{"id":80,"date":"2020-02-22T10:26:12","date_gmt":"2020-02-22T02:26:12","guid":{"rendered":"https:\/\/www.yemeng.site\/?p=80"},"modified":"2020-07-07T15:23:08","modified_gmt":"2020-07-07T07:23:08","slug":"small-box-fft%e7%ae%80%e4%bb%8b%ef%bc%9a%e7%90%86%e8%ae%ba%e7%af%87","status":"publish","type":"post","link":"https:\/\/ru.yemeng.site\/?p=80","title":{"rendered":"Small Box FFT\u5b66\u4e60\u7b14\u8bb0\uff08\u4e00\uff09\uff1a\u7406\u8bba"},"content":{"rendered":"\n

Small box fast Fourier transform\uff08SBFFT\uff09\u7b97\u6cd5\u662f\u4e00\u79cd\u5c06\u7a7a\u95f4\u5206\u4e3a\u5c0f\u5757\u5e76\u884c\u8fdb\u884c\u5085\u91cc\u53f6\u53d8\u6362\u7684\u7b97\u6cd5\uff0c\u5176\u4f18\u70b9\u5728\u4e8e\u76f8\u6bd4\u4f20\u7edf\u7684\u57fa\u4e8e\u4e00\u7ef4\u6570\u636e\u5212\u5206\u7684FFT\u7b97\u6cd5\u53ef\u4ee5\u4f7f\u7528\u66f4\u591a\u7684\u6838\u3002\u672c\u7cfb\u5217\u6587\u7ae0\u662f\u5b66\u4e60\u8fd9\u4e00\u7b97\u6cd5\u5728\u6c42\u89e3\u6cca\u677e\u65b9\u7a0b\u76f8\u5173\u4ee3\u7801\u7684\u7b14\u8bb0\uff0c\u672c\u6587\u662f\u5176\u4e2d\u7684\u7b2c\u4e00\u7bc7\uff0c\u4e3b\u8981\u53c2\u8003\u4e86\u6587\u732e [1]<\/a>\u3002<\/p>\n\n\n\n

\n

\u5728DFT\u8ba1\u7b97\u4e2d\uff0c\u4ece\u7535\u8377\u5bc6\u5ea6 \\rho(\\boldsymbol{r})<\/span> \u8ba1\u7b97\u5e93\u4ed1\u52bf \\phi(\\boldsymbol{r})<\/span> \u7684\u8fc7\u7a0b\uff0c\u662f\u901a\u8fc7\u5728\u7ed9\u5b9a\u7684\u8fb9\u754c\u6761\u4ef6\u4e0b\u6c42\u89e3\u6cca\u677e\u65b9\u7a0b<\/p>\n\n\n\n \\nabla^2\\phi(\\boldsymbol{r})=-4\\pi\\rho(\\boldsymbol{r}) <\/span>\n\n\n\n

\u4f7f\u7528\u683c\u6797\u51fd\u6570\u65b9\u6cd5\uff0c\u53ef\u4ee5\u5c06\u89e3\u5199\u6210<\/p>\n\n\n\n \\phi(\\boldsymbol{r})=\\int V(\\boldsymbol{r-r}_1)\\rho(\\boldsymbol{r}_1)\\,\\mathrm{d}^3\\boldsymbol{r}_1 <\/span>\n\n\n\n

\u5728\u5e38\u7528\u7684\u5468\u671f\u6027\u8fb9\u754c\u6761\u4ef6\uff0c\u8fd9\u4e00\u79ef\u5206\u662f\u5728\u6709\u9650\u7684\u683c\u5b50\uff08\u901a\u5e38\u662f\u5355\u80de\uff09\u4e2d\u8fdb\u884c\u7684\uff0c\u800c\u5176\u4e2d\u7684\u6838\u51fd\u6570V(\\boldsymbol{r-r}_1)=1\/|\\boldsymbol{r-r}_1|<\/span>\u3002\u6ce8\u610f\u5230\u8fd9\u4e00\u79ef\u5206\u662f\u4e00\u4e2a\u5377\u79ef\u7684\u5f62\u5f0f\uff0c\u90a3\u4e48\u5728\u5bf9\u76f8\u5e94\u7684\u51fd\u6570\u505a\u5085\u91cc\u53f6\u53d8\u6362f(\\boldsymbol{G})=\\mathscr{F}\\{f(\\boldsymbol{r})\\}<\/span>\uff0c\u5f97\u5230\u5728\u5012\u7a7a\u95f4\u4e2d\u7684\u5f62\u5f0f <\/p>\n\n\n\n \\phi(\\boldsymbol{G})=V(\\boldsymbol{G})\\rho(\\boldsymbol{G}) <\/span>\n\n\n\n

\u5176\u4e2dV(\\boldsymbol{G})=4\\pi\/|\\boldsymbol{G}|^2<\/span>\u3002\u56e0\u6b64\u5e93\u4f26\u52bf\\phi(\\boldsymbol{r})<\/span>\u53ef\u4ee5\u901a\u8fc7\u4e00\u4e2a\u5085\u91cc\u53f6\u53d8\u6362\uff08\\rho(\\boldsymbol{G})= \\mathscr{F}\\{\\rho(\\boldsymbol{G})\\}<\/span>\uff09\u548c\u4e00\u4e2a\u5085\u91cc\u53f6\u9006\u53d8\u6362 \uff08 \\phi(\\boldsymbol{r})= \\mathscr{F}^{-1}\\{\\phi(\\boldsymbol{G})\\} <\/span>\uff09\u5f97\u5230\u3002<\/p>\n<\/div><\/div>\n\n\n\n

\n

\u5728\u76ee\u524d\u5e38\u7528\u7684\uff08\u5e76\u884c\u7248\u672c\uff09FFTW\u4e2d\uff0c\u5bf9\u4e8e\u4e00\u4e2aN_1\\times N_2\\times N_3<\/span>\u7684\u4e09\u7ef4\u6570\u7ec4\u7684\u5085\u91cc\u53f6\u53d8\u6362\uff0c\u80fd\u4f7f\u7528\u7684\u6838\u6570\u6700\u591a\u4e3aN_3<\/span>\uff08Fortran\uff09\u6216 N_1<\/span>\uff08C\uff09 \u3002\u5bf9\u4e8e\u4e00\u4e2a\u5927\u6570\u7ec4\uff0c\u8fd9\u6837\u7684\u5e76\u884c\u5ea6\u663e\u7136\u8fd8\u4e0d\u591f\u9ad8\u3002\u56e0\u6b64\u6700\u597d\u662f\u80fd\u591f\u5b9e\u73b0\u4e00\u79cd\u5206\u5757\u7684FFT\u7b97\u6cd5\uff0c\u4e0d\u4ec5\u80fd\u591f\u4f7f\u7528\u66f4\u591a\u7684\u6838\uff0c\u800c\u4e14\u80fd\u591f\u5c3d\u91cf\u964d\u4f4e\u5168\u5c40\u901a\u4fe1\u3002\u7136\u800c\u8fd9\u91cc\u7684\u6838\u51fd\u6570V(\\boldsymbol{r-r}')<\/span>\u662f\u957f\u7a0b\u7684\uff0c\u5982\u679c\u76f4\u63a5\u4f7f\u7528\u5206\u5757\u7b97\u6cd5\u9700\u8981\u5927\u91cf\u7684\u5168\u5c40\u901a\u4fe1\uff0c\u5bf9\u6b64\u7684\u89e3\u51b3\u601d\u8def\u662f\u5c06\u8fd9\u4e00\u6838\u51fd\u6570\u5206\u89e3\u4e3a\u77ed\u7a0b\u9879\u548c\u957f\u7a0b\u9879\u4e24\u90e8\u5206<\/p>\n\n\n\n V(\\boldsymbol{r-r}_1) = V_\\text{S}(\\boldsymbol{r-r}_1)+ V_\\text{L}(\\boldsymbol{r-r}_1) <\/span>\n\n\n\n

\u8fd9\u91cc\u7684\u5173\u952e\u5728\u4e8e\u9009\u53d6\u5408\u9002\u7684\u5206\u89e3\u4f7f\u5f97\u77ed\u7a0b\u9879V_\\text{S}<\/span>\uff08\u5728\u5b9e\u7a7a\u95f4\uff09\u662f\u77ed\u7a0b\u7684\u800c\u957f\u7a0b\u9879V_\\text{L}<\/span>\u662f\u5e73\u6ed1\u7684\uff08\u56e0\u800c\u5728\u5012\u7a7a\u95f4\u4e2d\u662f\u77ed\u7a0b\u7684\uff09\uff0c\u4ece\u800c\u53ef\u4ee5\u5229\u7528\u5404\u81ea\u5728\u4e0d\u540c\u7a7a\u95f4\u7684\u77ed\u7a0b\u7279\u6027\u5206\u522b\u4f7f\u7528\u5206\u5757FFT\u7b97\u6cd5\u8fdb\u884c\u5904\u7406<\/p>\n\n\n\n \\begin{aligned} \\phi_\\text{S}(\\boldsymbol{r})&=\\int V_\\text{S}(\\boldsymbol{r-r}_1)\\rho(\\boldsymbol{r}_1)\\,\\mathrm{d}^3\\boldsymbol{r}_1 \\\\ \\phi_\\text{L}(\\boldsymbol{r})&=\\int V_\\text{L}(\\boldsymbol{r-r}_1)\\rho(\\boldsymbol{r}_1)\\,\\mathrm{d}^3\\boldsymbol{r}_1 \\end{aligned} <\/span>\n\n\n\n

\u6700\u540e\u5c06\u5f97\u5230\u7684\u77ed\u7a0b\u5e93\u4ed1\u52bf\\phi_\\text{S}(\\boldsymbol{r})<\/span>\u548c\u957f\u7a0b\u5e93\u4ed1\u52bf\\phi_\\text{L}(\\boldsymbol{r})<\/span>\u76f8\u52a0\u5373\u53ef\u5f97\u5230\u5e93\u4ed1\u52bf\\phi(\\boldsymbol{r})<\/span>\u3002<\/p>\n<\/div><\/div>\n\n\n\n

1. \u77ed\u7a0b\u9879\u7684\u9009\u53d6\u53ca\u5176\u5904\u7406 <\/h2>\n\n\n\n
\n

\u7531\u4e8e\u6838\u51fd\u6570V<\/span>\u65e0\u8bba\u5728\u5b9e\u7a7a\u95f4\u8fd8\u662f\u5012\u7a7a\u95f4\u4e2d\u90fd\u662f\u7403\u5bf9\u79f0\u7684\uff0c\u56e0\u6b64\u53ef\u4ee5\u628a\u5bf9\u5176\u7684\u5206\u89e3\u7b80\u5316\u4e3a\u4e00\u4e2a\u4e00\u7ef4\u95ee\u9898\uff0c\u5176\u4e00\u7ef4\u5085\u91cc\u53f6\u53d8\u6362\u7684\u5f62\u5f0f\u4e3a<\/p>\n\n\n\n V(r)=\\frac{1}{2\\pi^2r}\\int\\sin(Gr)GV(G)\\,\\mathrm{d}G <\/span>\n\n\n\n

\u6570\u503c\u8ba1\u7b97\u4e2d\u4f7f\u7528\u79bb\u6563\u7684\u957f\u5ea6\u4e3aN<\/span>\u6570\u7ec4r_i=iN\/L<\/span>\u548cG_j=j\\pi\/L<\/span>\uff0c\u90a3\u4e48\u8fd9\u4e00\u79ef\u5206\u7684\u79bb\u6563\u5f62\u5f0f\u4e3a<\/p>\n\n\n\n V(r_i)=\\sum_{j=1}^{N}\\frac{jN}{2iL^3} \\sin(ij\\pi\/N)V(G_j)= \\sum_{j=1}^{N}C_{ij}V(G_j) <\/span>\n\n\n\n

\u5bf9\u4e8e\u7ed9\u5b9a\u7684\u5b9e\u7a7a\u95f4\u548c\u5012\u7a7a\u95f4\u4e2d\u7684\u622a\u65ad\u957f\u5ea6r_\\text{c}<\/span>\u548cG_\\text{c}<\/span>\uff0c\u53ef\u4ee5\u5c06V_\\text{S}<\/span>\u5728\u5b9e\u7a7a\u95f4\u548cV_\\text{L}<\/span>\u5728\u5012\u7a7a\u95f4\u4e2d\u7684\u77ed\u7a0b\u7279\u6027\u5199\u6210<\/p>\n\n\n\n \\begin{aligned} &V_\\text{S}(r_i)=0, \\forall i>r_\\text{c}N\/L\\\\ &V_\\text{L}(G_j)=0, \\forall j>G_\\text{c}L\/\\pi \\end{aligned} <\/span>\n\n\n\n

\u7531\u4e8eV_\\text{S}+V_\\text{L}=V<\/span>\uff0c\u53ef\u4ee5\u901a\u8fc7\u4f7f\u5f97V_\\text{S}(r)<\/span>\u5728 r_\\text{c}<\/span>\u4ee5\u5916\u7684\u90e8\u5206\u4e3a0\u6765\u8fdb\u884c\u6c42\u89e3\uff0c\u5373\u6700\u5c0f\u5316\u5982\u4e0b\u7684\u91cf<\/p>\n\n\n\n \\begin{aligned} O=&\\sum_{i=r_\\text{c}N\/L}^N|V_\\text{S}(r_i)|^2r_i^2\\\\=&\\left(\\frac{L}{N}\\right)^2\\sum_{i=r_\\text{c}N\/L}^Ni^2\\left[\\sum_{j=1}^{N}C_{ij}\\frac{4\\pi}{G_j^2}- \\sum_{j=1}^{G_\\text{c}L\/\\pi}C_{ij}V_\\text{L}(G_j)\\right]^2 \\end{aligned} <\/span>\n\n\n\n

\u6c42\u89e3\u5bf9\u5e94\u7684\u7ebf\u6027\u65b9\u7a0b\u7ec4\u5373\u53ef\u6c42\u5f97V_\\text{L}(G_j)<\/span>\uff0c\u4ece\u800c\u53ef\u4ee5\u5c06\u6838\u51fd\u6570\u5206\u89e3\u4e3a\u77ed\u7a0b\u9879V_\\text{S}<\/span>\u548c\u957f\u7a0b\u9879V_\\text{L}<\/span>\uff0c\u793a\u4f8b\u7ed3\u679c\u5982\u4e0b\u56fe\u6240\u793a\u3002\u6ce8\u610f\u5230\u8fd9\u91cc\u5f97\u5230\u7684V_\\text{S}(r)<\/span>\u5728r_\\text{c}<\/span>\u4ee5\u5916\u5f88\u5c0f\uff08<10^{-5}<\/span>\uff09\u5e76\u4e0d\u4e25\u683c\u4e3a0\uff0c\u800cV_\\text{L}(G)<\/span>\u5728G_\\text{c}<\/span>\u4ee5\u5916\u4e25\u683c\u4e3a0\uff0c\u8fd9\u662f\u7531\u6c42\u89e3\u65b9\u5f0f\u51b3\u5b9a\u7684\u3002<\/p>\n<\/div><\/div>\n\n\n\n

\"\u5b9e\u7a7a\u95f4\u548c\u5012\u7a7a\u95f4\u7684\u6838\u51fd\u6570\u53ca\u5176\u5206\u89e3\"
\u5b9e\u7a7a\u95f4\u548c\u5012\u7a7a\u95f4\u7684\u6838\u51fd\u6570\u53ca\u5176\u5206\u89e3\uff0c\u5f15\u7528\u81ea\u6587\u732e [1]<\/a><\/figcaption><\/figure>\n\n\n\n

\u5bf9\u4e8e\u77ed\u7a0b\u9879\u90e8\u5206\u7684\u5904\u7406\u53ef\u4ee5\u4f7f\u7528SBFFT\u7b97\u6cd5\uff0c\u5c06\u4f53\u7cfb\u5728\u5b9e\u7a7a\u95f4\u4e2d\u5212\u5206\u6210\u4e00\u7cfb\u5217\u4e09\u7ef4\u7684\u5c0f\u5757\uff08\u79f0\u4e3a\u6838\u5fc3\u533a\u57df\uff09\uff0c\u5bf9\u4e8e\u6bcf\u4e00\u4e2a\u6838\u5fc3\u533a\u57df\\Omega_\\text{I}<\/span>\u5728\u5404\u4e2a\u65b9\u5411\u5411\u5916\u6269\u5c55\u957f\u5ea6L_\\text{c}<\/span>\uff08\u5176\u4e2dL_\\text{c}>r_\\text{c}<\/span>\uff09\u4f5c\u4e3a\u7f13\u51b2\u533a\uff0c\u4ece\u800c\u6210\u4e3a\u5b9e\u9645\u8ba1\u7b97\u7684\u5c0f\u5757\\Omega_\\text{II}<\/span>\u3002\u7531\u4e8eV_\\text{S}<\/span>\u5728\u7f13\u5b58\u533a\u5916\u4e3a0\uff0c\u56e0\u6b64\u5404\u4e2a\u5c0f\u5757\\Omega_\\text{II}<\/span>\u7684FFT\u53ef\u4ee5\u5e76\u884c\u5904\u7406\uff0c\u4ec5\u9700\u8981\u4e0e\u9644\u8fd126\u4e2a\u5c0f\u5757\u8fdb\u884c\u901a\u4fe1\uff08\u4ea4\u6362\u7f13\u51b2\u533a\u6570\u636e\uff09\uff0c\u907f\u514d\u4e86\u5168\u5c40\u901a\u4fe1\u3002\u8fd9\u6837\u7684\u8ba1\u7b97\u7b80\u8bb0\u4e3aDG-SBFFT\uff0c\u8fd9\u91cc\u7684DG\u8868\u793a\u539f\u59cb\u7684\u7a20\u5bc6\u5bc6\u7f51\u683c\uff0c\u4e0e\u4e0b\u6587\u4e2d\u7684\u7a00\u758f\u7f51\u683cSG\u5bf9\u5e94\u3002<\/p>\n\n\n\n

2. \u957f\u7a0b\u9879\u7684\u5904\u7406 <\/h2>\n\n\n\n
\n

\u5bf9\u4e8e\u957f\u7a0b\u5e93\u4ed1\u52bf\u7684\u8ba1\u7b97\u662f\u6bd4\u8f83\u590d\u6742\u7684\uff0c\u8fd9\u91cc\u5904\u7406\u7684\u5173\u952e\u7684\u5728\u4e8e\u5229\u7528\u957f\u7a0b\u9879V_\\text{L}(G)<\/span>\u5728G>G_\\text{c}<\/span>\u65f6\u4e3a0\u7684\u7279\u6027\u3002\u7531\u4e8e\u8ba1\u7b97\u4e2d\u4f7f\u7528\u7684\u6700\u5927\u7684\u5012\u683c\u77e2G_\\text{max}=N\\pi\/L<\/span>\u5927\u4e8eG_\\text{c}<\/span>\uff08\u53ef\u4ee5\u901a\u8fc7\u9009\u62e9\u5408\u9002\u7684G_\\text{c}<\/span>\u5b9e\u73b0\uff09\uff0c\u8fd9\u8868\u660e\u53ef\u4ee5\u7528\u4e00\u4e2a\u8f83\u7a00\u758f\u7684\u5b9e\u7a7a\u95f4\u7f51\u683c\uff08\u5bf9\u5e94\u8f83\u5c0fG'_\\text{max}<\/span>\uff09\u6765\u8fdb\u884c\u8ba1\u7b97\u800c\u5e76\u4e0d\u4f1a\u5f71\u54cd\u7cbe\u5ea6\u3002\u4e3a\u4e86\u65b9\u4fbf\u8d77\u89c1\uff0c\u4e0b\u9762\u7684\u8ba8\u8bba\u4e2d\u5206\u522b\u7528\u4e0d\u52a0\u4e0a\u6807\u548c\u52a0\u4e0a\u6807'<\/span>\u533a\u5206\u539f\u59cb\u7a20\u5bc6\u7f51\u683c\u548c\u65b0\u5f15\u5165\u7684\u7a00\u758f\u7f51\u683c\u4e0a\u7684\u53d8\u91cf\uff0c\u7528\u4e0d\u52a0\u4e0a\u6807\u548c\u52a0\u4e0a\u6807^\\text{s}<\/span>\u533a\u5206\u5168\u5c40\u548c\u5c0f\u5757\u4e2d\u7f51\u683c\u7684\u53d8\u91cf\u3002\u5728\u5012\u7a7a\u95f4\u4e2d\u5bf9\u4e8e\u957f\u7a0b\u9879\u6709\u5982\u4e0b\u5173\u7cfb<\/p>\n\n\n\n \\begin{aligned} \\phi_\\text{L}(\\boldsymbol{G})&=\\rho(\\boldsymbol{G})V_\\text{L}(\\boldsymbol{G})\\\\&=[\\rho(\\boldsymbol{G})m(\\boldsymbol{G})]\\cdot[V_\\text{L}(\\boldsymbol{G})\/m(\\boldsymbol{G})]\\\\&=\\rho_\\text{m}(\\boldsymbol{G})\\cdot[V_\\text{L}(\\boldsymbol{G})\/m(\\boldsymbol{G})] \\end{aligned} <\/span>\n\n\n\n

\u8fd9\u91cc\u7684m(\\boldsymbol{G})<\/span>\u662f\u4e00\u4e2a\uff08\u5012\u7a7a\u95f4\u4e2d\uff09\u77ed\u7a0b\u7684mask\u51fd\u6570\uff0c\u5f53G>G_\\text{mc}<\/span>\u65f6\u4e3a0\uff08\u8fd9\u91ccG_\\text{mc}>G_\\text{c}<\/span>\uff09\u4fdd\u8bc1\u4e86V_\\text{L}(\\boldsymbol{G})\/m(\\boldsymbol{G})<\/span>\u662f\u5b9a\u4e49\u826f\u597d\u7684\u3002\u5c06 \\rho_\\text{m}(\\boldsymbol{G})=\\rho(\\boldsymbol{G})m(\\boldsymbol{G})<\/span>\u53d8\u6362\u5230\u5b9e\u7a7a\u95f4\u4e2d\u6709<\/p>\n\n\n\n \\rho_\\text{m}(\\boldsymbol{G})=\\int m(\\boldsymbol{r-r}_1)\\rho(\\boldsymbol{r}_1)\\,\\mathrm{d}^3\\boldsymbol{r}_1 <\/span>\n\n\n\n

\u5982\u679cm(\\boldsymbol{r})<\/span>\u5728\u5b9e\u7a7a\u95f4\u4e2d\u4e5f\u662f\u662f\u77ed\u7a0b\u7684\uff0c\u5373r>r_\\text{c}<\/span>\u65f6\u5176\u503c\u4e3a0\uff0c\u90a3\u4e48\u5bf9\\rho_\\text{m}(\\boldsymbol{r})<\/span>\u53ef\u4ee5\u91c7\u7528\u4e0a\u4e00\u5c0f\u8282\u4e2d\u5bf9\u4e8e\u77ed\u7a0b\u9879\u76f8\u540c\u7684\u65b9\u6cd5\u8ba1\u7b97\u5f97\u5230\u3002\u8fd9\u91cc\u5f15\u5165\\rho_\\text{m}<\/span>\u7684\u76ee\u7684\u662f\u5229\u7528\u5176\u5728\u5012\u7a7a\u95f4\u4e2d\u7684\u77ed\u7a0b\u7279\u70b9\uff0c\u5bf9\u5176\u53ca\\phi_\\text{L}<\/span>\u7684FFT\u53ef\u4ee5\u5728\u4e00\u4e2a\u8f83\u4e3a\u7a00\u758f\u7684\u5168\u5c40\u7f51\u683c\u4e0a\u8fdb\u884c\uff0c\u4ece\u800c\u5927\u5927\u964d\u4f4e\u4e86\u8ba1\u7b97\u91cf\u3002\u6ce8\u610f\u5230\u8fd9\u91cc\u5bf9\u51fd\u6570m<\/span>\u7684\u8981\u6c42\u662f\u5728\u5b9e\u7a7a\u95f4\u548c\u5012\u7a7a\u95f4\u4e2d\u90fd\u662f\u77ed\u7a0b\u7684\uff0c\u4e0e\u4e0a\u4e00\u5c0f\u8282\u4e2d\u5206\u89e3\u6838\u51fd\u6570\u7684\u8981\u6c42\u76f8\u4f3c\uff0c\u53ef\u4ee5\u7c7b\u4f3c\u7684\u65b9\u6cd5\u5f97\u5230\u3002<\/p>\n<\/div><\/div>\n\n\n\n

\u56e0\u6b64\uff0c\u5bf9\u4e8e\uff08\u7a00\u758f\u7f51\u683c\u4e0a\u7684\uff09\u957f\u7a0b\u5e93\u4ed1\u52bf\\phi'_\\text{L}<\/span>\u7684\u8ba1\u7b97\u53ef\u4ee5\u5206\u4e3a\u4e09\u4e2a\u6b65\u9aa4\u3002\u7b2c\u4e00\u6b65\uff0c\u7528\u8ba1\u7b97\u77ed\u7a0b\u5e93\u4ed1\u52bf\u4e2d\u5f97\u5230\u5404\u5c0f\u5757\u4e2d\u7684\\rho^\\text{s}(\\boldsymbol{G}^\\text{s})<\/span>\u4e58\u4ee5mask\u51fd\u6570m(\\boldsymbol{G}^\\text{s})<\/span>\u5f97\u5230\\rho^\\text{s}_\\text{m}(\\boldsymbol{G}^\\text{s})<\/span> \uff0c\u7531\u4e8e\u5176\u5728G^\\text{s}>G_\\text{mc}<\/span>\u65f6\u4e3a0\uff0c\u56e0\u6b64\u53ef\u4ee5\u7b80\u5316\u4e3a\u4e00\u4e2aG_\\text{max}<\/span>\u8f83\u5c0f\u7684\\rho'^\\text{s}_\\text{m}(\\boldsymbol{G}^\\text{s})<\/span>\uff0c\u5bf9\u5e94\u5b9e\u7a7a\u95f4\u4e2d\u4e00\u4e2a\u8f83\u4e3a\u7a00\u758f\u7684\u7f51\u683c\u3002\u5bf9\\rho'^\\text{s}_\\text{m}(\\boldsymbol{G}^\\text{s})<\/span>\u505a\u9006SBFFT\u8ba1\u7b97\uff0c\u5f97\u5230\\rho'^\\text{s}_\\text{m}(\\boldsymbol{r}'^\\text{s})<\/span>\u3002\u7531\u4e8e\u8fd9\u4e00\u53d8\u6362\u662f\u5728\u7a00\u758f\u7f51\u683c\u4e0a\u8fdb\u884c\u7684\uff0c\u7b80\u8bb0\u4e3aSG-SBFFT\u3002\u7b2c\u4e8c\u6b65\uff0c\u5c06\u5404\u4e2a\u5c0f\u5757\u4e2d\u7684\\rho'^\\text{s}_\\text{m}(\\boldsymbol{r}'^\\text{s})<\/span>\u62fc\u63a5\u5f97\u5230\u5168\u5c40\u7684\\rho'_\\text{m}(\\boldsymbol{r}')<\/span>\uff0c\u5bf9\u5176\u8fdb\u884cFFT\u8ba1\u7b97\u5373\u53ef\u5f97\u5230\\rho'_\\text{m}(\\boldsymbol{G})<\/span>\uff0c\u8fd9\u4e00\u53d8\u6362\u7b80\u8bb0\u4e3aSG-FFT\u3002\u7b2c\u4e09\u6b65\uff0c\u7528\u4e0a\u4e00\u6b65\u5f97\u5230\u7684\\rho'_\\text{m}(\\boldsymbol{G})<\/span>\u4e58\u4ee5V_\\text{L}(\\boldsymbol{G})\/m(\\boldsymbol{G})<\/span>\u5f97\u5230\\Phi'_\\text{L}(\\boldsymbol{G})<\/span>\uff0c\u8fdb\u884c\u4e00\u6b21\u9006SG-FFT\u8ba1\u7b97\u5373\u53ef\u5f97\u5230\\Phi'_\\text{L}(\\boldsymbol{r}')<\/span>\u3002<\/p>\n\n\n\n

\u6700\u540e\u9700\u8981\u5c06\u7a00\u758f\u7f51\u683c\u4e0a\u7684\\Phi'_\\text{L}(\\boldsymbol{r}')<\/span>\u53d8\u6362\u4e3a\u539f\u59cb\u4e0a\u7f51\u683c\\Phi_\\text{L}(\\boldsymbol{r})<\/span>\u3002\u8fd9\u4e00\u95ee\u9898\u4e0e\u524d\u9762\u7684\u7a20\u5bc6\u7f51\u683c\u53d8\u6362\u4e3a\u7a00\u758f\u7f51\u683c\u662f\u9006\u95ee\u9898\uff0c\u56e0\u6b64\u540c\u6837\u53ef\u4ee5\u901a\u8fc7SBFFT\u6765\u89e3\u51b3\u3002\u8fd9\u4e00\u6b65\u540c\u6837\u9700\u8981\u5f15\u5165\u4e00\u4e2a\u5728\u5b9e\u7a7a\u95f4\u548c\u5012\u7a7a\u95f4\u4e2d\u5747\u4e3a\u77ed\u7a0b\u7684mask\u51fd\u6570M<\/span>\uff0c\u503c\u5f97\u6ce8\u610f\u7684\u662f\u8fd9\u91cc\u9700\u8981\u4f7f\u7528\u4e09\u4e2a\u4e00\u7ef4mask\u51fd\u6570\u76f8\u4e58\u7684\u5f62\u5f0f\u5e76\u5c06\u622a\u65ad\u957f\u5ea6L_\\text{s}\/2<\/span>\uff08L_\\text{c}\/2<\/span>\u4e3a\u5c0f\u5757\u6838\u5fc3\u533a\u57df\u7684\u957f\u5ea6\uff0c\u8fd9\u662f\u4e3a\u4e86\u4fdd\u8bc1\u5728\u8fb9\u754c\u4e0a\u4e0d\u5b58\u5728\u8df3\u53d8\uff09\u3002<\/p>\n\n\n\n

3. \u603b\u7ed3 <\/h2>\n\n\n\n

\u6839\u636e\u4e0a\u9762\u7684\u8ba8\u8bba\uff0c\u53ef\u4ee5\u5c06\u6574\u4e2a\u8ba1\u7b97\u8fc7\u7a0b\u7684\u6d41\u7a0b\u603b\u7ed3\u5982\u4e0b\uff1a<\/p>\n\n\n\n