Conference Agenda

In recent years fictitious domain methods have attracted attention since they do not require the generation of body-conforming finite element meshes. Therefore, fictitious domain methods are suited for problems with complex geometries. The finite cell method (FCM) is a combination of the fictitious domain approach with high-order finite elements. Thanks to the use of Cartesian meshes, the pre-processing, i.e. mesh generation is significantly simplified. However, due to the fact that the applied meshes do not conform to the geometry of the problem, special care has to be taken with respect to the numerical integration of the weak form, the local refinement of the approximation as well as the treatment of boundary conditions. The talk is intended to give an overview over the finite cell method and its application to solid mechanics. The examples to be presented will range from linear static and dynamic analysis to nonlinear problems including large deformations. We will also discuss the applicability of the FCM to image-based models stemming from CT-scans of micro-structured materials.

The Finite Cell Method (FCM) is a powerful tool for efficient and accurate simulation of complex geometries using higher-order shape functions and an unfitted discretization [1]. A widely used approach for the computation of the cut cell matrices intersected by the immersed boundaries is based on a quadtree/octree-decomposition of the cell domain into smaller integration domains called sub-cells. While the main advantage of this method is its robustness and simplicity, evaluating discontinuous integrals with this method can be computationally expensive if high accuracy is demanded. In this contribution, two novel approaches addressing this issue are presented, which inherit all the desired properties of the recursive sub-division-based integration schemes, however, they significantly reduce the computational effort while maintaining the same integration accuracy.
The first approach is based on merging the sub-cells using data compression techniques to form larger blocks and thereby reducing their number by a significant amount. An additional convenient feature of this approach lies in its modularity, as it can be simply added to FCM software without any major modification of the code. The proposed method has been investigated by numerical examples in the context of linear problems in 2D and non-linear problems in 3D. The results show a saving in computational time by up to 50% while yielding basically the same accuracy and stability of the simulation [2].
The second approach is an extension of the already existing Boolean FCM concept to multi-material problems. Here, the key idea lies in utilizing an overlapping structure of the integration sub-cells and using Boolean operations during the numerical integration to achieve a significantly more efficient distribution of the integration points. The proposed approach is combined with the local enrichment technique to accurately capture the weak discontinuity of the displacement field along material interfaces. The results show a significant reduction in the number of integration points and computational time by 70-80%, while maintaining the same accuracy as the standard quadtree/octree-based integration scheme [3].

References

[1] Parvizian, J., Düster, A. and Rank E. Finite cell method: h- and p-extension for embedded domain problems in solid mechanics. Computational Mechanics (2007) 41: 121–133.
[2] M. Petö, W. Garhuom, F. Duvigneau, S. Eisenträger, A. Düster, and D. Juhre. Octreebased integration scheme with merged sub-cells for the finite cell method: Application to non-linear problems in 3D. Computer Methods in Applied Mechanics and Engineering (2022) 401: 115565.
[3] M. Petö, S. Eisenträger, F. Duvigneau, and D. Juhre. Boolean finite cell method for multi-material problems including local enrichment of the ansatz space. Computational Mechanics (2023).

Without regularization, damage formulations are challenged with localization, mesh-dependency and the loss of ellipticity already at moderate damage levels. Through gradient enhancement, the damage solution field of corresponding formulations posesses a higher regularity leading to mesh-independent solutions and an increased numerical robustness. This contribution presents a corresponding Lagrange multiplier based mixed finite element formulation for finite strains. A suitable FE-interpolation scheme allowing for computational cost reduction through static condensation is introduced. In numerical tests mesh-independent solutions, robustness of the solution procedure for states of severe damage and under cyclic loading conditions are presented. Furthermore, an improved convergence behavior compared to the penalty approach is shown.

The relaxed micromorphic model [1] is an enriched continuum that can model materials with size-effects like metamaterials. It describes the kinematics of each material point using a displacement vector and a second-order micro-distortion field and has been shown to have many advantages over other higher-order continua. It utilizes fewer material parameters due to the simplified energy compared to classical micromorphic theory. Moreover, the relaxed micromorphic model operates between two scales, i.e., linear elasticity with the micro and macro elasticity tensors. The energy functional of the relaxed micromorphic model employs the curl of the micro-distortion field, necessitating an H(Curl)-conforming finite elements.

In our presentation, we will discuss the key aspects related to the finite element formulation of the relaxed micromorphic model and compare the tangential H(Curl)-conforming formulation against the classical nodal formulation using different numerical examples [2-3].

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] 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. To appear in Computational Mechanics, https://arxiv.org/abs/2210.17117 (2023).

Hybrid high-order (HHO) methods attach discrete unknowns to the mesh cells and faces. Their design is based on a local gradient reconstruction and a local stabilization operator weakly enforcing the matching of the trace of the cell unknowns to the face unknowns. HHO methods can be embedded into the framework of Hybridizable discontinuous Galerkin (HDG) methods, and they are closely related to Weak Galerkin (WG) and to nonconforming Virtual Element methods (ncVEM). HHO methods support polytopal meshes, lead to optimal error estimates, are locally conservative, and are computationally effective owing to the possibility of locally eliminating the cell unknowns.
In this talk, we first gently introduce the main ideas of HHO methods on the Poison model problem. Then, we consider the application of HHO methods for the space semi-discretization of the wave equation. We address both the second-order formulation in time of the wave equation and its reformulation as a first-order system, leading respectively to the use of Newmark and Runge-Kutta schemes for the time discretization. Numerical examples of wave propagation problems through heterogeneous media illustrate the benefits of using high-order approximations in space. The handling of curved interfaces will also be (briefly) discussed [4].
References:
[1] D. Di Pietro, A. Ern, and S. Lemaire, Comp. Methods Appl. Math., 14(4):461-472 (2014).
[2] M. Ciuttin, A. Ern, and N. Pignet, Hybrid high-order methods. A primer with applications to solid mechanics, SpringerBriefs in Mathematics (Springer, 2021).
[3] E. Burman, O. Duran and A. Ern, Commun. Appl. Math. Comput., 4(2):597-633 (2022).
[4] E. Burman, M. Cicuttin, G. Delay and A. Ern, SIAM J. Sci. Comput., 43(2):A859-A882 (2021).

We consider the Stokes problem involving one or two immiscible, viscous, incompressible fluids. Each fluid splits the computational domain into multiple subdomains either void or occupied by a fluid governed by the Stokes equations. At the interface between two fluids, the fluid velocities are continuous, whereas the jump of the normal stress is proportional to the curvature of the interface according to Laplace's law. These equations define the so-called Stokes immersed problem (fluid-void) and Stokes interface problem (fluid-fluid), respectively.

An unfitted NURBS-enhanced finite element method (NEFEM) combined with a hybridizable discontinuous Galerkin (HDG) approach, named unfitted HDG-NEFEM is proposed.

The approach is denoted as unfitted because the employed mesh does not conform to the interface. As a result, the interface can cut arbitrarily through the mesh elements. The adoption of unfitted meshes offers a substantial advantage in simplifying the meshing procedure, especially for high-order approximations. The enforcement of inhomogeneous Dirichlet boundary conditions is then achieved by means of a consistent penalty technique inspired by Nitsche’s method.

Given by a CAD model, NURBS define boundaries and interfaces; thus, the exact geometry of the computational domain is considered. As a consequence, geometric errors resulting from polynomial approximations of boundaries and interfaces are eliminated. The exact treatment of geometry is thus integrated into the functional approximation via the NEFEM paradigm.

HDG methods rely on a mixed formulation of the incompressible Stokes problem employing mesh skeleton-based hybrid unknowns for the velocity and only element-based unknowns for the pressure and the gradient of velocity. The element-based unknowns are then erased via static condensation to reduce the number of globally coupled degrees of freedom. Note that the HDG mixed formulation is particularly suited to accurately approximate the stress since both pressure and gradient of velocity converge with the same order of the primal variable.

The Stokes immersed problem is considered first to examine the influence of badly-cut elements and badly-cut faces on the condition number of the discrete system and the resulting accuracy of the method. On one hand, to counter the effects of badly-cut elements, we propose an element extension technique coupled with local static condensation of the element nodes lying outside the physical domain. On the other hand, the negative effect of badly-cut faces is handled by employing modal basis functions for the face variable.

Then, the Stokes interface problem is solved for complex geometries and interfaces. In particular, we investigate the equilibrium problem between two fluids, that is zero normal velocity at the fluid interface, when a shear flow is prescribed in the far field to assess the suitability of the method to solve physically-relevant problems in microfluidics.

In this study, the generalized finite difference method (GFDM) combined with an upwind scheme is applied to solve the compressible flow problem with discontinuities. GFDM is a meshless method developed in recent years. Based on Taylor series expansion and moving least squares (MLS) method, it converts the partial derivative into the linear summation of nodal values, avoiding complex grid generation and numerical integration. GFDM and the third order Runge-Kutta scheme are employed to discretize the Euler equations for compressible flow in space and time respectively. The upwind scheme in this paper is applied to overcome numerical oscillations when using high-order trial functions. Notably, the upwind scheme here is not based on the difference method but on Taylor series expansion and MLS. The present method avoids time-consuming mesh generation and numerical quadrature. Several typical numerical examples including one-dimensional and two-dimensional compressible flow problems are given to verify the effectiveness and accuracy of the proposed method. Numerical results also show that this method can accurately capture shockwaves.

We propose a methodology to implement generalised periodicity conditions in the solution of fourth-order PDE boundary value problems, in the framework of the C⁰ interior penalty method [1]. The methodology is developed for the analysis of flexoelectricity-based metamaterial unit cells, formalising the corresponding problem statement and weak form, and giving details on the implementation of the local and macro conditions for generalised periodicity. Numerical examples demonstrate the high-order convergence of the method and its applicability in realistic problem settings.

References

[1] O. Balcells-Quintana, D. Codony and S. Fernández-Méndez, “C0-IMP with generalized periodicity and application to flexoelectricity-based metamaterials”, Journal of Scientific Computing (2022)

This work was supported by the European Research Council (StG-679451 to I.A.), the Spanish Ministry of Economy and Competitiveness (RTI2018-101662-B-I00) and through the “Severo Ochoa Programme for Centres of Ex- cellence in R&D” (CEX2018-000797-S), and the Generalitat de Catalunya (ICREA Academia award for excellence in research to I.A., and Grant No. 2017-SGR-1278). D.C. acknowledges the support of the Spanish Ministry of Universities through the Margarita Salas fellowship (European Union-NextGenerationEU)

In the last decades, many researchers have focused on improving the efficiency of existing photovoltaic techniques. In that direction, the study of the flexoelectric phenomena [1] has opened the path to another line of research referred to as flexo-photovoltaics [2]. However, there is very scarce work done in its computational modeling and solution. In this work, a simple yet revealing model consisting on coupling the flexoelectricity [3] and the semiconductor modeling equations [4] is considered together with its Finite Element solution.
The photovoltaic part of the model is discretized by means of standard C0-FE. However, flexoelectricity is modeled by 4th order PDE and, consequently, standard finite element formulations cannot be used. Instead, its solution is carried out by means of the C0-Interior Penalty Method described in [5] for infinitesimal deformations. The extension of the C0-IPM formulation in [5] for the finite deformations framework [6] is currently under development.

References:

[1] Pavlo Zubko, Gustau Catalan, Alexander K. Tagantsev. Flexoelectric Effect in Solids. The Annual Review of Materials Research 33, 1, pp. 387-421. 2013.

[2] Ming-Min Yang, Jik Kim, Marin Alexe. ’Flexo-photovoltaic effect’. Science, vol. 360. 2018.

[3] D. Codony, O. Marco, S. Fernàndez-Méndez, I. Arias. An immersed boundary hierarchical B-spline method for flexoelectricity. Computer method in applied mechanics and engineering 354, 1, pp. 750-782. 2019.

[4] W. R. Van Roosbroeck. ’Theory of flow of electrons and holes in germanium and other semiconductors’. Bell System Technical J, 29: 560–607. 1950.

[5] Jordi Ventura, David Codony, Sonia Fernàndez-Méndez. ’A C0 Interior Penalty Finite Element Method for Flexoelectricity’. Journal of Scientific Computing. 2021

[6] D. Codony, P. Gupta, O. Marco, I. Arias. ‘Modeling flexoelectricity in soft dielectrics at finite deformation’. Journal of the Mechanics and Physics of Solids, 146. 2021

Continuum modelling of fexoelectricity, i.e. the ability of some dielectric materials to polarize under non-homogeneous deformations and viceversa, has been a historically challenging task due to the high-order nature of the PDE that describe their behaviour. Given the fact that the equations are a system of 4th order coupled PDE, we require high order continuity of the functional space used to describe the approximation of the solution. In order to face such intricacies, we use a complex computational framework which features an immersed hierarchical B-Spline approach [1], where the domain boundary is embedded within a Cartesian mesh. Then, the B-Spline basis which discretizes the unknowns is built on top of this. It is already known that such B-Spline bases present a Cartesian structure which makes them extremely useful when modelling structures that present orthogonal translational symmetry. By exploiting such symmetry, we are able to reduce the computational domain to one single periodic unit cell to which we apply generalised periodic boundary conditions [2,3]. Here in this work, we extend this methodology in order to apply the latter to structures with cyclic symmetry, that is, structures which are constructed by repeating a pattern around a fixed axis. To do so, we have rewritten the flexoelectric problem in a different curvilinear coordinate system which allows us to use the same computational framework, opening up to the possibility of exploiting other types of symmetries that might aid the design of flexoelectric metamaterials and devices.

References

[1] D. Codony, O. Marco, S. Fernández-Méndez, and I. Arias, “An immersed boundary hierarchical b-spline method for flexoelectricity,” Computer Methods in Applied Mechanics and Engineering, vol. 354, pp. 750–782, 2019.
[2] A. Mocci, J. Barceló-Mercader, D. Codony, and I. Arias, “Geometrically polarized architected dielectrics with apparent piezoelectricity,” Journal of the Mechanics and Physics of Solids, vol. 157, p. 104643, 2021.
[3] J. Barceló-Mercader, D. Codony, S. Fernández-Méndez, and I. Arias, “Weak enforcement of interface continuity and generalized periodicity in high-order electromechanical problems,” International Journal for Numerical Methods in Engineering, vol. 123, no. 4, pp. 901–923, 2022.