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Chap 3. Basic Concepts of Engineering Analysis and Introduction to FEM

Chap 3. Basic Concepts of Engineering Analysis and Introduction to FEM. Analysis of Engineering Systems Idealization of system Formulation of the governing equilibrium equation Solution of the equilibrium equation Interpretation of results Discrete system : algebraic equation

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Chap 3. Basic Concepts of Engineering Analysis and Introduction to FEM

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  1. Chap 3. Basic Concepts of Engineering Analysis and Introduction to FEM • Analysis of Engineering Systems • Idealization of system • Formulation of the governing equilibrium equation • Solution of the equilibrium equation • Interpretation of results • Discrete system : algebraic equation • Continuous system : differential equation (governing equlibrium eqn) Numerical procedure(Variational method)

  2. Ex 3.1 Elastic Spring System

  3. Ex 3.1(2)

  4. Ex 3.2 Heat Transfer(Heat conduction law ) Analogy to spring system

  5. Ex 3.3 Hydraulic Network : incompressible

  6. Ex 3.4 DC Network (Ohm’s law , Kirchhoff’s Current(Voltage) law)

  7. Ex 3.5 Nonlinear elastic spring Ex 3.8 Dynamic Problem ODE need I.C.

  8. k u P Variation Ex. 3.6 Variational formulation can generate governing equilibrium equation systematically but gives less physical insight.

  9. Eigenvalue Problems generalized EVP where A and B symmetric ( : standard EVP) EVP is used to check system stability (Critical load Pcr)

  10. Ex 3.11

  11. Ex 3.11

  12. Ex 3.12 Free Vibration For free vibration Let Eigenvalue Problem Nature of Solution – Linear System

  13. §3.3 Solution of Continuous System • Obtaining Differential Equations which express • Element equilibrium requirement • Constitutive relation (linear elasticity-generalized Hook’s law) • Element interconnectivity requirement (compatibility) To ensure single-valued, continuous solution Differential equation Static B.C. Dynamic B.C. & I.C. Intelligent System Design Lab. Dept. of Mechatronics, K-JIST

  14. Second-order general PDE Canonical forms of linear equation discriminant <0 elliptic: Laplace eqn (BVP) BCs =0 parabolic : heat conduction eqn BC & IC >0 hyperbolic : wave eqn BC & IC

  15. Variational Method for Continuous Systems Variational method is a very powerful method for CS since it easily generates natural BC. essential BC: geometry BC – prescribed disp and rotations natural BC : force BC – prescribed boundary force and moments Variation

  16. properties

  17. Ex 3.18 Integration by parts

  18. w E,L,I x P k Ex 3.20 governing DE : natural BC : governing DE and natural BC

  19. Note on Variational formulation 1. VF easily generates system-governing equation. Since VF deals scalar such as energies and potential, whereas DF deals vectors (displacement, force). 2. VF generates governing equation & natural BC. VF does not need to consider internal forces. 3. VF can be employed in general class of problems since natural BC is not specified in VF. ( natural BC is implicitly contained in VF) Simple BVP can be solved using separation of variable technique. For complex problem, trial functions are used to obtain an approximated solution.

  20. principle of minimum potential energy (for linear mechanical system) Minimum Functional Theorem Let be a positive definite linear operator with domain that is dense in and let have a solution. Then the solution is unique and minimizes the energy functional Let A be a symmetric positive definite matrix. Then is the solution of iff minimizes OR If there exists a function in that minimizes , then it is the unique solution of Proof MFT 1)  2) 

  21. Ritz Method Let and be linear independent (m<n) Approximate (3) (3)  (2), (4) (5) (6) Theorem : Let be a set of linear independent vectors in vector space with scalar product . Then the only vector in that is orthogonal to each of is the zero vector.

  22. Galerkin Method where Then residual is (7) (8) Iff are linear independent vectors. Let and substitute into (8) (9) Least Squares Method Minimizes norm of ,i.e. minimizes Let Ritz method : A sym & pos def Gelerkin & Least Squares : A may not be sym or pos def

  23. Overview of Chap. 3 E, A, L R DF: VF:

  24. DF  VF : since Principle of virtual displacement or principle of virtual work Find such that where is the space of square integrable function,

  25. E, A, L R n n+1 n-1 Finite Difference Forward FD Backward FD Central FD FD gives good approximation for simple geometry problem. FD cannot give natural BC for complex geometry problem. (Also symmetric matrix) FD Energy method FD is used for displacement derivatives in VF.

  26. Imposition of Constraints(1) Lagrange Multiplier Method (LMM) (additional varialbe required) Penalty Method (PM)

  27. k k Imposition of Constraints(2) Ex 3.31 LMM PM

  28. Imposition of Constraints(3) General Form (Bu=v) LMM PM  might generate ill-conditioned matrix Augmented LMM (Combined idea of LMM and PM)

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