Conference Agenda

Most of models encountered in applied physics can be nowadays numerically solved by using appropriate discretization techniques and adequate computational resources. However, sometimes, the predictions obtained from these physics-based simulations exhibit noticeable bias, and their solution is not compatible with real-time applications, compulsory in generative design or control of engineering systems.

In this work we propose the use of a hybrid paradigm, in which the alliance of physics-based and data-driven models, the last making use of artificial intelligence and machine learning technologies, enables improving the accuracy while ensuring real-time responses, enlarging the horizon of engineering.

Neural networks have recently gained attention in solving inverse problems. One prominent methodology are Physics-Informed Neural Networks (PINNs) which can solve both forward and inverse problems. In this presentation, we consider full waveform inversion as an example of an inverse problem. The performance of PINNs is compared against classical adjoint optimization, focusing on three key aspects: the forward-solver, the neural network Ansatz for the inverse field, and the sensitivity computation for the gradient-based minimization. Starting from PINNs, each of these key aspects is adapted individually until the classical adjoint optimization emerges. It is shown that it is beneficial to use the neural network only for the discretization of the unknown material field, where the neural network produces reconstructions without oscillatory artifacts as typically encountered in classical full waveform inversion approaches. Due to this finding, a hybrid approach is proposed. It exploits both the efficient gradient computation with the continuous adjoint method as well as the neural network Ansatz for the unknown material field. This new hybrid approach outperforms Physics-Informed Neural Networks and the classical adjoint optimization in settings of two and three-dimensional examples [1].
[1] https://arxiv.org/abs/2303.03260

In this work, we assess the stability of the lower-order mixed displacement-pressure formulation on polygonal and quadrilateral meshes in both linear and nonlinear analysis. We address the inf-sup stability and especially the occurrence of spurious pressure modes (checkerboard modes). It is shown that, in both linear and nonlinear analysis, the existence of spurious pressure modes is purely dependent on the chosen discretization technique. A comparison between quadrilateral and Voronoi discretizations demonstrates that spurious modes are suppressed on Voronoi meshes without the need for any type of stabilization method. To discretize Voronoi meshes, a polygonal mixed displacement-pressure element based on the scaled boundary parameterization [1] is used. Several numerical examples in both nearly-incompressible linear elasticity and nonlinear hyperelasticity are presented. In particular, the absence of spurious checkerboard modes on Voronoi meshes in each Newton iteration is shown in the large strain regime.

References

[1] B. Sauren, S. Klarmann, L. Kobbelt and S. Klinkel, A mixed polygonal finite element formulation for nearly-incompressible finite elasticity. Comput. Methods Appl. Mech. Engrg., Vol. 403, pp. 115656, 2023

High order immersed methods are advantageous to capture with high accuracy interface phenomena which often drives the physics of the application. For those, the numerical scheme is modified to solve the moving interface problem without having to remesh. Nonetheless, a refined mesh in the vicinity of the interface is usually required to capture stiff gradient which makes anistropic mesh adaptation very interesting for immersed method.

An anisotropic mesh adaptation method was developed by Coulaud et al. for high order finite element solution. The main principle of this method is to define an optimal metric field through the high order differential form of the solution. Drastic gains in terms of interpolation error and rate of convergence have been demonstrated for high order Discontinuous Galerkin (DG) simulations [1].

In this work, we propose to extend the use of anisotropic mesh adaptation for high order immersed boundary problems. A sharp interface method was integrated into the three dimensional high-order DG code Argo [2] to capture discontinuities on non-confirming mesh. This method ensures high-order of convergence of the DG scheme even near the interface for static and moving boundary problems [3].

To adapt the methodology for a discontinuous solution non conforming with the mesh, three key elements will be discussed in the paper. First, the solution is regularized to obtain a continuous field on the mesh. Secondly, the metric is computed based on this solution and a particular attention is paid on the mesh generated. Indeed, the intersection of the interface and the new mesh can lead to small cut cells which jeopardizes the robustness of the method. Hence some nodes are locally moved to ensure stability of the scheme. Finally, the projection method of the discontinuous solution on the adapted mesh will be discussed.

The mesh adaptation procedure will be illustrated on immersed hydrodynamic problem encountered in powder bed fusion modeling [4]. The powder bed fusion technology is a layer-by-layer manufacturing process which involves laser and powder interaction, melt pool formation and thermodynamics effects. This paper will show how these developments will help to investigate the interaction between laser and powder, melt pool formation and thermodynamics effects.

Neurodegenerative diseases (NDs) are complex disorders that primarily affect the neurons in the brain and nervous system, leading to progressive deterioration and loss of function over time. A common pathological hallmark among different NDs is the accumulation of disease–specific misfolded aggregated prionic proteins in different brain areas (Aβ and tau in Alzheimer’s disease, α–synuclein in Parkinson’s disease). In this talk, we discuss the numerical modeling of the misfolding process of α–synuclein in Parkinson’s disease. To characterize the progression of misfolded proteins across the brain we consider a suitable mathematical model (based on Fisher– Kolmogorov equations). For its numerical discretization, we propose and analyze a high-order discontinuous Galerkin method on polyhedral grids (PolyDG) for space discretization coupled with a Crank-Nicolson scheme to advance in time. Numerical simulations in patient-specific brain geometries reconstructed from magnetic resonance images are presented. In the second part of the talk, we introduce and analyze a PolyDG method for the semi-discrete numerical approximation of the equations of Multiple-Network Poroelastic Theory (MPET) in the dynamic formulation. The MPET model can comprehensively describe functional changes in the brain considering multiple scales of fluids and can be regarded as a preliminary attempt to model the perfusion in the brain. In this context, cerebrospinal fluid transport plays an essential role as a mechanism for waste removal (clearance) from the central nervous system. We present and analyze the numerical approach and we present simulations in three-dimensional patient-specific geometries.

Initially, the Space-Time Galerkin/ Least-Squares (ST-GLS) finite element method was proposed to solve the equations of motion for incompressible or compressible Newtonian fluids [1-2]. Based on the ST- GLS principle, we present a new, fully consistent and highly stable finite element method for non-Newtonian fluid flows, specifically designed for the three-field incompressible viscoelastic flows. Space and time are discretized using finite elements and all flow variables (velocity-pressure-stresses) are interpolated with polynomials of the same degree. The positive definiteness of the conformation tensor, which encapsulates the deformation history of the polymeric fluid, is enforced by a square-root reformulation of the constitutive equation [3]. The method is enriched with a consistent shock-capturing scheme that suppresses numerical oscillations around stress singularities. Multiphase flows are treated by coupling the present formulation with a quasi-elliptic mesh generator [4], providing accurate descriptions of interfacial dynamics. The accuracy, robustness, and generality of the method are validated in stationary and transient benchmark flows of viscoelastic fluids. We consider the creeping flow of an Oldroyd-B fluid past a cylinder in a straight channel. We then proceed to the axisymmetric capillary thinning of viscoelastic filament in which the bead-on-a-string formation appears. Finally, we present for the first time the symmetry breaking in 3-dimensional viscoelastic simulations of a falling sphere or a rising bubble. In all cases, we obtain numerically stable solutions for very high values of the Weissenberg number, which is defined as the shear rate times the relaxation time of the polymeric fluid and represents the importance of elastic effects over shear ones,that have never been accessed before by existing numerical methods.

References

[1] T. J. R. Hughes, L. P. Franca, G. M. Hulbert, Comp. meth. App. Mech. Eng., 73 (1989).

[2] T. E. Tezduyar, Advances in applied mechanics 28 (1991).

[3] N. Balci, B. Thomases, M. Renardy, C. R. Doering, J. Non-Newt. Fluid Mech., 166 (2011).

[4] Y. Dimakopoulos, J. Tsamopoulos, JCP, 192 (2003)

We present a coupled thermo-mechanical formulation for solving non-isothermal phase change problems, using a total Lagrangian description. For the mechanical problem, a mixed position-pressure Finite Element formulation is proposed to solve incompressible materials - both fluids and solids - in a unified framework. To overcome the numerical instabilities of the incompressibility circunventing Ladyzhenskaya-Babuška-Brezzi conditions, an adapted pressure stabilizing Petrov-Galerkin technique is applied. We consider the Newtonian constitutive model for fluids, a hyperelastic constitutive model for solids, and a mix of these models for the mushy phase. While fluid problems are most commonly expressed in terms of velocities, the use of velocity-based formulations on solid materials is generally limited to hypoelastic-derived constitutive models. Therefore, to account for hyperelastic solids while still applying a unified treatment of unknown variables, the fluid problem is also equationed in terms of positions. A partitioned method for thermo-mechanical coupling is employed, using a temperature-based Finite Element Method for solving the heat transfer problem, and the enthalpy method for phase change implementation. Finally, representative numerical examples are simulated to show the potentialities of the proposed formulation.

The Material Point Method (MPM) represents an alternative simulation technique as to, e.g., the Finite-Element-Method. Within the MPM, physical bodies are discretized as material points in a Lagrangian sense where all kinematic and constitutive quantities are stored on. The actual numerical solution of the balance equations is solved on the nodes of the Eulerian background grid, see [1], where it is common practice to use the same fixed grid geometry throughout the whole simulation. As the material points move independently of the background grid, mesh distortion as in, e.g., FEM simulations is completely avoided. In this contribution, numerical examples of dynamic processes subject to large deformations are analyzed and evaluated. These examples show an influence of the grid position in mechanical quantities like stresses. As the focus of this contribution is on the improvement of numerical results, a technique at almost no computational cost is presented. Within this technique, the origin of the background grid is shifted randomly for a small distance in each direction at the beginning of each time step. As a result, mesh-influences in stresses are averted as the grid translations can be interpreted as smearing the grid over time.

References

[1] D. Sulsky, Z. Chen und H. Schreyer. “A particle method for history-dependent materials”. In: Computer Methods in Applied Mechanics and Engineering. 118.1-2 (1994), S. 179–196. doi: 10.1016/0045-7825(94)90112-0.

The numerical modeling of fracture phenomena is of high relevance with respect to safety and durability of engineering structures. In this context, the phase field method (PFM) is a prominent approach for brittle fracture modeling. Due to its variational nature [1], it facilitates the simulation of complex fracture phenomena, such as crack branching and coalescence. As a regularized approach to crack propagation modeling, however, it is based on the use of a small length scale parameter which in turn leads to the requirement of very fine meshes. Particularly in 3D, the numerical effort associated with phase field modeling can be prohibitive. Therefore, adaptive solution schemes for phase field modeling of fracture have recently received attention.

In this contribution, an image-based scaled boundary finite element approach to three-dimensional phase field modeling will be presented, that has recently been proposed in Ref. [2]. The scaled boundary finite element method (SBFEM) is a semi-analytical technique that can be used on polyhedral elements with an arbitrary number of faces and thus facilitates numerical analyses on octree meshes. These hierarchical meshes stimulate rapid mesh size transition and lend themselves to adaptive refinement. Octree meshes can be generated and refined automatically from digital images, which is particularly advantageous for highly heterogeneous geometries [3]. In a balanced octree mesh, a limited number of unique cell configurations exist. This is exploited in the design of efficient numerical simulation techniques based on SBFEM. In addition, the SBFEM provides an energy-based error estimator, which is obtained as a result of the semi-analytical solution procedure without additional effort. In the context of phase field modeling, the latter is combined with a threshold value for the phase field variable. Following the derivation of a staggered solution scheme based on the scaled boundary finite element solutions of both the phase field equation and the balance of momentum, various examples are presented to illustrate the procedure.

References

1. C. Miehe, F. Welschinger, M. Hofacker, Thermodynamically consistent phase-field models of fracture: Variational principles and multi-field FE implementations, Int J Numer Methods Eng, 83 (2010), 1273-1311.

2. R. Assaf, C. Birk, S. Natarajan, H. Gravenkamp, Three-dimensional phase-field modeling of brittle fracture using an adaptive octree-based scaled boundary finite element approach, Comput Methods Appl Mech Engrg, 399 (2022), 115364.

3. A. Saputra, H. Talebi, D. Tran, C. Birk, C. Song, Automatic image-based stress analysis by the scaled boundary finite element method, Int J Numer Methods Eng, 109 (2017), 697-738

A new numerical model is proposed by combining the multiscale scaled boundary finite element method (MsSBFEM) and the temporally piecewise adaptive algorithm for the multiscale heterogeneous analysis of viscoelastic problems. In the time domain, temporally piecewise adaptive algorithm is adopted and the recursive equations for solving multiscale viscoelastic problems are derived. A spatial-time coupled multiscale viscoelastic problem is discretized into a series of multiscale elastic problems. When the different time step and error tolerance is adopted, the computation accuracy in time domain can be guaranteed by adaptive calculation. In the spatial domain, utilizing the quadtree or octree gridding, the construction of base functions can be conveniently and efficiently solved directly based on the 2D or 3D images, and the solution accuracy at the large-scale can be improved by increasing nodes of coarse elements only without increasing new nodes inside. Numerical examples are provided and the effectiveness of the proposed approach is stressed at both the large and small scales.

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.

The locking phenomenon, in the context of finite elements, gives rise to underestimated displacements and oscillating, parasitic stresses depending on a critical parameter, for instance, the slenderness of a beam. In the last eleven years, Long et al. (2012), Echter et al. (2013), and Oesterle et al. (2017) discovered that the transverse shear locking characteristics of a finite element can be avoided by reparametrizing the underlying kinematic equations. Later, Bieber et al. (2022) extended the idea of reparameterization of the kinematic equations to avoid membrane locking for the special case of curved beam formulations. Even though this technique aids in achieving a membrane-locking-free finite element formulation on a theoretical level, several difficulties are detected in terms of the practical application of the concept, the handling of boundary conditions like for the simply supported case due to the reparameterization of the primary variables, the extension of the theory to shell structures, and considering the case of large deformations.

Revisiting the idea of the discrete strain gap method (Bletzinger et al., 2000) within a variational framework, Bieber et al. (2018) developed the mixed displacement (MD) method, which includes additional degrees of freedom fulfilling a chosen kinematic law. This method can not only help to remove shear locking effects intrinsically but also to eliminate membrane locking in finite elements. In spite of having simpler implementational aspects, the MD method carries the challenge of handling certain additional constraints that are to be imposed on the additional degrees of freedom.

In this work, an overview of the above-mentioned two strategies to mitigate the locking characteristics of finite elements on a theoretical level is provided. This will be followed by a discussion of recent investigations on the applicability of the methods, considering the treatment of constraints and the handling of boundary conditions. Numerical examples demonstrating the locking-free characteristics of the proposed methodology will be addressed as well. The main focus will be on alleviating the membrane locking phenomenon on a theoretical level through these novel ideas.

The solution of the Lamé equations characterizes the displacement that minimizes the induced stored energy. The associated dual problem describes the stresses that satisfy the equilibrium condition and maximize the corresponding energy. When approximate solutions of both problems are known, then the theorem of Prager and Syngy provides error bounds for both of them. The name “two-energies-principle” is now preferred when it is used for getting a posteriori error bounds.

In 1959 Morgenstern started a proof of the justification of plate models, but it was not completed.
Now two changes enable a completion of the proof for the plate models by Mindlin-Reissner and Kirchhoff-Love. First, a hidden quadratic term is added to the displacement in the z-direction. Moreover, the regularity of the constructed stress tensor is determined in a sophisticated way. The error of the models depends on the thickness of the plate and on the mentioned regularity.

The details are given in a joint paper with Stefan Sauter and Christoph Schwab.

It is aimed here to propose triangular shell elements for the Kirchhoff-Love shell theory and the virtual element method (VEM), geometrically and physically in the linear range. The domain’s discretization by triangles allows the direct approximation of the shell’s geometry (mid-surface) and consequently a mapping approach as in [3] is needless. Furthermore, no curvilinear coordinate system is used. The superposition of the membrane and plate bending behaviors is such that it is considered the lowest-order case for each, i.e. a linear ansatz and linear projection (equating the constant strain triangle) for the first, and a cubic ansatz and quadratic projection for the second. Together with the element’s geometry it may thus represents a starting point for the development of several virtual shell elements. For the convergence analysis of a static boundary value problem (BVP), numerical examples with result’s comparisons involving the (classical) stabilization of [1], energy stabilization [4] and finite shell elements are performed.

References
[1] Franco Brezzi and L Donatella Marini. “Virtual element methods for plate bending problems”. In: Computer Methods in Applied Mechanics and Engineering 253 (2013), pp. 455–462.
[2] L Beirao Da Veiga, Franco Brezzi, and L Donatella Marini. “Virtual elements for linear elasticity problems”. In: SIAM Journal on Numerical Analysis 51.2 (2013), pp. 794–812.
[3] V Ivannikov, C Tiago, and PM Pimenta. “Meshless implementation of the geometrically exact Kirchhoff–Love shell theory”. In: International Journal for Numerical Methods in Engineering 100.1 (2014), pp. 1–39.
[4] P Wriggers, B Hudobivnik, and Olivier Allix. “On two simple virtual Kirchhoff-Love plate elements for isotropic and anisotropic materials”. In: Computational Mechanics 69.2 (2022), pp. 615–637.

The investigation and finite element simulation of damage and fracture processes in (quasi-)brittle and ductile materials plays an important role in many engineering applications. For this purpose, gradient-extended damage can be used to obtain mesh-objective results in simulations involving material softening. Unfortunately, these simulation may suffer from different locking phenomena. To this end, appropriate finite element technologies to avoid spurious numerical phenomena, such as volumetric and shear locking can be employed. Within this contribution, a novel family of continuum finite elements for gradient-extended models is presented, which has recently been developed (see [1], [2]). Here, a geometrically nonlinear modeling framework for gradient-extended damage and plasticity [3] is used at the material point level, while low-order displacement-based solid and solid-shell elements are used on the element level. A tailored combination of reduced integration with hourglass stabilization, the enhanced assumed strain (EAS) method, and the assumed natural strain (ANS) method cures these most dominant locking phenomena. Additionally, in order to increase numerical performance, a polynomial approximation of the kinematic and constitutively dependent quantities within the weak forms is used, and thus, the contribution of the hourglass stabilization can be analytically integrated. The accuracy and efficiency of the proposed framework is demonstrated by means of several structural examples under various loading conditions.

References

[1] O. Barfusz, T. Brepols, T. van der Velden, J. Frischkorn and S. Reese, Computer Methods in Applied Mechanics and Engineering, 373:113440, 2021.

[2] O. Barfusz, T. van der Velden, T. Brepols, H. Holthusen and S. Reese, Computer Methods in Applied Mechanics and Engineering, 382:113884, 2021.

[3] T. Brepols, S. Wulfinghoff and S. Reese, International Journal of Plasticity, 129:102635, 2020.

The importance of reconstructing conservative local fluxes from a primal discrete solution is widely recognized in the literature. One major application is in a posteriori error analysis: the difference between the numerical flux and a recovered equilibrated flux provides an a posteriori error indicator, with a reliability constant equal to 1, which is further used in adaptive mesh refinement.

In this talk, we consider an elliptic diffusion problem with discontinuous coefficients, approximated by conforming and nonconforming finite elements of arbitrary polynomial degree. We recover a conservative flux in the Raviart-Thomas space following the approach proposed by Becker, Capatina and Luce in 2016 for the Poisson equation. The idea is to introduce an equivalent hybrid mixed formulation and to use the Lagrange multiplier, which is defined on the sides of the mesh, as correction of the degrees of freedom of the flux. The main difficulty lies in the choice of the multiplier's space, which should allow to establish a uniform inf-sup condition and to compute the multiplier locally. Thus, we do not solve the mixed formulation but only use it for the error analysis. Contrarily to other approaches, no local nor global mixed problem needs to be solved. In addition, the previous approach provides a unified framework for several standard finite element methods and differential operators.

This study generalizes the methodology described above to diffusion problems, the main focus being on the robustness of the reconstruction and of the a posteriori error bounds with respect to the diffusion coefficients.

Firstly, we consider a conforming finite element approximation on triangular meshes. We express the inf-sup constant of the equivalent mixed formulation in terms of the coefficients and we also establish a local bound for the multiplier, with a constant whose dependence on the coefficients is given explicitly. This allows us to deduce, besides the usual sharp reliability of the a posteriori error indicator, its local efficiency with an explicit constant. This result is new, at the best of our knowledge; for quasi-monotone coefficients, we retrieve the complete robustness, which already exists in the literature in the quasi-monotone case.

Secondly, we consider a nonconforming finite element approximation of arbitrary polynomial degree, based on the space introduced by Matthies and Tobiska in 2005. The standard nonconforming space of odd degree rises no difficulty and the reconstruction of conservative fluxes in this case is well-known. Meanwhile, this is no longer true for even degree, due to the loss of unisolvence. Our contribution is the extension of the previous approach to the spaces of Matthies and L. Tobiska, which are well-defined for any degree and also inf-sup stable for the Stokes problem, in a robust way with respect to the diffusion coefficients.

Finally, we will present several numerical experiments illustrating the theoretical results.