From 30c01ee8d1c042173847724159f818d311be32b7 Mon Sep 17 00:00:00 2001 From: bangerth Date: Fri, 13 Apr 2012 08:24:21 +0000 Subject: [PATCH] Commit Sven's latest version. git-svn-id: https://svn.dealii.org/trunk@25409 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/examples/step-15/doc/intro.dox | 75 +++++++++++++----------- deal.II/examples/step-15/doc/results.dox | 28 +++++---- deal.II/examples/step-15/step-15.cc | 36 +++++------- 3 files changed, 70 insertions(+), 69 deletions(-) diff --git a/deal.II/examples/step-15/doc/intro.dox b/deal.II/examples/step-15/doc/intro.dox index d10cb2e0aa..526da2a527 100644 --- a/deal.II/examples/step-15/doc/intro.dox +++ b/deal.II/examples/step-15/doc/intro.dox @@ -1,15 +1,18 @@ +/** + @page mse Minimal surface equation +

Introduction

Foreword

This programm deals with an example of a non-linear elliptic pde, the minimal -surface equation. You can imagine the solution as a soap bubble inside a -closed wire, where the wire isn't smooth, but curved. The soap bubble will +surface equation. You can imagine the solution as a soap bubble inside a +closed wire, where the wire isn't smooth, but curved. The soap bubble will take a shape with minimal surface. The solution of the minimal surface equation -describes this shape with the wire as a boundary condition. +describes this shape with the wire as a boundary condition. -Because the equation is non-linear, we can't solve it directly, but have to use +Because the equation is non-linear, we can't solve it directly, but have to use the newton-method to compute the solution iterativly. @@ -26,10 +29,10 @@ In a classical sense, the problem posseses following form: @f[ u=g \qquad\qquad on ~ \partial \Omega @f] - + In this example, we choose the unitball as our domain $\Omega$. -As described above, we have to formulate the Newton-method for this problem +As described above, we have to formulate the Newton-method for this problem with a damping parameter $\lambda$ to have a better global convergence behaviour: @par @@ -39,13 +42,13 @@ with a damping parameter $\lambda$ to have a better global convergence behaviour @f[ u^{n+1}=u^{n}+\delta u^{n} @f] - + with: @f[ F(u):= -\nabla \cdot \left( \frac{1}{\sqrt{1+|\nabla u|^{2}}}\nabla u \right) @f] - + and $F'(u,\delta u)$ the derivative of F in direction of $\delta u$: @f[ @@ -57,26 +60,26 @@ So we have to solve a linear elliptic pde in every Newton-step, with $\delta u$ the solution of: @f[ - - \nabla \cdot \left( \frac{1}{(1+|\nabla u^{n}|^{2})^{\frac{1}{2}}}\nabla - \delta u^{n} \right) + - \nabla \cdot \left( \frac{\nabla u^{n} \cdot - \nabla \delta u^{n}}{(1+|\nabla u^{n}|^{2})^{\frac{3}{2}}} \nabla u^{n} - \right) = + - \nabla \cdot \left( \frac{1}{(1+|\nabla u^{n}|^{2})^{\frac{1}{2}}}\nabla + \delta u^{n} \right) + + \nabla \cdot \left( \frac{\nabla u^{n} \cdot + \nabla \delta u^{n}}{(1+|\nabla u^{n}|^{2})^{\frac{3}{2}}} \nabla u^{n} + \right) = -\left( - \nabla \cdot \left( \frac{1}{(1+|\nabla u^{n}|^{2})^{\frac{1}{2}}} \nabla u^{n} \right) \right) @f] In order to solve the minimal surface equation, we have to solve this equation in every -Newton step. To solve this, we have to take a look at the boundary condition of this -problem. Assuming that $u^{n}$ already has the right boundary values, the Newton update +Newton step. To solve this, we have to take a look at the boundary condition of this +problem. Assuming that $u^{n}$ already has the right boundary values, the Newton update $\delta u^{n}$ should have zero boundary conditions, in order to have the right boundary condition after adding both. -In the first Newton step, we are starting with the solution $u^{0}\equiv 0$, the Newton -update still has to deliever the right boundary condition to the solution $u^{1}$. +In the first Newton step, we are starting with the solution $u^{0}\equiv 0$, the Newton +update still has to deliever the right boundary condition to the solution $u^{1}$. @par -Summing up, we have to solve the pde above with the boundary condition $\delta u^{0}=g$ +Summing up, we have to solve the pde above with the boundary condition $\delta u^{0}=g$ in the first step and with $\delta u^{n}=0$ in all the other steps. @@ -86,14 +89,14 @@ Starting with the strong formulation above, we get the weak formulation by multi both sides of the pde with a testfunction $\varphi$ and integrating by parts on both sides: @f[ - \left( \nabla \varphi , \frac{1}{(1+|\nabla u^{n}|^{2})^{\frac{1}{2}}}\nabla - \delta u^{n} \right)-\left(\nabla \varphi ,\frac{\nabla u^{n} \cdot \nabla - \delta u^{n}}{(1+|\nabla u^{n}|^{2})^{\frac{3}{2}}}\nabla u^{n} \right) + \left( \nabla \varphi , \frac{1}{(1+|\nabla u^{n}|^{2})^{\frac{1}{2}}}\nabla + \delta u^{n} \right)-\left(\nabla \varphi ,\frac{\nabla u^{n} \cdot \nabla + \delta u^{n}}{(1+|\nabla u^{n}|^{2})^{\frac{3}{2}}}\nabla u^{n} \right) = -\left(\nabla \varphi , \frac{1}{(1+|\nabla u^{n}|^{2})^{\frac{1}{2}}} \nabla u^{n} \right) @f] - -Where the solution $\delta u^{n}$ is a function in the infinte space $H^{1}(\Omega)$. + +Where the solution $\delta u^{n}$ is a function in the infinte space $H^{1}(\Omega)$. Reducing this space to a finite space with basis $\left\{ \varphi_{0},\dots , \varphi_{N-1}\right\}$, we can write the solution: @@ -105,10 +108,10 @@ Using the basis functions as testfunctions and defining $a_{n}:=\frac{1} {\sqrt{1+|\nabla u^{n}|^{2}}}$, we can rewrite the weak formualtion: @f[ - \sum_{j=0}^{N-1}\left[ \left( \nabla \varphi_{i} , a_{n} \nabla \varphi_{j} \right) - - \left(\nabla u^{n}\cdot \nabla \varphi_{i} , a_{n}^{3} \nabla u^{n} \cdot \nabla - \varphi_{j} \right) \right] \cdot U_{j}=\left( \nabla \varphi_{i} , a_{n} - \nabla u^{n}\right) \qquad \forall i=0,\dots ,N-1 + \sum_{j=0}^{N-1}\left[ \left( \nabla \varphi_{i} , a_{n} \nabla \varphi_{j} \right) - + \left(\nabla u^{n}\cdot \nabla \varphi_{i} , a_{n}^{3} \nabla u^{n} \cdot \nabla + \varphi_{j} \right) \right] \cdot U_{j}=\left( \nabla \varphi_{i} , a_{n} + \nabla u^{n}\right) \qquad \forall i=0,\dots ,N-1 @f] where the solution $\delta u^{n}$ is given by the coefficents $\delta U^{n}_{j}$. @@ -121,8 +124,8 @@ This linear equation system can be rewritten as: where the entries of the matrix $A^{n}$ are given by: @f[ - A^{n}_{ij}:= \left( \nabla \varphi_{i} , a_{n} \nabla \varphi_{j} \right) - - \left(\nabla u^{n}\cdot \nabla \varphi_{i} , a_{n}^{3} \nabla u^{n} \cdot \nabla + A^{n}_{ij}:= \left( \nabla \varphi_{i} , a_{n} \nabla \varphi_{j} \right) - + \left(\nabla u^{n}\cdot \nabla \varphi_{i} , a_{n}^{3} \nabla u^{n} \cdot \nabla \varphi_{j} \right) @f] @@ -132,18 +135,20 @@ and the right hand side $b^{n}$ is given by: b^{n}_{i}:=\left( \nabla \varphi_{i} , a_{n} \nabla u^{n}\right) @f] -The matrix A is symmetric, but it is indefinite. So we have to take a better look -at the solver we choose for this linear system. The CG-method needs +The matrix A is symmetric, but it is indefinite. So we have to take a better look +at the solver we choose for this linear system. The CG-method needs positive-definiteness of the matrix A, which is not given, so it can't be used. Using the symmetry of the matrix we can choose the minimal residual method as a solver, which needs symmetry but no definiteness.

Summary

-Starting with the function $u^{0}\equiv 0$, the first Newton update is computed by +Starting with the function $u^{0}\equiv 0$, the first Newton update is computed by solving the system $A^{0}U^{0}=b^{0}$ with boundary condition $\delta u^{0}=g$ on - $\partial \Omega$. The new approximation of the solution is given by - $u^{1}=u^{0}+\delta u^{0}$. The next updates are given as solution of - the linear system $A^{n}U^{n}=b^{n}$ with boundary condition $\delta u^{n}=0$ on + $\partial \Omega$. The new approximation of the solution is given by + $u^{1}=u^{0}+\delta u^{0}$. The next updates are given as solution of + the linear system $A^{n}U^{n}=b^{n}$ with boundary condition $\delta u^{n}=0$ on $\partial \Omega$ and the new approximation given by $u^{n+1}=u^{n}+\delta u^{n}$. + +*/ \ No newline at end of file diff --git a/deal.II/examples/step-15/doc/results.dox b/deal.II/examples/step-15/doc/results.dox index df6c822069..74b107e326 100644 --- a/deal.II/examples/step-15/doc/results.dox +++ b/deal.II/examples/step-15/doc/results.dox @@ -42,7 +42,7 @@ mesh-refinement:9 -The final solution, as written by the program at the end of the +The last solution, as written by the program at the end of the %run() function, looks as follows: @@ -51,8 +51,10 @@ The final solution, as written by the program at the end of the -In each cycle, the program furthermore writes the grid in EPS -format. These are shown in the following: +After each refinement, the solution after the fifth Newton- +iteration is shown in the following pictures. On the left +the solution is visualized, on the right side the solution +with the mesh is shown: @@ -94,14 +96,16 @@ format. These are shown in the following: +It is clearly visible, that the solution minimizes the surface +after each refinement. The solution converges to a picture one +would imagine a soapbubble inside a wire, which is bended like +the boundary. Also it is visible, how the boundary +is smoothed out after each refinement. On the coarse mesh, +the boundary doesn't look like a sine, whereas it does the +finer the mesh gets. +The mesh is mostly refined near the boundary, where the solution +increases or decreases strongly, whereas it is coarsened on +the inside of the domain, where nothing interesting happens, +because there isn't much change in the solution. -It is clearly visible that the region where the solution has a kink, -i.e. the circle at radial distance 0.5 from the center, is -refined most. Furthermore, the central region where the solution is -very smooth and almost flat, is almost not refined at all, but this -results from the fact that we did not take into account that the -coefficient is large there. The region outside is refined rather -randomly, since the second derivative is constant there and refinement -is therefore mostly based on the size of the cells and their deviation -from the optimal square. diff --git a/deal.II/examples/step-15/step-15.cc b/deal.II/examples/step-15/step-15.cc index 0b4c4bd532..10318f62f4 100644 --- a/deal.II/examples/step-15/step-15.cc +++ b/deal.II/examples/step-15/step-15.cc @@ -492,6 +492,19 @@ void Step15::run () while(first_step || (res>1e-3)) { + // In the first step, we compute the solution on the two times globally + // refined mesh. After that the mesh will be refined + // adaptively, in order to not get too many cells. The refinement + // is the first thing done every time we restart the process in the while-loop. + if(!first_step) + { + refine_grid(); + + std::cout<<"********mesh-refinement:"<setup_system // function. @@ -542,29 +555,7 @@ void Step15::run () std::ofstream output (filename.c_str()); data_out.write_vtk (output); - // last thing to do is to refine the mesh: - - refine_grid(); - - std::cout<<"********mesh-refinement:"< data_out; - data_out.set_flags (vtk_flags); - - data_out.attach_dof_handler (dof_handler); - data_out.add_data_vector (present_solution, "solution"); - data_out.build_patches (6); - - std::ofstream output ("final-solution.vtk"); - data_out.write_vtk (output); } // @ sect4{The main function} @@ -608,3 +599,4 @@ int main () } return 0; } + -- 2.39.5