Inverse Discrete Elastic Rod
📝 Abstract
Inverse design of slender elastic structures underlies a wide range of applications in computer graphics, flexible electronics, biomedical devices, and soft robotics. Traditional optimization-based approaches, however, are often orders of magnitude slower than forward dynamic simulations and typically impose restrictive boundary conditions. In this work, we present an inverse discrete elastic rods (inverse-DER) method that enables efficient and accurate inverse design under general loading and boundary conditions. By reformulating the inverse problem as a static equilibrium in the reference configuration, our method attains computational efficiency comparable to forward simulations while preserving high fidelity. This framework allows rapid determination of undeformed geometries for elastic fabrication structures that naturally deform into desired target shapes upon actuation or loading. We validate the approach through both physical prototypes and forward simulations, demonstrating its accuracy, robustness, and potential for real-world design applications.
💡 Analysis
Inverse design of slender elastic structures underlies a wide range of applications in computer graphics, flexible electronics, biomedical devices, and soft robotics. Traditional optimization-based approaches, however, are often orders of magnitude slower than forward dynamic simulations and typically impose restrictive boundary conditions. In this work, we present an inverse discrete elastic rods (inverse-DER) method that enables efficient and accurate inverse design under general loading and boundary conditions. By reformulating the inverse problem as a static equilibrium in the reference configuration, our method attains computational efficiency comparable to forward simulations while preserving high fidelity. This framework allows rapid determination of undeformed geometries for elastic fabrication structures that naturally deform into desired target shapes upon actuation or loading. We validate the approach through both physical prototypes and forward simulations, demonstrating its accuracy, robustness, and potential for real-world design applications.
📄 Content
Inverse design of slender elastic structures underlies a wide range of applications in computer graphics, flexible electronics, biomedical devices, and soft robotics. Traditional optimization-based approaches, however, are often orders of magnitude slower than forward dynamic simulations and typically impose restrictive boundary conditions. In this work, we present an inverse discrete elastic rods (inverse-DER) method that enables efficient and accurate inverse design under general loading and boundary conditions. By reformulating the inverse problem as a static equilibrium in the reference configuration, our method attains computational efficiency comparable to forward simulations while preserving high fidelity. This framework allows rapid determination of undeformed geometries for elastic fabrication structures that naturally deform into desired target shapes upon actuation or loading. We validate the approach through both physical prototypes and forward simulations, demonstrating its accuracy, robustness, and potential for real-world design applications.
The simulation of elastic deformation is widely utilized in computer graphics [Bertails-Descoubes et al. 2018;Chen et al. 2014;Derouet-Jourdan et al. 2013, 2010;Ly et al. 2018], engineering science [Cheng et al. 2023;Fan et al. 2020;Zhang et al. 2022], and solid mechanics [Huang et al. 2025;Li et al. 2025a,b;Liu et al. 2020;Tong et al. 2025a;Yang et al. 2024Yang et al. , 2023]], and is increasingly important in emerging fields such as 3D printing and soft robotics. Compared with three-dimensional (3D) solid structures, one-dimensional (1D) slender structures such as beams, rod, and ribbons are more versatile and flexible, thereby exhibiting large elastic deformation behavior [Audoly and Pomeau 2000;Langer and Singer 1996;O’Reilly 2017]. Although the forward problem, solving the elastic deformation with boundary conditions to get the deformed configuration (DC), has been extensively studied over the past two decades [Antman 2005;Audoly and Pomeau 2000;Bergou et al. 2010Bergou et al. , 2008;;Huang et al. 2025;Love 1944;O’Reilly 2017], the inverse design problem, which seeks the undeformed configuration (UC) that produces a desired target shape under specific external loading conditions, remains a significant challenge.
Recently, a number of studies have focused on the inverse design problem: finding the UC of a target shape [Bertails-Descoubes et al. 2018;Derouet-Jourdan et al. 2013, 2010;Li et al. 2025a,b;Ly et al. 2018;Miller et al. 2014;Tong et al. 2025a]. Current approaches for inverse design can be broadly divided into two categories: numerical optimization and theoretical solutions. Inverse design is commonly formulated as a constrained minimization problem, owing to the absence of a comprehensive theoretical framework. The objective is to minimize the discrepancy between the forward simulation result and a target shape, with applications spanning animation control of 2D curves, 3D curves [Derouet-Jourdan et al. 2013, 2010]. However, when the problem is highly nonlinear and the initial guess is far from the true solution, these optimization-based methods tend to converge slowly and require substantial computational resources [Chen et al. 2014;Derouet-Jourdan et al. 2013;Ly et al. 2018]. For example, utilizing a Lev-Mar solver for inverse design incurs a computational cost nearly two orders of magnitude higher than that of a forward simulation [Chen et al. 2014]. In contrast, theoretical solutions are generally more efficient but are limited to relatively simple cases, e.g., a single rod system. Bertail et al. established a theoretical foundation for the inverse design of gravity actuated suspended Kirchhoff rods; however, their framework is restricted to clamped-free boundary conditions [Bertails-Descoubes et al. 2018]. Building on this, a unified theory, termed inverse elastica, was later developed, incorporating the geometric equations of the UC as presented in our previous work [Li et al. 2025a]. Although theoretical frameworks for the inverse design of slender structures exist, they struggle with geometrically complex systems such as nets, gridshells, and lattice-like systems [Bertails-Descoubes et al. 2018;Derouet-Jourdan et al. 2013, 2010;Li et al. 2025a]. Consequently, a general, optimization-free numerical method that leverages a unified theory is highly desired to efficiently tackle these challenging inverse design problems.
In this paper, we introduce the inverse discrete elastic rod (inverse-DER) method, built upon the discrete elastic rod (DER) framework, to address the inverse design of slender structures. The DER method is widely adopted for simulating the large-deformation dynamics of slender bodies [Bergou et al. 2010[Bergou et al. , 2008;;Huang et al. 2025], capturing their evolution from a UC to a DC. In a similar spirit, the proposed inverse-DER method enables the direct reconstruction of the UC through an inverse dynamic evolut
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