⚡ electrical engineering

Merging nonlinear finite element models and the floating frame of reference formulation

This paper presents a novel nonlinear finite element formulation relative to a body's floating frame that merges the floating frame of reference approach with nonlinear finite element models, enabling the inclusion of higher-order strain terms for moderate to large deformations while preserving the ability to apply model order reduction techniques.

Original authors: Marcel Ellenbroek, Jurnan Schilder, Quintus Kilsdonk

Published 2026-10-29

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CC BY 4.0

Original authors: Marcel Ellenbroek, Jurnan Schilder, Quintus Kilsdonk

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of engineering, predicting how a machine part will move and bend under stress is a constant balancing act. Engineers often rely on two different ways of thinking. One approach treats a part as a rigid object that moves through space, with only tiny, simple wiggles added on top. This is fast and efficient, perfect for systems where parts stay mostly straight. The other approach treats the part as a flexible, squishy object that can warp and twist in complex ways, capturing every subtle curve. This is accurate but computationally heavy, often requiring massive amounts of computer power to solve. The challenge has long been how to get the speed of the first method without sacrificing the accuracy of the second, especially when parts are subjected to forces that cause them to stretch or bend significantly.

Researchers at the University of Twente have found a way to bridge this gap. They developed a new method that merges these two distinct approaches, allowing engineers to simulate large, complex deformations with the efficiency usually reserved for simpler models. By rethinking how the reference point for a moving object is defined, they created a system that can handle the messy reality of large strains while still keeping the mathematical model lean enough for practical use. Their work suggests that for many real-world applications, it is possible to include the necessary complexity to describe severe bending without throwing away the speed advantages of the simpler models.

The core of this achievement lies in how the researchers defined the "frame of reference" for a moving object. In traditional simulations, engineers often fix their coordinate system to a point in empty space. As a part moves and deforms, the math must account for the fact that the part has traveled far from its starting point, which can make the equations difficult to solve. Alternatively, they can attach the coordinate system to the object itself. The standard way of doing this assumes the object only bends a little bit, keeping the math linear and simple. However, when a beam is pushed hard enough to stretch or buckle, those simple assumptions break down, and the simulation fails to predict the true behavior.

The team introduced a new way to attach this moving coordinate system. Instead of assuming the object stays nearly straight, they allowed the coordinate system to move with the object while still permitting the object to undergo large, complex strains relative to that system. They achieved this by carefully selecting a specific point on the deformed object to serve as the anchor. By ensuring that this anchor point does not experience any local elastic movement, they could mathematically separate the overall motion of the object from the complex internal stretching and twisting. This separation allowed them to keep the benefits of the moving frame—where the math stays manageable—while still capturing the high-order terms that describe large deformations.

To test this new formulation, the researchers simulated a long, thin beam clamped at one end and pushed at the other. They applied different types of forces, including a steady push, a force that changes direction as the beam bends, and a compressive force designed to cause buckling. In these simulations, they compared their new method against two established techniques: a highly detailed, computationally expensive method that tracks every twist from a fixed starting point, and the traditional, faster method that assumes small deformations.

The results showed that the new method produced the same accurate deformation patterns as the detailed, expensive method, even under extreme loads where the beam bent significantly. However, unlike the detailed method, the new approach did not require the same level of computational intensity. When the researchers analyzed the mathematical components driving the simulation, they found that in the new method, many of the complex, higher-order terms that usually complicate the equations could be ignored without losing accuracy. This was particularly true for the stiffness matrix, the part of the calculation that determines how resistant the object is to further bending. In the traditional detailed method, these complex terms were essential, but in the new moving frame approach, the simpler terms dominated the behavior.

This finding is significant because it suggests that engineers can use a more efficient model for problems that were previously thought to require heavy-duty computation. The study indicates that for many scenarios involving large deformations, the new method can capture the necessary physics by including just a few specific higher-order terms, rather than the full complexity of the traditional approach. The researchers noted that while the method showed great promise in these simulations, the impact on numerical stability for more complex, dynamic systems remains an area for future investigation.

The work demonstrates that by redefining how we look at a moving, deforming object, we can simplify the math without losing the truth of the physics. It offers a path forward for simulating flexible machinery, from robotic arms to aerospace components, with a level of efficiency that was previously out of reach for problems involving large strains. The researchers have shown that it is possible to have the best of both worlds: the speed of a simplified model and the accuracy of a complex one, provided the reference frame is chosen with the right perspective.

Technical Summary: Merging Nonlinear Finite Element Models and the Floating Frame of Reference Formulation

Problem Statement
In flexible multibody dynamics, the Floating Frame of Reference (FFR) formulation is the standard approach for simulating systems undergoing large rigid body motions but small elastic deformations. Traditionally, FFR relies on linear finite element models (FEM) defined relative to a moving frame, allowing for efficient model order reduction (e.g., Craig-Bampton method). However, this linear assumption becomes insufficient when bodies experience moderate to large local strains, such as in slender structures under high axial loads where stretching-bending coupling occurs.

Conversely, standard nonlinear FEM formulations (Total Lagrange or Updated Lagrange) accurately capture large strains by including higher-order terms in the strain definition. Yet, these methods often result in computationally expensive models that are difficult to reduce, as they typically use absolute nodal coordinates without a moving reference frame to keep elastic displacements small. The challenge lies in combining the computational efficiency and reduction capabilities of FFR with the geometric nonlinearity handling of nonlinear FEM.

Methodology
The authors propose a unified framework that merges nonlinear continuum mechanics with the floating frame of reference formulation. The methodology proceeds through the following steps:

Reformulation of Equilibrium: Starting from the standard equations of equilibrium in nonlinear continuum mechanics (using Green-Lagrange strain and second Piola-Kirchhoff stress), the authors reformulate the problem relative to an undeformed body that is allowed to undergo arbitrary rigid body motion (the "floating frame").
Discretization: The equations are discretized using the nonlinear finite element method. Crucially, the displacement field is defined relative to this moving undeformed body. This results in expressions for internal elastic loads and the tangent stiffness matrix that retain all higher-order terms in the local displacement field.
Coordinate Transformation: To bridge the gap between the nonlinear FEM (which uses absolute nodal coordinates) and the FFR (which uses absolute floating frame coordinates and local elastic coordinates), the authors utilize coordinate transformations derived in their previous work. These transformations express incremental changes in the floating frame coordinates and local elastic deformations in terms of incremental changes in absolute interface coordinates.
Derivation of Merged Formulation: By substituting these transformations into the nonlinear FEM expressions, a new "nonlinear floating frame" (NFF) formulation is obtained. This formulation expresses the internal loads and tangent stiffness matrix in the inertial frame but retains the structure of the FFR.
- The tangent stiffness matrix is decomposed into components: a linear part ( $K_I$ ), and higher-order parts ( $K_{II}, K_{III}, K_{IV}, K_V$ ) representing geometric nonlinearities.
- Additional terms ( $K_2, K_3$ ) arise from the coordinate transformation between the floating frame and absolute coordinates, accounting for stiffening effects due to internal loads.

Key Contributions

Unified Derivation: The paper demonstrates that the traditional FFR formulation can be derived directly as a simplification of nonlinear continuum mechanics by neglecting higher-order terms, rather than assuming linear mechanics a priori.
Explicit Higher-Order Terms: The work identifies the exact mathematical forms of the higher-order terms ignored in traditional FFR. This allows for the systematic inclusion of specific nonlinear terms (e.g., for stretching-bending coupling) into FFR models without abandoning model order reduction techniques.
Incremental Solution Strategy: The authors provide explicit expressions for the internal elastic loads and the tangent stiffness matrix suitable for incremental solution procedures (e.g., Newton-Raphson) in both static and dynamic simulations.
Computational Efficiency: Unlike the Updated Lagrange method, which requires updating the mesh configuration, the proposed NFF method calculates higher-order terms using the same initial mesh, offering a computationally attractive alternative for large deformation problems.

Results
The proposed method was verified using a benchmark problem: a clamped cantilever beam subjected to three loading cases—large vertical tip force, follower force, and compressive buckling force. The results were compared against:

Total Lagrange nonlinear FEM (NFE/2D).
The new nonlinear floating frame formulation (NFF/2D).
Traditional linear floating frame formulation (LFF/2D).
Commercial software (MSC/Nastran SOL400).

Findings:

Accuracy: The NFF formulation produced results identical to the Total Lagrange NFE method for all load cases, confirming that the merged formulation correctly captures large strains and geometric nonlinearities.
Term Significance: Analysis of the Frobenius norms of the stiffness matrix and internal load vector terms revealed that:
- In the Total Lagrange method (NFE), higher-order terms in both the internal load vector and stiffness matrix are critical, especially near the clamped end and during buckling.
- In the NFF formulation, the contribution of higher-order terms is significantly reduced. For the stiffness matrix, the linear term ( $K_I$ ) often dominates, and higher-order terms can frequently be neglected, particularly when using a modified Newton-Raphson scheme.
- The internal load vector in NFF still requires some higher-order terms to compensate for large geometric deformations, but less so than in the absolute coordinate formulation.
Stability: The Total Lagrange method exhibited instability at lower loads compared to the floating frame methods, suggesting that the moving frame formulation provides better numerical stability for large deformations.

Significance
The paper claims that this work enables the inclusion of carefully selected higher-order terms in the floating frame formulation for specific applications where linear assumptions fail. By merging the two approaches, the method allows for:

Model Order Reduction: Retaining the ability to use reduction techniques (like Craig-Bampton modes) even in the presence of moderate to large local strains.
Systematic Nonlinearity: Providing a clear path to add relevant nonlinear terms to standard FFR models, rather than relying on ad-hoc methods.
Efficiency: Offering a formulation that is computationally more attractive than standard Updated Lagrange methods for large deformation problems, as it avoids mesh updates while maintaining accuracy.

The authors conclude that the nonlinear floating frame method is a promising formulation for solving problems involving large deformations, balancing the accuracy of nonlinear FEM with the efficiency of multibody dynamics formulations.

Technical Summary: Merging Nonlinear Finite Element Models and the Floating Frame of Reference Formulation

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