本文主要研究内容
作者高鹏（2019）在《使用无相位数据反散射问题的数值方法研究》一文中研究指出：本文讨论了使用无相位数据重构裂缝和障碍物反散射问题的分析和计算,我们考虑的散射问题模型均为Helmholtz方程,由于在实际的应用中,获取散射场或者远场的相位信息是极其困难,因此,利用无相位数据求解反散射问题在数学和物理领域都得到了广泛的关注.本文的第一章为绪论,主要介绍了我们所做工作的背景以及研究现状,并且简单介绍本文的结构.第二章主要介绍本文用到的一些预备知识,包括声波散射的一些相关概念以及反问题正则化的方法和数值计算的Nystrom方法.第三章研究了利用无相位数据重构声软裂缝的问题,我们验证了平移不变性(Translation invariance),这表明利用无相位远场数据仅能重构散射体形状而不能重构散射体位置,对于反散射问题,我们采用非线性积分方程方法,此方法的优点是精度准确并且能减少计算复杂度,最后通过一些数值例子验证了算法的可行性.第四章研究了利用无相位数据重构加入参考球的裂缝问题,为了解决无相位数据无法重构裂缝位置的问题,我们考虑在散射系统中加入了声软参考球,利用非线性积分方程方法求解反问题,重构出声软裂缝的形状和位置,并且通过数值实验说明了方法的可行性.第五章研究了利用无相位数据重构加入参考球的障碍问题,我们考虑的是满足Neumann边界条件的障碍体,由平移不变性可知,仅能确定障碍的形状不能确定障碍的位置,我们利用加入声硬参考球的方法,与裂缝情况类似,使用非线性积分方程方法求解,重构出障碍体的形状和位置.本文的最后一章为结论,对全文内容进行了总结.本文几个主要工作如下:1.利用无相位数据重构裂缝问题设裂缝rc=c=:s ∈[-1,1]}为光滑弧线,z1:=z(1)和z-1:=z(-1)为裂缝Γc的端点.我们考虑如下模型问题:△u + k2uu = 0,in R2Γc(1)u = 0,on rc,(2)反散射问题:已知一个方向入射的平面波ui以及远场模态的模|u∞|,重构声软裂缝rc的形状.使用无相位远场数据重构裂缝的反散射问题的解不是唯一的,我们给出了如下结论.定理1.(Translation invariance)设u∞(x)为声软裂缝re散射的远场模态.对于裂缝Γcε={x+εh:x∈Γc}其中h∈R2,远场模态u∞ε有如下形式u∞ε(x)=eikεh·(d-x)u∞(x),x∈Ω,(4)也就是说,利用无相位远场模态重构声软裂缝的反散射问题具有平移不变性.由平移不变性可知,仅利用一个方向入射的平面波得到的无相位远场模态数据不能重构出声软裂缝的位置.对于反散射问题,我们通过非线性积分方程方法求解,首先引进单层位势算子Se:(Sc(φ)(x)=Φ(x,y)φφ(y)ds(y),x∈Γc以及远场算子Sc,∞:(Sc,∞φ)(X)=-γ∫Γc e-ikx·yφ(y)ds(y),x∈ Ω.则可得未知曲线Γc和密度函数φ满足下述方程组Scφ = ui|Γc,(5)|Sc,∞=|2=|U∞|2.(6)然后将积分算子Sc转换为参数化算子C,表示为C(z,ψ)(t)=i/4∫π0H01(k|z(cost)-z(cosτ)|)ψ(τ)dτ,t ∈[0,π],(7)其中ψ(t):= |sint||z’(cost)|φ(z(cost)),t∈[0,π].与上述相似,我们引进参数化远场算子C∞:其中z∞(t)=(cost,sint),t∈[0,2π].此外,我们也给出入射场ui以及远场u∞的参数化表示:ωc=uioz,ωc,∞=u∞oz∞.则方程组的参数化形式可以表示为C(z,ψ)= ωc(z),(9)C∞(z,ψ)C∞(z,ψ)= |ωc,∞|2.(10)C∞(z,)关于z的Frechet导数表示为C∞ C∞关于z的导数为将方程(10)线性化得到B[z,ψ]q=fz,ψ,(12)其中迭代过程的相对误差为反问题的迭代过程为:(ⅰ)发射波数为k>0,入射方向为d ∈ Ω的平面波,然后收集无相位远场数据|u∞|.(ⅱ)给裂缝Γc一个初始近似Γc0,设k=0.(ⅲ)对于曲线Γck,从方程(9)中得到密度函数ψ.(ⅳ)求解方程(12)得到q,可得裂缝曲线新的近似Γck+1=Γck+q,计算Ek.(ⅴ)若Ek≥∈则设k=+ 1,返回(ⅲ),否则我们将近似曲线Γck+1作为最后的重构曲线.值得注意的是,我们使用的迭代方法与经典Newton迭代法是有些不同的,我们的迭代法精确,可以简单的实现并且减少计算复杂度.我们给出若干数值例子证明了方法的可行性.2.无相位数据重构加入参考球的裂缝问题由于利用无相位远场模态重构裂缝的反散射问题解是不唯一的,仅能够重构出裂缝的形状而并不能确定裂缝的位置.为了解决这个问题,我们采用在散射系统中加入声软参考球的方法.我们假定裂缝r1={z(s)= ∈[-1,1]}为光滑弧线,参考球D(?)R2,并且D ∩ =(?),Γ2:=(?)D.反散射问题:已知由固定波数k和某一个入射方向d入射的平面波ui以及无相位远场数据|uD∪Γ1∞(x)|,x∈Ω,确定未知裂缝Γ1的形状以及位置.我们定义单层位势算子以及远场算子我们得到裂缝Γ1和密度函数φj满足下面的方程组S11φ1+S21φ2=ui|Γ1,(14)S12φ1+S22φ2 =ui|Γ2)(15)|s1∞φ1+S2∞φ2|2=|u∞|2(16)我们给出裂缝r1的参数化表示Γ1 = {P1(s)= c+z(s):c=(c1,c2),s∈[-1,1 },以及参考球边界Γ2的参数化表示Γ2 = yp-2(t)= b + Rx:b=(b1,b2),x∈ Ω}.将算子Sjl,Sj∞转换为参数化算子Cjl,Cj∞,右端项的参数化形式为ω1(t)=ui(p1(coSt)),ω2(t)=ui,(p2(t)),ω∞(t)=u∞(x(t)).则可以得到积分方程(14)-(16)的参数化形式将方程(19)线性化可以得到Bq=f,(20)其中我们给出了一些数值例子用以证明方法的有效性.3.无相位数据重构加入参考球的障碍体问题我们假定障碍散射体D(?)R2,考虑如下的障碍散射模型问题:△u + k2u = 0,in R2D,(21)(?)u/(?)v=0,on(?)D,(22)我们考虑的反散射问题是已知由固定波数k和某一个入射方向d入射的平面波ui以及无相位远场数据|uD∞(x)|,x∈Ω,确定位置障碍物D的形状以及位置.由于平移不变性|uDh∞(x,d)|=|uD∞(x,d)|,利用无相位远场数据仅能重构障碍物形状而不能够确定位置.为了解决这个问题,我们加入参考球B(?)R2,D ∩ B=(?).反散射问题:已知由固定波数kk和某一个入射方向d入射的平面波ui以及无相位远场数据|uD∪B(x)|,x∈Ω确定位置障碍物D的形状以及位置.定义法向导数算子和远场算子:可以得到障碍物D和密度函数φj满足下面的方程组我们给出D的边界Γ1的参数化表示Γ1={p1(x)=c+ r(x)x:c=(c1,C2),x ∈以及参考球边界Γ2的参数化表示Γ2 = {P2(x)= b + Rx:b=(b1,b2),x ∈ Ω}.将算子Tjl,Tj∞转换为参数化算子Ajl,Aj∞,右端项的参数化形式为我们可以得到方程组的参数化形式A11(p1,ψ1)+A21(p1,ψ2)=ω1,(27)A12(P2,ψ1)+A.22(P2,ψ2)=ω2,(28)|A1∞(p1,ψ1)+ A2∞(p2,ψ2)|2 =|ω∞|2.(29)将方程(29)线性化可以得到Bq=f,(30)其中最后我们给出了一些数值算例验证了方法的有效性。
Abstract
ben wen tao lun le shi yong mo xiang wei shu ju chong gou lie feng he zhang ai wu fan san she wen ti de fen xi he ji suan ,wo men kao lv de san she wen ti mo xing jun wei Helmholtzfang cheng ,you yu zai shi ji de ying yong zhong ,huo qu san she chang huo zhe yuan chang de xiang wei xin xi shi ji ji kun nan ,yin ci ,li yong mo xiang wei shu ju qiu jie fan san she wen ti zai shu xue he wu li ling yu dou de dao le an fan de guan zhu .ben wen de di yi zhang wei xu lun ,zhu yao jie shao le wo men suo zuo gong zuo de bei jing yi ji yan jiu xian zhuang ,bing ju jian chan jie shao ben wen de jie gou .di er zhang zhu yao jie shao ben wen yong dao de yi xie yu bei zhi shi ,bao gua sheng bo san she de yi xie xiang guan gai nian yi ji fan wen ti zheng ze hua de fang fa he shu zhi ji suan de Nystromfang fa .di san zhang yan jiu le li yong mo xiang wei shu ju chong gou sheng ruan lie feng de wen ti ,wo men yan zheng le ping yi bu bian xing (Translation invariance),zhe biao ming li yong mo xiang wei yuan chang shu ju jin neng chong gou san she ti xing zhuang er bu neng chong gou san she ti wei zhi ,dui yu fan san she wen ti ,wo men cai yong fei xian xing ji fen fang cheng fang fa ,ci fang fa de you dian shi jing du zhun que bing ju neng jian shao ji suan fu za du ,zui hou tong guo yi xie shu zhi li zi yan zheng le suan fa de ke hang xing .di si zhang yan jiu le li yong mo xiang wei shu ju chong gou jia ru can kao qiu de lie feng wen ti ,wei le jie jue mo xiang wei shu ju mo fa chong gou lie feng wei zhi de wen ti ,wo men kao lv zai san she ji tong zhong jia ru le sheng ruan can kao qiu ,li yong fei xian xing ji fen fang cheng fang fa qiu jie fan wen ti ,chong gou chu sheng ruan lie feng de xing zhuang he wei zhi ,bing ju tong guo shu zhi shi yan shui ming le fang fa de ke hang xing .di wu zhang yan jiu le li yong mo xiang wei shu ju chong gou jia ru can kao qiu de zhang ai wen ti ,wo men kao lv de shi man zu Neumannbian jie tiao jian de zhang ai ti ,you ping yi bu bian xing ke zhi ,jin neng que ding zhang ai de xing zhuang bu neng que ding zhang ai de wei zhi ,wo men li yong jia ru sheng ying can kao qiu de fang fa ,yu lie feng qing kuang lei shi ,shi yong fei xian xing ji fen fang cheng fang fa qiu jie ,chong gou chu zhang ai ti de xing zhuang he wei zhi .ben wen de zui hou yi zhang wei jie lun ,dui quan wen nei rong jin hang le zong jie .ben wen ji ge zhu yao gong zuo ru xia :1.li yong mo xiang wei shu ju chong gou lie feng wen ti she lie feng rc=c=:s ∈[-1,1]}wei guang hua hu xian ,z1:=z(1)he z-1:=z(-1)wei lie feng Γcde duan dian .wo men kao lv ru xia mo xing wen ti :△u + k2uu = 0,in R2Γc(1)u = 0,on rc,(2)fan san she wen ti :yi zhi yi ge fang xiang ru she de ping mian bo uiyi ji yuan chang mo tai de mo |u∞|,chong gou sheng ruan lie feng rcde xing zhuang .shi yong mo xiang wei yuan chang shu ju chong gou lie feng de fan san she wen ti de jie bu shi wei yi de ,wo men gei chu le ru xia jie lun .ding li 1.(Translation invariance)she u∞(x)wei sheng ruan lie feng resan she de yuan chang mo tai .dui yu lie feng Γcε={x+εh:x∈Γc}ji zhong h∈R2,yuan chang mo tai u∞εyou ru xia xing shi u∞ε(x)=eikεh·(d-x)u∞(x),x∈Ω,(4)ye jiu shi shui ,li yong mo xiang wei yuan chang mo tai chong gou sheng ruan lie feng de fan san she wen ti ju you ping yi bu bian xing .you ping yi bu bian xing ke zhi ,jin li yong yi ge fang xiang ru she de ping mian bo de dao de mo xiang wei yuan chang mo tai shu ju bu neng chong gou chu sheng ruan lie feng de wei zhi .dui yu fan san she wen ti ,wo men tong guo fei xian xing ji fen fang cheng fang fa qiu jie ,shou xian yin jin chan ceng wei shi suan zi Se:(Sc(φ)(x)=Φ(x,y)φφ(y)ds(y),x∈Γcyi ji yuan chang suan zi Sc,∞:(Sc,∞φ)(X)=-γ∫Γc e-ikx·yφ(y)ds(y),x∈ Ω.ze ke de wei zhi qu xian Γche mi du han shu φman zu xia shu fang cheng zu Scφ = ui|Γc,(5)|Sc,∞=|2=|U∞|2.(6)ran hou jiang ji fen suan zi Sczhuai huan wei can shu hua suan zi C,biao shi wei C(z,ψ)(t)=i/4∫π0H01(k|z(cost)-z(cosτ)|)ψ(τ)dτ,t ∈[0,π],(7)ji zhong ψ(t):= |sint||z’(cost)|φ(z(cost)),t∈[0,π].yu shang shu xiang shi ,wo men yin jin can shu hua yuan chang suan zi C∞:ji zhong z∞(t)=(cost,sint),t∈[0,2π].ci wai ,wo men ye gei chu ru she chang uiyi ji yuan chang u∞de can shu hua biao shi :ωc=uioz,ωc,∞=u∞oz∞.ze fang cheng zu de can shu hua xing shi ke yi biao shi wei C(z,ψ)= ωc(z),(9)C∞(z,ψ)C∞(z,ψ)= |ωc,∞|2.(10)C∞(z,)guan yu zde Frechetdao shu biao shi wei C∞ C∞guan yu zde dao shu wei jiang fang cheng (10)xian xing hua de dao B[z,ψ]q=fz,ψ,(12)ji zhong die dai guo cheng de xiang dui wu cha wei fan wen ti de die dai guo cheng wei :(ⅰ)fa she bo shu wei k>0,ru she fang xiang wei d ∈ Ωde ping mian bo ,ran hou shou ji mo xiang wei yuan chang shu ju |u∞|.(ⅱ)gei lie feng Γcyi ge chu shi jin shi Γc0,she k=0.(ⅲ)dui yu qu xian Γck,cong fang cheng (9)zhong de dao mi du han shu ψ.(ⅳ)qiu jie fang cheng (12)de dao q,ke de lie feng qu xian xin de jin shi Γck+1=Γck+q,ji suan Ek.(ⅴ)re Ek≥∈ze she k=+ 1,fan hui (ⅲ),fou ze wo men jiang jin shi qu xian Γck+1zuo wei zui hou de chong gou qu xian .zhi de zhu yi de shi ,wo men shi yong de die dai fang fa yu jing dian Newtondie dai fa shi you xie bu tong de ,wo men de die dai fa jing que ,ke yi jian chan de shi xian bing ju jian shao ji suan fu za du .wo men gei chu re gan shu zhi li zi zheng ming le fang fa de ke hang xing .2.mo xiang wei shu ju chong gou jia ru can kao qiu de lie feng wen ti you yu li yong mo xiang wei yuan chang mo tai chong gou lie feng de fan san she wen ti jie shi bu wei yi de ,jin neng gou chong gou chu lie feng de xing zhuang er bing bu neng que ding lie feng de wei zhi .wei le jie jue zhe ge wen ti ,wo men cai yong zai san she ji tong zhong jia ru sheng ruan can kao qiu de fang fa .wo men jia ding lie feng r1={z(s)= ∈[-1,1]}wei guang hua hu xian ,can kao qiu D(?)R2,bing ju D ∩ =(?),Γ2:=(?)D.fan san she wen ti :yi zhi you gu ding bo shu khe mou yi ge ru she fang xiang dru she de ping mian bo uiyi ji mo xiang wei yuan chang shu ju |uD∪Γ1∞(x)|,x∈Ω,que ding wei zhi lie feng Γ1de xing zhuang yi ji wei zhi .wo men ding yi chan ceng wei shi suan zi yi ji yuan chang suan zi wo men de dao lie feng Γ1he mi du han shu φjman zu xia mian de fang cheng zu S11φ1+S21φ2=ui|Γ1,(14)S12φ1+S22φ2 =ui|Γ2)(15)|s1∞φ1+S2∞φ2|2=|u∞|2(16)wo men gei chu lie feng r1de can shu hua biao shi Γ1 = {P1(s)= c+z(s):c=(c1,c2),s∈[-1,1 },yi ji can kao qiu bian jie Γ2de can shu hua biao shi Γ2 = yp-2(t)= b + Rx:b=(b1,b2),x∈ Ω}.jiang suan zi Sjl,Sj∞zhuai huan wei can shu hua suan zi Cjl,Cj∞,you duan xiang de can shu hua xing shi wei ω1(t)=ui(p1(coSt)),ω2(t)=ui,(p2(t)),ω∞(t)=u∞(x(t)).ze ke yi de dao ji fen fang cheng (14)-(16)de can shu hua xing shi jiang fang cheng (19)xian xing hua ke yi de dao Bq=f,(20)ji zhong wo men gei chu le yi xie shu zhi li zi yong yi zheng ming fang fa de you xiao xing .3.mo xiang wei shu ju chong gou jia ru can kao qiu de zhang ai ti wen ti wo men jia ding zhang ai san she ti D(?)R2,kao lv ru xia de zhang ai san she mo xing wen ti :△u + k2u = 0,in R2D,(21)(?)u/(?)v=0,on(?)D,(22)wo men kao lv de fan san she wen ti shi yi zhi you gu ding bo shu khe mou yi ge ru she fang xiang dru she de ping mian bo uiyi ji mo xiang wei yuan chang shu ju |uD∞(x)|,x∈Ω,que ding wei zhi zhang ai wu Dde xing zhuang yi ji wei zhi .you yu ping yi bu bian xing |uDh∞(x,d)|=|uD∞(x,d)|,li yong mo xiang wei yuan chang shu ju jin neng chong gou zhang ai wu xing zhuang er bu neng gou que ding wei zhi .wei le jie jue zhe ge wen ti ,wo men jia ru can kao qiu B(?)R2,D ∩ B=(?).fan san she wen ti :yi zhi you gu ding bo shu kkhe mou yi ge ru she fang xiang dru she de ping mian bo uiyi ji mo xiang wei yuan chang shu ju |uD∪B(x)|,x∈Ωque ding wei zhi zhang ai wu Dde xing zhuang yi ji wei zhi .ding yi fa xiang dao shu suan zi he yuan chang suan zi :ke yi de dao zhang ai wu Dhe mi du han shu φjman zu xia mian de fang cheng zu wo men gei chu Dde bian jie Γ1de can shu hua biao shi Γ1={p1(x)=c+ r(x)x:c=(c1,C2),x ∈yi ji can kao qiu bian jie Γ2de can shu hua biao shi Γ2 = {P2(x)= b + Rx:b=(b1,b2),x ∈ Ω}.jiang suan zi Tjl,Tj∞zhuai huan wei can shu hua suan zi Ajl,Aj∞,you duan xiang de can shu hua xing shi wei wo men ke yi de dao fang cheng zu de can shu hua xing shi A11(p1,ψ1)+A21(p1,ψ2)=ω1,(27)A12(P2,ψ1)+A.22(P2,ψ2)=ω2,(28)|A1∞(p1,ψ1)+ A2∞(p2,ψ2)|2 =|ω∞|2.(29)jiang fang cheng (29)xian xing hua ke yi de dao Bq=f,(30)ji zhong zui hou wo men gei chu le yi xie shu zhi suan li yan zheng le fang fa de you xiao xing 。
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论文作者分别是来自吉林大学的高鹏，发表于刊物吉林大学2019-06-25论文，是一篇关于反散射问题论文,方程论文,无相位数据论文,裂缝论文,障碍体论文,非线性积分方程方法论文,吉林大学2019-06-25论文的文章。本文可供学术参考使用，各位学者可以免费参考阅读下载，文章观点不代表本站观点，资料来自吉林大学2019-06-25论文网站，若本站收录的文献无意侵犯了您的著作版权，请联系我们删除。
标签：反散射问题论文; 方程论文; 无相位数据论文; 裂缝论文; 障碍体论文; 非线性积分方程方法论文; 吉林大学2019-06-25论文;