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1. Modular representation and control of floppy networks

   https://doi.org/10.1098/rspa.2022.0082  

   Chen, Siheng; Giardina, Fabio; Choi, Gary P.; Mahadevan, L. (August 2022, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   Geometric graph models of systems as diverse as proteins, DNA assemblies, architected materials and robot swarms are useful abstract representations of these objects that also unify ways to study their properties and control them in space and time. While much work has been done in the context of characterizing the behaviour of these networks close to critical points associated with bond and rigidity percolation, isostaticity, etc., much less is known about floppy, underconstrained 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. 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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3. 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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4. Applications of Polynomial Systems

   Cox, D; D'Andrea, C; Dickenstein, A; Hauenstein, J; Schenck, H; Sidman, J (January 2020, CBMS Lecture notes 134)

   null (Ed.)

   CBMS notes from a series of 10 Lectures by David Cox, with supplemental lectures by C. D'Andrea, A. Dickenstein, J. Hauenstein, H. Schenck, J. Sidman. Systems of polynomial equations can be used to model an astonishing variety of phenomena. This book explores the geometry and algebra of such systems and includes numerous applications. The book begins with elimination theory from Newton to the twenty-first century and then discusses the interaction between algebraic geometry and numerical computations, a subject now called numerical algebraic geometry. The final three chapters discuss applications to geometric modeling, rigidity theory, and chemical reaction networks in detail. Each chapter ends with a section written by a leading expert. 

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5. Improving Performance and Scalability of Algebraic Multigrid through a Specialized MATVEC

   Majid Rasouli, Vidhi Zala (September 2018, IEEE High Performance Extreme Computing Conference)

   Algebraic Multigrid (AMG) is an extremely popular linear system solver and/or preconditioner approach for matrices obtained from the discretization of elliptic operators. However, its performance and scalability for large systems obtained from unstructured discretizations seem less consistent than for geometric multigrid (GMG). To a large extent, this is due to loss of sparsity at the coarser grids and the resulting increased cost and poor scalability of the matrix-vector multiplication. While there have been attempts to address this concern by designing sparsification algorithms, these affect the overall convergence. In this work, we focus on designing a specialized matrix-vector multiplication (matvec) that achieves high performance and scalability for a large variation in the levels of sparsity. We evaluate distributed and shared memory implementations of our matvec operator and demonstrate the improvements to its scalability and performance in AMG hierarchy and finally, we compare it with PETSc. 

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