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dohrmann Episode 2 Part 2

Tham khảo tài liệu 'dohrmann episode 2 part 2', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | the material matrix D for plane stress can be expressed as 1 V 0 V 1 0 D 1-1 2 0 0 1 - 2 36 Six different element types shown in Figure 3 are considered in the example problems. These include the four-node quadrilateral Q4 eight-node quadrilateral Q8 twelve-node quadrilateral Q12 three-node triangle T3 six-node triangle T6 and ten-node triangle T10 . Stiffness matrices of the various elements are calculated using numerical integration. The quadrilateral elements use 2 by 2 3 by 3 and 4 by 4 Gaussian quadrature for Q4 Q8 and Q12 respectively. Numerical integration formulae for triangles see Ref. 4 with 1 3 and 7 points are used for T3 T6 and no respectively. Two meshes connected at a shared boundary are used in all the example problems. Mesh 1 is initially bounded by the the four sides Xi 0 Xi hỵ x2 0 and x2 h2 while Mesh 2 is initially bounded by the four sides Z1 hl Xi 2hỵ x2 0 and x2 h2. The two meshes consist of either quadrilateral or triangular elements as shown in Figure 4. The number of element edges in direction i for mesh m is designated as nim. Thus all the meshes in Figure 4 have nil 21 2 and Tii2 n22 3. Mesh configurations are designated by the element type for Mesh 1 followed by the element type for Mesh 2 see Figure 4 . Calculated values of the energy norm of the error are presented in the example problems for purposes of comparison and for the investigation of convergence rates. The energy norm of the error is a measure of the accuracy of a finite element approximation and is defined as __ exactxT fe - exact dA 1 2 37 where Qfc is the domain of element k and eỉe and eeiaci denote the finite element and exact strains respectively. The symbol 1 denotes the set of all element numbers for the two meshes. Calculation of energy norms for the quadrilateral and triangular elements is based on the integration schemes for element types Ọ12 and 7T0 respectively. Results are also presented for an energy norm density Cfc of the error defined as 38 where Ab .