The Experts below are selected from a list of 168 Experts worldwide ranked by ideXlab platform
Xiaodong Luo - One of the best experts on this subject based on the ideXlab platform.
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a numerical analysis of the shear behavior of granular soil with fines
Particuology, 2015Co-Authors: Beibing Dai, Jun Yang, Xiaodong LuoAbstract:Abstract Shear behavior of granular soil with fines is investigated using the discrete element method (DEM) and particle arrangements and inter-particle contacts during shear are examined. The DEM simulation reveals that fine particles play a vital role in the overall response of granular soil to shearing. The occurrence of liquefaction and temporary reduction of strength is ascribed mainly to the loss of support from the fine particle contacts (S–S) and fine particle-to-large particle contacts (S–L) as a consequence of the removal of fine particles from the load-carrying skeleton. The dilative strain-hardening response following the strain-softening response is associated with the migration of fine particles back into the load-carrying skeleton, which is thought to enhance the stiffness of the soil skeleton. During shear, the Unit Normal Vector of the large particle-to-large particle (L–L) contact has the strongest fabric anisotropy, and the S–S contact Unit Normal Vector possesses the weakest anisotropy, suggesting that the large particles play a dominant role in carrying the shear load. It is also found that, during shear, fine particles are prone to rolling at contacts while the large particles are prone to sliding, mainly at the S–L and L–L contacts.
Zhang Yibin - One of the best experts on this subject based on the ideXlab platform.
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Concentrating solutions for an anisotropic planar elliptic Neumann problem with Hardy-H\'{e}non weight and large exponent
2021Co-Authors: Zhang YibinAbstract:Let $\Omega$ be a bounded domain in $\mathbb{R}^2$ with smooth boundary, we study the following anisotropic elliptic Neumann problem with Hardy-H\'{e}non weight $$ \begin{cases} -\nabla(a(x)\nabla u)+a(x)u=a(x)|x-q|^{2\alpha}u^p,\,\,\,\, u>0\,\,\,\,\, \textrm{in}\,\,\,\,\, \Omega,\\[2mm] \frac{\partial u}{\partial\nu}=0\,\, \qquad\quad\qquad\qquad\qquad \qquad\qquad\qquad\qquad \,\ \ \,\,\,\, \textrm{on}\,\,\, \partial\Omega, \end{cases} $$ where $\nu$ denotes the outer Unit Normal Vector to $\partial\Omega$, $q\in\overline{\Omega}$, $\alpha\in(-1,+\infty)\setminus\mathbb{N}$, $p>1$ is a large exponent and $a(x)$ is a positive smooth function. We investigate the effect of the interaction between anisotropic coefficient $a(x)$ and singular source $q$ on the existence of concentrating solutions. We show that if $q\in\Omega$ is a strict local maximum point of $a(x)$, there exists a family of positive solutions with arbitrarily many interior spikes accumulating to $q$; while if $q\in\partial\Omega$ is a strict local maximum point of $a(x)$ and satisfies $\langle\nabla a(q),\,\nu(q)\rangle=0$, such a problem has a family of positive solutions with arbitrarily many mixed interior and boundary spikes accumulating to $q$. In particular, we find that concentration at singular source $q$ is always possible whether $q\in\overline{\Omega}$ is an isolated local maximum point of $a(x)$ or not.Comment: arXiv admin note: text overlap with arXiv:1904.0293
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Clustered boundary layer positive solutions for an elliptic Neumann problem with large exponent
2020Co-Authors: Zhang YibinAbstract:Let $\mathcal{D}$ be a smooth bounded domain in $\mathbb{R}^N$ with $N\geq3$, we study the existence and profile of positive solutions for the following elliptic Neumann problem $$\begin{cases}-\Delta \upsilon+\upsilon=\upsilon^p,\quad \upsilon>0 \quad\textrm{in}\ \mathcal{D},\\[1mm] \frac{\partial \upsilon}{\partial\nu}=0\qquad\textrm{on}\ \partial\mathcal{D}, \end{cases}$$ where $p>1$ is a large exponent and $\nu$ denotes the outer Unit Normal Vector to the boundary $\partial\mathcal{D}$. For suitable domains $\mathcal{D}$, by a constructive way we prove that, for any integers $l$, $m$ with $0\leq l\leq m$ and $m\geq1$, if $p$ is large enough, such a problem has a family of positive solutions with $l$ interior layers and $m-l$ boundary layers which concentrate along $m$ distinct $(N-2)$-dimensional minimal submanifolds of $\partial\mathcal{D}$, or collapse to the same $(N-2)$-dimensional minimal submanifold of $\partial\mathcal{D}$ as $p\rightarrow+\infty$
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Concentrating solutions for an anisotropic planar elliptic Neumann problem with H\'{e}non weight and large exponent
2020Co-Authors: Zhang YibinAbstract:Let $\Omega$ be a bounded domain in $\mathbb{R}^2$ with smooth boundary, we study the following anisotropic elliptic Neumann problem $$ \begin{cases} -\nabla(a(x)\nabla u)+a(x)u=a(x)|x-q|^{2\alpha}u^p,\,\,\,\, u>0\,\,\,\,\, \textrm{in}\,\,\,\,\, \Omega,\\[2mm] \frac{\partial u}{\partial\nu}=0\,\, \qquad\quad\qquad\qquad\qquad \qquad\qquad\qquad\qquad \,\ \ \,\,\,\, \textrm{on}\,\,\, \partial\Omega, \end{cases} $$ where $\nu$ denotes the outer Unit Normal Vector to $\partial\Omega$, $q\in\overline{\Omega}$, $\alpha\in(-1,+\infty)\setminus\mathbb{N}$, $p>1$ is a large exponent and $a(x)$ is a positive smooth function. We investigate the effect of the interaction between anisotropic coefficient $a(x)$ and singular source $q$ on the existence of concentrating solutions. We shows that if $q\in\Omega$ is a strict local maximum point of $a(x)$, there exists a family of positive solutions with arbitrarily many interior spikes accumulating to $q$; while if $q\in\partial\Omega$ is a strict local maximum point of $a(x)$ and satisfies $\langle\nabla a(q),\,\nu(q)\rangle=0$, such a problem has a family of positive solutions with arbitrarily many mixed interior and boundary spikes accumulating to $q$. In particular, we find that concentration at singular source $q$ is always possible whether $q\in\overline{\Omega}$ is an isolated local maximum point of $a(x)$ or not.Comment: arXiv admin note: text overlap with arXiv:1904.0293
Bao Weizhu - One of the best experts on this subject based on the ideXlab platform.
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A structure-preserving parametric finite element method for surface diffusion
2021Co-Authors: Bao Weizhu, Zhao QuanAbstract:We propose a structure-preserving parametric finite element method (SP-PFEM) for discretizing the surface diffusion of a closed curve in two dimensions (2D) or surface in three dimensions (3D). Here the "structure-preserving" refers to preserving the two fundamental geometric structures of the surface diffusion flow: (i) the conservation of the area/volume enclosed by the closed curve/surface, and (ii) the decrease of the perimeter/total surface area of the curve/surface. For simplicity of notations, we begin with the surface diffusion of a closed curve in 2D and present a weak (variational) formulation of the governing equation. Then we discretize the variational formulation by using the backward Euler method in time and piecewise linear parametric finite elements in space, with a proper approximation of the Unit Normal Vector by using the information of the curves at the current and next time step. The constructed numerical method is shown to preserve the two geometric structures and also enjoys the good property of asymptotic equal mesh distribution. The proposed SP-PFEM is "weakly" implicit (or almost semi-implicit) and the nonlinear system at each time step can be solved very efficiently and accurately by the Newton's iterative method. The SP-PFEM is then extended to discretize the surface diffusion of a closed surface in 3D. Extensive numerical results, including convergence tests, structure-preserving property and asymptotic equal mesh distribution, are reported to demonstrate the accuracy and efficiency of the proposed SP-PFEM for simulating surface diffusion in 2D and 3D.Comment: 25 pages, 13 figure
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An energy-stable parametric finite element method for anisotropic surface diffusion
2020Co-Authors: Li Yifei, Bao WeizhuAbstract:We propose an energy-stable parametric finite element method (ES-PFEM) to discretize the motion of a closed curve under surface diffusion with an anisotropic surface energy $\gamma(\theta)$ -- anisotropic surface diffusion -- in two dimensions, while $\theta$ is the angle between the outward Unit Normal Vector and the vertical axis. By introducing a positive definite surface energy (density) matrix $G(\theta)$, we present a new and simple variational formulation for the anisotropic surface diffusion and prove that it satisfies area/mass conservation and energy dissipation. The variational problem is discretized in space by the parametric finite element method and area/mass conservation and energy dissipation are established for the semi-discretization. Then the problem is further discretized in time by a (semi-implicit) backward Euler method so that only a linear system is to be solved at each time step for the full-discretization and thus it is efficient. We establish well-posedness of the full-discretization and identify some simple conditions on $\gamma(\theta)$ such that the full-discretization keeps energy dissipation and thus it is unconditionally energy-stable. Finally the ES-PFEM is applied to simulate solid-state dewetting of thin films with anisotropic surface energies, i.e. the motion of an open curve under anisotropic surface diffusion with proper boundary conditions at the two triple points moving along the horizontal substrate. Numerical results are reported to demonstrate the efficiency and accuracy as well as energy dissipation of the proposed ES-PFEM.Comment: 27 pages, 10 figure
Yinghui Zhang - One of the best experts on this subject based on the ideXlab platform.
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singularity of the extremal solution for supercritical biharmonic equations with power type nonlinearity
Chinese Annals of Mathematics Series B, 2017Co-Authors: Baishun Lai, Zhengxiang Yan, Yinghui ZhangAbstract:Let B ⊂ ℝn be the Unit ball centered at the origin. The authors consider the following biharmonic equation: $$\left\{ {\begin{array}{*{20}{c}} {{\Delta ^2}u = \lambda {{\left( {1 + u} \right)}^p}}&{in \mathbb{B},} \\ {u = \frac{{\partial u}}{{\partial \nu }} = 0}&{on\partial \mathbb{B},} \end{array}} \right.$$ where \(p > \frac{{n + 4}}{{n - 4}}\) and v is the outward Unit Normal Vector. It is well-known that there exists a λ* > 0 such that the biharmonic equation has a solution for λ ∈ (0, λ*) and has a unique weak solution u* with parameter λ = λ*, called the extremal solution. It is proved that u* is singular when n ≥ 13 for p large enough and satisfies \(u* \leqslant {r^{ - \frac{4}{{p - 1}}}} - 1\) on the Unit ball, which actually solve a part of the open problem left in [Davila, J., Flores, I., Guerra, I., Multiplicity of solutions for a fourth order equation with power-type nonlinearity, Math. Ann., 348(1), 2009, 143–193].
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singularity of the extremal solution for supercritical biharmonic equations with power type nonlinearity
arXiv: Analysis of PDEs, 2011Co-Authors: Baishun Lai, Zhengxiang Yan, Yinghui ZhangAbstract:Let $\lambda^{*}>0$ denote the largest possible value of $\lambda$ such that $$ \{{array}{lllllll} \Delta^{2}u=\lambda(1+u)^{p} & {in}\ \ \B, %0 \frac{n+4}{n-4}$ and $n$ is the exterior Unit Normal Vector. We show that for $\lambda=\lambda^{*}$ this problem possesses a unique weak solution $u^{*}$, called the extremal solution. We prove that $u^{*}$ is singular when $n\geq 13$ for $p$ large enough, in which case $u^{*}(x)\leq r^{-\frac{4}{p-1}}-1$ on the Unit ball.
Beibing Dai - One of the best experts on this subject based on the ideXlab platform.
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a numerical analysis of the shear behavior of granular soil with fines
Particuology, 2015Co-Authors: Beibing Dai, Jun Yang, Xiaodong LuoAbstract:Abstract Shear behavior of granular soil with fines is investigated using the discrete element method (DEM) and particle arrangements and inter-particle contacts during shear are examined. The DEM simulation reveals that fine particles play a vital role in the overall response of granular soil to shearing. The occurrence of liquefaction and temporary reduction of strength is ascribed mainly to the loss of support from the fine particle contacts (S–S) and fine particle-to-large particle contacts (S–L) as a consequence of the removal of fine particles from the load-carrying skeleton. The dilative strain-hardening response following the strain-softening response is associated with the migration of fine particles back into the load-carrying skeleton, which is thought to enhance the stiffness of the soil skeleton. During shear, the Unit Normal Vector of the large particle-to-large particle (L–L) contact has the strongest fabric anisotropy, and the S–S contact Unit Normal Vector possesses the weakest anisotropy, suggesting that the large particles play a dominant role in carrying the shear load. It is also found that, during shear, fine particles are prone to rolling at contacts while the large particles are prone to sliding, mainly at the S–L and L–L contacts.
