本文選題：計(jì)算動(dòng)力學(xué) + 偽弧長(zhǎng)　； 參考：《力學(xué)學(xué)報(bào)》2017年03期

【摘要】：動(dòng)力學(xué)問(wèn)題通常采用微分方程來(lái)描繪,但由于工程實(shí)際問(wèn)題的復(fù)雜性,微分方程模型常伴隨著解的不連續(xù)性、剛性或激波間斷奇異性特點(diǎn),傳統(tǒng)方法很難求解,奇異性問(wèn)題是計(jì)算動(dòng)力學(xué)難點(diǎn),同時(shí)也是國(guó)內(nèi)外學(xué)者研究的熱點(diǎn).偽弧長(zhǎng)數(shù)值算法是針對(duì)計(jì)算動(dòng)力學(xué)中的奇異性問(wèn)題所提出的,其基本思想為通過(guò)在解曲線上引入偽弧長(zhǎng)參數(shù),并增加一個(gè)約束方程,在偽弧長(zhǎng)參數(shù)作用下,使得原始離散單元發(fā)生扭曲形變,從而達(dá)到消除或減弱奇異性的目的.本文首先介紹偽弧長(zhǎng)方法求解定常對(duì)流-擴(kuò)散方程的奇異性問(wèn)題,并提出針對(duì)雙曲守恒定律的局部偽弧長(zhǎng)算法,其思想在于首先通過(guò)間斷解的梯度變換來(lái)確定強(qiáng)間斷所處位置,進(jìn)而通過(guò)局部網(wǎng)格點(diǎn)重構(gòu)以及數(shù)值修正來(lái)達(dá)到強(qiáng)間斷處奇異性消除與降低的目的.針對(duì)高維問(wèn)題,提出全局偽弧長(zhǎng)方法,通過(guò)對(duì)整個(gè)計(jì)算區(qū)域內(nèi)的網(wǎng)格點(diǎn)進(jìn)行重構(gòu),使得所有網(wǎng)格點(diǎn)向奇異間斷點(diǎn)處移動(dòng),從而降低間斷點(diǎn)的影響域,達(dá)到降低奇異性的目的.重點(diǎn)討論了三維全局偽弧長(zhǎng)算法問(wèn)題的計(jì)算難點(diǎn),即三維空間網(wǎng)格扭曲大變形導(dǎo)致的數(shù)值算法不收斂,并提出在算法設(shè)計(jì)過(guò)程中采用分塊重構(gòu)與整體計(jì)算相結(jié)合的策略,實(shí)現(xiàn)了三維空間中的偽弧長(zhǎng)數(shù)值算法,最后通過(guò)數(shù)值實(shí)驗(yàn)來(lái)驗(yàn)證偽弧長(zhǎng)算法對(duì)于奇異性問(wèn)題的有效性.
[Abstract]:Dynamic problems are usually described by differential equations, but because of the complexity of practical problems in engineering, the model of differential equations is often accompanied by discontinuity, rigidity or shock wave singularity, so the traditional method is difficult to solve.Singularity is a difficult problem in computational dynamics, and it is also a hot topic for scholars at home and abroad.The pseudo-arc length numerical algorithm is proposed to solve the singularity problem in computational dynamics. The basic idea is to introduce pseudo-arc length parameters into the solution curve and add a constraint equation under the action of pseudo-arc length parameters.The original discrete element is distorted and the singularity is eliminated or weakened.In this paper, we first introduce the pseudo-arc length method to solve the singularity problem of steady convection-diffusion equation, and propose a local pseudo-arc length algorithm for hyperbolic conservation law. The idea is to determine the position of strong discontinuity by gradient transformation of discontinuous solution.Then the singularity of strong discontinuity is eliminated and reduced by local mesh reconstruction and numerical correction.A global pseudo-arc length method is proposed to solve the high dimensional problem. By reconstructing the grid points in the whole computing area, all the grid points move to the singular discontinuity points, thus reducing the influence region of the discontinuous points and reducing the singularity.In this paper, the computational difficulty of 3D global pseudo-arc length algorithm is discussed, that is, the numerical algorithm does not converge due to the distortion and deformation of three-dimensional mesh, and the strategy of combining block reconstruction and global computation in the design process of the algorithm is proposed.The pseudo arc length numerical algorithm in three dimensional space is implemented. Finally, the validity of the pseudo arc length algorithm for singularity problem is verified by numerical experiments.
【作者單位】： 北京理工大學(xué)爆炸科學(xué)與技術(shù)國(guó)家重點(diǎn)實(shí)驗(yàn)室;
【基金】：國(guó)家自然科學(xué)基金資助項(xiàng)目(11390363,11532012)
【分類(lèi)號(hào)】：O175

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