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Introduction to Finite Element Methods

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Introduction to Finite Element Methods

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  1. Introduction to Finite Element Methods UNIT I

  2. Numerical Methods – Definition and Advantages • Definition: Methods that seek quantitative approximations to the solutions of mathematical problems • Advantages:

  3. What is a Numerical Method – An Example Example 1:

  4. What is a Numerical Method – An Example Example 1:

  5. What is a Numerical Method – An Example Example 2:

  6. What is a Numerical Method – An Example Example 2:

  7. What is a Numerical Method – An Example Example 3:

  8. What is a Finite Element Method

  9. Discretization 1-D 2-D ?-D 3-D Hybrid

  10. Approximation Numerical Interpolation Non-exact Boundary Conditions

  11. Applications of Finite Element Methods • Structural & Stress Analysis • Thermal Analysis • Dynamic Analysis • Acoustic Analysis • Electro-Magnetic Analysis • Manufacturing Processes • Fluid Dynamics

  12. Lecture 2 Review

  13. Matrix Algebra • Row and column vectors • Addition and Subtraction – must have the same dimensions • Multiplication – with scalar, with vector, with matrix • Transposition – • Differentiation and Integration

  14. Matrix Algebra • Determinant of a Matrix: • Matrix inversion - • Important Matrices • diagonal matrix • identity matrix • zero matrix • eye matrix

  15. Numerical Integration Calculate: • Newton – Cotes integration • Trapezoidal rule – 1st order Newton-Cotes integration • Trapezoidal rule – multiple application

  16. Numerical Integration Calculate: • Newton – Cotes integration • Simpson 1/3 rule – 2nd order Newton-Cotes integration

  17. Numerical Integration Calculate: • Gaussian Quadrature Trapezoidal Rule: Gaussian Quadrature: Choose according to certain criteria

  18. Numerical Integration Calculate: • Gaussian Quadrature • 2pt Gaussian Quadrature • 3pt Gaussian Quadrature Let:

  19. Numerical Integration - Example Calculate: • Trapezoidal rule • Simpson 1/3 rule • 2pt Gaussian quadrature • Exact solution

  20. Linear System Solver Solve: • Gaussian Elimination: forward elimination + back substitution Example:

  21. Linear System Solver Solve: • Gaussian Elimination: forward elimination + back substitution Pseudo code: Forward elimination: Back substitution: Do k = 1, n-1 Do i = k+1,n Do j = k+1, n Do ii = 1, n-1 i = n – ii sum = 0 Do j = i+1, n sum = sum +

  22. UNIT II Finite Element Analysis (F.E.A.) of 1-D Problems

  23. Historical Background • Hrenikoff, 1941 – “frame work method” • Courant, 1943 – “piecewise polynomial interpolation” • Turner, 1956 – derived stiffness matrice for truss, beam, etc • Clough, 1960 – coined the term “finite element” Key Ideas: - frame work method piecewise polynomial approximation

  24. Axially Loaded Bar Review: Stress: Stress: Strain: Strain: Deformation: Deformation:

  25. Axially Loaded Bar Review: Stress: Strain: Deformation:

  26. Axially Loaded Bar – Governing Equations and Boundary Conditions • Differential Equation • Boundary Condition Types • prescribed displacement (essential BC) • prescribed force/derivative of displacement (natural BC)

  27. Axially Loaded Bar –Boundary Conditions • Examples • fixed end • simple support • free end

  28. Potential Energy • Elastic Potential Energy (PE) - Spring case Unstretched spring Stretched bar x - Axially loaded bar undeformed: deformed: - Elastic body

  29. Potential Energy • Work Potential (WE) f P f: distributed force over a line P: point force u: displacement B A • Total Potential Energy • Principle of Minimum Potential Energy For conservative systems, of all the kinematically admissible displacement fields, those corresponding to equilibrium extremize the total potential energy. If the extremum condition is a minimum, the equilibrium state is stable.

  30. Potential Energy + Rayleigh-Ritz Approach Example: f P B A Step 1: assume a displacement field f is shape function / basis function n is the order of approximation Step 2: calculate total potential energy

  31. Potential Energy + Rayleigh-Ritz Approach Example: f P B A Step 3:select ai so that the total potential energy is minimum

  32. Galerkin’s Method Example: f P B A Seek an approximation so In the Galerkin’s method, the weight function is chosen to be the same as the shape function.

  33. Galerkin’s Method Example: f P B A 3 2 1 1 2 3

  34. Finite Element Method – Piecewise Approximation u x u x

  35. FEM Formulation of Axially Loaded Bar – Governing Equations • Differential Equation • Weighted-Integral Formulation • Weak Form

  36. Approximation Methods – Finite Element Method Example: Step 1: Discretization Step 2: Weak form of one element P2 P1 x1 x2

  37. Approximation Methods – Finite Element Method Example (cont): Step 3: Choosing shape functions - linear shape functions x x x=0 x=-1 x=1 x2 x1 l

  38. Approximation Methods – Finite Element Method Example (cont): Step 4: Forming element equation E,A are constant Let , weak form becomes Let , weak form becomes

  39. Approximation Methods – Finite Element Method Example (cont): Step 5: Assembling to form system equation Approach 1: Element 1: Element 2: Element 3:

  40. Approximation Methods – Finite Element Method Example (cont): Step 5: Assembling to form system equation Assembled System:

  41. Approximation Methods – Finite Element Method Example (cont): Step 5: Assembling to form system equation Approach 2: Element connectivity table global node index (I,J) local node (i,j)

  42. Approximation Methods – Finite Element Method Example (cont): Step 6: Imposing boundary conditions and forming condense system Condensed system:

  43. Approximation Methods – Finite Element Method Example (cont): Step 7: solution Step 8: post calculation

  44. Summary - Major Steps in FEM • Discretization • Derivation of element equation • weak form • construct form of approximation solution over one element • derive finite element model • Assembling – putting elements together • Imposing boundary conditions • Solving equations • Postcomputation

  45. Exercises – Linear Element Example 1: E = 100 GPa, A = 1 cm2

  46. Linear Formulation for Bar Element f(x) u2 u u1 x L = x2-x1 x= x2 x=x1 1 1 f1 f2 x=x2 x=x1

  47. Higher Order Formulation for Bar Element u2 u3 u u u u1 x x x 3 2 1 u4 u2 u3 u1 3 2 1 4 un u1 u4 u3 …………… u2 1 n …………… 2 3 4

  48. Natural Coordinates and Interpolation Functions x x x x x=-1 x=1 x=-1 x=1 2 1 3 1 2 x=-1 x=1 4 2 1 3 x=1 x=-1 x x=x1 x= x2 Natural (or Normal) Coordinate:

  49. Quadratic Formulation for Bar Element f3 f1 f2 x=1 x=0 x=-1

  50. Quadratic Formulation for Bar Element u2 f(x) u3 u1 P2 P1 P3 x=1 x=0 x=-1