The effect of the mesh refinement on a multiscale domain decomposition method for the non-linear simulation of composite structures

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==The effect of the mesh refinement on a multiscale domain decomposition method for the non-linear simulation of composite structures==

'''Jorge Fernández<sup>a,b</sup>, Karin Saavedra<sup>b</sup>, Jorge Hinojosa<sup>b</sup>, Paulo Flores<sup>c</sup>'''

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|<sup>a</sup> Dpt. Ingeniería Mecánica, Universidad del Bío-Bío, Av. Collao 1202, Concepción, Chile 
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|<sup>b</sup> Dpt. Tecnologías Industriales, Universidad de Talca, Los Niches km 1, Curicó, Chile
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|<sup>c</sup> Dpt. Ingeniería Mecánica, Universidad de Concepción,  Edmundo Larenas 270, Concepción, Chile
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==Abstract==

This article is focused on the study of a micro-macro LaTIn based Domain Decomposition Method (LaTIn-DDM) for the prediction of the nonlinear behavior of slender composite structures subjected to bending, buckling and delamination. Previous studies have shown that an adequate selection of the iterative parameters (search directions and macroscopic space) allow to improve the convergence rate and ensure scalability (i.e. number of iterations is independent of the number of subdomains) of the iterative schema. To obtain precise solutions, only the size reduction of the subdomains' discretization has been addressed (<math display="inline">h</math>-refinement), disregarding the option of increasing the polynomial degree of the finite elements (<math display="inline">p</math>-refinement) and ignoring their underlying effects on the information's transmission through the interfaces between subdomains. In this work and using linear and quadratic finite elements, <math display="inline">h</math> and <math display="inline">p</math> refinements on the subdomains and local <math display="inline">h</math>-refinement only along the edges of the subdomains were investigated. It is conclude that the <math display="inline">p</math>-refinement in the whole subdomain not only enables to reach more exact solutions than using global or local <math display="inline">h</math>-refinement, but also the convergence rate is improved. These enhancements allow more complex simulations but using less degrees of freedom and less calculation time, even up to 97% faster.

'''Keywords''': Domain decomposition method, quadratic finite elements, composites, delamination, buckling

==1. Introduction==

Since the middle of the last century, composite materials have been widely used in several industrial applications, showing advantages over materials such as steel and aluminum due to their specific properties. Furthermore, scientists and engineers have made efforts to understand their behavior and to predict them <span id='citeF-1'></span><span id='citeF-2'></span>[[#cite-1|[1,2]]]. The usage of models which are defined at micro-length scales would be ideal, but numerical complexity and computational limitations (memory and time) appear when simulations are performed <span id='citeF-3'></span>[[#cite-3|[3]]]. Instead, Domain Decomposition Methods (DDM) <span id='citeF-4'></span><span id='citeF-5'></span><span id='citeF-6'></span>[[#cite-4|[4,5,6]]] are suitable to face these issues taking advantage of the power of parallel and distributed computations (high performance computing). By partitioning the structure into smaller subdomains connected by interfaces, these algorithms lead with local problems defined in each subdomain and a condensed interface problem. Then computational limitations are overcome because they are numerically cheaper and adapted to the parallel architecture of hardwares. The scalability of these methods (i.e. convergence rate does not depends on the number of subdomains) is often managed using a coarse problem ensuring a global communication between subdomains (i.e. a second calculation's scale) <span id='citeF-7'></span><span id='citeF-8'></span><span id='citeF-9'></span>[[#cite-7|[7,8,9]]].

The LaTIn method <span id='citeF-10'></span>[[#cite-10|[10]]] is in principle a non-incremental schema to solve non-linear problems; however its extension as a multiscale mixed DDM has been easily done <span id='citeF-11'></span>[[#cite-11|[11]]]. Contact <span id='citeF-12'></span>[[#cite-12|[12]]], crack propagation <span id='citeF-13'></span><span id='citeF-14'></span><span id='citeF-15'></span>[[#cite-13|[13,14,15]]], buckling-delamination interaction <span id='citeF-16'></span><span id='citeF-17'></span>[[#cite-16|[16,17]]] and heterogeneous structures <span id='citeF-11'></span>[[#cite-11|[11]]] are among the different nonlinear and complex problems solved with this method. It distinguishes the no-linear equations defined on the subdomains from the non-linear ones defined on the interfaces, in order to define two groups of partial solutions over which a fixed point is applied to reach the intersection of them at convergence. This algorithm is configured with two search directions linking the force to the displacement distributions on the interfaces (i.e. mixed or Robin conditions). The LaTIn method aims to improve its convergence rate by introducing a second space scale at the interface level (classically called the “macroscopic problem” in contrast to the local problems solved in each subdomain which are called “microscopic problems”). For this reason, the global equilibrium over the structure is enforced in a few interface's unknowns by defining a macroscopic basis of the interface's displacements. Additionally, the Robin conditions need to be optimized (large-wavelength components converge rapidly whereas small-wavelength components converge slowly) as shown in <span id='citeF-18'></span><span id='citeF-19'></span><span id='citeF-20'></span>[[#cite-18|[18,19,20]]].

Delamination is one of the main degradation mechanisms of laminated composite materials. This phenomenon is generally initiated by large interlaminar stresses and can be accelerated under geometric instabilities as buckling, leading eventually to a structural failure. For simulating the buckling-delamination interaction in composite laminates, the work of <span id='citeF-18'></span>[[#cite-18|[18]]] uses the mesoscopic scale defining two constituents: the plies (3D elements) and the interfaces (2D elements), which are naturally linked to the domain partitioning. The geometrically nonlinear evolution is handled through a total Lagrangian formulation as proposed by <span id='citeF-16'></span>[[#cite-16|[16]]], while delamination is modeled using a Cohesive Zone Model (CZM) which are based on damage mechanics <span id='citeF-21'></span>[[#cite-21|[21]]]. Conditions of unilateral contact are considered to avoid interpenetration between delaminated surfaces.

The simulations previously carried out need a large amount of degrees of freedom (dof) to correctly capture the different local and global phenomena. Until now, the strategy has privileged to use sufficiently refined meshes (<math display="inline">h</math>-refinement), but this has implyied expensive computations. The other classical technique to reach more exact solutions rather than dividing elements into smaller ones is to increase the polynomial degree of the finite element approximation (<math display="inline">p</math>-refinement), as proposed in <span id='citeF-22'></span><span id='citeF-23'></span>[[#cite-22|[22,23]]] for problems using direct solvers (without parallel computations). Therefore, this work studies the influence of the <math display="inline">p</math>-refinement not only on the accuracy of the results, but also on the iterative solver (LaTIn-DDM). The numerical implementation is made using the C++ research code MULTI developed at the Laboratoire de Mécanique et Technologies de Cachan<span id="fnc-1"></span>[[#fn-1|<sup>1</sup>]] using MPI and METIS libraries for the parallel assignments.

This work proposes to use second order finite elements in the LaTIn-DDM and to study their effects on the convergence rate and on the calculation time with respect to the LaTIn-DDM previously defined in <span id='citeF-18'></span>[[#cite-18|[18]]]. To achieve this goal, Section [[#2 The micro-macro LaTIn based Domain Decomposition Method|2]] shows the general aspects of the multiscale strategy, then the reference problem, the domain partitioning, governing equations and the multiscale iterative algorithm are detailed. Subsequently, in Section [[#3 Influence of the discretization on the LaTIn-DDM|3]], different <math display="inline">h</math>- and <math display="inline">p</math>-refinements are compared in 3D beam problems: bending, buckling and delamination. Finally, the conclusions and ongoing work are presented in Section [[#4 Conclusions|4]].

<span id="fn-1"></span>
<span style="text-align: center; font-size: 75%;">([[#fnc-1|<sup>1</sup>]]) LMT-Cachan (ENS Paris-Saclay/CNRS/Université Paris-Saclay)</span>

==2. The micro-macro LaTIn based Domain Decomposition Method==

Let us consider a laminate composite (see Figure [[#img-1|1]]) occupying the domain <math display="inline">\Omega </math> bounded by <math display="inline">\partial \Omega </math> in the current configuration, and consisting of <math display="inline">N_P</math> plies. Each ply <math display="inline">P</math> is connected to an adjacent ply <math display="inline">P' </math> through the interface <math display="inline">{\Gamma _{PP'}}</math>. The structure is subjected to an external surface traction field <math display="inline">{\underline{F}_d}</math> on the part <math display="inline">{\partial \Omega _{F_d}}</math> and to a displacement field <math display="inline">{\underline{U}_d}</math> on the complementary part <math display="inline">{\partial \Omega _{U_d}}</math>. The body force per unit of mass is written <math display="inline">{\underline{f}_d}</math>. The relevant quantities are described in reference to the undeformed configuration using the index <math display="inline">\cdot _\mathit{0}</math>. The geometrically nonlinear evolution is handled through a total Lagrangian formulation and delamination (damageable interfaces) is modeled using CZM and unilateral contact conditions. For the sake of simplicity, an extensive description of the CZM is found in <span id='citeF-24'></span>[[#cite-24|[24]]], while <span id='citeF-25'></span>[[#cite-25|[25]]] describe contact inequalities.

To propose the partitioned problem, the whole domain <math display="inline">\Omega </math> is split into subdomains which are connected by interfaces with mechanical behaviors. Two possibilities are considered: “material” interfaces between plies with localized non-linearities (damage, contact) that are compatible with the mesomodeling, and “numerical” interfaces (the perfect ones) within the plies to conceive smaller problems that are suited for parallelism, as schematized in Figure [[#img-1|1]]. A subdomain <math display="inline">{S^{\quad }_\mathit{0}}</math> defined in the undeformed domain <math display="inline">{\Omega _{S_\mathit{0}}}</math> is connected to an adjacent undeformed subdomain <math display="inline">{{S'}_\mathit{0}}</math> through an undeformed interface <math display="inline">{\Gamma _{S_\mathit{0}{S'}_\mathit{0}}}=\partial {\Omega _{S_\mathit{0}}}\cap \partial {\Omega _{S_\mathit{0}'}}</math>. The surface entity <math display="inline">{\Gamma _{S_\mathit{0}{S'}_\mathit{0}}}</math> applies force distributions <math display="inline">{\underline{F}_{{S^{\quad }_\mathit{0}}}}</math>, <math display="inline">{\underline{F}_{S_\mathit{0}'}}</math> and displacement distributions <math display="inline">{\underline{W}_{{S^{\quad }_\mathit{0}}}}</math>, <math display="inline">{\underline{W}_{{S'}_\mathit{0}}}</math> to <math display="inline">{S^{\quad }_\mathit{0}}</math> and <math display="inline">{{S'}_\mathit{0}}</math> respectively (see Figure [[#img-2|2]]). Let us define <math display="inline">{\Gamma _{S_\mathit{0}}}= \cup _{{{S'}_\mathit{0}}\in {\mathbf{E}}} {\Gamma _{S_\mathit{0}{S'}_\mathit{0}}}</math>. For a subdomain <math display="inline">{S^{\quad }_\mathit{0}}</math> such that <math display="inline">{\Gamma _{S_\mathit{0}}}\cap ({\partial \Omega _{F_{d_\mathit{0}}}}\cup {\partial \Omega _{U_{d_\mathit{0}}}}) \neq \emptyset </math>, the boundary condition <math display="inline">({\underline{F}_{d_\mathit{0}}},{\underline{U}_{d_\mathit{0}}})</math> is applied through a boundary interface <math display="inline">{\Gamma _{{S}_{d_\mathit{0}}}}</math>.


<div id='img-1'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
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|[[Image:Draft_Saavedra_368043899-reference_pb.png|600px|The reference problem, the mesomodel and its partitioning <span id='citeF-18'></span>[[#cite-18|[18]]]]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 1:''' The reference problem, the mesomodel and its partitioning <span id='citeF-18'></span>[[#cite-18|[18]]]
|}

<div id='img-2'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
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|[[Image:Draft_Saavedra_368043899-sst_v2.png|540px|Subdomains and interfaces <span id='citeF-16'></span>[[#cite-16|[16]]]]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 2:''' Subdomains and interfaces <span id='citeF-16'></span>[[#cite-16|[16]]]
|}


The purpose of the method is to find the subdomain fields <math display="inline">{\underline{u}_{{S^{\quad }_\mathit{0}}}}</math> (displacement) and <math display="inline">{\underline{\underline{\pi}}_{{S^{\quad }_\mathit{0}}}}</math> (second Piola-Kirchhoff stress), and the interface fields <math display="inline">{\underline{W}_{{S^{\quad }_\mathit{0}}}}</math> (displacement) and <math display="inline">{\underline{F}_{{S^{\quad }_\mathit{0}}}}</math> (inter-forces), where the index <math display="inline">\cdot _{S_\mathit{0}}</math> ranges over all subdomains. After partitioning, the governing equations are separated into two groups:

1. Non-linear equations in subdomains and macroscopic admissibility of interfaces, whose solutions are elements of the manifold <math display="inline">\mathbf{A_d}</math>:

* non-linear kinematic admissibility of the subdomains:

<span id="eq-1"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: left; margin:auto;width: 100%;" 
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| style="text-align: center;" | <math>

{\underline{\underline{E}}_{S_\mathit{0}}}= \frac{1}{2}\left({{\nabla }}{\underline{u}_{{S^{\quad }_\mathit{0}}}}+ {}^{t}{{\nabla }}{\underline{u}_{{S^{\quad }_\mathit{0}}}}+ {{\nabla }}{\underline{u}_{{S^{\quad }_\mathit{0}}}}{}^{t}{{\nabla }}{\underline{u}_{{S^{\quad }_\mathit{0}}}}\right)  , \hbox{on}  {\Omega _{S_\mathit{0}}} </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (1)
|}

{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: left; margin:auto;width: 100%;" 
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| style="text-align: center;" | <math>

\underline{u} _{S_{0}|\partial \Omega _{S_0}}= \underline{W} _{S_{0}|\Gamma _{S_0}} \hbox{, on } \Gamma _{{S_0 S'}_0} </math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (2)
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* non-linear static admissibility of the subdomains:

<span id="eq-3"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: left; margin:auto;width: 100%;" 
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| style="text-align: center;" | <math>

\forall ({\underline{u}^{\star }_{S_\mathit{0}}},{{\underline{W}^{\star }_{S_\mathit{0}}}}) \in {\mathcal{U}_{S_\mathit{0}}^0}\times {\mathcal{W}_{S_\mathit{0}}^0}\quad \mathit{such\; that } \quad {\underline{u}^{\star }_{S_\mathit{0}}}_{| {\partial \Omega _{S_\mathit{0}}}} = {{\underline{W}^{\star }_{S_\mathit{0}}}}_{| {\Gamma _{S_\mathit{0}}}},  </math>
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| style="text-align: center;" | <math> \int _{\Omega _{S_\mathit{0}}} {\underline{\underline{\pi }}_{{S^{\quad }_\mathit{0}}}}: {\underline{\underline{\dot E}}(\underline{u}^{\star }_{{S^{\quad }_\mathit{0}}})} d{\Omega _\mathit{0}} = \int _{\Omega _{S_\mathit{0}}}\rho _{{S^{\quad }_\mathit{0}}} \; {\underline{f}_d}\cdot {\underline{u}^{\star }_{S_\mathit{0}}}\, d {\Omega _\mathit{0}}+ \int _{{\Gamma _{S_\mathit{0}}}} {\underline{F}_{{S^{\quad }_\mathit{0}}}}\cdot {{\underline{W}^{\star }_{S_\mathit{0}}}}\, d {\Gamma _\mathit{0}} </math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (3)
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where <math display="inline">{\underline{\underline{\dot E}}(\underline{u}^{\star }_{{S^{\quad }_\mathit{0}}})}=\frac{1}{2}({{\nabla }}{\underline{u}^{\star }_{S_\mathit{0}}}+ {}^{t}{{\nabla }}{\underline{u}^{\star }_{S_\mathit{0}}}+ {}^{t}{{\nabla }}{\underline{u}_{{S^{\quad }_\mathit{0}}}}{{\nabla }}{\underline{u}^{\star }_{S_\mathit{0}}}+ {}^{t}{{\nabla }}{\underline{u}^{\star }_{S_\mathit{0}}}{{\nabla }}{\underline{u}_{{S^{\quad }_\mathit{0}}}})</math>.
* behavior of the subdomains:

<span id="eq-4"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: left; margin:auto;width: 100%;" 
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| style="text-align: center;" | <math>

{\underline{\underline{\pi }}_{{S^{\quad }_\mathit{0}}}}= \frac{\partial \psi }{\partial {\underline{\underline{E}}_{S_\mathit{0}}}} \;,\; \hbox{on} \; {\Omega _{S_\mathit{0}}}\;,\; </math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (4)
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where <math display="inline">\psi </math> is the stored energy function per unit of undeformed volume. For this work, <math display="inline">\psi =\frac{1}{2} \mathbf{K}_{S_\mathit{0}}{\underline{\underline{E}}_{S_\mathit{0}}}:{\underline{\underline{E}}_{S_\mathit{0}}}</math> has been used.
* macroscopic admissibility of the interfaces (after the linearization of the previous equations), which is a partial verification of the action-reaction principle:

<span id="eq-5"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: left; margin:auto;width: 100%;" 
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| style="text-align: center;" | <math>

\forall {{\underline{W}^M_{{S^{\quad }_\mathit{0}}}}}^\star \in {\mathcal{W}_{{S^{\quad }_\mathit{0}}{{S'}_\mathit{0}}}^M},  \quad \int _{{\Gamma _{S_\mathit{0}{S'}_\mathit{0}}}}  ({\underline{F}_{{S^{\quad }_\mathit{0}}}}+ {\underline{F}_{S_\mathit{0}'}})\cdot{{\underline{W}^M_{{S^{\quad }_\mathit{0}}}}}^\star \, d {\Gamma _\mathit{0}}= 0 \, , </math>
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| style="width: 5px;text-align: right;white-space: nowrap;" | (5)
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where the subspace <math display="inline">{\mathcal{W}_{{S^{\quad }_\mathit{0}}{{S'}_\mathit{0}}}^M}</math> of interface macroscopic admissible displacements is a parameter described in Section [[#2.1 Separation of the micro-macro scales|2.1]].

2. Local (non-linear) equations in the interfaces whose solutions belong to the manifold <math display="inline">{\mathbf{L}_\Gamma}</math>:

* constitutive equation <math display="inline">\displaystyle {\mathcal{R}}_{{S^{\quad }_\mathit{0}}{{S'}_\mathit{0}}} (   [{\underline{W}_{{S^{\quad }_\mathit{0}}}}] \, , \, {\underline{F}_{{S^{\quad }_\mathit{0}}}}\, , \, {\underline{F}_{S_\mathit{0}'}}) = \underline{0} \; \hbox{over} \; {\Gamma _{S_\mathit{0}{S'}_\mathit{0}}}\in {\Gamma _{S_\mathit{0}}}\;,</math> where <math display="inline">[{\underline{W}_{{S^{\quad }_\mathit{0}}}}] ={\underline{W}_{{S'}_\mathit{0}}}-{\underline{W}_{{S^{\quad }_\mathit{0}}}}</math> is the interface displacement gap.
* boundary behavior <math display="inline">\mathcal{R}_{S_{d_\mathit{0}}}( {\underline{W}_{{S^{\quad }_\mathit{0}}}}, {\underline{F}_{{S^{\quad }_\mathit{0}}}}) = \underline{0} \;  \hbox{over the boundary} \; {\Gamma _{{S}_{d_\mathit{0}}}}\;</math>. The relation <math display="inline">\mathcal{R}_{{S^{\quad }_\mathit{0}}{{S'}_\mathit{0}}}=\underline{0}</math> can be made explicit for:

- Perfect interface: <span id="eq-6"></span>
<math display="inline"> \left\{\begin{array}{l} {\underline{F}_{{S^{\quad }_\mathit{0}}}}+ {\underline{F}_{S_\mathit{0}'}}= \underline{0} \\[] [{\underline{W}_{{S^{\quad }_\mathit{0}}}}]  = \underline{0} \end{array} \right.  </math>

- Cohesive interface: <span id="eq-6"></span>
<math display="inline"> \left\{\begin{array}{l} {\underline{F}_{{S^{\quad }_\mathit{0}}}}+ {\underline{F}_{S_\mathit{0}'}}= \underline{0} \\ \displaystyle \mathcal{A}_{PP'}([{\underline{W}_{{S^{\quad }_\mathit{0}}}}] \, ,\, {\underline{F}_{{S^{\quad }_\mathit{0}}}}) = \underline{0} \end{array} \right.  </math>

- Unilateral contact interface: <span id="eq-6"></span>
<math display="inline"> \left\{\begin{array}{l} {\underline{F}_{{S^{\quad }_\mathit{0}}}}+ {\underline{F}_{S_\mathit{0}'}}= \underline{0} \\ {\underline{n}}\cdot [{\underline{W}_{{S^{\quad }_\mathit{0}}}}] \geq 0 \; \; \mathbf{or} \;\; {\underline{n}}\cdot {\underline{F}_{{S^{\quad }_\mathit{0}}}}\geq 0  \\ ({\underline{n}}\cdot [{\underline{W}_{{S^{\quad }_\mathit{0}}}}] )({\underline{n}}\cdot {\underline{F}_{{S^{\quad }_\mathit{0}}}}) = 0 \\ \mathbf{P} {\underline{F}_{{S^{\quad }_\mathit{0}}}}= \mathbf{P} {\underline{F}_{S_\mathit{0}'}}= \underline{0} \\ \end{array} \right.  </math>

where <math display="inline">{\underline{n}}</math> is the unit normal to <math display="inline">\Gamma _{SS'}</math> and <math display="inline">\mathbf{P}</math> is the corresponding tangential projection operator.

The algorithm consists in seeking the interface solution <math display="inline">s_{ref}</math> alternatively in these two spaces: first, one finds a solution <math display="inline">s = (s_{S^{\quad }_\mathit{0}})_{{S^{\quad }_\mathit{0}}\in {\mathbf{E}}} = ({\underline{W}_{{S^{\quad }_\mathit{0}}}}, {\underline{F}_{{S^{\quad }_\mathit{0}}}})_{{S^{\quad }_\mathit{0}}\in {\mathbf{E}}}</math> in <math display="inline">\mathbf{A_d}</math>, then a solution <math display="inline">\widehat{s} = (\widehat{s}_{S^{\quad }_\mathit{0}})_{{S^{\quad }_\mathit{0}}\in {\mathbf{E}}} = ({\underline{\widehat{W}}_{{S^{\quad }_\mathit{0}}}}, {\underline{\widehat{F}}_{{S^{\quad }_\mathit{0}}}})_{{S^{\quad }_\mathit{0}}\in {\mathbf{E}}}</math> in <math display="inline">{\mathbf{L}_\Gamma }</math>. In order for the two problems to be well-posed, two search directions <math display="inline">{{k^+}}</math> and <math display="inline">{{k^-}}</math>, which link the solutions <math display="inline">s</math> and <math display="inline">\widehat{s}</math> during the iterative process, are introduced. The converged interface solution <math display="inline">s_{ref}</math> is such that <math display="inline">s_{ref} \in \mathbf{A_d}\cap {\mathbf{L}_\Gamma }</math>. A complete description of this iterative method can be found in&nbsp;<span id='citeF-16'></span>[[#cite-16|[16]]].

In order to evaluate the convergence, a criterion based on the verification of the constitutive laws of the interfaces quantities issued from the admissibility stage has been implemented, as proposed by <span id='citeF-26'></span>[[#cite-26|[26]]].

===2.1 Separation of the micro-macro scales===

The definition of the macroscopic quantities is done through an orthogonal projector <math display="inline">\Pi </math> for each interface <math display="inline">{\Gamma _{S_\mathit{0}{S'}_\mathit{0}}}</math>, such as the macroscopic fields are <math display="inline">{\underline{F}^M_{{S^{\quad }_\mathit{0}}}}~=~\Pi  {\underline{F}_{{S^{\quad }_\mathit{0}}}}</math> and <math display="inline">{\underline{W}_{{S^{\quad }_\mathit{0}}}^M}~=~\Pi  {\underline{W}_{{S^{\quad }_\mathit{0}}}}</math>. Consequently, the microscopic spaces, <math display="inline">{\mathcal{W}_{{S^{\quad }_\mathit{0}}{{S'}_\mathit{0}}}^m}</math> and <math display="inline">{\mathcal{F}_{{S^{\quad }_\mathit{0}}{{S'}_\mathit{0}}}^m}</math>, are orthogonal to the macroscopic spaces, <math display="inline">{\mathcal{W}_{{S^{\quad }_\mathit{0}}{{S'}_\mathit{0}}}^M}</math> and <math display="inline">{\mathcal{F}_{{S^{\quad }_\mathit{0}}{{S'}_\mathit{0}}}^M}</math>, with respect to the inner product <math display="inline">L^2({\Gamma _{S_\mathit{0}{S'}_\mathit{0}}})</math>, so the scales can be uncoupled with respect to the interface virtual work as follows:

{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: left; margin:auto;width: 100%;" 
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| style="text-align: center;" | <math>\begin{align}&{\underline{F}^M_{{S^{\quad }_\mathit{0}}}}=  \Pi  {\underline{F}_{{S^{\quad }_\mathit{0}}}} \quad ;\quad {\underline{W}_{{S^{\quad }_\mathit{0}}}^M}= \Pi  {\underline{W}_{{S^{\quad }_\mathit{0}}}} \\  & {\underline{F}^m_{{S^{\quad }_\mathit{0}}}}=  (\mathbf{I} - \Pi ){\underline{F}_{{S^{\quad }_\mathit{0}}}} \quad ;\quad {\underline{W}_{{S^{\quad }_\mathit{0}}}^m}= (\mathbf{I} - \Pi ){\underline{W}_{{S^{\quad }_\mathit{0}}}}   \end{align} </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (6)
|}

<span id="eq-7"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
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{| style="text-align: left; margin:auto;width: 100%;" 
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| style="text-align: center;" | <math>\begin{align}\int \limits _{{\Gamma _{S_\mathit{0}{S'}_\mathit{0}}}} {\underline{F}_{{S^{\quad }_\mathit{0}}}}\cdot {\underline{W}_{{S^{\quad }_\mathit{0}}}}\, d {\Gamma _\mathit{0}} & = \int \limits _{{\Gamma _{S_\mathit{0}{S'}_\mathit{0}}}} ({\underline{F}^M_{{S^{\quad }_\mathit{0}}}}+{\underline{F}^m_{{S^{\quad }_\mathit{0}}}})\cdot ({\underline{W}_{{S^{\quad }_\mathit{0}}}^M}+{\underline{W}_{{S^{\quad }_\mathit{0}}}^m}) \, d {\Gamma _\mathit{0}} \\  & = \int \limits _{{\Gamma _{S_\mathit{0}{S'}_\mathit{0}}}} (\Pi  {\underline{F}_{{S^{\quad }_\mathit{0}}}}+(\mathbf{I} - \Pi ){\underline{F}_{{S^{\quad }_\mathit{0}}}})\cdot (\Pi  {\underline{W}_{{S^{\quad }_\mathit{0}}}}+(\mathbf{I} - \Pi ){\underline{W}_{{S^{\quad }_\mathit{0}}}}) \, d {\Gamma _\mathit{0}} \\ &=  \int \limits _{{\Gamma _{S_\mathit{0}{S'}_\mathit{0}}}} {\underline{F}^M_{{S^{\quad }_\mathit{0}}}}\cdot {\underline{W}_{{S^{\quad }_\mathit{0}}}^M}\, d {\Gamma _\mathit{0}}    + \int \limits _{{\Gamma _{S_\mathit{0}{S'}_\mathit{0}}}} {\underline{F}^m_{{S^{\quad }_\mathit{0}}}}\cdot {\underline{W}_{{S^{\quad }_\mathit{0}}}^m}\, d {\Gamma _\mathit{0}}   \end{align}  </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (7)
|}

In order to ensure the scalability of the iterative scheme, a global linear coarse grid problem [[#eq-5|5]] has been introduced. It is fully characterized by the set of interface macroscopic spaces <math display="inline">({\mathcal{W}_{{S^{\quad }_\mathit{0}}{{S'}_\mathit{0}}}^M})</math> which is a parameter of the method as well as its dual <math display="inline">({\mathcal{F}_{{S^{\quad }_\mathit{0}}{{S'}_\mathit{0}}}^M})</math>. Usually, a common macroscopic basis for both the traction and displacement macroscopic fields is chosen so that the uniqueness of the micro-macro descomposition is ensured. Classically, the macroscopic space contain at least the affine part of the interface displacements; this corresponds to verify the balance of the first moments of forces at the interface. However, if complex structures are involved, the geometry and its partitioning degrade the convergence rate. To palliate this problem, <span id='citeF-18'></span>[[#cite-18|[18]]] propose to find an optimized search direction instead to enrich the macroscopic space.

In this paper, simulations have been carried out using the standard linear macroscopic space and the local search direction previously used in&nbsp;<span id='citeF-16'></span>[[#cite-16|[16]]].

===2.2 Discretization===

To solve both equations' sets of the multiscale algorithm, interfaces and subdomains are discretized in space using classical finite elements. At an interface <math display="inline">{\Gamma _{S_\mathit{0}{S'}_\mathit{0}}}</math>, the displacements <math display="inline">{\underline{W}_{{S^{\quad }_\mathit{0}}}}</math> and forces <math display="inline">{\underline{F}_{{S^{\quad }_\mathit{0}}}}</math> belong to the approximation spaces <math display="inline">{\mathcal{W}_{{S^{\quad }_\mathit{0}}{{S'}_\mathit{0}},h}}</math> and <math display="inline">{\mathcal{F}_{{S^{\quad }_\mathit{0}}{{S'}_\mathit{0}},h}}</math>. These spaces are chosen such that the bilinear form (in the sense of the interface mechanical work) is non-degenerate. Additionally, a wrong discretization for <math display="inline">{\mathcal{F}_{{S^{\quad }_\mathit{0}}{{S'}_\mathit{0}},h}}</math> could generate spurious oscillating modes leading to numerical instability. These inconveniences can be evaded using a common space for the displacements and forces and including a local refinement of the mesh near the boundary (over-discretization) of the subdomains (approximation space <math display="inline">{\mathcal{U}_{S_\mathit{0}}}</math>). Two manners are possible: to increase the number of elements (<math display="inline">h</math>-version) or to use a higher degree of approximation (<math display="inline">p</math>-version) for the field <math display="inline">{\underline{u}_{{S^{\quad }_\mathit{0}}}}</math> near to the interface, as illustrated in Figure [[#img-3|3]] <span id='citeF-27'></span>[[#cite-27|[27]]]. Classically, the code MULTI has considered linear elements <math display="inline">\mathcal{P}_1</math> with local <math display="inline">h</math>-refinement along the subdomain's boundary, while the spaces <math display="inline">{\mathcal{W}_{{S^{\quad }_\mathit{0}}{{S'}_\mathit{0}},h}}</math> and <math display="inline">{\mathcal{F}_{{S^{\quad }_\mathit{0}}{{S'}_\mathit{0}},h}}</math> are piecewise constant functions <math display="inline">\mathcal{P}_0</math>.

In this work, the <math display="inline">p</math>-refinement is also explored. Indeed, it is here proposed to use the second order approximation for <math display="inline">{\underline{u}_{{S^{\quad }_\mathit{0}}}}</math> not only in the boundary but in the whole subdomain.

<div id='img-3'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Review_265042547183_2078_Fig3.png|600px|Modification of the classical approximations of the inter-force and local displacement along the edge of a subdomain: h-and p-versions <span id='citeF-13'></span>[[#cite-13|[13]]]]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 3:''' Modification of the classical approximations of the inter-force and local displacement along the edge of a subdomain: <math>h</math>-and <math>p</math>-versions <span id='citeF-13'></span>[[#cite-13|[13]]]
|}

==3. Influence of the discretization on the LaTIn-DDM==

This section analyses the influence of the subdomains' discretization for three different problems: bending, buckling and delamination. The following three meshes <math display="inline">{\mathcal{U}_{S_\hbox{0}}}</math> are considered:

* <math display="inline">i</math>-version: linear six-node wedge elements of equal size in the whole subdomain (“initial” version without local over discretization, see Figure [[#img-3|3]]) ;
* <math display="inline">h</math>-version: linear six-node wedge elements, where the elements along the subdomain's boundary are divided into three four-node tetrahedral elements as shown in Figure [[#img-4|4]] (local <math display="inline">h</math> over discretization);
* <math display="inline">p</math>-version: quadratic fifteen-node wedge elements of equal size in the whole subdomain (global <math display="inline">p</math> over discretization).

For all cases, the approximation functions of the interfaces are <math display="inline">\mathcal{P}_0</math>. Displacements, convergence rate and time are used to compare the refinements.

<div id='img-4'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Review_265042547183_2934_Fig4.png|600px|before and after local h-refinement: a) wedge element b) mesh of the subdomain's boundary (the inner mesh remains unchanged)]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 4:''' before and after local <math>h</math>-refinement. (a) Wedge element. (b) Mesh of the subdomain's boundary (the inner mesh remains unchanged)
|}

===3.1 Bending===

The problem consists of a thin 3D beam whose length is <math display="inline">L=160</math> [mm] (<math display="inline">x</math>-direction), width is <math display="inline">a=120</math> [mm] (<math display="inline">y</math>-direction) and thickness is <math display="inline">h=4</math> [mm] (<math display="inline">z</math>-direction). Regarding the boundary conditions, the cantilever beam is embedded at one end and subjected to a surface load <math display="inline">\underline{F}_{d_0}=0.1</math> [N/m<math display="inline">^2</math>] on its upper face. Here, the hypothesis of small displacements is considered. First an isotropic material is studied, then an orthotropic one is analysed to study the influence of material heterogeneities.

====3.1.1 Isotropic material====

It is considered an elastic modulus <math display="inline">E=210</math> [GPa] and a poisson coefficient <math display="inline">\nu=0.3</math> [-]. The geometry is partitioned into <math display="inline">64</math> identical subdomains and <math display="inline">184</math> interfaces (see Figure [[#img-5|5]]); each subdomain has length <math display="inline">L_{sst}=20</math> [mm], width <math display="inline">a_{sst}= 15</math> [mm] and thickness <math display="inline">h_{sst}= 4</math> [mm]. Five meshes are considered: two different initial discretization with their respective <math display="inline">h</math>-refinement while the last one is a <math display="inline">p</math>-version. Each subdomain has <math display="inline">n_x</math>(<math display="inline">n_y</math>,<math display="inline">n_z</math>) elements in the <math display="inline">x</math>(<math display="inline">y</math>,<math display="inline">z</math>)-direction, respectively, as shown in Table [[#table-1|1]] as well as the total number of elements and the total degrees of freedom used for each mesh. In order to estimate the solution's error (see Table [[#table-1|1]]), the  maximum vertical displacement is compared with the theoretical elastic curve <span id='citeF-28'></span>[[#cite-28|[28]]].

<div id='img-5'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Review_265042547183_5579_Fig5.png|600px|Partitioning of the geometry (bending problem with isotropic material)]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 5:''' Partitioning of the geometry (bending problem with isotropic material)
|}


{|  class="floating_tableSCP wikitable" style="text-align: center; margin: 1em auto;min-width:50%;font-size: 85%;"
|+ <span id='table-1'></span>Table 1. Results according to the discretization (bending problem with isotropic material); '''+''' ratio respect to simulation (b.1)
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;" |  mesh
| discretization   
| <math>n_x</math>-<math display="inline">n_y</math>-<math display="inline">n_z</math> 
| total 
| total dof
| time+
| style="border-right: 2px solid;" | displ.
|-
| style="border-left: 2px solid;" | 
| / element 
| 
| elements 
| 
| 
| style="border-right: 2px solid;" | error [mm] 
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;" |  a.1) 
| <math>i</math>/ wedge 6 
| <math>10</math>-<math display="inline">8</math>-<math display="inline">2</math> 
| <math>22\,272</math>
| <math>61\,056</math>
| <math>0.046</math>
| style="border-right: 2px solid;" | <math>1.182</math>
|-
| style="border-left: 2px solid;" | a.2) 
| <math>h</math>/ wedge 6 
| <math>10</math>-<math display="inline">8</math>-<math display="inline">2</math> 
| <math>121\,088</math>
| <math>141\,696</math>
| <math>0.089</math>
| style="border-right: 2px solid;" | <math>3.438</math>
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;" |  b.1) 
| <math>i</math>/ wedge 6 
| <math>15</math>-<math display="inline">20</math>-<math display="inline">4</math> 
| <math>173\,568</math>
| <math>360\,000</math>
| <math>1</math>
| style="border-right: 2px solid;" | <math>0.607</math>
|-
| style="border-left: 2px solid;" | b.2) 
| <math>h</math>/ wedge 6 
| <math>15</math>-<math display="inline">20</math>-<math display="inline">4</math> 
| <math>574\,464</math>
| <math>674\,112</math>
| <math>2.6</math>
| style="border-right: 2px solid;" | <math>1.166</math>
|- style="border-top: 2px solid;border-bottom: 2px solid;"
| style="border-left: 2px solid;" |  c) 
| <math>p</math>/ wedge 15 
| <math>10</math>-<math display="inline">8</math>-<math display="inline">2</math> 
| <math>22\,272</math>
| <math>262\,464</math>
| <math>0.193</math>
| style="border-right: 2px solid;" | <math>0.281</math>

|}

In Figure [[#img-6|6]], theoretical and numerical displacements of the neutral line are compared. As naturally expected, when increasing the number of linear elements in the thickness, the solution's error decreases, but the dof and the computational cost increase. However, it is important to notice that for the <math display="inline">h</math>-refinements (a.2) and (b.2), the displacement's error and calculation time increase (see Table [[#table-1|1]]) while the convergence rate decrease respect to the corresponding <math display="inline">i</math>-versions (a.1) and (b.1). Even after 1000 iterations, the iterative LaTIn error for mesh (a.2) is twice the detention criteria. This phenomena could be explained by the fact that <math display="inline">h</math>-version has only a localized refinement along the edge of a subdomain, while the element's size inside the subdomain remains the same as the <math display="inline">i</math>-version. This choice could induce different stiffness through a subdomain, implying additional difficulties to transfer informationn between subdomains.

Finally, the mesh (d) (<math display="inline">p</math>-version) is twice more accurate, has 73% less dof and is 19,3% more quickly than mesh (b.1). Differences in the computation time (see Table [[#table-1|1]]) are mainly related to the mesh size, because convergence rates are similar, as shown in Figure fig:res_iso_b.

<div id='img-6a'></div>
<div id='img-6'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Review_265042547183_4713_Fig6.png|600px|]]
|- style="text-align: center; font-size: 75%;"
| colspan="2" | '''Figure 6:''' Bending problem with isotropic material. (a) Deflection. (b) Evolution of the iterative LaTIn error
|}

====3.1.2 Orthotropic material====

The precedent problem is now studied considering a composite laminate made of four plies <math display="inline">[0^o,90^o]_S</math>, each 1 [mm] a thick. A <math display="inline">0^o</math>-layer is transversely isotropic with the following elastic properties: <math display="inline">E_1=165</math> [GPa], <math display="inline">E_2=E_3=9</math> [GPa], <math display="inline">\nu _{12}=\nu _{13}=0.3</math> [-], <math display="inline">\nu _{23}=0.5</math> [-], <math display="inline">G_{12}=G_{13}=5.6</math> [GPa] and <math display="inline">G_{23}=2.8</math> [GPa]. The geometry is divided into 256 identical subdomains of <math display="inline">L_{sst}=20</math> [mm], <math display="inline">a_{sst}= 15</math> [mm] and <math display="inline">h_{sst}= 1</math> [mm], generating 736 interfaces. Therefore, for each ply there is one subdomain in the <math display="inline">z</math>-direction (four in total through the thickness). Table&nbsp;[[#table-2|2]] compares the different discretizations.


{|  class="floating_tableSCP wikitable" style="text-align: center; margin: 1em auto;min-width:50%;font-size: 85%;"
|+ <span id='table-2'></span>Table 2. Results according to the discretization (bending problem with an orthotropic material); '''+''' ratio respect to simulation (b.1)
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;" |  mesh
| disc. / element
| <math>n_x</math>-<math display="inline">n_y</math>-<math display="inline">n_z</math>
| total elements 
| total dof 
| style="border-right: 2px solid;" | time+
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;" |  a.1)
| <math>i</math>/ wedge 6 
| <math>10</math>-<math display="inline">10</math>-<math display="inline">2</math> 
| <math>113\,664</math>
| <math>574\,128</math>
| style="border-right: 2px solid;" | <math>0.402</math>
|-
| style="border-left: 2px solid;" | a.2)
| <math>h</math>/ wedge 6 
| <math>10</math>-<math display="inline">10</math>-<math display="inline">2</math> 
| <math>611\,328</math>
| <math>706\,560</math>
| style="border-right: 2px solid;" | <math>0.814</math>
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;" |  b.1)
| <math>i</math>/ wedge 6 
| <math>15</math>-<math display="inline">15</math>-<math display="inline">3</math> 
| <math>393\,216</math>
| <math>881\,664</math>
| style="border-right: 2px solid;" | <math>1</math>
|-
| style="border-left: 2px solid;" | b.2)
| <math>h</math>/ wedge 6 
| <math>15</math>-<math display="inline">15</math>-<math display="inline">3</math> 
| <math>1\,565\,696</math>
| <math>1\,806\,336</math>
| style="border-right: 2px solid;" | <math>3.707</math>
|- style="border-top: 2px solid;border-bottom: 2px solid;"
| style="border-left: 2px solid;" |  c) 
| <math>p</math>/ wedge 15 
| <math>10</math>-<math display="inline">10</math>-<math display="inline">2</math>
| <math>113\,664</math>
| <math>1\,320\,192</math>
| style="border-right: 2px solid;" | <math>0.542</math>

|}

In Figure [[#img-7|7]] is observed that the vertical displacement of the neutral axis is similar except for simulations (a.1) and (a.2). In these two cases, the iterative LaTIn error (see Figure [[#img-7|7]]) is twice the detention criteria, even after 1000 iterations. Using the <math display="inline">p</math>-version mesh (c), the LaTIn error is less than <math display="inline">10^{-5}</math> in only 68 iteraciones, less than half of the iterations made by the curve (b.1) to converge. If the upper stresses <math display="inline">\sigma _{xx}</math> are compared respect to the theoretical ones (see Figure [[#img-8|8]]), it is possible to confirm that (a.1), (a.2) and (b.2) do not fit the desired solution.

<div id='img-7a'></div>
<div id='img-7'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
| [[Image:Review_265042547183_6563_Fig7.png|600px]]
|- style="text-align: center; font-size: 75%;"
| colspan="2" | '''Figure 7:''' Bending problem with orthotropic material. (a) Deflection. (b) Evolution of the iterative LaTIn error
|}

Table [[#table-2|2]] compares calculation time, total number of elements and total dof. It is verified that using <math display="inline">p</math>-version, mesh (c), it is possible to obtain the same results as in (b.1) but consuming only 54.2% of the time, even considering that the number of dof increases in a 50%. This is explained because (c) has the best convergence rate (see Figure [[#img-8|8]]).

<div id='img-8a'></div>
<div id='img-8'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
| [[Image:Review_265042547183_4572_Fig8.png|600px]]
|- style="text-align: center; font-size: 75%;"
| colspan="2" | '''Figure 8:''' Results for the orthotropic material: normal stresses <math>\sigma _{xx}</math>
|}

Similar to the precedent example, the local <math display="inline">h</math>-version does not ensure better results. Therefore, the meshes considered for the next examples will be <math display="inline">i</math> and <math display="inline">p</math>-versions.

===3.2 Buckling===

The problem to be addressed is a slender 3D beam built-in at both ends, with one of them subjected to a negative displacement <math display="inline">\underline{u}_d</math> to produce uniaxial compression, while a perpendicular perturbation <math display="inline">\underline{F}_d</math> induces buckling out of the plane (see Figure [[#img-9|9]]). The structure has the following geometry: length <math display="inline">L=10</math> [mm] (<math display="inline">x</math>-direction), width <math display="inline">a=1</math> [mm] (<math display="inline">y</math>-direction) and thickness <math display="inline">h=0.1</math> [mm] (<math display="inline">z</math>-direction). The properties of the material are <math display="inline">E= 135</math> [GPa] and <math display="inline"> \nu = 0.3 </math> [-]. The geometry is divided into 100 subdomains and 156 perfect interfaces, therefore, each subdomain has <math display="inline">L_{sst}=0.2</math> [mm], width <math display="inline">a_{sst}= 0.5</math> [mm] and thickness <math display="inline">h_{sst}= 0.1</math> [mm]. Three meshes are considered, the first two are linear without over discretization (<math display="inline">i</math>-version), while the mesh (c) is a <math display="inline">p</math>-refinement. More details of the meshes and their results are shown in Table [[#table-3|3]].


{|  class="floating_tableSCP wikitable" style="text-align: center; margin: 1em auto;min-width:50%;font-size: 85%;"
|+ <span id='table-3'></span>Table 3. Results according to the discretization (buckling problem); '''+''' ratio respect to simulation (b)
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;" |  mesh
| disc. / element
| <math>n_x</math>-<math display="inline">n_y</math>-<math display="inline">n_z</math>
| total elements 
| total dof 
| style="border-right: 2px solid;" | time+ 
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;" |  a)
| <math>i</math>/ wedge 6 
| <math>5</math>-<math display="inline">10</math>-<math display="inline">4</math> 
| <math>41\,600</math>
| <math>102\,000</math>
| style="border-right: 2px solid;" | <math>0.028</math>
|-
| style="border-left: 2px solid;" | b)
| <math>i</math>/ wedge 6 
| <math>10</math>-<math display="inline">20</math>-<math display="inline">10</math> 
| <math>422\,000</math>
| <math>798\,600</math>
| style="border-right: 2px solid;" | <math>1</math>
|- style="border-top: 2px solid;border-bottom: 2px solid;"
| style="border-left: 2px solid;" |  c)
| <math>p</math>/ wedge 15 
| <math>2</math>-<math display="inline">5</math>-<math display="inline">2</math> 
| <math>4\,000</math>
| <math>60\,300</math>
| style="border-right: 2px solid;" | <math>0.029</math>

|}

The simulation was performed in 1000 steps. Figure [[#img-9|9a]] shows the initial configuration and the final deformation at the last time step. The evolution of the compression axial load (<math display="inline">P/P_{crit}</math>) is obtained as a function of the transverse displacement in <math display="inline">x=L/2</math>, as shown in Figure [[#img-9|9b]].

It is noticed that simulations (a) and (b) has respectively <math display="inline">8.68%</math> and <math display="inline">2.61%</math> of error in the critical load when the transverse displacement over <math display="inline">L_0</math> is 0.005 [-], while mesh (c) is the closest (only <math display="inline">1.46 %</math> of error at the same point). In addition, the time spent for mesh (c) is only the <math display="inline">2.9 %</math> used in (b).

<div id='img-9'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Review_265042547183_1362_Fig9.png|600px|(a) The initial configuration and the final deformation after the last time step (b) the load-displacement curve ]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 9:''' (a) The initial configuration and the final deformation after the last time step. (b) The load-displacement curve 
|}

===3.3 Delamination===

In this section we study the effect of discretization when problems involve CZM. The example to be simulated is a 3D double cantilever beam (DCB), whose length is <math display="inline">L=20</math> [mm] (<math display="inline">x</math>-direction), width <math display="inline">a=2</math> [mm] (<math display="inline">y</math>-direction), thickness <math display="inline">h=1</math> [mm] (<math display="inline">z</math>-direction) and pre-crack <math display="inline">a_0=10</math> [mm] located at the end of the beam along the <math display="inline">x</math>-direction (see Figure [[#img-10|10]]). The properties of the material are <math display="inline">E= 135</math> [GPa] and <math display="inline"> \nu = 0.3 </math> [-]; the cohesive interface parameters are <math display="inline">k_n=100\cdot 10^3</math> [N/mm<math display="inline">^3</math>], <math display="inline">\alpha=1</math> [-], <math display="inline">Y_c=0.4</math> [N/mm] and <math display="inline">n=0.5</math> [-].

The geometry is divided into 160 subdomains and 324 interfaces such as each subdomain has <math display="inline">L_{sst}=0.5</math> [mm], width <math display="inline">a_{sst}= 1</math> [mm] and thickness <math display="inline">h_{sst}= 0.5</math> [mm]. Four meshes were considered (see details in Table [[#table-4|4]]) and the simulation was performed in 50 time steps.


{|  class="floating_tableSCP wikitable" style="text-align: center; margin: 1em auto;min-width:50%;font-size: 85%;"
|+ <span id='table-4'></span>Table 4. Results according to the discretization (DCB problem); '''+''' ratio respect to simulation (a)
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;" |  mesh
| disc. / element
| <math>n_x</math>-<math display="inline">n_y</math>-<math display="inline">n_z</math>
| total elements 
| total dof 
| style="border-right: 2px solid;" | time+ 
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;" |  a)
| <math>i</math>/ wedge 6 
| <math>5</math>-<math display="inline">10</math>-<math display="inline">6</math> 
| <math>109\,440</math>
| <math>245\,280</math>
| style="border-right: 2px solid;" | <math>1</math>
|-
| style="border-left: 2px solid;" | b)
| <math>p</math>/ wedge 15 
| <math>3</math>-<math display="inline">5</math>-<math display="inline">3</math> 
| <math>14\,400</math>
| <math>182\,400</math>
| style="border-right: 2px solid;" | <math>0.240</math>
|- style="border-bottom: 2px solid;"
| style="border-left: 2px solid;" | c)
| <math>p</math>/ wedge 15 
| <math>4</math>-<math display="inline">8</math>-<math display="inline">5</math> 
| <math>54\,400</math>
| <math>576\,480</math>
| style="border-right: 2px solid;" | <math>2.174</math>

|}

Results are compared to the theoretical solution <span id='citeF-29'></span>[[#cite-29|[29]]] in Figure [[#img-10|10]]. It is possible to observe three areas: the first is the bending mode (without delamination); the second zone appears for the crack's propagation (softening curve) and the third one is the second bending mode (when the beam has been completely delaminated). For bending, mesh (b) with three non-linear elements in the thickness is satisfactory, but it does not correctly represent delamination due to the visible zigzag. If the discretization (c) is studied, with a greater number of elements in the <math display="inline">y</math>-direction, the entire curve is correctly predicted, but the time used for the calculation is double that used for the linear discretization (a). The lack of accuracy in the response of mesh (b) could be related to the fact that the forces calculated to evaluate damage are performed at the interfaces which are discretize by constant functions <math display="inline">\mathcal{P}_0</math> (see Figure [[#img-3|3]]), although subdomains have finite elements of higher order. Figure [[#img-11|11]] shows the crack's front at the beginning of the propagation.

<div id='img-10'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Review_265042547183_8931_Fig10.png|600px|The load-displacement curve of the DCB test]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 10:''' The load-displacement curve of the DCB test
|}

<div id='img-11'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Review_265042547183_2690_Fig11.png|600px|DCB problem: (a) subdomains and interfaces (b) crack's front after the 11<sup>th</sup> step where pre-crack is in black and d is the damage variable ranging from 0 (healthy point) to 1 (completely damaged interface point)]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 11:''' DCB problem. (a) subdomains and interfaces. (b) Crack's front after the 11<math>^{th}</math> step where pre-crack is in black and d is the damage variable ranging from 0 (healthy point) to 1 (completely damaged interface point)
|}

==4. Conclusions==

In this article, the influence of the discretization on a micro-macro LaTIn-based Domain Decomposition Method have been investigated. Three subdomains's discretizations have been analyzed: initial linear mesh without localized over discretization on the boundary (<math display="inline">i</math>-version); linear mesh with local <math display="inline">h</math>-refinement on the boundary; and <math display="inline">p</math>-version with quadratic elements on the whole subdomain.

Bending results show that the <math display="inline">h</math>-refinement on the boundary elements increase the calculation time and decrease the convergence rate with respect to the meshes without over discretization (so-called initial version). The best results were obtained when using <math display="inline">p</math>-refinement on the whole subdomain, reaching even 97% faster simulations for the buckling example.

However, <math display="inline">p</math>-refinement is not enough to well represent delamination due to the <math display="inline">\mathcal{P}_0</math> polynomial degree used for the approximation space of the interfaces. Therefore, to have a correct representation of the phenomenon, a greater number of elements are required in the crack's front (<math display="inline">y</math>-direction). A posible solution will be to use linear or higher order finite elements for the interfaces' discretization.

==Acknowledgements==

K. Saavedra acknowledges the financial support from CONICYT, FONDECYT Initiation into research project No 11130623. J. Fernández acknowledges the financial support from CONICYT-PFCHA, National Doctorat 2017, No 21171988. The authors also wish to thanks LMT-Cachan (ENS Paris-Saclay/CNRS/Université Paris-Saclay) for enabling to use the research code MULTI.

==References==
<div class="auto" style="width: auto; margin-left: auto; margin-right: auto;font-size: 85%;">

<div id="cite-1"></div>
[[#citeF-1|[1]]] Carl T. Herakovich. (2012) "Mechanics of composites: A historical review", Volume 41. Mechanics Research Communications 0 1 - 20

<div id="cite-2"></div>
[[#citeF-2|[2]]] LLorca, J and González, C and Molina-Aldareguía, Jon M and Lópes, CS. (2013) "Multiscale modeling of composites: toward virtual testing… and beyond", Volume 65. Springer. JOM 2 215&#8211;225

<div id="cite-3"></div>
[[#citeF-3|[3]]] M.I. Okereke and A.I. Akpoyomare and M.S. Bingley. (2014) "Virtual testing of advanced composites, cellular materials and biomaterials: A review", Volume 60. Composites Part B: Engineering 637 - 662

<div id="cite-4"></div>
[[#citeF-4|[4]]] Jan Mandel. (1993) "Balancing Domain Decomposition", Volume 9. Communications in Numerical Methods in Engineering 233&#8211;241

<div id="cite-5"></div>
[[#citeF-5|[5]]] C. Farhat and F.&#8211;X. Roux. (1991) "A method of finite element tearing and interconnecting and its parallel solution algorithm", Volume 32. International Journal for Numerical Methods in Engineering 1205&#8211;1227

<div id="cite-6"></div>
[[#citeF-6|[6]]] Roux, Francois-Xavier. (2009) "A FETI-2LM method for non-matching grids". Domain Decomposition Methods in Science and Engineering XVIII. Springer 121&#8211;128

<div id="cite-7"></div>
[[#citeF-7|[7]]] P. Gosselet and C. Rey. (2006) "Non-overlapping domain decomposition methods in structural mechanics", Volume 13. Archives of Computational Methods in Engineering 515-572

<div id="cite-8"></div>
[[#citeF-8|[8]]] A. Klawonn and P. Radtke and O. Rheinbach. (2015) "FETI-DP Methods with an Adaptive Coarse Space", Volume 53. SIAM Journal on Numerical Analysis 1 297-320

<div id="cite-9"></div>
[[#citeF-9|[9]]] R. Haferssas and P. Jolivet and F. Nataf. (2015) "A robust coarse space for Optimized Schwarz methods SORAS-GenEO-2", Volume 353. C. R. Math. Acad. Sci. Paris 959-963

<div id="cite-10"></div>
[[#citeF-10|[10]]] Ladeveze, Pierre. (1999) "Nonlinear computational structural mechanics: new approaches and non-incremental methods of calculation". Springer-Verlag

<div id="cite-11"></div>
[[#citeF-11|[11]]] Ladeveze, P and Loiseau, O and Dureisseix, D. (2001) "A micro-macro and parallel computational strategy for highly heterogeneous structures", Volume 52. International Journal for Numerical Methods in Engineering 1-2 121-138

<div id="cite-12"></div>
[[#citeF-12|[12]]] V. Roulet and L. Champaney and P.-A. Boucard. (2011) "A parallel strategy for the multiparametric analysis of structures with large contact and friction surfaces", Volume 42. Advances in Engineering Software 6 347 - 358

<div id="cite-13"></div>
[[#citeF-13|[13]]] Guidault, P. -A. and Allix, O. and Champaney, L. and Cornuault, C. (2008) "A multiscale extended finite element method for crack propagation", Volume 197. Computer Methods in Applied Mechanics and Engineering 5 381&#8211;399

<div id="cite-14"></div>
[[#citeF-14|[14]]] Kerfriden, P. and Allix, O. and Gosselet, P. (2009) "A three-scale domain decomposition method for the 3D analysis of debonding in laminates", Volume 44. Computational Mechanics 3 343&#8211;362

<div id="cite-15"></div>
[[#citeF-15|[15]]] E. I. Saavedra Flores and K. Saavedra and J. Hinojosa and Y. Chandra and R. Das. (2016) "Multi-scale modelling of rolling shear failure in cross-laminated timber structures by homogenisation and cohesive zone models", Volume 81. International Journal of Solids and Structures 219 - 232

<div id="cite-16"></div>
[[#citeF-16|[16]]] Saavedra, Karin and Allix, Olivier and Gosselet, Pierre. (2012) "On a multiscale strategy and its optimization for the simulation of combined delamination and buckling", Volume 91. Wiley Online Library. International Journal for Numerical Methods in Engineering 7 772&#8211;798

<div id="cite-17"></div>
[[#citeF-17|[17]]] Allix, O. and Gosselet, P. and Kerfriden, P. and Saavedra, K. (2012) "Virtual delamination testing through non-linear multi-scale computational methods: some recent progress", Volume 32. Computers, Materials & Continua 2 107-132

<div id="cite-18"></div>
[[#citeF-18|[18]]] Saavedra, K and Allix, O and Gosselet, P and Hinojosa, J and Viard, A. (2017) "An enhanced nonlinear multi-scale strategy for the simulation of buckling and delamination on 3D composite plates", Volume 317. Elsevier. Computer Methods in Applied Mechanics and Engineering 952&#8211;969

<div id="cite-19"></div>
[[#citeF-19|[19]]] Dubois, Olivier and Gander, Martin J. (2009) "Domain Decomposition Methods in Science and Engineering XVIII". Springer 177&#8211;184

<div id="cite-20"></div>
[[#citeF-20|[20]]] Negrello, Camille and Gosselet, Pierre and Rey, Christian. (2018) "A new impedance accounting for short‐ and long‐range effects in mixed substructured formulations of nonlinear problems", Volume 114. International Journal for Numerical Methods in Engineering 7 675-693

<div id="cite-21"></div>
[[#citeF-21|[21]]] J. Mosler and I. Scheider. (2011) "A thermodynamically and variationally consistent class of damagetype cohesive models", Volume 59. Journal of the Mechanics and Physics of Solids 8 1647 - 1668

<div id="cite-22"></div>
[[#citeF-22|[22]]] Babuska, Ivo and Szabo, Barna. (1982) "On the rates of convergence of the finite element method", Volume 18. Wiley Online Library. International Journal for Numerical Methods in Engineering 3 323&#8211;341

<div id="cite-23"></div>
[[#citeF-23|[23]]] Babuska, Ivo and Suri, Manil. (1990) "The p-and hp versions of the finite element method, an overview", Volume 80. Elsevier. Computer Methods in Applied Mechanics and Engineering 1-3 5&#8211;26

<div id="cite-24"></div>
[[#citeF-24|[24]]] Allix, O. and Léveque, D. and Perret, L. (1998) "Identification and forecast of delamination in composite laminates by an interlaminar interface model", Volume 58. Composites Science and Technology 5 671&#8211;678

<div id="cite-25"></div>
[[#citeF-25|[25]]] Duvant, Georges and Lions, Jacques Louis. (2012) "Inequalities in Mechanics and Physics", Volume 219. Springer Science & Business Media

<div id="cite-26"></div>
[[#citeF-26|[26]]] Allix, O. and Kerfriden, P. and Gosselet, P. (2010) "On the control of the load increments for a proper description of multiple delamination in a domain decomposition framework", Volume 83. International Journal for Numerical Methods in Engineering 11 1518&#8211;1540

<div id="cite-27"></div>
[[#citeF-27|[27]]] Ladeveze, Pierre and Nouy, Anthony and Loiseau, O. (2002) "A multiscale computational approach for contact problems", Volume 191. Elsevier. Computer Methods in Applied Mechanics and Engineering 43 4869&#8211;4891

<div id="cite-28"></div>
[[#citeF-28|[28]]] Timoshenko, Stephen. (1970) "Theory of Elastic Stability". McGraw-Hill Education

<div id="cite-29"></div>
[[#citeF-29|[29]]] Kanninen, MF. (1973) "An augmented double cantilever beam model for studying crack propagation and arrest", Volume 9. Springer. International Journal of fracture 1 83&#8211;92
</div>