THE USE OF ORTHOGONAL PROJECTIONS TO HANDLE CONSTRAINTS WITH APPLICATIONS TO INCOMPRESSIBLE 4-NODE QUADRILATERAL ELEMENTS

Linear equality multipoint constraint conditions define a vector space which is used to construct an orthogonal projection operator. Another orthogonal projection operator follows from the complement to the constrained vector space. A procedure is developed in which these operators are used in a systematic fashion to solve a set of algebraic equations subject to a set of multipoint constraints. In addition to elementary examples to illustrate the procedure, a problem associated with an incompressible, isotropic, elastic material is solved using four-node quadrilateral finite elements. The method of orthogonal projections provides the insight to show that, with the application of the incompressibility constraint, convergence is obtained for any arbitrary value of Poisson's ratio chosen sufficiently far from the value of one-half, which is the value normally associated with incompressibility. The bending performance of the incompressible four-node quadrilateral can be adjusted with artificial values of Poisson's ratio. The result is that calculations are performed with full numerical quadrature of the element, and mesh locking, hourglass instabilities and subsequent modifications to the element stiffness matrix are avoided.