Shell Benchmark Problems Research Articles

An isogeometric continuum shell element for modeling the nonlinear response of functionally graded material structures

A geometrically nonlinear continuum shell element using a NURBS-based isogeometric analysis (IGA) approach is presented for the analysis of functionally graded material (FGM) structures. IGA offers a computationally efficient and geometrically exact representation of the original shell geometry and its underlying basis functions provide high-order continuity for the solution variables. The use of high-order smooth basis functions also alleviates shear and membrane locking phenomena that commonly occurred in shell structures. In addition, the developed continuum shell element features a precise description of the thickness-varying material properties in FGM in the sense that a set of desired high-order B-spline basis functions with sufficient number of quadrature points are employed for accurate through-thickness numerical integration. A simple power-law distribution function of the FGM is adopted in the current study. The performance of the proposed IGA solid shell element is demonstrated via a variety of nonlinear shell benchmark problems. The effect of the FGM power-law exponent on the geometrically nonlinear response of the shell structures is investigated as well.

Kirchhoff–Love shell formulation based on triangular isogeometric analysis

This article presents application of rational triangular Bézier splines (rTBS) for developing Kirchhoff–Love shell elements in the context of isogeometric analysis. Kirchhoff–Love shell formulation requires high continuity between elements because of higher order PDEs in the description of the problem. Non-uniform rational B-spline (NURBS)-based IGA has been extensively used for developing Kirchhoff–Love shell elements, as NURBS-based IGA can provide high continuity between and within elements. However, NURBS-based IGA has some limitations; such as, analysis of a complex geometry might need multiple NURBS patches and imposing higher continuity constraints over interfaces of patches is a challenging issue. Addressing these limitations, isogeometric analysis based on rTBS can provide C1 continuity over the mesh including element interfaces, a necessary condition in finite elements formulation of Kirchhoff–Love shell theory. Based on this technology, we use Cr smooth rational triangular Bézier spline as the basis functions for representing both geometry and solution field. In addition to providing higher continuity for Kirchhoff–Love formulation, using rTBS elements we can achieve three significant challenging goals: optimal convergence rate, efficient local mesh refinement and analysis of geometric models of complex topology. The proposed method is applied on several examples; first, this technique is verified against multiple plate and shell benchmark problems; investigating the convergence rate on the benchmark problems demonstrates that the optimal convergence rate can be obtained by the proposed technique. We also apply our method on geometric models of complex topology or geometric models in which efficient local refinement is required. Moreover, a car hood is modeled with rTBS and structurally analyzed by using the proposed framework.

Geometrically nonlinear analysis of thin-shell structures based on an isogeometric-meshfree coupling approach

This paper develops a novel coupling approach of the isogeometric analysis (IGA) method and the meshfree method for geometrically nonlinear analysis of thin-shell structures. The Kirchhoff–Love (KL) thin-shell theory is employed without the consideration of rotational degrees of freedom. Both parametric domain and physical domain are utilized for the thin-shell structures, and the former one is used to couple the IGA and meshfree methods and to obtain the later one via mapping. The domain is divided into three subdomains: the subdomain described by the IGA method to ensure geometry exactness, the subdomain described by the meshfree method to achieve local refinement, and the coupling subdomain described by both methods. In the coupling subdomain, the reproducing points are obtained based on the consistency conditions to realize smoothness between the IGA and meshfree subdomains. The coupling approach can achieve a higher convergence rate than the IGA and meshfree methods because of the realization of local refinement. The accuracy and robustness of the coupling approach are validated by solving shell benchmark problems.

A new MITC4+ shell element

We presented in Ko et al. (2016) an MITC4+ shell element that shows a much improved convergence behavior when compared to the widely used MITC4 (Dvorkin and Bathe, 1984) shell element. However, the element does still not show optimal convergence behavior when used in distorted meshes in the analysis of some shell problems. In the present paper, we establish a new MITC4+ shell element which shows significantly improved and indeed an almost optimal convergence behavior. In this new continuum mechanics-based shell element, the shear locking is alleviated using the well-known assumed transverse shear strain field of the original MITC4 shell element and the membrane locking is alleviated using a new assumed membrane strain field. The new MITC4+ shell element passes all basic tests: the isotropy, zero energy mode and patch tests. The excellent performance of the shell element is demonstrated through convergence studies in the solution of well-selected behavior-encompassing shell benchmark problems.

Testing of the Kriging-based Finite Element to Shell Structures with Varying Thickness

A variant of the finite element method with Kriging basis functions has been recently developed and applied to plane, plate bending, and shell elastostatic problems. The main advantage of this novel method is that high degree of basis functions can be easily constructed without additional finite element nodes (such as mid-side and inner nodes). This paper revisits the formulation of the Kriging-based finite element method for analysis of curved shell structures, referred to as K-Shell, and presents new numerical tests to shell structures with varying thickness. The K-Shell was formulated based on degenerated 3D elasticity theory. The basis functions were constructed using a set of nodes covering several layers of triangular elements and were employed to approximate both the displacement and geometry. The quartic polynomial basis was chosen to alleviate shear and membrane locking. The K-Shell has been tested using a series of shell benchmark problems with constant shell thickness. The tests showed that K-Shell performed very well in analyzing moderately thick smooth shells. In this paper the K-Shell is tested using two modified shell benchmark problems with varying thickness, i.e. the modified pinched cylinder and the modified hemispherical shell with 18 degrees cut-off. The converged results of three-dimensional finite element model using commercial software Abaqus were used to assess the convergence of the results. The tests show that the performance of K-Shell in analyzing shells with varying thickness is satisfactory.

Mechanical form-Finding of the Timber Fabric Structures with Dynamic Relaxation Method

Timber Fabric structures (TFS) initiate from a correspondence between textile principles and recent industrial developments in producing cross laminated timber panels. Several individual timber strips are interlaced according to a pattern and result in an innovative space structure. The obtained three-dimensional geometry can be regarded as the relaxed configuration of deformed panels under the imposed boundary conditions. We herein propose a form-finding procedure, which reproduces this deformed configuration as the steady state of a pseudo transient constrained dynamic problem. The corresponding nonlinear problem involves finite rotation regime and contact handling through the cross section and on both panel faces. To effectively deal with nonlinear constraints, a new modified dynamic relaxation method is herein used which combines elastic material behavior with a fictitious stiffness proportional damping into an equivalent fictitious viscous material model. The procedure is implemented as an ABAQUS/Explicit user subroutine VUMAT and the overall accuracy of the numerical results has been studied for a number of geometrically nonlinear shell benchmark problems. This numerical approach is then employed to simulate the assembly process for a Timber Fabric Module (TFM), an interlaced assembly of two timber strips. The simulated geometry for the deformed surfaces is then extracted and is compared with a 3D processed surface mesh obtained from scanning a built-in prototype with noncontact Laser scanner arm to validate the simulation procedure.

A large deformation, rotation-free, isogeometric shell

Conventional finite shell element formulations use rotational degrees of freedom to describe the motion of the fiber in the Reissner–Mindlin shear deformable shell theory, resulting in an element with five or six degrees of freedom per node. These additional degrees of freedom are frequently the source of convergence difficulties in implicit structural analyses, and, unless the rotational inertias are scaled, control the time step size in explicit analyses. Structural formulations that are based on only the translational degrees of freedom are therefore attractive. Although rotation-free formulations using C 0 basis functions are possible, they are complicated in comparison to their C 1 counterparts. A C k -continuous, k ⩾ 1, NURBS-based isogeometric shell for large deformations formulated without rotational degrees of freedom is presented here. The effect of different choices for defining the shell normal vector is demonstrated using a simple eigenvalue problem, and a simple lifting operator is shown to provide the most accurate solution. Higher order elements are commonly regarded as inefficient for large deformation analyses, but a traditional shell benchmark problem demonstrates the contrary for isogeometric analysis. The rapid convergence of the quadratic element is demonstrated for the NUMISHEET S-rail benchmark metal stamping problem.

A unified approach to nonlinear buckling optimization of composite structures

A unified approach to nonlinear buckling fiber angle optimization of laminated composite shell structures is presented. The method includes loss of stability due to bifurcation and limiting behaviour. The optimization formulation is formulated as a mathematical programming problem and solved using gradient-based techniques. Buckling of a well-known cylindrical shell benchmark problem is studied and the solutions found in literature are proved to be incorrect. The nonlinear buckling optimization formulation is benchmarked against the traditional linear buckling optimization formulation through several numerical optimization cases of a composite cylindrical shell panel which clearly illustrates the advantage and potential of the presented approach.

Backstepping-enhanced decentralised PID control for MIMO processes with an experimental study

A novel decentralised PID controller design procedure based on backstepping principles is presented to operate multiple-input multiple-output (MIMO) dynamic processes. First, a control Lyapunov function (CLF) and virtual control variable based on the backstepping method are derived recursively for each loop and a multivariable controller is obtained. Appropriate selection of auxiliary control variables can then eliminate coupling effects between subprocesses, and a decentralised controller can be derived. By analysing the regulation dynamics, external disturbances are rejected via appropriate tuning of the backstepping design parameters. Sufficient stability conditions are then derived using the small-gain theorem for two-input two-output and MIMO closed-loop systems, respectively. Simulation results for the control of the Shell benchmark problem and experimental results with a real-time water-level control system are provided to demonstrate the effectiveness and practicality of the proposed design procedure.

Asymptotic deformation modes of benchmark problems suitable for evaluating shell elements

An analytical approach introduced recently concerning the definition of the limiting cases of the asymptotic behaviour of shell elements is critically reviewed. A simple analytical approach for defining the asymptotic behaviour of shell elements is proposed. This approach is employed to define and analyse the asymptotic deformation modes of selected benchmark problems suitable for the evaluation of the asymptotic behaviour of shell elements. A new limit test problem, the hyperbolic paraboloid shell problem, proposed as a strong test for the pure bending asymptotic behaviour of shell elements, is analysed in the context of the proposed approach. Explicit expressions for characteristic asymptotic displacements are derived for the shell benchmark problems considered. These expressions are proposed for testing the asymptotic behaviour of finite shell elements in membrane-dominated, bending-dominated and mixed membrane-bending mode shell problems.

Nash-optimization enhanced distributed model predictive control applied to the Shell benchmark problem

This paper presents an efficient distributed model predictive control scheme based on Nash optimality, in which the on-line optimization of the whole system is decomposed into that of several small co-operative agents in distributed structures, thus it can significantly reduce computational complexity in model predictive control of large-scale systems. The relevant nominal stability and the performance on single-step horizon under the communication failure are investigated. The Shell heavy oil fractionator benchmark control problem is illustrated to verify the effectiveness of the proposed control algorithm.

Identification of dynamic models for the Shell benchmark problem

Dynamic process and disturbance models are identified for the 2×2 distillation column described in the summary paper by Cott. Process models are identified using both classical least squares and the newer AUDI identification algorithm developed at the University of Alberta. The identified models are validated and gain-adjusted in the frequency domain by comparing the magnitude spectrum of the models, | G (e −iω )|, versus the spectrum obtained by fast Fourier transformation of the original plant I/O time series. The noise/disturbance models are identified using standard time series tools and all models are verified by showing that the filtered residuals (measured process output minus model based estimate of the output) are essentially white noise. The estimated SISO models are all first-order and very close to the actual process whether compared on the basis of step-response coefficients, model parameters or frequency domain plots. The control problem is so simple that two standard PID feedback loops with one-way (lead/lag) interaction compensation produce results that are essentially identical to a 2×2 multivariable, model-based (GPC) controller.