Research On The Validity Of Isogeometric Method Used In The Mechanics Analysis Of Typical Structures

Isogeometric analysis (IGA) was first proposed in 2005 with the goal of integrating design into analysis and optimization. The root idea is that the basis functions, e.g., the Non Uniform Rational B-Spline (NURBS), used to exactly model the geometry will also serve as the basis functions for the solution space of the numerical analysis. And the uniform expression form makes an accessible interaction between the analysis model and the geometry model. As the model in engineering practice becomes more and more complicated, the mesh generation can be time-consuming in FEA. The integration of Computer Aided Design (CAD) and Computer Aided Engineering (CAE) can deal with this issue. The study of IGA has become a developing tendency in engineering analysis and IGA will have effect on CAD and CAE.Owning to the wide use of Fortran in some mature software of FEA, e.g. ANSYS, the implementation is hopeful to integrate IGA to the finite element environment. The author did a study on IGA with a Fortran implementation. Several mechanical problems were illustrated to demonstrate the efficiency, accuracy, and reliability of the implementation. In the Fortran implementation of IGA, the modularization design is adopted. For the compatibility of Fortran implementation in different numerical examples, the maximized amount of subroutines are contained in the communal module. Furthermore,for the large sparse stiffness matrix, the existing solvers in IGA may be time-consuming which is an obstacle in the development of IGA for more complex problems. An effective solver GSS (Grus Sparse Solver) was used to implement IGA with Fortran in this study. The Fortran code of the related examples can be downloaded freely in this thesis.Generally, NURBS basis functions are non-interpolated, for example, they do not satisfy the Kronecker-Delta property. Special treatments of essential boundary conditions need to be employed. The penalty method was used as an efficient way to resolve IGA boundary problems. However, the influence of penalty coefficients on the results is ambiguous. Therefore, it is difficult to choose the appropriate ones. And the inappropriate selection of values of penalty coefficients may lead to errors. Through the data summary, a suggestion of the values of penalty coefficients was given based on the maximums of global stiffness matrix K (MOK). A value that is 102-103 higher than the magnitude of MOK is better. The influence of penalty coefficients can be reduced by refinements. However, the computational cost was increased.We improved the Kirchhoff-Love theory with the penalty method. The NURBS basis functions were not only used for the presentation of element shapes and displacement field, but also provided the high order continuous functions needed by the Kirchhoff-Love theory. Several plate and shell problems were illustrated to demonstrate the accuracy and quick convergency of the Fortran implementation. And it could explain that even at the coarsest mesh level, IGA can produce accurate results.Extended finite element analysis (XFEA) were combined with Isogeometric analysis to solve the fracture mechanic problems. It was demonstrated the availability of the Fortran based extended Isogeometric analysis (XIGA) used to simulate the discontinuity. The selections of enriched control points were provided. The crack discontinuous field and the tip displacement field were enriched by Heavisde functions and crack tip enrichment functions, respectively. In comparison with the XFEA used for the same crack model, it was illustrated that the XIGA results were more accurate with less elements than XFEA. The high order continuity made the expressions of displacement and stress contour smooth and continuous in different examples.In this research, the multiple patches were used to realize the local refinements of the model with a circular hole. Displacement and stress variables were computed on each patch and accurate results were achieved. The insertion of linear knot values created meshes with a certain level of refinement at region around cracks in fracture mechanical problems and the errors of the results were reduced. By improving the quality of the mesh refinement appropriately, the results can be more accurate, the convergence can be faster, and the stress field can be more continuous.To conclude, we developed a Fortran based Isogeometric analysis tool and it was used for different mechanical problems to demonstrate its accuracy and efficiency. The related theories of IGA were perfected. Meanwhile, we analyzed the problems in IGA and provided proper suggestions. Due to constraints in time and environment, the procedures of IGA need to be optimized and the Fortran codes need to be used in more IGA algorithms.