Finite Element Method - The Basis (Volume 1)

xContents9.15 A computational advantage of numerically integrated ®niteelements 2369.16 Some practical examples of two-dimensional stress analysis 2379.17 Three-dimensional stress analysis 2389.18 Symmetry and repeatability 244References 24610. The patch test, reduced integration, and non-conforming elements 25010.1 Introduction 25010.2 Convergence requirements 25110.3 The simple patch test (tests A and B) ± a necessary condition forconvergence 25310.4 Generalized patch test (test C) and the single-element test 25510.5 The generality of a numerical patch test 25710.6 Higher order patch tests 25710.7 Application of the patch test to plane elasticity elements with`standard' and `reduced' quadrature 25810.8 Application of the patch test to an incompatible element 26410.9 Generation of incompatible shape functions which satisfy thepatch test 26810.10 The weak patch test ± example 27010.11 Higher order patch test ± assessment of robustness 27110.12 Conclusion 273References 27411. Mixed formulation and constraints± complete ®eld methods 27611.1 Introduction 27611.2 Discretization of mixed forms ± some general remarks 27811.3 Stability of mixed approximation. The patch test 28011.4 Two-®eld mixed formulation in elasticity 28411.5 Three-®eld mixed formulations in elasticity 29111.6 An iterative method solution of mixed approximations 29811.7 Complementary forms with direct constraint 30111.8 Concluding remarks ± mixed formulation or a test of element`robustness' 304References 30412. Incompressible materials, mixed methods and other procedures ofsolution 30712.1 Introduction 30712.2 Deviatoric stress and strain, pressure and volume change 30712.3 Two-®eld incompressible elasticity (u±p form) 30812.4 Three-®eld nearly incompressible elasticity (u±p±" v form) 31412.5 Reduced and selective integration and its equivalence to penalizedmixed problems 31812.6 A simple iterative solution process for mixed problems: Uzawamethod 323