Step3：空間微分の離散化の定義
Diff1d.hに名前空間fdとして空間微分の定義を行います。
- gradX：x方向1階微分後退差分$\frac{\partial u}{\partial x}=\frac{u_{i,j}-u_{i-1,j}}{dx}$
- gradY：y方向1階微分後退差分$\frac{\partial u}{\partial y}=\frac{u_{i,j}-u_{i,j-1}}{dy}$
- gradXCDS：x方向1階微分中心差分$\frac{\partial u}{\partial x}=\frac{u_{i+1,j}-u_{i-1,j}}{2dx}$
- gradYCDS：y方向1階微分中心差分$\frac{\partial u}{\partial y}=\frac{u_{i,j+1}-u_{i,j-1}}{2dy}$
- poissonX：x方向2階微分中心差分$\frac{\partial^2 u}{\partial x^2}=\frac{u_{i+1,j}-2u_{i,j}+u_{i-1,j}}{dx^2}$
- poissonY：y方向2階微分中心差分$\frac{\partial^2 u}{\partial y^2}=\frac{u_{i,j+1}-2u_{i,j}+u_{i,j-1}}{dy^2}$
Diff1d.h
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#include <vector> #include "Fields.h" #include <cmath> //add using std::vector; namespace fd1d { // U U U U //*-----*----*----* // i-1 i Fields::vectorField1d gradX1d(Fields::vectorField1d& vec) //typedef vector<Fields> vectorField1d; { Fields::vectorField1d temp = vec; for (int i = 1; i < vec.size(); i++) { temp[i].value = (vec[i].value - vec[i - 1].value) / (vec[i].x - vec[i - 1].x); } return temp; } Fields::vectorField1d laplacianX1d(Fields::vectorField1d& vec) //typedef vector<Fields> vectorField1d; { Fields::vectorField1d temp = vec; for (int i = 1; i < vec.size() - 1; i++) { temp[i].value = (vec[i + 1].value - 2.0 * vec[i].value + vec[i - 1].value) / pow((vec[i].x - vec[i - 1].x), 2.0); } return temp; } } namespace fd { Fields::vectorField2d gradX(Fields::vectorField2d& vec) { Fields::vectorField2d temp = vec; for (int i = 1; i < vec.size() - 1; i++) { for (int j = 1; j < vec[i].size() - 1; j++) { temp[i][j].value = (vec[i][j].value - vec[i - 1][j].value) / (vec[i][j].x - vec[i - 1][j].x); } } return temp; } Fields::vectorField2d gradY(Fields::vectorField2d& vec) { Fields::vectorField2d temp = vec; for (int i = 1; i < vec.size() - 1; i++) { for (int j = 1; j < vec[i].size() - 1; j++) { temp[i][j].value = (vec[i][j].value - vec[i][j - 1].value) / (vec[i][j].y - vec[i][j - 1].y); } } return temp; } Fields::vectorField2d gradXCDS(Fields::vectorField2d& vec) { Fields::vectorField2d temp = vec; for (int i = 1; i < vec.size() - 1; i++) { for (int j = 1; j < vec[i].size() - 1; j++) { temp[i][j].value = (vec[i + 1][j].value - vec[i - 1][j].value) / (2.0 * (vec[i + 1][j].x - vec[i - 1][j].x)); } } return temp; } Fields::vectorField2d gradYCDS(Fields::vectorField2d& vec) { Fields::vectorField2d temp = vec; for (int i = 1; i < vec.size() - 1; i++) { for (int j = 1; j < vec[i].size() - 1; j++) { temp[i][j].value = (-vec[i][j - 1].value) / (2.0 * (vec[i][j + 1].y - vec[i][j - 1].y)); } } return temp; } Fields::vectorField2d laplacian2d(Fields::vectorField2d& vec) //typedef vector<Fields> vectorField1d; { //Fields::vectorField2d temp(vec.size(), vector<Fields>(vec[0].size())); temp[i][j].x が空[0]になる Fields::vectorField2d temp = vec; //上書きする.もしくは参照渡しにする. for (int i = 1; i < vec.size() - 1; i++) { for (int j = 1; j < vec[j].size() - 1; j++) { temp[i][j].value = (vec[i + 1][j].value - 2.0 * vec[i][j].value + vec[i - 1][j].value) / pow((vec[i][j].x - vec[i - 1][j].x), 2.0) + (vec[i][j + 1].value - 2.0 * vec[i][j].value + vec[i][j - 1].value) / pow((vec[i][j].y - vec[i][j - 1].y), 2.0); } } return temp; } Fields::vectorField2d lapOp(Fields::vectorField2d& vec) //typedef vector<Fields> vectorField1d; { //Fields::vectorField2d temp(vec.size(), vector<Fields>(vec[0].size())); temp[i][j].x が空[0]になる Fields::vectorField2d temp = vec; //上書きする.もしくは参照渡しにする. for (int i = 1; i < vec.size() - 1; i++) { for (int j = 1; j < vec[i].size() - 1; j++) { double dxsq = pow((vec[i][j].x - vec[i - 1][j].x), 2.0); double dysq = dxsq; temp[i][j].value = ( (vec[i + 1][j].value + vec[i - 1][j].value) * dxsq + (vec[i][j + 1].value + vec[i][j - 1].value) * dysq ) / (2.0 * (dxsq + dysq)); } } for (int i = 1; i < vec.size() - 1; i++) { for (int j = 1; j < vec[i].size() - 1; j++) { //std::cout << "temp[i][j].value = " << temp[i][j].value << std::endl; } } return temp; } Fields::vectorField2d poissonX(Fields::vectorField2d& vec) { Fields::vectorField2d temp = vec; for (int i = 1; i < vec.size() - 1; i++) { for (int j = 1; j < vec[i].size() - 1; j++) { double dxsq = pow((vec[i][j].y - vec[i][j - 1].y), 2); double dysq = pow((vec[i][j].x - vec[i - 1][j].x), 2); temp[i][j].value = (dysq * (vec[i + 1][j].value + vec[i - 1][j].value)) / (2.0 * (dxsq + dysq)); } } return temp; } Fields::vectorField2d poissonY(Fields::vectorField2d& vec) { Fields::vectorField2d temp = vec; for (int i = 1; i < vec.size() - 1; i++) { for (int j = 1; j < vec[i].size() - 1; j++) { double dxsq = pow((vec[i][j].y - vec[i][j - 1].y), 2); double dysq = pow((vec[i][j].x - vec[i - 1][j].x), 2); temp[i][j].value = (dxsq * (vec[i][j + 1].value + vec[i][j - 1].value)) / (2.0 * (dxsq + dysq)); } } return temp; } }// ene namespace fd |
こちらの微分の離散化を使ってmain.cppでステップ数の分だけ計算をしてみましょう。
解く方程式は以下です。
\begin{align*}
\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y} = \nu \left(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2} \right) \tag{15}
\end{align*}
\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y} = \nu \left(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2} \right) \tag{15}
\end{align*}
\begin{align*}
\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y} = \nu \left(\frac{\partial^2 v}{\partial x^2}+\frac{\partial^2 v}{\partial y^2} \right) \tag{16}
\end{align*}
\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y} = \nu \left(\frac{\partial^2 v}{\partial x^2}+\frac{\partial^2 v}{\partial y^2} \right) \tag{16}
\end{align*}
\begin{align*}
\frac{\partial^2 p}{\partial x^2}+\frac{\partial^2 p}{\partial y^2}=0 \tag{18}
\end{align*}
\frac{\partial^2 p}{\partial x^2}+\frac{\partial^2 p}{\partial y^2}=0 \tag{18}
\end{align*}
テスト計算なので流速ベクトル$\boldsymbol{u}$と圧力$p$は独立の式として解くようにしています。
main.cppでのステップ数のループは以下の部分です。
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for (int nt = 0; nt < ntimestep; nt++) { cout << "Iteration no : " << nt << endl; //outcome if (nt % 10 == 0) { filewriter2d_.write2dUVFields(Unew, Vnew, Pnew, nt, name); } U = Unew; V = Vnew; P = Pnew; /* du/dt + udu/dx + vdu/dy= nu ∇^2 u dv/dt + udv/dx + vdv/dy= nu ∇^2 v */ Unew = U - dt * (U && fd::gradX(U)) - dt * (V && fd::gradY(U)) + dt * visc * (fd::laplacian2d(U)); Vnew = V - dt * (U && fd::gradX(V)) - dt * (V && fd::gradY(V)) + dt * visc * (fd::laplacian2d(V)); /* ∇^2 P = 0 */ Pnew = (fd::poissonX(P)) + (fd::poissonY(P)); //ポアソン方程式 } |
初期状態は以下としています。
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// initial condition for (int i = 0; i < U.size(); i++) { for (int j = 0; j < U[i].size(); j++) { if ((U[i][j].x > 0.75 && U[i][j].x < 1.25) && (U[i][j].y > 0.75 && U[i][j].y < 1.25)) { U[i][j].value = 2.0; } } } Unew = U; |
今回はUだけを計算して結果を確認します。
main.cpp
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#include <iostream> #include <vector> #include "Mesh.h" //メッシュ設定 #include "Fields.h" //場の定義 #include "Diff1d.h" //空間微分 #include "FileWriter.h" //ファイル出力 #include "PrintVector.h" //場のprint文 using namespace std; using std::vector; int main() { int nx = 20, ny = 20; double lengthx = 2.0, lengthy = 2.0; Mesh mesh2d_(nx, ny, lengthx, lengthy);//Meshクラスのインスタンス化 /* U,V,P,bの定義 */ #include "creatFields.h" print2dfield(P, "P"); // solution int ntimestep = 500; double dt = 0.001; double rho = 1.0; double visc = 1e-3; // file write FileWriter filewriter2d_; string name = "2DConvectional_diffusion"; // initial condition for (int i = 0; i < U.size(); i++) { for (int j = 0; j < U[i].size(); j++) { if ((U[i][j].x > 0.75 && U[i][j].x < 1.25) && (U[i][j].y > 0.75 && U[i][j].y < 1.25)) { U[i][j].value = 2.0; } } } Unew = U; for (int nt = 0; nt < ntimestep; nt++) { cout << "Iteration no : " << nt << endl; //outcome if (nt % 10 == 0) { filewriter2d_.write2dUVFields(Unew, Vnew, Pnew, nt, name); } U = Unew; V = Vnew; P = Pnew; /* du/dt + udu/dx + vdu/dy= nu ∇^2 u dv/dt + udv/dx + vdv/dy= nu ∇^2 v */ Unew = U - dt * (U && fd::gradX(U)) - dt * (V && fd::gradY(U)) + dt * visc * (fd::laplacian2d(U)); Vnew = V - dt * (U && fd::gradX(V)) - dt * (V && fd::gradY(V)) + dt * visc * (fd::laplacian2d(V)); /* ∇^2 P = 0 */ Pnew = (fd::poissonX(P)) + (fd::poissonY(P)); //ポアソン方程式 } return 0; } |
結果を出力するためにFileWriter.hで
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void write2dUVFields(Fields::vectorField2d&, Fields::vectorField2d&, Fields::vectorField2d&, int, std::string&); |
を定義しています。
FileWriter.h
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#ifndef FILEWRITWE_H #define FILEWRITWE_H #include <stdio.h> #include <iostream> #include <vector> #include <cmath> #include <string> #include <cstring> #include <sstream> #include <fstream> //ファイルの書き込みに必要 #include "Fields.h" using namespace std; using std::vector; using std::string; class FileWriter { public: FileWriter();//constructor virtual ~FileWriter(); //destructor void write1dField(Fields::vectorField1d&, string&); void write2dUVFields(Fields::vectorField2d&, Fields::vectorField2d&, Fields::vectorField2d&, int, std::string&); }; #endif |
FileWriter.cppに処理内容を書きます。
FileWriter::write1dFieldは1次元の計算結果を出力する際には必要でしたが今回は使いません。
FileWriter.cpp
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#include "FileWriter.h" #include <iostream> FileWriter::FileWriter() { } void FileWriter::write1dField(Fields::vectorField1d& vec, string& name_) { string myfileType = ".dat"; //拡張子 string path = "C:\\work\\C_lang\\20220413_diff_C_puls\\004_NS_Eq_1D_blog\\1DNS\\1DNS\\"; //絶対パス string pathName = path.append(name_).append(myfileType); //"myOutput;" + ".dat" => "myOutput.dat" std::cout << "pathName = " << pathName << " : pathName.size() = " << pathName.size() << std::endl; FILE* outfile; fopen_s(&outfile, pathName.c_str(), "w+t"); //https://www.paveway.info/entry/2018/12/25/cpp_std_string_char std::cout << "outfile = " << &outfile << std::endl; for (int i = 0; i < vec.size(); i++) { double xpos = vec[i].x; double uvel = vec[i].value; if (outfile) { fprintf(outfile, "%5.8lf\t%5.8lf\n", xpos, uvel); } } fclose(outfile); } void FileWriter::write2dUVFields(Fields::vectorField2d& Utemp, Fields::vectorField2d& Vtemp, Fields::vectorField2d& Ptemp, int time_, std::string& name_) { string myfileType = ".dat"; //拡張子 string path = "C:\\work\\C_lang\\20220413_diff_C_puls\\005_NS_eq_2D\\NS2d\\NS2d\\"; //絶対パス ostringstream temp; temp << time_; string pathName = path.append(name_).append(temp.str()).append(myfileType); //std::cout << "pathName = " << pathName << " : pathName.size() = " << pathName.size() << std::endl; //char* cstr = new char[pathName.size() + 2]; // メモリ確保 //std::cout << "cstr = " << &cstr << std::endl; //strcpy_s(cstr, name2.size() + 2, name2.c_str()); FILE* outfile; fopen_s(&outfile, pathName.c_str(), "w+t"); //https://www.paveway.info/entry/2018/12/25/cpp_std_string_char //std::cout << "outfile = " << &outfile << std::endl; // tcplot format - Paraview is an Open Source Software for visualization int NXtemp = Utemp.size(); int NYtemp = Utemp[0].size(); fprintf(outfile, "VARIABLES=\"X\",\"Y\",\"U\",\"V\"\"P\"\n"); fprintf(outfile, "ZONE F=POINT\n"); fprintf(outfile, "I=%d, J=%d\n", NXtemp, NYtemp); for (int i = 0; i < NYtemp; i++) { for (int j = 0; j < NXtemp; j++) { double xpos, ypos, UU, VV, PP; xpos = Utemp[i][j].x; ypos = Utemp[i][j].y; UU = Utemp[i][j].value; VV = Vtemp[i][j].value; PP = Ptemp[i][j].value; //%5.8lf\t fprintf(outfile, "%5.8lf\t%5.8lf\t%5.8lf\t%5.8lf\t%5.8lf\n", xpos, ypos, UU, VV, PP); } } fclose(outfile); } FileWriter::~FileWriter() { } |
結果をParaViewで確認してみましょう。
【結果】
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