Finite Element Modeling Techniques (2) دانشگاه صنعتي اصفهان- دانشكده مكانيك

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Reginald Richards
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1 Finite Element Modeling Techniques (2) 1

2 Where Finer Meshes Should be Used GEOMETRY MODELLING 2

3 GEOMETRY MODELLING Reduction of a complex geometry to a manageable one. 3D? 2D? 1D? Combination? Bulky solids 3-D solid element Neutral surface z y x Neutral surface Shell x z y h 2-D shell element mesh (Using 2D or 1D makes meshing much easier) z f y1 f y2 x Centroid Beam member 1-D beam element mesh 3

Mesh density To minimize the number of DOFs, have fine mesh at important areas.

4 GEOMETRY MODELLING Detailed modelling of areas where critical results are expected. Use of CAD software to aid modelling. Can be imported into FE software for meshing. Mesh density To minimize the number of DOFs, have fine mesh at important areas. In FE packages, mesh density can be controlled by mesh seeds. (Image courtesy of Institute of High Performance Computing and Sunstar Logistics(s) Pte Ltd (s)) 4

5 Element distortion Use of distorted elements in irregular and complex geometry is common but there are some limits to the distortion. The distortions are measured against the basic shape of the element Square Quadrilateral elements Isosceles triangle Triangle elements Cube Hexahedron elements Isosceles tetrahedron Tetrahedron elements 5

6 Element distortion Aspect ratio distortion a Rule of thumb: b a b 3 Stress analysis 10 Displacement analysis 6

7 Element distortion Angular distortion skew b Taper a b<5a Curvature distortion 7

8 Element distortion Volumetric distortion y Area outside distorted element maps into an internal area negative volume integration Volumetric distortion (Cont d) 1 x

9 Element distortion Avoid 2D/3D Elements of Bad Aspect Ratio 9

Elements Must Not Cross Interfaces A physical interface, resulting from example from a change in material, should also be an interelement boundary.

10 Elements Must Not Cross Interfaces A physical interface, resulting from example from a change in material, should also be an interelement boundary. Element Geometry Preferences Other things being equal, prefer in 2D: Quadrilaterals over Triangles in 3D: Bricks over Wedges, Wedges over Tetrahedral 10

11 Direct Lumping of Distributed Loads In practical structural problems, distributed loads are more common than concentrated (point) loads. In fact, one of the objectives of a good design is to avoid or alleviate stress concentrations produced by concentrated forces. Whatever their nature or source, distributed loads must be converted to consistent nodal forces for FEM analysis. These forces eventually end up in the right-hand side of the master stiffness equations. The meaning of consistent can be made precise through variational arguments, by requiring that the distributed loads and the nodal forces produce the same external work. However, a simpler approach called direct load lumping, or simply load lumping, is often used by structural engineers in lieu of the more mathematically impeccable but complicated variational approach. Two variants of this technique are described below for distributed surface loads. 11

12 Node by Node (NbN) Distributed Load Lumping 12

13 Element by Element (EbE) Distributed Load Lumping 13

14 Boundary Conditions (BCs) The most difficult topic for FEM program users Essential Two types Natural 1. If a BC involves one or more DOF in a direct way, it is essential and goes to the Left Hand Side (LHS) of Ku = f 2. Otherwise it is natural and goes to the Right Hand Side (RHS) of Ku =f 14

15 Boundary Conditions in Structural Problems In mechanical problems, essential boundary conditions are those that involve displacements (but not strain-type displacement derivatives). The support conditions for the truss problem furnish a particularly simple example. But there are more general boundary conditions that occur in practice. A structural engineer must be familiar with displacement B.C. of the following types. Ground or support constraints. Directly restraint the structure against rigid body motions. Symmetry conditions. To impose symmetry or antisymmetry restraints at certain points, lines or planes of structural symmetry. Ignorable freedoms. To suppress displacements that are irrelevant to the problem. (In classical dynamics these are called ignorable coordinates.) Even experienced users of finite element programs are sometimes baffled by this kind. An example are rotational degrees of freedom normal to shell surfaces. Connection constraints. To provide connectivity to adjoining structures or substructures, or to specify relations between degrees of freedom. Many conditions of this type can be subsumed under the label multipoint constraints or multifreedom constraints, which can be notoriously difficult to handle from a numerical standpoint. 15

16 Boundary Conditions in Structural Problems In structural problems, the distinguishes between essential and natural BC is: if it directly involves the nodal freedoms, such as displacements or rotations, it is essential. Otherwise it is natural. Conditions involving applied loads are natural. Essential BCs take precedence over natural BCs. The simplest essential boundary conditions are support and symmetry conditions. These appear in many practical problems. More exotic types, such as multifreedom constraints, require more advanced mathematical tools. 16

17 Boundary Conditions in Structural Problems Minimum Support Conditions to Suppress Rigid Body Motions in 2D 2D 3D 17

18 Boundary Conditions in Structural Problems Example: Modelling of Supports Beam with built-in end a) Full constraint only in the horizontal direction b) Support provides full constraint only on the lower surface c) Fully clamped support 18

19 Boundary Conditions in Structural Problems Example: Modelling of Supports (Prop support of beam) 19

20 Symmetry A structure possesses symmetry if its components are arranged in a periodic or reflective manner. Types of Symmetry: Mirror (Reflective, bilateral) symmetry cyclical (Rotational) symmetry Axisymmetry Translational (Repetitive) symmetry... Cautions: In vibration and buckling analyses, symmetry concepts, in general, should not be used in FE solutions (works fine in modeling), since symmetric structures often have antisymmetric vibration or buckling modes. 20

21 Symmetry Mirror symmetry Consider a 2D symmetric solid: u 1x = 0 y 1 1 u 2x = 0 u 3x = Single point constraints (SPC) 3 x 3 21

22 Symmetry Mirror symmetry Symmetric loading y P P x a b a b Deflection = Free Rotation = 0 P 22

23 Symmetry Mirror symmetry Anti-symmetric loading y P P x Deflection = 0 Rotation = Free a b a b P 23

24 Symmetry Mirror symmetry Symmetric No translational displacement normal to symmetry plane No rotational components with respect to the axis parallel to symmetry plane Plane of symmetry xy yz zx u v w x y z Free Free Fix Fix Fix Free Fix Free Free Free Fix Fix Free Fix Free Fix Free Fix 24

25 Mirror symmetry Anti-symmetric Symmetry No translational displacement parallel to symmetry plane No rotational components with respect to the axis normal to symmetry plane Plane of symmetry u v w x y z xy Fix Fix Free Free Free Fix yz Free Fix Fix Fix Free Free zx Fix Free Fix Free Fix Free 25

26 Symmetry Mirror symmetry Any load can be decomposed to a symmetric and an anti-symmetric load y P/2 P/2 y Symmetric loading x Asymmetric loading P x a b a b = + y a b a b P/2 P/2 Anti-Symmetric loading x a b a b 26

27 Symmetry Mirror symmetry Y P/2 P/2 P Sym. Full frame structure X = P/2 + P/2 Anti-sym. 27

28 Symmetry Mirror symmetry Y P 2 Y P 2 Properties are halved for this member Sym. X Anti- Sym. X All nodes on this line fixed against the horizontal displacement and rotation. All nodes on this line fixed against vertical displacement. 28

29 Mirror symmetry Symmetry Dynamic problems (e.g. two half models to get full set of eigenmodes in eigenvalue analysis) motion symmetric about this node Rotation dof = 0 at this node motion antisymmetric about this node translational dof v = 0 at this node 29

30 Axial symmetry Symmetry Use of 1D or 2D axisymmetric elements Formulation similar to 1D and 2D elements except the use of polar coordinates w 1 w 2 z y x w = W sin Cylindrical shell using 1D axisymmetric elements 30 3D structure using 2D axisymmetric elements

31 Symmetry Cyclic symmetry u An = u Bn F u An u At u At = u Bt F Side A u Bn F u Bt Multipoint constraints (MPC) F Side B F Representative cell 31

32 Symmetry Repetitive symmetry u Ax = u Bx P P A u Ax B u Bx P Representative cell P 32

33 Symmetry Example of Symmetry Py P Y P Px y x Px P X X Py Plane structure having Reflective symmetry Py/2 Beam under symmetric load P Y P X P Px/2 33 Beam under unsymmetric load X

34 Example of Symmetry and Antisymmetry Symmetry 34

35 Example of Application of Symmetry BCs Symmetry 35

36 Symmetry Example of Application of Antisymmetry BCs 36

37 Symmetry Example of Symmetry 37

Global-Local Analysis (an instance of Multiscale Analysis)

fashion like substructure, A related, but not identical,

1- The whole system is first analyzed as a global entity,

38 Global-Local Analysis (an instance of Multiscale Analysis) Complex engineering systems are often modeled in a multilevel fashion like substructure, A related, but not identical, technique is multiscale analysis. 1- The whole system is first analyzed as a global entity, neglecting the detail 2- Local details are then analyzed using the results of the global analysis as boundary conditions. 38

39 Nature of Finite Element Solutions FE Model A mathematical model of the real structure, based on many approximations. Real Structure Infinite number of nodes (physical points or particles), thus infinite number of DOF s. FE Model finite number of nodes, thus finite number of DOF s. Displacement field is controlled (or constrained) by the values at a limited number of nodes. 39

40 Nature of Finite Element Solutions Stiffening Effect: FE Model is stiffer than the real structure. In general, displacement results are smaller in magnitudes than the exact values. Hence, FEM solution of displacement provides a lower bound of the exact solution. 40

41 Numerical Error Error Mistakes in FEM (modeling or solution). Types of Error: Modeling Error (beam, plate theories) Discretization Error (finite, piecewise ) Numerical Error ( in solving FE equations) 41

42 Modeling Error Modeling error Error that arise from the description of the boundary value problem (BVP): Geometric description, material description, loading, boundary conditions, type of analysis. What physical details are important in the BVP description? Should a mechanically fastened joint be modeled as a pin joint, welded joint, or a flexible joint. 42

43 Modeling Error How should the load be modeled? Should the properties of the adhesive be included or ignored in a bonded joint? 43

44 Modeling Error Should the material be modeled as isotropic or orthotropic? How should the support be modeled? i.e., what are the appropriate boundary conditions. 44

45 Modeling Error What type of analysis should be conducted? Should you conduct a linear or non-linear analysis? 1.Material non-linearity: Stress and strain are non-linearly related. 2.Geometric non-linearity: Strain and displacement non-linearly related. (large deformation or strain) 3.Contact problem: The contact length changes with load. (i) No friction. (ii) With friction need the slip (F f =mm) and no slip boundary(f f <mm). Should buckling analysis be conducted? For time dependent problems should you conduct a dynamic or quasistatic analysis? Should the material be modeled as elastic or viscoelastic? 45

46 Errors that arises from creation of the mesh. Elements in FEM are based on analytical models. All assumptions that are made in the analytical models are applicable to FEM elements. What type of elements should be used? Should 1-d element be used? Discretization Error 46

47 Discretization Error Should beam element, which is based on symmetric bending, be used? What type of 2-d (plane stress, plane strain) or 3-d element should you use.? What mesh density should you use? Too fine a mesh results in large computer time that may prevent optimization or parametric studies or non-linear analysis. Too coarse a mesh may result in high inaccuracies. Start with a coarse mesh, study the results and then refine the mesh as needed. 47

48 Discretization Error How accurately should the geometry be modeled? Errors from modeling of geometric are generally small. For the same computational effort higher returns in accuracy are obtained in better modeling of displacement-isoparametric elements are adequate. 48

49 Numerical Error Errors that arise from finite digit arithmetic and use of numerical methods. Integration error Few Gauss points leads to numerical instabilities. Large number of Gauss points are computationally expensive and may result in overly stiff elements leading to higher errors. Round off error The finite digit arithmetic causes these errors, but the growth of round off errors are dictated by several factors. Need to avoid: adding or subtracting very large and very small numbers; dividing by small numbers. (i) The manner in which algorithms are written in the computer codes. Non-dimensionalizing the problem will always help. (ii) Large differences in physical dimensions. 49

50 Numerical Error (iii) Large differences in stiffness caused my large differences in material properties (or dimensions). (iv) Elements with poor aspect ratio: ratio of largest to smallest dimension in an element. 50

51 Numerical Error Example (numerical error): 51

52 Numerical Error FE Equations: The system will be singular if k 2 is small compared with k 1. 52

53 Numerical Error u 6 u u u u 6 u u u

54 Numerical Error Large difference in stiffness of different parts in FE model may cause ill-conditioning in FE equations. Hence giving results with large errors. Ill-conditioned system of equations can lead to large changes in solution with small changes in input (right hand side vector). 54

55 Convergence Requirements for Finite Element Discretization Convergence: discrete (FEM) solution approaches the analytical (math model) solution in some sense Convergence = Consistency + Stability Further Breakdown of Convergence Requirements Convergence Requirements Consistency Completeness individual elements Compatibility element patches Stability Rank Sufficiency individual elements Positive Jacobian individual elements 55

56 Convergence Requirements The Variational Index m 56

57 Element Patches Nonconforming elements and the patch test Conforming = compatible Nonconforming = incompatible Ideal: Conforming elements Observation: Certain nonconforming elements also give good results, at the expense of nonmonotonic convergence Nonconforming elements: satisfy completeness do not satisfy compatibility result in at least nonmonotonic convergence if the element assemblage as a whole is complete, i.e., they satisfy the PATCH TEST 57

58 Element Patches PATCH TEST: 1. A patch of elements is subjected to the minimum displacement boundary conditions to eliminate all rigid body motions 2. Apply to boundary nodal points forces or displacements which should result in a state of constant stress within the assemblage 3. Nodes not on the boundary are neither loaded nor restrained. 4. Compute the displacements of nodes which do not have a prescribed value 5. Compute the stresses and strains The patch test is passed if the computed stresses and strains match the expected values to the limit of computer precision. 58

59 Element Patches 59

60 Element Patches Patch Test - Procedure Build a simple FE model Consists of a Patch of Elements At least one internal node Load by nodal equivalent forces consistent with state of constant stress Internal Node is unloaded and unsupported 60

61 Element Patches Patch Test - Procedure 1 F s xht 2 Compute results of FE patch If (computed s x ) = (assumed s x ) test passed 61

62 Element Patches NOTES: 1. This is a great way to debug a computer code 2. Conforming elements ALWAYS pass the patch test 3. Nodes not on the boundary are neither loaded nor restrained. 4. Since a patch may also consist of a single element, this test may be used to check the completeness of a single element 5. The number of constant stress states in a patch test depends on the actual number of constant stress states in the mathematical model (3 for plane stress analysis. 6 for a full 3D analysis) 62

63 Completeness & Compatibility in Terms of m Convergence Requirements Completeness The element shape functions must represent exactly all polynomial terms of order m in the Cartesian coordinates. A set of shape functions that satisfies this condition is call m-complete Compatibility The patch trial functions must be ( m 1) C continuous between (m) elements, and C piecewise differentiable inside each Element. 63

64 Plane Stress: m = 1 in Two Dimensions Convergence Requirements Completeness The element shape functions must represent exactly all polynomial terms of order <=1 in the Cartesian coordinates. That means any linear polynomial in x, y with a constant as special case Compatibility The patch trial functions must be 0 C continuous between elements, and 1 piecewise differentiable inside each element C 64

65 Stability Rank Sufficiency The discrete model must possess the same solution uniqueness attributes of the mathematical model For displacement finite elements: the rigid body modes (RBMs) must be preserved no zero-energy modes other than RBMs can be tested by the rank of the stiffness matrix Positive Jacobian Determinant The determinant of the Jacobian matrix that relates cartesian and natural coordinates must be everywhere positive within the element Rank Sufficiency The element stiffness matrix must not possess any zero-energy kinematic modes other than rigid body modes This can be checked by verifying that the element stiffness matrix has the correct rank: A stiffness matrix that has proper rank is called rank sufficient 65

Rank Sufficiency for Numerically Integrated Finite Elements Notation for Rank Analysis of Element Stiffness n F n R n G n E r C r d number of element DOF number of independent rigid body modes

66 Rank Sufficiency for Numerically Integrated Finite Elements Notation for Rank Analysis of Element Stiffness n F n R n G n E r C r d number of element DOF number of independent rigid body modes number of Gauss points in integration rule for K order of E (stress-strain) matrix correct (proper) rank n F n R actual rank of stiffness matrix rank deficiency r C r General case Plane Stress, n nodes 66

67 Rank Sufficiency for Numerically Integrated Finite Elements The element stiffness matrix must not possess any zero-energy kinematic mode other than rigid body modes. This can be mathematically expressed as follows. Let n F be the number of element degrees of freedom, and n R be the number of independent rigid body modes. Let r denote the rank of K(e). The element is called rank sufficient if r = n F n R and rank deficient if r < n F n R. In the latter case, d = (n F n R ) r is called the rank deficiency. 67

68 Rank Sufficiency for Numerically Integrated Finite Elements If an isoparametric element is numerically integrated, let n G be the number of Gauss points, while n E denotes the order of the stress-strain matrix E. Two additional assumptions are made: (i) The element shape functions satisfy completeness in the sense that the rigid body modes are exactly captured by them. (ii) Matrix E is of full rank. Then each Gauss point adds n E to the rank of K(e), up to a maximum of n F n R. Hence the rank of K(e) will be r =min( n F n R, n E n G ) 68

69 Rank Sufficiency for Numerically Integrated Finite Elements To attain rank sufficiency, n E n G must equal or exceed n F n R : n E n G >=n F n R from which the appropriate Gauss integration rule can be selected. In the plane stress problem, n E = 3 because E is a 3 3 matrix of elastic moduli; Also n R = 3. Consequently r = min(n F 3, 3n G ) and 3n G n F 3. 69

70 Rank Sufficiency for Numerically Integrated Finite Elements EXAMPLE Consider a plane stress 6-node quadratic triangle. Then n F = 2 6 = 12. To attain the proper rank of 12 n R = 12 3 = 9, n G 3. A 3-point Gauss rule makes the element rank sufficient. EXAMPLE Consider a plane stress 9-node biquadratic quadrilateral. Then n F = 2 9 = 18. To attain the proper rank of 18 n R = 18 3 = 15, n G 5. The 2 2 product Gauss rule is insufficient because n G = 4. Hence a 3 3 rule, which yields n G = 9, is required to attain rank sufficiency. 70

71 Rank Sufficiency for Numerically Integrated Finite Elements 71

72 Positive Jacobian Requirement Displacing a Corner Node of 4-Node Quad 72

73 Positive Jacobian Requirement Displacing a Midside Node of 9-Node Quad 73

74 Positive Jacobian Requirement 74

75 Convergence of FE Solution As the mesh in an FE model is refined repeatedly, the FE solution will converge to the exact solution of the mathematical model of the problem (the model based on bar, beam, plane stress/strain, plate, shell, or 3-D elasticity theories or assumptions). Types of Refinement: h-refinement: reduce the size of the element ( h refers to the typical size of the elements); p-refinement: Increase the order of the polynomials on an element (linear to quadratic, etc.; h refers to the highest order in a polynomial); r-refinement: re-arrange the nodes in the mesh; hp-refinement: Combination of the h- and p-refinements (better results!). 75

76 Adaptivity (h-, p-, and hp-methods) Future of FE applications Automatic refinement of FE meshes until converged results are obtained User s responsibility reduced: only need to generate a good initial mesh Error Indicators: Define, s : element by element stress field (discontinuous), s * : averaged or smooth stress (continuous), s E = s -s * : the error stress field. 76

77 Adaptivity (h-, p-, and hp-methods) Compute strain energy, where M is the total number of elements, V i is the volume of the element i. One error indicator --- the relative energy error: 77

78 Adaptivity (h-, p-, and hp-methods) The indicator is computed after each FE solution. Refinement of the FE model continues until, say <= => converged FE solution. 78

FEM Convergence Requirements

19 FEM Convergence Requirements IFEM Ch 19 Slide 1 Convergence Requirements for Finite Element Discretization Convergence: discrete (FEM) solution approaches the analytical (math model) solution in some