Introduction to Finite Elements


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Finite Element Method is explained taking a simple example
Essential concepts in this technique are introduced
Top-down approach and bottom-up approach are used to present a holistic picture of FEM

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Introduction to Finite Elements

  1. 1. An introduction to FEM V.S.S.Srinivas
  2. 2. Plan <ul><li>Day 1 </li></ul><ul><li>A brief introduction to Finite Difference Method (FDM) using a simple example </li></ul><ul><li>Introduction to the concept of Finite Element Method (FEM) using a top-down approach </li></ul><ul><ul><li>Weighted residuals—illustration using an example </li></ul></ul><ul><ul><li>Interpolation functions—illustration using an example </li></ul></ul><ul><ul><li>Comparison of the numerical solutions obtained using FEM and FDM against the analytical solution—discussion </li></ul></ul><ul><ul><li>Applying boundary conditions—Natural boundary conditions </li></ul></ul><ul><ul><li>Concept of element </li></ul></ul><ul><li>Day 2 </li></ul><ul><li>Revisiting FEM using a bottom-up approach—The Standard Procedure </li></ul><ul><ul><li>Element shape functions </li></ul></ul><ul><ul><li>Natural coordinates—Geometric coordinates </li></ul></ul><ul><ul><li>Coordinate transformation—Jacobian </li></ul></ul><ul><ul><li>Numerical integration—Gauss quadrature </li></ul></ul><ul><ul><li>Nodal connectivity—assembly of element matrices—global matrix </li></ul></ul><ul><ul><li>Applying boundary conditions—Essential boundary conditions </li></ul></ul><ul><li>Questions session </li></ul>
  3. 3. A glance at Finite Difference Method <ul><li>Consider a steady one dimensional heat conduction case </li></ul><ul><li>Approximate the governing equations and boundary conditions with algebraic equivalents </li></ul><ul><li>Impose the equivalent conditions at select locations </li></ul>q
  4. 4. Illustration by example 1 2 3 4 q
  5. 5. Solution of FDM
  6. 6. Summary <ul><li>Derivatives are approximated using Taylor series </li></ul><ul><li>The resultant difference (algebraic) equations are imposed at nodes </li></ul><ul><li>The set of linear algebraic equations are solved </li></ul><ul><li>A solution is obtained for the approximated system of equations </li></ul><ul><li>At the outset, Finite Element Method differs from FDM in the above aspect </li></ul>
  7. 7. Finite Element Method <ul><li>We approximate the solution </li></ul><ul><li>Interpolation functions </li></ul><ul><li>Let l=1 </li></ul>
  8. 8. Finite Element Method-Galerkin Weighted Residuals <ul><li>Analytical solution is the exact solution for a system of differential equations </li></ul><ul><li>We seek approximate solution when there is no exact one </li></ul><ul><li>How do we go about it </li></ul><ul><li>Can we satisfy the equations in an average sense? </li></ul><ul><li>How can we improve upon the solution we are seeking </li></ul>
  9. 9. FEM-Galerkin Weighted Residuals
  10. 10. Analytical solution and comparison <ul><li>Analytical solution </li></ul><ul><li>Comparison of all the three solutions </li></ul>
  11. 11. Other Weighted Residual Methods <ul><li>Least squares method </li></ul><ul><li>Point collocation method </li></ul><ul><li>Subdomain collocation method </li></ul>
  12. 12. Concept of assembly <ul><li>The total integral can be considered as the sum of integrals over a set of sub-domains </li></ul><ul><li>In finite element terminology, they are called elements </li></ul>1 2 3 1 2 3 4
  13. 13. Concept of Assembly Assembled matrix
  14. 14. Contd.. <ul><li>Boundary term can also be decomposed into sum of integrals over each subdomain </li></ul><ul><li>If you notice, except at the end points, the integral cancels at every other point or node in the domain. </li></ul><ul><li>Essentially, this is to say that the whole integral can be seen as the sum of integral over each subdomain </li></ul><ul><li>Till now, we dissected the whole integral and saw the details. We depart at this point and resume FEM by assembling the integrals of every subdomain (element) </li></ul><ul><li>In this process, we will visit the standard procedure of finite element method </li></ul>
  15. 15. Revisiting interpolation functions Element point of view <ul><li>Non-zero functions in element 1: N1, N2 </li></ul><ul><li>Non-zero functions in element 2: N2, N3 </li></ul><ul><li>Non-zero functions in element 3: N3, N4 </li></ul><ul><li>For every element, the components of interpolation functions are presenting a common picture </li></ul><ul><li>It is easy to obtain the matrix for every element and then assemble them to obtain the global matrix </li></ul>
  16. 16. Interpolation functions from an element point of view
  17. 17. FEM-Standard Procedure <ul><li>Reconsider the example discussed before, resuming from the last point of departure </li></ul><ul><li>The integrand in the equation cannot be always analytically integrated </li></ul><ul><li>For example, if </li></ul><ul><li>Or k can also be a function of Temperature. </li></ul><ul><li>What is the way out? </li></ul>
  18. 18. Element shape functions <ul><li>Most of the times, the integrand is not numerically integrable </li></ul><ul><li>We resort to numerical integration then </li></ul>
  19. 19. FEM Standard Procedure- Coordinate Transformation <ul><li>Numerical integration, popularly known as gauss quadrature </li></ul><ul><li>This rule is for a generic element </li></ul><ul><li>Limits of the integration are from -1 to 1 instead of x e1 and x e2 </li></ul><ul><li>Necessitates a coordinate transformation </li></ul><ul><li>Old coordinates ‒G eometric coordinates </li></ul><ul><li>New non-dimensional coordinates ‒N atural coordinates </li></ul><ul><li>The coordinate transformation brings in a scaling factor named Jacobian </li></ul>
  20. 20. Pictorial representation-coordinate transformation <ul><li>Notion of isoparametric formulation </li></ul>Jacobian
  21. 21. Assembly of element matrices nodal connectivity-1D 1 2 3 4 1 2 3 1 2 Local node no. Global node no. 1 - 3, 2 – 4 3 1 - 2, 2 – 3 2 1 - 1, 2 - 2 1 Local to global Element
  22. 22. Nodal Connectivity-2D Global node no. Local node no. (i,j) entry in every element conductivity matrix goes to (I,J) entry in global conductivity matrix (i,j)—local node nos, (I,J)—Global node nos. 1 2 3 4 1 5 6 7 8 9 2 3 4 1 2 3 4 1 – 4, 2 – 5,3 – 8, 4 – 7 3 1 – 5, 2 – 6, 3 – 9, 4 – 8 3 2 – 2, 2 – 3,3 – 6,4 – 5, 2 1 – 1,2– 2, 3 – 5, 4 – 4 1 Local to global Element
  23. 23. Applying Boundary Conditions <ul><li>Natural or neuman boundary conditions are applied in the integral form </li></ul><ul><li>Number of ways to impose essential (dirichlet) conditions </li></ul><ul><li>Revisiting the example,T 4 is known, T 1 , T 2 , T 3 have to be solved </li></ul><ul><li>Considering the assembled system of equations </li></ul>
  24. 24. Contd.. <ul><li>We can take any set of three equations </li></ul><ul><li>Consider the first three equations </li></ul><ul><li>Subtract the term associated with T 4 from both sides </li></ul><ul><li>Solve for the unknowns </li></ul>
  25. 25. Contd.. <ul><li>Subtract the fourth column multiplied by T 4 from the right hand side </li></ul><ul><li>Remove the fourth row and column </li></ul><ul><li>Remove the fourth entry from the right hand side </li></ul><ul><li>Solve for T 1 , T 2 , T 3 using the resulting set of linear equations </li></ul><ul><li>Other popular methods are lagrange multiplier, penalty etc. </li></ul>
  26. 26. Summary <ul><li>Considered a steady state heat conduction problem as the example problem to illustrate the concepts of FDM and FEM </li></ul><ul><li>To lay a platform for the comparison of FDM and FEM, the problem is solved using FDM </li></ul><ul><li>Next, obtained solution using FEM. In the process, explained the important concepts </li></ul><ul><ul><li>Weighted residuals </li></ul></ul><ul><ul><li>Integral form </li></ul></ul><ul><ul><li>Interpolation functions </li></ul></ul><ul><ul><li>Imposition of natural boundary conditions </li></ul></ul><ul><ul><li>Notion of element </li></ul></ul><ul><li>Compared the FDM and FEM solutions against the analytical solution </li></ul><ul><li>Finite element method is explained by using a dissection approach </li></ul><ul><li>Next, the standard approach of assembly starting from the element stiffness matrices is explained </li></ul><ul><ul><li>Natural or intrinsic coordinates, spatial coordinates are explained </li></ul></ul><ul><ul><li>Local-global nodal connectivity, gauss quadrature, applying essential boundary conditions are explained </li></ul></ul><ul><ul><li>The concepts of Jacobian and Gauss Quadrature are introduced </li></ul></ul>
  27. 27. References <ul><li>An Introduction to Finite Element Method, J.N.Reddy, McGraw-Hill Science Engineering </li></ul><ul><li>Introduction to Finite Elements in Engineering (3rd Edition) by Tirupathi R. Chandrupatla and Ashok D. Belegundu, </li></ul><ul><li>Differential equations with exact solutions: </li></ul>
  28. 28. Interpolation functions