Rubio et al 2023a - Revision history

Ferminotero: Ferminotero moved page Review 602617684658 to Rubio et al 2023a

2023-10-10T08:33:02Z

Ferminotero moved page Review 602617684658 to Rubio et al 2023a

RaulRubio: RaulRubio moved page Draft Rubio 812965691 to Review 602617684658

2023-05-19T14:09:04Z

RaulRubio moved page Draft Rubio 812965691 to Review 602617684658

RaulRubio at 13:12, 19 May 2023

2023-05-19T13:12:25Z

RaulRubio at 13:09, 19 May 2023

2023-05-19T13:09:55Z

RaulRubio at 09:08, 19 May 2023

2023-05-19T09:08:54Z

RaulRubio at 09:05, 19 May 2023

2023-05-19T09:05:45Z

RaulRubio at 08:03, 18 May 2023

2023-05-18T08:03:13Z

RaulRubio at 19:00, 17 May 2023

2023-05-17T19:00:30Z

RaulRubio: /* Resumen */

2023-05-17T18:57:54Z

‎Resumen

RaulRubio at 18:57, 17 May 2023

2023-05-17T18:57:05Z

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 |}
-Before moving onto the condensation and regression stages, we must ensure the ROM can reproduce FE solutions. To do so, we have run a full-scale FE simulation and compared the solution with the ROM. The simulation is a 2m long clamped beam under a uniform load of 1 MPa in the other end, as shown in Figure [[#img-3|3]]. The surface dimensions are 0.0256 <math display="inline">\times </math> 0.01m
+Before moving onto the condensation and regression stages, we must ensure the ROM can reproduce FE solutions. To do so, we have solved a full-scale FE simulation and compared the solution with the ROM. The simulation is a 2m long clamped beam under a uniform load of 1 MPa in the other end, as shown in Figure [[#img-3|3]]. The surface dimensions are 0.0256 <math display="inline">\times </math> 0.01m
 <div id='img-3a'></div>
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 |}
-Figure [[#img-4|4]] presents the solution for FE and ROM. It can be seen that the patterns are very similar, although the ROM is more flexible than the FE. This discrepancy is expected as the ROM cannot capture boundary effects, but as we move away from the edge of the beam, both solutions converge. On top of this, the training was made prescribing periodic boundary conditions on the slice, hence softer elements in ROM are expected.
+Figure [[#img-4|4]] presents the solution for FE and ROM. It can be seen that the results are very similar, although the ROM is a bit more flexible than the FE. This discrepancy is expected as the ROM cannot capture boundary effects, but as we move away from the edge of the beam, both solutions converge. On top of this, the training was made prescribing periodic boundary conditions on the slice, hence softer elements in ROM are expected.
 <div id='img-4a'></div>
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 Once checked the ROM can reproduce FE simulations, we can proceed to the condensation stage. The results of the condensation must be analysed carefully. Cross terms appear due to two reasons: material orthotropy and geometrical coupling. The latter is due to the fact that the shear centre and centroid, in profiles with only one axis of symmetry, do not coincide <span id="fnc-1"></span>[[#fn-1|<sup>1</sup>]] and thus coupling between torsion and bending and shear in the non-symmetric plane appears; the former is caused by the material behavior itself.
-To discern the geometrical coupling, the same geometry with an isotropic material has been trained and run. Figure [[#img-5|5]] represents the evolution of the stiffness matrix values (obtained with the condensation procedure) as a function of the beam length. The non-zero values of terms 2,3 and 1,5 prove the geometrical coupling is non-negligible.
+To discern the geometrical coupling, the same geometry with an isotropic material has been trained and computed. Figure [[#img-5|5]] represents the evolution of the stiffness matrix values (obtained with the condensation procedure) as a function of the beam length. The non-zero values of terms 2,3 and 1,5 prove the geometrical coupling is non-negligible.
 Since geometrical coupling is a property of the cross-sectional shape, we can make sure any other coupling is caused by the orthotropy of the material, as it is the case of the terms <math display="inline">k_{23}</math> and <math display="inline">k_{15}</math> (see Figure [[#img-6|6]]). This material orthotropy is precisely the one that cannot be captured by classical beam theories but naturally appear following <span id='citeF-1'></span>[[#cite-1|[1]]] and the strategy presented in section [[#2 Methodology|2]].
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 ===3.1 Validation===
-Lastly, we must check whether to process of condensation plus regression gives similar solutions than the full ROM. To so do we have run the same case with the full ROM and the condensed model with 3 different element sizes. Figure [[#img-7|7]] shows the displacements and rotations along the beam for the loading case studied (see Figure [[#img-3|3]]). It can be observed that, as the element size gets closer to 10cm (a point in the database), the 6 DoF solution is able to reproduce the full ROM results. In addition, no instabilities appear in any of the 3 element sizes so we can consider the methodology presented in this work is validated for orthotropic materials.
+Lastly, we must check whether to process of condensation plus regression gives similar solutions than the full ROM. To so do we have computed the same case with the full ROM and the condensed model with 3 different element sizes. Figure [[#img-7|7]] shows the displacements and rotations along the beam for the loading case studied (see Figure [[#img-3|3]]). It can be observed that, as the element size gets closer to 10cm (a point in the database), the 6 DoF solution is able to reproduce the full ROM results. In addition, no instabilities appear in any of the 3 element sizes so we can consider the methodology presented in this work is validated for orthotropic materials.
 <div id='img-7'></div>

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 Among the several options available, we will make use of the multiscale method for periodic structures using domain decomposition and ECM-hyperreduction proposed by the third author in Ref. <span id='citeF-1'></span>[[#cite-1|[1]]]. The reason for favouring this approach is its outcome is the stiffness matrix of a beam element that can be directly assembled in the same fashion as standard FE elements. Since this stiffness matrix is obtained from a set of numerical experiments it naturally captures the orthotropy present in composite materials, but its integration in FE codes poses a challenging framework which we tried to address. The difficulty resides in two distinctive aspects: the first is the number of DoF resulting from <span id='citeF-1'></span>[[#cite-1|[1]]] may differ from the classical beam theories; and the second is that the element size is fixed.
-The integration strategy consist of two different stages. The first is the so-called condensation, from which the <math display="inline">n \times n</math> stiffness matrix is condensed into a matrix of size <math display="inline">12 \times 12</math> for a given beam length. Condensation will be used to create a database of stiffness matrices over which a regression is then performed.
+The integration strategy consists of two different stages. The first is the so-called condensation, from which the <math display="inline">n \times n</math> stiffness matrix is condensed into a matrix of size <math display="inline">12 \times 12</math> for a given beam length. Condensation will be used to create a database of stiffness matrices over which a regression is then performed.
 In the present work, the methodology will be briefly presented, but the main focus is on its application in a real case scenario under the scope of the EU funded project FIBRE4YARDS.
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 |}
-where <math display="inline">u_D</math> are the prescribed displacements, c characterizes the linear relation between each boundary condition and the Lagrange multipliers <math display="inline">\lambda </math> associated. Periodic boundary conditions over the deformational modes are prescribed as they are in agreement with the classical theories for the case of isotropic materials. Following this strategy, we can account for the deformation inside the domain with the maximum resolution the method provides (conversely, should we truncate the stiffness matrix, we would get a stiffer response than the actual one is and so, truncation was discarded).
+where <math display="inline">u_D</math> are the prescribed displacements, c characterizes the linear relation between each boundary condition and the Lagrange multipliers <math display="inline">\lambda </math> associated. Periodic boundary conditions over the deformational modes are prescribed as they are in agreement with the classical theories for the case of isotropic materials. Following this strategy, we can account for the deformation inside the domain with the maximum resolution the method provides (conversely, should we truncate the stiffness matrix, we would get a stiffer response than the actual; for this reason, we discarded the option of using truncation).
 ===2.3 Interpolation===

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 -->
 ==1 Introduction==

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 |[[Image:Draft_Rubio_812965691-k_omega_iso_2.png|600px|Results of the condensation for the omega profile with isotropic material.]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure 5:''' Results of the condensation for the omega profile with isotropic material. Units in SI
+| colspan="1" | '''Figure 5:''' Results of the condensation for the omega profile with isotropic material. Units in IS
 |}
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 |[[Image:Draft_Rubio_812965691-k_omega_2.png|600px|Results of the condensation for the omega profile with the properties shown in table [[#table-1|1]]. ]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure 6:''' Results of the condensation for the omega profile with the properties shown in table [[#table-1|1]]. Units in SI
+| colspan="1" | '''Figure 6:''' Results of the condensation for the omega profile with the properties shown in table [[#table-1|1]]. Units in IS
 |}

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 Estructuras tipo viga ayudan a proporcionar integridad estructural en un gran número de aplicaciones. Anteriormente fabricados de madera o acero, los procesos de fabricación actuales permiten manufacturar estos componentes de material compuesto dotándolos de grandes propiedades mecánicas por unidad de massa. El comportamiento ortótropo de los materiales compuestos, sin embargo, supone un desafío al tratar de simularlos numéricamente.
-Este trabajo presenta la integraciṕn, en códigos de elementos finitos, de un método basado en machine learning capaz de capturar el comportamiento ortótropo de los materiales compuestos: the multiscale method for periodic structures using domain decomposition and ECM-hyper reduction. El método proporciona un nuevo tipo de elemento finito que puede ser ensamblado con librerías ya existentes en dichos códigos. La difultad de la integración de este nuevo elemento finito reside en que su cinemática puede estar descrita por más grados de libertad del que los códigos de vigas estándar consideran. La estrategia presentada consiste en una condensación estática seguida de una regresión. Condiciones de borde periódicas se aplican a los grados de libertad "extra" mientras que la regresión consiste en una interpolación lineal con soporte local.
+Este trabajo presenta la integración, en códigos de elementos finitos, de un método basado en machine learning capaz de capturar el comportamiento ortótropo de los materiales compuestos: the multiscale method for periodic structures using domain decomposition and ECM-hyper reduction. El método proporciona un nuevo tipo de elemento finito que puede ser ensamblado con librerías ya existentes en dichos códigos. La difultad de la integración de este nuevo elemento finito reside en que su cinemática puede estar descrita por más grados de libertad del que los códigos de vigas estándar consideran. La estrategia presentada consiste en una condensación estática seguida de una regresión. Condiciones de borde periódicas se aplican a los grados de libertad "extra" mientras que la regresión consiste en una interpolación lineal con soporte local.
-El artículo se centra en el desempeño del elemento integrado mediante el análisis de un perfil pultruído cuya anisotropia requiere el uso de formulaciones complejas para capturar su comporatamiento mecánico. La convergencia de resultados entre el modelo propuesto y forumaciones más generales valida la metodologia y su uso en códigos de elementos finitos para caracterizar vigas fabricadas de material compuesto
+El artículo se centra en el desempeño del elemento integrado mediante el análisis de un perfil pultruído cuya anisotropia requiere el uso de formulaciones complejas para capturar su comporatamiento mecánico. La convergencia de resultados entre el modelo propuesto y formulaciones más generales valida la metodologia y su uso en códigos de elementos finitos para caracterizar vigas fabricadas de material compuesto.

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	Este trabajo presenta la integraciṕn, en códigos de elementos finitos, de un método basado en machine learning capaz de capturar el comportamiento ortótropo de los materiales compuestos: the multiscale method for periodic structures using domain decomposition and ECM-hyper reduction. El método proporciona un nuevo tipo de elemento finito que puede ser ensamblado con librerías ya existentes en dichos códigos. La difultad de la integración de este nuevo elemento finito reside en que su cinemática puede estar descrita por más grados de libertad del que los códigos de vigas estándar consideran. La estrategia presentada consiste en una condensación estática seguida de una regresión. Condiciones de borde periódicas se aplican a los grados de libertad "extra" mientras que la regresión consiste en una interpolación lineal con soporte local.		Este trabajo presenta la integraciṕn, en códigos de elementos finitos, de un método basado en machine learning capaz de capturar el comportamiento ortótropo de los materiales compuestos: the multiscale method for periodic structures using domain decomposition and ECM-hyper reduction. El método proporciona un nuevo tipo de elemento finito que puede ser ensamblado con librerías ya existentes en dichos códigos. La difultad de la integración de este nuevo elemento finito reside en que su cinemática puede estar descrita por más grados de libertad del que los códigos de vigas estándar consideran. La estrategia presentada consiste en una condensación estática seguida de una regresión. Condiciones de borde periódicas se aplican a los grados de libertad "extra" mientras que la regresión consiste en una interpolación lineal con soporte local.

−	El artículo se centra en el desempeño del elemento integrado mediante el análisis de un perfil pultruído cuya anisotropia requiere el uso de formulaciones complejas para capturar su comporatamiento mecánico. La convergencia de resultados entre el modelo propuesto y forumaciones más generales valida la metodologia y su uso en códigos de elementos finitos para caracterizar vigas fabricadas ~~con~~ material compuesto	+	El artículo se centra en el desempeño del elemento integrado mediante el análisis de un perfil pultruído cuya anisotropia requiere el uso de formulaciones complejas para capturar su comporatamiento mecánico. La convergencia de resultados entre el modelo propuesto y forumaciones más generales valida la metodologia y su uso en códigos de elementos finitos para caracterizar vigas fabricadas de material compuesto