本文選題：無網(wǎng)格方法　切入點(diǎn)：Signorini問題　出處：《重慶師范大學(xué)》2015年碩士論文　論文類型：學(xué)位論文

【摘要】：Signorini問題是一類重要的數(shù)學(xué)物理問題,該問題的Signorini邊界條件是由一個(gè)等式約束下的兩個(gè)不等式以互補(bǔ)形式給出的,其邊界條件的這種不確定性給求解帶來一定的麻煩,因此Signorini問題形成了一類特殊的橢圓邊值問題。由于該問題的Signorini條件是定義在求解區(qū)域的邊界上,因此基于邊界積分方程的無網(wǎng)格方法很適合求解這類問題。本文利用投影算法,將移動(dòng)最小二乘逼近法和邊界積分方程相結(jié)合,提出了求解Signorini問題的一種無網(wǎng)格邊界點(diǎn)投影迭代算法。該方法首先利用一個(gè)簡(jiǎn)單的投影算子將Signorini邊界條件轉(zhuǎn)化為與之等價(jià)的投影不動(dòng)點(diǎn)方程,然后分別構(gòu)造出Signorini問題的隱式和顯式投影迭代格式。隱式投影迭代格式是在不動(dòng)點(diǎn)方程的基礎(chǔ)上直接構(gòu)造的,而顯式投影迭代格式則是通過引入了一個(gè)與不動(dòng)點(diǎn)方程等價(jià)的殘量函數(shù),在殘量函數(shù)的基礎(chǔ)上構(gòu)造的。這樣在每一次的迭代過程中,我們就將一個(gè)特殊的橢圓邊值問題轉(zhuǎn)化成為一個(gè)可求解的一般橢圓邊值問題并且采用無網(wǎng)格邊界點(diǎn)方法進(jìn)行數(shù)值求解,并且對(duì)隱式投影迭代格式收斂且唯一收斂到Signorini問題的唯一解進(jìn)行了證明。最后,對(duì)于隱式和顯式投影算法,我們分別以數(shù)值算例對(duì)其求解效率給予驗(yàn)證,結(jié)果表明了本文方法在求解Signorini問題時(shí)的可行性和有效性,相對(duì)于邊界元方法也具有更好的精度和收斂速度。
[Abstract]:The Signorini problem is an important mathematical and physical problem. The Signorini boundary condition of the problem is given in the form of complementarity by two inequalities under an equality constraint. The uncertainty of the boundary condition brings some trouble to the solution of the problem. So the Signorini problem forms a kind of special elliptic boundary value problem. Because the Signorini condition of the problem is defined on the boundary of the solution region, the meshless method based on the boundary integral equation is very suitable for solving this kind of problem. The moving least square approximation method is combined with the boundary integral equation. In this paper, a meshless boundary point projection iterative algorithm for solving Signorini problem is proposed. Firstly, a simple projection operator is used to transform the Signorini boundary condition into an equivalent projective fixed point equation. Then the implicit and explicit projection iterative schemes of Signorini problem are constructed respectively. The implicit projection iterative schemes are constructed directly on the basis of fixed point equations. The explicit projection iterative scheme is constructed on the basis of the residual function by introducing a residue function equivalent to the fixed point equation. We transform a special elliptic boundary value problem into a general elliptic boundary value problem that can be solved and numerically solve it by using the meshless boundary point method. The convergence of implicit projection iterative scheme and its unique convergence to the unique solution of Signorini problem are proved. Finally, the efficiency of implicit and explicit projection algorithms is verified by numerical examples. The results show that the proposed method is feasible and effective in solving Signorini problem and has better accuracy and convergence speed than BEM.
【學(xué)位授予單位】：重慶師范大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2015
【分類號(hào)】：O241.82

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