4Displacement interpolation 2 1 1 1 1 s 1 Consider a special 4-noded rectangle in its local coordinate system (s,t)tDisplacement interpolation21111s1Shape functions in local coord system34
5Recall thatRigid body modesConstant strain states
6Global coordinate system Local coordinate system Goal is to map this element from local coords to a general quadrilateral element in global coord systemsy1234xtGlobal coordinate systemst1234Local coordinate system
71. Kronecker delta property In the mapped coordinates, the shape functions need to satisfy1. Kronecker delta propertyThen2. Polynomial completeness
8The relationshipProvides the required mapping from the local coordinate systemTo the global coordinate system and is known as isoparametric mapping(s,t): isoparametric coordinates(x,y): global coordinates
11NOTES1. Given a point in the isoparametric coordinates, I can obtain the corresponding mapped point in the global coordinates using the isoparametric mapping equationQuestionx=? at s=0.5?
122. The shape functions themselves get mapped In the isoparametric coordinates (s) they are polynomials.In the global coordinates (x) they are in general nonpolynomialsLets consider the following numerical example42x132Isoparametric mapping x(s)Simple polynomialInverse mapping s(x)Complicated function
13Now lets compute the shape functions in the global coordinates
14Now lets compute the shape functions in the global coordinates N2(x)N2(s)1132142xs13211N2(x) is a complicated functionN2(s) is a simple polynomialHowever, thanks to isoparametric mapping, we always ensure1. Knonecker delta property2. Rigid body and constant strain states
15Element matrices and vectors for a mapped 1D bar element 32xs13211Displacement interpolationStrain-displacement relationStressThe strain-displacement matrix
16The only difference from before is that the shape functions are in the isoparametric coordinates We know the isoparametric mappingAnd we will not try to obtain explicitly the inverse map.How to compute the B matrix?
17Using chain rule(*)Do I knowDo I knowI knowHenceFrom (*)
18What does the Jacobian do? Maps a differential element from the isoparametric coordinates to the global coordinates
19The strain-displacement matrix For the 3-noded element
20The element stiffness matrix NOTES1. The integral on ANY element in the global coordinates in now an integral from -1 to 1 in the local coodinates2. The jacobian is a function of ‘s’ in general and enters the integral. The specific form of ‘J’ is determined by the values of x1, x2 and x3. Gaussian quadrature is used to evaluate the stiffness matrix3. In general B is a vector of rational functions in ‘s’
22Shape functions of parent element in isoparametric coordinates Isoparametric mapping
23NOTES:The isoparametric mapping provides the map (s,t) to (x,y) , i.e., if you are given a point (s,t) in isoparametric coordinates, then you can compute the coordinates of the point in the (x,y) coordinate system using the equations2. The inverse map will never be explicitly computed.
258-noded Serendipity element: element shape functions in isoparametric coordinates
26NOTES 1. Ni(s,t) is a simple polynomial in s and t NOTES 1. Ni(s,t) is a simple polynomial in s and t. But Ni(x,y) is a complex function of x and y.2. The element edges can be curved in the mapped coordinates3. A “midside” node in the parent element may not remain as a midside node in the mapped element. An extreme examplet5y211111582,6,386sx17347
274. Care must be taken to ensure UNIQUENESS of mapping 2yt2111131xs1344
28Isoparametric mapping in 2D: Triangular parent elements Parent element: a right angled triangles with arms of unit lengthKey is to link the isoparametric coordinates with the area coordinates21P(s,t)sts311
29Isoparametric mapping Parent shape functions Now replace L1, L2, L3 in the formulas for the shape functions of triangular elements to obtain the shape functions in terms of (s,t)Example: 3-noded trianglest2 (x2,y2)t211yx3 (x3,y3)s1 (x1,y1)31Isoparametric mappingParent shape functions
30Element matrices and vectors for a mapped 2D element Recall: For each elementDisplacement approximationStrain approximationStress approximationElement stiffness matrixElement nodal load vectorSTee
31In isoparametric formulation Shape functions first expressed in (s,t) coordinate systemi.e., Ni(s,t)2. The isoparamtric mapping relates the (s,t) coordinates with the global coordinates (x,y)3. It is laborious to find the inverse map s(x,y) and t(x,y) and we do not do that. Instead we compute the integrals in the domain of the parent element.
32NOTE1. Ni(s,t) s are already available as simple polynomial functions2. The first task is to find andUse chain rule
33In matrix formThis is known as the Jacobian matrix (J) for the mapping(s,t) → (x,y)We want to compute these for the B matrixCan be computed