A nonintrusive global/local approach applied to phasefield modeling of brittle fracture
 Tymofiy Gerasimov^{1}Email authorView ORCID ID profile,
 Nima Noii^{1},
 Olivier Allix^{2} and
 Laura De Lorenzis^{1}
https://doi.org/10.1186/s4032301801058
© The Author(s) 2018
Received: 22 December 2017
Accepted: 21 April 2018
Published: 18 May 2018
Abstract
This paper aims at investigating the adoption of nonintrusive global/local approaches while modeling fracture by means of the phasefield framework. A successful extension of the nonintrusive global/local approach to this setting would pave the way for a wide adoption of phasefield modeling of fracture, already well established in the research community, within legacy codes for industrial applications. Due to the extreme difference in stiffness between the global counterpart of the zone to be analized locally and its actual response when undergoing extensive cracking, the main foreseen issues are robustness, accuracy and efficiency of the fixed point iterative algorithm which is at the core of the method. These issues are tackled in this paper. We investigate the convergence performance when using the native global/local algorithm and show that the obtained results are identical to the reference phasefield solution. We also equip the global/local solution update procedure with relaxation/acceleration techniques such as Aitken’s \(\Delta ^2\)method, the Symmetric Rank One and Broyden’s methods and show that the iterative convergence can be improved significantly. Results indicate that Aitken’s \(\Delta ^2\)method is probably the most convenient choice for the implementation of the approach within legacy codes, as this method needs only tools already available for the socalled submodeling approach, a strategy routinely used in industrial contexts.
Keywords
Introduction
The use of phasefield approaches in the case of structures of industrial complexity has been the subject of limited investigations thus far and poses a number of challenges. In this paper, in order to move forward in this direction we advocate the use of nonintrusive global/local strategies initially proposed in [6]. When dealing with large structures, fracture phenomena most often occur in regions of limited extent only. Moreover, in the case of brittle fracture most of the structure behaves elastically. These features are particularly appealing for global/local approaches as they make it possible to first compute the global model elastically, and then determine the critical areas to be reanalyzed, while storing the factorization of the decomposition of the structural stiffness. The local models are then iteratively substituted within the unchanged global one, which has the advantage of avoiding the reconstruction of the mesh of the whole structure. In fact, this is the main motivation of nonintrusive global/local approaches: to avoid the modification of the finite element model used by engineers, the creation of a complex global model being by far the most timeconsuming task, a task which is more and more externalized.
In the past decade, both phasefield and nonintrusive global/local approaches have been extended to deal with a growing number of situations of interest for engineers. The currently available phasefield formulations of brittle fracture encompass static and dynamic models. We mention the papers by Amor et al. [7], Miehe et al. [8, 9], Kuhn and Müller [10], Pham et al. [11], Borden et al. [12], Mesgarnejad et al. [13], Kuhn et al. [14], Ambati et al. [15], Wu et al. [16], where various formulations are developed and validated. Recently, the framework has been also extended to ductile (elastoplastic) fracture [17–22], pressurized fracture in elastic and porous media [23, 24], fracture in films [25] and shells [26–28], and multifield fracture [29–36]. Nonintrusive global/local approaches have also been applied to a quite large number of situations: the computation of the propagation of cracks in a sound model using the extended finite element method (XFEM) [37], the computation of assembly of plates introducing realistic nonlinear 3D modeling of connectors [38], the extension to nonlinear domain decomposition methods [39] and to explicit dynamics [40, 41] with an application to the prediction of delamination under impact using Abaqus [42]. Alternative strategies can be derived from the Partition of Unity Method [43, 44].
The phasefield simulation of fracture processes with legacy codes bears a number of advantages which fit perfectly within the framework of nonintrusive coupling approaches using predefined ‘fixed’ meshes. The most obvious advantage is the ability to track automatically a cracking process by the evolution of the smooth crack field on a ‘fixed’ mesh which, in the proposed procedure, is the mesh of the local model. This is a significant advantage over the discrete fracture description, whose numerical implementation requires explicit (in the classical finite element method, FEM) or implicit (within XFEM) handling of the discontinuities. The possibility to avoid the tedious task of tracking complicated crack surfaces in 3D significantly simplifies the implementation. The second advantage is the ability to simulate complicated processes, including crack initiation (also in the absence of a crack tip singularity), propagation, coalescence and branching without the need for additional adhoc criteria and with very few parameters to be identified. This feature is particularly attractive for industrial applications, as it minimizes the need for timeconsuming and expensive calibration tests.
Due to the extreme difference in stiffness between the global counterpart of the zone to be reanalyzed locally and its actual response when undergoing extensive cracking, the foreseen fundamental issues associated with the use of the global/local strategy in combination with phasefield fracture modeling are robustness, accuracy and efficiency of the fixed point iterative algorithm which is at the core of the method. Also, the finite element treatment of the phasefield formulation of brittle fracture is known to be computationally demanding, mainly due to the nonconvexity of the energy functional to be minimized with respect to both arguments (the displacement and the phase field) simultaneously [45–47]. As a result, the socalled monolithic approach manifests major iterative convergence issues of the Newton–Raphson procedure. A new linesearch scheme [46] and modified Newton methods [47] have been recently proposed to tackle this problem. Alternatively, staggered (also termed partitioned, or alternate minimization) solution scheme is widely used. This is based on decoupling of the strongly nonlinear weak formulation into a system and then iterating between the equations [2–5, 7–9, 11–13, 15, 16]. The staggered scheme is proved to be robust, but typically has a very slow convergence behavior of the iterative solution process, see e.g. [15, 46, 48]. In view of the above, a central question that arises when combining nonintrusive global/local approaches with phasefield modeling of fracture is how additional global/local iterations affect and possibly deteriorate the highly sensitive iterative behavior of the staggered scheme used to solve the phasefield equations. In this paper, we make a first attempt to address these questions.
The paper is organized as follows. In “The phasefield approach to brittle fracture” section, we outline the main concepts of phasefield modeling of brittle fracture and illustrate the specific formulation used in the present paper. “Global/local approach in a nonintrusive setting” section introduces the nonintrusive global/local approach for the solution of the reference phasefield model considered in “The phasefield approach to brittle fracture” section. This is done in several steps. We start by illustrating an intrusive global/local scheme through a domain decomposition formulation in a variational setting well adapted to the phase field formulation. Several options are considered, including the socalled primal, dual and localized Lagrange multipliers based versions. This domain decomposition framework is used afterwards to define some convergence indicators in terms both of incompatibility of the reaction forces and of displacement jumps at the interface between the unchanged global model and the reanalyzed local one. The motivation here is to see which indicator or combination of indicators are the most suited to an appropriate estimation of the quality of the global/local iteration results with respect to the phasefield determination. The third version is then extended to the global/local setting, for which a nonintrusive computational procedure is devised. The numerical results which illustrate the performance of the proposed nonintrusive global/local approach as well as their qualitative and quantitative comparison with the reference solution are reported in “Results and discussion” section. Therein, we also outline and apply three relaxation/acceleration techniques, which are incorporated into the global/local iterative procedure and aim at improving its efficiency. Conclusions and outlook finalize the paper.
The phasefield approach to brittle fracture
In this section, we consider a mechanical system undergoing a brittle fracture process modeled with the phasefield formulation, and term this the reference problem. For this problem, we develop in “Global/local approach in a nonintrusive setting” section a global/local formulation, which is dissected numerically in “Results and discussion” section.
Staggered iterative solution process for (11) at a fixed loading step l

Note that the equation \({\mathcal {E}}_{\varvec{u}}=0\) in Table 1 is strongly nonlinear due to the nonlinearity of \(\varvec{\sigma }(\varvec{u},d):=(1d)^2\frac{\partial \Psi ^+}{\partial \varvec{\varepsilon }}(\varvec{\varepsilon }(\varvec{u}))+\frac{\partial \Psi ^}{\partial \varvec{\varepsilon }}(\varvec{\varepsilon }(\varvec{u}))\), see equation (8). Therefore, at every staggered iteration \(k\ge 1\) with given \(d^{k1}\), a Newton–Raphson procedure is needed to compute \(\varvec{u}^k\), with e.g. \(\varvec{u}^{k1}\) being taken as the initial guess, and \(\texttt {TOL}_\mathrm {NR}\) as a userdefined tolerance. Owing to the ‘nested in’ nature of the Newton–Raphson process, it has to be \(\texttt {TOL}_\mathrm {NR}<\texttt {TOL}_\mathrm {Stag}\). In the presented numerical examples we take \(\texttt {TOL}_\mathrm {NR}:=10^{8}<\texttt {TOL}_\mathrm {Stag}:=10^{5}\).
Global/local approach in a nonintrusive setting
The starting point towards a nonintrusive global/local approach to the phasefield problem (4) with \(\mathcal {E}\) defined by (1) is a standard nonoverlapping domain decomposition procedure applied to \(\mathcal {E}\). The resulting formulation is then extended to a global/local one in the spirit of [39, 51], for which the nonintrusive computational scheme is devised.
Domain decomposition formulation
Two standard ways to proceed with (19) and (20), and obtaining a variational formulation equivalent to the original one in (4) are as follows.
Domain decomposition formulations of the reference problem (4)
Formulation  Imposed continuity between  Unknowns  

\( \varvec{u}_{{\varvec{C}}}\,\varvec{ \& }\,\varvec{u}_{{\varvec{L}}}\)  \( \varvec{\lambda }_{{\varvec{C}}}\,\varvec{ \& }\,\varvec{\lambda }_{{\varvec{L}}}\)  
Primal, (18)  Strong  –  \((\varvec{u}_C,\varvec{u}_L,d_L)\) 
Dual, (23)  Weak  Strong  \((\varvec{u}_C,\varvec{u}_L,d_L,\varvec{\lambda })\) 
LLM, (26)  Weak  Weak  \((\varvec{u}_C,\varvec{u}_L,d_L,\varvec{u}_\Gamma ,\varvec{\lambda }_C,\varvec{\lambda }_L)\) 
Formulation (23) is seemingly less computationally demanding than (26), since there is only one extra field \(\varvec{\lambda }\) to be solved for in the former case, versus the triple \((\varvec{u}_\Gamma ,\varvec{\lambda }_C,\varvec{\lambda }_L)\) of unknown fields in the latter one. The potential advantage of (26) over (23) is a greater flexibility, at the finite element discretization stage, of handling the interface between complementary and local domains.
As follows, we move on with the LLM formulation (26) and extend it to the global/local setting, for which, in turn, a nonintrusive solution procedure is devised. This will lead to a nonintrusive global/local approach to the phasefield formulation (4).
Global/local formulation
As a first step, a socalled fictitious domain \(\Omega _F\) is introduced to ‘fill the gap’ obtained in \(\Omega \) by removing \(\Omega _L\) from it, see Fig. 3b. It is assumed that \(\Omega _F\) is constituted by a material with the same linear elastic behaviour as in \(\Omega _C\). It is also assumed that \(\Omega _F\) is open (i.e. \(\Gamma \not \subset \Omega _F\)). Unification of \(\Omega _F\) with \(\Gamma \) and \(\Omega _C\) forms the global domain \(\Omega _G\), that is, \(\Omega _G:=\Omega _F\cup \Gamma \cup \Omega _C\). The fictitious domain \(\Omega _F\) is furthermore assumed free of geometrical ‘imperfections’ which may be present in \(\Omega _L\), see Fig. 3b. Therefore, it is in general \(\Omega _G\ne \Omega \), and the constructed global domain \(\Omega _G\) should not be confused with the original reference domain \(\Omega \).
Summing up the above, the role of the fictitious domain \(\Omega _F\) is twofold: it replaces the “subregions” of a structure (reference domain) containing geometric details (e.g. holes, inclusions etc.) and/or constitutive nonlinearity by there detailsfree and linearly elastic “counterparts”. The obtained global domain \(\Omega _G\) is then straightforwardly suitable for meshing and solving procedures within legacy codes. As it will be also seen below, the use of \(\Omega _F\) is essential to realize the concept of nonintrusiveness of the computational scheme for solving the coupled global/local formulation.
Next to this, it is assumed that there exists a continuous prolongation of \(\varvec{u}_C\) into \(\Omega _F\). That is, we introduce a function \(\varvec{u}_G\in \mathbf H ^1(\Omega _G)\) such that \(\varvec{u}_G_{\Omega _C}\equiv \varvec{u}_C\) and \(\varvec{u}_G=\varvec{u}_C\) on \(\Gamma \) in the sense of trace. The former also implies that \(\varvec{u}_G=\varvec{0}\) on \(\Gamma _{D,0}\) and \(\varvec{u}_G=\bar{\varvec{u}}_l\) on \(\Gamma _{D,1}\).
Coupled system in weak form
For the presented system of equations, a computational scheme can already be devised. We should notice, however, that equation (G) in the current form does not fit in the notion of nonintrusiveness yet. Indeed, being a linear one, it can naturally be solved for \(\varvec{u}_G\) ‘straightforwardly’. But the presence of the two domain integrals, namely, over \(\Omega _G\) and \(\Omega _F\subset \Omega _G\) would imply in this case the need to simultaneously access the corresponding stiffness matrices (in the following, \(\mathsf {K}_G\) and \(\mathsf {K}_F\)), or, in other words, a necessity of modifying \(\mathsf {K}_G\)—a situation that contradicts the concept of nonintrusiveness. Avoiding this can be done in two steps: first, by introducing a partitioning of equation (G), and then, devising the appropriate iterative solution procedure. The former will be presented here, and the latter is addressed in “Nonintrusive computational scheme” section.

\(\varvec{\sigma }(\varvec{u}_G)\cdot \varvec{n}_{\Gamma _{N,0}}=0\) on \(\Gamma _{N,0}\),

\(\varvec{v}_G=0\) on \(\Gamma _{D,0}\) and on \(\Gamma _{D,1}\),

\(\varvec{\sigma }(\varvec{u}_G)\cdot \varvec{n}_{\Gamma _{N,1}}=\bar{\varvec{t}}_l\) on \(\Gamma _{N,1}\),
Equations (\(\hbox {G}_1\)), (\(\hbox {G}_2\)), system (L) and coupling equations (\(\hbox {C}_1\)), (\(\hbox {C}_2\)), (\(\hbox {C}_3\)) constitute what we term global/local coupled system, which is to be solved for the vector \((\varvec{u}_G,\varvec{u}_L,d_L,\varvec{u}_\Gamma ,\varvec{\lambda }_C,\varvec{\lambda }_L)\).
Nonintrusive computational scheme
 (a)
Since the data \((\bar{\varvec{u}}_{l},\bar{\varvec{t}}_{l})\) are posed on \(\Gamma _{D,1},\Gamma _{N,1}\subset \partial \Omega _G\), the process initialization (i.e. iteration \(n=0\)) is started with the solution of global problem (\(\hbox {G}_1\)), (\(\hbox {G}_2\)).
 (b)
In order to fit equation (\(\hbox {G}_1\)) with \(\varvec{\lambda }_F=\varvec{\lambda }_F(\varvec{u}_G)\) in the concept of nonintrusiveness, \(\varvec{\lambda }_F\) must be treated as a known quantity. This defines the order in which equations (\(\hbox {G}_1\)) and (\(\hbox {G}_2\)) are solved at any iteration \(n\ge 0\): the solution of (\(\hbox {G}_2\)) precedes the solution of (\(\hbox {G}_1\)). In this case, as desired, the stiffness matrix \(\mathsf {K}_G\) remains unaltered; the access to \(\mathsf {K}_F\) is still required, but only at the stage of solving (\(\hbox {G}_2\)), not (\(\hbox {G}_1\)).
 (c)
For solving (\(\hbox {G}_1\)), \(\varvec{\lambda }_C\) must be also known. At \(n=0\), \(\varvec{\lambda }_C\) can simply be taken from the previous loading step. At \(n\ge 1\), we use coupling equation (\(\hbox {C}_1\)) for the extraction of \(\varvec{\lambda }_C\), assuming \(\varvec{\lambda }_L\) is already known. This defines the order in which the global and local problems are solved: at any iteration starting from \(n=1\), the solution of (L) precedes the solution of (\(\hbox {G}_1\)).
We also notice that:
 (d)
Coupling equation (\(\hbox {C}_3\)) provides the boundary condition for \(\varvec{u}_L\) of the local problem (L).
 (e)
Coupling equation (\(\hbox {C}_2\)) is used for the recovery of \(\varvec{u}_\Gamma \).

solution of local problem (L) coupled with (\(\hbox {C}_3\)),

recovery phase using (\(\hbox {C}_1\)) and (\(\hbox {G}_2\)),

solution of global problem (\(\hbox {G}_1\)),

recovery phase using (\(\hbox {C}_2\)).
Nonintrusive iterative solution process for (\(\hbox {G}_1\)), (\(\hbox {G}_2\)), (L), and (\(\hbox {C}_1\)), (\(\hbox {C}_2\)), (\(\hbox {C}_3\)) at a fixed loading step l

Staggered process for the local problem
Solution of the local system in Table 3 at the given global/local iteration \(n\ge 1\) requires an additional nested iterative solution process. In our case, this is the staggered procedure from Table 1, which is adjusted to handle an extra variable \(\varvec{\lambda }_L\), and is also equipped with the appropriate definition of the input (initial guess) data and of the stopping criterion.
The initial guess for the staggered loop (with the iteration index \(k\ge 0\)) is chosen as follows. At iteration \(n=1\) (and staggered iteration \(k=0\)) the values \((\varvec{u}_{L,l1},d_{L,l1},\varvec{\lambda }_{L,l1})\) known from the previous loading step are used as the initial guess. At \(n\ge 2\) (and staggered iteration \(k=0\)), we naturally take \((\varvec{u}_L^{n1},d_L^{n1},\varvec{\lambda }_L^{n1})\).
Accuracy/convergence check
Since the quantity \(\eta \) naturally stems from the global/local solution accuracy check \(\widetilde{\mathcal {E}}_{\varvec{z}}(\varvec{z}^n;\varvec{y})=0\), it represents not only the iterative convergence indicator, but also the solution accuracy indicator—a very desired property, since the former is only suitable for tracing the convergence of the corresponding iterative solution process, but, clearly, is not adequate for stopping criterion. The corresponding ingredients \(\eta _{\varvec{u}}\) and \(\eta _{\varvec{\lambda }}\) are only iterative convergence indicators, but none of them provides an adequate check of the solution accuracy. In particular, since \(\eta _{\varvec{u}}\) measures, though implicitly, the displacement continuity—a match between \(\varvec{u}_G\) and \(\varvec{u}_L\)across \(\Gamma \) (recall that the traction continuity—a match between \(\varvec{\lambda }_C\) and \(\varvec{\lambda }_L\) on \(\Gamma \)—is, in our case, fulfilled automatically), it is also the indicator of a good “gluing” between the two models.
Incremental setting
Finite element discretization
In the following, for the sake of simplicity, we assume the dimension of the reference problem is 2. Let \(\mathcal {P}\) be a finite element partition of \(\Omega \) into triangles or quadrilaterals, I be the number of nodes in \(\mathcal {P}\), and \(N_i\), \(i=1,\ldots ,I\) be the nodal shape function associated with the node i and supported on the collection of elements in \(\mathcal {P}\) that share i. Finally, let a scalarvalued quantity \(\hat{\cdot }_i\) represent the nodal value.

the partitions \(\mathcal {T}_G\), \(\mathcal {T}_L\) and \(\mathcal {T}_\Gamma \) match (this is usually termed a ‘matching case’);

the basis in the global and local domains is identical, that is, \(\varvec{N}_u^G=\varvec{N}_u^L=:\varvec{N}_u\);

the basis on the interface is obtained from \(\varvec{N}_u\) by the corresponding restriction, that is, \(\varvec{N}_\lambda ^G=\varvec{N}_\lambda ^L=\varvec{N}_u^\Gamma =\varvec{N}_u_\Gamma \);

the nodal shape functions \(N_i\) composing bases \(\varvec{N}_u\) and \(\varvec{N}_d\) are piecewise linear.
Using expressions (47), (48) and (49) along with the above assumptions, the matrix representation of all equations in Table 3 is straightforward.
Results and discussion
To illustrate the proposed approach, we consider the following benchmark problem. A square specimen with two holes of different diameters is subjected to tension loading, see Fig. 5a. The holes are introduced to weaken the structure and to facilitate the specimen cracking in absence of a stronger singularity such as a preexisting crack. The holes location is chosen such that prediction of the subregion where cracking occurs (hence, the local domain for the forthcoming global/local analysis) is feasible. Taking a different size of the holes is intended to obtain a geometrically nontrivial crack pattern, as depicted in Fig. 5b. This, moreover, results in a multistage crack propagation process to be manifested by a loaddisplacement response with two peak points, see Fig. 5c for a sketch, and Figs. 7 and 12 for the actually obtained results. We believe that the present setup, being neither extremely complex, nor trivial, is suitable for the purpose of a qualitative and quantitative comparison between the reference results and results obtained with the proposed global/local approach.
The geometric data are as follows (all given in mm): \(a=1\), \(b_1=0.197\), \(b_2=0.210\), \(b_3=0.490\) with the hole diameters \(c_1=0.247\) and \(c_2=0.0806\). The material data are: Young’s modulus \(E=210\,\mathrm {GPa}\), Poisson’s ratio \(\nu =0.3\) and the critical energy release rate \(G_c=2.7\cdot 10^{3}\,\mathrm {kN/mm}\). The characteristic length in the phasefield formulation is \(\ell =1.5\cdot 10^{2}\,\mathrm {mm}\). We consider the planestrain situation.
The algorithmic parameters are: the loading \(\bar{u}_l=l\Delta u\) with \(l\in [1,110)\) and the increment size \(\Delta u:= 0.06\cdot 10^{3}\,\mathrm {mm}\), the tolerance magnitudes are \(\mathtt {TOL}_\mathrm {NR}:=10^{8}\), \(\mathtt {TOL}_\mathrm {Stag}:=10^{5}\), and \(\widetilde{\texttt {TOL}}_\mathrm {GL}:=10^{6}\).
Reference and global/local results
The first important observation is that \(\eta _{\varvec{u}}\), which implicitly measures the displacement discontinuity between the solutions of the global and local problem across the interface, is two orders of magnitude less than \(\eta _{\varvec{\lambda }}\). Thus, its contribution to \(\eta \), which is used not only for tracing the convergence of the iterative solution process, but also for the solution accuracy check, is negligible. This means that a stopping criterion based solely on the use of \(\eta _{\varvec{u}}\) (what seems typical for the global/local approaches in e.g. plasticity) will yield, in our case, erroneous results. Secondly, it can be noted that a quite large amount of global/local iterations is needed, especially at loading steps corresponding to the peak loads of the loaddisplacement curves in Fig. 7 (the points of interest 2 and 4 from Fig. 5c).
It can be grasped that the rapid increase of cumulative time in Fig. 10 for both formulation appears at loading steps related to the peak points 2 and 4. Also, regardless of the formulation, the computational time per step in Fig. 11 at these points is significantly higher (by almost two orders of magnitude, to be more precise) than at the prepeak loading steps. These observations correlate with the convergence results from Fig. 9.
Nonconvexity and nonlinearity of the global/local formulation, as well as the complicated multilevel iterative nature of the related iterative solution procedure result in a generically slow convergence of the approach. Another impacting factor that should be noted is that the stiffness matrix of the global problem \(\mathsf {K}_G\) is never updated within the global/local computation process. Incorporation of an incremental update relaxation in this process is thus our next goal, with the objective to obtain an acceleration of the convergence process.
Relaxation/acceleration techniques: Aitken’s, SR1, Broyden et al.
Following [39] and [51], we will consider and incorporate two types of relaxation/acceleration techniques into our approach: Aitken’s \(\Delta ^2\)method (also known as dynamic relaxation, whose efficient implementation in fluidstructure interaction computations has already been reported [60, 61]) and QuasiNewton correction. Within the family of QuasiNewton correction formulae, we restrict ourselves to the Symmetric Rank One (SR1) and the Broyden update versions.
Conclusions
We combined the adoption of nonintrusive global/local approaches with phasefield modeling of brittle fracture, with the main objective to pave the way for a wide adoption of this framework for industrial applications within legacy codes. We investigate the convergence performance of the fixedpoint scheme used for the global/local iterations and showed that the obtained results are identical to the reference phasefield solution. In order to accelerate the observably quite slow convergence behavior, especially close to and beyond the peak point(s) of the loaddisplacement response, we also equipped the global/local solution update procedure with relaxation/acceleration techniques such as Aitken’s \(\Delta ^2\)method, the Symmetric Rank One and Broyden’s methods. Findings showed that the iterative convergence can be improved significantly, to a similar extent for all investigated methods. Aitken’s \(\Delta ^2\)method is probably the most convenient choice for the implementation of the approach within legacy codes, as this method needs only tools which are often used for the socalled submodeling strategy, which is well known and widely used in industrial contexts.
Declarations
Author's contributions
TG elaborated the concepts proposed and discussed in the manuscript, contributed to the numerical studies and wrote the draft of this paper, NN performed the numerical implementation and experiments, OA contributed to writing the part of the introductory section, and LDL contributed to the numerical experiments and paper revisions. All authors read and approved the final manuscript.
Acknowledgements
Not applicable.
Competing interests
The authors declare that they have no competing interests.
Ethics approval and consent to participate
Not applicable.
Funding
TG was partially funded by the ERC through the Starting Researcher Grant INTERFACES, GA 279439. NN was partially funded by the DFG through the IRTG 1627 “Virtual materials and their validation”.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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