A posteriori error estimates for the virtual element method

An a posteriori error analysis for the virtual element method (VEM) applied to general elliptic problems is presented. The resulting error estimator is of residual-type and applies on very general polygonal/polyhedral meshes. The estimator is fully computable as it relies only on quantities available from the VEM solution, namely its degrees of freedom and element-wise polynomial projection. Upper and lower bounds of the error estimator with respect to the VEM approximation error are proven. The error estimator is used to drive adaptive mesh refinement in a number of test problems. Mesh adaptation is particularly simple to implement since elements with consecutive co-planar edges/faces are allowed and, therefore, locally adapted meshes do not require any local mesh post-processing

Association of non-synonymous SNPs of <i>OPN</i> gene with litter size traits in pigs

Osteopontin (<i>OPN</i>) gene is a secreted phosphoprotein which appears to play a key function in the conceptus implantation, placentation and maintenance of pregnancy in pigs. The objectives of this study were to verify the non-synonymous single nucleotide polymorphisms (SNPs) and their association with litter size traits in commercial Thai Large White pigs. A total of 320 Thai Large White sows were genotyped using the polymerase chain reaction-restriction fragment length polymorphism (PCR-RFLP) method. Three SNPs at c.425G> A, c.573T> C and c.881C> T revealed amino acid exchange rates of p.110Ala> Thr, p.159Val> Ala and p.262Pro> Ser, respectively, and were then segregated. These three SNPs were significantly associated with total number born (TNB) and number born alive (NBA) traits. No polymorphisms of the two SNP markers (c.278A> G and c.452T> G) were observed in this study. Moreover, the SNPs at c.425G> A and c.573T> C were found to be in strong linkage disequilibrium. The association of <i>OPN</i> with litter size emphasizes the importance of porcine <i>OPN</i> as a candidate gene for reproductive traits in pig breeding

Damped Response of a Cantilever Sensor Embedded in Passive Muscle under Impact

在生活中，容易因為外物的碰撞使得體內之微懸臂梁感測器的量測失準。為了排除因撞擊而產生的位移量，並準確地偵測CRP蛋白質濃度，我們從文獻中蒐集生物軟組織(皮膚、肌肉)的單軸拉伸實驗曲線和鬆弛試驗曲線，並以曲線擬合的方式擬合出超彈性(hyperelastic) (以Ogden form表示)和黏彈性(viscoelastic)(以Prony級數表示)的材料常數。 　　在計算微懸臂梁的位移量時會因為充斥感測器內的液體(即血液)產生的剪力和正向壓力，使得梁不會很迅速地回到原來的位置，而類似於阻尼的效果。其間的過程在於固液耦合的物理現象，在每個時間點固體結構的變形和液體的流速互相影響。經由有限元素法的分析，可以得到微懸臂梁上產生最大位移的地方(梁末端)衰減所需要的時間。梁末端位移衰減至因衝擊造成最大位移的10%所需要的時間，經由二維以及三維分析得到的結果分別為11.5秒和6.2秒。目錄 摘要 i 目錄 ii 圖目錄 v 表目錄 viii 第一章、導論 1 1.1、研究動機 1 1.2、文獻回顧 2 1.3、研究方法 8 1.4、論文架構 10 第二章、理論與數值方法 11 2.1、黏彈性之本構律(viscoelasticity) 11 2.1.1、積分形式的應力應變關係 11 2.1.2、微分形式的應力應變關係 15 2.1.3、模型理論 16 2.2、大應變時的本構關係(超彈性(hyperelastic)) 20 2.3、幾何非線性(大變形之情況) 21 2.3.1、大變形下的應變 22 2.3.2、大變形下的應力 23 2.4、流場統御域方程式 24 2.5、懸臂梁變形理論 25 第三章、模擬結果與分析(2D) 27 3.1、第一階段 28 3.1.1、建立模型 28 3.1.2、網格生成 28 3.1.3、材料屬性 29 3.1.4、邊界條件和接觸條件 34 3.1.5、模擬結果 35 3.1.6、結果分析 39 3.2、第二階段 41 3.2.1、建立模型 41 3.2.2、網格生成 42 3.2.3、材料屬性 42 3.2.4、邊界條件 43 3.2.5、求解與數值結果 43 3.2.6、結果分析 47 3.3、不同位置下的變形情況 52 3.3.1、第一階段 52 3.3.2、第二階段 54 3.3.3、結果分析 55 第四章、模擬結果與分析(3D) 56 4.1、第一階段 56 4.1.1、模型建立 56 4.1.2、網格生成 57 4.1.3、材料屬性 57 4.1.4、邊界條件 57 4.1.5、模擬結果 58 4.1.6、結果分析 60 4.2、第二階段 61 4.2.1、模型建立 61 4.2.2、網格生成、材料屬性、邊界條件 61 4.2.3、模擬結果 61 4.2.4、結果分析 62 第五章、結論與未來展望 64 5.1、結論 64 5.2、未來展望 65 參考文獻 66 附錄A、數值模擬中使用的材料參數 6

Primer of adaptive finite element methods

Adaptive finite element methods (AFEM) are a fundamental numerical instrument in science and engineering to approximate partial differential equations. In the 1980s and 1990s a great deal of effort was devoted to the design of a posteriori error estimators, following the pioneering work of Babuska. These are computable quantities, depending on the discrete solution(s) and data, that can be used to assess the approximation quality and improve it adaptively. Despite their practical success, adaptive processes have been shown to converge, and to exhibit optimal cardinality, only recently for dimension d > 1 and for linear elliptic PDE. These series of lectures presents an up-to-date discussion of AFEM encompassing the derivation of upper and lower a posteriori error bounds for residual-type estimators, including a critical look at the role of oscillation, the design of AFEM and its basic properties, as well as a complete discussion of convergence, contraction property and quasi-optimal cardinality of AFEM