Herrera et al 2018b - Revision history

Rimni at 11:17, 28 April 2020

2020-04-28T11:17:49Z

Rimni at 11:16, 28 April 2020

2020-04-28T11:16:01Z

Rimni at 11:13, 28 April 2020

2020-04-28T11:13:07Z

Rimni at 11:11, 28 April 2020

2020-04-28T11:11:35Z

Rimni at 12:30, 24 April 2020

2020-04-24T12:30:00Z

Rimni at 12:28, 24 April 2020

2020-04-24T12:28:11Z

Rimni at 12:25, 24 April 2020

2020-04-24T12:25:58Z

Rimni at 12:11, 24 April 2020

2020-04-24T12:11:36Z

Rimni at 12:02, 24 April 2020

2020-04-24T12:02:37Z

Rimni at 11:59, 24 April 2020

2020-04-24T11:59:16Z

@@ Line 1,159: / Line 1,159: @@
 |}
-{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
+{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 30%"
 |-
 |style="padding:10px;"| [[Image:Review_239195285364_5542_Fig_6_circle_prb.png|260px|]]

@@ Line 1,270: / Line 1,270: @@
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
 |-
-|  style="text-align: center;padding:10px;"| [[Image:Review_239195285364_7846_Fig_8_circle_contour_temp.png|200px|]]
+|  style="text-align: center;padding:10px;vertical-align:center;"| [[Image:Review_239195285364_7846_Fig_8_circle_contour_temp.png|200px|]]
-|  style="text-align: center;vertical-align: top;padding:10px;"| [[Image:Review_239195285364_1490_Fig_8_circle_contour_flux.png|200px|]]
+|  style="text-align: center;padding:10px;vertical-align:center;"| [[Image:Review_239195285364_1490_Fig_8_circle_contour_flux.png|200px|]]
 |-
 |  style="text-align: center;font-size: 75%;"|(a)

@@ Line 68: / Line 68: @@
 Note that if we list the vectors in the opposite order, we have a different orientation and, therefore, the algebraic objects must satisfy <math display="inline">e_2\wedge e_1=-e_1\wedge e_2</math> (Figure [[#lb-2.1|2]]).
-{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
+{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width:35%;"
 |-
 |style="padding:10px;"| [[Image:Review_239195285364_6648_text-Square01-eps-converted-to.png|250px|Change of orientation of the parallelogram implies anticommutativity in the wedge product of two vectors.]]

@@ Line 7: / Line 7: @@
 ==1. Introduction==
-The purpose of this paper is to introduce the theory of Discrete Exterior Calculus (DEC)  to the widest possible audience and, therefore, we will  rely mainly on Vector Calculus and Matrix Algebra. Discrete Exterior Calculus is a relatively new method for solving partial differential equations <span id='citeF-8'></span>[[#cite-8|[8]]] based on the idea of discretizing the mathematical theory of Exterior Differential Calculus, a theory that goes back to E. Cartan <span id='citeF-3'></span>[[#cite-3|[3]]] and is fundamental  in the areas of Differential Geometry and Differential Topology. Although Exterior Differential Calculus is an abstract mathematical theory, it has been introduced in various fields such as in digital geometry processing <span id='citeF-4'></span>[[#cite-4|[4]]],  numerical schemes for partial differential equations <span id='citeF-8'></span><span id='citeF-1'></span>[[#cite-8|[1,8]]], etc.
+The purpose of this paper is to introduce the theory of Discrete Exterior Calculus (DEC)  to the widest possible audience and, therefore, we will  rely mainly on Vector Calculus and Matrix Algebra. Discrete Exterior Calculus is a relatively new method for solving partial differential equations <span id='citeF-8'></span>[[#cite-8|[8]]] based on the idea of discretizing the mathematical theory of Exterior Differential Calculus, a theory that goes back to Cartan <span id='citeF-3'></span>[[#cite-3|[3]]] and is fundamental  in the areas of Differential Geometry and Differential Topology. Although Exterior Differential Calculus is an abstract mathematical theory, it has been introduced in various fields such as in digital geometry processing <span id='citeF-4'></span>[[#cite-4|[4]]],  numerical schemes for partial differential equations <span id='citeF-8'></span><span id='citeF-1'></span>[[#cite-8|[1,8]]], etc.
 In his PhD thesis <span id='citeF-8'></span>[[#cite-8|[8]]], Hirani laid down the fundamental concepts  of Discrete Exterior Calculus (DEC), using discrete combinatorial and geometric  operations on simplicial complexes (in any dimension), proposing discrete equivalents for differential forms, vector fields, differential and geometric operators, etc. Perhaps the first numerical application of DEC to PDE was given in <span id='citeF-9'></span>[[#cite-9|[9]]] in order to solve Darcy flow and Poisson's equation.  In <span id='citeF-7'></span>[[#cite-7|[7]]], the authors develop a modification of DEC and show that in simple cases (e.g. flat geometry and regular meshes), the equations resulting from DEC are equivalent to classical numerical schemes such as finite difference or finite volume discretizations. In <span id='citeF-11'></span>[[#cite-11|[11]]], the authors used DEC to solve the Navier-Stokes equations and,  in <span id='citeF-5'></span>[[#cite-5|[5]]] DEC was used with a discrete lattice model to simulate  elasticity, plasticity and failure of isotropic materials.

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 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
 |-
-| style="padding:10px;"| [[Image:Review_239195285364_6808_text-boundary04-eps-converted-to.png|250px|]]
+| style="padding:10px;"| [[Image:Review_239195285364_6808_text-boundary04-eps-converted-to.png|220px|]]
-| style="padding:10px;"|[[Image:Review_239195285364_2587_text-Dual00-eps-converted-to.png|250px|]]
+| style="padding:10px;"|[[Image:Review_239195285364_2587_text-Dual00-eps-converted-to.png|220px|]]
-| style="padding:10px;"|[[File:Review_239195285364_6445_Dual01-eps-converted-to.jpg|220px|]]
+| style="padding:10px;"|[[File:Review_239195285364_6445_Dual01-eps-converted-to.jpg|200px|]]
 |- style="text-align: center; font-size: 75%;"
 | (a)
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 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
 |-
-|style="padding:10px;"|  [[Image:Review_239195285364_3618_text-TwoTriangles01-eps-converted-to.png|220px|]]
+|style="padding:10px;"|  [[Image:Review_239195285364_3618_text-TwoTriangles01-eps-converted-to.png|120px|]]
-| style="padding:10px;"|  [[Image:Review_239195285364_1851_text-TwoTriangles02-eps-converted-to.png|220px|]]
+| style="padding:10px;"|  [[Image:Review_239195285364_1851_text-TwoTriangles02-eps-converted-to.png|120px|]]
 |- style="text-align: center; font-size: 75%;"
 | (a)
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 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
 |-
-| style="padding:10px;"|  [[Image:Review_239195285364_5980_text-Dual01a-eps-converted-to.png|160px|]]
+| style="padding:10px;"|  [[Image:Review_239195285364_5980_text-Dual01a-eps-converted-to.png|220px|]]
-| style="padding:10px;"| [[Image:Review_239195285364_7382_text-Dual01b-eps-converted-to.png|160px|n]]
+| style="padding:10px;"| [[Image:Review_239195285364_7382_text-Dual01b-eps-converted-to.png|220px|n]]
 |- style="text-align: center; font-size: 75%;"
 | colspan="2" style="padding:10px;"| '''Figure 13'''. Oriented dual mesh
@@ Line 886: / Line 886: @@
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
 |-
-| style="padding:10px;"|  [[Image:Review_239195285364_9611_text-Dual00a-eps-converted-to.png|160px]]
+| style="padding:10px;"|  [[Image:Review_239195285364_9611_text-Dual00a-eps-converted-to.png|220px]]
-| style="padding:10px;"| [[Image:Review_239195285364_4015_text-Dual00aa-eps-converted-to.png|160px]]
+| style="padding:10px;"| [[Image:Review_239195285364_4015_text-Dual00aa-eps-converted-to.png|220px]]
 |- style="text-align: center; font-size: 75%;"
 | colspan="2" style="padding:10px;"| '''Figure 14'''. Oriented dual mesh

@@ Line 1,340: / Line 1,340: @@
 The numerical results for the maximum temperature and flux magnitude values are exemplified in Table [[#5.2 Second example|2]].
 <div class="center" style="font-size: 75%;">'''Table 2'''. Numerical simulation results</div>
@@ Line 1,399: / Line 1,398: @@
 | style="text-align: center;border-bottom: 1pt solid black;"|602.19
 | style="text-align: center;border-bottom: 1pt solid black;"|602.19
 |}

← Older revision		Revision as of 12:11, 24 April 2020
Line 1,340:		Line 1,340:

	The numerical results for the maximum temperature and flux magnitude values are exemplified in Table [[#5.2 Second example\|2]].		The numerical results for the maximum temperature and flux magnitude values are exemplified in Table [[#5.2 Second example\|2]].
		+
		+
		+	<div class="center" style="font-size: 75%;">'''Table 2'''. Numerical simulation results</div>
		+
		+	{\| style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:75%;"
		+	\|-
		+	\| style="border-top: 1pt solid black;text-align: left;"\|'''Mesh'''
		+	\| style="border-top: 1pt solid black;text-align: center;"\|'''#nodes'''
		+	\| style="border-top: 1pt solid black;text-align: center;"\| '''#elements'''
		+	\| colspan="2" style="border-top: 1pt solid black;text-align: center;"\|'''Max. Temp. Value'''
		+	\| colspan="2" style="border-top: 1pt solid black;text-align: center;"\|'''Max. Flux Magnitude'''
		+	\|-
		+	\|
		+	\|
		+	\|
		+	\|style="text-align: center;"\|DEC
		+	\| style="text-align: center;"\|FEML
		+	\|style="text-align: center;"\|DEC
		+	\| style="text-align: center;"\|FEML
		+	\|-
		+	\| style="border-top: 1pt solid black;text-align: left;"\| Figure 21(a)
		+	\| style="border-top: 1pt solid black;text-align: center;"\|162
		+	\| style="border-top: 1pt solid black;text-align: center;"\|268
		+	\| style="border-top: 1pt solid black;text-align: center;"\|129.07
		+	\| style="border-top: 1pt solid black;text-align: center;"\|128.92
		+	\| style="border-top: 1pt solid black;text-align: center;"\|467.22
		+	\| style="border-top: 1pt solid black;text-align: center;"\|467.25
		+	\|-
		+	\| style="text-align: left;"\| Figure 21(b)
		+	\| style="text-align: center;"\|678
		+	\| style="text-align: center;"\|1223
		+	\| style="text-align: center;"\|129.71
		+	\| style="text-align: center;"\|129.70
		+	\| style="text-align: center;"\|547.87
		+	\| style="text-align: center;"\|548.42
		+	\|-
		+	\| style="text-align: left;"\| Figure 21(c)
		+	\| style="text-align: center;"\|9489
		+	\| style="text-align: center;"\|18284
		+	\| style="text-align: center;"\|130.00
		+	\| style="text-align: center;"\|130.00
		+	\| style="text-align: center;"\|591.58
		+	\| style="text-align: center;"\|591.63
		+	\|-
		+	\| style="text-align: left;"\|
		+	\| style="text-align: center;"\|27453
		+	\| style="text-align: center;"\|53532
		+	\| style="text-align: center;"\|130.01
		+	\| style="text-align: center;"\|130.01
		+	\| style="text-align: center;"\|597.37
		+	\| style="text-align: center;"\|597.38
		+	\|-
		+	\| style="text-align: left;border-bottom: 1pt solid black;"\|
		+	\| style="text-align: center;border-bottom: 1pt solid black;"\|651406
		+	\| style="text-align: center;border-bottom: 1pt solid black;"\|1295960
		+	\| style="text-align: center;border-bottom: 1pt solid black;"\|130.02
		+	\| style="text-align: center;border-bottom: 1pt solid black;"\|130.02
		+	\| style="text-align: center;border-bottom: 1pt solid black;"\|602.19
		+	\| style="text-align: center;border-bottom: 1pt solid black;"\|602.19
		+	\|}

	{\| class="formulaSCP" style="width: 100%; text-align: left;"		{\| class="formulaSCP" style="width: 100%; text-align: left;"

@@ Line 1,209: / Line 1,209: @@
 <div class="center" style="font-size: 75%;">'''Table 1'''. Maximum temperature and Flux magnitude values in the numerical simulation</div>
-{| style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:80%;"
+{| style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:75%;"
 |-
 |  style="border-top: 1pt solid black;text-align: left;"|'''Mesh'''
@@ Line 1,225: / Line 1,225: @@
 | style="text-align: center;"|FEML
 |-
-| style="border-top: 1pt solid black;text-align: center;"| Figure (a)
+| style="border-top: 1pt solid black;text-align: left;"| Figure (a)
 | style="border-top: 1pt solid black;text-align: center;"|9
 | style="border-top: 1pt solid black;text-align: center;"|8
@@ Line 1,233: / Line 1,233: @@
 | style="border-top: 1pt solid black;text-align: center;"|0.307
 |-
-| style="text-align: center;"| Figure (b)
+| style="text-align: left;"| Figure (b)
 |  style="text-align: center;"|17
 | style="text-align: center;"|20
@@ Line 1,241: / Line 1,241: @@
 | style="text-align: center;"|0.405
 |-
-| style="text-align: center;"| Figure (c)
+| style="text-align: left;"| Figure (c)
 |  style="text-align: center;"|41
 |  style="text-align: center;"|56
@@ Line 1,249: / Line 1,249: @@
 | style="text-align: center;"|0.453
 |-
-| style="text-align: center;"| Figure (d)
+| style="text-align: left;"| Figure (d)
 |  style="text-align: center;"|713
 |  style="text-align: center;"|1304
@@ Line 1,257: / Line 1,257: @@
 | style="text-align: center;"|0.492
 |-
-| style="text-align: center;border-bottom: 1pt solid black;"| Figure (e)
+| style="text-align: left;border-bottom: 1pt solid black;"| Figure (e)
 |  style="text-align: center;border-bottom: 1pt solid black;"|42298
 |  style="text-align: center;border-bottom: 1pt solid black;"|83346
@@ Line 1,264: / Line 1,264: @@
 | style="text-align: center;border-bottom: 1pt solid black;"|0.496
 | style="text-align: center;border-bottom: 1pt solid black;"|0.496
 |}

@@ Line 1,212: / Line 1,212: @@
 |-
 |  style="border-top: 1pt solid black;text-align: left;"|'''Mesh'''
-|  style="border-top: 1pt solid black;text-align: center;"|'''nodes'''
+|  style="border-top: 1pt solid black;text-align: center;"|'''#nodes'''
-|  style="border-top: 1pt solid black;text-align: center;"| '''elements'''
+|  style="border-top: 1pt solid black;text-align: center;"| '''#elements'''
 | colspan="2" style="border-top: 1pt solid black;text-align: center;"|'''Max. Temp. Value'''
-| colspan="2"style="border-top: 1pt solid black;text-align: center;"|'''Max. Flux Magnitude'''
+| colspan="2" style="border-top: 1pt solid black;text-align: center;"|'''Max. Flux Magnitude'''
 |-
+|
@@ Line 1,264: / Line 1,264: @@
 | style="text-align: center;border-bottom: 1pt solid black;"|0.496
 | style="text-align: center;border-bottom: 1pt solid black;"|0.496
+|}