Introduction

Origami structures, as deployable structures and metamaterials, have been widely applied in numerous engineering fields, including aerospace, architecture, medicine, and robotics. However, traditional design methods rely on experience-driven forward design or iteration-driven inverse design. When faced with complex design objectives and constraints, the inverse design process is computationally expensive and difficult to generalize. How to overcome the computational cost bottleneck of iterative optimization and achieve direct inverse design without the need for an external optimizer is a critical issue that urgently needs to be addressed in the field of origami engineering.

Recently, Professor Chen Yan’s group from the School of Mechanical Engineering at Tianjin University proposed a direct inverse design method for origami structures called Ori-DID (Origami Direct Inverse Design). Based on the dynamic relaxation method, this approach maintains geometric compatibility among the panels of an origami structure through a bar length coupling mechanism. It describes objectives such as developability, flat-foldability, and complex surface fitting using equivalent external forces, and establishes a contact force model covering all collision scenarios to ensure generated structures are interference-free. Consequently, the design optimization problem is fully translated into a dynamic relaxation problem. Without the need for iterative calculations, the simultaneous design of the origami structure’s dimensions and configuration can be completed in a single forward process, as shown in Figure 1. Typical designs can be completed on a personal computer within minutes, significantly improving computational efficiency compared to traditional iterative optimization methods.

Figure 1 Comparison of traditional inverse design methods and the direct inverse design method. (a) Traditional methods rely on iterative external optimization and is computationally expensive. (b) The direct inverse design method is completed in a single forward computation using dynamic relaxation, making it highly efficient.

Key Methods

The Ori-DID software is based on a bar-hinge model and enables the unified handling of both rigid-foldable and non-rigid origami structures. For both forward simulation and inverse form-finding of structures, it utilizes an implicit Euler solver to achieve unconditional numerical stability and accurately captures highly nonlinear behaviors such as collisions and discontinuities. To address contact issues between non-adjacent triangular panels, a comprehensive contact force modeling scheme was developed, covering three typical collision scenarios: edge-edge, point-face, and point-edge. Finally, based on these methods, a dynamic relaxation method incorporating the bar length coupling mechanism was developed.

As shown in Figure 2, the bar length coupling mechanism is at the core of multi-state design: by setting the equilibrium length of the members in multiple target states (such as the unfolded and folded states) to the average of the instantaneous lengths in each state, and by applying corresponding equivalent forces that describe the design objectives and constraints, the structure naturally converges to the optimal configuration that satisfies the design objectives for each state during its dynamic evolution, thereby avoiding excessive computational costs.

Figure 2: Computational workflow for direct inverse design based on dynamic relaxation. (a) Computational scheme incorporating bar length coupling. (b–c) Normal forces on surfaces describe the surface fitting objectives for parametric and rotational surfaces.

Example Designs

This research demonstrates the method’s capabilities in both dimensional inverse design and configuration inverse design. As shown in Figure 3, the design based on the Waterbomb origami achieved simultaneous optimization of the cylindrical unfolded state and the convex folded state. Comprehensive contact force modeling ensured that all panels in the final design may admit contact, but do not have physical interference. As shown in Figure 4, by adjusting the target surface constraints, the same initial configuration can converge to a cylindrical folded state and a concave unfolded state, and evolve into a twisted folded configuration to satisfy geometric coordination constraints. This configuration innovation is difficult to achieve using traditional dimensional synthesis methods.br />

Figure 3. Dimensional synthesis of the Waterbomb origami structure: cylindrical unfolded state and convex folded state. (a) Evolution of the unfolded state. (b) Evolution of the folded state. (c) Evolution of relative bar length differences and total strain energy. (d) Schematic of constraint application. (e) Panel interference when contact modelling is not included.

Figure 4. Configuration synthesis of the Waterbomb origami structure: cylindrical folded state and concave unfolded state, forming a twisted folded configuration. (a) Evolution of the unfolded state. (b) Evolution of the folded state. (c) Differences in relative bar lengths and evolution of strain energy.

An even greater challenge lies in the design of multi-configuration morphing origami structures. As shown in Figure 5, a four-configuration morphing structure (planar–spherical–saddle–cylindrical) based on a triangular Resch origami pattern enables a single pattern to freely morph between the four target surfaces.

Figure 5. Multi-configuration morphing Resch origami design. (a) Target surfaces. (b) Design results after dynamic relaxation. (c) Overlay of the crease pattern design results and the initial pattern. (d) Differences in relative bar lengths and the evolution of strain energy. (e) Paper physical prototype. (f) Deviation between the 3D-scanned prototype and the ideal design.

The paper detailing this research was published in 2026 in the International Journal of Mechanical Sciences, a top journal in mechanical engineering. The corresponding authors are Professors Chen Yan and Ma Jiayao of Tianjin University. The first author is Wang Tianshu, a postdoctoral researcher at Tianjin University, and the co-authors include students Su Zhitao and Liu Wenli. The direct inverse design method proposed in this study breaks the reliance of origami structure design on iterative optimization. Through key techniques such as the equivalent force representation of design objectives and constraints, a bar length coupling mechanism, and comprehensive contact modeling, the study achieved a unified direct inverse design method for both rigid-foldable and non-rigid origami structures. The research findings provide a new tool for the rapid design of origami structures in fields such as deployable aerospace structures, medical devices, and metamaterials. Please find the full paper at https://authors.elsevier.com/a/1nb444jpxmw6I. The Ori-DID software is now open source, please click here to download.

Tianshu Wang, Zhitao Su, Wenli Liu, Jiayao Ma*, Yan Chen*. Ori-DID: a Direct Inverse Design Method for Origami Structures. International Journal of Mechanical Sciences. 2026.
(https://doi.org/10.1016/j.ijmecsci.2026.111864)