Onate et al 2015a - Revision history

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← Older revision		Revision as of 13:40, 5 October 2018
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−	Published in ''Comput. Meth. Appl. Mech. Engng.'', Vol. 293, pp. 191-206, 2015	+	Published in ''Comput. Meth. Appl. Mech. Engng.'', Vol. 293, pp. 191-206, 2015<br />
		+	10.1016/j.cma.2013.12.009

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 |}
-==References==
+==REFERENCES==
 <div id="cite-1"></div>

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 |}
-Figures 7a and 7c show the pressure solution <math display="inline">p^h</math> obtained by the P1-P0+ triangle using a structured mesh (<math display="inline">2 \times 30 \times 30</math> elements) and an unstructured mesh (<math display="inline">2 \times 30 \times 30</math> elements), respectively. The corresponding best approximations in the <math display="inline">\hbox{L}^{2}</math> norm <math display="inline">\mathcal{P}^{h}_{\hbox{L}^{2}}p</math> are shown in figures  fig_ex4_smesh_tri_grid_30_p_oL2 and [[#img-7|7]], respectively.
+Figures 7a and 7c show the pressure solution <math display="inline">p^h</math> obtained by the P1-P0+ triangle using a structured mesh (<math display="inline">2 \times 30 \times 30</math> elements) and an unstructured mesh (<math display="inline">2 \times 30 \times 30</math> elements), respectively. The corresponding best approximations in the <math display="inline">\hbox{L}^{2}</math> norm <math display="inline">\mathcal{P}^{h}_{\hbox{L}^{2}}p</math> are shown in Figures  7b and [[#img-7|7d]], respectively.
 Figure [[#img-6|6b]] shows the convergence of the error <math display="inline">(p - p^{h})</math> in the <math display="inline">\hbox{L}^{2}</math> norm with respect to mesh refinement. This error is compared with the best approximation error <math display="inline">(p - \mathcal{P}^{h}_{\hbox{L}^{2}}p)</math>. As expected for a piecewise constant pressure approximation, the convergence rate is found to be one for both solutions. Further, the error line for the numerical solution <math display="inline">p^{h}</math> is found to be close to the error line of the best approximation <math display="inline">\mathcal{P}^{h}_{\hbox{L}^{2}}p</math>, indicating the good accuracy of the obtained solution. Additionally, in Figure [[#img-6|6b]] the convergence in the <math display="inline">\hbox{L}^{2}</math> norm of the divergence of the velocity is also shown.  As expected for a piecewise linear velocity approximation, a first-order convergence rate is found for <math display="inline">\| \boldsymbol \nabla  \cdot {\bf v}^{h} \| </math>.

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 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>{v}(x,y) = \begin{bmatrix}1 - x \\  y \end{bmatrix} , \quad  p(x,y) = \left\{ \begin{array}{ll}\rho \left[x - \dfrac{1}{2}(x^{2} + y^{2}) -\dfrac{7}{24}\right]+ (\mu _{1} - \mu _{2}) & \forall  y > 0
+| style="text-align: center;" | <math>{\bf v}(x,y) = \begin{bmatrix}1 - x \\  y \end{bmatrix} , \quad  p(x,y) = \left\{ \begin{array}{ll}\rho \left[x - \dfrac{1}{2}(x^{2} + y^{2}) -\dfrac{7}{24}\right]+ (\mu _{1} - \mu _{2}) & \forall  y > 0
 \\  \rho \left[x - \dfrac{1}{2}(x^{2} + y^{2}) -\dfrac{7}{24}\right]- (\mu _{1} - \mu _{2}) & \forall  y < 0 \end{array} \right. </math>

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 where <math display="inline">N</math> is the number of nodes in the mesh. The different matrices and vectors in Eqs.([[#6 FEM DISCRETIZATION|6]]a,b) are assembled from the element contributions given in Box 1 for 3-noded triangles.
-<div class="center"
+<div class="center">
 {| class="formulaSCP" style="width: 80%; text-align: center;"
 |-style="text-align: center;

← Older revision	Revision as of 14:14, 10 October 2018
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-{| class="formulaSCP" style="width: 80%; text-align: left;"
+{| class="formulaSCP" style="width: 80%; text-align: center;"
 |-
+|
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 \left[\!\!\left[ 2\mu \frac{v_n}{n}+ ({\bf x}_c^T - {\bf x}_{er}^T)  {\bf b}-{t}_n  \right]\!\!\right]\right)
 \end{array}</math>
 |-style="text-align: left; margin:auto;width: 80%;font-size:85%;"
 |where <math>l_{er}</math> is the length of the side connecting elements ''e'' and ''r'' and <math>{\bf x}_{er}</math>  is the mid-point of the side. For continuous body forces between elements <math>\left[\!\left[{\bf x}_{er}^T {\bf b} \right]\!\right]=0</math>
 |-style="text-align: left; margin:auto;width: 80%;font-size:85%;"
 |The expression of <math>\tau^{er}</math> is given in Eqs.(45) and (46).
 |}