An Energy–Momentum Conserving Algorithm for Co-Rotational Quadrilateral Shell Elements in Nonlinear Multibody Dynamics

Abstract

A new computational framework for nonlinear dynamic analysis of smooth shell structures is presented in this paper. The new framework is based on Simo & Tarnow’s energy–momentum conservation algorithm. A novel co-rotational nine-node quadrilateral shell element is embedded in the new framework. The dynamic equilibrium differential equations are derived using the Hamilton principle and solved by the Newmark algorithm. At each step, midpoint interpolation is applied to both nodal variables and their time derivatives. The average value of strains at the beginning and the end of each step is used to evaluate strain energy to obtain a symmetric tangent stiffness matrix. When deriving the kinetic energy functional, the first-order derivatives of vectorial rotational variables are embedded into equivalent nodal forces. Therefore, a symmetric equivalent mass matrix is generated. The symmetric stiffness and mass matrices significantly reduce the workload in solving the nonlinear governing equations. Benchmark validations reveal close agreement with results in the existing literature. The proposed algorithm is applicable for solving smooth shell structures undergoing large displacements and rotations within spatial domains, while maintaining unconditional stability and geometric exactness.

1. Introduction

Multibody system dynamics are widely used in aerospace and mechanical engineering, as they can predict the dynamic response of structures and their components which undergo large deformations during long-period simulation times. In contrast to the conventional rigid-body dynamics, which neglect elastic deformation, multibody system dynamics focus on developing nonlinear transient solutions, in which large rigid-body motion and elastic deformation occur simultaneously. Thus, it can predict system dynamics with unprecedented accuracy [1].

Geometric nonlinearity needs to be addressed in solving transient problems of flexible structures with finite rotations and large displacements. Besides rigid body motion, elastic deformation often appears during analysis. An accurate measurement on the strain is required as strain energy produced by the elastic deformation is usually not negligible. Therefore, the partition of total deformation into elastic deformation and rigid body motion needs to be studied. Due to the complexity of the nature of the problem, together with the various complex shapes of structures, analytical solutions to flexible multibody dynamics are usually not available. Application engineers must use numerical methods to explore reliable solutions. The combination of the Finite Element Method (FEM) with time integration algorithms is the predominant solution [2].

When using FEM to solve a flexible multibody system, kinematic equations of flexible bodies and formulation of global dynamic equations are the two challenges [3]. Three approaches to model flexible bodies can be identified according to the selection of the coordinate system: floating frame formulation, inertial frame formulation, and co-rotational frame formulation. Despite the equivalence in the dynamic governing equation derivation, the different selections of coordinate systems show remarkable differences in computational performance.

In this paper, the co-rotational method is adopted. The success of the co-rotational method lies in the explicit incorporation of geometric nonlinearity stemming from rigid-body motions. Compared to the other two approaches, the co-rotational method sets up a local coordinate system in each element at the beginning of analysis, and they are consistently updated during the analysis. The local coordinate system needs a proper selection so that the element deformation at each step is decomposed into a rigid-body motion and an elastic deformation. The elimination of rigid-body motion is achieved by a transformation matrix. The transformation matrix is formulated to connect the local and the global coordinate systems. When constructing element tangent stiffness matrices and internal force vectors, the co-rotational method naturally eliminates the parasitic coupling effects induced by rigid-body motion through location and orientation updates of the local system. Thus, a balance between computational efficiency and numerical accuracy is achieved when formulating the element kinematic equations.

In contrast to static problems, in the study of flexible multibody systems, a robust time stepping algorithm is demanded to accurately simulate the transient response of the flexible bodies. The generalized-$\alpha$ method and high-order backward differentiation formulas [4] are widely used as time discretization schemes due to their robustness and excellent performance in linear transient analyses. However, these algorithms exhibit numerical instability in nonlinear problems [5].

Energy conservation is a criterion in stability assessment for nonlinear time-history integration methods. Following this principle, researchers have pioneered approaches to derive governing equations based on energy-conserving principles since the 1960s. In 1978, Hughes et al. [6] developed the Constraint Energy Method by incorporating Lagrange multipliers into the trapezoidal rule, yet this algorithm fails to preserve momentum conservation. To resolve this limitation, Simo and coworkers [7] introduced an energy–momentum conserving algorithm, which ensures exact conservation of both energy and momentum within the Newmark algorithm framework by incorporating temporal averaging of stress tensors to define midpoint stress states. This advancement not only demonstrates unconditional stability in strongly nonlinear problems but also reveals design principles for conservative algorithms addressing geometric nonlinearity and establishes a theoretical foundation for further development of higher-order conservation algorithms.

In 2019, Zhang H M et al. took energy conservation as a constraint condition and selected more than two interpolation points in the calculation of internal forces to im-prove calculation accuracy [8]. Their computational framework is similar to Kuhl’s generalized energy–momentum conservation algorithm. Zhang R et al. took the geometrically exact shell model as the basis [9], introduced Simo’s energy–momentum conserving algorithm, and successfully calculated the mechanical responses of shell structures under dynamic loads.

Recent research in flexible multibody system dynamics has shown enhancements to classical numerical algorithms, specifically addressing the geometric nonlinearities in governing dynamic equations. Among them, heterogenous asynchronous time integrators [10], composite time integration algorithms [11], and symplectic structure-preserving Hamilton algorithms [12] are the most successful ones. However, progressive stability deterioration in long-duration simulations of dynamic systems with geometric nonlinearity is still a challenge in flexible multibody system dynamics. The balance between computational accuracy and energy dissipation needs to be explored.

To address the challenge of progressive stability deterioration, this study develops a numerical framework for large displacement dynamic analysis of thin-wall structures, which integrates a novel co-rotational nine-node quadrilateral element and the energy–momentum conserving algorithm. The proposed framework exhibits the following key features:

(1)

The formulation achieves symmetric tangent stiffness matrices in both global and local coordinate systems by introducing vectorial rotational variables, which ensures direct additivity and updates [13].

(2)

The nonlinear equation solving is significantly simplified by an incremental iterative algorithm, where nodal variable increments are directly superimposed onto the accumulated variables through algebraic summation operations. Therefore, complex iterative updates typically involved in conventional incremental approaches are avoided.

(3)

Extending Simo’s energy–momentum conservation algorithm, a midpoint interpolation strategy is rigorously implemented for all strain measures in the strain energy functional. Conservation of energy and momentum is achieved with symmetric element tangent stiffness matrices.

(4)

The algorithm incorporates two parameters—spectral radius ${\rho }_{\infty }$ and dissipation coefficient $\xi$—to control energy dissipation in generalized midpoint variables and their time derivatives. It prevents high-frequency energy oscillations while maintaining numerical stability through energy dissipation controls.

2. Formulation of Novel Co-Rotational Quadrilateral Elements

A nine-node quadrilateral shell element is presented in this section, with three coordinate systems defined at its mid-surface [13]: the global coordinate system $\left(X,Y,Z\right)$, the local coordinate system $\left(x,y,z\right)$, and the natural coordinate system $\left(\xi ,\eta ,\zeta \right)$. In both the global and local coordinate systems, the element nodal variables $u_G$ and $u_L$ incorporate translational displacements and vectorial rotational variables, employing 45 degrees of freedom at each element:

$$u_G^T=〈d_1^T\hspace{1em}{n}_{g1}^T\hspace{1em}\dots \hspace{1em}{d}_i^T\hspace{1em}{n}_{gi}^T\hspace{1em}\dots \hspace{1em}{d}_9^T\hspace{1em}{n}_{g9}^T〉$$

(1)

$$u_L^T=〈t_1^T\hspace{1em}{\theta }_1^T\hspace{1em}\dots \hspace{1em}{t}_i^T\hspace{1em}{\theta }_i^T\hspace{1em}\dots \hspace{1em}{t}_9^T\hspace{1em}{\theta }_9^T〉$$

(2)

where $d_i^T=〈\begin{array}{ccc}{U}_i& V_i& W_i\end{array}〉$ and $t_i^T=〈\begin{array}{ccc}{u}_i& v_i& w_i\end{array}〉$ represent the translational displacements vector of node i in the global and local coordinate systems, respectively; $n_{gi}^T=〈\begin{array}{cc}{p}_{i,n}& p_{i,m}\end{array}〉$ and ${\theta }_i^T=〈\begin{array}{cc}{r}_{ix}& r_{iy}\end{array}〉$ represent the vectorial rotational variables at node i of the element in the global and local coordinate systems, respectively; $p_{i,n}$ and $p_{i,m}$ denote the two minor components of the nodal normal vector $p_i$ of the shell mid-surface along the global coordinate axis; and $r_{ix}$ and $r_{iy}$ denote the components of $p_i$ along the x-axis and y-axis of the local coordinate system, respectively. The specific meanings of all symbols appearing in this paper can be found in the Nomenclature section.

Figure 1 illustrates the co-rotational frame of the quadrilateral element. In the initial configuration, the local coordinate system of the element is defined with its origin at the diagonal intersection point, where the initial triad vectors are derived from the unit vectors of the diagonals:

$$\left\{\begin{array}{l}{e}_{x0}={\displaystyle \frac{e_{130}-e_{240}}{\left|e_{130}-e_{240}\right|}}\hfill \\ e_{y0}={\displaystyle \frac{e_{130}+e_{240}}{\left|e_{130}+e_{240}\right|}}\hfill \\ e_{z0}=e_{x0}\times e_{y0}\hfill \end{array}\right.$$

(3)

where $e_{130}$ and $e_{240}$ are the unit vectors of the quadrilateral diagonals:

$$\left\{\begin{array}{l}{e}_{130}={\displaystyle \frac{X_{30}-X_{10}}{\left|X_{30}-X_{10}\right|}}\hfill \\ e_{240}={\displaystyle \frac{X_{40}-X_{20}}{\left|X_{40}-X_{20}\right|}}\hfill \end{array}\right.$$

(4)

where $X_{i0}$ denotes the global coordinates of node i in the initial configuration. Upon rigid-body motion, the unit vectors of the quadrilateral diagonals in the current configuration are updated as follows:

$$\left\{\begin{array}{l}{e}_{13}={\displaystyle \frac{X_{30}-X_{10}+d_3-d_1}{\left|X_{30}-X_{10}+d_3-d_1\right|}}\hfill \\ e_{24}={\displaystyle \frac{X_{40}-X_{20}+d_4-d_2}{\left|X_{40}-X_{20}+d_4-d_2\right|}}\hfill \end{array}\right.$$

(5)

The triad vectors $\left(e_x,e_y,e_z\right)$ of the local coordinate axes in the current configuration are updated as follows:

$$\left\{\begin{array}{l}{e}_x={\displaystyle \frac{e_{13}-e_{24}}{\left|e_{13}-e_{24}\right|}}\hfill \\ e_y={\displaystyle \frac{e_{13}+e_{24}}{\left|e_{13}+e_{24}\right|}}\hfill \\ e_z=e_x\times e_y\hfill \end{array}\right.$$

(6)

For a quadrilateral shell element, the normal vector at node i in the initial configuration is computed as follows:

$${\overline{p}}_{i0}={\displaystyle \frac{\partial \left[{\displaystyle \sum _{i=1}^9{N}_i\left(\xi ,\eta \right)X_{i0}}\right]}{\partial \xi }}\times {\left.{\displaystyle \frac{\partial \left[{\displaystyle \sum _{i=1}^9{N}_i\left(\xi ,\eta \right)X_{i0}}\right]}{\partial \eta }}\right|}_{\left({\xi }_i,{\eta }_i\right)}$$

(7)

where $N_i\left(\xi ,\eta \right)$ denotes the Lagrangian interpolation functions of node i. The normal vectors of adjacent elements at shared nodes are averaged to ensure geometric continuity across element boundaries.

Coordinates of any point on the element neutral surface can be interpolated by means of shape functions from the element node coordinates:

$$x=\left\{\begin{array}{c}x\\ y\\ z\end{array}\right\}={\displaystyle \sum _{i=1}^m{N}_i{x}_{i0}}$$

(8)

Similarly, the dynamic variables of any point in the element under the local coordinate system can also be obtained by interpolation:

$$t=\left\{\begin{array}{c}u\\ v\\ w\end{array}\right\}=\left\{\begin{array}{c}{\displaystyle \sum _{i=1}^9{N}_i\left(\xi ,\eta \right)u_i}\\ {\displaystyle \sum _{i=1}^9{N}_i\left(\xi ,\eta \right)v_i}\\ {\displaystyle \sum _{i=1}^9{N}_i\left(\xi ,\eta \right)w_i}\end{array}\right\}$$

(9)

$$\left\{\begin{array}{c}{r}_x\\ r_y\end{array}\right\}=\left\{\begin{array}{c}{\displaystyle \sum _{i=1}^9{N}_i\left(\xi ,\eta \right)r_{ix}}\\ {\displaystyle \sum _{i=1}^9{N}_i\left(\xi ,\eta \right)r_{iy}}\end{array}\right\}$$

(10)

Therefore, the displacement and coordinate functions of any point P in the element can be obtained:

$$\left\{\begin{array}{l}{t}_P=\left\{\begin{array}{c}{u}_P\\ v_P\\ w_P\end{array}\right\}={\displaystyle \sum _{i=1}^9{N}_i\left(\xi ,\eta \right)\left\{\begin{array}{c}{u}_i\\ v_i\\ w_i\end{array}\right\}}+{\displaystyle \frac{1}{2}}\zeta h{\displaystyle \sum _{i=1}^9{N}_i\left(\xi ,\eta \right)\left\{\begin{array}{c}{r}_{ix}-r_{ix0}\\ r_{iy}-r_{iy0}\\ r_{iz}-r_{iz0}\end{array}\right\}}\hfill \\ x_P=\left\{\begin{array}{c}{x}_P\\ y_P\\ z_P\end{array}\right\}={\displaystyle \sum _{i=1}^9{N}_i(\xi ,\eta )\left\{\begin{array}{c}{x}_{i0}\\ y_{i0}\\ z_{i0}\end{array}\right\}+\frac{1}{2}\zeta h{\displaystyle \sum _{i=1}^9{N}_i(\xi ,\eta )\left\{\begin{array}{c}{r}_{ix0}\\ r_{iy0}\\ r_{iz0}\end{array}\right\}}}\hfill \end{array}\right.$$

(11)

where $\left\{\begin{array}{ccc}{x}_{i0}& y_{i0}& z_{i0}\end{array}\right\}$ is the local coordinate value of node i in the initial configuration; and $\left\{\begin{array}{ccc}{r}_{ix0}& r_{iy0}& r_{iz0}\end{array}\right\}$ represents the three components of the normal vector of node i in the local coordinate system under the initial configuration.

The Green–Lagrange strains are adopted to describe the elastic deformation of the shell element, partitioned into three components: membrane strain vector ${\epsilon }_m$, shear strain vector $\gamma$, and bending strain vector $z_l\chi$.

$$\epsilon ={\epsilon }_m+z_l\chi ={\epsilon }_m+{\displaystyle \frac{1}{2}}\zeta h\chi$$

(12)

where $\epsilon$ represents the membrane strain components of the element; $h$ denotes the thickness of the shell element; and $\zeta$ is the coordinate along the thickness direction in the natural coordinate system.

The potential energy functional of the elemental is expressed as follows:

$$\Pi ={\displaystyle \frac{1}{2}}{\displaystyle \underset{V}{\int }{\left({\epsilon }_m+z_l\chi \right)}^T{D}_1\left({\epsilon }_m+z_l\chi \right)dV+\frac{1}{2}{\displaystyle \underset{V}{\int }{\gamma }^T{D}_2\gamma dV-W_{ext}}}$$

(13)

where $D_1$ and $D_2$ are the elastic-moduli matrices, and $W_{ext}$ represents the external forces.

The local internal force vector of the element is derived through the first derivatives of the strain energy functional with respect to nodal variable $u_L$ under the stationary condition, expressed as follows:

$$\delta \Pi ={\displaystyle \underset{V}{\int }{\left(z_l\chi +{\epsilon }_m\right)}^T{D}_1\left(z_l{B}_b+B_m\right)\delta u_{L}dV+}{\displaystyle \underset{V}{\int }{\gamma }^T{D}_2{B}_{\gamma }\delta u_{L}dV}-f_{ext}^T\delta u_L=0$$

(14)

The element tangent stiffness matrix is derived by taking the partial derivative of the element internal force vector again:

$$k_T={\displaystyle \underset{V}{\int }\left[B_m^T{D}_1{B}_m+{\left(z_l\right)}^2{B}_b^T{D}_1{B}_b+\frac{\partial B_m^T}{\partial u_L^T}{D}_1{\epsilon }_m+B_{\gamma }^T{D}_2{B}_{\gamma }\right]dV}$$

(15)

where $B_m$, $z_l{B}_b$, and $B_{\gamma }$ denote the first derivatives of strain vector ${\epsilon }_m$, $z_l\chi$ and $\gamma$ with respect to the nodal variable $u_L$, respectively. Since the term ${\displaystyle \frac{\partial B_m^T}{\partial u_L^T}}{D}_1{\epsilon }_m$ can be regarded as the derivative of a scalar with respect to nodal variables, and the interchange of derivatives order leaves the result invariant, the resulting tangent stiffness matrix of the element exhibits symmetry.

The derivation process of the element formulation is well established, and the detailed calculation procedures and content can be found in Reference [13].

3. Solution Formulation Based on Energy–Momentum Conservation Algorithm

We define $H$ as the total energy of the system. According to the work–energy principle, it is defined as follows:

$$H=K+V_i+V_{ext}=\mathrm{Constant}$$

(16)

Taking the derivative of the total energy of the system with respect to time yields the following:

$${\displaystyle \frac{dH}{dt}}={\displaystyle \frac{dK}{dt}}+{\displaystyle \frac{d{V}_i}{dt}}+{\displaystyle \frac{d{V}_{ext}}{dt}}=0$$

(17)

where $H$ denotes the total energy of the system; and $K$, $V_i$, and $V_{\mathrm{ext}}$ represent the kinetic energy, strain energy, and external forces of the system, respectively.

The nodal velocities of the element can be computed as follows:

$${\dot{X}}_i={\displaystyle \sum _{i=1}^9{N}_i\left[{\dot{d}}_i+\frac{1}{2}\zeta h{\dot{p}}_i\right]}$$

(18)

The kinetic energy distribution function of a quadrilateral element can be formulated by integrating the nodal kinetic energies over the element volume [13]:

$$\begin{array}{l}{K}^e={\displaystyle \underset{V}{\int }\frac{1}{2}{\dot{X}}_i^T{\dot{X}}_{i}dV}={\displaystyle \underset{V}{\int }\frac{1}{2}\rho {\left[{\displaystyle \sum _{i=1}^9{N}_i\left({\dot{d}}_i+\frac{1}{2}\zeta h{\dot{p}}_i\right)}\right]}^T\left[{\displaystyle \sum _{j=1}^9{N}_j\left({\dot{d}}_j+\frac{1}{2}\zeta h{\dot{p}}_j\right)}\right]dV}\\ \hspace{1em}\hspace{0.17em}={\displaystyle \frac{1}{2}}\rho h{\displaystyle \underset{V}{\int }{\left({\displaystyle \sum _{i=1}^9{N}_i{\dot{d}}_i}\right)}^T\left({\displaystyle \sum _{j=1}^9{N}_j{\dot{d}}_j}\right)+\frac{h^2}{12}{\left({\displaystyle \sum _{i=1}^9{N}_i{\dot{p}}_i}\right)}^T\left({\displaystyle \sum _{j=1}^9{N}_j{\dot{p}}_j}\right)}dV\end{array}$$

(19)

By taking the derivative of the kinetic energy expression with respect to time t, we obtain the following:

$${\displaystyle \frac{d{K}^e}{dt}}=\rho h{\displaystyle \underset{V}{\int }{\left({\displaystyle \sum _{i=1}^9{N}_i{\dot{d}}_i}\right)}^T\left({\displaystyle \sum _{j=1}^9{N}_j{\ddot{d}}_j}\right)+\frac{h^2}{12}{\left({\displaystyle \sum _{i=1}^9{N}_i{\dot{p}}_i}\right)}^T\left({\displaystyle \sum _{j=1}^9{N}_j{\ddot{p}}_j}\right)dV}$$

(20)

where

$${\dot{p}}_i={\displaystyle \frac{d}{dt}}\left\{\begin{array}{c}{p}_{i,X}\\ p_{i,Y}\\ p_{i,Z}\end{array}\right\}={\displaystyle \frac{\partial p_i}{\partial n_{gi}^T}}{\dot{n}}_{gi}=\left[\begin{array}{cc}{\displaystyle \frac{\partial p_{i,X}}{\partial p_{i,n}}}& {\displaystyle \frac{\partial p_{i,X}}{\partial p_{i,m}}}\\ {\displaystyle \frac{\partial p_{i,Y}}{\partial p_{i,n}}}& {\displaystyle \frac{\partial p_{i,Y}}{\partial p_{i,m}}}\\ {\displaystyle \frac{\partial p_{i,Z}}{\partial p_{i,n}}}& {\displaystyle \frac{\partial p_{i,Z}}{\partial p_{i,m}}}\end{array}\right]\left\{\begin{array}{c}{\dot{p}}_{i,n}\\ {\dot{p}}_{i,m}\end{array}\right\}$$

(21)

$${\ddot{p}}_i={\displaystyle \frac{d{\dot{p}}_i}{dt}}=\left[\begin{array}{cc}{\displaystyle \frac{\partial p_{i,X}}{\partial p_{i,n}}}& {\displaystyle \frac{\partial p_{i,X}}{\partial p_{i,m}}}\\ {\displaystyle \frac{\partial p_{i,Y}}{\partial p_{i,n}}}& {\displaystyle \frac{\partial p_{i,Y}}{\partial p_{i,m}}}\\ {\displaystyle \frac{\partial p_{i,Z}}{\partial p_{i,n}}}& {\displaystyle \frac{\partial p_{i,Z}}{\partial p_{i,m}}}\end{array}\right]\left\{\begin{array}{c}{\ddot{p}}_{i,n}\\ {\ddot{p}}_{i,m}\end{array}\right\}+\left[\begin{array}{cc}{\displaystyle \frac{{\partial }^2{p}_i}{\partial p_{i,n}^2}}& {\displaystyle \frac{{\partial }^2{p}_i}{\partial p_{i,n}\partial p_{i,m}}}\\ {\displaystyle \frac{{\partial }^2{p}_i}{\partial p_{i,n}\partial p_{i,m}}}& {\displaystyle \frac{{\partial }^2{p}_i}{\partial p_{i,m}^2}}\end{array}\right]\left\{\begin{array}{c}{\dot{p}}_{i,n}\\ {\dot{p}}_{i,m}\end{array}\right\}\otimes \left\{\begin{array}{c}{\dot{p}}_{i,n}\\ {\dot{p}}_{i,m}\end{array}\right\}$$

(22)

where $m\ne n\ne l$ and $m,n,l\in \left\{X,Y,Z\right\}$.

To facilitate the mathematical formulation, we define the following quantities:

$$\left\{\begin{array}{l}{A}_{1i}^T=\left[\begin{array}{lll}{\displaystyle \frac{\partial p_{i,X}}{\partial p_{i,n}}}\hfill & {\displaystyle \frac{\partial p_{i,Y}}{\partial p_{i,n}}}\hfill & {\displaystyle \frac{\partial p_{i,Z}}{\partial p_{i,n}}}\hfill \\ {\displaystyle \frac{\partial p_{i,X}}{\partial p_{i,m}}}\hfill & {\displaystyle \frac{\partial p_{i,Y}}{\partial p_{i,m}}}\hfill & {\displaystyle \frac{\partial p_{i,Z}}{\partial p_{i,m}}}\hfill \end{array}\right]\hfill \\ A_{2i}=\left[\begin{array}{cc}{\displaystyle \frac{{\partial }^2{p}_i}{\partial p_{i,n}^2}}& {\displaystyle \frac{{\partial }^2{p}_i}{\partial p_{i,n}\partial p_{i,m}}}\\ {\displaystyle \frac{{\partial }^2{p}_i}{\partial p_{i,n}\partial p_{i,m}}}& {\displaystyle \frac{{\partial }^2{p}_i}{\partial p_{i,m}^2}}\end{array}\right]\left\{\begin{array}{c}{\dot{p}}_{i,n}\\ {\dot{p}}_{i,m}\end{array}\right\}\otimes \left\{\begin{array}{c}{\dot{p}}_{i,n}\\ {\dot{p}}_{i,m}\end{array}\right\}\hfill \end{array}\right.$$

(23)

Substituting Equations (16)–(18) into Equation (15), extracting terms containing $A_{2i}$ to define the element equivalent load matrix $F_m^e$, and combining terms related to the global nodal variables $u_G$ to assemble the element equivalent mass matrix $M^e$, we obtain the following:

$${\displaystyle \frac{d{K}^e}{dt}}={\displaystyle \frac{d{u}_G^T}{dt}}\left(M^e{\displaystyle \frac{d^2{u}_G}{d{t}^2}}+F_m^e\right)$$

(24)

where

$$F_m^e={\displaystyle \frac{\rho h^3}{12}}\left\{\begin{array}{c}0\\ {\displaystyle {\int }_{-1}^1{\displaystyle {\int }_{-1}^1{N}_1{\displaystyle \sum _{j=1}^9{N}_j{A}_{11}^T{A}_{2j}Jd\xi d\eta }}}\\ ⋮\\ 0\\ {\displaystyle {\int }_{-1}^1{\displaystyle {\int }_{-1}^1{N}_9{\displaystyle \sum _{j=1}^9{N}_j{A}_{19}^T{A}_{2j}Jd\xi d\eta }}}\end{array}\right\}$$

(25)

$$M^e=\left[\begin{array}{cccccccc}{M}_{11}^u& 0& \dots & M_{1j}^u& 0& \dots & M_{19}^u& 0\\ 0& M_{11}^r& \dots & 0& M_{1j}^r& \dots & 0& M_{19}^r\\ ⋮& ⋮& \ddots & ⋮& ⋮& ⋰& ⋮& ⋮\\ M_{i1}^u& 0& \dots & M_{ij}^u& 0& \dots & M_{i9}^u& 0\\ 0& M_{i1}^r& \dots & 0& M_{ij}^r& \dots & 0& M_{i9}^r\\ ⋮& ⋮& ⋰& ⋮& ⋮& \ddots & ⋮& ⋮\\ M_{91}^u& 0& \dots & M_{9j}^u& 0& \dots & M_{99}^u& 0\\ 0& M_{91}^r& \dots & 0& M_{9j}^r& \dots & 0& M_{99}^r\end{array}\right]$$

(26)

$$M_{ij}^u=\rho h\left[\begin{array}{ccc}{\displaystyle \underset{A}{\int }{N}_i{N}_{j}dA}& 0& 0\\ 0& {\displaystyle \underset{A}{\int }{N}_i{N}_{j}dA}& 0\\ 0& 0& {\displaystyle \underset{A}{\int }{N}_i{N}_{j}dA}\end{array}\right]$$

(27)

$$M_{ij}^r={\displaystyle \frac{\rho h^3}{12}}{A}_{1i}^T{\displaystyle \underset{A}{\int }{N}_i{N}_{j}dA}{A}_{1j}$$

(28)

The first-order derivatives of the element strain energy and external forces with respect to time are expressed as follows:

$${\displaystyle \frac{d{V}_i^e}{dt}}={\displaystyle \frac{d{V}_i^e}{d{u}_G^T}}{\displaystyle \frac{d{u}_G}{dt}}={\displaystyle \frac{d{u}_G^T}{dt}}{F}_{\mathrm{int}}^e$$

(29)

$${\displaystyle \frac{d{V}_{ext}}{dt}}=-{\displaystyle \frac{d{u}_G^T}{dt}}{F}_{ext}$$

(30)

Substituting the kinetic energy, strain energy, and external forces of all elements in the system into Equation (12) yields the following:

$${\displaystyle \frac{d{u}_G^T}{dt}}\left(M{\displaystyle \frac{d^2{u}_G}{d{t}^2}}+F_m\right)+{\displaystyle \frac{d{u}_G^T}{dt}}{F}_{\mathrm{int}}={\displaystyle \frac{d{u}_G^T}{dt}}{F}_{ext}$$

(31)

where $M$, $F_m$ and $F_{\mathrm{int}}$ are assembled from the equivalent mass matrix $M^e$, equivalent load matrix $F_m^e$, and internal forces $F_{\mathrm{int}}^e$ of all system elements, respectively.

Within time step $\left[t_n,t_{n+1}\right]$, two generalized midpoints are formulated using the generalized energy–momentum-conserving algorithm [14]:

$$\left\{\begin{array}{l}{t}_{n+1-{\alpha }_m}=\left(1-{\alpha }_m\right)t_{n+1}+{\alpha }_m{t}_n\hfill \\ t_{n+1-{\alpha }_f}=\left(1-{\alpha }_f\right)t_{n+1}+{\alpha }_f{t}_n\hfill \end{array}\right.$$

(32)

Referring to the Newmark-$\beta$ method, generalized midpoint interpolation formulas for nodal velocities and accelerations are developed:

$$\left\{\begin{array}{l}\dot{u}\left(t_{n+1-{\alpha }_f}\right)={\displaystyle \frac{1-{\alpha }_f}{\beta \Delta \mathrm{t}}}\left[u\left(t_{n+1}\right)-u\left(t_n\right)\right]-{\displaystyle \frac{(1-{\alpha }_f)\gamma -\beta }{\beta }}\dot{u}\left(t_n\right)-{\displaystyle \frac{(\gamma -2\beta )(1-{\alpha }_f)}{2\beta }}\Delta t\ddot{u}\left(t_n\right)\hfill \\ \ddot{u}\left(t_{n+1-{\alpha }_m}\right)={\displaystyle \frac{1-{\alpha }_m}{\beta \Delta t^2}}\left[u\left(t_{n+1}\right)-u\left(t_n\right)\right]-{\displaystyle \frac{1-{\alpha }_m}{\beta \Delta t}}\dot{u}\left(t_n\right)-{\displaystyle \frac{1-{\alpha }_m-2\beta }{2\beta }}\ddot{u}\left(t_n\right)\hfill \end{array}\right.$$

(33)

where ${\alpha }_m$, ${\alpha }_f$, $\beta$ and $\gamma$ are integration parameters governed by the spectral radius ${\rho }_{\infty }$.

Applying the variational operations to both sides of the motion equilibrium equation (Equation (26)) and substituting the formulations of velocities and accelerations at generalized midpoint instant (Equation (28)), all known quantities at time $t_n$ are extracted to yield the equivalent force vector $F_a\left(t_{n+1-{\alpha }_f}\right)$:

$$F_a\left(t_{n+1-{\alpha }_f}\right)=M\left(t_{n+1-{\alpha }_f}\right)\left[{\displaystyle \frac{\left(1-{\alpha }_m\right)}{\beta \Delta t^2}}{u}_G\left(t_n\right)+{\displaystyle \frac{\left(1-{\alpha }_m\right)}{\beta \Delta t}}{\dot{u}}_G\left(t_n\right)+{\displaystyle \frac{\left(1-{\alpha }_m-2\beta \right)}{2\beta }}{\ddot{u}}_G\left(t_n\right)\right]$$

(34)

This yields a system of governing differential equations for motion equilibrium, with $u_G\left(t_{n+1}\right)$ as the primary unknown:

$$M\left(t_{n+1-{\alpha }_f}\right){\displaystyle \frac{\left(1-{\alpha }_m\right)}{\beta \Delta t^2}}{u}_G\left(t_{n+1}\right)+F_{\mathrm{int}}\left(t_{n+1-{\alpha }_f}\right)=F_{ext}\left(t_{n+1-{\alpha }_f}\right)+F_a\left(t_{n+1-{\alpha }_f}\right)-F_m\left(t_{n+1-{\alpha }_f}\right)$$

(35)

where the internal force vector $F_{\mathrm{int}}\left(t_{n+1-{\alpha }_f}\right)$ and tangent stiffness matrix $k_G\left(t_{n+1-{\alpha }_f}\right)$ at the generalized midpoint configuration in the global coordinate system are interpolated from quantities at time steps $t_n$ and $t_{n+1}$:

$$\begin{array}{l}{F}_{\mathrm{int}}\left(t_{n+1-{\alpha }_f}\right)=T^T{f}_{\mathrm{int}}\left(t_{n+1-{\alpha }_f}\right)=T^{T}h{\displaystyle {\int }_V\left[B_m^T\left(t_{n+1-{\alpha }_f}\right)D_1{\epsilon }_m\left(t_{n+1-{\alpha }_f}\right)\right.}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\left.{\displaystyle \frac{h^2}{12}}{B}_b^T\left(t_{n+1-{\alpha }_f}\right)D_1\chi \left(t_{n+1-{\alpha }_f}\right)+B_{\gamma }^T\left(t_{n+1-{\alpha }_f}\right)D_2\gamma \left(t_{n+1-{\alpha }_f}\right)\right]dV\end{array}$$

(36)

$$\begin{array}{l}{k}_G\left(t_{n+1-{\alpha }_f}\right)=h{\displaystyle {\int }_V{\left[B_m^T\left(t_{n+1-{\alpha }_f}\right)D_1{B}_m\left(t_{n+1-{\alpha }_f}\right)+\left.\frac{\partial B_m^T}{\partial u_L^T}\right|\right.}_{t_{n+1-{\alpha }_f}}{D}_1{\epsilon }_m\left(t_{n+1-{\alpha }_f}\right)}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\left.{\displaystyle \frac{h^2}{12}}{B}_b^T\left(t_{n+1-{\alpha }_f}\right)D_1{B}_b\left(t_{n+1-{\alpha }_f}\right)+B_{\gamma }^T\left(t_{n+1-{\alpha }_f}\right)D_2{B}_{\gamma }\left(t_{n+1-{\alpha }_f}\right)\right]dV\end{array}$$

(37)

The strain of the element at the generalized midpoint time instant can be interpolated by the following:

$$\left\{\begin{array}{l}{\epsilon }_m\left(t_{n+1-{\alpha }_f}\right)=\left(1-{\alpha }_f+\xi \right){\epsilon }_m\left(t_{n+1}\right)+\left({\alpha }_f-\xi \right){\epsilon }_m\left(t_n\right)\hfill \\ \gamma \left(t_{n+1-{\alpha }_f}\right)=\left(1-{\alpha }_f+\xi \right)\gamma \left(t_{n+1}\right)+\left({\alpha }_f-\xi \right)\gamma \left(t_n\right)\hfill \\ \chi \left(t_{n+1-{\alpha }_f}\right)=\left(1-{\alpha }_f+\xi \right)\chi \left(t_{n+1}\right)+\left({\alpha }_f-\xi \right)\chi \left(t_n\right)\hfill \end{array}\right.$$

(38)

where $\xi$ represents the numerical dissipation parameter. Setting $\xi =0$ disables artificial damping, ensuring a non-dissipative system.

In contrast to Simo’s energy–momentum conservation algorithm [13], this study performs interpolation calculations on both strain components adjacent to the elastic-moduli matrix in the strain energy functional formulation, ensuring that the symmetry of the tangent stiffness matrix remains unaffected before and after interpolation.

The structural configuration variables at each time step are determined via the Newton–Raphson method [15], with convergence governed by an energy-based criterion.

Considering the displacement increment, the equilibrium equation for the (k + 1)-th iteration step is reformulated as follows:

$$\begin{array}{l}{M}^k\left(t_{n+1-{\alpha }_f}\right){\displaystyle \frac{\left(1-{\alpha }_f\right)}{\beta \Delta t^2}}\left[u_G^k\left(t_{n+1}\right)+\Delta u^{k+1}\left(t_{n+1}\right)\right]+\left(1-{\alpha }_f\right)k_G^k\left(t_{n+1-{\alpha }_f}\right)\Delta u^k\left(t_{n+1}\right)\\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=F_{ext}\left(t_{n+1-{\alpha }_f}\right)+F_a^k\left(t_{n+1-{\alpha }_f}\right)-F_m^k\left(t_{n+1-{\alpha }_f}\right)-F_{\mathrm{int}}^k\left(t_{n+1-{\alpha }_f}\right)\end{array}$$

(39)

where the superscripts k and k + 1 denote the iteration steps in the time step $\left[t_n,t_{n+1}\right]$.

To simplify the presentation, the equivalent stiffness matrix ${\overline{K}}_T^k\left(t_{n+1-{\alpha }_f}\right)$ and equivalent load matrix ${\overline{P}}^k\left(t_{n+1-{\alpha }_f}\right)$ are defined as follows:

$${\overline{K}}_T^k\left(t_{n+1-{\alpha }_f}\right)=M^k\left(t_{n+1-{\alpha }_f}\right){\displaystyle \frac{\left(1-{\alpha }_f\right)}{\beta \Delta t^2}}+\left(1-{\alpha }_f\right)k_G^k\left(t_{n+1-{\alpha }_f}\right)$$

(40)

$$\begin{array}{l}{\overline{P}}^k\left(t_{n+1-{\alpha }_f}\right)=F_{ext}\left(t_{n+1-{\alpha }_f}\right)+F_a^k\left(t_{n+1-{\alpha }_f}\right)-F_m^k\left(t_{n+1-{\alpha }_f}\right)\\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-F_{\mathrm{int}}^k\left(t_{n+1-{\alpha }_f}\right)-M^k\left(t_{n+1-{\alpha }_f}\right){\displaystyle \frac{1-{\alpha }_f}{\beta \Delta t^2}}{u}_G^k\left(t_{n+1}\right)\end{array}$$

(41)

Since the equivalent mass matrix $M$ derived above is symmetric, and the global tangent stiffness matrix at the generalized midpoint time obtained through special interpolation techniques is also symmetric, the equivalent stiffness matrix calculated by Equation (35) is symmetric.

During the first iteration process of the current time step, the equation can be simplified as follows:

$${\overline{K}}_T\left(t_n\right)\Delta u^1\left(t_{n+1}\right)=\overline{P}\left(t_n\right)$$

(42)

At the end of the first iteration step, the displacements can be updated as

$$u_G^1\left(t_{n+1}\right)=u_G\left(t_n\right)+\Delta u^1\left(t_{n+1}\right)$$

(43)

Then, at the (k + 1)-th iteration, the equilibrium equation is updated to

$${\overline{K}}_T^k\left(t_{n+1-{\alpha }_f}\right)\Delta u^{k+1}\left(t_{n+1}\right)={\overline{P}}^k\left(t_{n+1-{\alpha }_f}\right)$$

(44)

After each iteration is completed, the equivalent stiffness matrix and equivalent load vector need to be updated based on the displacement changes.

After the completion of the (K + 1)-th iteration step, the work done by the residual load is $E^k$, and the ratio of $E^k$ to the residual load work $E$ in the first iteration step are calculated as follows:

$$cc={\displaystyle \frac{{{\rm E}}^k}{{\rm E}}}={\displaystyle \frac{\Delta u^{k+1}\left(t_{n+1}\right){\overline{P}}^k\left(t_{n+1-{\alpha }_f}\right)}{\Delta u^1\left(t_{n+1}\right){\overline{P}}^1\left(t_{n+1-{\alpha }_f}\right)}}$$

(45)

When $cc\le 1\times {10}^{-6}$ and $k\ge 3$, the calculation converges, terminates the iteration, and proceeds to the iteration process of the next time step; otherwise, we set $k=k+1$ to continue the iteration.

For clarity, the complete iterative calculation flow chart is presented in Figure 2.

4. Examples

Three classical benchmark cases are selected for computational analysis. All quantities in this chapter employ base SI units. All cases were calculated using programs written in FORTRAN language. Curves were plotted with Origin 95 software, and model schematics were drawn with AutoCAD 2021 and Visio 2019 software.

4.1. Flying L-Shaped Plate

The mesh discretization and loading conditions of the flying L-shaped plate are illustrated in Figure 3. The geometric and material parameters are defined as follows: thickness $\mathrm{h}=0.1 \mathrm{m}$, density $\rho =1 \mathrm{kg}/{\mathrm{m}}^3$, Young’s modulus ${\mathrm{E}=10}^6 \mathrm{Pa}$ and Poisson’s ratio $\nu =0.3$. We set the quadrilateral element size to 1 m × 1 m, resulting in a total of 30 elements. A time step size $\Delta t={10}^{-3} \mathrm{s}$ is adopted with a total simulation duration of 20 s.

Figure 4 presents the total energy time history of the flying L-shaped plate over a 20 s simulation period, while Figure 5 illustrates its linear and angular momentum histories. After the termination of external loads, the system demonstrates excellent conservation of total energy and momentum. The total energy curve shows good agreement with numerical results from Lavrenčič & Brank [16], verifying the algorithm’s energy-conserving capability.

Figure 6 illustrates the motion trajectory of the flying L-shaped plate, exhibiting substantial displacements and bending deformations within the spatial domain.

4.2. Free Large Overall Motion of a Cylindrical Panel

The curved shell, as illustrated in Figure 7, is discretized into $4\times 6$ quadrilateral meshes. The structural parameters are defined as follows: radius $\mathrm{R}=150 \mathrm{m}$, thickness $\mathrm{t}=1 \mathrm{m}$, height $\mathrm{h}=30 \mathrm{m}$, elastic modulus $\mathrm{E}=31027.5 \mathrm{Pa}$, Poisson’s ratio $\nu =0.3$, density ${\rho =10}^{-8} \mathrm{kg}/{\mathrm{m}}^3$, and curvature $\theta =0.2 \mathrm{rad}$. A time step size of $\Delta t={10}^{-5} \mathrm{s}$ is adopted for numerical integration, with a total simulation duration of 0.1 s. Nodal loads $P\left(t\right)={\left[0 1 1\right]}^{T}f\left(t\right)$ (unit: N) are applied in two orthogonal directions at the corner nodes of the curved shell.

Figure 8 provides the energy–time history of the cylindrical panel over a 0.1 s duration. In the absence of external forces, the total energy exhibits strict conservation, with the energy curve showing close agreement with those reported in references [17,18]. These results confirm the computational accuracy of the proposed methodology.

Figure 9 depicts the linear and angular momentum time histories of the cylindrical panel. The computed results exhibit excellent agreement with the benchmark data reported by Sansour et al. [17], therefore validating the accuracy of the proposed numerical scheme.

Figure 10 shows the deformation configurations of the cylindrical panel at various time instants. Significant rigid-body motion is observed in three-dimensional space, with the work by external loads predominantly converted into the system’s kinetic energy.

4.3. Pinched Hemispherical Shell

The pinched hemispherical shell with a central aperture is discretized into 256 quadrilateral finite elements, with the mesh configuration and loading conditions depicted in Figure 11. The geometric and material properties are specified as follows: radius $\mathrm{R}=10 \mathrm{m}$, top opening angle ${\theta =18}^{°}$, thickness $\mathrm{h}=0.04 \mathrm{m}$, Poisson’s ratio $\nu =0.3$, Young’s modulus $\mathrm{E}=6.825\times {10}^7 \mathrm{Pa}$, and mass density $\rho =1000 \mathrm{kg}/{\mathrm{m}}^3$. A constant time step $\Delta t=1.25\times {10}^{-4} \mathrm{s}$ is employed for dynamic simulation over a total duration of 1.25 s. External loads $P\left(t\right)$ (unit: N) are applied to the four base nodes as follows:

$$P(t)=\left\{\begin{array}{l}{\displaystyle \frac{20000}{3}}t,\hspace{1em}\hspace{1em}\hspace{1em} t<0.015 \mathrm{s}\hfill \\ 200-{\displaystyle \frac{20000}{3}}t,\hspace{1em}\hspace{1em}0.015 \mathrm{s}\le t<0.03 \mathrm{s}\hfill \\ 0,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} 0.03 \mathrm{s}\le t\hfill \end{array}\right.$$

(46)

Figure 12a presents the time history curve of system energy over a 1.25 s duration. The strain energy and kinetic energy of the hemispherical shell with a central aperture exhibit periodic variations, while the total energy remains conserved throughout the computational period. To facilitate clear observation, the data at 0.3 s are extracted for plotting, as shown in Figure 12b.

Figure 13 depicts the deformation of the apertured hemispherical shell at 0.01 s intervals. Subjected to concentrated nodal loads, the shell undergoes significant geometrically nonlinear deformation without measurable rigid-body translation or rotation. The structural vibrations exhibit distinct periodicity with a fundamental period of 0.04 s, consistent with numerical predictions by Zhang et al. [19]. Analysis of radial displacements at Nodes A and B reveals a phase difference of a half cycle, accompanied by equal vibration amplitudes.

5. Conclusions

This paper proposes an efficient framework for nonlinear dynamic analysis of smooth thin-wall structures with a co-rotational nine-node quadrilateral shell element formulation. An enhanced conservative discretization scheme is developed based on Simo’s classical energy–momentum algorithm. The governing dynamic differential equations are derived via Hamilton’s principle, the midpoint theorem and generalized variational principles. The generalized midpoint interpolation strategy is integrated with the generalized-$\alpha$ method to discretize nodal variables and their time derivatives. The average of the Green–Lagrange strains at the beginning and the end of each step is adopted to replace the midpoint configuration strains in the strain energy functional evaluation, which produces a symmetric tangent stiffness matrix. In the dynamic differential equations, the asymmetric terms of the equivalent tangent stiffness matrix are incorporated into the equivalent force vector together with the other load terms. Therefore, computational effort is saved significantly without any loss of total energy and momentum conservation during time integration.

Three test cases demonstrate that the proposed algorithm captures geometric nonlinear dynamic responses of thin shells, achieves excellent agreement with results reported in the existing literature, and exhibits superior performance in both computational accuracy and efficiency. Overall, this work establishes a valuable reference for analyzing long-duration dynamic behaviors in multibody system dynamics.