[笔记2.1]学习AFEPack的例程

 [学习例程1]:/usr/local/AFEPack/example/possion_equation/
问题:-/Delta u = f     u |_{/partial /Omega} = u_b

***** $ls
CVS  D.d  Makefile  possion_equation.cfg  possion_equation.cpp  possion_equation.net  README  run
------------------------------------------------------------------------------------
***** README (运行步骤)
This example will solve a Possion equation on an easymesh generated data.
The obtained solution 
is  saved  in  file  " u.dx " .

Usage:
    
1 . compile with make [ - e debug = no];
    
2 . generate the mesh with easymesh  as :
         easymesh D
    
3 . run with . / run;
    
4 . view the obtained numerical solution with OpenDX; 

[补充]
3. 本目录的例子只需运行 $ ./run 或者 $ ./main
(后者需要先设置环境变量 $ export AFEPACK_TEMPLATE_PATH=/usr/local/AFEPack/template/triangle,否则找不到读入文件的路径)
4. OpenDX ——> Edit Visual Programs...——> 选 possion_equation.net
(possion_equation.cfg 是OpenDx用到的配置文件)
双击FileSelector,(若需要)修改输出的数值解位置,保存
Execute ——> 选Execute Once

***** Makefile(略)

***** possion_equation.cpp(略)
可用 $ cat -n possion_equation.cpp 加上行号显示

***** D.d(EasyMesh输入文件,略)

***** run(shell脚本)
# !   / bin / sh

export AFEPACK_PATH
= " /usr/local/AFEPack "
export AFEPACK_TEMPLATE_PATH
= " $AFEPACK_PATH/template/triangle "

exec .
/ main $ *

[补充](shell编程和环境变量,略)
$*:执行shell程序时,传递给程序的所有参数组成的一个字符串。
比如:
$ ./run D 实际是运行了 ./main D
$ ./run parameters are three
   此时 $* = parameters are three; $1 = parameters; $2 = are; $3 = three; $# = 3.
------------------------------------------------------------------------------------
***** 关于possion_equation.cpp的补充解释(详见[笔记2.2])
(1) Vector类(实现数值的 STL:vector)
65           Vector < double >  right_hand_side;

template class Vector;
位置:/usr/local/deal.II/lac/include/lac/vector.h 和 vector.templates.h

(2) 求解方程
73     AMGSolver solver(stiff_matrix);
74     solver.solve(solution, right_hand_side);    

[运行提示信息] 其中 failed to converge...,无大碍,唯不顺眼也
AMGSolver initializing ... OK! grid levels: 3
AMGSolver begin with initial residual 0.458239 ...
        failed to converge with residual 9.01090e-11 at step 21.

L2 error = 0.000629244

[原因] AMGSolver::solve是 默认参数的函数,可将 20放大,比如:
74行改为:solver.solve(solution, right_hand_side, 0, 0, 50);
class  AMGSolver  {
...
 
public:
  
void solve(Vector<double>& x,
         
const Vector<double>& r,
         
double tol = 0.0,
         unsigned 
int s = 0,
         unsigned 
int s1 = 20const;
...
}
;
参数:
    x     the solution vector
    r     the right hand side vector
    tol     tolerence
    s       0: solve mode; 1: precondition mode
    s1     upper bound of iteration step

AMGSolver::solve 函数定义在:/usr/local/AFEPack/library/src/AMGSolver.cpp

[修改后结果]
73    AMGSolver solver(stiff_matrix);
74    solver.solve(solution, right_hand_side,0,0,50);    

AMGSolver initializing ... OK! grid levels: 3
AMGSolver begin with initial residual 0.458239 ...
        converge with residual 4.32514e-13 at step 27.

L2 error = 0.000629244

(3) 计算和精确解的L2误差(本例Dirichlet边界条件由精确解给出)
78            double  error  =  Functional::L2Error(solution, FunctionFunction < double > ( & u),  3 );

(4) 计算解和近似精确解的误差(后述)

[学习例程2]:/usr/local/AFEPack/example/coefficient_possion_equation/
问题:-/nabla/cdot(A /nabla u) = f, 其中:A=(a_{ij})_{2*2}  
      u |_{/partial /Omega} = u_b

***** $ls
coefficient_possion_equation.cfg  coefficient_possion_equation.net  D.d       README
coefficient_possion_equation.cpp  CVS                               Makefile  run

如上例:.cfg .net是OpenDx文件。本例只需关注 .cpp文件
------------------------------------------------------------------------------------
***** 比较coefficient_possion_equation.cpp和possion_equation.cpp的不同之处:
(1) 增加2*2矩阵A的定义
(2) 两问题:possion_equation 和 coefficient_possion_equation 各自的单元刚度矩阵不同
故定义刚度矩阵的一个新派生类,虚函数 getElementMatrix(...) 重载基类的函数
class  Matrix :  public  StiffMatrix < 2 double >
{
    
public:
        Matrix(FEMSpace
<double,2>& sp) :
            StiffMatrix
<2,double>(sp) {};
        
virtual ~Matrix() {};
public:
    
virtual void getElementMatrix(const Element<double,2>& element0,
        
const Element<double,2>& element1,
        
const ActiveElementPairIterator<2>::State state);
}
;

[补充]:
1) C++允许子类的成员函数重载基类的成员函数
2) 运行时,依据类型确认调用哪个函数的能力称为多态性(polymorphism)
3) 为指明某个成员函数具有多态性,用virtual来标志其为虚函数

(3) 函数 void Matrix::getElementMatrix 的定义和 BilinearOperator.templates.h 文件中 StiffMatrix::getElementMatrix 定义类似,只是加进了系数A,可对比看

(4) 函数getElementMatrix是提供给stiff_matrix.build();内部调用的,所以:
possion_equation.cpp
    StiffMatrix < 2 , double >  stiff_matrix(fem_space);
    stiff_matrix.algebricAccuracy() 
=   3 ;
    stiff_matrix.build();
改为 coefficient_possion_equation.cpp
    Matrix stiff_matrix(fem_space);
    stiff_matrix.algebricAccuracy() 
=   3 ;
    stiff_matrix.build();

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