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1、1,Finite Element Method,MODELLING TECHNIQUES,CHAPTER 8:,2,CONTENTS,INTRODUCTIONCPU TIME ESTIMATIONGEOMETRY MODELLINGMESHINGMesh densityElement distortionMESH COMPATIBILITYDifferent order of elementsStraddling elements,3,CONTENTS,USE OF SYMMETRYMirror symmetryAxial symmetryCyclic symmetryRepetitive s
2、ymmetryMODELLING OF OFFSETSCreation of MPC equations for offsetsMODELLING OF SUPPORTSMODELLING OF JOINTS,4,CONTENTS,OTHER APPLICATIONS OF MPC EQUATIONSModelling of symmetric boundary conditionsEnforcement of mesh compatibilityModelling of constraints by rigid body attachmentIMPLEMENTATION OF MPC EQU
3、ATIONSLagrange multiplier methodPenalty method,5,INTRODUCTION,Ensure reliability and accuracy of results.Improve efficiency and accuracy.,6,INTRODUCTION,Considerations:Computational and man power resource that limit the scale of the FEM model.Requirement on results that defines the purpose and hence
4、 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 bandwidth,Aim:,To create a FEM model with
5、 minimum DOFs by using elements of as low dimension as possible,and To use as coarse a mesh as possible,and use fine meshes only for important areas.,8,GEOMETRY MODELLING,Reduction of complex geometry to a manageable one.3D?2D?1D?Combination?,(Using 2D or 1D makes meshing much easier),9,GEOMETRY MOD
6、ELLING,Detailed modelling of areas where critical results are expected.Use of CAD software to aid modelling.Can be imported to FE software for meshing.,10,MESHING,To minimize the number of DOFs,have fine mesh at important areas.,In FE packages,mesh density can be controlled by mesh seeds.,Mesh densi
7、ty,(Image courtesy of Institute of High Performance Computing and Sunstar Logistics(s)Pte Ltd(s),11,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
8、Square Quadrilateral elements Isosceles triangle Triangle elements Cube Hexahedron elements Isosceles tetrahedron Tetrahedron elements,12,Element distortion,Aspect ratio distortion,Rule of thumb:,13,Element distortion,Angular distortion,14,Element distortion,Curvature distortion,15,Element distortio
9、n,Volumetric distortion,Area outside distorted element maps into an internal area negative volume integration,16,Element distortion,Volumetric distortion(Contd),17,Element distortion,Mid-node position distortion,Shifting of nodes beyond limits can result in singular stress field(see crack tip elemen
10、ts),18,MESH COMPATIBILITY,Requirement of Hamiltons 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 throug
11、houtUse transition elementsUse MPC equations,21,Straddling elements,Avoid straddling of elements in mesh,22,USE OF SYMMETRY,Different types of symmetry:,Mirror symmetry,Axial symmetry,Cyclic symmetry,Repetitive symmetry,Use of symmetry reduces number of DOFs and hence computational time.Also reduces
12、 numerical error.,23,Mirror symmetry,Symmetry about a particular plane,24,Mirror symmetry,Consider a 2D symmetric solid:,u1x=0,u2x=0,u3x=0,Single point constraints(SPC),25,Mirror symmetry,Deflection=FreeRotation=0,Symmetric loading,26,Mirror symmetry,Anti-symmetric loading,Deflection=0Rotation=Free,
13、27,Mirror symmetry,SymmetricNo translational displacement normal to symmetry planeNo rotational components w.r.t.axis parallel to symmetry plane,28,Mirror symmetry,Anti-symmetricNo translational displacement parallel to symmetry planeNo rotational components w.r.t.axis normal to symmetry plane,29,Mi
14、rror symmetry,Any load can be decomposed to a symmetric and an anti-symmetric load,30,Mirror symmetry,31,Mirror symmetry,32,Mirror symmetry,Dynamic problems(E.g.2 half models to get full set of eigen modes in eigenvalue analysis),33,Axial symmetry,Use of 1D or 2D axisymmetric elementsFormulation sim
15、ilar to 1D and 2D elements except the use of polar coordinates,Cylindrical shell using 1D axisymmetric elements,3D structure using 2D axisymmetric elements,34,Cyclic symmetry,uAn=uBn,uAt=uBt,Multipoint constraints(MPC),35,Repetitive symmetry,uAx=uBx,36,MODELLING OF OFFSETS,offset can be safely ignor
16、ed,offset need to be modelled,ordinary beam,plate,and shell elements should not be used.Use 2-D or 3-D solid elements.,Guidelines:,37,MODELLING OF OFFSETS,Three methods:Very stiff elementRigid elementMPC equations,38,Creation of MPC equations for offsets,Eliminate q1,q2,q3,39,Creation of MPC equatio
17、ns for offsets,40,Creation of MPC equations for offfsets,d6=d1+d5 or d1+d5-d6=0d7=d2-d4 or d2-d4-d7=0d8=d3 or d3-d8=0 d9=d5 or d5-d9=0,41,MODELLING OF SUPPORTS,42,MODELLING OF SUPPORTS,(Prop support of beam),43,MODELLING OF JOINTS,Perfect connection ensured here,44,MODELLING OF JOINTS,Mismatch betwe
18、en 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),45,MODELLING OF JOINTS,Using MPC equations,46,MODELLING OF JOINTS,Similar for plate connected to 3D solid,47,OTHER APPLICATIONS
19、OF MPC EQUATIONS,Modelling of symmetric boundary conditions,dn=0,ui cos+vi sin=0 or ui+vi tan=0 for i=1,2,3,48,Enforcement of mesh compatibility,dx=0.5(1-)d1+0.5(1+)d3,dy=0.5(1-)d4+0.5(1+)d6,Substitute value of at node 3,0.5 d1-d2+0.5 d3=0,0.5 d4-d5+0.5 d6=0,Use lower order shape function to interpo
20、late,49,Enforcement of mesh compatibility,Use shape function of longer element to interpolate,dx=-0.5(1-)d1+(1+)(1-)d3+0.5(1+)d5,Substituting the values of for the two additional nodes,d2=0.251.5 d1+1.50.5 d3-0.250.5 d5,d4=-0.250.5 d1+0.51.5 d3+0.251.5 d5,50,Enforcement of mesh compatibility,In x di
21、rection,0.375 d1-d2+0.75 d3-0.125 d5=0,-0.125 d1+0.75 d3-d4+0.375 d5=0,In y direction,0.375 d6-d7+0.75 d8-0.125 d10=0,-0.125 d6+0.75 d8-d9+0.375 d10=0,51,Modelling of constraints by rigid body attachment,d1=q1 d2=q1+q2 l1 d3=q1+q2 l2 d4=q1+q2 l3,(l2/l1-1)d1-(l2/l1)d2+d3=0(l3/l1-1)d1-(l3/l1)d2+d4=0,E
22、liminate q1 and q2,(DOF in x direction not considered),52,IMPLEMENTATION OF MPC EQUATIONS,(Matrix form of MPC equations),(Global system equation),Constant matrices,53,Lagrange multiplier method,(Lagrange multipliers),Multiplied to MPC equations,Added to functional,The stationary condition requires t
23、he derivatives of p with respect to the Di and i to vanish.,Matrix equation is solved,54,Lagrange multiplier method,Constraint equations are satisfied exactly Total number of unknowns is increased Expanded stiffness matrix is non-positive definite due to the presence of zero diagonal termsEfficiency
24、 of solving the system equations is lower,55,Penalty method,(Constrain equations),=1 2.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 vanish,Penalty matrix,56,Penalty method,Zienkiewicz,et.al.2000:,=c
25、onstant(1/h)p+1,Characteristic size of element,P is the order of element used,max(diagonal elements in the stiffness matrix),or,Youngs 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.,
