Presentation on theme: "Finite Element Method CHAPTER 7: FEM FOR 2D SOLIDS"— Presentation transcript:
1 Finite Element Method CHAPTER 7: FEM FOR 2D SOLIDS for readers of all backgroundsG. R. Liu and S. S. QuekCHAPTER 7:FEM FOR 2D SOLIDS
2 CONTENTS INTRODUCTION LINEAR TRIANGULAR ELEMENTS Field variable interpolationShape functions constructionUsing area coordinatesStrain matrixElement matricesLINEAR RECTANGULAR ELEMENTSGauss integrationEvaluation of me
3 CONTENTS LINEAR QUADRILATERAL ELEMENTS HIGHER ORDER ELEMENTS Coordinate mappingStrain matrixElement matricesRemarksHIGHER ORDER ELEMENTSCOMMENTS (GAUSS INTEGRATION)
4 INTRODUCTION2D solid elements are applicable for the analysis of plane strain and plane stress problems.A 2D solid element can have a triangular, rectangular or quadrilateral shape with straight or curved edges.A 2D solid element can deform only in the plane of the 2D solid.At any point, there are two components in the x and y directions for the displacement as well as forces.
5 INTRODUCTIONFor plane strain problems, the thickness of the element is unit, but for plane stress problems, the actual thickness must be used.In this course, it is assumed that the element has a uniform thickness h.Formulating 2D elements with a given variation of thickness is also straightforward, as the procedure is the same as that for a uniform element.
30 Evaluation of meE.g.Note: In practice, Gauss integration is often used
31 LINEAR QUADRILATERAL ELEMENTS Rectangular elements have limited applicationQuadrilateral elements with unparallel edges are more usefulIrregular shape requires coordinate mapping before using Gauss integration
32 Coordinate mapping Physical coordinates Natural coordinates (Interpolation of displacements)(Interpolation of coordinates)
36 Strain matrix Therefore, (Relationship between differentials of shape functions w.r.t. physical coordinates and differentials w.r.t. natural coordinates)Therefore,Replace differentials of Ni w.r.t. x and y with differentials of Ni w.r.t. and
37 Element matricesMurnaghan (1951) :dA=det |J | dxdh
38 RemarksShape functions used for interpolating the coordinates are the same as the shape functions used for interpolation of the displacement field. Therefore, the element is called an isoparametric element.Note that the shape functions for coordinate interpolation and displacement interpolation do not have to be the same.Using the different shape functions for coordinate interpolation and displacement interpolation, respectively, will lead to the development of so-called subparametric or superparametric elements.
39 HIGHER ORDER ELEMENTS Higher order triangular elements nd = (p+1)(p+2)/2Node i,Argyris, 1968 :
40 HIGHER ORDER ELEMENTS Higher order triangular elements (Cont’d) Cubic elementQuadratic element
41 HIGHER ORDER ELEMENTS Higher order rectangular elements Lagrange type: [Zienkiewicz et al., 2000]
42 HIGHER ORDER ELEMENTS Higher order rectangular elements (Cont’d) (nine node quadratic element)
43 HIGHER ORDER ELEMENTS Higher order rectangular elements (Cont’d) Serendipity type:(eight node quadratic element)
44 HIGHER ORDER ELEMENTS Higher order rectangular elements (Cont’d) (twelve node cubic element)
46 COMMENTS (GAUSS INTEGRATION) When the Gauss integration scheme is used, one has to decide how many Gauss points should be used.Theoretically, for a one-dimensional integral, using m points can give the exact solution for the integral of a polynomial integrand of up to an order of (2m-1).As a general rule of thumb, more points should be used for a higher order of elements.
47 COMMENTS (GAUSS INTEGRATION) Using a smaller number of Gauss points tends to counteract the over-stiff behaviour associated with the displacement-based method.Displacement in an element is assumed using shape functions. This implies that the deformation of the element is somehow prescribed in a fashion of the shape function. This prescription gives a constraint to the element. The so- constrained element behaves stiffer than it should. It is often observed that higher order elements are usually softer than lower order ones. This is because using higher order elements gives fewer constraint to the elements.
48 COMMENTS ON GAUSS INTEGRATION Two Gauss points for linear elements, and two or three points for quadratic elements in each direction should be sufficient for most cases.Most of the explicit FEM codes based on explicit formulation tend to use one-point integration to achieve the best performance in saving CPU time.
50 Elastic Properties of Polysilicon CASE STUDY10N/mElastic Properties of PolysiliconYoung’s Modulus, E169GPaPoisson’s ratio, 0.262Density, 2300kgm-310N/m10N/m
51 CASE STUDYAnalysis no. 1: Von Mises stress distribution using 24 bilinear quadrilateral elements (41 nodes)
52 CASE STUDYAnalysis no. 2: Von Mises stress distribution using 96 bilinear quadrilateral elements (129 nodes)
53 CASE STUDYAnalysis no. 3: Von Mises stress distribution using 144 bilinear quadrilateral elements (185 nodes)
54 CASE STUDYAnalysis no. 4: Von Mises stress distribution using 24 eight-nodal, quadratic elements (105 nodes)
55 CASE STUDYAnalysis no. 5: Von Mises stress distribution using 192 three-nodal, triangular elements (129 nodes)
56 CASE STUDY Analysis no. Number / type of elements Total number of nodes in modelMaximum Von Mises Stress (GPa)124 bilinear, quadrilateral410.0139296 bilinear, quadrilateral1290.01803144 bilinear, quadrilateral1850.0197424 quadratic, quadrilateral1050.01915192 linear, triangular0.0167