Flexel ecosystem: simulating mechanical systems from entities with arbitrarily complex mechanical responses

TL;DR

Nonlinearities and instabilities in mechanical structures challenge traditional finite element analysis. The paper introduces flexels, energy-based nonlinear elements parameterized by a geometric measure $α$ and a generalized force–displacement curve $v(α,t)$, enabling intrinsic nonlinear responses with reduced DOF. A decoupled flexel ecosystem pairs geometry and behavior to model diverse phenomena—including snapping, contact, and actuation—using arc-length continuation to trace equilibrium paths through turning points. Demonstrations include a workflow linking Bezier-fitted experimental data to flexel blocks and multiple use cases (tensegrity, tape-spring grippers, buckling beams, metafluid-actuated grippers), with good agreement to experiments and an open-source library springable for rapid adoption. This framework reduces computational cost, improves interpretability, and provides a modular path to design nonlinear mechanical structures in compliant mechanisms, soft robotics, and metamaterials.

Abstract

Nonlinearities and instabilities in mechanical structures have shown great promise for embedding advanced functionalities. However, simulating structures subject to nonlinearities can be challenging due to the complexity of their behavior, such as large shape changes, effect of pre-tension, negative stiffness and instabilities. While traditional finite element analysis is capable of simulating a specific nonlinear structure quantitatively, it can be costly and cumbersome to use due to the high number of degrees of freedom involved. We propose a framework to facilitate the exploration of highly nonlinear structures under quasistatic conditions. In our framework, models are simplified by introducing `flexels', elements capable of intrinsically representing the complex mechanical responses of compound structures. By extending the concept of nonlinear springs, flexels can be characterized by multi-valued response curves, and model various mechanical deformations, interactions and stimuli, e.g., stretching, bending, contact, pneumatic actuation, and cable-driven actuation. We demonstrate that the versatility of the formulation allows to model and simulate, with just a few elements, complex mechanical systems such as pre-stressed tensegrities, tape spring mechanisms, interaction of buckled beams and pneumatic soft gripper actuated using a metafluid. With the implementation of the framework in an easy-to-use Python library, we believe that the flexel formulation will provide a useful modeling approach for understanding and designing nonlinear mechanical structures.

Figures (5)

• Figure 1: Construction of flexels. (a-b) Von-Mises truss composed of a pair of inclined linear springs (non-dimensional stiffness of 0.6 for (a), 1.0 for (b)) driven from their connection point through a third, vertical linear spring (stiffness of 20 for (a), 0.33 for (b)). The force-displacement response of the system is either (a) non-monotonic or (b) multi-valued. (c-d) Flexel equivalents of the Von-Mises trusses shown in (a) and (b), whose mechanical behavior has been tuned to mimic their force-displacement response. (e-f) Assemblies composed of nonmonotonic and multi-valued flexels as shown in (c) and (d) loaded in series (e) or at an angle (f), exhibiting complex force-displacement curves. (g-h) Flexel equivalents of the systems shown in (e) and (f). Black (gray) lines refer to states stable (unstable) under force-controlled conditions. Solid (dashed) lines refer to states stable (unstable) under displacement-driven conditions. The full descriptions of the models are provided in SI section 7.1.
• Figure 2: Flexel ecosystem. (a) Various geometric measures $\alpha$ computed from a list of node coordinates $z_i$: length, angle, area, total length of a polygonal chain, distance point-line. (b) Various generalized force-displacement curves (linear, nonlinear, multi-valued) defining intrinsic mechanical behaviors via energy potentials $v(\alpha, t)$. (c) Examples of pairs of geometric measure and behavior forming the flexel ecosystem. The energy of a flexel is given by $v(\alpha(\bm{z}), t)$. More details on the definition of the geometric measures $\alpha$ and energy potentials $v$ are provided in SI section 3 and 4.
• Figure 3: Examples of assemblies of flexels (top) that collectively produce force-displacement responses $F(U)$ (bottom). Black (gray) lines refer to states stable (unstable) under force-controlled conditions. Solid (dashed) lines refer to states stable (unstable) under displacement-driven conditions. Gray flexels are characterized by a linear generalized force-displacement curve. (a) An angular flexel with a multi-valued torque-angular displacement curve modeling a snapping flexure. (b) An area flexel with a softening pressure-areal displacement curve modeling pneumatic actuation. (c) A path flexel with a bilinear stiffening force-displacement curve, modeling cable-driven actuation. (d) A distance flexel with a force-displacement curve yielding a high nonzero force only for small distance values, modeling contact between a point and a line. The full descriptions of the models are provided in SI section 7.2.
• Figure 4: Workflow to simulate an experimental assembly of nonlinear building blocks ducarme_exotic_2025. (a) Two building blocks fabricated in silicone rubber. (b) Experimental force-displacement curves obtained by performing a tensile test on the building blocks. (c) Left: Bezier curves (solid green lines) fitting the experimental force-displacement curves (orange lines). The control polygons and the control points are depicted by the black dashed lines and the black dots. Right: Equivalent flexels with tensile behaviors defined by Bezier curves, specified by a string of text listing the coordinates of the control points (green text). (d) Flexel model describing the assembly of the serially-coupled building blocks (Right) and the input text file representing the model (Left). (e) Force-displacement curve of the serially-coupled assembly. Experimental data obtained during the loading (unloading) phase is depicted by light (bright) orange dots. Simulated data is depicted by black and gray curves, where black (gray) refers to states stable (unstable) under force-controlled conditions, and solid (dashed) curves refer to states stable (unstable) under displacement-driven conditions. Force and displacement are measured with respect to the preloaded configuration. The full descriptions of the models are provided in SI section 7.3.
• Figure 5: Examples of use cases. (a) Left: Tensegrity structures without prestress (top) and with prestress (bottom) subject to an external loading. Flexels in compression, at rest and in tension are colored in blue, gray and red respectively. Right: Force-displacement curves of the tensegrity structures with and without prestress. (b) Left: Model of a tape-spring gripper he_grasping_2025 loaded in two steps. The first one applies compression to the tape spring, eventually triggering buckling and creating a kink (top). The second deforms the buckled tape spring by driving the kink (bottom). Right: Force-displacement curves corresponding to the first (top) and second (bottom) loadsteps. (c) Left: model of a pair of buckling beams separated by a distance $d$, loaded in compression and interacting through the contact of their middle node kwakernaak_counting_2023guerra_selfordering_2023kwakernaak_collective_2024. Center: deformation sequence of the beams when $d=0.25$ (top) and $d=0.30$ (bottom). Right: Deformation paths when $d=0.25$ (top) and $d=0.30$ (bottom). $\bar{x}:=(x_1+x_2)/2$. (d) Top: Model of a gripper actuated using a metafluid djellouli_shell_2024 to grasp an object. Bottom: Intrinsic behavior of the area flexel modeling the metafluid (left) and deformation path of the system (right), shown as the gripping force $f_\text{grip}$ as a function of the applied displacement $U$. The full descriptions of the models are provided in SI section 7.4.