Cardoso Nungaray et al 2019a - Revision history

Rimni at 10:30, 3 March 2021

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Rimni at 11:56, 18 March 2020

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Rimni at 11:44, 18 March 2020

2020-03-18T11:44:22Z

@@ Line 445: / Line 445: @@
 |}
-where <math display="inline">\mathbf{e}_q</math> is the <math display="inline">q^{th}</math> standard basis vector. Figures [[#img-6|6]] and [[#img-7|7]] illustrates the original and the normalized simplices with the corresponding node numeration for 2D and 3D respectively. <div id='img-6'></div>
+where <math display="inline">\mathbf{e}_q</math> is the <math display="inline">q^{th}</math> standard basis vector. Figures [[#img-6|6]] and [[#img-7|7]] illustrates the original and the normalized simplices with the corresponding node numeration for 2D and 3D respectively.
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
 |[[Image:Review_101239056093_1743_image6.jpg|600px|(a) The simplex formed by the points x₁, x₂ and   x₃ in the original space contains an interior point   x<sub>g</sub> that is mapped to            (b) ξ<sub>g</sub> into the normalized 2D-simplex formed by the points   ξ₁, ξ₂ and ξ₃.]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure 6'''. (a) The simplex formed by the points <math>\mathbf{x}_1</math>, <math>\mathbf{x}_2</math> and   <math>\mathbf{x}_3</math> in the original space contains an interior point   <math>\mathbf{x}_g</math> that is mapped to            (b) <math>\boldsymbol{\xi }_g</math> into the normalized 2D-simplex formed by the points   <math>\boldsymbol{\xi }_1</math>, <math>\boldsymbol{\xi }_2</math> and <math>\boldsymbol{\xi }_3</math>
+| colspan="1" style="padding:10px;"| '''Figure 6'''. (a) The simplex formed by the points <math>\mathbf{x}_1</math>, <math>\mathbf{x}_2</math> and   <math>\mathbf{x}_3</math> in the original space contains an interior point   <math>\mathbf{x}_g</math> that is mapped to            (b) <math>\boldsymbol{\xi }_g</math> into the normalized 2D-simplex formed by the points   <math>\boldsymbol{\xi }_1</math>, <math>\boldsymbol{\xi }_2</math> and <math>\boldsymbol{\xi }_3</math>
 |}
 <div id='img-7'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 70%;"
 |-
 |[[Image:Review_101239056093_8671_image7.jpg|600px|(a) The 3D-simplex formed by the points x₁, x₂,   x₃ and x₄  in the original space contains an interior            point x<sub>g</sub> that is mapped to            (b) ξ<sub>g</sub> into the normalized 3D-simplex formed by the points   ξ₁, ξ₂, ξ₃ and ξ₄.]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure 7'''. (a) The 3D-simplex formed by the points <math>\mathbf{x}_1</math>, <math>\mathbf{x}_2</math>,   <math>\mathbf{x}_3</math> and <math>\mathbf{x}_4</math>  in the original space contains an interior            point <math>\mathbf{x}_g</math> that is mapped to            (b) <math>\boldsymbol{\xi }_g</math> into the normalized 3D-simplex formed by the points   <math>\boldsymbol{\xi }_1</math>, <math>\boldsymbol{\xi }_2</math>, <math>\boldsymbol{\xi }_3</math> and <math>\boldsymbol{\xi }_4</math>
+| colspan="1" style="padding:10px;"| '''Figure 7'''. (a) The 3D-simplex formed by the points <math>\mathbf{x}_1</math>, <math>\mathbf{x}_2</math>,   <math>\mathbf{x}_3</math> and <math>\mathbf{x}_4</math>  in the original space contains an interior            point <math>\mathbf{x}_g</math> that is mapped to            (b) <math>\boldsymbol{\xi }_g</math> into the normalized 3D-simplex formed by the points   <math>\boldsymbol{\xi }_1</math>, <math>\boldsymbol{\xi }_2</math>, <math>\boldsymbol{\xi }_3</math> and <math>\boldsymbol{\xi }_4</math>
 |}

@@ Line 416: / Line 416: @@
 |}
-where <math display="inline">w_g</math> is the corresponding quadrature weight and <math display="inline">(\mathbf{S}\boldsymbol{u})_{ijk}|_{\mathbf{x}_g}</math> is the strain evaluation of the Gauss point with the proper change of interval, denoted <math display="inline">\mathbf{x}_{g}</math>. Figure [[#img-5|5]] shows the change of interval required for a 2D face. A 3D face (a polygon) must be subdivided to be integrated with a triangular quadrature. <div id='img-5'></div>
+where <math display="inline">w_g</math> is the corresponding quadrature weight and <math display="inline">(\mathbf{S}\boldsymbol{u})_{ijk}|_{\mathbf{x}_g}</math> is the strain evaluation of the Gauss point with the proper change of interval, denoted <math display="inline">\mathbf{x}_{g}</math>. Figure [[#img-5|5]] shows the change of interval required for a 2D face. A 3D face (a polygon) must be subdivided to be integrated with a triangular quadrature.
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
 |[[Image:Review_101239056093_3659_image5.jpg|600px|(a) The shaded volume V<sub>i</sub> is being integrated.            The integral over the subface e<sub>ijk</sub> is calculated using the            polynomial approximation of the shaded simplex.            The integration point must be mapped to            (b) the normalized space [-1,1] in order to use the Gauss-            Legendre quadrature.]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure 5'''. (a) The shaded volume <math>V_i</math> is being integrated.            The integral over the subface <math>e_{ijk}</math> is calculated using the            polynomial approximation of the shaded simplex.            The integration point must be mapped to            (b) the normalized space <math>[-1,1]</math> in order to use the Gauss-            Legendre quadrature
+| colspan="1" style="padding:10px;"| '''Figure 5'''. (a) The shaded volume <math>V_i</math> is being integrated.            The integral over the subface <math>e_{ijk}</math> is calculated using the            polynomial approximation of the shaded simplex.            The integration point must be mapped to            (b) the normalized space <math>[-1,1]</math> in order to use the Gauss-            Legendre quadrature
 |}

@@ Line 368: / Line 368: @@
 <div id='img-4'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 80%;"
 |-
 |[[Image:Review_101239056093_7534_image4.jpg|600px|(a) The dotted line illustrates the triangulation of the calculation            points of adjacent volumes to V<sub>i</sub>, used by most of the FV methods.            (b) The dotted line shows the simplices forming the piece-wise            approximation used to solve the integrals H<sub>ij</sub> of the            control volume V<sub>i</sub>.]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure 4'''. (a) The dotted line illustrates the triangulation of the calculation            points of adjacent volumes to <math>V_i</math>, used by most of the FV methods. (b) The dotted line shows the simplices forming the piece-wise approximation used to solve the integrals <math>\mathbf{H}_{ij}</math> of the control volume <math>V_i</math>
+| colspan="1" style="padding:10px;"| '''Figure 4'''. (a) The dotted line illustrates the triangulation of the calculation            points of adjacent volumes to <math>V_i</math>, used by most of the FV methods. (b) The dotted line shows the simplices forming the piece-wise approximation used to solve the integrals <math>\mathbf{H}_{ij}</math> of the control volume <math>V_i</math>
 |}
@@ Line 388: / Line 388: @@
 |}
-these subfaces result from the intersection between <math display="inline">P_\alpha </math> and the control volume <math display="inline">V_i</math>. The Figure [[#img-4|4]].b illustrates six key points of this approach, 1) the simplices are used to create a polynomial interpolation of <math display="inline">\boldsymbol{u}(\mathbf{x})</math> over the boundary of the control volume, 2) most of the faces are intersected by several simplices, such faces must be divided into subfaces to be integrated, 3) some few faces are inside a single simplex, as illustrated in the face formed by <math display="inline">V_i</math> and <math display="inline">V_k</math>, 4) there are volumes that require information of non-adjacent volumes to calculate its face integrals, such as <math display="inline">V_i</math> requires <math display="inline">V_k</math>, 5) the dependency between volumes is not always symmetric, which means that if <math display="inline">V_i</math> requires <math display="inline">V_k</math> does not implies that <math display="inline">V_k</math> requires <math display="inline">V_i</math>, and 6) non conforming meshes are supported, as shown in the faces formed by <math display="inline">V_a</math>, <math display="inline">V_b</math>, <math display="inline">V_c</math>, <math display="inline">V_d</math> and <math display="inline">V_j</math>.
+these subfaces result from the intersection between <math display="inline">P_\alpha </math> and the control volume <math display="inline">V_i</math>. Figure [[#img-4|4]] (b) illustrates six key points of this approach: 1) the simplices are used to create a polynomial interpolation of <math display="inline">\boldsymbol{u}(\mathbf{x})</math> over the boundary of the control volume, 2) most of the faces are intersected by several simplices, such faces must be divided into subfaces to be integrated, 3) some few faces are inside a single simplex, as illustrated in the face formed by <math display="inline">V_i</math> and <math display="inline">V_k</math>, 4) there are volumes that require information of non-adjacent volumes to calculate its face integrals, such as <math display="inline">V_i</math> requires <math display="inline">V_k</math>, 5) the dependency between volumes is not always symmetric, which means that if <math display="inline">V_i</math> requires <math display="inline">V_k</math> does not implies that <math display="inline">V_k</math> requires <math display="inline">V_i</math>, and 6) non conforming meshes are supported, as shown in the faces formed by <math display="inline">V_a</math>, <math display="inline">V_b</math>, <math display="inline">V_c</math>, <math display="inline">V_d</math> and <math display="inline">V_j</math>.
 The integral [[#eq-23|(23)]] is now rewritten in terms of the subfaces

@@ Line 167: / Line 167: @@
 <div id='img-3'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 40%;"
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 60%;"
 |-
 |[[Image:Review_101239056093_1413_image3.jpg|600px|The boundary ퟃV<sub>i</sub> of the three dimensional control volume V<sub>i</sub> is            subdivided into N<sub>i</sub> flat faces, denoted e<sub>ij</sub>. The unit vector   n<sub>ij</sub> is normal to the face e<sub>ij</sub>.]]

@@ Line 167: / Line 167: @@
 <div id='img-3'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 40%;"
 |-
 |[[Image:Review_101239056093_1413_image3.jpg|600px|The boundary ퟃV<sub>i</sub> of the three dimensional control volume V<sub>i</sub> is            subdivided into N<sub>i</sub> flat faces, denoted e<sub>ij</sub>. The unit vector   n<sub>ij</sub> is normal to the face e<sub>ij</sub>.]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure 3'''. The boundary <math>\partial V_i</math> of the three dimensional control volume <math>V_i</math> is            subdivided into <math>N_{i}</math> flat faces, denoted <math>e_{ij}</math>. The unit vector   <math>\mathbf{n}_{ij}</math> is normal to the face <math>e_{ij}</math>
+| colspan="1" style="padding:10px;"| '''Figure 3'''. The boundary <math>\partial V_i</math> of the three dimensional control volume <math>V_i</math> is            subdivided into <math>N_{i}</math> flat faces, denoted <math>e_{ij}</math>. The unit vector   <math>\mathbf{n}_{ij}</math> is normal to the face <math>e_{ij}</math>
 |}

@@ Line 156: / Line 156: @@
 <div id='img-2'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 70%;"
 |-
 |[[Image:Review_101239056093_8669_image2.jpg|600px|The partition Pₕ is the discretization of the domain Ω into   N control volumes.            The boundary of the control volumes, ퟃV<sub>i</sub>, is            conformed by N<sub>i</sub> flat faces, denoted e<sub>ij</sub>. The unit vector   n<sub>ij</sub> is normal to the face e<sub>ij</sub>.            The faces of the volumes adjacent to the boundary Γ<sub>N</sub> are            integrated using the condition b<sub>N</sub>.]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure 2'''. The partition <math>P_h</math> is the discretization of the domain <math>\Omega </math> into   <math>N</math> control volumes.            The boundary of the control volumes, <math>\partial V_i</math>, is            conformed by <math>N_{i}</math> flat faces, denoted <math>e_{ij}</math>. The unit vector   <math>\mathbf{n}_{ij}</math> is normal to the face <math>e_{ij}</math>.            The faces of the volumes adjacent to the boundary <math>\Gamma _N</math> are            integrated using the condition <math>\boldsymbol{b}_N</math>
+| colspan="1" style="padding:10px;"| '''Figure 2'''. The partition <math>P_h</math> is the discretization of the domain <math>\Omega </math> into   <math>N</math> control volumes.            The boundary of the control volumes, <math>\partial V_i</math>, is            conformed by <math>N_{i}</math> flat faces, denoted <math>e_{ij}</math>. The unit vector   <math>\mathbf{n}_{ij}</math> is normal to the face <math>e_{ij}</math>.            The faces of the volumes adjacent to the boundary <math>\Gamma _N</math> are            integrated using the condition <math>\boldsymbol{b}_N</math>
 |}

@@ Line 112: / Line 112: @@
 <div id='img-1'></div>
-{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 80%;"
+{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 70%;"
 |-
 |[[Image:Review_101239056093_4049_image1.jpg|600px|(a) The initial body, Ω, with its boundary conditions   ퟃΩ= Γ<sub>N</sub>∪Γ<sub>D</sub>. (b) The distorted body            resulting from solving equilibrium in the elasticity equation,            with the boundary conditions given by b<sub>N</sub> and u<sub>D</sub>.            The boundaries where there are not conditions indicated explicitly,            correspond to Neumann conditions b<sub>N</sub>= 0.]]
 |- style="text-align: center; font-size: 75%;"
-| colspan="1" | '''Figure 1'''. (a) The initial body, <math>\Omega </math>, with its boundary conditions   <math>\partial \Omega = \Gamma _N \cup \Gamma _D</math>. (b) The distorted body            resulting from solving equilibrium in the elasticity equation,            with the boundary conditions given by <math>\boldsymbol{b}_N</math> and <math>\boldsymbol{u}_D</math>.  The boundaries where there are not conditions indicated explicitly, correspond to Neumann conditions <math>\boldsymbol{b}_N = 0</math>
+| colspan="1" style="padding:10px;"| '''Figure 1'''. (a) The initial body, <math>\Omega </math>, with its boundary conditions   <math>\partial \Omega = \Gamma _N \cup \Gamma _D</math>. (b) The distorted body            resulting from solving equilibrium in the elasticity equation,            with the boundary conditions given by <math>\boldsymbol{b}_N</math> and <math>\boldsymbol{u}_D</math>.  The boundaries where there are not conditions indicated explicitly, correspond to Neumann conditions <math>\boldsymbol{b}_N = 0</math>
 |}

@@ Line 92: / Line 92: @@
 |}
-The domain boundary <math display="inline">\partial \Omega = \Gamma _N \cup \Gamma _D</math> is the union of the boundary with Neumann conditions, denoted <math display="inline">\Gamma _N</math>, and the boundary with Dirichlet conditions, denoted <math display="inline">\Gamma _D</math>. The equation [[#eq-5|(5)]] must be satisfied for the given forces <math display="inline">\boldsymbol{b}_N</math> and displacements <math display="inline">\boldsymbol{u}_D</math> along the boundary. The following equations remark these user defined boundary conditions,
+The domain boundary <math display="inline">\partial \Omega = \Gamma _N \cup \Gamma _D</math> is the union of the boundary with Neumann conditions, denoted <math display="inline">\Gamma _N</math>, and the boundary with Dirichlet conditions, denoted <math display="inline">\Gamma _D</math>. Equation [[#eq-5|(5)]] must be satisfied for the given forces <math display="inline">\boldsymbol{b}_N</math> and displacements <math display="inline">\boldsymbol{u}_D</math> along the boundary. The following equations remark these user defined boundary conditions,
 <span id="eq-6.a"></span>
@@ Line 109: / Line 109: @@
 |}
-The Figure [[#img-1|1]] illustrates the initial body with the boundary conditions, and the distorted body after equilibrium is solved in the elasticity equation.
+Figure [[#img-1|1]] illustrates the initial body with the boundary conditions, and the distorted body after equilibrium is solved in the elasticity equation.
 <div id='img-1'></div>
@@ Line 122: / Line 122: @@
 ==3. Numerical method==
-On this section we go into the details of the numerical procedure by  discussing 1) the discretization with CVFA, 2) the control volumes integration, 3) the subfaces integrals, 4) the simplex-wise polynomial approximation, 5) the pair-wise polynomial approximation, 6) the homeostatic splines used within the shape functions, 7) the linear system assembling,  8) how to impose boundary conditions and 9) two special cases of the formulation.
+On this section we go into the details of the numerical procedure by  discussing: 1) the discretization with CVFA, 2) the control volumes integration, 3) the subfaces integrals, 4) the simplex-wise polynomial approximation, 5) the pair-wise polynomial approximation, 6) the homeostatic splines used within the shape functions, 7) the linear system assembling,  8) how to impose boundary conditions and 9) two special cases of the formulation.
 For the sake of legibility, in some parts of the text we unfold the matrices only for the bidimensional case, but the very same procedures must be followed for the 3D case.
@@ Line 154: / Line 154: @@
 |}
-The Figure [[#img-2|2]] illustrates the partition <math display="inline">P_h</math> of <math display="inline">\Omega </math> into <math display="inline">N</math> control volumes defined in the equations [[#eq-7|(7)]] and [[#eq-8|(8)]]. <div id='img-2'></div>
+Figure [[#img-2|2]] illustrates the partition <math display="inline">P_h</math> of <math display="inline">\Omega </math> into <math display="inline">N</math> control volumes defined in the equations [[#eq-7|(7)]] and [[#eq-8|(8)]].
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
@@ Line 163: / Line 165: @@
-The Figure [[#img-3|3]] shows a three dimensional control volume.
+Figure [[#img-3|3]] shows a three dimensional control volume.
 <div id='img-3'></div>
@@ Line 364: / Line 366: @@
 ===3.3 Calculating face integrals===
-The surface integrals <math display="inline">\mathbf{H}_{ij}</math> along the flat faces <math display="inline">e_{ij}</math> are calculated using an auxiliary piece-wise polynomial approximation of the displacement field. This approximation is based on the simplices (triangles in 2D or tetrahedra in 3D) resulting from the Delaunay triangulation of the calculation points <math display="inline">\mathbf{x}_i</math> from the neighborhood of <math display="inline">V_i</math>. The Delaunay triangulation is the best triangulation for numerical interpolation, since it maximizes the minimum angle of the simplices, which means that its quality is maximized as well. We define the neighborhood <math display="inline">{B}_i</math> of volume <math display="inline">V_i</math> as the minimum set of calculation points <math display="inline">\mathbf{x}_j</math> such that the simplices intersecting <math display="inline">V_i</math> does not change if we add another calculation point to the set. Observe that the neighborhood <math display="inline">{B}_i</math> does not always coincide with the set of calculation points in volumes adjacent to <math display="inline">V_i</math>, as in most of the FV formulations. Once the neighborhood <math display="inline">{B}_i</math> is triangulated, we ignore the simplices with angles smaller than <math display="inline">10</math> degrees, and the simplices formed outside the domain, which commonly appear in concavities of the boundary <math display="inline">\partial \Omega </math>. The local set of simplices resulting from the neighborhood of <math display="inline">V_i</math> is denoted <math display="inline">P_\alpha </math>. The Figure [[#img-4|4]] illustrates the difference between (a) the simplices resulting from the triangulation of the calculation points in adjacent volumes and (b) those resulting from the triangulation of the proposed neighborhood <math display="inline">{B}_i</math>.
+The surface integrals <math display="inline">\mathbf{H}_{ij}</math> along the flat faces <math display="inline">e_{ij}</math> are calculated using an auxiliary piece-wise polynomial approximation of the displacement field. This approximation is based on the simplices (triangles in 2D or tetrahedra in 3D) resulting from the Delaunay triangulation of the calculation points <math display="inline">\mathbf{x}_i</math> from the neighborhood of <math display="inline">V_i</math>. The Delaunay triangulation is the best triangulation for numerical interpolation, since it maximizes the minimum angle of the simplices, which means that its quality is maximized as well. We define the neighborhood <math display="inline">{B}_i</math> of volume <math display="inline">V_i</math> as the minimum set of calculation points <math display="inline">\mathbf{x}_j</math> such that the simplices intersecting <math display="inline">V_i</math> does not change if we add another calculation point to the set. Observe that the neighborhood <math display="inline">{B}_i</math> does not always coincide with the set of calculation points in volumes adjacent to <math display="inline">V_i</math>, as in most of the FV formulations. Once the neighborhood <math display="inline">{B}_i</math> is triangulated, we ignore the simplices with angles smaller than <math display="inline">10</math> degrees, and the simplices formed outside the domain, which commonly appear in concavities of the boundary <math display="inline">\partial \Omega </math>. The local set of simplices resulting from the neighborhood of <math display="inline">V_i</math> is denoted <math display="inline">P_\alpha </math>. Figure [[#img-4|4]] illustrates the difference between (a) the simplices resulting from the triangulation of the calculation points in adjacent volumes and (b) those resulting from the triangulation of the proposed neighborhood <math display="inline">{B}_i</math>.
 <div id='img-4'></div>
@@ Line 415: / Line 417: @@
 |}
-where <math display="inline">w_g</math> is the corresponding quadrature weight and <math display="inline">(\mathbf{S}\boldsymbol{u})_{ijk}|_{\mathbf{x}_g}</math> is the strain evaluation of the Gauss point with the proper change of interval, denoted <math display="inline">\mathbf{x}_{g}</math>. The Figure [[#img-5|5]] shows the change of interval required for a 2D face. A 3D face (a polygon) must be subdivided to be integrated with a triangular quadrature. <div id='img-5'></div>
+where <math display="inline">w_g</math> is the corresponding quadrature weight and <math display="inline">(\mathbf{S}\boldsymbol{u})_{ijk}|_{\mathbf{x}_g}</math> is the strain evaluation of the Gauss point with the proper change of interval, denoted <math display="inline">\mathbf{x}_{g}</math>. Figure [[#img-5|5]] shows the change of interval required for a 2D face. A 3D face (a polygon) must be subdivided to be integrated with a triangular quadrature. <div id='img-5'></div>
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-
@@ Line 442: / Line 444: @@
 |}
-where <math display="inline">\mathbf{e}_q</math> is the <math display="inline">q^{th}</math> standard basis vector. The Figures [[#img-6|6]] and [[#img-7|7]] illustrates the original and the normalized simplices with the corresponding node numeration for 2D and 3D respectively. <div id='img-6'></div>
+where <math display="inline">\mathbf{e}_q</math> is the <math display="inline">q^{th}</math> standard basis vector. Figures [[#img-6|6]] and [[#img-7|7]] illustrates the original and the normalized simplices with the corresponding node numeration for 2D and 3D respectively. <div id='img-6'></div>
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-