On The Locking Problem And The Nitsche's Method In Isogeometric Analysis

Finite element method(FEM)is one of the most significant achievements in the field of mechanics in the 20th century.After more than 50 years of development,FEM has formed a profound foundation of mathematics and mechanics,numerous researchers have constructed a large number of elements and developed sophisticated static and dynamic analysis methods and software,which have been widely used in many fields.In various element formulations developed in FEM,the basic idea of the quasi-conforming(QC)element is instructive for the construction of many elements,the QC element releases the conforming conditions between elements by "integral weakness".The QC element is simple to construct,and the element stiffness matrix is expressed explicitly,therefore research on construction of QC elements helps to analysis practial problems in a simple and quick manner.For difficulties in bridging computer aided design(CAD)and computer aided engineering(CAE)caused by mesh generation,isogeometric analysis(IGA),as a newly developed finite element framework,adopts non-uniform rational B-splines(NURBS)as shape functions and can be used for both design and analysis in an uniform expression.IGA is believed to have the potential to seamlessly merge CAD and CAE and has become a very fast-growing direction.Thus relevant research on IGA has important theoretical significance and engineering application value.In the construction of high-order elements,the iso-parametric transformation usually used in QC for element domain integrations shows sensitive to element shapes and fails to achieve the theoretical maximum algebraci precision,which lead to elements with limited performance.Similar to traditional FEM,the Timoshenko beam and Reissner-Mindlin plate/shell elements in IGA suffer from numerical locking phenomenon,the meaning of research on locking consists in generalized elements usable for both thin and thick thickneses,high-accuracy results obtained from coarse meshes and less computational efforts.In IGA the boundary conditions imposing issues are hot topics,for instance,the structural displacement boundary conditions are difficult to be applied directly,it is inconvenient to control the rotational boundary conditions of the Kirchhoff-Love thin plate element,moreover the coupling boundary conditions of complex multi-patches structures needs to be studied,these formulaitons and corresponding influence still remain underdeveloped.NURBS are capable of exactly describing boundaries of structures and has unique advantages for solving contact problems,therefore the study of contact boundary conditions enforcing and the simulation of structural contact problems are important to IGA.The following studies are carried out focusing on QC and IGA:(1)Development of a high-accuracy quasi-conforming element.The usually used isoparametric transformation for element domain integrations limits the performance and precision of the high-order QC element under construction,which explains the necessity of a proper method for element domain integrations.To address this issue,the B-net method,which has good properties such as high-accuracy and anti-distortion,is employed within QC formulation to develop a QC plane quadrilateral 8-node element.The newly developed element by adopting the B-net method for element domain integrations saves computational time and more importantly achieves the 2nd order precision.Thanks to the properties of the B-net method,the present element performs stable for mesh distortion and calculable even for concave quadrilateral meshes.(2)Research on element formulations of beam and plate/shell and locking problems within IGA.Beam and plate/shell elements in IGA still suffer from the locking phenomenon,the element accuracy drops down further especially when control mesh distortion happens simultaneously.The order reduction method(OR)of shape functons is proposed for plane Timoshenko curved beam element and Reissner-Mindlin plate/shell element,in order to solve the field inconsistency problem by projecting the locking strains onto lower order spaces.In addition the OR strategy is also discussed,the computation cost is reduced and the accuracy is improved effectively by employing projecting basis functions of different orders in different elements.For the spatial curved rod element and solid-shell element,the formulations and locking phenomenon are studied,and the reduced integration method is adopted to alleviate the locking phenomenon.(3)Research on formulations of displacement,rotational and interface coupling boundary conditions imposing.It is a common case in engineering practice to apply the displacement and rotational boundary conditions,as well as simulate complex structures with multi-patches.However in IGA these boundary conditions are difficult to impose directly,the Nitsche's method is generally used to impose boundary conditions into the weak form of the original problem by "integral weakness".The Nitsche's formulations are expressed in different forms for different boundary conditions,these formulations are integrated into an unified framework,which helps to complement the formulation theory.The present skew-symmetric Nitsche's formulation is parameter-free from solving the stabilization parameter,in this research the skew-symmetric Nitsche's formulation is also introduced into the unified framework.Numerical examples show the numerical performance and prove the effectiveness of the Nitsche's formulations,by which the influence of the Nitsche's coupling process is also studied.(4)Formualtions of contact conditions imposing and simulation of contact problems in IGA framework.For frictionless contact in linear elasticity,after an equivalent transformation of the contact conditions into a projection operator,the Nitsche's method takes over to apply these contact conditions.The formualtion derivation starts from the contact between an elastic body and a rigid body,then it is expanded for master-slave contact and unbiased contact between two elastic bodies,the unbiased formulation is suitable for self-contact.The frictional conditions are further introduced into the formulation,then the frictional contact in large deformation is simulated by the Nitsche's method based on the large deformation formulation of the solid-shell element.In addition the linearization process of the contact formulatons is introduced,as well as an efficient and robust contact search algorithm.The Nitsche's contact formulations are tested and studied by several examples,numerical results indicate that the Nitsche's method can effectively apply contact conditions and simulate contact problems.