Presentation on theme: "Finite Element Method CHAPTER 11: MODELLING TECHNIQUES"— Presentation transcript:
1 Finite Element Method CHAPTER 11: MODELLING TECHNIQUES for readers of all backgroundsG. R. Liu and S. S. QuekCHAPTER 11:MODELLING TECHNIQUES
2 CONTENTS INTRODUCTION CPU TIME ESTIMATION GEOMETRY MODELLING MESHING Mesh densityElement distortionMESH COMPATIBILITYDifferent order of elementsStraddling elements
3 CONTENTS USE OF SYMMETRY MODELLING OF OFFSETS MODELLING OF SUPPORTS Mirror symmetryAxial symmetryCyclic symmetryRepetitive symmetryMODELLING OF OFFSETSCreation of MPC equations for offsetsMODELLING OF SUPPORTSMODELLING OF JOINTS
4 CONTENTS OTHER APPLICATIONS OF MPC EQUATIONS Modelling of symmetric boundary conditionsEnforcement of mesh compatibilityModelling of constraints by rigid body attachmentIMPLEMENTATION OF MPC EQUATIONSLagrange multiplier methodPenalty method
5 INTRODUCTION Ensure reliability and accuracy of results. Improve efficiency and accuracy.
6 INTRODUCTION Considerations: Computational and manpower resources that limit the scale of the FEM model.Requirement on results that defines the purpose and hence the methods of the analysis.Mechanical characteristics of the geometry of the problem domain that determine the types of elements to use.Boundary conditions.Loading and initial conditions.
7 CPU TIME ESTIMATION ( ranges from 2 – 3) Bandwidth, b, affects - minimize bandwidthAim:To create a FEM model with minimum DOFs by using elements of as low dimension as possible, andTo use as coarse a mesh as possible, and use fine meshes only for important areas.
8 GEOMETRY MODELLINGReduction of a complex geometry to a manageable one.3D? 2D? 1D? Combination?(Using 2D or 1D makes meshing much easier)
9 GEOMETRY MODELLINGDetailed modelling of areas where critical results are expected.Use of CAD software to aid modelling.Can be imported into FE software for meshing.
10 MESHINGMesh densityTo 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))
11 Element distortionUse 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 elementSquare Quadrilateral elementsIsosceles triangle Triangle elementsCube Hexahedron elementsIsosceles tetrahedron Tetrahedron elements
12 Element distortionAspect ratio distortionRule of thumb:
15 Element distortion Volumetric distortion Area outside distorted element maps into an internal area – negative volume integration
16 Element distortionVolumetric distortion (Cont’d)
17 Element distortion Mid-node position distortion Shifting of nodes beyond limits can result in singular stress field (see crack tip elements)
18 MESH COMPATIBILITYRequirement of Hamilton’s principle – admissible displacementThe displacement field is continuous along all the edges between elements
19 Different order of elements Crack like behaviour – incorrect results
20 Different order of elements Solution:Use same type of elements throughoutUse transition elementsUse MPC equations
21 Straddling elementsAvoid straddling of elements in mesh
22 USE OF SYMMETRY Different types of symmetry: Use of symmetry reduces number of DOFs and hence computational time. Also reduces numerical error.Mirror symmetryAxial symmetryCyclic symmetryRepetitive symmetry
23 Mirror symmetrySymmetry about a particular plane
24 Mirror symmetry Consider a 2D symmetric solid: u1x = 0 u2x = 0 u3x = 0 Single point constraints (SPC)
32 Mirror symmetryDynamic problems (e.g. two half models to get full set of eigenmodes in eigenvalue analysis)
33 Axial symmetry Use of 1D or 2D axisymmetric elements Formulation similar to 1D and 2D elements except the use of polar coordinatesCylindrical shell using 1D axisymmetric elements3D structure using 2D axisymmetric elements
43 MODELLING OF JOINTSPerfect connection ensured here
44 MODELLING OF JOINTSMismatch between DOFs of beams and 2D solid – beam is free to rotate (rotation not transmitted to 2D solid)Perfect connection by artificially extending beam into 2D solid (Additional mass)
46 MODELLING OF JOINTSSimilar for plate connected to 3D solid
47 OTHER APPLICATIONS OF MPC EQUATIONS Modelling of symmetric boundary conditionsdn = 0ui cos + vi sin=0 or ui+vi tan =0for i=1, 2, 3
48 Enforcement of mesh compatibility Use lower order shape function to interpolatedx = 0.5(1-) d (1+) d3dy = 0.5(1-) d (1+) d6Substitute value of at node 30.5 d1 - d d3 =00.5 d4 - d d6 =0
49 Enforcement of mesh compatibility Use shape function of longer element to interpolatedx = -0.5 (1-) d1 + (1+)(1-) d (1+) d5Substituting the values of for the two additional nodesd2 = 0.251.5 d 0.5 d 0.5 d5d4 = -0.250.5 d 1.5 d 1.5 d5
50 Enforcement of mesh compatibility In x direction,0.375 d1 - d d d5 = 0d d3 - d d5 = 0In y direction,0.375 d6- d d d10 = 0d d8 - d d10 = 0
51 Modelling of constraints by rigid body attachment d1 = q1d2 = q1+q2 l1d3=q1+q2 l2d4=q1+q2 l3Eliminate q1 and q2(l2 /l1-1) d1 - ( l2 /l1) d2 + d3 = 0(l3 /l1-1) d1 - ( l3 /l1) d2 + d4 = 0(DOF in x direction not considered)
52 IMPLEMENTATION OF MPC EQUATIONS (Global system equation)(Matrix form of MPC equations)Constant matrices
53 Lagrange multiplier method (Lagrange multipliers)Multiplied to MPC equationsAdded to functionalThe stationary condition requires the derivatives of p with respect to the Di and i to vanish.Matrix equation is solved
54 Lagrange multiplier method Constraint equations are satisfied exactlyTotal number of unknowns is increasedExpanded stiffness matrix is non-positive definite due to the presence of zero diagonal termsEfficiency of solving the system equations is lower
55 Penalty method (Constrain equations) =1 m is a diagonal matrix of ‘penalty numbers’Stationary condition of the modified functional requires the derivatives of p with respect to the Di to vanishPenalty matrix
56 Penalty method [Zienkiewicz et al., 2000] : = constant (1/h)p+1 P is the order of element usedCharacteristic size of elementmax (diagonal elements in the stiffness matrix)orYoung’s modulus
57 Penalty method The total number of unknowns is not changed. System equations generally behave well.The constraint equations can only be satisfied approximately.Right choice of may be ambiguous.