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1. Modular representation and control of floppy networks

   Siheng Chen, Fabio Giardina (January 2022, Proceedings of the Royal Society of London)

   Geometric graph models of systems as diverse as proteins, robots, and mechanical structures from DNA assemblies to architected materials point towards a unified way to represent and control them in space and time. While much work has been done in the context of characterizing the behavior of these networks close to critical points associated with bond and rigidity percolation, isostaticity, etc., much less is known about floppy, under-constrained networks that are far more common in nature and technology. Here we combine geometric rigidity and algebraic sparsity to provide a framework for identifying the zero-energy floppy modes via a representation that illuminates the underlying hierarchy and modularity of the network, and thence the control of its nestedness and locality. Our framework allows us to demonstrate a range of applications of this approach that include robotic reaching tasks with motion primitives, and predicting the linear and nonlinear response of elastic networks based solely on infinitesimal rigidity and sparsity, which we test using physical experiments. Our approach is thus likely to be of use broadly in dissecting the geometrical properties of floppy networks using algebraic sparsity to optimize their function and performance. 

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2. Stress–stress correlations reveal force chains in gels

   https://doi.org/10.1063/5.0131473  

   Vinutha, H A; Diaz_Ruiz, Fabiola_Doraly; Mao, Xiaoming; Chakraborty, Bulbul; Del_Gado, Emanuela (March 2023, The Journal of Chemical Physics)

   We investigate the spatial correlations of microscopic stresses in soft particulate gels using 2D and 3D numerical simulations. We use a recently developed theoretical framework predicting the analytical form of stress–stress correlations in amorphous assemblies of athermal grains that acquire rigidity under an external load. These correlations exhibit a pinch-point singularity in Fourier space. This leads to long-range correlations and strong anisotropy in real space, which are at the origin of force-chains in granular solids. Our analysis of the model particulate gels at low particle volume fractions demonstrates that stress–stress correlations in these soft materials have characteristics very similar to those in granular solids and can be used to identify force chains. We show that the stress–stress correlations can distinguish floppy from rigid gel networks and that the intensity patterns reflect changes in shear moduli and network topology, due to the emergence of rigid structures during solidification. 

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3. Validating an algebraic approach to characterizing resonator networks

   https://doi.org/10.1038/s41598-023-50089-1  

   Horowitz, Viva R; Carter, Brittany; Hernandez, Uriel F; Scheuing, Trevor; Alemán, Benjamín J (December 2024, Scientific Reports)

   Abstract Resonator networks are ubiquitous in natural and engineered systems, such as solid-state materials, electrical circuits, quantum processors, and even neural tissue. To understand and manipulate these networks it is essential to characterize their building blocks, which include the mechanical analogs of mass, elasticity, damping, and coupling of each resonator element. While these mechanical parameters are typically obtained from response spectra using least-squares fitting, this approach requires a priori knowledge of all parameters and is susceptible to large error due to convergence to local minima. Here we validate an alternative algebraic means to characterize resonator networks with no or minimal a priori knowledge. Our approach recasts the equations of motion of the network into a linear homogeneous algebraic equation and solves the equation with a set of discrete measured network response vectors. For validation, we employ our approach on noisy simulated data from a single resonator and a coupled resonator pair, and we characterize the accuracy of the recovered parameters using high-dimension factorial simulations. Generally, we find that the error is inversely proportional to the signal-to-noise ratio, that measurements at two frequencies are sufficient to recover all parameters, and that sampling near the resonant peaks is optimal. Our simple, powerful tool will enable future efforts to ascertain network properties and control resonator networks in diverse physical domains. 

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4. Explosive rigidity percolation in kirigami

   https://doi.org/10.1098/rspa.2022.0798  

   Choi, Gary P.; Liu, Lucy; Mahadevan, L. (March 2023, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   Controlling the connectivity and rigidity of kirigami, i.e. the process of cutting paper to deploy it into an articulated system, is critical in the manifestations of kirigami in art, science and technology, as it provides the resulting metamaterial with a range of mechanical and geometric properties. Here, we combine deterministic and stochastic approaches for the control of rigidity in kirigami using the power of k choices, an approach borrowed from the statistical mechanics of explosive percolation transitions. We show that several methods for rigidifying a kirigami system by incrementally changing either the connectivity or the rigidity of individual components allow us to control the nature of the explosive transition by a choice of selection rules. Our results suggest simple lessons for the design of mechanical metamaterials. 

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5. Geometric Methods for Adjoint Systems

   https://doi.org/10.1007/s00332-023-09999-7  

   Tran, Brian Kha; Leok, Melvin (December 2023, Journal of Nonlinear Science)

   Abstract Adjoint systems are widely used to inform control, optimization, and design in systems described by ordinary differential equations or differential-algebraic equations. In this paper, we explore the geometric properties and develop methods for such adjoint systems. In particular, we utilize symplectic and presymplectic geometry to investigate the properties of adjoint systems associated with ordinary differential equations and differential-algebraic equations, respectively. We show that the adjoint variational quadratic conservation laws, which are key to adjoint sensitivity analysis, arise from (pre)symplecticity of such adjoint systems. We discuss various additional geometric properties of adjoint systems, such as symmetries and variational characterizations. For adjoint systems associated with a differential-algebraic equation, we relate the index of the differential-algebraic equation to the presymplectic constraint algorithm of Gotay et al. (J Math Phys 19(11):2388–2399, 1978). As an application of this geometric framework, we discuss how the adjoint variational quadratic conservation laws can be used to compute sensitivities of terminal or running cost functions. Furthermore, we develop structure-preserving numerical methods for such systems using Galerkin Hamiltonian variational integrators (Leok and Zhang in IMA J. Numer. Anal. 31(4):1497–1532, 2011) which admit discrete analogues of these quadratic conservation laws. We additionally show that such methods are natural, in the sense that reduction, forming the adjoint system, and discretization all commute, for suitable choices of these processes. We utilize this naturality to derive a variational error analysis result for the presymplectic variational integrator that we use to discretize the adjoint DAE system. Finally, we discuss the application of adjoint systems in the context of optimal control problems, where we prove a similar naturality result. 

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