In this paper, we study the problem of shape-programming of incompressible hyperelastic shells through differential growth. The aim of the current work is to determine the growth tensor (or growth functions) that can produce the deformation of a shell to the desired shape. First, a consistent finite-strain shell theory is introduced. The shell equation system is established from the 3D governing system through a series expansion and truncation approach. Based on the shell theory, the problem of shape-programming is studied under the stress-free assumption. For a special case in which the parametric coordinate curves generate a net of curvature lines on the target surface, the sufficient condition to ensure the vanishing of the stress components is analyzed, from which the explicit expression of the growth tensor can be derived. In the general case, we conduct the variable changes and derive the total growth tensor by considering a two-step deformation of the shell. With these obtained results, a general theoretical scheme for shape-programming of thin hyperelastic shells through differential growth is proposed. To demonstrate the feasibility and efficiency of the proposed scheme, several nature-inspired examples are studied. The derived growth tensors in these examples have also been implemented in the numerical simulations to verify their correctness and accuracy. The simulation results show that the target shapes of the shell samples can be recovered completely. The scheme for shape-programming proposed in the current work is helpful in designing and manufacturing intelligent soft devices.
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In a constructive setting, no concrete formulation of ordinal numbers can simultaneously have all the properties one might be interested in; for example, being able to calculate limits of sequences is constructively incompatible with deciding extensional equality. Using homotopy type theory as the foundational setting, we develop an abstract framework for ordinal theory and establish a collection of desirable properties and constructions. We then study and compare three concrete implementations of ordinals in homotopy type theory: first, a notation system based on Cantor normal forms (binary trees); second, a refined version of Brouwer trees (infinitely-branching trees); and third, extensional well-founded orders. Each of our three formulations has the central properties expected of ordinals, such as being equipped with an extensional and well-founded ordering as well as allowing basic arithmetic operations, but they differ with respect to what they make possible in addition. For example, for finite collections of ordinals, Cantor normal forms have decidable properties, but suprema of infinite collections cannot be computed. In contrast, extensional well-founded orders work well with infinite collections, but almost all properties are undecidable. Brouwer trees take the sweet spot in the middle by combining a restricted form of decidability with the ability to work with infinite increasing sequences. Our three approaches are connected by canonical order-preserving functions from the "more decidable" to the "less decidable" notions. We have formalised the results on Cantor normal forms and Brouwer trees in cubical Agda, while extensional well-founded orders have been studied and formalised thoroughly by Escardo and his collaborators. Finally, we compare the computational efficiency of our implementations with the results reported by Berger.
The use of high order fully implicit Runge-Kutta methods is of significant importance in the context of the numerical solution of transient partial differential equations, in particular when solving large scale problems due to fine space resolution with many millions of spatial degrees of freedom and long time intervals. In this study we consider strongly A-stable implicit Runge-Kutta methods of arbitrary order of accuracy, based on Radau quadratures, for which efficient preconditioners have been introduced. A refined spectral analysis of the corresponding matrices and matrix-sequences is presented, both in terms of localization and asymptotic global distribution of the eigenvalues. Specific expressions of the eigenvectors are also obtained. The given study fully agrees with the numerically observed spectral behavior and substantially improves the theoretical studies done in this direction so far. Concluding remarks and open problems end the current work, with specific attention to the potential generalizations of the hereby suggested general approach.
In this note, we formulate a ``one-sided'' version of Wormald's differential equation method. In the standard ``two-sided'' method, one is given a family of random variables which evolve over time and which satisfy some conditions including a tight estimate of the expected change in each variable over one time step. These estimates for the expected one-step changes suggest that the variables ought to be close to the solution of a certain system of differential equations, and the standard method concludes that this is indeed the case. We give a result for the case where instead of a tight estimate for each variable's expected one-step change, we have only an upper bound. Our proof is very simple, and is flexible enough that if we instead assume tight estimates on the variables, then we recover the conclusion of the standard differential equation method.
Optical backbone networks are required to be highly dynamic in supporting requests with flexible bandwidth granularities to cope with the demands of new broadband wireless and fixed access networks. To provide this flexibility, services are offered by taking requested bandwidth profile into consideration, instead of assigning a fixed amount of bandwidth to each request. New techniques are developed for the resource management of the elastic optical networks to realize services with a specified bandwidth profile, consisting of minimum, average, and maximum required number of spectrum slots, in addition to holding time. In this work, two new schemes are proposed to realize such services, exploiting a probabilistic spectrum partitioning approach. This new probabilistic spectrum partitioning scheme is devised to enhance the chance of accommodating requests and consequently lower request blocking probability. It enforces different probabilities to contributing spectrum partitions in a certain service realization. Taking advantage of this probabilistic spectrum partitioning and a profile-based routing, we introduce two multistage spectrum assignment methods to make a certain lightpath meet the requested service profile constraints, considering the time-weighted average of the assigned spectrum slots. The results indicate that our algorithms can successfully realize the requests with the probability of 0.993, for erlangs lower than 400.
In the last decade, parameter-free approaches to shape optimization problems have matured to a state where they provide a versatile tool for complex engineering applications. However, sensitivity distributions obtained from shape derivatives in this context cannot be directly used as a shape update in gradient-based optimization strategies. Instead, an auxiliary problem has to be solved to obtain a gradient from the sensitivity. While several choices for these auxiliary problems were investigated mathematically, the complexity of the concepts behind their derivation has often prevented their application in engineering. This work aims at an explanation of several approaches to compute shape updates from an engineering perspective. We introduce the corresponding auxiliary problems in a formal way and compare the choices by means of numerical examples. To this end, a test case and exemplary applications from computational fluid dynamics are considered.
The classical algorithms used in tabular reinforcement learning (Value Iteration and Policy Iteration) have been shown to converge linearly with a rate given by the discount factor $\gamma$ of a discounted Markov Decision Process. Recently, there has been an increased interest in the study of gradient based methods. In this work, we show that the dimension-free linear $\gamma$-rate of classical reinforcement learning algorithms can be achieved by a general family of unregularised Policy Mirror Descent (PMD) algorithms under an adaptive step-size. We also provide a matching worst-case lower-bound that demonstrates that the $\gamma$-rate is optimal for PMD methods. Our work offers a novel perspective on the convergence of PMD. We avoid the use of the performance difference lemma beyond establishing the monotonic improvement of the iterates, which leads to a simple analysis that may be of independent interest. We also extend our analysis to the inexact setting and establish the first dimension-free $\varepsilon$-optimal sample complexity for unregularised PMD under a generative model, improving upon the best-known result.
Partial Least Squares (PLS) refer to a class of dimension-reduction techniques aiming at the identification of two sets of components with maximal covariance, to model the relationship between two sets of observed variables $x\in\mathbb{R}^p$ and $y\in\mathbb{R}^q$, with $p\geq 1, q\geq 1$. Probabilistic formulations have recently been proposed for several versions of the PLS. Focusing first on the probabilistic formulation of the PLS-SVD proposed by el Bouhaddani et al., we establish that the constraints on their model parameters are too restrictive and define particular distributions for $(x,y)$, under which components with maximal covariance (solutions of PLS-SVD) are also necessarily of respective maximal variances (solutions of principal components analyses of $x$ and $y$, respectively). We propose an alternative probabilistic formulation of PLS-SVD, no longer restricted to these particular distributions. We then present numerical illustrations of the limitation of the original model of el Bouhaddani et al. We also briefly discuss similar limitations in another latent variable model for dimension-reduction.
Cricket, "a Gentleman's Game", is a prominent sport rising worldwide. Due to the rising competitiveness of the sport, players and team management have become more professional with their approach. Prior studies predicted individual performance or chose the best team but did not highlight the batter's potential. On the other hand, our research aims to evaluate a player's impact while considering his control in various circumstances. This paper seeks to understand the conundrum behind this impactful performance by determining how much control a player has over the circumstances and generating the "Effective Runs",a new measure we propose. We first gathered the fundamental cricket data from open-source datasets; however, variables like pitch, weather, and control were not readily available for all matches. As a result, we compiled our corpus data by analyzing the commentary of the match summaries. This gave us an insight into the particular game's weather and pitch conditions. Furthermore, ball-by-ball inspection from the commentary led us to determine the control of the shots played by the batter. We collected data for the entire One Day International career, up to February 2022, of 3 prominent cricket players: Rohit G Sharma, David A Warner, and Kane S Williamson. Lastly, to prepare the dataset, we encoded, scaled, and split the dataset to train and test Machine Learning Algorithms. We used Multiple Linear Regression (MLR), Polynomial Regression, Support Vector Regression (SVR), Decision Tree Regression, and Random Forest Regression on each player's data individually to train them and predict the Impact the player will have on the game. Multiple Linear Regression and Random Forest give the best predictions accuracy of 90.16 percent and 87.12 percent, respectively.
We present a convergence analysis of an unconditionally energy-stable first-order semi-discrete numerical scheme designed for a hydrodynamic Q-tensor model, the so-called Beris-Edwards system, based on the Invariant Energy Quadratization Method (IEQ). The model consists of the Navier-Stokes equations for the fluid flow, coupled to the Q-tensor gradient flow describing the liquid crystal molecule alignment. By using the Invariant Energy Quadratization Method, we obtain a linearly implicit scheme, accelerating the computational speed. However, this introduces an auxiliary variable to replace the bulk potential energy and it is a priori unclear whether the reformulated system is equivalent to the Beris-Edward system. In this work, we prove stability properties of the scheme and show its convergence to a weak solution of the coupled liquid crystal system. We also demonstrate the equivalence of the reformulated and original systems in the weak sense.
The growing energy and performance costs of deep learning have driven the community to reduce the size of neural networks by selectively pruning components. Similarly to their biological counterparts, sparse networks generalize just as well, if not better than, the original dense networks. Sparsity can reduce the memory footprint of regular networks to fit mobile devices, as well as shorten training time for ever growing networks. In this paper, we survey prior work on sparsity in deep learning and provide an extensive tutorial of sparsification for both inference and training. We describe approaches to remove and add elements of neural networks, different training strategies to achieve model sparsity, and mechanisms to exploit sparsity in practice. Our work distills ideas from more than 300 research papers and provides guidance to practitioners who wish to utilize sparsity today, as well as to researchers whose goal is to push the frontier forward. We include the necessary background on mathematical methods in sparsification, describe phenomena such as early structure adaptation, the intricate relations between sparsity and the training process, and show techniques for achieving acceleration on real hardware. We also define a metric of pruned parameter efficiency that could serve as a baseline for comparison of different sparse networks. We close by speculating on how sparsity can improve future workloads and outline major open problems in the field.
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