Nonlinear shell analysis has long been a subject of extensive research [1], owing to its critical role in a wide range of engineering fields such as aerospace, civil architecture, and automotive design. Although the mathematical foundations of shell theory were laid well before the advent of modern computational mechanics, its notoriously intricate formulations have historically posed considerable challenges for numerical implementation [2]. A major breakthrough came in 1989 when Simo and Fox [3] developed the geometrically exact shell theory based on Cosserat’s hypothesis [4] and Mindlin-Reissner theory [5]. This shell model only requires a mid-surface’s position vector and an inextensible director to describe the configuration, with the exclusion of drilling rotations. Since the introduction of the geometrically exact shell model, considerable efforts have been devoted to advancing its theoretical framework. Ibrahimbegović established a finite rotation-based stress resultant framework and addressed singularity issues through vector-like parameterizations [6]. Zhang et al. [7] developed a nonlocal geometrically exact shell theory for fracture modeling under finite deformations. Ma et al. [8] introduced a phase-field fracture model for stress-resultant geometrically exact shells within the finite deformation regime. Parallel to these, the development of the finite element method has been a central theme for improving numerical accuracy and mitigating locking effects. In this context, Zhang et al. [9] introduced a weak-form quadrature element method (QEM), which improves computational efficiency through direct numerical integration at collocation points. Lavrenčič et al. [10] systematically compared mixed and hybrid finite element formulations based on variational principles such as Assumed Natural Strain and Hu–Washizu functionals, aiming to identify optimal approaches for approximating shell behavior. More recently, Kim et al. [11] applied isogeometric analysis with Bézier extraction and assumed natural strain methods, further bridging geometric modeling and high-order finite element analysis.

In practical engineering, the finite element method (FEM) based on geometrically exact formulations has been extensively applied to model the nonlinear response of shells under large deformations [12]. Representative applications include structural stability in advanced materials and structures [13], the nonlinear dynamic analysis of fiber-reinforced composite shell [14], the modeling of curved thin-walled structures with deformable cross-sections [15,16], the simulation of the contact of composite laminates [17], as well as complex failure phenomena in large deformation [18]. Moreover, contemporary shell modeling is being reshaped by broader computational mechanics trends. One major trend is the incorporation of machine learning into the finite element analysis workflow, for purposes ranging from integrating nonlinear shell analysis into design frameworks [19] to merge pattern recognition with a finite element model [20]. Simultaneously, the push for higher computational efficiency has established GPU-accelerated solvers as a critical performance enabler, with demonstrated successes in stamping simulation using shell elements [21] and in explicit dynamics for thin shells [22].

Despite these advancements, practical engineering applications face notable challenges. Low-order elements are widely used but hindered by high computational costs and locking effects. To address these limitations, high-order methods have emerged. In the past decades, the hierarchical finite element method [23], meshfree method [24,25], weak form quadrature element method [26,27], iso-geometric analysis (IGA) [28,29], spectral elements [30,31] and other high-order schemes have begun to emerge for large-scale engineering analyses with high accuracy. However, high-order methods often introduce a new set of challenges, such as meshing difficulties for complex geometries and cumbersome stiffness formulations, presenting a hurdle for practical application. In this context, the Hierarchical Quadrature Element Method (HQEM) was developed to address the implementation difficulties of the hierarchical finite element method, particularly in imposing boundary conditions and assembling elements, by introducing interpolative bases based on differential quadrature nodes along element boundaries [32]. Consequently, the HQEM offers a great balance between computational efficiency and implementation practicality. Further compounding these numerical challenges is the growing need for formulations capable of handling laminated composites, which are increasingly used in lightweight and high-performance structures [33,34]. Extending existing models to laminate materials, however, introduces further complexities in constitutive integration and stress resultant computations, necessitating a mathematically concise and computationally efficient formulation for the nonlinear analysis of complex shell structures undergoing large deformations.

To meet this need, this work presents an extension of the Hierarchical Quadrature Element Method (HQEM) [32] to the nonlinear analysis of both isotropic and laminated shells. By leveraging high-order approximation [35] and hierarchical structure [32], HQEM inherently mitigates shear and membrane locking and readily enables local p-refinement. Notably, the HQEM has demonstrated versatility across various engineering applications. Its applications range from the dynamic analysis of rotor systems [36] and the multi-physics reliability assessment of conical shells [37], to thermo-mechanical fracture simulation with efficient remeshing [38,39]. The key contributions of this work are summarized as follows: (1) a unified HQEM formulation for geometrically exact isotropic and laminated shells; (2) significant simplification of residual force and stiffness matrix computations; (3) inherent mitigation of shear and membrane locking; (4) convenient support for local p-refinement; (5) compared with the weak-form quadrilateral element (QEM) approach [9], accurate capture of nonlinear behavior under high loads; and (6) flexible mesh generation through the combination of quadrilateral and triangular elements. While the current formulations are applied to linear elastic laminates, the HQEM framework can be extended to accommodate more complex scenarios, such as those involving plasticity and functionally graded materials, by introducing appropriate inelastic material models. These extensions will be explored in future work.

The organization of this paper is as follows: Section 2 introduces a brief review of geometrically exact shell theory and constitutive relations of laminated shells. Section 3 introduces the rotation description, updating of the current director with quaternions. Section 4 introduces the shape function of quadrilateral/trilateral HQEM and develops the stress resultant formulation within the HQEM framework, subsequently deriving the corresponding residual force vectors and tangent stiffness matrices. In Section 5, seven benchmark problems are analyzed to confirm the accuracy and convergence of the HQEM. Finally, Section 6 presents the conclusions of the study.

In the geometrically exact shell model, the shell’s configuration is defined by its mid-surface position and an associated unit director field. As illustrated in Fig. 1, a Cartesian reference frame {ei}(i=1,2,3) is introduced, where r denotes the mid-surface position vector, and t represents the unit director. The current and initial configurations are labeled as Ω and Ω0, respectively. The current configuration of a quadrilateral shell patch is then fully described by (1)Ω={ϕ⊂R3|ϕ=r(xα)+x3t(xα),α=1,2 and x3∈[h−,h+]}where x^𝛼 (𝛼 = 1, 2) are convected coordinates describing the midsurface of the shell, and the x³ is the thickness coordinate. The thickness of the shell is defined by h=h+−h−. For the ease of representing the deformation gradient of the shell, the current covariant frames G_i (i = 1, 2, 3) and initial covariant frames g_i (i = 1, 2, 3) are defined as (2){G1G2G3}={r,1+x3t,1r,2+x3t,2t}and (3){g1g2g3}={r0,1+x3t0,1r0,2+x3t0,2t0}where r₀ and r denote the mid-surface position vectors in the initial and current configurations, respectively; t₀ and t are the corresponding unit directors. Subscript, (α) indicates partial derivatives with respect to the coordinates x^α (α = 1, 2). The differentiation of the mid-surface in the initial configuration is defined as (4)dΩ0=j0dx1dx2in which j0=‖r01×r02‖ is the projection parameter in the initial configuration. With the definition of deformation gradient F (5)F=∇ϕ(∇ϕ)−1=Gi⊗giwhere gⁱ is the contravariant basis vector of g_i, satisfying g_i·g^j = δ_ij, with δ_ij being the Kronecker delta. The Green–Lagrange strain tensor ε can be obtained as (6)ε=12(FTF−I)=12(Gi⋅Gj−gi⋅gj)gi⊗gj=εijgi⊗gjand the strain component ε_ij can also be written as (7)εij=εij(0)+x3εij(1)+(x3)2εij(2)

Given the negligible effect of transverse shear gradients in this context [40], the quadratic terms in x³ can be omitted from Eq. (7). Meanwhile, the retained strain components align with those established by Simo et al. [41]. The explicit expressions of the strain component are (8)εαβ(0)=12(r,α⋅r,β−r,α0⋅r,β0)εα3(0)=ε3α(0)=12r,α⋅tεαβ(1)=12(r,α⋅t,β+r,β⋅t,α−r,α0⋅t,β0−r,β0⋅t,α0)

The relationship between strain ε and stress s can be expressed as: (9)s(α)=Dε(α),α=0,1in which the vector forms of strain ε measures and stress s resultants are given as (10)s=[s11s22s12s13s23]Tε=[ε11ε222ε122ε132ε23]T

A laminated composite shell is composed of a finite number of stacked thin layers with individual material properties along its thickness direction. As shown in Fig. 2, a local Cartesian material frame {eα} (α=1,2,3) is defined on the layer’s initial configuration, with its base vectors e₁ and e₂ aligned with the fiber’s longitudinal and transverse directions, respectively. For a certain single layer, the homogeneous orthotropic linear elastic constitutive relationship is assumed for material modeling. The constitutive matrix D of pth layer is thus obtained as (11)D(p)=LD¯LTwhere (12)D¯=[1/E1−υ12/E1000−υ12/E11/E2000001/G12000001/G13000001/G23]−1 (13)L=[(g11)2(g21)22g11g2100(g12)2(g22)22g12g2200g11g12g21g22g12g21+g22g1100000g11g21000g12g22]in which gαβ = e_α·g^β. E, υ and G denote the Young’s modulus, Poisson’s ratio, and shear modulus of the material. As depicted in Fig. 2, the vectors e_α and the base vectors g_α, g^α all lie in the plane normal to t₀.

Although the current formulations are applied to linear elastic laminates, the HQEM framework can be extended to accommodate more complex scenarios, such as those involving plasticity and functionally graded materials, by introducing appropriate inelastic material models. These extensions will be explored in future work.

In classical shell theory, the individual element nodal variable vector δu is defined as (14)δu=[δrTδθ1δθ2]Twhere δθ1 and δθ2 represent the two rotation components, respectively. The current director t can be obtained through quaternions [42] as (15)t=q∘e3∘q∗where q∗ is the conjugated quaternion, and e3=[001]T denotes one of the base vectors of the Cartesian coordinate. The current quaternion q can be updated from the initial quaternion q0 as (16)q=exp⁡(δθ/2)∘q0in which (17)exp⁡(δθ/2)=[cos⁡(‖δθ‖/2)sin⁡(‖δθ‖/2)⋅δθ‖δθ‖]and δθ represents the incremental rotation vector. The quaternion representation adopted here has proven to be a stable and efficient tool for algebraic operations [43], enabling a convenient and robust description of spatial rotations and the corresponding update procedures. Besides the quaternion, other rotation parameterizations, such as updating the rotation vector via the rotation tensor [44,45], are also feasible.

The rotation from the initial director to the current director can also be described by the rotation tensor Λ as (18)t=Λe3

In classical shell theory, the rotation of the director vector t is constrained to remain within its normal plane (i.e., no drilling rotation). In addition, the director orthogonal frame is denoted by {tα} (α = 1, 2, 3). The {tα} (α = 1, 2, 3) is defined as (19)tα=Λαeα(α=1,2),t3=Λe3=twhere ei (i=1,2,3) represents the inertial (fixed) basis of the Cartesian reference frame {eα} (α=1,2,3), and defined as (20)e1=[100]T,e2=[010]T,e3=[001]T

Because the drilling rotation of director t is omitted, the rotations at a point are described by only two independent variables (δθ₁ and δθ₂). Therefore, the infinitesimal incremental rotation vector δθ is constrained to be orthogonal to t and its variation δt. In this way, the infinitesimal incremental rotation vector δθ can be obtained as (21)δθ=t×δt=[t1t2][δθ1δθ2]

For the orthogonality of the rotation tensor Λ, the variation of Λ is given by introducing a rotation vector variation δθ as (22)δΛ=δθ^Λwhere ^ denotes the skew-symmetric tensor form of a vector. The variation of t thus can be obtained from Eq. (18) as (23)δt=δΛt0=−t^δθ=−t^[t1t2][δθ1δθ2]