The Kinematics and Dynamics Theories of a Total Lagrangian Finite Element Analysis Framework for Finite Deformation Multibody Dynamics

This work presents a Total Lagrangian finite element formulation for deformable body dynamics. We employ the TL-FEA framework to simulate the time evolution of collections of bodies whose motion is constrained by kinematic constraints and which mutually interact through contact and friction. These bodies experience large displacements, large deformations, and large rotations. A systematic approach is proposed for classifying and posing kinematic constraints acting between the bodies present in the system. We derive the governing equations for ANCF beam, ANCF shell, and tetrahedral elements, and present hyperelastic material models including St. Venant-Kirchhoff and Mooney-Rivlin formulations with their corresponding internal force contributions and consistent tangent stiffness matrices. A finite-strain Kelvin-Voigt viscous damping model is incorporated in the TL-FEA formulation for numerical stability.

💡 Research Summary

The paper introduces a comprehensive Total Lagrangian (TL) finite‑element framework for simulating deformable bodies undergoing large displacements, rotations, and strains, and for handling collections of such bodies that interact through kinematic constraints, contact, and friction. The authors adopt a deformation‑gradient‑centric view: all strain measures, stresses, and constitutive relations are expressed with respect to the reference configuration, and the deformation gradient F = ∂r/∂X becomes the primary kinematic variable.

A key methodological shift is the re‑formulation of the Absolute Nodal Coordinate Formulation (ANCF). Instead of the classic r(X,t)=S(X) e(t) representation, the position field is written as r(X,t)=N(t) s(X), where s(X) contains scalar shape functions and N(t) stores the nodal vectors. This yields an explicit factorization F = N(t) H(X) with H = ∂s/∂X, allowing beam, shell, and solid elements to share a unified algebraic structure.

Three element families are derived: (1) the 2‑node 3243 ANCF beam element (12 DOF per node, fully parameterized with position and three spatial gradients), (2) the 4‑node 3443 ANCF shell element (also 12 DOF per node, high‑order shape functions for bending and shear), and (3) a 10‑node quadratic tetrahedral solid (T10) using barycentric coordinates and quadratic Lagrange shape functions. For each element the internal force vector and consistent tangent stiffness matrix are expressed directly in terms of F and its variations, guaranteeing a consistent linearization across all element types.

Material models include the linear St‑Venant‑Kirchhoff hyperelasticity and the nonlinear incompressible Mooney‑Rivlin formulation. Both models provide internal forces f_int = ∂Ψ/∂F · Hᵀ and a consistent tangent K_tan = ∂²Ψ/∂F² · (H Hᵀ). To improve numerical stability in dynamic simulations, a finite‑strain Kelvin‑Voigt viscous damping term η · \dot{F} is added, yielding a viscous stiffness contribution that damps high‑frequency modes without compromising the energy‑consistent TL formulation.

The multibody aspect is handled through an Augmented Lagrangian Method (ALM). Position‑level constraints c(r,A,t)=0 are enforced by augmenting the Lagrangian with linear multiplier terms λᵢcᵢ and quadratic penalty terms (ρᵢ/2)‖cᵢ‖². The resulting equations of motion, derived from the principle of virtual work, take the form

M \ddot{q} + C \dot{q} + K q + G = F_ext + Jᵀλ,

where J is the Jacobian of the constraints. The ALM enables simultaneous updates of the primal variables (nodal positions, velocities) and the dual variables (Lagrange multipliers) using either first‑order or second‑order optimization schemes (e.g., Newton‑Raphson). Contact and friction are incorporated as additional constraints (gap ≤ 0, Coulomb friction law) and treated within the same augmented framework, ensuring a unified solution strategy for all interactions.

Implementation details emphasize the advantage of the deformation‑gradient‑centric formulation: element‑specific code is reduced to defining the shape‑function matrix s(X) and its gradient H(X); the rest of the assembly, linearization, and constraint handling follows a common pipeline. Numerical integration uses Gaussian quadrature (five‑point rule for T10) that is exact for polynomials up to degree three, guaranteeing consistent mass and internal‑force evaluation.

Overall, the paper delivers a mathematically coherent, algorithmically unified TL‑FEA platform that integrates ANCF beam, shell, and solid elements, hyperelastic and viscous material models, and a robust augmented Lagrangian treatment of constraints, making it suitable for large‑deformation multibody dynamics involving contact, friction, and complex kinematic couplings.