Isogeometric thickness stretchable shell: Efficient formulation for nonlinear dynamic problems

Abstract

An isogeometric shell element with through-thickness stretch is introduced and applied to quasi-static and dynamic problems. The shell element supports full three-dimensional constitutive laws, ie, the plane stress assumption is not required. An updated Lagrangian rate formulation is adopted, and biquadratic spline-based interpolation functions are used for in-plane interpolation. The concept of fiber mass scaling is proposed to lower the highest eigenfrequencies to improve the performance of the formulation. Clamped and unclamped knot vectors are compared, and the advantages of using unclamped knot vectors are demonstrated. The shell element is validated using several benchmark tests, which indicate good performance of the proposed formulation. KEYWORDS explicit dynamics, isogeometric analysis, large deformations, thick-shell element, unclamped knot vectors Recently, the focus in the shell finite element technology has shifted toward more advanced formulations, such as thickness stretchable shell elements and solid shells. Two isogeometric solid-shell elements have been proposed by Hosseini et al 20,21 ; the latter formulation was extended to model propagation of delamination in composite materials. 22 Int J Numer Methods Eng. 2019;119:105-127.wileyonlinelibrary.com/journal/nme

Isogeometric shell analysis of incremental sheet forming

“…The performance of the thickness stretchable shell element was found to benefit from using bicubic interpolation functions combined with a recently proposed patchwise quadrature rule. In comparison to the interpolation functions and the quadrature employed in the original paper (Hokkanen and Pedroso, 2019a), the results showed an improved convergence rate and less locking without introducing spurious modes or increasing the total computational cost of the ISF simulation. Additionally, an irregular patchwise quadrature recently proposed in the literature was found to result in a wavy pattern leading to a poor solution.…”

“…Concepts such as using the principle of virtual power instead of the principle of virtual work, evaluating calculations in appropriate local coordinate systems, and neglecting certain higher order terms for efficiency are adopted from the aforementioned classical formulations. Based on these ideas, an isogeometric solid-like shell element was developed by Hokkanen and Pedroso (2019a) and is given as Appendix A. Several Kirchhoff-Love formulations are proposed in the literature.…”

“…An efficient technique compatible with diagonal mass matrices was proposed by Hokkanen and Pedroso (2019a) to enforce Dirichlet boundary conditions at arbitrary locations within a patch. A boundary condition of a prescribed acceleration along…”

“…Furthermore, the developed solid-like shell formulation (Hokkanen and Pedroso, 2019a) which is capable of directly representing thickness changes and supports complete three-dimensional constitutive laws is referred to as "S7". This formulation does not require plane stress enforcement nor it assumes normality of the shell fiber vector to the reference surface.…”

“…In (3.76), the derivatives of the shape functions N A,χ are approximated by the constant and linear terms in ζ, denoted by N 0 A,χ and N 1 A,χ , respectively. A similar approach is suggested by Belytschko et al (1989) and was adopted in the developments of the S7 element (Hokkanen and Pedroso, 2019a). As the shell formulation does not consider quadratic variation in ζ for the velocity strains, N 1 A,χ is not considered for…”