210.17117 (2022).

In the present contribution, we deal with a second-order computational homogenization of fluid flow in porous materials. Similar to the first-order computational homogenization in [1], the microscopic problem is formulated employing a minimization-type variational formulation at small strains; see also [2]. While a first-order Darcy-Biot-type fluid transport is considered at the microscale [2], the macroscopic problem is characterized by a second-order material response [3]. Hence, the present formulation allows the relaxation of the scale-separation assumption and the incorporation of the macroscopic second-order terms associated with deformation and fluid-flux fields at the microscale. The macro- and microscale boundary value problems are then bridged via an extended form of the Hill-Mandel condition, which results in suitable boundary conditions at the microscale and a set of constraints [4,5]. Finally, we present numerical examples that provide further insights into the presented formulation.

References:

[1] E. Polukhov and M.-A. Keip. Computational homogenization of transient chemo-mechanical processes based on a variational minimization principle. Advanced Modeling and Simulation in Engineering Sciences, 7, 1-26 (2020).

[2] C. Miehe, S. Mauthe, and S. Teichtmeister. Minimization principles for the coupled problem of Darcy--Biot-type fluid transport in porous media linked to phase field modeling of fracture. Journal of the Mechanics and Physics of Solids, 82, 186-217 (2015).

[3] G. Sciarra, F. dell'Isola, and O. Coussy. Second gradient poromechancis. International Journal of Solids and Structures, 44, 6607-6629 (2007).

[4] V.G. Kouznetsova, M.G.D. Geers and W.A.M. Brekelmans. Multi-scale second-order computational homogenization of multi-phase materials: a nested finite element solution strategy. Computer methods in Applied Mechanics and Engineering, 193, 5525-5550 (2020).

[5] I. A. Rodrigues Lopez, and F. M. Andrade Pires. Unlocking the potential of second-order computational homogenisation: An overview of distinct formulations and a guide for their implementation. Archives of Computational Methods in Engineering, 1-55 (2021).