Onate et al 2007c - Revision history

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-<!-- metadata commented in wiki content
+Published in ''Computational Plasticity'', E. Oñate and R. Owen (Eds.), pp. 239-265, 2007<br />
-==Enhanced Rotation-Free Basic Shell Triangle. Applications to Sheet   Metal Forming==
+DOI: 10.1007/978-1-4020-6577-4_14
-−
-'''Eugenio Oñate1, Fernando G. Flores2 Laurentiu Neamtu3'''
--->
-−
 ==Abstract==

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 ==1 Introduction==
-Triangular shell elements are very useful for the solution of large scale shell problems occurring in many practical engineering situations. Typical examples are the analysis of shell roofs under static and dynamic loads, sheet stamping processes, vehicle dynamics and crash-worthiness situations. Many of these problems involve high geometrical and material non linearities and time changing frictional contact conditions. These difficulties are usually increased by the need of discretizing complex geometrical shapes. Here the use of shell triangles and non-structured meshes becomes a critical necessity. Despite recent advances in the field <span id='citeF-1'></span>[[#cite-1|[1]]-<span id='citeF-6'></span>[[#cite-6|6]]] there are not so many simple shell triangles which are capable of accurately modelling the deformation of a shell structure under arbitrary loading conditions.
+Triangular shell elements are very useful for the solution of large scale shell problems occurring in many practical engineering situations. Typical examples are the analysis of shell roofs under static and dynamic loads, sheet stamping processes, vehicle dynamics and crash-worthiness situations. Many of these problems involve high geometrical and material non linearities and time changing frictional contact conditions. These difficulties are usually increased by the need of discretizing complex geometrical shapes. Here the use of shell triangles and non-structured meshes becomes a critical necessity. Despite recent advances in the field <span id='citeF-1'></span><span id='citeF-2'></span><span id='citeF-3'></span><span id='citeF-4'></span><span id='citeF-5'></span>[[#cite-1|[1]]-<span id='citeF-6'></span>[[#cite-6|6]]] there are not so many simple shell triangles which are capable of accurately modelling the deformation of a shell structure under arbitrary loading conditions.
 A promising line to derive simple shell triangles is to use the nodal displacements as the only unknowns for describing the shell kinematics. This idea goes back to the original attempts to solve thin plate bending problems using finite difference schemes with the deflection as the only nodal variable <span id='citeF-7'></span>[[#cite-7|[7]]-<span id='citeF-9'></span>[[#cite-9|9]]].

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 A promising line to derive simple shell triangles is to use the nodal displacements as the only unknowns for describing the shell kinematics. This idea goes back to the original attempts to solve thin plate bending problems using finite difference schemes with the deflection as the only nodal variable <span id='citeF-7'></span>[[#cite-7|[7]]-<span id='citeF-9'></span>[[#cite-9|9]]].
-In past years some authors have derived a number of thin plate and shell triangular elements free of rotational degrees of freedom (d.o.f.) based on Kirchhoff's theory <span id='citeF-10'></span>[[#cite-10|[10]]-<span id='citeF-26'></span>[[#cite-26|26]]]. In essence all methods attempt to express the curvatures field over an element in terms of the displacements of a collection of nodes belonging to a patch of adjacent elements. Oñate and Cervera <span id='citeF-14'></span>[[#cite-14|[14]]] proposed a general procedure of this kind combining finite element and finite volume concepts for deriving thin plate triangles and quadrilaterals with the deflection as the only nodal variable and presented a simple and competitive rotation-free three d.o.f. triangular element termed BPT (for Basic Plate Triangle). These ideas were extended in <span id='citeF-20'></span>[[#cite-20|[20]]] to derive a number of rotation-free thin plate and shell triangles. The basic ingredients of the method are a mixed Hu-Washizu formulation, a standard discretization into three node triangles, a linear finite element interpolation of the displacement field within each triangle and a finite volume type approach for computing constant curvature and bending moment fields within appropriate non-overlapping control domains. The so called cell-centered and cell-vertex triangular domains yield different families of rotation-free plate and shell triangles. Both the BPT plate element and its extension to shell analysis (termed BST for Basic Shell Triangle) can be derived from the cell-centered formulation. Here the control domain is an individual triangle. The constant curvatures field within a triangle is computed in terms of the displacements of the six nodes belonging to the four elements patch formed by the chosen triangle and the three adjacent triangles. The cell-vertex approach yields a different family of rotation-free plate and shell triangles. Details of the derivation of both rotation-free triangular shell element families can be found in <span id='citeF-20'></span>[[#cite-20|[20]]].
+In past years some authors have derived a number of thin plate and shell triangular elements free of rotational degrees of freedom (d.o.f.) based on Kirchhoff's theory <span id='citeF-10'></span><span id='citeF-11'></span><span id='citeF-12'></span><span id='citeF-13'></span><span id='citeF-14'></span><span id='citeF-15'></span><span id='citeF-16'></span><span id='citeF-17'></span><span id='citeF-18'></span><span id='citeF-19'></span><span id='citeF-20'></span><span id='citeF-21'></span><span id='citeF-22'></span><span id='citeF-23'></span><span id='citeF-24'></span><span id='citeF-25'></span>[[#cite-10|[10]]-<span id='citeF-26'></span>[[#cite-26|26]]]. In essence all methods attempt to express the curvatures field over an element in terms of the displacements of a collection of nodes belonging to a patch of adjacent elements. Oñate and Cervera <span id='citeF-14'></span>[[#cite-14|[14]]] proposed a general procedure of this kind combining finite element and finite volume concepts for deriving thin plate triangles and quadrilaterals with the deflection as the only nodal variable and presented a simple and competitive rotation-free three d.o.f. triangular element termed BPT (for Basic Plate Triangle). These ideas were extended in <span id='citeF-20'></span>[[#cite-20|[20]]] to derive a number of rotation-free thin plate and shell triangles. The basic ingredients of the method are a mixed Hu-Washizu formulation, a standard discretization into three node triangles, a linear finite element interpolation of the displacement field within each triangle and a finite volume type approach for computing constant curvature and bending moment fields within appropriate non-overlapping control domains. The so called cell-centered and cell-vertex triangular domains yield different families of rotation-free plate and shell triangles. Both the BPT plate element and its extension to shell analysis (termed BST for Basic Shell Triangle) can be derived from the cell-centered formulation. Here the control domain is an individual triangle. The constant curvatures field within a triangle is computed in terms of the displacements of the six nodes belonging to the four elements patch formed by the chosen triangle and the three adjacent triangles. The cell-vertex approach yields a different family of rotation-free plate and shell triangles. Details of the derivation of both rotation-free triangular shell element families can be found in <span id='citeF-20'></span>[[#cite-20|[20]]].
 An extension of the BST element to the non linear analysis of shells was implemented in an explicit dynamic code by Oñate ''et al.'' <span id='citeF-25'></span>[[#cite-25|[25]]] using an updated Lagrangian formulation and a hypo-elastic constitutive model. Excellent numerical results were obtained for non linear dynamics of shells involving frictional contact situations and sheet stamping problems <span id='citeF-17'></span><span id='citeF-18'></span><span id='citeF-19'></span><span id='citeF-25'></span>[[#cite-17|[17]],[[#cite-18|18]],[[#cite-19|19]],[[#cite-25|25]]].

← Older revision		Revision as of 09:38, 8 May 2019
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	==References==		==References==

		+	<div id="cite-1"></div>
	[[#citeF-1\|[1]]] Oñate E (1994) A review of some finite element families for thick and thin plate and shell analysis. Publication CIMNE N. 53, May		[[#citeF-1\|[1]]] Oñate E (1994) A review of some finite element families for thick and thin plate and shell analysis. Publication CIMNE N. 53, May

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 <div id="cite-2"></div>
-[2]  Flores FG, Oñate E, Zárate F (1995) New assumed strain triangles for non-linear shell analysis. Computational Mechanics 17:107&#8211;114
+[[#citeF-2|[2]]]  Flores FG, Oñate E, Zárate F (1995) New assumed strain triangles for non-linear shell analysis. Computational Mechanics 17:107&#8211;114
 <div id="cite-3"></div>
-[3]  Argyris JH, Papadrakakis M, Apostolopoulou C, Koutsourelakis S (2000) The TRIC element. Theoretical and numerical investigation. Comput Meth Appl Mech Engrg 182:217&#8211;245
+[[#citeF-3|[3]]]  Argyris JH, Papadrakakis M, Apostolopoulou C, Koutsourelakis S (2000) The TRIC element. Theoretical and numerical investigation. Comput Meth Appl Mech Engrg 182:217&#8211;245
 <div id="cite-4"></div>
@@ Line 1,210: / Line 1,210: @@
 <div id="cite-5"></div>
-[5]  Stolarski H, Belytschko T, Lee S-H  (1995) A review of shell finite elements and corotational theories. Computational Mechanics Advances vol. 2 (2), North-Holland
+[[#citeF-5|[5]]]  Stolarski H, Belytschko T, Lee S-H  (1995) A review of shell finite elements and corotational theories. Computational Mechanics Advances vol. 2 (2), North-Holland
 <div id="cite-6"></div>
@@ Line 1,219: / Line 1,219: @@
 <div id="cite-8"></div>
-[8] Timoshenko SP (1971)  Theory of Plates and Shells, McGraw Hill, New York
+[[#citeF-8|[8]]] Timoshenko SP (1971)  Theory of Plates and Shells, McGraw Hill, New York
 <div id="cite-9"></div>
@@ Line 1,225: / Line 1,225: @@
 <div id="cite-10"></div>
-[10]  Nay RA, Utku S (1972) An alternative to the finite element method. Variational Methods Eng vol. 1
+[[#citeF-10|[10]]]  Nay RA, Utku S (1972) An alternative to the finite element method. Variational Methods Eng vol. 1
 <div id="cite-11"></div>
-[11]  Hampshire JK, Topping BHV, Chan HC (1992) Three node triangular elements with one degree of freedom per node. Engng Comput  9:49&#8211;62
+[[#citeF-11|[11]]]  Hampshire JK, Topping BHV, Chan HC (1992) Three node triangular elements with one degree of freedom per node. Engng Comput  9:49&#8211;62
 <div id="cite-12"></div>
-[12] Phaal R, Calladine CR (1992) A simple class of finite elements for plate and shell problems. I: Elements for beams and thin plates. Int J Num Meth Engng 35:955&#8211;977
+[[#citeF-12|[12]]] Phaal R, Calladine CR (1992) A simple class of finite elements for plate and shell problems. I: Elements for beams and thin plates. Int J Num Meth Engng 35:955&#8211;977
 <div id="cite-13"></div>
-[13] Phaal R, Calladine CR (1992)  A simple class of finite elements for plate and shell problems. II: An element for thin shells with only translational degrees of freedom. Int J Num Meth Engng 35:979&#8211;996
+[[#citeF-13|[13]]] Phaal R, Calladine CR (1992)  A simple class of finite elements for plate and shell problems. II: An element for thin shells with only translational degrees of freedom. Int J Num Meth Engng 35:979&#8211;996
 <div id="cite-14"></div>
-[14] Oñate E, Cervera M (1993) Derivation of thin plate bending elements with one degree of freedom per node. Engineering Computations 10:553&#8211;561
+[[#citeF-14|[14]]] Oñate E, Cervera M (1993) Derivation of thin plate bending elements with one degree of freedom per node. Engineering Computations 10:553&#8211;561
 <div id="cite-15"></div>
-[15] Brunet M, Sabourin F (1994) Prediction of necking and wrinkles with a simplified shell element in sheet forming. Int Conf of Metal Forming Simulation in Industry II:27&#8211;48, Kröplin B (ed)
+[[#citeF-15|[15]]] Brunet M, Sabourin F (1994) Prediction of necking and wrinkles with a simplified shell element in sheet forming. Int Conf of Metal Forming Simulation in Industry II:27&#8211;48, Kröplin B (ed)
 <div id="cite-16"></div>
-[16] Rio G, Tathi B, Laurent H (1994) A new efficient finite element model of shell with only three degrees of freedom per node. Applications to industrial deep drawing test. In Recent Developments in Sheet Metal Forming Technology, Barata Marques MJM (ed), 18th IDDRG Biennial Congress, Lisbon  <div id="cite-17"></div>
+[[#citeF-16|[16]]] Rio G, Tathi B, Laurent H (1994) A new efficient finite element model of shell with only three degrees of freedom per node. Applications to industrial deep drawing test. In Recent Developments in Sheet Metal Forming Technology, Barata Marques MJM (ed), 18th IDDRG Biennial Congress, Lisbon
 [17]  Rojek J, Oñate E (1998)  Sheet springback analysis using a simple shell triangle with translational degrees of freedom only. Int J of Forming Processes 1(3):275&#8211;296  <div id="cite-18"></div>
-[18]  Rojek J, Oñate E, Postek E  (1998) Application of explicit FE codes to simulation of sheet and bulk forming processes. J of Materials Processing Technology 80-81:620&#8211;627  <div id="cite-19"></div>
+<div id="cite-17"></div>
-[19]  Jovicevic J, Oñate E (1999) Analysis of beams and shells using a rotation-free finite element-finite volume formulation, Monograph 43, CIMNE, Barcelona
+[[#citeF-17|[17]]]  Rojek J, Oñate E (1998)  Sheet springback analysis using a simple shell triangle with translational degrees of freedom only. Int J of Forming Processes 1(3):275&#8211;296
 <div id="cite-20"></div>
@@ Line 1,252: / Line 1,258: @@
 <div id="cite-21"></div>
-[21] Cirak F, Ortiz M  (2000) Subdivision surfaces: A new paradigm for thin-shell finite element analysis. Int J Num Meths in Engng 47:2039-2072
+[[#citeF-21|[21]]] Cirak F, Ortiz M  (2000) Subdivision surfaces: A new paradigm for thin-shell finite element analysis. Int J Num Meths in Engng 47:2039-2072
 <div id="cite-22"></div>
-[22] Cirak F, Ortiz M  (2001) Fully <math display="inline">C^{1}</math>-conforming subdivision elements for finite deformations thin-shell analysis. Int J Num Meths in Engng 51:813-833
+[[#citeF-22|[22]]] Cirak F, Ortiz M  (2001) Fully <math display="inline">C^{1}</math>-conforming subdivision elements for finite deformations thin-shell analysis. Int J Num Meths in Engng 51:813-833
 <div id="cite-23"></div>
@@ Line 1,261: / Line 1,267: @@
 <div id="cite-24"></div>
-[24]  Engel G, Garikipati K, Hughes TJR,  Larson MG, Mazzei L, Taylor RL (2002). Continuous/discontinuous finite element approximation of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity. Comput Methods Appl Mech Engrg 191:3669&#8211;3750
+[[#citeF-24|[24]]]  Engel G, Garikipati K, Hughes TJR,  Larson MG, Mazzei L, Taylor RL (2002). Continuous/discontinuous finite element approximation of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity. Comput Methods Appl Mech Engrg 191:3669&#8211;3750
 <div id="cite-25"></div>
-[25]  Oñate E, Cendoya P, Miquel J (2002) Non linear explicit dynamic analysis of shells using the BST rotation-free triangle. Engineering Computations 19(6):662&#8211;706
+[[#citeF-25|[25]]]  Oñate E, Cendoya P, Miquel J (2002) Non linear explicit dynamic analysis of shells using the BST rotation-free triangle. Engineering Computations 19(6):662&#8211;706
 <div id="cite-26"></div>
@@ Line 1,273: / Line 1,279: @@
 <div id="cite-28"></div>
-[28]   Zienkiewicz OC, Oñate E (1991)  Finite Elements vs. Finite Volumes. Is there really a choice?. Nonlinear Computational Mechanics. State of the Art. Wriggers P, Wagner R (eds), Springer Verlag, Heidelberg
+[[#citeF-28|[28]]]   Zienkiewicz OC, Oñate E (1991)  Finite Elements vs. Finite Volumes. Is there really a choice?. Nonlinear Computational Mechanics. State of the Art. Wriggers P, Wagner R (eds), Springer Verlag, Heidelberg
 <div id="cite-29"></div>
-[29] Oñate E, Cervera M,  Zienkiewicz OC (1994) A finite volume format for structural mechanics. Int J Num Meth Engng 37:181&#8211;201
+[[#citeF-29|[29]]] Oñate E, Cervera M,  Zienkiewicz OC (1994) A finite volume format for structural mechanics. Int J Num Meth Engng 37:181&#8211;201
 <div id="cite-30"></div>

← Older revision	Revision as of 11:35, 27 May 2019
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 |-
 |[[Image:Draft_Samper_105745998-chapa_map_thick.png|351px|]]
 |[[Image:Draft_Samper_105745998-chapa_map_form.png|351px|Car tail gate. Map of relative thickness distribution and forming zones on the   stamped part]]
 |- style="text-align: center; font-size: 75%;"

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 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
-|[[Image:Draft_Samper_105745998-guardabarros_2.png|351px|Front fender. Results of the stamping analysis using an   initial mesh of   121960 EBST1 elements. The final adapted mesh had 389870 elements]]
+|[[Image:Draft_Samper_105745998-guardabarros_2.png|500px|Front fender. Results of the stamping analysis using an   initial mesh of   121960 EBST1 elements. The final adapted mesh had 389870 elements]]
 |- style="text-align: center; font-size: 75%;"
 | colspan="1" | '''Figure 12:''' Front fender. Results of the stamping analysis using an   initial mesh of   121960 EBST1 elements. The final adapted mesh had 389870 elements

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 |-
 |[[Image:Draft_Samper_105745998-f_zone_237.png|550px|]]
 |[[Image:Draft_Samper_105745998-s_zone_237.png|550px|]]
 |-
 |[[Image:Draft_Samper_105745998-mesh_det_2_237.png|300px|]]
-|[[Image:Draft_Samper_105745998-relth_det_237.png|300px|Lateral panel of a car. Results of the stamping analysis ]]
+|[[Image:Draft_Samper_105745998-relth_det_237.png|300px|]]
 |- style="text-align: center; font-size: 75%;"
 | colspan="2" | '''Figure 11:''' Lateral panel of a car. Results of the stamping analysis