本文關(guān)鍵詞：二維三維彈性問題混合元　出處：《鄭州大學(xué)》2016年博士論文　論文類型：學(xué)位論文

  更多相關(guān)文章： 線彈性方程 混合有限元 彈性復(fù)形 各向異性 仿射等價 誤差估計(jì)

【摘要】：本文主要討論線彈性方程,在Hellinger-Reissner變分形式的基礎(chǔ)上,系統(tǒng)的構(gòu)造了二維空間下的矩形和三角形單元,三維空間下的立方體和四面體單元等一系列簡單穩(wěn)定的單元.對單元的適定性,收斂性,誤差估計(jì),以及二維三維矩形和立方體單元的各向異性特征進(jìn)行了深入的分析和系統(tǒng)的研究.并對二維協(xié)調(diào)的矩形單元和非協(xié)調(diào)的三角形單元進(jìn)行了相應(yīng)的數(shù)值實(shí)驗(yàn).在這篇論文中,首先,我們構(gòu)造了具有最少自由度的協(xié)調(diào)的二維矩形單元R8-2和三維立方體單元C18-3.即矩形單元的應(yīng)力和位移空間分別為8個和2個自由度.立方體單元的應(yīng)力和位移空間中分別為18和3個自由度.由于所構(gòu)造單元不滿足關(guān)于散度的投影性質(zhì),因此我們采用了構(gòu)造的方法證明了離散BB條件,即離散混合問題的唯一可解性條件.在此基礎(chǔ)上進(jìn)一步分析,發(fā)現(xiàn)了單元的各向異性特征,并由此得到了單元的誤差估計(jì).據(jù)我們所知這是首次構(gòu)造的各向異性彈性問題混合元。其次,在構(gòu)造最簡單矩形單元的基礎(chǔ)上,進(jìn)一步構(gòu)造了一系列矩形高階單元,當(dāng)次數(shù)大于等于4時滿足散度的投影性質(zhì),據(jù)此得到單元的適定性和離散問題的唯一可解性及誤差估計(jì),并得到相應(yīng)的彈性復(fù)形.再次,在構(gòu)造矩形類單元的基礎(chǔ)上,又對三維四面體單元構(gòu)造過程中的空間Mκ(K)進(jìn)行了研究和討論,因?yàn)樵贖ellinge-Reissner變分形式下,應(yīng)力是屬于H(diυ,Ω；S)空間,協(xié)調(diào)元構(gòu)造要求應(yīng)力的法向分量跨過單元邊界連續(xù),在協(xié)調(diào)元構(gòu)造中計(jì)算空間Mk(K)={τ∈Pk(K;S)|divτ=0,τn|(?)K=0)的維數(shù)是個難點(diǎn)問題.在本部分中給出了任意階空間維數(shù)計(jì)算的一股方法,并且此方法能夠很方便的求出相應(yīng)的顯式基,同時證明了κ=3時,集合Mκ(K)的維數(shù)為0,并且給出κ=4時空間的一組基.最后,構(gòu)造了關(guān)于線彈性問題的一系列新的從低階到高階的三角形和四面體非協(xié)調(diào)單元及相應(yīng)剛體運(yùn)動下的簡化單元,這里構(gòu)造的二維三維單元區(qū)別于之前文獻(xiàn)中構(gòu)造的單元,構(gòu)造簡單,自由度少.這些單元定義在參考單元上,形函數(shù)空間顯式給出,我們嚴(yán)格證明了這類單元的仿射等價性,易于進(jìn)行數(shù)值實(shí)驗(yàn).當(dāng)κ=1時簡化的三角形單元的應(yīng)力空間和位移空間具有12+3個自由度,簡化的四面體單元的應(yīng)力空間和位移空間分別具有42+12個自由度.這些單元滿足散度的投影性質(zhì).本文得到了這些單元的適定性,離散問題的唯一可解性及相應(yīng)的誤差估計(jì).
[Abstract]:This paper mainly discusses the linear elastic equations in Hellinger-Reissner based on the form, system structure and rectangular two dimensional triangular element, 3D cube and tetrahedron and a series of simple and stable unit. For unit well posedness, convergence, error estimation, and 2D and 3D rectangular cube unit of the anisotropic characteristics are analyzed and systematically. And the two-dimensional rectangular unit coordination and non coordination triangle unit the corresponding numerical experiments. In this thesis, firstly, we construct a minimum degree of freedom of the coordination of the two-dimensional rectangular unit R8-2 and unit C18-3. rectangular three-dimensional cube the unit of stress and displacement of space are respectively 8 and 2 degrees of freedom. A cubic element of stress and displacement in the space are 18 and 3 degrees of freedom. Because of the single structure Element does not satisfy the projection properties on the divergence, so we adopt the construction method show that the discrete BB condition, namely the discrete mixed problem of unique solvability conditions. On the basis of further analysis, found the anisotropic characteristics of unit, and obtained the error estimation unit. To our knowledge this is the first structure the anisotropic elastic problem of mixed element. Secondly, based on the structure of simple rectangular elements, and further construct a series of high order rectangular unit, when the projection properties of divergence times greater than or equal to 4, then get the well posedness and uniqueness of the solution of the discrete problem and error estimation unit, and the corresponding elastic complex. Thirdly, based on constructing the rectangle class unit, and the three-dimensional tetrahedron structure in the process of space M (K) were studied and discussed, because in Hellinge-Reissner variational form 涓，

本文編號：1409233