On stiffness matrices for C0 continuous tetrahedra
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...Thus, these linear substitute shape functions return the corner nodes of the 2-square in the transform plane as a set of corner nodes for a parallelogram instead of the original quadrilateral (Subramanian and Bose, 1982; 1983) for a similar observation)....
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...Simple exactly integrated numerical universal matrices like those for triangles and tetrahedrons (Subramanian and Bose, 1982; 1983) also are possible only for parallelograms....
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Cites methods from "On stiffness matrices for C0 contin..."
...1.5.1 Closed-form Stiffness Matrices Tinawi [23] used closed-form integration to obtain the stiffness matrix of non- hierarchic triangular elements in 1972; Subramanian and Bose [24, 25] developed stiffness matrices without the use of numerical approximation for the family of plane triangular elements in 1982 and for C0 continuous tetrahedra in 1983....
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...113 nn[[11]] = Simplify[L2 L3 Ei[2,L2,L3]]; nn[[12]] = Simplify[L1 L3 Ei[2,L1,L3]]; nn[[13]] = Simplify[L1 L2 Ei[2,L1,L2]]; nn[[14]] = Simplify[L1 L4 Ei[2,L1,L4]]; nn[[15]] = Simplify[L2 L4 Ei[2,L2,L4]]; nn[[16]] = Simplify[L3 L4 Ei[2,L3,L4]]; (* Face modes *) nn[[17]] = Simplify[ L2 L3 L4 Fi[0, 0, L2, L3, L4 ]]; nn[[18]] = Simplify[ L3 L4 L1 Fi[0, 0, L1, L3, L4 ]]; nn[[19]] = Simplify[ L4 L1 L2 Fi[0, 0, L1, L2, L4 ]]; nn[[20]] = Simplify[ L1 L2 L3 Fi[0, 0, L1, L2, L3 ]]; nn[[21]] = Simplify[L2 L3 Ei[3,L2,L3]]; nn[[22]] = Simplify[L1 L3 Ei[3,L1,L3]]; nn[[23]] = Simplify[L1 L2 Ei[3,L1,L2]]; nn[[24]] = Simplify[L1 L4 Ei[3,L1,L4]]; nn[[25]] = Simplify[L2 L4 Ei[3,L2,L4]]; nn[[26]] = Simplify[L3 L4 Ei[3,L3,L4]]; (* Face modes *) nn[[27]] = Simplify[ L2 L3 L4 Fi[1, 0, L2, L3, L4 ]]; nn[[28]] = Simplify[ L3 L4 L1 Fi[1, 0, L1, L3, L4 ]]; nn[[29]] = Simplify[ L4 L1 L2 Fi[1, 0, L1, L2, L4 ]]; nn[[30]] = Simplify[ L1 L2 L3 Fi[1, 0, L1, L2, L3 ]]; nn[[31]] = Simplify[ L2 L3 L4 Fi[0, 1, L2, L3, L4 ]]; nn[[32]] = Simplify[ L3 L4 L1 Fi[0, 1, L1, L3, L4 ]]; nn[[33]] = Simplify[ L4 L1 L2 Fi[0, 1, L1, L2, L4 ]]; nn[[34]] = Simplify[ L1 L2 L3 Fi[0, 1, L1, L2, L3 ]]; (* Bubble mode *) nn[[35]] = L1 L2 L3 L4; (* From Shiakolas *) (* Put into appropriate format for use with developed equations *) NT = Flatten[Table[i3 * nn[[i]], {i,1,nTot}],1]; NN = Transpose[NT]; NN = Simplify[NN]; (* Form the R matrix *) RL1 = D[NN,L1];RL2 = D[NN,L2];RL3 = D[NN,L3]; R = Flatten[{RL1, RL2, RL3}, 1];...
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...1 Closed-form Stiffness Matrices Tinawi [23] used closed-form integration to obtain the stiffness matrix of nonhierarchic triangular elements in 1972; Subramanian and Bose [24, 25] developed stiffness matrices without the use of numerical approximation for the family of plane triangular elements in 1982 and for C0 continuous tetrahedra in 1983....
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...Subramanian, G., and Bose, C.J....
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...Subramanian, G., and Bose, C.J. Convenient generation of stiffness matrices for the family of plane triangular elements, Computers and Structures 1982; 10:119 – 124....
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