Ferrer et al 2017a - Revision history

Scipediacontent at 09:02, 22 March 2018

2018-03-22T09:02:45Z

Cinmemj at 11:11, 21 March 2018

2018-03-21T11:11:23Z

Cinmemj: Cinmemj moved page Draft Samper 118254298 to Ferrer et al 2017a

2018-03-20T13:56:07Z

Cinmemj moved page Draft Samper 118254298 to Ferrer et al 2017a

Cinmemj at 13:49, 20 March 2018

2018-03-20T13:49:16Z

Cinmemj at 13:35, 20 March 2018

2018-03-20T13:35:59Z

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Cinmemj at 12:35, 20 March 2018

2018-03-20T12:35:56Z

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Cinmemj at 12:08, 20 March 2018

2018-03-20T12:08:01Z

Cinmemj at 11:51, 20 March 2018

2018-03-20T11:51:38Z

Cinmemj at 11:45, 20 March 2018

2018-03-20T11:45:00Z

Cinmemj at 11:39, 20 March 2018

2018-03-20T11:39:48Z

@@ Line 795: / Line 795: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\chi =\begin{array}{ccc}1 &  & x\in \Omega ^{+}\\ 0 &  & x\in \Omega ^{-} \end{array}</math>
+| style="text-align: center;" | <math>\chi =\begin{cases}\begin{array}{ccc}1 &  & x\in \Omega ^{+}\\ 0 &  & x\in \Omega ^{-} \end{array}\end{cases} </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (2.42)
@@ Line 866: / Line 866: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\mathbb{C}\mbox{( }\chi \mbox{)}=\left\{\begin{array}{ccc}\mathbb{C\hbox{+}} &  & \chi=1,\\ \mathbb{C}^{-} &  & \chi=0, \end{array}\right.</math>
+| style="text-align: center;" | <math>\mathbb{C}\mbox{( }\chi \mbox{)}=\begin{cases}\begin{array}{ccc}\mathbb{C\hbox{+}} &  & \chi=1,\\ \mathbb{C}^{-} &  & \chi=0, \end{array}\end{cases} </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (2.47)

@@ Line 1,854: / Line 1,854: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>(z^{*})^{2}=z_{1}^{2}  c(\psi )>-\frac{\lambda }{\rho }</math>
+| style="text-align: center;" | <math>(z^{*})^{2}=\left\{ \begin{array}{ll}z_{1}^{2}  & c(\psi )>-\frac{\lambda }{\rho }\\
-|-
+ z_{2}^{2}  & c(\psi )<-\frac{\lambda }{\rho }\end{array}\right.</math>
-| style="text-align: center;" | <math> z_{2}^{2}  c(\psi )<-\frac{\lambda }{\rho } </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (3.47)

@@ Line 1,381: / Line 1,381: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>x\in \Omega ^{+}:D_{T}\mathcal{J}(\hat{x})=\mathbb{P}\sigma (u)(\widehat{x})\cdot \nabla ^{s}u(\widehat{x})\geq{0},\qquad ({\bf Case~A})} </math>
+| style="text-align: center;" | <math>x\in \Omega ^{+}:D_{T}\mathcal{J}(\hat{x})=\mathbb{P}\sigma (u)(\widehat{x})\cdot \nabla ^{s}u(\widehat{x})\geq{0},\qquad ({\bf Case~A}) </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (3.19)
@@ Line 1,392: / Line 1,392: @@
 {| style="text-align: left; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>x\in \Omega ^{-}:D_{T}\mathcal{J}(\hat{x})=\mathbb{P}\sigma (u)(\widehat{x})\cdot \nabla ^{s}u(\widehat{x})\leq{0.}\qquad \hbox{'''({\bf Case~B})'''} </math>
+| style="text-align: center;" | <math>x\in \Omega ^{-}:D_{T}\mathcal{J}(\hat{x})=\mathbb{P}\sigma (u)(\widehat{x})\cdot \nabla ^{s}u(\widehat{x})\leq{0.}\qquad ({\bf Case~B}) </math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | (3.20)
@@ Line 4,166: / Line 4,166: @@
 |-
 | style="text-align: center;" | <math> \begin{array}{lll} |\widetilde{u}_{\epsilon }|_{H^{1}(\Omega ;R^{2})}  & = & |u_{\epsilon }-u-\epsilon w|_{H^{1}(\Omega ;R^{2})}\\
-&  \leq  |u_{\epsilon }-u|_{H^{1}(\Omega ;R^{2})}+\epsilon |w|_{H^{1}(\Omega ;R^{2})}\\
+&  \leq  & |u_{\epsilon }-u|_{H^{1}(\Omega ;R^{2})}+\epsilon |w|_{H^{1}(\Omega ;R^{2})}\\
-&  & \leq  \| u_{\epsilon }-u\| _{H^{1}(\Omega ;R^{2})}+\epsilon |w|_{H^{1}(R^{2};R^{2})}\\
+& \leq  & \| u_{\epsilon }-u\| _{H^{1}(\Omega ;R^{2})}+\epsilon |w|_{H^{1}(R^{2};R^{2})}\\
 & \leq & C_{1}\epsilon \,,\end{array} </math>
 |}

@@ Line 7,530: / Line 7,530: @@
 |}
-===BIBLIOGRAPHY===
+==BIBLIOGRAPHY==
 <div id="cite-1"></div>

@@ Line 2,027: / Line 2,027: @@
 As an advantage, this scheme allows using the same implementation for both cases; equalities or inequalities; by means of exchanging the constraint <math display="inline">c(\psi _{n})</math> with the constraint <math display="inline">r(\psi _{n},\lambda _{n})</math>. For further information, the reader is referred to the reference book <span id='citeF-71'></span>[[#cite-71|[71]]] for a rigorous description and work <span id='citeF-72'></span>[[#cite-72|[72]]] for applying to topological optimization problem.
-==Penalty value and algorithm==
+'''Penalty value and algorithm'''. It is worth mentioning that giving an adequate value to the penalty variable is not an easy task. In the reference book <span id='citeF-71'></span>[[#cite-71|[71]]], it is suggested increasing the penalty during the iterations (when the constraint is not tightened enough). Similarly, in work <span id='citeF-77'></span>[[#cite-77|[77]]], it is proposed to increase the penalty every five iterations. However, our experience shows us that the penalty cannot increase unlimitedly. A very small value of the penalty will make the problem converge very slowly but if, on the contrary, is very high the solution oscillates with the risk of non-convergence. So, as it was mentioned before, the appropiate value of this parameter will depend on the problem and on the sensibility of the cost when varying the constraint. The numerical experiments, further presented, suggest to normalize the cost function and take a small value of the penalty.
-−
 It is worth mentioning that giving an adequate value to the penalty variable is not an easy task. In the reference book <span id='citeF-71'></span>[[#cite-71|[71]]], it is suggested increasing the penalty during the iterations (when the constraint is not tightened enough). Similarly, in work <span id='citeF-77'></span>[[#cite-77|[77]]], it is proposed to increase the penalty every five iterations. However, our experience shows us that the penalty cannot increase unlimitedly. A very small value of the penalty will make the problem converge very slowly but if, on the contrary, is very high the solution oscillates with the risk of non-convergence. So, as it was mentioned before, the appropiate value of this parameter will depend on the problem and on the sensibility of the cost when varying the constraint. The numerical experiments, further presented, suggest to normalize the cost function and take a small value of the penalty.
 A detailed scheme of the Augmented Lagrangian Slerp algorithm is presented in Algorithm [[#algorithm-1|1]].
+<span id='algorithm-1'></span>
 {| style="margin: 1em auto;border: 1px solid darkgray;font-size:85%;"
 |-
@@ Line 2,066: / Line 2,064: @@
 ::     Compute <math display="inline">\hat{g}_{n+1}</math> from ([[#eq-3.58|3.58]]) with <math display="inline">g</math> computed from ([[#eq-3.18|3.18]]) and set <math display="inline">n = n+1</math>.
 |}
 ==3.4 Treatment of the interface ==

← Older revision	Revision as of 13:56, 20 March 2018
(No difference)

@@ Line 2,067: / Line 2,067: @@
 ::     Compute <math display="inline">\hat{g}_{n+1}</math> from ([[#eq-3.58|3.58]]) with <math display="inline">g</math> computed from ([[#eq-3.18|3.18]]) and set <math display="inline">n = n+1</math>.
 |-
-| style="text-align: center; font-size: 75%;"|
+| style="text-align: center; "|
 <span id='algorithm-1'></span>'''Algorithm. 1''' Augmented Lagrangian slerp algorithm. The Lagrange multiplier is updated in every topology iteration, i.e.,'' Usawa''-like scheme is used.
 |}

@@ Line 2,037: / Line 2,037: @@
 |-
+|
+: '''Init:''' choose initial values of <math display="inline">\psi _0</math>, <math display="inline">\theta _{min}</math>, tol, <math display="inline">\kappa _{min}</math>, <math display="inline">\lambda _0</math> and <math display="inline">\rho </math>
-: '''Init:''' choose initial values of <math display="inline">\psi _0</math>, <math display="inline">\theta _{min}</math>, tol, <math display="inline">\kappa _{min}</math>, <math display="inline">\lambda _0</math> and <math display="inline">\rho </math>
+: Compute <math display="inline">\sigma _0</math> and <math display="inline">u_0</math> from ([[#eq-2.46|2.46]]).
-: Compute <math display="inline">\sigma _0</math> and <math display="inline">u_0</math> from ([[#eq-2.46|2.46]]).
+: Compute <math display="inline">r_0</math> from ([[#eq-3.56|3.56]]) with <math display="inline">c_0</math> from ([[#eq-3.10|3.10]]).
-: Compute <math display="inline">r_0</math> from ([[#eq-3.56|3.56]]) with <math display="inline">c_0</math> from ([[#eq-3.10|3.10]]).
+: Compute <math display="inline">\hat{g}_0</math> from ([[#eq-3.58|3.58]]) with <math display="inline">g</math> computed from ([[#eq-3.18|3.18]]).
-: Compute <math display="inline">\hat{g}_0</math> from ([[#eq-3.58|3.58]]) with <math display="inline">g</math> computed from ([[#eq-3.18|3.18]]).
+: '''while''' <math>\theta _{n} \geq \theta _{min} \mathbf{or} r_n \geq tol</math> '''do'''
-: '''while''' <math>\theta _{n} \geq \theta _{min} \mathbf{or} r_n \geq tol</math> '''do'''
+::      Set <math display="inline">\kappa = 1</math>, <math display="inline">k=1</math>, <math display="inline">\mathcal{L}_k = \mathcal{L}_n</math>, <math display="inline">\theta _k = \theta _n</math>.
-:      Set <math display="inline">\kappa = 1</math>, <math display="inline">k=1</math>, <math display="inline">\mathcal{L}_k = \mathcal{L}_n</math>, <math display="inline">\theta _k = \theta _n</math>.
+::   '''while''' <math>\mathcal{L}_k \geq \mathcal{L}_n</math> (line search) '''do'''
-:   '''while''' <math>\mathcal{L}_k \geq \mathcal{L}_n</math> (line search) '''do'''
+:::       Update <math display="inline">\psi _k</math> from ([[#eq-3.33|3.33]]) with <math display="inline">\kappa </math>, <math display="inline">\theta _k</math> and <math display="inline">\hat{g}_n</math>, and update <math display="inline">\chi _k</math> from ([[#eq-3.24|3.24]]).
-:       Update <math display="inline">\psi _k</math> from ([[#eq-3.33|3.33]]) with <math display="inline">\kappa </math>, <math display="inline">\theta _k</math> and <math display="inline">\hat{g}_n</math>, and update <math display="inline">\chi _k</math> from ([[#eq-3.24|3.24]]).
+:::       Compute <math display="inline">\sigma _k</math> and <math display="inline">u_k</math> from ([[#eq-2.46|2.46]]).
-:       Compute <math display="inline">\sigma _k</math> and <math display="inline">u_k</math> from ([[#eq-2.46|2.46]]).
+:::      Compute <math display="inline">\mathcal{L}_k</math> from ([[#eq-3.50|3.50]]), with <math display="inline">\mathcal{J}_k</math> from ([[#eq-3.16|3.16]]) and set <math display="inline">\kappa = \kappa /2</math> and <math display="inline">k = k+1</math>.
-:      Compute <math display="inline">\mathcal{L}_k</math> from ([[#eq-3.50|3.50]]), with <math display="inline">\mathcal{J}_k</math> from ([[#eq-3.16|3.16]]) and set <math display="inline">\kappa = \kappa /2</math> and <math display="inline">k = k+1</math>.
+::     Set <math display="inline">\sigma _{n+1} = \sigma _k</math>, <math display="inline">u_{n+1} = u_k</math>, <math display="inline">\mathcal{L}_{n+1} = \mathcal{L}_k</math>, <math display="inline">\psi _{n+1}=\psi _k</math>
-:     Set <math display="inline">\sigma _{n+1} = \sigma _k</math>, <math display="inline">u_{n+1} = u_k</math>, <math display="inline">\mathcal{L}_{n+1} = \mathcal{L}_k</math>, <math display="inline">\psi _{n+1}=\psi _k</math>
+::     Compute <math display="inline">\theta _{n+1}</math> from ([[#eq-3.34|3.34]])
-:     Compute <math display="inline">\theta _{n+1}</math> from ([[#eq-3.34|3.34]])
+::     Compute <math display="inline">r_{n+1}</math> from ([[#eq-3.56|3.56]]) with <math display="inline">r_0</math> from ([[#eq-3.10|3.10]]).
-−
-:     Compute <math display="inline">r_{n+1}</math> from ([[#eq-3.56|3.56]]) with <math display="inline">r_0</math> from ([[#eq-3.10|3.10]]).
-−
-:     Update <math display="inline">\lambda _{n+1}</math> from ([[#eq-3.57|3.57]]).
-−
-:     Compute <math display="inline">\hat{g}_{n+1}</math> from ([[#eq-3.58|3.58]]) with <math display="inline">g</math> computed from ([[#eq-3.18|3.18]]) and set <math display="inline">n = n+1</math>.
 |-
 | style="text-align: center; font-size: 75%;"|