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Auto-ejection of liquid from a nozzle

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Physical Review E 2024年第4期109卷 045302-045302页

作者： Fang Shan Zhenhua Chai Baochang Shi School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan 430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan 430074 China Institute of Interdisciplinary Research for Mathematics and Applied Science Huazhong University of Science and Technology Wuhan 430074 China

Auto-ejection of liquid is an important process in engineering applications, and is also very complicated since it involves interface moving, deforming, and jet breaking up. In this work, a theoretical velocity of meniscus at nozzle exit is first derived, which can be used to analyze the critical condition for auto-ejection of liquid. Then a consistent and conservative axisymmetric lattice Boltzmann (LB) method is proposed to study the auto-ejection process of liquid jet from a nozzle. We test the LB model by conducting some simulations, and find that the numerical results agree well with the theoretical and experimental data. We further consider the effects of contraction ratio, length ratio, contact angle, and nozzle structure on the auto-ejection, and observe some distinct phenomena during the ejection process, including the deformation of meniscus, capillary necking, and droplet pinch off. Finally, the results reported in the present work may play an instructive role on the design of droplet ejectors and the understanding of jetting dynamics in microgravity environment.

关键词： Lattice-Boltzmann methods

GENERALIZED TRUNCATED MOMENT PROBLEMS WITH UNBOUNDED SETS

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arXiv 2022年

作者： Huang, Lei Nie, Jiawang Yuan, Ya-Xiang Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China Department of Mathematics University of California 9500 Gilman Drive La Jolla San DiegoCA92093 United States Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100049 China

This paper studies generalized truncated moment problems with unbounded sets. First, we study geometric properties of the truncated moment cone and its dual cone of nonnegative polynomials. By the technique of homogenization, we give a convergent hierarchy of Moment-SOS relaxations for approximating these cones. With them, we give a Moment-SOS method for solving generalized truncated moment problems with unbounded sets. Finitely atomic representing measures, or certificates for their nonexistence, can be obtained by the proposed method. Numerical experiments and applications are also given. Copyright © 2022, The Authors. All rights reserved.

关键词： Polynomials

Random ISAC Signals Deserve Dedicated Precoding

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arXiv 2023年

作者： Lu, Shihang Liu, Fan Dong, Fuwang Xiong, Yifeng Xu, Jie Liu, Ya-Feng Jin, Shi Southern University of Science and Technology Shenzhen518055 China The School of Information and Communication Engineering Beijing University of Posts and Telecommunications Beijing100876 China Shenzhen518172 China The State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China National Mobile Communications Research Laboratory Southeast University Nanjing210096 China

Radar systems typically employ well-designed deterministic signals for target sensing, while integrated sensing and communications (ISAC) systems have to adopt random signals to convey useful information. This paper analyzes the sensing and ISAC performance relying on random signaling in a multi-antenna system. Towards this end, we define a new sensing performance metric, namely, ergodic linear minimum mean square error (ELMMSE), which characterizes the estimation error averaged over random ISAC signals. Then, we investigate a data-dependent precoding (DDP) scheme to minimize the ELMMSE in sensing-only scenarios, which attains the optimized performance at the cost of high implementation overhead. To reduce the cost, we present an alternative data-independent precoding (DIP) scheme by stochastic gradient projection (SGP). Moreover, we shed light on the optimal structures of both sensing-only DDP and DIP precoders. As a further step, we extend the proposed DDP and DIP approaches to ISAC scenarios, which are solved via a tailored penalty-based alternating optimization algorithm. Our numerical results demonstrate that the proposed DDP and DIP methods achieve substantial performance gains over conventional ISAC signaling schemes that treat the signal sample covariance matrix as deterministic, which proves that random ISAC signals deserve dedicated precoding designs. © 2023, CC BY.

关键词： Numerical methods

POINTWISE ESTIMATES FOR THE FUNDAMENTAL SOLUTIONS OF HIGHER ORDER SCHRÖDINGER EQUATIONS IN ODD DIMENSIONS II: HIGH DIMENSIONAL CASE

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arXiv 2024年

作者： Cheng, Han Huang, Shanlin Huang, Tianxiao Zheng, Quan Institute of Applied Physics and Computational Mathematics Beijing100088 China School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Hubei Wuhan430074 China Sun Yat-sen University Guangdong Zhuhai519082 China School of Mathematics and Statistics Huazhong University of Science and Technology Hubei Wuhan430074 China

In this paper, for any odd n and any integer m ≥ 1 with n > 4m, we study the fundamental solution of the higher order Schrödinger equation i∂tu(x, t) = ((−∆)m + V(x))u(x, t), t ∈ R, x ∈ Rn, where V is a real-valued C n+21 −2m potential with certain decay. Let Pac(H) denote the projection onto the absolutely continuous spectrum space of H = (−∆)m+V, and assume that H has no positive embedded eigenvalue. Our main result says that e−itHPac(H) has integral kernel K(t, x, y) satisfying n(m−1) |K(t, x, y)| ≤ C(1 + |t|)−(2nm −σ)(1 + |t|− 2nm ) (1 + |t|− 21m |x − y|)− 2m−1 , t , 0, x, y ∈ Rn, where σ = 2 if 0 is an eigenvalue of H, and σ = 0 otherwise. A similar result for smoothing operators H 2αm e−itHPac(H) is also given. The regularity condition V ∈ C n+21 −2m is optimal in the second order case, and it also seems optimal when m > 1. © 2024, CC0.

关键词： Schrodinger equation

PARTIALLY ADJOINT DISCRETIZATIONS OF ADJOINT OPERATORS AND NONCONFORMING FINITE ELEMENT EXTERIOR CALCULUS

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arXiv 2022年

作者： Zhang, Shuo LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and System Sciences Chinese Academy of Sciences Beijing100190 China University of Chinese Academy of Sciences Beijing100049 China

This paper concerns the discretizations in pair of adjoint operators between Hilbert spaces so that the adjoint properties can be preserved. Due to the finite-dimensional essence of discretized operators, a new framework, theory of partially adjoint operators, is motivated and presented in this paper, so that adjoint properties can be figured out for finite-dimensional operators which can not be non-trivially densely defined in other background spaces. A formal methodology is presented under the framework to construct partially adjoint discretizations by a conforming discretization (CD) and an accompanied-by-conforming discretization (ABCD) for each of the operators. Moreover, the methodology leads to an asymptotic uniformity of an infinite family of finite-dimensional operators. A theory of nonconforming finite element exterior calculus (NCFEEC) is developed by constructing a systematic family of nonconforming finite element differential forms with certain structures by the theoretical framework and the formal construction of discretizations;by the NCFEEC, together with the classical FEEC, adjoint properties are reconstructed between discretizations of adjoint operators and between discretizations of variational *** Codes 47A05, 47A65, 47N40, 65J10, 65N30 Copyright © 2022, The Authors. All rights reserved.

关键词： Calculations

A hybrid FEM-PINN method for time-dependent partial differential equations

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arXiv 2024年

作者： Feng, Xiaodong Shangguan, Haojiong Tang, Tao Wan, Xiaoliang Zhou, Tao Division of Science and Technology BNU-HKBU United International College Zhuhai519087 China School of Electrical and Computer Engineering Guangzhou Nanfang College Guangzhou510970 China Department of Mathematics Center for Computation and Technology Louisiana State University Baton Rouge70803 United States Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China

In this work, we present a hybrid numerical method for solving evolution partial differential equations (PDEs) by merging the time finite element method with deep neural networks. In contrast to the conventional deep learning-based formulation where the neural network is defined on a spatiotemporal domain, our methodology utilizes finite element basis functions in the time direction where the space-dependent coefficients are defined as the output of a neural network. We then apply the Galerkin or collocation projection in the time direction to obtain a system of PDEs for the space-dependent coefficients which is approximated in the framework of PINN. The advantages of such a hybrid formulation are twofold: statistical errors are avoided for the integral in the time direction, and the neural network’s output can be regarded as a set of reduced spatial basis functions. To further alleviate the difficulties from high dimensionality and low regularity, we have developed an adaptive sampling strategy that refines the training set. More specifically, we use an explicit density model to approximate the distribution induced by the PDE residual and then augment the training set with new time-dependent random samples given by the learned density model. The effectiveness and efficiency of our proposed method have been demonstrated through a series of numerical experiments. © 2024, CC BY.

关键词： Partial differential equations

A PRIMAL FINITE ELEMENT SCHEME OF THE HODGE LAPLACE PROBLEM

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arXiv 2022年

In this paper, a unified family, for any n ≽ 2 and 1 ≼ k ≼ n − 1, of nonconforming finite element schemes are presented for the primal weak formulation of the n-dimensional Hodge-Laplace equation on (Equation presented) and on the simplicial subdivisions of the domain. The finite element scheme possesses an O(h)-order convergence rate for sufficiently regular data, and an O(hs)order rate on any s-regular domain, 0 Copyright © 2022, The Authors. All rights reserved.

关键词： Finite element method

LOW-DEGREE ROBUST HELLINGER-REISSNER FINITE ELEMENT SCHEMES FOR PLANAR LINEAR ELASTICITY WITH SYMMETRIC STRESS TENSORS

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arXiv 2022年

作者： Zhang, Shuo Lsec Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and System Sciences Chinese Academy of Sciences Beijing100190 China University of Chinese Academy of Sciences Beijing100049 China

In this paper, we study the construction of low-degree robust finite element schemes for planar linear elasticity on general triangulations. Firstly, we present a low-degree nonconformingHelling-Reissner finite element scheme. For the stress tensor space, the piecewise polynomial shape function space is (Formula presented) the dimension of the total space is asymptotically 8 times of the number of vertices, and the supports of the basis functions are each a patch of an edge. The piecewise rigid body space is used for the displacement. Robust error estimations in L2and broken H(div) norms are presented. Secondly, we investigate the theoretical construction of schemes with lowest-degree polynomial shape function spaces. Specifically, a Hellinger-Reissner finite element scheme is constructed, with the local shape function space for the stress tensor being (Formula presented) which is of the lowest degree for the local approximation of H(div;S), and the space for the displacement is piecewise constants. Robust error estimations in L2and broken H(div) norms are presented for regular solutions and data. Meanwhile, accompanied with this Hellinger-Reissner finite element scheme, a minimal Navier-Lamé finite element scheme is constructed, with the local shape function space for the displacement being (Formula presented) which is of lowest degree for the local approximation in the norm ||·||0+ ||ϵ(·)||0. Robust error estimations in broken energy norms are presented for regular solutions and data. Copyright © 2022, The Authors. All rights reserved.

关键词： Stress tensor

A PRIMAL FINITE ELEMENT SCHEME OF THE H(d) ∩ H(δ) ELLIPTIC PROBLEM

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arXiv 2022年

In this paper, a unified family, for n > 2 and 1 6 k 6 n − 1, of finite element schemes are presented for the primal weak formulation of the n-dimensional (Equation presented) elliptic problem. Copyright © 202... 详细信息

关键词： Finite element method