Noninvasive global–local coupling as a Schwarz domain decomposition method: acceleration and generalization
 Pierre Gosselet^{1}Email authorView ORCID ID profile,
 Maxime Blanchard^{1, 2},
 Olivier Allix^{1}View ORCID ID profile and
 Guillaume Guguin^{1}
https://doi.org/10.1186/s4032301800974
© The Author(s) 2018
Received: 5 October 2017
Accepted: 13 February 2018
Published: 27 February 2018
Abstract
The noninvasive global–local coupling algorithm is revisited and shown to realize a simple implementation of the optimized nonoverlapping Schwarz domain decomposition method. This connection is used to propose and compare several acceleration techniques, and to extend the approach to non conforming meshes.
Keywords
Mathematics Subject Classification
Introduction
The noninvasive local–global coupling technique proposed by Allix [1] is an iterative method which aims at making accurate the well known submodeling technique [2–4]. It is strongly related to many reanalysis techniques [5–7] and domain decomposition methods [8].
The aim of this technique is to evaluate the effect of local modifications inside a computational model (geometry, material and load) without requiring heavy developments. More precisely the objective is to use an industrial model with a given commercial software and to simulate the presence of local alterations by iteratively spawning computations with only extra traction loads inside the model. Moreover, the alterations can be computed on any chosen software including dedicated research codes.
This philosophy was successfully applied in many different contexts like: the introduction of local plasticity and geometrical refinements [1], the computation of the propagation of cracks in a sound model [9], the evaluation of stochastic effects with deterministic computations [10], the taking into account of the exact geometry of connectors in an assembly of plates [11]. In [12] the method was used in order to implement a nonlinear domain decomposition method [13–16] in a noninvasive manner with Code_aster. The extension of the approach to explicit dynamics was proposed in [17], improved in [18] and applied to the prediction of delamination under impact in [19]. Alternative noninvasive strategies can be derived from the extended finite element method [20, 21].
After a description of the method (“Derivation of the noninvasive algorithm” section), this paper provides several contributions. First the noninvasive coupling algorithm is proved to realize a simple implementation of the optimized nonoverlapping Schwarz domain decomposition method (“Connexion with alternate nonoverlapping Schwarz method” section). Several accelerations techniques are proposed (“Analysis and acceleration of the global/local algorithm” section), some are classical but the linear and nonlinear conjugate gradient is new in this framework. The algorithms are described in a very programmerfriendly manner. Last, an overlapping version of the method is proposed (“Overlapping version” section) which can be used to handle fully nonconforming meshes.
Derivation of the noninvasive algorithm
The algorithm we study is very general and applies to the study of many PDEs; in order to fix the ideas, we consider problems of nonlinear quasistatic structure mechanics under the small strain hypothesis. We note u the displacement field, \(\varepsilon \) the symmetric part of the gradient, \(\sigma \) the Cauchy stress tensor. The domain \(\Omega \) is submitted to given body force f, Dirichlet condition \(u_d\) on the part \(\partial _d \Omega \) of the boundary and Neumann condition g on the complement part \(\partial _n \Omega \). In order to manage viscous materials, the study is conducted over a time interval \(\mathcal {T}=[0,T]\), and the following equations are meant to be satisfied at any time \(t\in \mathcal {T}\), which we omit to write except when necessary.
Remark 1

Assuming the fine and auxiliary problems were solved exactly, we have:the corrective load p is then an immersed surface traction. In the following, we always assume the exactness of the computations; note that using inexact solvers was investigated in [22] where the method is identified with a localized multigrid iteration.$$\begin{aligned} p_{n+1} = \left( \lambda {}^A_{n}\lambda {}^F_{n} \right) \in V_\Gamma ^* \end{aligned}$$(9)

Because the Auxiliary problem corresponds to the restriction of the Global problem on the zone of interest with global displacement imposed, we directly have:The introduction of the Auxiliary problem is thus not mandatory, it is just a workaround in case of software unable to compute the reaction in an immersed surface. Of course, the Auxiliary problem can be solved in parallel with the Fine problem.$$\begin{aligned} u{}^A_{n}= u{}^G_{n\Omega {}^A} \end{aligned}$$(10)

We can also define the reaction from the Complement zone for a given \(u{}^G_{n}\):Then we see that:$$\begin{aligned} \langle \lambda {}^C_{n} , v \rangle _\Gamma = a{}^C(u{}^G_{n},v)  l{}^C(v), \quad \forall v \in V(\Omega {}^C) \end{aligned}$$(11)The surface traction \(p_{n}\) generates a discontinuity in the normal stress of the Global problem.$$\begin{aligned} \lambda {}^C_{n} + \lambda {}^A_{n} = p_{n} \end{aligned}$$(12)

If we replace the auxiliary reaction by the complement one, we have:in words, the correction brought to \(p_{n+1}\) corresponds to the lack of balance between the Complement zone and the Fine representation of the zone of interest. This lack of balance is the residual r of the algorithm. The algorithm converges when the two representations are in equilibrium (\(r=0\), in which case the extra load p shall not evolve anymore).$$\begin{aligned} p_{n+1} = p_n + r_{n} \text { with } r_{n+1} = \left( \lambda {}^F_{n}+\lambda {}^C_{n}\right) \end{aligned}$$(13)

The algorithm makes no use of domain integrals to communicate between subdomains; only interface data (on \(\Gamma \)) are exchanged, namely the displacement \(u{}^G\) and the reactions \(\lambda {}^F\) and \(\lambda {}^A\) (or \(\lambda {}^C\)). As long as the interface \(\Gamma \) is well represented in all models, it is not necessary to use the exact Fine domain \(\Omega {}^F\) in the Auxiliary problem, any coarser representation is possible (\(\Omega {}^A\)). Typically microperforations or micro cracks need not be represented in the Auxiliary problem. Of course modifying the representation of the zone of interest may have consequences on the convergence of the algorithm (but not on its limit which is the reference solution).
Connexion with alternate nonoverlapping Schwarz method
The question of linking the noninvasive global–local coupling method to the many variants of domain decomposition and associated algorithms, like chimera, was studied in other publications like [8]. Here we propose to connect the method with the iterations of a nonoverlapping optimized Schwarz method. The theoretical framework of Schwarz method will allow us natural extensions to the method, in particular the use of overlaps to treat mesh incompatibilities.
The global–local algorithm corresponds to the choice \(Q{}^C=\mathcal {S}{}^A\) and formally \(Q{}^F=\infty \) (the Dirichlet condition being seen as the limit case of an infinite interface impedance). The choice \(Q{}^C=\mathcal {S}{}^A\) is extremely strong because we can expect \(\mathcal {S}{}^A\) to be a good approximation of \(\mathcal {S}{}^F\), not only in term of stiffness (\(a{}^A\) vs \(a{}^F\)) but also in term of load (\(l{}^A\) vs \(l{}^F\)) which corresponds to providing a good initialization to the algorithm.

Krylov acceleration: replacing stationary iterations by Krylov solvers is classical in Schwarz methods [30]. The Dirichlet condition \(Q{}^F=\infty \) preserves some symmetry so that we can derive a conjugate gradient algorithm, see “Conjugate gradient” section.

Mixed approach: the condition \(Q{}^F=\infty \) is a poor approximation of the optimal choice. In [31] a twoscale approximation of \(\mathcal {S}{}^C\) was proposed for the global–local coupling.

Parallel processing: the global–local method corresponds to the alternate version of the optimized Schwarz method. The parallel version could be tried in the noninvasive context. Note that this would only make sense in the presence of multiple Fine zones with finite Fine impedance \(Q{}^F<\infty \).

Nonlinearity: stationary iterations can directly be transferred to nonlinear problems, in particular the ones with monotone operators (positive hardening) [32, 33]. The local–global method was successfully applied in many nonlinear problems like plasticity or fracture [1, 22]

Overlapping version: optimized Schwarz methods also exist with overlaps. In [34], the overlap was used as a buffer zone to dampen edge effects in plate/3D coupling. In “Overlapping version” section, we present another application, the handling of nonmatching meshes.
Analysis and acceleration of the global/local algorithm
Notations
In order to further analyze the algorithm and be more practical, we now consider the finite element discretization of the problem. We use the following notations: \(\mathbf {f}\) for the generalized forces, \(\mathbf {u}\) for the nodal displacement and \(\varvec{\lambda }\) for the nodal reactions and \(\mathbf {p}\) for the nodal component on the immersed surface effort. When indexing degrees of freedom, \(F,\ A,\ C\) stand for the internal degrees of freedom whereas \(\Gamma \) stands for nodes on the interface (whose description is identical in all models). We tried to use minimal notations, but sometimes a quantity defined on the interface is issued from one side specifically, in which case we make it clear by an extra superscript. In the linear(ized) case notation \(\mathbf {K}\) is used for the stiffness matrices.
Remark 2
Remark 3

\([\mathbf {u}{}^G]=\mathtt {SolveGlobal}(\mathbf {p};\mathbf {f}{}^G)\), \(\mathbf {u}{}^G\) is defined on the whole Global model and in particular we have \(\mathbf {u}{}^G_\Gamma = \mathcal {S}{}^{G^{1}}(\mathbf {p};\mathbf {f}{}^G)\).

\([\mathbf {u}{}^F,\varvec{\lambda }{}^F]=\mathtt {SolveFine}(\mathbf {u}{}^G;\mathbf {f}{}^F)\), \(\mathbf {u}{}^F\) is defined in the Fine model and we have \(\varvec{\lambda }{}^F=\mathcal {S}{}^F(\mathbf {u}{}^G_\Gamma ;\mathbf {f}{}^F)\).

\([\varvec{\lambda }{}^A]=\mathtt {SolveAux}(\mathbf {u}{}^G;\mathbf {f}{}^A)\), which in corresponds to \(\varvec{\lambda }{}^A=\mathcal {S}{}^A(\mathbf {u}{}^G_\Gamma ;\mathbf {f}{}^A)\). When authorized by the software, it can be replaced by the postprocessing of the stress (19).
Stationary iterations
Remark 4
The results of the existence of sufficient and of optimal relaxations can be extended to the case of a monotone problem. Indeed in that case, the method can be interpreted as an operator splitting technique [35] on the condensed problem which inherits the useful properties of the original system (in particular monotonicity and coercivity). Reader may refer to [36] for detailed proof with weak assumptions.
In practice, it is convenient to have \(\omega \) adapted at each step. A good heuristic for the sequence \((\omega _n)\) is provided by Aitken’s \(\Delta ^2\). It was first tried in the global/local framework in [37]. The strategy is summedup in Algorithm 1.
QuasiNewton’s approaches for linear Global model
Conjugate gradient
Full linear case
This case occurs when all models are linear. Noninvasive global/local coupling can still be of interest in order to introduce complex local heterogeneities, stochastic behaviors or complex geometries in the Fine model.
For linear problems, it is rather classical to use Krylov accelerators on stationary iterations. In our case, the problem to solve (29) is governed by the operator \(\mathbf {S}{}^R{\mathbf {S}{}^G}^{1}\) which is symmetric in the \({\mathbf {S}{}^G}^{1}\) innerproduct. We then can derive a rightpreconditioned conjugate gradient. The algorithm being not so standard, it is given in Algorithm 3.
Beside the improved convergence compared to stationary iterations, using conjugate gradient allows an unconditional convergence (without necessity for the Auxiliary model to be sufficiently stiff).
Nonlinear case

A line search algorithm to optimize the length of the steps. For a given search direction \(\underline{\mathbf {p}}\), one tries to find the optimal length \(\alpha \) in term of the minimization of some norm of the residual. This can be done in a noninvasive manner by a sampling technique with several lengths \((\alpha _i)\) being tested in parallel. Classically these samples are used to interpolate the objective function and decide the final \(\alpha \). Because of the cost of the estimation of one configuration (one global solve followed by one local solve), we prefer to use directly the best sample already computed (except if the interpolated minimal let us expect a significantly better configuration).

A “conjugation” technique for the new search direction \(\underline{\mathbf {p}}_{j+1} = \mathbf {r}_{j+1} + \beta _j \underline{\mathbf {p}}_{j}\) given by a heuristic (using the notations of Algorithm 4) like:Moreover it is often chosen to avoid negative steps by using \(\beta _j \leftarrow \max (0,\beta _j)\). The reader may refer to [39] and associated bibliography for more details. In our examples, the Polac–Ribière formula appeared to be more stable.$$\begin{aligned} \begin{aligned}&\text {FletcherReeves:}\, \beta _j=\frac{\mathbf {r}_{j+1}^T \mathbf {r}_{j+1}}{\mathbf {r}_{j}^T\mathbf {r}_{j} }&\text {PolacRibi}\grave{\hbox {e}}\text {re:} \ \beta _j=\frac{\mathbf {r}_{j+1}^T (\mathbf {r}_{j+1}\mathbf {r}_{j})}{\mathbf {r}_{j}^T\mathbf {r}_{j} } \\&\text {DaiYuan:}\ \beta _j=\frac{\mathbf {r}_{j+1}^T \mathbf {r}_{j+1}}{\underline{\mathbf {p}}_{j}^T(\mathbf {r}_{j+1}\mathbf {r}_{j}) }&\text {HestenesStiefel:}\ \beta _j=\frac{\mathbf {r}_{j+1}^T (\mathbf {r}_{j+1}\mathbf {r}_{j})}{\underline{\mathbf {p}}_{j}^T(\mathbf {r}_{j+1}\mathbf {r}_{j}) } \end{aligned} \end{aligned}$$(34)
Noninvasive implementation
Numerical illustration
The method is illustrated on an academic 2D test case modeling a high pressure turbine blade of a plane engine (see Fig. 3) with an approximate size of 10 cm. The Reference model possesses local perforations with adapted mesh which are not present in the Global model. Note that, the Fine model is naturally more flexible than the Auxiliary model (because of the holes and of the refined mesh). The mechanical behavior is either isotropic linear elastic (Young’s modulus E and Poisson’s coefficient \(\nu \) are given in Table 1) or elastoviscoplastic of the form of [40] modeling a realistic IN100 material at hot temperature (\(\simeq 800\,^\circ \)C, parameters are reported in Table 1 using the notations of [40]).
Material parameters for IN100 at \(800^{\circ }\) from [44]
E [MPa]  \(\varvec{\nu }\)  C [MPa]  \(\varvec{D}\)  R  \(\varvec{n_f,\;K_f}\)  \(\varvec{n_s,\;K_s}\) 

154,000  0.28  615,000  1870  80  14, 630  17.2, 1300 
Linear models
We study the various acceleration strategies presented above in the case where all models are linear elastic. Figure 4a presents the evolution of the Euclidean norm of the residual on the interface. As expected conjugate gradient is faster than other acceleration techniques. Figure 4b presents the evolution of the error measured by the Mises stress on the most loaded element with respect to the Reference model (which should not be available in production cases). We observe an important practical difficulty: Abaqus’ truncation of Gauss point data makes it impossible to observe convergence beyond a relative precision of \(10^{6}\). This problem would appear much later on the residual which only involves nodal computations (which can be manipulated in double precision). Note that this problem was encountered in other studies based on Abaqus software [37] but not in studies which used Code_Aster for example [41].
CPU time for various methods in the full linear case
Method  Stat.  SR1  Aitken  CG 

CPU time  2.55  2.11  1.57  1.47 
Nonlinear models
In that case, all models are granted the same nonlinear elastoviscoplastic behavior. As a consequence, plasticity may spread in the Complement zone. In that case the Fine and Auxiliary models only differ by their topology and their mesh.
Before comparing the acceleration techniques, we specifically study the choice of the parameters of the nonlinear conjugate gradient. Regarding the line search, the stationary iteration corresponds to \(\alpha =1\) (at least at the first iteration), and it behaves rather well (thanks to \(\mathbf {S}{}^A\simeq \mathbf {S}{}^F\)), so it is not absurd to take samples near 1. Of course the size of the sampled interval is also a question of experience for a given class of problems. In the studied case, prior experiments showed that, the optimal linesearch always belonged to the interval [.8, 1.4], we thus use either 4 sampling points \(\{.8,1.,1.2,1.4\}\), 9 sampling points \(\{.8,1.,1.1,1.15,1.2,1.25,1.3,1.35,1.4\}\) or 13 sampling points \(\{.8,1.,1.1,1.15,1.2, 1.22, 1.24, 1.25, 1.26, 1.28,1.3,1.35,1.4\}\).
Overlapping version
In previous sections, we had assumed that the interface was described as the boundary of elements for all models. In practice this hypothesis is not so restrictive because most often the zone of interest is detected after an initial computation on the coarse Global model, and it is constituted as a set of coarse elements satisfying a certain criterion. Even after remeshing, the boundary of the Fine description of the zone of interest matches a set of coarse faces (edges in 2D). Then a “simple” transfer matrix \(\mathbf {T}\) can be sufficient to communicate between models on the interface (25). In particular, the easy choice of \(\mathbf {T}\) being the interpolation matrix of the coarse kinematics in the fine kinematics can be implemented in most software. More evolved choices like mortar connections can also be employed in certain software [12].
We propose an alternative strategy which makes use of the possibility to have the models overlap. In that case, there is no restriction on the definition of the meshes. This idea can directly be connected to overlapping optimized Schwarz methods, yet we propose a mechanical interpretation of it.
Note that the use of the overlap can be advantageous in the situations where edge effects may affect the Fine model, even if meshes are conforming at the interfaces [34].
Handling of incompatible patches
The starting point is the observation that the method can be formulated as the search for \(\mathbf {p}\) which is the stress discontinuity on the Global model between the Complement zone and the Auxiliary description of the zone of interest. This discontinuity must be such that the Complement zone is in equilibrium with the Fine description of zone of interest loaded with Dirichlet conditions (29).
We thus propose to follow Fig. 8. The Fine subdomain \(\Omega {}^F\) is positioned where needed in the zone of interest, its mesh is independent from the coarse mesh. We note \(\Gamma {}^F=\partial \Omega {}^F\) the boundary of the Fine subdomain. The Auxiliary subdomain is the largest set of coarse elements fully contained in the zone of interest. We note \(\Gamma {}^A=\partial \Omega {}^A\) the boundary of the Auxiliary zone. The two interfaces \(\Gamma {}^F\) and \(\Gamma {}^A\) thus do not coincide. \(\Omega {}^C\) is defined as the Complement to \(\Omega {}^A\) in the Global model \(\Omega {}^C=\Omega {}^G\setminus \Omega {}^A\).
The Fine and Global models’ displacements are connected on \(\Gamma {}^F\); \(\mathbf {p}\) is applied to the Global model on \(\Gamma {}^A\) with the aim to reach balance between the nodal reaction from Complement model and the normal from the Fine model projected on \(\Gamma {}^A\). In the end, the coupled solution is \(u{}^F\) in \(\Omega {}^A\) and \(u{}^G\) in \(\Omega {}^G\setminus \Omega {}^F\). In the overlap, also called buffer zone \(\Omega {}^B=\Omega {}^F\setminus \Omega {}^A\), the Complement and the Fine model coexist.
Algorithm 5 gives the basic stationary iteration in the presence of overlap, all acceleration techniques could be considered. In order to distinguish between the interfaces, the Auxiliary and the Fine problems are written on separate lines even if they can be solved in parallel.
The main practical difficulty of this algorithm is the computation of the Fine reaction on \(\Gamma {}^A\) with \(\Omega {}^F\) not exactly represented on the coarse grid. This computation mixes the Fine stress \(\sigma _h{}^F\) and the coarse shape functions \(\phi {}^G_i\). Even if complex, this computation is feasible in certain software. Anyhow in the nonlinear case, \(\sigma _h{}^F\) is only known at Fine Gauss points and the integral can only be approximated.
Note that the conceptual difficulty caused by the coexistence of two models in the overlap is common to other coupling methods [42]. It seems wise not to introduce strong dissimilarities between the Fine and the Global models in the buffer zone, so that only the nonconformity of meshes lead to (hopefully only slightly) different solutions in it.
Usually enlarging the overlap leads to improved converge rate in Schwarz methods. This is not as clear in our method where the Fine and Global models may differ in the buffer zone. Since a good convergence rate is already ensured by the good approximation of the Fine model provided by the Auxiliary model, and because of the ambiguity of the model in the overlap, we see no interest in considering large overlaps (unless it also plays a role from a mechanical point of view, see the discussion on the decay of edge effects below).
Illustration of the coupling with overlap
For now, the version with overlap has only been implemented in the context of the coupling between a Global heterogeneous plate model and a Fine 3D model, for the study of bolted assemblies of thin structures [43], in Code_aster.
As said earlier, in the case of nonmatching meshes, the difficulty for the coupling with overlap is the computation of the fine reaction on \(\Gamma ^A\) written \(\varvec{\lambda }{}^F_{j,i}\) in Algorithm 5. In [43], it was proposed to extract a band of Auxiliary elements connected to \(\Gamma ^A\) and project on it the Fine stress (defined at the Gauss point of the Fine mesh). This was implemented in Code_Aster using existing routines (PROJ_CHAMP() with keyword ECLA_PG).
Conclusion
The global/local noninvasive coupling technique is a convenient way to enrich a global coarse model, handled by a commercial software, with local features, handled by the most adapted software. In this paper we proposed to interpret the method as an alternate optimized nonoverlapping Schwarz domain decomposition method. In this framework the coarse representation of the zone of interest is a clever way to build an approximation of the DirichlettoNeumann operator of the Fine model, which includes the effects of the imposed load. Belonging to the Schwarz family of domain decomposition method allows to benefit many theoretical results and practical shrewdness. We then derive a conjugate gradient solver in the linear and nonlinear cases, in that later case the line search is realized by a sampling which can be conducted in parallel in order not to penalize the wallclock time. Finally we show that an overlapping version can also be applied which enables to connect nonmatching meshes.
Declarations
Author's contributions
This work on the method initiated by OA is in the line of previous contributions by OA and PG. PG proposed the Schwarz’ interpretation. All authors contributed to the study of the acceleration methods. MB implemented the method and conducted the numerical experiments in the absence of overlap. Numerical experiments in the presence of overlap are extracted from the Ph.D. of GG. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Availability of data and materials
Not applicable.
Consent for publication
Not applicable.
Ethics approval and consent to participate
Not applicable.
Funding
This work was partially funded by the French National Research Agency as part of project ICARE (ANR12MONU000204).
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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