清华大学有限元分析课件2-4 Solution of Equilibrium Equations_569506475.pdfVIP

第2 章 直接刚度法 2.4 平衡方程组的求解 Solution of Equilibrium Equations [Bathe]: Chapter 8 Finite Element Method Solution of Equilibrium Equations   Direct solution techniques Ka = R   Gauss elimination Equations are solved using a number of steps and T operations that are predetermined in an exact manner.   LDL solution   Cholesky factorization   Frontal solution   Iterative solution methods   Gauss-Seidel Method   Conjugate Gradient Method with Preconditioning   The number of iterations required for convergence depends on the condition number of the matrix K and whether the acceleration schemes used are effective for the particular case considered   For large system, iterative methods can be much more effective. 2/14 School of Aerospace, Tsinghua University Finite Element Method Solution of Equilibrium Equations Gauss elimination ⎡ u ⎤ ⎡ 5 −4 1 0 ⎤ ⎢ 1 ⎥ ⎡ 0 ⎤ ⎢ −4 6 −4 1 ⎥ u2 ⎢ 1 ⎥ Ka = R ⎢ ⎥= ⎢ 1 −4 6 −4 ⎥ u ⎢ 0 ⎥ ⎢ 3 ⎥ ⎢ 0 1 −4 5 ⎥ ⎢ 0 ⎥ ⎣ ⎦ ⎢ u ⎥ ⎣ ⎦ ⎣ 4 ⎦ Forward reduction 4 1 1.  Subtract − 5 times the first equation from the second and 5 times the first equation from third equations to obtain zero elements in the first column of K ⎡ 5 −4 1 0 ⎤ ⎡ a ⎤ ⎢ 14 16 ⎥ 1 ⎡ 0 ⎤ 0 − 1 ⎢ ⎥ ⎢ 5 5 ⎥ a2 ⎢ 1 ⎥