Overcoming locking in NURBS-based discretizations of rods and nearly-incompressible solids: Continuous-assumed-strain (CAS) elements

Abstract

In finite element analysis (FEA), locking refers to a numerical pathology that results in the finite-element solution becoming overly stiff. Specifically, the numerical solutions tend to exhibit significantly smaller displacements than expected (hence the term “locking”) and large-amplitude spurious oscillations of the stresses. There are different types of locking. Rod structural theories that take into account transverse shear deformation suffer from membrane and shear locking, which get worse as the slenderness ratio of the structure increases. Nearly- incompressible solids suffer from volumetric locking, which gets worse as the Poisson ratio approaches 0.5. In order to overcome locking, a locking treatment needs to be added to the standard FEA discretization based on the Galerkin method. The two types of locking treatments whose use is widespread in commercial FEA software are reduced integration and assumed strains. Conventional FEA software uses Lagrange polynomials as basis functions. However, computer-aided design (CAD) software uses nonuniform rational B-splines (NURBS) as basis functions. The fact that FEA and CAD programs use two different geometric representa- tions causes multiple interoperability issues in the design-through-analysis cycle. In order to eliminate these interoperability issues, isogeometric analysis (IGA) proposes to use the same geometric representation in both CAD and FEA programs. Specifically, IGA proposes to use either NURBS or other types of smooth splines in both CAD and FEA programs. When applied to structural theories that take into account transverse shear deformation (e.g., Timoshenko rods), NURBS-based discretizations of the Galerkin method suffer from the same types of locking as conventional FEA discretizations based on Lagrange polynomials. When applied to structural theories that neglect transverse shear deformation (e.g., Kirch- hoff rods), NURBS-based discretizations of the Galerkin method still suffer from membrane locking. When the material considered is nearly-incompressible, NURBS-based discretiza- tions suffer from volumetric locking as it is the case for conventional FEA discretizations based on Lagrange polynomials. However, a direct deployment of the numerical schemes used to overcome locking when using Lagrange polynomials is not an effective strategy to vanquish locking when using NURBS since the levels of continuity across element boundaries of Lagrange polynomials and NURBS are different. Thus, the higher inter-element continuity of NURBS requires the development of new numerical schemes to overcome locking. The goal of this thesis is to develop the first assumed-strain locking treatment that ef- fectively overcomes locking from NURBS-based discretizations while satisfying the following important characteristics: (1) No additional unknowns are added, (2) No additional systems of algebraic equations need to be solved, (3) The same elements are used to approximate the displacements and the assumed strains, (4) No additional matrix operations such as matrix inversions or matrix multiplications are needed to obtain the stiffness matrix, and (5) The nonzero pattern of the stiffness matrix is preserved. In order to do so, this thesis proposes Continuous-Assumed-Strain (CAS) elements. CAS elements take advantage of the C 1 inter- element continuity of the displacement vector given by NURBS to interpolate the compatible strains using Lagrange polynomials while preserving the C 0 inter-element continuity of the compatible strains. Both symmetric and non-symmetric versions of CAS elements are devel- oped. Linear plane Kirchhoff rods, linear plane Timoshenko rods, and nearly-incompressible linear-elastic solids are used as model problems to study how to overcome membrane locking in fourth-order structural theories, membrane and shear locking in second-order structural theories, and volumetric locking, respectively. We solve benchmark problems with known exact solutions so that we can compute the relative errors in L2 norm of all the quantities of interest. Our results show that CAS elements effectively overcome locking, namely, the numerical approximations of the unknowns become more accurate for coarse meshes and the numerical approximations of the quantities of interest that depend on the derivatives of the unknowns (i.e., stresses and stress resultants) are free from spurious oscillations. The sym- metric and non-symmetric versions of CAS elements result in essentially the same accuracy when solving fourth-order theories while the non-symmetric version of CAS elements is more accurate than its symmetric counterpart when solving second-order theories. Both the sym- metric and non-symmetric versions of CAS elements are computationally efficient since these locking treatments can be applied by only modifying how the element stiffness matrices are computed. p Gauss-Legendre quadrature points per direction can be used to speed up the simulations without sacrificing accuracy, where p is the degree of the basis functions being used. In summary, CAS elements provide a robust, accurate, and computationally efficient numerical scheme for overcoming locking in NURBS-based discretizations. Thus, advancing the state-of-the-art in numerical methods within the IGA framework for structural and solid mechanics.