On the development of compatible elements for the solution of three-dimensional elasticity problems

As an extension of work presented for two-dimensional elements (1-7), the derivations for generalized compatible elements useful for solving three-dimensional problems in elasticity are presented. The formulations are derived by using displacement functions which are compatible on the interior and "average compatible" on the boundaries. In addition, an approach to make the solution auto-adaptive to local regions for high strain rates by increasing the order of interpolating functions (as opposed to re-meshing) is presented. Compatible Displacement Functions Compatible displacement functions for elastic deformations can be constructed by the use of the Boussinesq-Papkovich-Neuber stress function (8) which is adopted for threedimensiona form: elasticity problems in the following 1 a=u = Poisson's Ratio 4(1-u) ' In this paper the vector L$o. @29 b 3 1 is called the BPN stress function or vector of stress functions. For cases where the elastic body is of convex type, on1.y three out of four of the stress functions as shown above will be adequate for complete representation of the displacement field [9-111. In most cases of the compositions of stress functions for the construction of compatible displacement functions, the four stress functlons $ 9 , $ , and @ all are included in each vector.''~oT co?wenienc2 of expression of the results developed, a set of harmonic functions in terms of x,y, and z, up to 6-th order, is listed as follows: 3 2 3 2 3 2 xyz, (x -3xv ) , !X -3xz ) , (y -3x y), 3 2 3 2 (z -3x z), (z -3y z), 2 2 2 2 y (x -z ) , 2 (x -y ) ; 3 2 3 2 ) , (x z-3xv 2). (xy -3xyz ) , 3 2 3 2 ) 9 (xz -3xy z), (YZ -3x yz), 2 2 4 4 2 2 ( X ~ + ~ ~ ~ X y ) , (x +z -6x z ) , 4 4 2 2 3 3 (y +Z -6y z ) ; (x yz-xyz ) , 3 3 4 4 2 2 (xy z-xyz ) , (x z+y z-6x v z), 4 4 2 2 4 4 7 2 (x y+yz -6x yz ) . (xy +xz -6xy z ) , 5 3 2 5 4 3 2 (x +5xy4-lox y ) , (x +5xz -1Ox z ) , 5 4 2 3 4 3 2 (y +5x v-lox y ) , (y5+5yz -1oy z 1, 5 4 2 3 5 4 2 3 (Z +5x Z-IOX z ) , (Z + 5 ~ Z-10y z ) J. (2) Let the vector of BPN stress functions be In matrix form equation (3) Is written where n is the number of the degrees of freedom. The j-th column vector of the stress function in the matrix I Y ] reads: Note that each conponent of the stress function b . is a proper composition of harmonic functions a:Jording to certajn weighting coefficients selected in order to accommodate the desired shape function in the form: In this equation, m is the total number of harmonic polynomials used, and b,. I k is the coefficient of the k-th harmonic fuaction, fk' It is proposed that each vector of stress functions corresponding to some simple displacement, strain or stress field be as explained below. 1 ,,Prof. of Civil Engg. ;Prof. of Electrical and Computer En@. Doctoral candidate, Civil Engg. Note that there are many ways to compose the harmonic function to obtain desired vectors of stress functions. Also note that the coefficients presented are for the case when the Possion's ratio is 113. For higher order elements they can be derived and solved according to the sample steps given in this part, except that higher order polynomials must be used. Compatible Elements IJsing Nodal Parameters Using a proper selection of harmonic functions as derived, one can construct { @ I = [@jl{aj} where I a. I is the vector of unknown coefficients: 'dter substituting Eq. 3 into Eq. 1, where n is the total number of degrees of freedomewhich characterize the field displacement vector for each point of the e-th element; a 1' a2, ..., a is a vector of unknown coefficients; and q areethe corresponding functions. i j Using this displacement function, the calculated stresses satisfy the equilibrium conditions at all the interior points if the body forces are ignored (Ref. 8). When a standard finite element approach is used, the total number of unknown coefficients equal where n = number of boundary surfaces of the 3-D element, and n = number of points per boundary surface. P Let be the displacement vector at the k-th nodal point. Then establishes the relationship between the unknown coefficients and the unknown nodal parameters {ue}. In short, this relationship can be written as Solving for the {A) vector, Substituting Eq. 17 into Eq . 13 gives