Elliptic Quadratic Operator Equations

The paper is devoted to the study of elliptic quadratic operator equations over the finite dimensional Euclidean space. We provide necessary and sufficient conditions for the existence of solutions of elliptic quadratic operator equations. The iterative Newton-Kantorovich method for stable solutions is also presented.

     

Harnack's inequality for parabolic nonlocal equations

The main result of this paper is a nonlocal version of Harnack's inequality for a class of parabolic nonlocal equations. We additionally establish a weak Harnack inequality as well as local boundedness of solutions. None of the results require the solution to be globally positive.     

Generalized fractional maximal and integral operators on Orlicz and generalized Orlicz–Morrey spaces of the third kind

Positivity - In the present paper, we will characterize the boundedness of the generalized fractional integral operators $$I_{\rho }$$ and the generalized fractional maximal operators $$M_{\rho }$$...     

An interpolating element-free Galerkin scaled boundary method applied to structural dynamic analysis

An efficient semi-analytical method, namely the interpolating element-free Galerkin scaled boundary method (IEFG-SBM) is developed for structural dynamic analysis in this paper, which is based on boundary scattered nodes with no need of element connectivity. Since the shape functions of the improved interpolating moving least-squares (IIMLS) method satisfy the delta function property, the essential boundary conditions, as a result, can be enforced accurately without any additional efforts. Based on the improved continued fraction, the dynamic properties of a bounded domain are expressed by the high order static stiffness and mass matrices. This continued fraction solution is computationally more robust and the transient response can be obtained directly in the time domain using standard procedures in structural dynamics. It is testified from the computational results that the present method for structural dynamic analysis is quite easy to be implemented with high accuracy.     

Time‐dependent finite element simulations of a shear‐thinning viscoelastic fluid with application to blood flow

A new finite element method is developed to simulate time‐dependent viscoelastic shear‐thinning flows characterized by the generalized Oldroyd‐B model. The focus of the algorithm is improved stability through a free‐energy dissipative scheme by using low‐order piecewise‐constant finite element approximations for stress. The algorithm is further modified by incorporating a pressure‐projection method, a DG‐upwinding scheme, a symmetric interior penalty DG method to solve the elliptic pressure‐update equation and a geometric multigrid preconditioner. The improved stability and cost to accuracy is compared when using higher order discontinuous bilinear approximation, where in addition, we consider the influence of a slope limiter for these elements. The algorithm is applied to the 2D start‐up‐driven cavity problem, and the stability of the free energy is illustrated and compared between element choices. An application of the model to modelling blood in small arterioles and channels is considered by simulating pulsatile blood flow through a stenotic arteriole. The individual influences of viscoelasticity and shear‐thinning within the generalized Oldroyd‐B model are investigated by comparing results to the Newtonian, generalized Newtonian and Oldroyd‐B models. Copyright © 2014 John Wiley & Sons, Ltd.     

Korn inequality for a thin rod with rounded ends

We consider an elastic rod with rounded ends and diameter proportional to a small parameter h > 0. The roundness of the ends is described by an exponent m ∈ (0,1). We derive for the rod an asymptotically sharp Korn inequality with a special weighted anisotropic norm. Weight factors with m‐dependent powers of h appear both in the anisotropic norm and the Korn inequality itself, and we discover three critical values m = 1 ∕ 4, m = 1 ∕ 2 and m = 3 ∕ 4 at which these exponents are crucially changed. Copyright © 2014 John Wiley & Sons, Ltd.     

Two-phase flow simulation for distinguishing deformable particles with a LiMCA system

In the metallurgical industry, Liquid Metal Cleanliness Analyser (LiMCA) commercial equipment cannot distinguish between hard particles (e.g., oxides, borides) and deformable particles (e.g., bubbles, molten salts). Therefore, hard particle concentrations can sometimes be grossly overestimated, which reduces the measurement accuracy. However, the method could potentially discriminate between deformable particles and hard particles by evaluating the particle's ability to deform. In this work, the coupled multiphysics problem of a particle deforming within current-carrying aluminium metal passing through the electric sensing zone (ESZ) is simulated using the conservative level-set (CLS) method. An emphasis is placed on understanding the transient deformation history, and the effect of the capillary number, Reynolds number, and confinement ratio on deformation are studied. Furthermore, a computational basis is given to estimate the influence of particle deformation on electrical resistance pulses (ERP). It is found that ERP features of deformation particles, including the peak magnitude and the pulse width, are different from those of hard particles. Based on the results, the effect of a particle's deformation and the feasibility to discriminate it from non-deformable particles in the LiMCA system is evaluated.