(→‎2. The micro-macro LaTIn based Domain Decomposition Method)
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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:
 
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">i</math>-version: linear six-node wedge elements of equal size in the whole subdomain (“initial” version without local over discretization, ([[#img-3|Figure 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">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 [[#img-4|Figure 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).
 
* <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).
  
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|[[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)]]
 
|[[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%;"
 
|- style="text-align: center; font-size: 75%;"
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| colspan="1" style="padding-bottom:10px;"| '''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)
+
| colspan="1" style="padding-bottom:10px;"| '''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)
 
|}
 
|}
  
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====3.1.1 Isotropic material====
 
====3.1.1 Isotropic material====
  
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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]]].
+
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 [[#img-5|Figure 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 ([[#table-1|Table 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>
 
<div id='img-5'></div>
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|[[Image:Review_265042547183_5579_Fig5.png|600px|Partitioning of the geometry (bending problem with isotropic material)]]
 
|[[Image:Review_265042547183_5579_Fig5.png|600px|Partitioning of the geometry (bending problem with isotropic material)]]
 
|- style="text-align: center; font-size: 75%;"
 
|- style="text-align: center; font-size: 75%;"
−
| colspan="1" style="padding-bottom:10px;"| '''Figure 5:''' Partitioning of the geometry (bending problem with isotropic material)
+
| colspan="1" style="padding-bottom:10px;"| '''Figure 5'''. Partitioning of the geometry (bending problem with isotropic material)
 
|}
 
|}
  
  
 +
<div class="center" style="font-size: 75%;">'''Table 1'''. Results according to the discretization (bending problem with isotropic material); '''+''' ratio respect to simulation (b.1)</div>
 +
 +
<div id='tab-1'></div>
 
{|  class="floating_tableSCP wikitable" style="text-align: center; margin: 1em auto;min-width:50%;font-size: 85%;"
 
{|  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-top: 2px solid;"
 
| style="border-left: 2px solid;" |  mesh
 
| style="border-left: 2px solid;" |  mesh
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|}
 
|}
  
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In Figure [[#img-6|6a]], 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.
+
In [[#img-6|Figure 6a]], 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 ([[#table-1|Table 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 [[#img-6|6b]].
+
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 ([[#table-1|Table 1]]) are mainly related to the mesh size, because convergence rates are similar, as shown in [[#img-6|Figure 6b]].
  
 
<div id='img-6a'></div>
 
<div id='img-6a'></div>
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|[[Image:Review_265042547183_4713_Fig6.png|700px|]]
 
|[[Image:Review_265042547183_4713_Fig6.png|700px|]]
 
|- style="text-align: center; font-size: 75%;"
 
|- style="text-align: center; font-size: 75%;"
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| colspan="2" style="padding-bottom:10px;"| '''Figure 6:''' Bending problem with isotropic material. (a) Deflection. (b) Evolution of the iterative LaTIn error
+
| colspan="2" style="padding-bottom:10px;"| '''Figure 6'''. Bending problem with isotropic material. (a) Deflection. (b) Evolution of the iterative LaTIn error
 
|}
 
|}
  
 
====3.1.2 Orthotropic material====
 
====3.1.2 Orthotropic material====
  
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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.
+
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-2|Table 2]] compares the different discretizations.
  
 +
<div class="center" style="font-size: 75%;">'''Table 2'''. Results according to the discretization (bending problem with an orthotropic material); '''+''' ratio respect to simulation (b.1)</div>
  
 +
<div id='table-2'></div>
 
{|  class="floating_tableSCP wikitable" style="text-align: center; margin: 1em auto;min-width:50%;font-size: 85%;"
 
{|  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-top: 2px solid;"
 
| style="border-left: 2px solid;" |  mesh
 
| style="border-left: 2px solid;" |  mesh
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|}
 
|}
  
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In Figure [[#img-7|7a]] 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|7b]]) 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.
+
In [[#img-7|Figure 7a]] 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 ([[#img-7|Figure 7b]]) 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 ([[#img-8|Figure 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-7a'></div>
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| [[Image:Review_265042547183_6563_Fig7.png|700px]]
 
| [[Image:Review_265042547183_6563_Fig7.png|700px]]
 
|- style="text-align: center; font-size: 75%;"
 
|- style="text-align: center; font-size: 75%;"
−
| colspan="2" style="padding-bottom:10px;"| '''Figure 7:''' Bending problem with orthotropic material. (a) Deflection. (b) Evolution of the iterative LaTIn error
+
| colspan="2" style="padding-bottom:10px;"| '''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-7|7b]]).
+
[[#table-2|Table 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 ([[#img-7|Figure 7b]]).
  
 
<div id='img-8a'></div>
 
<div id='img-8a'></div>
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| [[Image:Review_265042547183_4572_Fig8.png|700px]]
 
| [[Image:Review_265042547183_4572_Fig8.png|700px]]
 
|- style="text-align: center; font-size: 75%;"
 
|- style="text-align: center; font-size: 75%;"
−
| colspan="2" style="padding-bottom:10px;"| '''Figure 8:''' Results for the orthotropic material: normal stresses <math>\sigma _{xx}</math>
+
| colspan="2" style="padding-bottom:10px;"| '''Figure 8'''. Results for the orthotropic material: normal stresses <math>\sigma _{xx}</math>
 
|}
 
|}
  
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===3.2 Buckling===
 
===3.2 Buckling===
  
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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|9a]]). 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]].
+
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 ([[#img-9|Figure 9a]]). 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-3|Table 3]].
  
 +
<div class="center" style="font-size: 75%;">'''Table 3'''. Results according to the discretization (buckling problem); '''+''' ratio respect to simulation (b)</div>
  
 +
<div id='table-3'></div>
 
{|  class="floating_tableSCP wikitable" style="text-align: center; margin: 1em auto;min-width:50%;font-size: 85%;"
 
{|  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-top: 2px solid;"
 
| style="border-left: 2px solid;" |  mesh
 
| style="border-left: 2px solid;" |  mesh
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|}
 
|}
  
−
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]].
+
The simulation was performed in 1000 steps. [[#img-9|Figure 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 [[#img-9|Figure 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).
 
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).
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|[[Image:Review_265042547183_1362_Fig9.png|700px|(a) The initial configuration and the final deformation after the last time step (b) the load-displacement curve ]]
 
|[[Image:Review_265042547183_1362_Fig9.png|700px|(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%;"
 
|- style="text-align: center; font-size: 75%;"
−
| colspan="1" style="padding-bottom:10px;"| '''Figure 9:''' (a) The initial configuration and the final deformation after the last time step. (b) The load-displacement curve  
+
| colspan="1" style="padding-bottom:10px;"| '''Figure 9'''. (a) The initial configuration and the final deformation after the last time step. (b) The load-displacement curve  
 
|}
 
|}
  
 
===3.3 Delamination===
 
===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-11|11a]]). 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> [-].
+
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 ([[#img-11|Figure 11a]]). 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.
+
  
 +
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-4|Table 4]]) and the simulation was performed in 50 time steps.
  
 +
<div class="center" style="font-size: 75%;">'''Table 4'''. Results according to the discretization (DCB problem); '''+''' ratio respect to simulation (a)</div>
 +
<div id='table-1'></div>
 
{|  class="floating_tableSCP wikitable" style="text-align: center; margin: 1em auto;min-width:50%;font-size: 85%;"
 
{|  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-top: 2px solid;"
 
| style="border-left: 2px solid;" |  mesh
 
| style="border-left: 2px solid;" |  mesh
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|}
 
|}
  
−
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.
+
Results are compared to the theoretical solution <span id='citeF-29'></span>[[#cite-29|[29]]] in [[#img-10|Figure 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 [[#img-3|Figure 3]]), although subdomains have finite elements of higher order. [[#img-11|Figure 11]] shows the crack's front at the beginning of the propagation.
  
 
<div id='img-10'></div>
 
<div id='img-10'></div>
Line 490: Line 494:
 
|[[Image:Review_265042547183_8931_Fig10.png|700px|The load-displacement curve of the DCB test]]
 
|[[Image:Review_265042547183_8931_Fig10.png|700px|The load-displacement curve of the DCB test]]
 
|- style="text-align: center; font-size: 75%;"
 
|- style="text-align: center; font-size: 75%;"
−
| colspan="1" style="padding-bottom:10px;"| '''Figure 10:''' The load-displacement curve of the DCB test
+
| colspan="1" style="padding-bottom:10px;"| '''Figure 10'''. The load-displacement curve of the DCB test
 
|}
 
|}
  
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|[[Image:Review_265042547183_2690_Fig11.png|650px|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)]]
 
|[[Image:Review_265042547183_2690_Fig11.png|650px|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%;"
 
|- style="text-align: center; font-size: 75%;"
−
| colspan="1" style="padding-bottom:10px;"| '''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)
+
| colspan="1" style="padding-bottom:10px;"| '''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)
 
|}
 
|}
  

Revision as of 14:19, 2 July 2021

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 (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h} -refinement), disregarding the option of increasing the polynomial degree of the finite elements (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p} -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, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h} 

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p}
refinements on the subdomains and local Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h}

-refinement only along the edges of the subdomains were investigated. It is conclude that the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p} -refinement in the whole subdomain not only enables to reach more exact solutions than using global or local Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h} -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 [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 [3]. Instead, Domain Decomposition Methods (DDM) [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) [7,8,9].

The LaTIn method [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 [11]. Contact [12], crack propagation [13,14,15], buckling-delamination interaction [16,17] and heterogeneous structures [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 [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 [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 [16], while delamination is modeled using a Cohesive Zone Model (CZM) which are based on damage mechanics [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 (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h} -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 (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p} -refinement), as proposed in [22,23] for problems using direct solvers (without parallel computations). Therefore, this work studies the influence of the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p} -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 Cachan1 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 [18]. To achieve this goal, Section 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, different Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h} - and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p} -refinements are compared in 3D beam problems: bending, buckling and delamination. Finally, the conclusions and ongoing work are presented in Section 4.

(1) LMT-Cachan (ENS Paris-Saclay/CNRS/Université Paris-Saclay)

2. The micro-macro LaTIn based Domain Decomposition Method

Let us consider a laminate composite (see Figure 1) occupying the domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Omega } 

bounded by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \partial \Omega }
in the current configuration, and consisting of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N_P}
plies. Each ply Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P}
is connected to an adjacent ply Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P' }
through the interface Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\Gamma _{PP'}}}

. The structure is subjected to an external surface traction field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\underline{F}_d}} 

on the part Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\partial \Omega _{F_d}}}
and to a displacement field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\underline{U}_d}}
on the complementary part Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\partial \Omega _{U_d}}}

. The body force per unit of mass is written Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\underline{f}_d}} . The relevant quantities are described in reference to the undeformed configuration using the index Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \cdot _\mathit{0}} . 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 [24], while [25] describe contact inequalities.

To propose the partitioned problem, the whole domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Omega } 

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 1. A subdomain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {S^{\quad }_\mathit{0}}}
defined in the undeformed domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\Omega _{S_\mathit{0}}}}
is connected to an adjacent undeformed subdomain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {{S'}_\mathit{0}}}
through an undeformed interface Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\Gamma _{S_\mathit{0}{S'}_\mathit{0}}}=\partial {\Omega _{S_\mathit{0}}}\cap \partial {\Omega _{S_\mathit{0}'}}}

. The surface entity Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\Gamma _{S_\mathit{0}{S'}_\mathit{0}}}} 

applies force distributions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\underline{F}_{{S^{\quad }_\mathit{0}}}}}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\underline{F}_{S_\mathit{0}'}}} 

and displacement distributions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\underline{W}_{{S^{\quad }_\mathit{0}}}}}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\underline{W}_{{S'}_\mathit{0}}}} 

to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {S^{\quad }_\mathit{0}}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {{S'}_\mathit{0}}}
respectively (see Figure 2). Let us define Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\Gamma _{S_\mathit{0}}}= \cup _{{{S'}_\mathit{0}}\in {\mathbf{E}}} {\Gamma _{S_\mathit{0}{S'}_\mathit{0}}}}

. For a subdomain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {S^{\quad }_\mathit{0}}} 

such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\Gamma _{S_\mathit{0}}}\cap ({\partial \Omega _{F_{d_\mathit{0}}}}\cup {\partial \Omega _{U_{d_\mathit{0}}}}) \neq \emptyset }

, the boundary condition Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle ({\underline{F}_{d_\mathit{0}}},{\underline{U}_{d_\mathit{0}}})} 

is applied through a boundary interface Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\Gamma _{{S}_{d_\mathit{0}}}}}

.


The reference problem, the mesomodel and its partitioning [18
Figure 1. The reference problem, the mesomodel and its partitioning [18]
Subdomains and interfaces [16
Figure 2. Subdomains and interfaces [16]


The purpose of the method is to find the subdomain fields Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\underline{u}_{{S^{\quad }_\mathit{0}}}}} 

(displacement) and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\underline{\underline{\pi}}_{{S^{\quad }_\mathit{0}}}}}
(second Piola-Kirchhoff stress), and the interface fields Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\underline{W}_{{S^{\quad }_\mathit{0}}}}}
(displacement) and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\underline{F}_{{S^{\quad }_\mathit{0}}}}}
(inter-forces), where the index Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \cdot _{S_\mathit{0}}}
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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{A_d}}

  • non-linear kinematic admissibility of the subdomains:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\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}}}
(1)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \underline{u} _{S_{0}|\partial \Omega _{S_0}}= \underline{W} _{S_{0}|\Gamma _{S_0}} \hbox{, on } \Gamma _{{S_0 S'}_0}
(2)
  • non-linear static admissibility of the subdomains:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \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}}}},
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \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}}
(3)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\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}}}})} .

  • behavior of the subdomains:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\underline{\underline{\pi }}_{{S^{\quad }_\mathit{0}}}}= \frac{\partial \psi }{\partial {\underline{\underline{E}}_{S_\mathit{0}}}} \;,\; \hbox{on} \; {\Omega _{S_\mathit{0}}}\;,\;
(4)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \psi } 

is the stored energy function per unit of undeformed volume. For this work, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \psi =\frac{1}{2} \mathbf{K}_{S_\mathit{0}}{\underline{\underline{E}}_{S_\mathit{0}}}:{\underline{\underline{E}}_{S_\mathit{0}}}}
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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \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 \, ,
(5)

where the subspace Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{W}_{{S^{\quad }_\mathit{0}}{{S'}_\mathit{0}}}^M}} 

of interface macroscopic admissible displacements is a parameter described in Section 2.1.

2. Local (non-linear) equations in the interfaces whose solutions belong to the manifold Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathbf{L}_\Gamma}}

  • constitutive equation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \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}}}\;,}
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle [{\underline{W}_{{S^{\quad }_\mathit{0}}}}] ={\underline{W}_{{S'}_\mathit{0}}}-{\underline{W}_{{S^{\quad }_\mathit{0}}}}}
is the interface displacement gap.
  • boundary behavior Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \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}}}}\;}

. The relation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{R}_{{S^{\quad }_\mathit{0}}{{S'}_\mathit{0}}}=\underline{0}} 

can be made explicit for:

- Perfect interface: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \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. }


- Cohesive interface: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \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. }


- Unilateral contact interface: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \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. }


where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\underline{n}}} 

is the unit normal to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Gamma _{SS'}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{P}}
is the corresponding tangential projection operator.

The algorithm consists in seeking the interface solution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle s_{ref}} 

alternatively in these two spaces: first, one finds a solution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 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}}}}
in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{A_d}}

, then a solution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \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}}}} 

in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathbf{L}_\Gamma }}

. In order for the two problems to be well-posed, two search directions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {{k^+}}} 

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {{k^-}}}

, which link the solutions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle s} 

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \widehat{s}}
during the iterative process, are introduced. The converged interface solution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle s_{ref}}
is such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle s_{ref} \in \mathbf{A_d}\cap {\mathbf{L}_\Gamma }}

. A complete description of this iterative method can be found in [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 [26].

2.1 Separation of the micro-macro scales

The definition of the macroscopic quantities is done through an orthogonal projector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Pi } 

for each interface Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\Gamma _{S_\mathit{0}{S'}_\mathit{0}}}}

, such as the macroscopic fields are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\underline{F}^M_{{S^{\quad }_\mathit{0}}}}~=~\Pi {\underline{F}_{{S^{\quad }_\mathit{0}}}}} 

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\underline{W}_{{S^{\quad }_\mathit{0}}}^M}~=~\Pi  {\underline{W}_{{S^{\quad }_\mathit{0}}}}}

. Consequently, the microscopic spaces, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{W}_{{S^{\quad }_\mathit{0}}{{S'}_\mathit{0}}}^m}} 

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{F}_{{S^{\quad }_\mathit{0}}{{S'}_\mathit{0}}}^m}}

, are orthogonal to the macroscopic spaces, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{W}_{{S^{\quad }_\mathit{0}}{{S'}_\mathit{0}}}^M}} 

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{F}_{{S^{\quad }_\mathit{0}}{{S'}_\mathit{0}}}^M}}

, with respect to the inner product Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle L^2({\Gamma _{S_\mathit{0}{S'}_\mathit{0}}})} , so the scales can be uncoupled with respect to the interface virtual work as follows:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \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}
(6)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \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}
(7)

In order to ensure the scalability of the iterative scheme, a global linear coarse grid problem 5 has been introduced. It is fully characterized by the set of interface macroscopic spaces Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle ({\mathcal{W}_{{S^{\quad }_\mathit{0}}{{S'}_\mathit{0}}}^M})} 

which is a parameter of the method as well as its dual Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle ({\mathcal{F}_{{S^{\quad }_\mathit{0}}{{S'}_\mathit{0}}}^M})}

. 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, [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 [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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\Gamma _{S_\mathit{0}{S'}_\mathit{0}}}} , the displacements Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\underline{W}_{{S^{\quad }_\mathit{0}}}}} 

and forces Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\underline{F}_{{S^{\quad }_\mathit{0}}}}}
belong to the approximation spaces Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{W}_{{S^{\quad }_\mathit{0}}{{S'}_\mathit{0}},h}}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{F}_{{S^{\quad }_\mathit{0}}{{S'}_\mathit{0}},h}}}

. 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{F}_{{S^{\quad }_\mathit{0}}{{S'}_\mathit{0}},h}}} 

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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{U}_{S_\mathit{0}}}}

). Two manners are possible: to increase the number of elements (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h} -version) or to use a higher degree of approximation (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p} -version) for the field Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\underline{u}_{{S^{\quad }_\mathit{0}}}}} 

near to the interface, as illustrated in  Figure 3 [27]. Classically, the code MULTI has considered linear elements Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{P}_1}
with local Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h}

-refinement along the subdomain's boundary, while the spaces Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{W}_{{S^{\quad }_\mathit{0}}{{S'}_\mathit{0}},h}}} 

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{F}_{{S^{\quad }_\mathit{0}}{{S'}_\mathit{0}},h}}}
are piecewise constant functions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{P}_0}

.

In this work, the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p} -refinement is also explored. Indeed, it is here proposed to use the second order approximation for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\underline{u}_{{S^{\quad }_\mathit{0}}}}} 

not only in the boundary but in the whole subdomain.
Modification of the classical approximations of the inter-force and local displacement along the edge of a subdomain: h-and p-versions [13
Figure 3. Modification of the classical approximations of the inter-force and local displacement along the edge of a subdomain: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): h

-and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): p -versions [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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle {\mathcal{U}_{S_\hbox{0}}}} 

are considered:
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i}

-version: linear six-node wedge elements of equal size in the whole subdomain (“initial” version without local over discretization, (Figure 3);

  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h}

-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 4 (local Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h} 

over discretization);
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p}

-version: quadratic fifteen-node wedge elements of equal size in the whole subdomain (global Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p} 

over discretization).

For all cases, the approximation functions of the interfaces are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{P}_0} . Displacements, convergence rate and time are used to compare the refinements.

before and after local h-refinement: a) wedge element b) mesh of the subdomain's boundary (the inner mesh remains unchanged)
Figure 4. before and after local Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): h

-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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle L=160} 

[mm] (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle x}

-direction), width is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle a=120} 

[mm] (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle y}

-direction) and thickness is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h=4} 

[mm] (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle z}

-direction). Regarding the boundary conditions, the cantilever beam is embedded at one end and subjected to a surface load Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \underline{F}_{d_0}=0.1} 

[N/mFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle ^2}

] 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle E=210} 

[GPa] and a poisson coefficient Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \nu=0.3}
[-]. The geometry is partitioned into Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 64}
identical subdomains and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 184}
interfaces (see  Figure 5); each subdomain has length Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle L_{sst}=20}
[mm], width Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle a_{sst}= 15}
[mm] and thickness Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h_{sst}= 4}
[mm]. Five meshes are considered: two different initial discretization with their respective Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h}

-refinement while the last one is a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p} -version. Each subdomain has Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n_x} (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n_y} ,Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n_z} ) elements in the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle x} (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle y} ,Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle z} )-direction, respectively, as shown in Table 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 (Table 1), the maximum vertical displacement is compared with the theoretical elastic curve [28].

Partitioning of the geometry (bending problem with isotropic material)
Figure 5. Partitioning of the geometry (bending problem with isotropic material)


Table 1. Results according to the discretization (bending problem with isotropic material); + ratio respect to simulation (b.1)
mesh discretization Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): n_x

-Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n_y} -Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n_z}

total total dof time+ displ.
/ element elements error [mm]
a.1) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): i

/ wedge 6

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 10

-Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 8} -Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 2}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 22\,272 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 61\,056 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 0.046 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 1.182
a.2) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): h

/ wedge 6

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 10

-Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 8} -Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 2}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 121\,088 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 141\,696 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 0.089 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 3.438
b.1) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): i

/ wedge 6

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 15

-Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 20} -Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 4}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 173\,568 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 360\,000 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 1 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 0.607
b.2) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): h

/ wedge 6

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 15

-Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 20} -Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 4}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 574\,464 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 674\,112 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 2.6 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 1.166
c) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): p

/ wedge 15

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 10

-Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 8} -Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 2}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 22\,272 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 262\,464 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 0.193 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 0.281


In Figure 6a, 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h} -refinements (a.2) and (b.2), the displacement's error and calculation time increase (Table 1) while the convergence rate decrease respect to the corresponding Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i} -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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h} -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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i} -version. This choice could induce different stiffness through a subdomain, implying additional difficulties to transfer informationn between subdomains.

Finally, the mesh (d) (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p} -version) is twice more accurate, has 73% less dof and is 19,3% more quickly than mesh (b.1). Differences in the computation time (Table 1) are mainly related to the mesh size, because convergence rates are similar, as shown in Figure 6b.

Review 265042547183 4713 Fig6.png
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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle [0^o,90^o]_S} , each 1 [mm] a thick. A Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 0^o} -layer is transversely isotropic with the following elastic properties: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle E_1=165} 

[GPa], Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle E_2=E_3=9}
[GPa], Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \nu _{12}=\nu _{13}=0.3}
[-], Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \nu _{23}=0.5}
[-], Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle G_{12}=G_{13}=5.6}
[GPa] and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle G_{23}=2.8}
[GPa]. The geometry is divided into 256 identical subdomains of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle L_{sst}=20}
[mm], Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle a_{sst}= 15}
[mm] and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h_{sst}= 1}
[mm], generating 736 interfaces. Therefore, for each ply there is one subdomain in the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle z}

-direction (four in total through the thickness). Table 2 compares the different discretizations.

Table 2. Results according to the discretization (bending problem with an orthotropic material); + ratio respect to simulation (b.1)
mesh disc. / element Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): n_x

-Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n_y} -Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n_z}

total elements total dof time+
a.1) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): i

/ wedge 6

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 10

-Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 10} -Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 2}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 113\,664 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 574\,128 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 0.402
a.2) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): h

/ wedge 6

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 10

-Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 10} -Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 2}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 611\,328 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 706\,560 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 0.814
b.1) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): i

/ wedge 6

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 15

-Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 15} -Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 3}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 393\,216 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 881\,664 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 1
b.2) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): h

/ wedge 6

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 15

-Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 15} -Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 3}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 1\,565\,696 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 1\,806\,336 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 3.707
c) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): p

/ wedge 15

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 10

-Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 10} -Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 2}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 113\,664 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 1\,320\,192 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 0.542


In Figure 7a 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 (Figure 7b) is twice the detention criteria, even after 1000 iterations. Using the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p} -version mesh (c), the LaTIn error is less than Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 10^{-5}} 

in only 68 iteraciones, less than half of the iterations made by the curve (b.1) to converge. If the upper stresses Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \sigma _{xx}}
are compared respect to the theoretical ones (Figure 8), it is possible to confirm that (a.1), (a.2) and (b.2) do not fit the desired solution.
Review 265042547183 6563 Fig7.png
Figure 7. Bending problem with orthotropic material. (a) Deflection. (b) Evolution of the iterative LaTIn error

Table 2 compares calculation time, total number of elements and total dof. It is verified that using Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p} -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 (Figure 7b).

Review 265042547183 4572 Fig8.png
Figure 8. Results for the orthotropic material: normal stresses Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \sigma _{xx}

Similar to the precedent example, the local Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h} -version does not ensure better results. Therefore, the meshes considered for the next examples will be Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i} 

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p}

-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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \underline{u}_d} 

to produce uniaxial compression, while a perpendicular perturbation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \underline{F}_d}
induces buckling out of the plane (Figure 9a). The structure has the following geometry: length Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle L=10}
[mm] (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle x}

-direction), width Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle a=1} 

[mm] (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle y}

-direction) and thickness Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h=0.1} 

[mm] (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle z}

-direction). The properties of the material are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle E= 135} 

[GPa] and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle  \nu = 0.3 }
[-]. The geometry is divided into 100 subdomains and 156 perfect interfaces, therefore, each subdomain has Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle L_{sst}=0.2}
[mm], width Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle a_{sst}= 0.5}
[mm] and thickness Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h_{sst}= 0.1}
[mm]. Three meshes are considered, the first two are linear without over discretization (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i}

-version), while the mesh (c) is a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p} -refinement. More details of the meshes and their results are shown in Table 3.

Table 3. Results according to the discretization (buckling problem); + ratio respect to simulation (b)
mesh disc. / element Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): n_x

-Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n_y} -Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n_z}

total elements total dof time+
a) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): i

/ wedge 6

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 5

-Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 10} -Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 4}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 41\,600 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 102\,000 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 0.028
b) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): i

/ wedge 6

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 10

-Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 20} -Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 10}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 422\,000 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 798\,600 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 1
c) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): p

/ wedge 15

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 2

-Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 5} -Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 2}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 4\,000 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 60\,300 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 0.029


The simulation was performed in 1000 steps. Figure 9a shows the initial configuration and the final deformation at the last time step. The evolution of the compression axial load (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle P/P_{crit}} ) is obtained as a function of the transverse displacement in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle x=L/2} , as shown in Figure 9b.

It is noticed that simulations (a) and (b) has respectively Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 8.68%} 

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 2.61%}
of error in the critical load when the transverse displacement over Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle L_0}
is 0.005 [-], while mesh (c) is the closest (only Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 1.46 %}
of error at the same point). In addition, the time spent for mesh (c) is only the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 2.9 %}
used in (b).
(a) The initial configuration and the final deformation after the last time step (b) the load-displacement curve
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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle L=20} 

[mm] (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle x}

-direction), width Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle a=2} 

[mm] (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle y}

-direction), thickness Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h=1} 

[mm] (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle z}

-direction) and pre-crack Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle a_0=10} 

[mm] located at the end of the beam along the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle x}

-direction (Figure 11a). The properties of the material are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle E= 135} 

[GPa] and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle  \nu = 0.3 }
[-]; the cohesive interface parameters are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k_n=100\cdot 10^3}
[N/mmFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle ^3}

], Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \alpha=1} 

[-], Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Y_c=0.4}
[N/mm] and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n=0.5}
[-].

The geometry is divided into 160 subdomains and 324 interfaces such as each subdomain has Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle L_{sst}=0.5} 

[mm], width Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle a_{sst}= 1}
[mm] and thickness Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h_{sst}= 0.5}
[mm]. Four meshes were considered (see details in  Table 4) and the simulation was performed in 50 time steps.
Table 4. Results according to the discretization (DCB problem); + ratio respect to simulation (a)
mesh disc. / element Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): n_x

-Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n_y} -Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n_z}

total elements total dof time+
a) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): i

/ wedge 6

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 5

-Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 10} -Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 6}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 109\,440 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 245\,280 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 1
b) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): p

/ wedge 15

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 3

-Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 5} -Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 3}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 14\,400 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 182\,400 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 0.240
c) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): p

/ wedge 15

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 4

-Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 8} -Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 5}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 54\,400 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 576\,480 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 2.174


Results are compared to the theoretical solution [29] in Figure 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle y} -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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{P}_0} 

(see  Figure 3), although subdomains have finite elements of higher order. Figure 11 shows the crack's front at the beginning of the propagation.
The load-displacement curve of the DCB test
Figure 10. The load-displacement curve of the DCB test
DCB problem: (a) subdomains and interfaces (b) crack's front after the 11th step where pre-crack is in black and d is the damage variable ranging from 0 (healthy point) to 1 (completely damaged interface point)
Figure 11. DCB problem. (a) Subdomains and interfaces. (b) Crack's front after the 11Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): ^{th} 
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 (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i} -version); linear mesh with local Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h} -refinement on the boundary; and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p} -version with quadratic elements on the whole subdomain.

Bending results show that the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h} -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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p} -refinement on the whole subdomain, reaching even 97% faster simulations for the buckling example.

However, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p} -refinement is not enough to well represent delamination due to the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{P}_0} 

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 (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle y}

-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.

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Published on 13/03/19
Accepted on 27/02/19
Submitted on 03/06/18

Volume 35, Issue 1, 2019
DOI: 10.23967/j.rimni.2019.02.002
Licence: CC BY-NC-SA license

Web of Science Core Collection® Times cited: 0

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