Onate Flores 2005a - Revision history

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← Older revision		Revision as of 14:50, 12 November 2018
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−	Published in ''Comput. Meth. Appl. Mech. Engng.'', Vol. 194 (21-24), pp. 2406-2443<br />	+	Published in ''Comput. Meth. Appl. Mech. Engng.'', Vol. 194 (21-24), pp. 2406-2443, 2005<br />
	doi: 10.1016/j.cma.2004.07.039		doi: 10.1016/j.cma.2004.07.039

← Older revision		Revision as of 14:48, 12 November 2018
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		+	Published in ''Comput. Meth. Appl. Mech. Engng.'', Vol. 194 (21-24), pp. 2406-2443<br />
		+	doi: 10.1016/j.cma.2004.07.039
		+

	==ABSTRACT==		==ABSTRACT==

@@ Line 2,543: / Line 2,543: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\delta {\boldsymbol \kappa } ={\boldsymbol B}_b \times \mathbf{t}_3 \delta {\boldsymbol a}^p  </math>
+| style="text-align: center;" | <math>\delta {\boldsymbol \kappa } ={\mathbf B}_b \times \mathbf{t}_3 \delta {\mathbf a}^p  </math>
 |}
 |}

@@ Line 2,492: / Line 2,492: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\delta {\boldsymbol \varepsilon }_m ={\boldsymbol B}_m \delta {\boldsymbol a}^p  </math>
+| style="text-align: center;" | <math>\delta {\boldsymbol \varepsilon }_m ={\mathbf B}_m \delta {\mathbf a}^p  </math>
 |}
 |}

@@ Line 87: / Line 87: @@
 where <math display="inline">\Gamma _{M}</math> is the boundary of the central triangle and <math display="inline">\mathbf{n}=\left(n_{x},n_{y} \right)</math> is the boundary normal. Equation ([[#eq-4|4]]) defines the assumed constant curvature field within the central triangle in terms of the deflection gradient along the sides of the triangle. Equation (4) can be found to be equivalent to the standard conservation laws used in finite volume procedures as described in <span id='citeF-27'></span>[[#cite-27|[27,28]]].
-The computation of the line integral in Eq.([[#eq-4|4]]) poses a difficulty as the deflection gradient is discontinuous along the element sides. A simple method to overcome this problem is to compute the deflection gradient at the element sides as the average values of the gradient contributed by the two triangles sharing the side [20,28]. Following this idea the constant curvature field with the element is computed as
+The computation of the line integral in Eq.([[#eq-4|4]]) poses a difficulty as the deflection gradient is discontinuous along the element sides. A simple method to overcome this problem is to compute the deflection gradient at the element sides as the average values of the gradient contributed by the two triangles sharing the side <span id='citeF-20'></span>[[#cite-20|[20,28]]]. Following this idea the constant curvature field with the element is computed as
 <span id="eq-5"></span>
@@ Line 178: / Line 178: @@
 |}
-Note that <math display="inline">\mathbf{K}^{e}</math> is  a <math display="inline">6\times{6}</math> matrix, whereas <math display="inline">\mathbf{f}^{e}</math> has the same structure than for the standard linear triangle. The explicit form of <math display="inline">\mathbf{K}^{e}</math> and <math display="inline">\mathbf{f}^{e}</math> can be found in [14].
+Note that <math display="inline">\mathbf{K}^{e}</math> is  a <math display="inline">6\times{6}</math> matrix, whereas <math display="inline">\mathbf{f}^{e}</math> has the same structure than for the standard linear triangle. The explicit form of <math display="inline">\mathbf{K}^{e}</math> and <math display="inline">\mathbf{f}^{e}</math> can be found in <span id='citeF-14'></span>[[#cite-14|[14]]].
 The resulting Basic Plate Triangle (BPT) has one degree of freedom per node and a wider bandwidth than the standard three node triangles as each triangular element is linked to its three neighbours through Eq.([[#eq-5|5]]).

← Older revision	Revision as of 09:40, 19 November 2018
(No difference)

@@ Line 85: / Line 85: @@
 |}
-where <math display="inline">\Gamma _{M}</math> is the boundary of the central triangle and <math display="inline">\mathbf{n}=\left(n_{x},n_{y} \right)</math> is the boundary normal. Equation ([[#eq-4|4]]) defines the assumed constant curvature field within the central triangle in terms of the deflection gradient along the sides of the triangle. Equation (4) can be found to be equivalent to the standard conservation laws used in finite volume procedures as described in [27,28].
+where <math display="inline">\Gamma _{M}</math> is the boundary of the central triangle and <math display="inline">\mathbf{n}=\left(n_{x},n_{y} \right)</math> is the boundary normal. Equation ([[#eq-4|4]]) defines the assumed constant curvature field within the central triangle in terms of the deflection gradient along the sides of the triangle. Equation (4) can be found to be equivalent to the standard conservation laws used in finite volume procedures as described in <span id='citeF-27'></span>[[#cite-27|[27,28]]].
 The computation of the line integral in Eq.([[#eq-4|4]]) poses a difficulty as the deflection gradient is discontinuous along the element sides. A simple method to overcome this problem is to compute the deflection gradient at the element sides as the average values of the gradient contributed by the two triangles sharing the side [20,28]. Following this idea the constant curvature field with the element is computed as
@@ Line 115: / Line 115: @@
 |}
-Note also that the constant curvature field is expressed in terms of the six nodes of the four element patch linked to the element <math display="inline">M</math>. The expression of the <math display="inline">3\times{6}</math> <math display="inline">\mathbf{B}_{b}</math> matrix can be found in [14,20].
+Note also that the constant curvature field is expressed in terms of the six nodes of the four element patch linked to the element <math display="inline">M</math>. The expression of the <math display="inline">3\times{6}</math> <math display="inline">\mathbf{B}_{b}</math> matrix can be found in <span id='citeF-20'></span>[[#cite-20|[14,20]]].
 The virtual work expression is written as
@@ Line 182: / Line 182: @@
 The resulting Basic Plate Triangle (BPT) has one degree of freedom per node and a wider bandwidth than the standard three node triangles as each triangular element is linked to its three neighbours through Eq.([[#eq-5|5]]).
-Examples of the good performance of the BPT element for analysis of thin plates can be found in [14,20]. The extension of the BPT element to the analysis of shells yields the Basic Shell Triangle (BST) [20]. Different applications of the BST element to linear and non linear analysis of shells are reported in [14,17&#8211;20,23,25,26].
+Examples of the good performance of the BPT element for analysis of thin plates can be found in <span id='citeF-20'></span>[[#cite-20|[14,20]]]. The extension of the BPT element to the analysis of shells yields the Basic Shell Triangle (BST) <span id='citeF-20'></span>[[#cite-20|[20]]]. Different applications of the BST element to linear and non linear analysis of shells are reported in <span id='citeF-20'></span>[[#cite-20|[14,17&#8211;20,23,25,26]]].
 The ideas used to derive the BPT element will now be extended to derive two families of Basic Shell Triangles using a total Lagrangian description.

@@ Line 10: / Line 10: @@
 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]]]&#8211;<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]]]&#8211;<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 [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 and formalized 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>[[#cite-10|[10]]]&#8211;<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 [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 and formalized 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>[[#cite-17|[17,18,19,25]]].

@@ Line 10: / Line 10: @@
 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]]]&#8211;<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 [10]&#8211;<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 [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 and formalized in [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 [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 [10]&#8211;<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 [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 and formalized 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.'' [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 [17,18,19,25].
+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>[[#cite-17|[17,18,19,25]]].
-A large strain formulation for the BST element using a total Lagrangian description was presented by Flores and Oñate [23]. A recent extension of this formulation is based on a quadratic interpolation of the geometry of the patch formed by the BST element and the three adjacent triangles [26]. This yields a linear displacement gradient field over the element from which linear membrane strains and  constant curvatures  can be computed within the BST element.
+A large strain formulation for the BST element using a total Lagrangian description was presented by Flores and Oñate <span id='citeF-23'></span>[[#cite-23|[23]]]. A recent extension of this formulation is based on a quadratic interpolation of the geometry of the patch formed by the BST element and the three adjacent triangles [26]. This yields a linear displacement gradient field over the element from which linear membrane strains and  constant curvatures  can be computed within the BST element.
-In this paper the formulation of the BST element is revisited using an assumed strain approach. The content of the paper is the following. First some basic concepts of the formulation of the original BST element using an assumed constant curvature field are given. Next, the basic equations of the non linear thin shell theory chosen based on a total Lagrangian description are presented. Then the non linear formulation of the BST element is presented. This is based on an assumed constant membrane field derived from the linear displacement interpolation and an assumed constant curvature field expressed in terms of the displacements of the nodes of the four element patch using a finite volume type approach. An enhanced version of the BST element is derived using an assumed linear field for the membrane strains and an assumed constant curvature field. Both assumed fields are obtained from the quadratic interpolation of the patch geometry following the ideas presented in [26]. Details of the derivation of the tangent stiffness matrix needed  for a quasi-static implicit solution are given for both the BST and EBST elements. An efficient version of the  EBST element using one single quadrature point for integration of the tangent matrix is  presented. An explicit  scheme adequate for dynamic analysis is  briefly described.
+In this paper the formulation of the BST element is revisited using an assumed strain approach. The content of the paper is the following. First some basic concepts of the formulation of the original BST element using an assumed constant curvature field are given. Next, the basic equations of the non linear thin shell theory chosen based on a total Lagrangian description are presented. Then the non linear formulation of the BST element is presented. This is based on an assumed constant membrane field derived from the linear displacement interpolation and an assumed constant curvature field expressed in terms of the displacements of the nodes of the four element patch using a finite volume type approach. An enhanced version of the BST element is derived using an assumed linear field for the membrane strains and an assumed constant curvature field. Both assumed fields are obtained from the quadratic interpolation of the patch geometry following the ideas presented in <span id='citeF-26'></span>[[#cite-26|[26]]]. Details of the derivation of the tangent stiffness matrix needed  for a quasi-static implicit solution are given for both the BST and EBST elements. An efficient version of the  EBST element using one single quadrature point for integration of the tangent matrix is  presented. An explicit  scheme adequate for dynamic analysis is  briefly described.
 The efficiency and accuracy of the standard and enhanced versions of the BST element is validated in a number of examples of application including linear and non linear analysis of shells under static and dynamic loads, the inflation and de-inflation of membranes and a sheet stamping problem.