Nonlinear rotation‐free three‐node shell finite element formulation

Abstract

SUMMARYA new triangular thin‐shell finite element formulation is presented, which employs only translational degrees of freedom. The formulation allows for large deformations, and it is based on the nonlinear Kirchhoff thin‐shell theory. A number of static and dynamic test problems are considered for which analytical or benchmark solutions exist. Comparisons between the predictions of the new model and these solutions show that the new model accurately reproduces complex nonlinear analytical solutions as well as solutions obtained using existing, more complex finite element formulations. Copyright © 2013 John Wiley & Sons, Ltd.

Comparative hemodynamics in an aorta with bicuspid and trileaflet valves

“…2. The leaflets of a heart valve are represented as three-dimensional surfaces and modeled as thin flexible shells with stretched and bending stiffness [28]. The fluid and solid domains intersect at the interface: Γ fsi = Ω s Ω f .…”

“…The fluid solver is based on the CURVIB approach of Ge and Sotiropoulos [27], which uses the hybrid stagger/non-staggered approach originally proposed by Gilmanov and Sotiropoulos [30] to solve the governing equations in generalized curvilinear grids with complex immersed boundaries [27]. For the structural solver, we use the FE model proposed by Stolarski et al [28], which approximates the shell curvature tensor, associated with a main element and three surrounding elements, using a complete quadratic polynomial in a moving with the shell rectilinear coordinate system attached to that element. This approach permits accurate approximations in the evaluation of the element curvature tensor.…”

“…Most importantly, since the above approach is used only to compute the bending strains within the elementcalculation of membrane strains is based on the flat geometry of the element-non-physical membrane locking is automatically avoided. For the derivation and the details of our method, the reader is referred to Stolarski et al [28]. We use the sharp interface immersed boundary method of Gilmanov and Sotiropoulos [30], Ge and Sotiropoulos [27] and Borazjani et al [31], which is able to deal with the arbitrarily complex, moving/deformable solid bodies in a curvilinear background grid where the fluid equations are solved.…”

“…Therefore, the main contribution of our paper is to take the first step in this direction by reporting FSI simulations of both TAV and BAV in a 3D anatomic aorta and analyzing the computed results to elucidate important differences between the aortic hemodynamics induced by each valve type. We employ our previously developed and validated curvilinear immersed boundary (CURVIB) method [27] coupled with an efficient, rotation-free thin-shell FE formulation [28] to carry out FSI simulations of a TAV and a BAV placed in the same anatomic aorta. The specific BAV morphology we consider in this work is the R-L type (Fig.…”

A 3-Node Co-Rotational Triangular Elasto-Plastic Shell Element Using Vectorial Rotational Variables

“…Existing 3-node triangular shell elements can be categorized into 4 types: Type 1 with only 3 displacement degrees-of-freedom (dofs) per node [14][15][16][17][18][19][20][21][22]; Type 2 with 3 displacement dofs and 2 rotational dofs per node [23][24][25]; Type 3 with 3 displacement dofs at the vertices and the rotational dofs at side nodes [26][27]; Type 4 with 3 displacement dofs and 3 rotational dofs per node [28][29][30][31][32][33]. Most existing 3-node rotation-free triangular shell elements are based on the Kirchhoff assumption, ignoring shear deformation, can therefore only be employed in modeling thin plate and shell problems [21].…”