From: frohne Date: Sun, 24 Feb 2013 18:51:12 +0000 (+0000) Subject: changes in section 7 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=7ddff44f3e9143c7a277b51d468ca07502b7b302;p=dealii-svn.git changes in section 7 git-svn-id: https://svn.dealii.org/trunk@28538 0785d39b-7218-0410-832d-ea1e28bc413d --- diff --git a/deal.II/examples/step-42/doc/intro-step-42.tex b/deal.II/examples/step-42/doc/intro-step-42.tex index d7f35467d3..efc55f8804 100644 --- a/deal.II/examples/step-42/doc/intro-step-42.tex +++ b/deal.II/examples/step-42/doc/intro-step-42.tex @@ -283,12 +283,12 @@ method for the contact. It sums up the results of the sections before and works \mathcal{F}_k = \emptyset$ and set $k = 1$. The start value $\hat U^0 := P_{\mathcal{A}_k}(0)$ fullfills our obstacle condition. \item[(1)] Assemble the Newton matrix $A := a'(\hat - U^{k-1};\varphi_i,\varphi_j)$ and the right-hand-side $F(\hat U^{k-1})$. + U^{k-1};\varphi_p,\varphi_q)$ and the right-hand-side $F(\hat U^{k-1})$. \item[(2)] Find the primal-dual pair $(\bar U^k,\Lambda^k)$ that satisfies \begin{align*} A\bar U^k + B\Lambda^k & = F, &\\ - \left[B^T\bar U^k\right]_i & = G_i & \forall i\in\mathcal{A}_k,\\ - \Lambda^k_i & = 0 & \forall i\in\mathcal{F}_k. + \left[B^T\bar U^k\right]_p & = G_p & \forall p\in\mathcal{A}_k,\\ + \Lambda^k_p & = 0 & \forall p\in\mathcal{F}_k. \end{align*} \item[(3)] Damping for $k>2$ by applying a line search and calculating a linear combination of $U^{k-1}$ and $\bar U^k$. Find an @@ -296,10 +296,10 @@ method for the contact. It sums up the results of the sections before and works (1-\alpha_i)U^{k-1}$$ yields $$\vert F\left(U^{k}\right) \vert < \vert F\left(U^{k-1}\right) \vert.$$ \item[(4)] Define the new active and inactive sets by - $$\mathcal{A}_{k+1}:=\lbrace i\in\mathcal{S}:\Lambda^k_i + - c\left(\left[B^TU^k\right]_i - G_i\right) > 0\rbrace,$$ - $$\mathcal{F}_{k+1}:=\lbrace i\in\mathcal{S}:\Lambda^k_i + - c\left(\left[B^TU^k\right]_i - G_i\right) \leq 0\rbrace.$$ + $$\mathcal{A}_{k+1}:=\lbrace p\in\mathcal{S}:\Lambda^k_p + + c\left(\left[B^TU^k\right]_p - G_p\right) > 0\rbrace,$$ + $$\mathcal{F}_{k+1}:=\lbrace p\in\mathcal{S}:\Lambda^k_p + + c\left(\left[B^TU^k\right]_p - G_p\right) \leq 0\rbrace.$$ Projection $U^k$ so that it holds the second equation in (2) $$\hat U^K := P_{\mathcal{A}_{k+1}}(U^k).$$ \item[(5)] If $\mathcal{A}_{k+1} = \mathcal{A}_k$ and $\vert @@ -307,32 +307,46 @@ method for the contact. It sums up the results of the sections before and works step (1). \end{itemize} \noindent -The meaning of the decorated and none decorated $U$ is as follows: $\bar U$ -denotes the solution of the linear system of equations in (2), $U$ is the +The subscript $p$ denotes a vertex and the meaning of the +decorated and none decorated $U$ is as follows: $\bar U$ denotes the solution of the linear system of equations in (2), $U$ is the damped solution and equals to $\bar U$ if the damping parameter $\alpha_0 = 1$ and $\hat U := P_{\mathcal{A}}(U)$ is the projection of the active components in $\mathcal{A}$ to the gap $$P_{\mathcal{A}}(U):=\begin{cases} -U_i, & \textrm{if}\quad i\notin\mathcal{A}\\ -{}^{G_i}/_{B^T_i}, & \textrm{if}\quad -i\in\mathcal{A}. +U_p, & \textrm{if}\quad p\notin\mathcal{A}\\ +g_{h,p}, & \textrm{if}\quad +p\in\mathcal{A}. \end{cases}$$\\ -The mass matrix $B\in\mathbb{R}^{n\times m}$, $n>m$, is quadratic in our situation since $\Lambda^k$ is only defined on $\Gamma_C$: -$$B_{ij} = \begin{cases} -\int\limits_{\Gamma_C}\varphi_i^2(x)dx, & \text{if}\quad i=j\\ -0, & \text{if}\quad i\neq j. +The matrix $B\in\mathbb{R}^{n\times m}$, $n>m$ describes the coupling of the +bases for the displacements and lagrange multiplier (contact forces) +and it is not quadratic in our situation since $\Lambda^k$ is only defined on +$\Gamma_C$. Due to the ansatz functions $\psi_i$ (scalar valued) of the +lagrange multiplier are fullfilling the following biorthogonal condition (see Hüeber, Wohlmuth: A primal–dual active +set strategy for non-linear multibody contact problems, Comput. Methods Appl. Mech. Engrg. +194, 2005, pp. 3147-3166) +$$ \int\limits_{\Gamma_C}\psi_i(x)\varphi_j(x)dx = +\delta_{ij}\int\limits_{\Gamma_C}\varphi_j(x)dx$$ +this yields +$$B_{pq} = \begin{cases} +\int\limits_{\Gamma_C}\varphi_p(x)dxI_3, & \text{if}\quad p=q,\quad +p,q\in\mathcal{S}\\ +0I_3, & \text{if}\quad p\neq q,\quad p\textrm{ or }q\notin\mathcal{S}. \end{cases}$$ -So $m$ denotes the size of $\Lambda^k$ and $i$ a degree of freedom. In our -programm we use the structure of a quadratic sparse for $B\in\mathbb{R}^{n\times -n}$ and the length of $\Lambda^k$ is $n$ with $\Lambda^k_i = 0$ for $i>m$. +Here $I_3$ denotes the threedimensional identity matrix. +In our programm we use the structure of a quadratic sparse for +$B\in\mathbb{R}^{n\times n}$ and for $\Lambda^k$ a vector with length $n$ where +$\Lambda^k_p = 0$ for $p\notin \mathcal{S}$. The vector $G$ is defined by a suitable approximation $g_h$ of the gap $g$ -$$G_i = \begin{cases} -\int\limits_{\Gamma_C}g_h(x)\varphi_i(x)dx, & \text{if}\quad i\leq m\\ -0, & \text{if}\quad i>m. +$$G_p = \begin{cases} +g_{h,p}\int\limits_{\Gamma_C}\varphi_p(x)dx, & \text{if}\quad p\in\mathcal{S}\\ +0, & \text{if}\quad p\notin\mathcal{S}. \end{cases}$$\\ -Compared to step-41, step (1) is added but it should be clear -from the sections above that we only linearize the problem. In step (2) we have -to solve a linear system of equations again. And now the solution has to fulfill two stopping +Note that $G_p$ is a threedimensional vector and that again we applied the +biorthogonal property of the lagrange multiplier ansatz functions to the +integral $\int\limits_{\Gamma_C}g_h(x)\varphi_p(x)dx$ with $g_h(x)=\sum\limits_i +g_{h,p}\varphi_p(x)$ (see the reference mentioned above).\\ +Compared to step-41, step (1) is added but it should be clear from the sections +above that we only linearize the problem. In step (2) we have to solve a linear system of equations again. And now the solution has to fulfill two stopping criteria. $\mathcal{A}_{k+1} = \mathcal{A}_k$ makes sure that the contact zones are iterated out and the second ensures an accurate enough residual which means that the plastic zones are also iterated out.\\ A similar method can also be found in Brunssen, Schmid, Schaefer, Wohlmuth: A diff --git a/deal.II/examples/step-42/doc/intro.dox b/deal.II/examples/step-42/doc/intro.dox index 94dd0077be..96c08feac8 100644 --- a/deal.II/examples/step-42/doc/intro.dox +++ b/deal.II/examples/step-42/doc/intro.dox @@ -296,12 +296,12 @@ method for the contact. It sums up the results of the sections before and works \mathcal{F}_k = \emptyset$ and set $k = 1$. The start value $\hat U^0 := P_{\mathcal{A}_k}(0)$ fullfills our obstacle condition.
  • Assemble the Newton matrix $A := a'(\hat - U^{k-1};\varphi_i,\varphi_j)$ and the right-hand-side $F(\hat U^{k-1})$. + U^{k-1};\varphi_p,\varphi_q)$ and the right-hand-side $F(\hat U^{k-1})$.
  • Find the primal-dual pair $(\bar U^k,\Lambda^k)$ that satisfies @f{align*} A\bar U^k + B\Lambda^k & = F, &\\ - \left[B^T\bar U^k\right]_i & = G_i & \forall i\in\mathcal{A}_k,\\ - \Lambda^k_i & = 0 & \forall i\in\mathcal{F}_k. + \left[B^T\bar U^k\right]_p & = G_p & \forall p\in\mathcal{A}_k,\\ + \Lambda^k_p & = 0 & \forall p\in\mathcal{F}_k. @f}
  • Damping for $k>2$ by applying a line search and calculating a linear combination of $U^{k-1}$ and $\bar U^k$. Find an @@ -311,10 +311,10 @@ method for the contact. It sums up the results of the sections before and works yields @f{gather*}\vert F\left(U^{k}\right) \vert < \vert F\left(U^{k-1}\right) \vert.\f}
  • Define the new active and inactive sets by - @f{gather*}\mathcal{A}_{k+1}:=\lbrace i\in\mathcal{S}:\Lambda^k_i + - c\left(\left[B^TU^k\right]_i - G_i\right) > 0\rbrace,@f} - @f{gather*}\mathcal{F}_{k+1}:=\lbrace i\in\mathcal{S}:\Lambda^k_i + - c\left(\left[B^TU^k\right]_i - G_i\right) \leq 0\rbrace.@f} + @f{gather*}\mathcal{A}_{k+1}:=\lbrace p\in\mathcal{S}:\Lambda^k_p + + c\left(\left[B^TU^k\right]_p - G_p\right) > 0\rbrace,@f} + @f{gather*}\mathcal{F}_{k+1}:=\lbrace p\in\mathcal{S}:\Lambda^k_p + + c\left(\left[B^TU^k\right]_p - G_p\right) \leq 0\rbrace.@f} Projection $U^k$ so that it holds the second equation in (2) @f{gather*}\hat U^K := P_{\mathcal{A}_{k+1}}(U^k).@f}
  • If $\mathcal{A}_{k+1} = \mathcal{A}_k$ and $\vert @@ -322,31 +322,47 @@ method for the contact. It sums up the results of the sections before and works step (1). -The meaning of the decorated and none decorated $U$ is as follows: $\bar U$ -denotes the solution of the linear system of equations in (2), $U$ is the +The subscript $p$ denotes a vertex and the meaning of the +decorated and none decorated $U$ is as follows: $\bar U$ denotes the solution of the linear system of equations in (2), $U$ is the damped solution and equals to $\bar U$ if the damping parameter $\alpha_0 = 1$ and $\hat U := P_{\mathcal{A}}(U)$ is the projection of the active components in $\mathcal{A}$ to the gap @f{gather*}P_{\mathcal{A}}(U):=\begin{cases} -U_i, & \textrm{if}\quad i\notin\mathcal{A}\\ -{}^{G_i}/_{B^T_i}, & \textrm{if}\quad -i\in\mathcal{A}. +U_p, & \textrm{if}\quad p\notin\mathcal{A}\\ +g_{h,p}, & \textrm{if}\quad +p\in\mathcal{A}. \end{cases}@f}\\ -The mass matrix $B\in\mathbb{R}^{n\times m}$, $n>m$, is quadratic in our situation since $\Lambda^k$ is only defined on $\Gamma_C$: -@f{gather*}B_{ij} = \begin{cases} -\int\limits_{\Gamma_C}\varphi_i^2(x)dx, & \text{if}\quad i=j\\ -0, & \text{if}\quad i\neq j. +The matrix $B\in\mathbb{R}^{n\times m}$, $n>m$ describes the coupling of the +bases for the displacements and lagrange multiplier (contact forces) +and it is not quadratic in our situation since $\Lambda^k$ is only defined on +$\Gamma_C$. Due to the ansatz functions $\psi_i$ (scalar valued) of the +lagrange multiplier are fullfilling the following biorthogonal condition (see Hüeber, Wohlmuth: A primal–dual active +set strategy for non-linear multibody contact problems, Comput. Methods Appl. Mech. Engrg. +194, 2005, pp. 3147-3166) +@f{gather} \int\limits_{\Gamma_C}\psi_i(x)\varphi_j(x)dx = +\delta_{ij}\int\limits_{\Gamma_C}\varphi_j(x)dx@f} +this yields +@f{gather*}B_{pq} = \begin{cases} +\int\limits_{\Gamma_C}\varphi_p(x)dxI_3, & \text{if}\quad p=q,\quad +p,q\in\mathcal{S}\\ +0I_3, & \text{if}\quad p\neq q,\quad p\textrm{ or }q\notin\mathcal{S}. \end{cases}@f} -So $m$ denotes the size of $\Lambda^k$ and $i$ a degree of freedom. In our -programm we use the structure of a quadratic sparse for $B\in\mathbb{R}^{n\times -n}$ and the length of $\Lambda^k$ is $n$ with $\Lambda^k_i = 0$ for $i>m$. +Here $I_3$ denotes the threedimensional identity matrix. +In our programm we use the structure of a quadratic sparse for +$B\in\mathbb{R}^{n\times n}$ and for $\Lambda^k$ a vector with length $n$ where +$\Lambda^k_p = 0$ for $p\notin \mathcal{S}$. The vector $G$ is defined by a suitable approximation $g_h$ of the gap $g$ -@f{gather*}G_i = \begin{cases} -\int\limits_{\Gamma_C}g_h(x)\varphi_i(x)dx, & \text{if}\quad i\leq m\\ -0, & \text{if}\quad i>m. +@f{gather*}G_p = \begin{cases} +g_{h,p}\int\limits_{\Gamma_C}\varphi_p(x)dx, & \text{if}\quad p\in\mathcal{S}\\ +0, & \text{if}\quad p\notin\mathcal{S}. \end{cases}@f} +Note that $G_p$ is a threedimensional vector and that again we applied the +biorthogonal property of the lagrange multiplier ansatz functions to the +integral $\int\limits_{\Gamma_C}g_h(x)\varphi_p(x)dx$ with $g_h(x)=\sum\limits_i +g_{h,p}\varphi_p(x)$ (see the reference mentioned above). + Compared to step-41, step (1) is added but it should be clear from the sections above that we only linearize the problem. In step (2) we have to solve a linear system of equations again. And now the solution has to fulfill two stopping