Finite Element Analysis - The Basics
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Finite Element Analysis is a widely used computational method in most of the engineering domains. But still, its considered as a difficult topic by most students. This presentation is an effort to introduce the very basics of FEA so as to build an intuitive feel for the method. Enjoy !
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Finite Element Analysis - The Basics
- 1. Finite Element Analysis Sujith Jose Introduction Steps in Finite Element Analysis Finite Element Discretization Elementary Governing Equations Assembling of all elements Solving the resulting equations i.Iteration Method ii.Band Matrix Method Example References Finite Element Analysis - The Basics Sujith Jose University of California, Los Angeles sujithjose5@ucla.edu May 31, 2016
- 2. Finite Element Analysis Sujith Jose Introduction Steps in Finite Element Analysis Finite Element Discretization Elementary Governing Equations Assembling of all elements Solving the resulting equations i.Iteration Method ii.Band Matrix Method Example References Overview 1 Introduction 2 Steps in Finite Element Analysis Finite Element Discretization Elementary Governing Equations Assembling of all elements Solving the resulting equations i.Iteration Method ii.Band Matrix Method 3 Example 4 References
- 3. Finite Element Analysis Sujith Jose Introduction Steps in Finite Element Analysis Finite Element Discretization Elementary Governing Equations Assembling of all elements Solving the resulting equations i.Iteration Method ii.Band Matrix Method Example References Introduction Origin in structural analysis Mathematical treatment - 1948 Applied to Electromagnetic problems - 1968 Can handle complex geometries Used in almost all engineering disciplines including electrical, aeronautical, biomedical and civil
- 4. Finite Element Analysis Sujith Jose Introduction Steps in Finite Element Analysis Finite Element Discretization Elementary Governing Equations Assembling of all elements Solving the resulting equations i.Iteration Method ii.Band Matrix Method Example References Steps in Finite Element Analysis 1 Discretize the solution region into elements 2 Derive governing equations for one element 3 Assemble all elements 4 Solving system of equations obtained
- 5. Finite Element Analysis Sujith Jose Introduction Steps in Finite Element Analysis Finite Element Discretization Elementary Governing Equations Assembling of all elements Solving the resulting equations i.Iteration Method ii.Band Matrix Method Example References Finite Element Discretization Figure: A typical finite element subdivision of an irregular 2D domain
- 6. Finite Element Analysis Sujith Jose Introduction Steps in Finite Element Analysis Finite Element Discretization Elementary Governing Equations Assembling of all elements Solving the resulting equations i.Iteration Method ii.Band Matrix Method Example References Finite Element Discretization Consider a single element (triangular or quadrilateral) Let Ve = Potential at any point (x,y) Ve = 0, inside element Ve = 0, outside element For triangular element (used here) Ve = a + bx + cy For quadrilateral element Ve = a + bx + cy + dxy
- 7. Finite Element Analysis Sujith Jose Introduction Steps in Finite Element Analysis Finite Element Discretization Elementary Governing Equations Assembling of all elements Solving the resulting equations i.Iteration Method ii.Band Matrix Method Example References Finite Element Discretization Consider triangular element, Ve(x, y) = a + bx + cy Linear variation of potential is the same as assuming that electric field is uniform within the element.i.e Ee = − Ve = −(bax + cay )
- 8. Finite Element Analysis Sujith Jose Introduction Steps in Finite Element Analysis Finite Element Discretization Elementary Governing Equations Assembling of all elements Solving the resulting equations i.Iteration Method ii.Band Matrix Method Example References Elementary Governing Equations At any point (x,y), Ve(x, y) = a + bx + cy. We can find potential at any point if we can find values of a, b and c. Ve1(x1, y1) = a + bx1 + cy1 Ve2(x2, y2) = a + bx2 + cy2 Ve3(x3, y3) = a + bx3 + cy3 Ve1 Ve2 Ve3 = 1 x1 y1 1 x2 y2 1 x3 y3 a b c Coefficients a, b and c can be found by inverting matrix Figure: Typical triangular element
- 9. Finite Element Analysis Sujith Jose Introduction Steps in Finite Element Analysis Finite Element Discretization Elementary Governing Equations Assembling of all elements Solving the resulting equations i.Iteration Method ii.Band Matrix Method Example References Elementary Governing Equations a b c = 1 x1 y1 1 x2 y2 1 x3 y3 −1 Ve1 Ve2 Ve3 a b c = 1 DET (x2y3 − x3y2) (x3y1 − x1y3) (x1y2 − x2y1) (y2 − y3) (y3 − y1) (y1 − y2) (x3 − x2) (x1 − x3) (x2 − x1) Ve1 Ve2 Ve3 Let DET = 1 x1 y1 1 x2 y2 1 x3 y3 = 2A
- 10. Finite Element Analysis Sujith Jose Introduction Steps in Finite Element Analysis Finite Element Discretization Elementary Governing Equations Assembling of all elements Solving the resulting equations i.Iteration Method ii.Band Matrix Method Example References Elementary Governing Equations a b c = 1 2A (x2y3 − x3y2) (x3y1 − x1y3) (x1y2 − x2y1) (y2 − y3) (y3 − y1) (y1 − y2) (x3 − x2) (x1 − x3) (x2 − x1) Ve1 Ve2 Ve3 Ve(x, y) = a + bx + cy = 1 x y a b c Ve(x, y) = 1 x y 1 2A (x2y3 − x3y2) (x3y1 − x1y3) (x1y2 − x2y1) (y2 − y3) (y3 − y1) (y1 − y2) (x3 − x2) (x1 − x3) (x2 − x1) Ve1 Ve2 Ve3 Ve(x, y) = 1 2A α1 α2 α3 Ve1 Ve2 Ve3 = 3 i=1 αi (x, y)Vei
- 11. Finite Element Analysis Sujith Jose Introduction Steps in Finite Element Analysis Finite Element Discretization Elementary Governing Equations Assembling of all elements Solving the resulting equations i.Iteration Method ii.Band Matrix Method Example References Elementary Governing Equations Potential Ve(x, y) at any point (x,y) within the element (provided the potential at vertices) Ve(x, y) = 3 i=1 αi (x, y)Vei where α1 = 1 2A [(x2y3 − x3y2) + (y2 − y3)x + (x3 − x2)y] (1) α2 = 1 2A [(x3y1 − x1y3) + (y3 − y1)x + (x1 − x3)y] (2) α3 = 1 2A [(x1y2 − x2y1) + (y1 − y2)x + (x2 − x1)y] (3) αi are called linear interpolation functions or element shape functions
- 12. Finite Element Analysis Sujith Jose Introduction Steps in Finite Element Analysis Finite Element Discretization Elementary Governing Equations Assembling of all elements Solving the resulting equations i.Iteration Method ii.Band Matrix Method Example References Elementary Governing Equations - Energy term Energy density = 1 2 E2 Energy per unit length We = 1 2 |E|2 dS = 1 2 | Ve|2 dS (4) Ve = 3 i=1 αi (x, y)Vei ⇒ Ve = 3 i=1 Vei αi (5) Substituting We = 1 2 3 i=1 3 j=1 Vei | αi . αj dS|Vei (6)
- 13. Finite Element Analysis Sujith Jose Introduction Steps in Finite Element Analysis Finite Element Discretization Elementary Governing Equations Assembling of all elements Solving the resulting equations i.Iteration Method ii.Band Matrix Method Example References Elementary Governing Equations - Energy term Let coupling term between nodes i and j be C (e) ij = αi . αj dS We = 1 2 3 i=1 3 j=1 Vei | αi . αj dS|Vei = 1 2 3 i=1 3 j=1 Vei |C (e) ij |Vej (7) Writing in matrix form, energy per unit length is We = 1 2 [Ve]T [C(e) ][Ve] (8) where [Ve] = Ve1 Ve2 Ve3 and [C(e)] = C (e) 11 C (e) 12 C (e) 13 C (e) 21 C (e) 22 C (e) 23 C (e) 31 C (e) 32 C (e) 33 called element coefficient matrix or stiffness matrix
- 14. Finite Element Analysis Sujith Jose Introduction Steps in Finite Element Analysis Finite Element Discretization Elementary Governing Equations Assembling of all elements Solving the resulting equations i.Iteration Method ii.Band Matrix Method Example References Assembling of all elements The energy associated with all the N elements in the solution region W = N e=1 We = 1 2 [V ]T [C][V ] (9) where [V ] = V1 V2 . . Vn (10) n is the number of nodes [C] is called the over-all or global coefficient matrix which is the assemblage of individual element coefficient matrices.
- 15. Finite Element Analysis Sujith Jose Introduction Steps in Finite Element Analysis Finite Element Discretization Elementary Governing Equations Assembling of all elements Solving the resulting equations i.Iteration Method ii.Band Matrix Method Example References Global coefficient matrix - an example Consider a 3 element finite element mesh. 5 nodes give a 5x5 global coefficient matrix. [C] = C11 C12 C13 C14 C15 C21 C22 C23 C24 C25 C31 C32 C33 C34 C35 C41 C42 C43 C44 C45 C51 C52 C53 C54 C55 Figure: Assembly of three elements
- 16. Finite Element Analysis Sujith Jose Introduction Steps in Finite Element Analysis Finite Element Discretization Elementary Governing Equations Assembling of all elements Solving the resulting equations i.Iteration Method ii.Band Matrix Method Example References Global coefficient matrix - an example Cij is the coupling term between global nodes i and j. Cij = αi . αj dS Contribution to Cij comes from all elements containing nodes i and j. Write global coefficient elements in terms of contributing element coefficient elements Figure: Assembly of three elements
- 17. Finite Element Analysis Sujith Jose Introduction Steps in Finite Element Analysis Finite Element Discretization Elementary Governing Equations Assembling of all elements Solving the resulting equations i.Iteration Method ii.Band Matrix Method Example References Global coefficient matrix - an example Elements 1 and 2 have node 1 in common C11 = C (1) 11 + C (2) 11 Node 2 belongs to element 1 only C22 = C (1) 33 Node 4 belongs to elements 1, 2 and 3 C44 = C (1) 22 + C (2) 33 + C (3) 33 Nodes 1 and 4 belong simultaneously to elements 1 and 2 C14 = C (1) 12 + C (2) 13 No coupling between nodes 2 and 3 C23 = C32 = 0 Figure: Assembly of three elements
- 18. Finite Element Analysis Sujith Jose Introduction Steps in Finite Element Analysis Finite Element Discretization Elementary Governing Equations Assembling of all elements Solving the resulting equations i.Iteration Method ii.Band Matrix Method Example References Global coefficient matrix - an example The global coefficient matrix Symmetric (Cij = Cji ) Sparse Singular [C] = C (1) 11 + C (2) 11 C (1) 13 C (2) 12 C (1) 12 + C (2) 13 0 C (1) 31 C (1) 33 0 C (1) 32 0 C (2) 21 0 C (2) 22 + C (3) 11 C (2) 23 + C (3) 13 C (3) 13 C (1) 21 + C (2) 31 C (1) 23 C (2) 32 + C (3) 31 C (1) 22 + C (2) 33 + C (3) 33 C (3) 32 0 0 C (3) 21 C (3) 23 C (3) 22
- 19. Finite Element Analysis Sujith Jose Introduction Steps in Finite Element Analysis Finite Element Discretization Elementary Governing Equations Assembling of all elements Solving the resulting equations i.Iteration Method ii.Band Matrix Method Example References Global coefficient matrix - an example Energy associated with assemblage of 3 elements W = 1 2 [V ]T [C][V ] [C] = C (1) 11 + C (2) 11 C (1) 13 C (2) 12 C (1) 12 + C (2) 13 0 C (1) 31 C (1) 33 0 C (1) 32 0 C (2) 21 0 C (2) 22 + C (3) 11 C (2) 23 + C (3) 13 C (3) 13 C (1) 21 + C (2) 31 C (1) 23 C (2) 32 + C (3) 31 C (1) 22 + C (2) 33 + C (3) 33 C (3) 32 0 0 C (3) 21 C (3) 23 C (3) 22 , [V ] = V1 V2 V3 V4 V5
- 20. Finite Element Analysis Sujith Jose Introduction Steps in Finite Element Analysis Finite Element Discretization Elementary Governing Equations Assembling of all elements Solving the resulting equations i.Iteration Method ii.Band Matrix Method Example References Solving the resulting equations Laplace’s (or Poisson’s ) equation is satisfied when the total energy in the solution region is minimum Hence, ∂W ∂V1 = ∂W ∂V2 = ... = ∂W ∂Vn = 0
- 21. Finite Element Analysis Sujith Jose Introduction Steps in Finite Element Analysis Finite Element Discretization Elementary Governing Equations Assembling of all elements Solving the resulting equations i.Iteration Method ii.Band Matrix Method Example References Solving the resulting equations For example, ∂W ∂V1 = 0 ⇒ 0 = 2V1C11+V2C12+V3C13+V4C14+V5C15+V2C21+V3C31+V4C41+V5C5 Or 0 = V1C11 + V2C12 + V3C13 + V4C14 + V5C15 In general, ∂W ∂Vk = 0 leads to 0 = n i=1 Vi Cki where n is the number of nodes in the mesh. Writing for all nodes k = 1, 2, ..., n → set of simultaneous equations. From these equations, V1, V2, .., Vn can be found.
- 22. Finite Element Analysis Sujith Jose Introduction Steps in Finite Element Analysis Finite Element Discretization Elementary Governing Equations Assembling of all elements Solving the resulting equations i.Iteration Method ii.Band Matrix Method Example References Solving the resulting equations Iteration method Suppose node 0 is connected to m nodes. 0 = V0C00 +V1C01 +V2C02 +...+VmC0m or V0 = − 1 C00 m k=1 VkC0k V0 can be calculated if the potentials at nodes connected to 0 are known. Figure: Node 0 connected to m other nodes
- 23. Finite Element Analysis Sujith Jose Introduction Steps in Finite Element Analysis Finite Element Discretization Elementary Governing Equations Assembling of all elements Solving the resulting equations i.Iteration Method ii.Band Matrix Method Example References Solving the resulting equations Iteration method Free nodes - Nodes whose potential are unknown Fixed nodes - Nodes where the potential V is prescribed or known Iteration process: 1 Set free node potential initial value equal to 1 Zero 2 Or average potential of fixed nodes Vave = 1 2 (Vmin + Vmax ), where Vmin and Vmax are the minimum and maximum values of V at the fixed nodes. 2 Calculate value for free node using V0 = − 1 C00 m k=1 VkC0k 3 Use these as fixed node potential for next iteration 4 Repeat until change between subsequent iterations is negligible.
- 24. Finite Element Analysis Sujith Jose Introduction Steps in Finite Element Analysis Finite Element Discretization Elementary Governing Equations Assembling of all elements Solving the resulting equations i.Iteration Method ii.Band Matrix Method Example References Solving the resulting equations Band Matrix Method If all free nodes (f) are numbered first and fixed/prescribed nodes (p) last, W = 1 2 [Ve]T [C(e)][Ve] can be written as W = 1 2 Vf Vp Cff Cfp Cpf Cpp Vf Vp Differentiating wrt Vf , Cff Cfp Vf Vp = 0
- 25. Finite Element Analysis Sujith Jose Introduction Steps in Finite Element Analysis Finite Element Discretization Elementary Governing Equations Assembling of all elements Solving the resulting equations i.Iteration Method ii.Band Matrix Method Example References Solving the resulting equations Band Matrix Method Cff Cfp Vf Vp = 0 ⇒ [Cff ][Vf ] = −[Cfp][Vp] This equation can be written as [A][V ] = [B] or [V ] = [A]−1 [B] where [V ] = [Vf ], [A] = [Cff ], [B] = −[Cfp][Vp] Thus, we can solve for [V ] using matrix techniques.
- 26. Finite Element Analysis Sujith Jose Introduction Steps in Finite Element Analysis Finite Element Discretization Elementary Governing Equations Assembling of all elements Solving the resulting equations i.Iteration Method ii.Band Matrix Method Example References Review of steps 1 Discretize the solution region into elements: Ve = a + bx + cy 2 Derive governing equations for one element: Ve(x, y) = 3 i=1 αi (x, y)Vei 3 Assemble all elements: W = 1 2 [V ]T [C][V ] 4 Solving system of equations obtained: [Cff ][Vf ] = −[Cfp][Vp]
- 27. Finite Element Analysis Sujith Jose Introduction Steps in Finite Element Analysis Finite Element Discretization Elementary Governing Equations Assembling of all elements Solving the resulting equations i.Iteration Method ii.Band Matrix Method Example References Example - Potential on a 2D surface Voltage at nodes 1 and 3 are known. Can we find potential at any point within the mesh using FEM ? Using x1, x2, x3, x4, y1, y2, y3 and y4, element [C(1) ] = 1.236 −0.7786 −0.4571 −0.7786 0.6929 0.0857 −0.4571 0.0857 0.3714 [C(2) ] = 0.5571 −0.4571 0.1 −0.4571 0.8238 −0.3667 −0.1 0.3667 0.4667 Figure: Two element mesh
- 28. Finite Element Analysis Sujith Jose Introduction Steps in Finite Element Analysis Finite Element Discretization Elementary Governing Equations Assembling of all elements Solving the resulting equations i.Iteration Method ii.Band Matrix Method Example References Example - Potential on a 2D surface Using band matrix method, [Cff ][Vf ] = −[Cfp][Vp] C22 C24 C42 C44 V2 V4 = C21 C23 C41 C43 V1 V3 1 0 0 0 0 1.25 0 −0.0143 0 0 1 0 1 −0.0143 0 0.8381 V1 V2 V3 V4 = 0 3.708 10.0 4.438 Figure: Two element mesh
- 29. Finite Element Analysis Sujith Jose Introduction Steps in Finite Element Analysis Finite Element Discretization Elementary Governing Equations Assembling of all elements Solving the resulting equations i.Iteration Method ii.Band Matrix Method Example References Example - Potential on a 2D surface We get V1 = 0, V2 = 3.708, V3 = 10 and V4 = 4.438 Now voltage at any point inside each element can be found using linear interpolation functions Ve(x, y) = 3 i=1 αi (x, y)Vei Figure: Two element mesh
- 30. Finite Element Analysis Sujith Jose Introduction Steps in Finite Element Analysis Finite Element Discretization Elementary Governing Equations Assembling of all elements Solving the resulting equations i.Iteration Method ii.Band Matrix Method Example References References Matthew Sadiku (1989) A Simple Introduction to Finite Element Analysis of Electromagnetic Problems IEEE Transactions on Education 32(2), 85 - 93. Jianming Jin (2002) The Finite Element Method in Electromagnetics Second Edition
- 31. Finite Element Analysis Sujith Jose Introduction Steps in Finite Element Analysis Finite Element Discretization Elementary Governing Equations Assembling of all elements Solving the resulting equations i.Iteration Method ii.Band Matrix Method Example References Questions?
- 32. Finite Element Analysis Sujith Jose Introduction Steps in Finite Element Analysis Finite Element Discretization Elementary Governing Equations Assembling of all elements Solving the resulting equations i.Iteration Method ii.Band Matrix Method Example References Appendix - Boundary value problems A boundary value problem can be defined by a governing differential equation in a domain Ω: Lφ = f together with boundary conditions on the boundary that encloses the domain. Approximate solutions to boundary value problems can be found using Ritz or Galerkin’s method.
- 33. Finite Element Analysis Sujith Jose Introduction Steps in Finite Element Analysis Finite Element Discretization Elementary Governing Equations Assembling of all elements Solving the resulting equations i.Iteration Method ii.Band Matrix Method Example References Appendix - Ritz method Boundary value problem is formulated in terms of a variational expression called functional. Minimum of this functional corresponds to the governing differential equation under the given boundary conditions. Approximate solution is then obtained by minimizing the functional with respect to variables that define a certain approximation to the solution.
- 34. Finite Element Analysis Sujith Jose Introduction Steps in Finite Element Analysis Finite Element Discretization Elementary Governing Equations Assembling of all elements Solving the resulting equations i.Iteration Method ii.Band Matrix Method Example References Appendix - Galerkin’s method This method is one of the weighted residual methods i.e. seek the solution by weighting the residual of the differential equation. Assume that φ is an approximate solution to boundary value problem. Then, nonzero residual r = Lφ − f = 0 The best approximation for φ will be the one that reduces residual r to least value at all points of Ω. Ri = wi rdΩ = 0 where Ri denote weighted residual integrals and wi are chosen weighting functions.