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Chap 8. FE Analysis- Solution

Chap 8. FE Analysis- Solution. FE Model (DOF) is getting large for better accuracy. Needs good solution algorithm of the equilibrium equations for efficiency K x = R Accuracy, Efficiency, Stability Direct Soln - Gauss Elimination, LDL T (LU), Cholesky factorization ( ), QR

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Chap 8. FE Analysis- Solution

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  1. Chap 8. FE Analysis- Solution FE Model (DOF) is getting large for better accuracy. Needs good solution algorithm of the equilibrium equations for efficiency Kx=R Accuracy, Efficiency, Stability Direct Soln - Gauss Elimination, LDLT(LU), Cholesky factorization ( ), QR Iterative Soln - Gauss-Seidel, Conjugate Gradient Nonlinear - Newton-Raphson, BFGS Intelligent System Design Lab. Dept. of Mechatronics, K-JIST

  2. Direct Solution • Gauss Elimination , LDLT(LU) • Assumption : ① Symmetric ② Nonzero Diagonal : true for displacement-based FE • : not always true for mixed formulation Intelligent System Design Lab. Dept. of Mechatronics, K-JIST

  3. Direct Solution Large system : use of back-up storage , effective out-of-core solutionFE K -> symmetric, positive definite, banded (depend on node numbering) Operation count for LDLT= Intelligent System Design Lab. Dept. of Mechatronics, K-JIST

  4. Cholesky Factorization Directly calculate rather than from LDLT Slightly more operation count than LDLT Main use is to transform from general eigen problem ( ) to standard form ( ) Substructure Using static condensation to reduce DOF Frontal Solution Method Statically condensing out one DOF after the other and always assembling only those element matrices required Nodal numbering will decide half-bandwidth Direct Solution

  5. For very large FE system, direct methods require nmk for storage and nmk for operation Iterative Solution

  6. Lanczos Algorithm (AQ=QT) Krylov Sequence Iterative Solution

  7. Lanczos Algorithm (AQ=QT) Load Dependent Ritz Vectors (LDRV) Iterative Solution

  8. References Lanczos Algorithm Nour-Omid, B. and Clough, K.W.,”Dynamic Analysis of Structure using Lanczos Co-ordinates”, Earthquake Eng. And structure Dyn.Vol. 12, pp 565-577 ,1984 Load Dependent Ritz Vectors (LDRV) Kline, K.A., ”Dynamic Analysis Using a Reduced Basis of Exact Modes and Ritz Vectors”, AIAA J, Vol. 24, pp2022-2029, 1986 Wang, S. and Choi, K.K. ,”Continuum Design Sensitivity of Transient Response Using Ritz and Mode Acceleration Methods”, AIAA J ,Vol. 30, pp1099-1109, 1992. Iterative Solution

  9. Conjugate Gradient Method : A must be positive definite Iterative Solution

  10. Newton-Raphson’s Method Modified Newton Steepest Descent Method Nonlinear Equations Q

  11. Quasi-Newton Nonlinear Equations

  12. Explicit : Central Difference , Runge-Kunta Implicit : Houbolt , Newmark β , wilson θ , Hughes α • Direct Integration MDM MAM LDRV , Lanczos , Krylov • Superposition Transient System Analysis • Superposition & Direct Integration

  13. Mode Displacement Method (MDM)

  14. Mode Acceleration Method (MAM)

  15. Load Depedent Ritz Vector Method (LDRV)

  16. Direct Integration

  17. Central Difference Method Direct Integration Methods

  18. Houbolt Method Acceleration and velocity are approximated using finite difference Wilson Method ( = 1.4 ) Direct Integration Methods Modified Linear acceleration

  19. Wilson Method Direct Integration Methods

  20. Newmark Method Direct Integration Methods Constant-average acceleration

  21. Newmark Method Direct Integration Methods

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