arXiv subjects

Xu Song

Publications and source records attributed to Xu Song.

9 recordsLinked to original sources

arXiv2026

Limiting Pointwise Decay for the compressible isentropic Navier-Stokes equations

We study the long-time pointwise behavior of small localized perturbations of a constant state for the one-dimensional compressible isentropic Navier-Stokes equations. After subtracting the two Burgers diffusion waves, the convergent sum of all higher-order diffusion waves, and the cross-family viscous corrections, we prove a cone-preserving pointwise estimate for the exact physical remainder and, in particular, \[ |R_i(x,t)|\leq C E_N\log(2+t)\Psi_i(x,t), \qquad \sup_{x\in\mathbb R}\Psi_i(x,t)\leq C(1+t)^{-1}. \] Here $E_N$ measures the size of the initial data and $\Psi_i$ is the cone-resolved weight; both are defined precisely in the main theorem below. Thus $\|R_i(t)\|_{L^\infty}\leq C E_N(1+t)^{-1}\log(2+t)$. The key new idea is to apply a familywise Cole--Hopf transformation to the spatial antiderivative of the remainder, which exactly eliminates the critical same-family first-order feedback. We further construct an approximate Green function adapted to the two characteristic families and combine it with Gaussian-mode extraction and a Kawashima-type energy argument. This yields a cone-preserving estimate at the limiting decay rate, up to a logarithmic loss.

math.AP↗

arXiv2026

A new construction of permutation polynomials over $\mathbb{F}_{q^3}$

We determine all permutation polynomials among several families of polynomials over $\mathbb{F}_{q^3}$ for arbitrary prime powers $q$. We obtain some new families of permutation polynomials over $\mathbb{F}_{q^3}$ with simple coefficients for infinitely many characteristics. As a specific consequence, our results resolve the generalization of conjectures of Zhang, Zheng, Wang, Peng, and Li in the even characteristic. Our proofs are conceptually short and involve no complicated computations, in contrast to the proofs of results on permutation polynomials which were published previously. Moreover, we develop a totally new systematic method in this paper for the study of permutation polynomials.

math.CO↗

arXiv2024

Learning Based Toolpath Planner on Diverse Graphs for 3D Printing

This paper presents a learning based planner for computing optimized 3D printing toolpaths on prescribed graphs, the challenges of which include the varying graph structures on different models and the large scale of nodes & edges on a graph. We adopt an on-the-fly strategy to tackle these challenges, formulating the planner as a Deep Q-Network (DQN) based optimizer to decide the next `best' node to visit. We construct the state spaces by the Local Search Graph (LSG) centered at different nodes on a graph, which is encoded by a carefully designed algorithm so that LSGs in similar configurations can be identified to re-use the earlier learned DQN priors for accelerating the computation of toolpath planning. Our method can cover different 3D printing applications by defining their corresponding reward functions. Toolpath planning problems in wire-frame printing, continuous fiber printing, and metallic printing are selected to demonstrate its generality. The performance of our planner has been verified by testing the resultant toolpaths in physical experiments. By using our planner, wire-frame models with up to 4.2k struts can be successfully printed, up to 93.3% of sharp turns on continuous fiber toolpaths can be avoided, and the thermal distortion in metallic printing can be reduced by 24.9%.

cs.RO↗

arXiv2024

Deep Reinforcement Learning Based Toolpath Generation for Thermal Uniformity in Laser Powder Bed Fusion Process

Laser powder bed fusion (LPBF) is a widely used metal additive manufacturing technology. However, the accumulation of internal residual stress during printing can cause significant distortion and potential failure. Although various scan patterns have been studied to reduce possible accumulated stress, such as zigzag scanning vectors with changing directions or a chessboard-based scan pattern with divided small islands, most conventional scan patterns cannot significantly reduce residual stress. The proposed adaptive toolpath generation (ATG) algorithms, aiming to minimize the thermal gradients, may result in extremely accumulated temperature fields in some cases. To address these issues, we developed a deep reinforcement learning (DRL)-based toolpath generation framework, with the goal of achieving uniformly distributed heat and avoiding extremely thermal accumulation regions during the LPBF process. We first developed an overall pipeline for the DRL-based toolpath generation framework, which includes uniformly sampling, agent moving and environment observation, action selection, moving constraints, rewards calculation, and the training process. To accelerate the training process, we simplified the data-intensive numerical model by considering the turning angles on the toolpath. We designed the action spaces with three options, including the minimum temperature value, the smoothest path, and the second smoothest path. The reward function was designed to minimize energy density to ensure the temperature field remains relatively stable. To verify the effectiveness of the proposed DRL-based toolpath generation framework, we performed numerical simulations of polygon shape printing domains. In addition, four groups of thin plate samples with different scan patterns were compared using the LPBF process.

cs.CE↗

arXiv2022

D2-LRR: A Dual-Decomposed MDLatLRR Approach for Medical Image Fusion

In image fusion tasks, an ideal image decomposition method can bring better performance. MDLatLRR has done a great job in this aspect, but there is still exist some space for improvement. Considering that MDLatLRR focuses solely on the detailed parts (salient features) extracted from input images via latent low-rank representation (LatLRR), the basic parts (principal features) extracted by LatLRR are not fully utilized. Therefore, we introduced an enhanced multi-level decomposition method named dual-decomposed MDLatLRR (D2-LRR) which effectively analyzes and utilizes all image features extracted through LatLRR. Specifically, color images are converted into YUV color space and grayscale images, and the Y-channel and grayscale images are input into the trained parameters of LatLRR to obtain the detailed parts containing four rounds of decomposition and the basic parts. Subsequently, the basic parts are fused using an average strategy, while the detail part is fused using kernel norm operation. The fused image is ultimately transformed back into an RGB image, resulting in the final fusion output. We apply D2-LRR to medical image fusion tasks. The detailed parts are fused employing a nuclear-norm operation, while the basic parts are fused using an average strategy. Comparative analyses among existing methods showcase that our proposed approach attains cutting-edge fusion performance in both objective and subjective assessments.

eess.IV↗

arXiv2021

Res2NetFuse: A Novel Res2Net-based Fusion Method for Infrared and Visible Images

The fusion of visible light and infrared images has garnered significant attention in the field of imaging due to its pivotal role in various applications, including surveillance, remote sensing, and medical imaging. Therefore, this paper introduces a novel fusion framework using Res2Net architecture, capturing features across diverse receptive fields and scales for effective extraction of global and local features. Our methodology is structured into three fundamental components: the first part involves the Res2Net-based encoder, followed by the second part, which encompasses the fusion layer, and finally, the third part, which comprises the decoder. The encoder based on Res2Net is utilized for extracting multi-scale features from the input image. Simultaneously, with a single image as input, we introduce a pioneering training strategy tailored for a Res2Net-based encoder. We further enhance the fusion process with a novel strategy based on the attention model, ensuring precise reconstruction by the decoder for the fused image. Experimental results unequivocally showcase our method's unparalleled fusion performance, surpassing existing techniques, as evidenced by rigorous subjective and objective evaluations.

cs.CV↗

arXiv2021

Elastically-isotropic open-cell minimal surface shell lattices with superior stiffness via variable thickness design

Triply periodic minimal surface (TPMS) shell lattices are attracting increasingly attention due to their unique combination of geometric and mechanical properties, and their open-cell topology. However, uniform thickness TPMS shell lattices are usually anisotropic in stiffness, namely having different Young's moduli along different lattice directions. To reduce the elastic anisotropy, we propose a family of variable thickness TPMS shell lattices with isotropic stiffness designed by a strain energy-based optimization algorithm. The optimization results show that all the six selected types of TPMS lattices can be made to achieve isotropic stiffness by varying the shell thickness, among which N14 can maintain over 90% of the Hashin-Shtrikman upper bound of bulk modulus. All the optimized shell lattices exhibit superior stiffness properties and significantly outperform elastically-isotropic truss lattices of equal relative densities. Both uniform and optimized types of N14 shell lattices along [100], [110] and [111] directions are fabricated by the micro laser powder bed fusion techniques with stainless steel 316L and tested under quasi-static compression loads. Experimental results show that the elastic anisotropy of the optimized N14 lattices is reduced compared to that of the uniform ones. Large deformation compression results reveal different failure deformation behaviors along different directions. The [100] direction shows a layer-by-layer plastic buckling failure mode, while the failures along [110] and [111] directions are related to the shear deformation. The optimized N14 lattices possess a reduced anisotropy of plateau stresses and can even attain nearly isotropic energy absorption capacity.

math.NA↗

arXiv2019

Selective Diffraction with Complex Amplitude Modulation by Dielectric Metasurfaces

Metasurfaces have attracted extensive interests due to their ability to locally manipulate optical parameters of light and easy integration to complex optical systems. Particularly, metasurfaces can provide a novel platform for splitting and diffracting light into several beams with desired profile, which is in contrast to traditional gratings. Here, we propose and experimentally demonstrate a novel method for generating independently selective diffraction orders. Our method is based on complex amplitude modulation with ultrathin dielectric metasurfaces. By tailoring the geometric parameters of silicon nanofin structures, we can spatially control the geometric and dynamic phase as well as the amplitude simultaneously. We compare the results with a metasurface that uses a phase-only modulation, to verify such selective diffraction can be solely efficiently achieved with complex amplitude modulation. Besides, the diffraction angles of each order have been measured, which are consistent with standard grating theory. Our developed method for achieving selective diffraction with metasurfaces has potential applications in beam shaping, parallel laser fabrication, and nanoscale optical detection.

physics.optics↗

arXiv2018

Cube sums of form $3p$ and $3p^2$

Let $p\equiv 2,5\mod 9$ be an odd prime. In this paper, we prove that at least one of $3p$ and $3p^2$ is a cube sum by constructing certain nontrivial Heegner points. We also establish the explicit Gross-Zagier formulae for these Heegner points and give variants of the Birch and Swinnerton-Dyer conjecture of the related elliptic curves.

math.NT↗