A global/local approach based on CUF for the accurate and efficient analysis of metallic and composite structures

Highlights

• A novel global/local finite element approach is developed.

• The method makes use of the Carrera unified formulation to enhance the solution in critical areas.

• Global analyses are performed with available finite element tools, such as Nastran and Abaqus.

• By using higher-order models in critical areas, it is possible to recover accurate stress/strain fields.

• The method is demonstrated to be very effective for both metallic and composite structures.

Abstract

The design and analysis of aerospace structures requires a detailed evaluation of stresses. Nevertheless, the complexity of large structures and the use of composite materials can significantly increase the computational costs of the models. The computational burden of such analyses can be reduced by a suitable global/local approach developed in a very general Finite Element framework. Generally, a global/local modelling approach aims at using a finer mesh in the “local” zones where a detailed evaluation of stress/strain field is required, whereas a coarse mesh is used in the rest of the structure. This work proposes a global/local methodology to set up a high-order beam model in the Carrera Unified Formulation framework only for a reduced region of the global model. The methodology makes use of two steps. In the first step, a static analysis of the global structure is done by means of a commercial software in order to identify the critical regions deserving more accurate investigations. In the second step, thus, a high-order beam model is employed for the local region based on the information from the previous global analysis. Linear elastic static analysis are considered in this work, and the attention is mainly focussed on the capability of the method to provide stable solutions and accurate 3D stress fields in the local region, even in the case of laminated composite structures. Hence, the effectiveness of the proposed approach is proven through some meaningful benchmarks.

Introduction

In the aeronautical field, when dealing with the design of an aircraft structure the finite element (FE) model of the system is usually built by combining 1D and 2D elements, which opportunely discretize mathematical domains of stringers, panels, ribs, and other components. Clearly, this discretization results in a simplification of the reality. In fact, it may be necessary to determine 3D stress fields in certain regions of the model. To accurately capture these localized 3D stress fields, solid models or high-order theories are often necessary. However, in order to make the model more efficient, i.e. to balance computational cost and results accuracy, a global/local approach is often employed. Three main approaches are available in the literature to deal with a global/local analysis: (1) refining the mesh or the shape functions within critical regions [1], [2], [3], [4], [5]; (2) formulating multi-model methods, in which different subregions of the structure are analysed with different mathematical models [6], [7], [8], [9], [10], [11], [12], [13]; (3) using models based on the Static Condensation also known as “Super-elements Methods” [14], [15].

The first method listed above mainly faces convergence problems in those regions where singularities occur. Adaptive techniques are often used to couple coarse and refined subregions of a structure. The h-adaption method [1] is used when the structure subregions differ in mesh size, whereas the p-adaption method [2] can be applied when the subregions differ in the polynomial order of the shape functions. Moreover, the hp-adaption [3] can allow the implementation of subregions differing in both mesh size and shape functions. Other techniques allowing for the coupling of different meshes are, for instance, the multi-grid method [4], and the extended finite element method (XFEM) [5]. All these methods can be addressed as single-model methods. In the case of multiple-model methods, where different subregions of the structure are modelled with kinematically incompatible elements, the compatibility of displacements and equilibrium of stresses at the interface between dissimilar elements have to be achieved. In the s-version of the finite element method (FEM) [6], [7], the resolution in a certain subregion of the structure is increased by superimposing additional meshes of high-order hierarchical elements. Shim et al. [8] combined 1D and 2D elements with 3D solid elements via multipoint constraint equations evaluated by equating the work done on both sides of the dimensional interface. In [9], the coupling of structural models with different dimensionalities was achieved by exploiting conditions derived from the governing variational principle formulated at the continuum level. Ben Dhia [11] proposed the Arlequin method to couple different numerical models. This method was adopted by Hu et al. [12] for the linear analysis of sandwich beams modelled via 1D and 2D finite elements.

Among the multiple-model methods, there are the so-called “Multi-steps methods” in which the analysis of the critical region requires the boundary conditions (BCs) at the interface level that are extracted by the analysis on the global structure. For instance, in the global/local method proposed by Mao et al. [10], a coarse mesh was used to analyse the entire structure to obtain the nodal displacements which were subsequently used as boundary conditions for the refined local analysis. According to [10], the application of the boundary conditions in the local region unavoidably introduces errors. To minimize the effect of such errors, the local analysis generally requires a region larger than the critical region where accurate stress fields are to be evaluated. Ransom and Knight [16] presented a method for performing a global/local stress analysis. The method makes use of spline interpolation functions which satisfy the linear plate bending equation to determine displacements and rotations from a global model, which are then used as BCs for the local model. The local analysis is done in a second step and it is completely independent of the global one. This method can be used to determine detailed stress states for specific structural regions using independent, refined local models which exploit information from less-refined global models, thus reducing the computational effort.

Haryadi et al. presented a two-step global/local methodology to compute the static response of a simply supported composite plate with cutouts [17] and small cracks [18]. In these works, the Ritz method is used for the computations of the kinematic BCs of the local region and subsequently standard finite element method for the analysis of the local model. Their method resulted in accurate prediction of stresses with considerable computational cost savings. The work of Thompson et al. [19] is one of the first examples of global/local analysis from a 2D global model to 3D local one. As a first step, they realised a 2D global model of the laminate composite plate using a zooming technique to refine the mesh in the proximity of the hole to avoid the displacements interpolation in the interfaces between global/local model. In recent works [13], the Arlequin method was formulated in the context of the Carrera Unified Formulation (CUF) to couple 1D finite elements differing in the approximation order of the displacement field. The global mechanical problem was solved by merging two sub-domains via the Arlequin method. An overlapping zone was thus necessary to guarantee the structural integrity via a Lagrangian multiplier field and a coupling operator that links the degrees of freedom (DOFs) of each sub-domain within the overlapping zone. Similar results were reproduced by Carrera et al. [20] by coupling models with different kinematics by using point-wise Lagrange multipliers. The main difference between the Arlequin-based and the Lagrange multipliers-based variable kinematic models is that the former includes an overlapping region, in which two solutions coexist. Nevertheless, both methods are suitable for building variable kinematic models. Recently, CUF has been extended in [21] to deal with the global/local analysis of laminates by employing its intrinsic variable-kinematics capability.

A different approach is used in the case of the super-elements method where a large structure is divided into many small ones (super-elements) which are then processed individually. The processing of each super-element results in a reduced set of matrices that represent the properties of the super-element as seen at its connections to adjacent structures. The reduced matrices are computed using the static condensation by the Guyan’s method [14], [15] and they are assembled with the residual structure.

This work proposes a global/local methodology that consists of a two-step procedure for the evaluation of accurate stress fields in critical regions of structures. In the proposed method, the first step is devoted to the static analysis of a global model of the structure and it could be done by commercial software using 1D/2D elements. A criterion is established to identify the most critical region, which is subsequently analyzed in the second step by using high-order models, to obtain accurate stress fields. The refined theories used in the detailed analysis are implemented in the CUF framework. Over the last few years, CUF models have been demonstrated to be very efficient and effective for evaluating complex strain/stress fields of composite structures [22], [23] and also successful in the elastoplastic and progressive failure analyses [24], [25]. The main advantage of the proposed global/local methodology is that by exploiting the information of the static analysis on a global model composed 1D/2D elements, it is possible to obtain a detailed description of the stress field using high-order beam theories in the CUF framework in critical region of the structure, at reduced computational costs. Recently, the global/local methodology has been extended to the elastoplastic analysis of compact and thin-walled structures via refined models [26] and following works may address localized buckling analysis and global/local optimization processes for aerospace structure design. Fig. 1 shows some of the possible applications of the proposed global/local methodology.

This paper is structured as follows: a brief introduction of 1D models based on the CUF is given in Section 2, followed by a description of the global/local methodology in Section 3, where the application of BCs and coupling effects are also discussed. In Section 4, meaningful benchmarks are presented to assess the validity of the proposed global/local methodology. Finally, the conclusions of this work are presented in Section 5.