Rotation-free Shell Formulation Research Articles

An integrated design approach for simultaneous shape and topology optimization of shell structures

In this work, a novel design approach is developed to carry out shape and topology optimization of shell structures simultaneously. This approach effectively integrates the IsoGeometric Analysis (IGA) method, the Adaptive Bubble Method (ABM) and the Finite Cell Method (FCM) to make full use of their respective advantages in shell structure optimization. To be specific, the IGA method using Non-Uniform Rational B-Splines (NURBS) elements is adopted to analyze the shell structure and realize shape optimization with merits of exact shell surface representation geometrically and rotation-free shell formulation physically; the ABM is utilized to insert deformable holes adaptively in the parametric domain of the NURBS surface for topology optimization of the shell structure; the FCM is employed to facilitate the optimization process by simplifying significantly the discrimination and the numerical integration processes of trimmed elements involved in fixed-mesh analysis of the holed shell structure. Based on this integrated design approach of IGA/ABM/FCM, shape optimization can be implemented by directly optimizing the control-point coordinates of the NURBS surface in the physical space, while topology optimization could be carried out simultaneously in the parametric domain as easily as for 2D planar structures. Finally, representative examples are investigated to demonstrate the effectiveness and advantages of the proposed design approach.

A nonlinear rotation-free shell formulation with prestressing for vascular biomechanics

We implement a nonlinear rotation-free shell formulation capable of handling large deformations for applications in vascular biomechanics. The formulation employs a previously reported shell element that calculates both the membrane and bending behavior via displacement degrees of freedom for a triangular element. The thickness stretch is statically condensed to enforce vessel wall incompressibility via a plane stress condition. Consequently, the formulation allows incorporation of appropriate 3D constitutive material models. We also incorporate external tissue support conditions to model the effect of surrounding tissue. We present theoretical and variational details of the formulation and verify our implementation against axisymmetric results and literature data. We also adapt a previously reported prestress methodology to identify the unloaded configuration corresponding to the medically imaged in vivo vessel geometry. We verify the prestress methodology in an idealized bifurcation model and demonstrate the significance of including prestress. Lastly, we demonstrate the robustness of our formulation via its application to mouse-specific models of arterial mechanics using an experimentally informed four-fiber constitutive model.

Dynamic contact model of shell for multibody system applications

In a multibody system consisting of shell structures, the contact may appear in any area of shells. It is difficult to simulate the contact of shells with large deformation because of the geometric nonlinearity of deformation and the boundary nonlinearity of contact. This study presents a rotation-free shell formulation and an extended contact discretization in multibody systems using a corotational frame. This model is different from previous formulations in the definition of the local frame and the processing of local large curvature. In order to deal with the shell contact, a unified contact discretization scheme including edge-to-edge contact for facet triangle shell elements is proposed to solve the large penetration problem. A series of numerical examples of multibody dynamics have validated the approach of the nonlinear shell model and contact treatments. Moreover, a practical application of deployment of solar cells shows the capability of the proposed formulation in solving large-scale problems of flexible multibody system with large deformation and contact.

Multibody dynamic analysis using a rotation-free shell element with corotational frame

Rotation-free shell formulation is a simple and effective method to model a shell with large deformation. Moreover, it can be compatible with the existing theories of finite element method. However, a rotation-free shell is seldom employed in multibody systems. Using a derivative of rigid body motion, an efficient nonlinear shell model is proposed based on the rotation-free shell element and corotational frame. The bending and membrane strains of the shell have been simplified by isolating deformational displacements from the detailed description of rigid body motion. The consistent stiffness matrix can be obtained easily in this form of shell model. To model the multibody system consisting of the presented shells, joint kinematic constraints including translational and rotational constraints are deduced in the context of geometric nonlinear rotation-free element. A simple node-to-surface contact discretization and penalty method are adopted for contacts between shells. A series of analyses for multibody system dynamics are presented to validate the proposed formulation. Furthermore, the deployment of a large scaled solar array is presented to verify the comprehensive performance of the nonlinear shell model.

A new shell formulation for graphene structures based on existing ab-initio data

An existing hyperelastic membrane model for graphene calibrated from ab-initio data (Kumar and Parks, 2014) is adapted to curvilinear coordinates and extended to a rotation-free shell formulation based on isogeometric finite elements. Therefore, the membrane model is extended by a hyperelastic bending model that reflects the ab-inito data of Kudin et al. (2001). The proposed formulation can be implemented straight-forwardly into an existing finite element package, since it does not require the description of molecular interactions. It thus circumvents the use of interatomic potentials that tend to be less accurate than ab-initio data. The proposed shell formulation is verified and analyzed by a set of simple test cases. The results are in agreement to analytical solutions and satisfy the FE patch test. The performance of the shell formulation for graphene structures is illustrated by several numerical examples. The considered examples are indentation and peeling of graphene and torsion, bending and axial stretch of carbon nanotubes. Adhesive substrates are modeled by the Lennard–Jones potential and a coarse grained contact model. In principle, the proposed formulation can be extended to other 2D materials.

Isogeometric shape optimization of shells using semi-analytical sensitivity analysis and sensitivity weighting

We present isogeometric shape optimization for shell structures applying sensitivity weighting and semi-analytical analysis. We use a rotation-free shell formulation and all involved geometry models, i.e., initial design, analysis model, optimization model, and final design use the same geometric basis, in particular NURBS. A sensitivity weighting scheme is presented which eliminates certain effects of the chosen discretization on the design update. A multilevel design approach is applied such that the design space can be chosen independently from the analysis space. The use of semi-analytical sensitivities allows having different polynomial degrees for design and analysis model. Different numerical examples are performed which confirm the applicability of the proposed method. Furthermore, a shape optimization example with an exact solution is presented which can serve as general benchmark for shape optimization methods.