Tangential Boundary Research Articles

Isogeometric finite element analysis of composite sandwich plates using a higher order shear deformation theory

We present in this paper a simple and effective formulation based on a fifth-order shear deformation theory (FiSDT) in combination with isogeometric finite element analysis (IGA) for composite sandwich plates. The FiSDT yields non-linear distribution of the transverse shear stresses through the plate thickness and ensures a prior tangential stress-free boundary condition. The IGA uses same basis functions, namely B-splines or non-uniform rational B-splines (NURBS), for preserving the precisely geometric representation and providing the numerical solution. It enables to achieve easily the smoothness with arbitrary continuity order and in the present method the C1-continuity requirement of higher order shear deformation models is fulfilled. The static, dynamic and buckling analysis of rectangular and circular plates is investigated for different boundary conditions. Numerical examples are given to show high accuracy of the proposed method.

Nematic liquid crystal boojums with handles on colloidal handlebodies

Topological defects that form on surfaces of ordered media, dubbed boojums, are ubiquitous in superfluids, liquid crystals (LCs), Langmuir monolayers, and Bose-Einstein condensates. They determine supercurrents in superfluids, impinge on electrooptical switching in polymer-dispersed LCs, and mediate chemical response at nematic-isotropic fluid interfaces, but the role of surface topology in the appearance, stability, and core structure of these defects remains poorly understood. Here, we demonstrate robust generation of boojums by controlling surface topology of colloidal particles that impose tangential boundary conditions for the alignment of LC molecules. To do this, we design handlebody-shaped polymer particles with different genus g. When introduced into a nematic LC, these particles distort the nematic molecular alignment field while obeying topological constraints and induce at least 2g - 2 boojums that allow for topological charge conservation. We characterize 3D textures of boojums using polarized nonlinear optical imaging of molecular alignment and explain our findings by invoking symmetry considerations and numerical modeling of experiment-matching director fields, order parameter variations, and nontrivial handle-shaped core structure of defects. Finally, we discuss how this interplay between the topologies of colloidal surfaces and boojums may lead to controlled self-assembly of colloidal particles in nematic and paranematic hosts, which, in turn, may enable reconfigurable topological composites.

Spine-fan reconnection

Context. From observations, the atmosphere of the Sun has been shown to be highly dynamic with perturbations of the magnetic field often lacking temporal or spatial symmetry. Despite this, studies of the spine-fan reconnection mode at 3D nulls have so far focused on the very idealised case with symmetric driving of a fixed spatial extent. Aims. We investigate the spine-fan reconnection process for less idealised cases, focusing on asymmetric driving and drivers with different length scales. We look at the initial current sheet formation and whether the scalings developed in the idealised models are robust in more realistic situations. Methods. The investigation was carried out by numerically solving the resistive compressible 3D magnetohydrodynamic equations in a Cartesian box containing a linear null point. The spine-fan collapse was driven at the null through tangential boundary driving of the spine foot points. Results. We find significant differences in the initial current sheet formation with asymmetric driving. Notable is the displacement of the null point position as a function of driving velocity and resistivity (η). However, the scaling relations developed in the idealised case are found to be robust (albeit at reduced amplitudes) despite this extra complexity. Lastly, the spatial variation is also shown to play an important role in the initial current sheet formation through controlling the displacement of the spine foot points. Conclusions. We conclude that during the early stages of spine-fan reconnection both the temporal and spatial nature of the driving play important roles, with the idealised symmetrically driven case giving a “best case” for the rate of current development and connectivity change. As the most interesting eruptive events occur in relatively short time frames this work clearly shows the need for high temporal and spatial knowledge of the flows for accurate interpretation of the reconnection scenario. Lastly, since the scalings developed in the idealised case remain robust with more complex driving we can be more confident of their use in interpreting reconnection in complex magnetic field structures.

Simulation of turbulent swirling flow in convergent nozzles

This work simulates the turbulent boundary layer of an incompressible viscous swirling flow through a conical chamber. To model the pressure gradient normal to the wall, the radial and tangential velocity components across the boundary layer have been calculated by both the integral and numerical methods. The numerical solution is accomplished by finite difference, based on the finite volume method. The results show that the radial and tangential boundary layer thicknesses depend on the velocity ratios, Reynolds number and nozzle angle. The peak of radial and tangential boundary layer thicknesses are located at z/L≈0.2 and z/L≈0.8 from the nozzle inlet, respectively. Due to the short length of the nozzle, the contribution of momentum change on pressure loss is more significant than that on the shear stress. Also, the pressure gradient normal to the wall had been considered more accurately than that of the previous attempts.

Generalized hybrid quasi-3D shear deformation theory for the static analysis of advanced composite plates

This paper presents a generalized hybrid quasi-3D shear deformation theory for the bending analysis of advanced composite plates such as functionally graded plates (FGPs). Many 6DOF hybrid shear deformation theories with stretching effect included, can be derived from the present generalized formulation. All these theories account for an adequate distribution of transverse shear strains through the plate thickness and tangential stress-free boundary conditions on the plate boundary surfaces not requiring thus a shear correction factor. The generalized governing equations of a functionally graded (FG) plate and boundary conditions are derived by employing the principle of virtual work. Navier-type analytical solution is obtained for FGP subjected to transverse load for simply supported boundary conditions. Numerical examples of the new quasi-3D HSDTs (non-polynomial, polynomial and hybrid) derived by using the present generalized formulation are compared with 3D exact solutions and with other HSDTs. Results show that some of the new HSDTs are more accurate than, for example, the well-known trigonometric HSDT, having the same 6DOF.

New artificial boundary condition for saturated soil foundations

A new artificial boundary model based on multi-directional transmitting and viscous-spring artificial boundary theories is proposed to absorb stress waves in a saturated soil foundation in dynamic analysis. Since shear waves (S-waves) are the same in a saturated soil foundation and a single-phase medium foundation, a tangential visco-elastic boundary condition for a single-phase medium foundation can also be used for saturated soil foundations. Thus, the purpose of the artificial boundary proposed in this paper is primarily to absorb two types of P-waves in a saturated soil foundation. The main idea is that the stress of the P-waves in the saturated soil foundation is decomposed into two types. The first type of stress, σra, is absorbed by the first artificial boundary. The second type of stress, σrb, is balanced by the stress generated by the second artificial boundary. Ultimately, both types of P-waves (fast-P-waves and slow-P-waves) are absorbed by the artificial boundary model proposed in this paper. In particular, note that the fast-P-waves and slow-P-waves are absorbed at the position of the first boundary. Thus, the artificial boundary model proposed herein can simultaneously absorb P-fast waves, P-slow waves and shear waves. Finally, a numerical example is given to examine the proposed artificial boundary model, and the results show that it is very accurate.

Problem of Second Grade Fluids in Convex Polyhedrons

This paper studies the solutions of a three-dimensional grade-two fluid model with a tangential boundary condition in a polyhedron. We begin to split the problem into a system with a generalized Stokes problem and a transport equation, as Girault and Scott have done in the two-dimensional case. But, compared to the two-dimensional problem, we have an additional term that is difficult to bound which requires regularity of the solutions and we have to prove that the solutions of the transport equation, which is no longer scalar, are divergence-free. In order to deal with these specific difficulties of the three dimensions, we establish a new system which implies the previous one without being equivalent. A substantial part of the article is devoted to proving that any solution in $L^2(\Omega)^3$ of the transport equation is divergence-free if the right-hand side is divergence-free and the velocity small enough in $W^{1,\infty}(\Omega)^3$. Existence is proven in a convex polyhedron with adequate restrictions on the size of the data and parameters. Uniqueness requires, in addition, inner angles smaller than $\frac{3\pi}{4}$.

Boundary values in range spaces of co-analytic truncated Toeplitz operators

Functions in backward shift invariant subspaces have nice analytic continuation properties outside the spectrum of the inner function de ning the space. Inside the spectrum of the inner function, Ahern and Clark showed that under some distribution condition on the zeros and the singular measure of the inner function, it is possible to obtain non-tangential boundary values of every function in the backward shift invariant subspace as well as for their derivatives up to a certain order. Here we will investigate, at least when the inner function is a Blaschke product, the non- tangential boundary values of the functions of the backward shift invariant subspace after having applied a co-analytic (truncated) Toeplitz operator. There appears to be a smoothing effect.

On Simultaneous Identification of the Shape and Generalized Impedance Boundary Condition in Obstacle Scattering

We consider the inverse obstacle scattering problem of determining both the shape and the “equivalent impedance” from far field measurements at a fixed frequency. In this work, the surface impedance is represented by a second order surface differential operator (referred to as generalized impedance boundary condition) as opposed to a scalar function. The generalized impedance boundary condition can be seen as a more accurate model for effective impedances and is widely used in the scattering problem for thin coatings. Our approach is based on a least square optimization technique. A major part of our analysis is to characterize the derivative of the cost function with respect to the boundary and this complex surface impedance configuration. In particular, we provide an extension of the notion of shape derivative to the case where the involved impedance parameters do not need to be surface traces of given functions, which leads (in general) to a nonvanishing tangential boundary perturbation. The efficiency of considering this type of derivative is illustrated by several two-dimensional numerical experiments based on a (classical) steepest descent method. The feasibility of retrieving both the shape and the impedance parameters is also discussed in our numerical experiments.

Efficient electromagnetic boundary conditions to accelerate optimization of RF devices

To achieve efficient field formulations and fast numerical computations, the reciprocal relations and equivalence between tangential and normal boundary conditions for electromagnetic fields are discussed in terms of the Maxwell's differential equations. Using the equivalence of each boundary condition, we propose the six essential boundary conditions, which may be applicable to matching electromagnetic discontinuities to efficiently design RF devices. In order to verify our approach, the reflection characteristics of a rectangular waveguide step are compared with respect to six essential boundary conditions.

Off-Axis Gaussian Beam Scattering by an Anisotropic Coated Sphere

An analytical solution to the scattering of an off-axis Gaussian beam incident on an anisotropic coated sphere is proposed. Based on the local approximation of the off-axis beam shape coefficients, the field of the incident Gaussian beam is expanded using first spherical vector wave functions. By introducing the Fourier transform, the electromagnetic fields in the anisotropic layer are expressed as the addition of the first and the second spherical vector wave functions. The expansion coefficients are analytically derived by applying the continuous tangential boundary conditions to each interface among the internal isotropic dielectric or conducting sphere, the anisotropic shell, and the free space. The influence of the beam widths, the beam waist center positioning, and the size parameters of the spherical structure on the field distributions are analyzed. The applications of this theoretical development in the fields of biomedicine, target shielding, and anti-radar coating are numerically discussed. The accuracy of the theory is verified by comparing the numerical results reduced to the special cases of a plane wave incidence and the case of a homogeneous anisotropic sphere with results from a CST simulation and references.

A robust weakly compressible SPH method and its comparison with an incompressible SPH

SUMMARYThis paper presents a comparative study for the weakly compressible (WCSPH) and incompressible (ISPH) smoothed particle hydrodynamics methods by providing numerical solutions for fluid flows over an airfoil and a square obstacle. Improved WCSPH and ISPH techniques are used to solve these two bluff body flow problems. It is shown that both approaches can handle complex geometries using the multiple boundary tangents (MBT) method, and eliminate particle clustering‐induced instabilities with the implementation of a particle fracture repair procedure as well as the corrected SPH discretization scheme. WCSPH and ISPH simulation results are compared and validated with those of a finite element method (FEM). The quantitative comparisons of WCSPH, ISPH and FEM results in terms of Strouhal number for the square obstacle test case, and the pressure envelope, surface traction forces, and velocity gradients on the airfoil boundaries as well as the lift and drag values for the airfoil geometry indicate that the WCSPH method with the suggested implementation produces numerical results as accurate and reliable as those of the ISPH and FEM methods. Copyright © 2011 John Wiley & Sons, Ltd.

On a p-curl system arising in electromagnetism

We prove existence of solution of a $p$-curl type evolutionary system arising inelectromagnetism with a power nonlinearity of order $p$, $1 < p < \infty$, assumingnatural tangential boundary conditions.We consider also the asymptotic behaviour in thepower obtaining, when $p$ tends to infinity, a variational inequality with acurl constraint.We also discuss the existence, uniqueness and continuous dependence on the data of the solutions togeneral variational inequalities with curl constraints dependent on time, aswell as the asymptotic stabilization in time towards the stationary solutionwith and without constraint.

A new trigonometric shear deformation theory for isotropic, laminated composite and sandwich plates

A new trigonometric shear deformation theory for isotropic and composite laminated and sandwich plates, is developed. The new displacement field depends on a parameter “ m”, whose value is determined so as to give results closest to the 3D elasticity bending solutions. The theory accounts for adequate distribution of the transverse shear strains through the plate thickness and tangential stress-free boundary conditions on the plate boundary surface, thus a shear correction factor is not required. Plate governing equations and boundary conditions are derived by employing the principle of virtual work. The Navier-type exact solutions for static bending analysis are presented for sinusoidally and uniformly distributed loads. The accuracy of the present theory is ascertained by comparing it with various available results in the literature. The results show that the present model performs as good as the Reddy’s and Touratier’s shear deformation theories for analyzing the static behavior of isotropic and composite laminated and sandwich plates.

Bending response of functionally graded plates by using a new higher order shear deformation theory

This paper presents an analytical solution to the static analysis of functionally graded plates, using a recently developed higher order shear deformation theory (HSDT) and provides detailed comparisons with other HSDT’s available in the literature. These theories account for adequate distribution of the transverse shear strains through the plate thickness and tangential stress-free boundary conditions on the plate boundary surfaces, thus a shear correction factor is not required. The mechanical properties of the plates are assumed to vary in the thickness direction according to a power-law distribution in terms of the volume fractions of the constituents. The governing equations of a functionally graded (FG) plate and boundary conditions are derived by employing the principle of virtual work. Navier-type analytical solution is obtained for FG plates subjected to transverse bi-sinusoidal and distributed loads for simply supported boundary conditions. Results are provided for thick to thin FG plates and for different volume fraction distributions. The accuracy of the present code is verified by comparing it with known results in the literature.

The unique continuation property of eigenfunctions to Stokes–Oseen operator is generic with respect to the coefficients

The unique continuation property of eigenfunctions φ to a Stokes–Oseen operator in Ω⊂R2 with boundary conditions {φ=0,∂∂νφ=0 on ∂Ω} is generic with respect to C2-coefficients. This implies that generically each C2 equilibrium solution to a 2-D Navier–Stokes equation is exponentially stabilizable by a tangential boundary controller.

Static and dynamic analysis of laminated composite and sandwich plates and shells by using a new higher-order shear deformation theory

A new higher order shear deformation theory for elastic composite/sandwich plates and shells is developed. The new displacement field depends on a parameter “ m”, whose value is determined so as to give results closest to the 3D elasticity bending solutions. The present theory accounts for an approximately parabolic distribution of the transverse shear strains through the shell thickness and tangential stress-free boundary conditions on the shell boundary surface. The governing equations and boundary conditions are derived by employing the principle of virtual work. These equations are solved using Navier-type, closed form solutions. Static and dynamic results are presented for cylindrical and spherical shells and plates for simply supported boundary conditions. Shells and plates are subjected to bi-sinusoidal, distributed and point loads. Results are provided for thick to thin as well as shallow and deep shells. The accuracy of the present code is verified by comparing it with various available results in the literature.

Alignment of high-aspect ratio colloidal gold nanoplatelets in nematic liquid crystals

We study elasticity-mediated alignment of anisotropic gold colloids in liquid crystals. Colloidal gold particles of controlled shapes (spheres, rods, and polygonal platelets) and sizes are prepared using well-established biosynthesis techniques with varying solvent conditions. When introduced into liquid crystalline structured solvents, these gold particles impose tangential or vertical surface boundary conditions for the liquid crystal molecules or building blocks such as chromonic molecular aggregates. This allows for multiple types of their controlled alignment in both lyotropic and thermotropic liquid crystals and is of interest for self-assembly-based fabrication of tunable nanostructured composite materials.

A new higher order shear deformation theory for sandwich and composite laminated plates

A new shear deformation theory for sandwich and composite plates is developed. The proposed displacement field, which is “m” parameter dependent, is assessed by performing several computations of the plate governing equations. Therefore, the present theory, which gives accurate results, is relatively close to 3D elasticity bending solutions. The theory accounts for adequate distribution of the transverse shear strains through the plate thickness and tangential stress-free boundary conditions on the plate boundary surface, thus a shear correction factor is not required. Plate governing equations and boundary conditions are derived by employing the principle of virtual work. The Navier-type exact solutions for static bending analysis are presented for sinusoidally and uniformly distributed loads. The accuracy of the present theory is ascertained by comparing it with various available results in the literature.

SU‐E‐T‐450: Can Low Frequency Electric Fields Be Generally Used as a Sensitizer for Radiation Therapy and Chemo‐Therapy?

Purpose: A low frequency electric field applied to a tumor is called a Tumor Treatment Field (TTF) and works by arresting cells during mitosis (M‐ phase). I have numerically simulated E‐field distributions to determine if a TTF can be applied to arbitrary tumor masses. This modality is currently under clinical trail with chemotherapy as a sensitizer. Because cells in M‐phase are also sensitive to radiation this method may also work as a radiotherapy sensitizer. Methods: Previous published results showed a TTF applied in to an intracranial glioma in a rat brain and subsequent tumor shrinkage. The TTF was numerically modeled. I have also modeled several therapies. The E‐field is calculated in inhomogeneous tissue using the Finite Element Method with normal and tangential boundary conditions applied to the exterior of the patient. These boundary conditions correspond to electrodes that may be manipulated to design treatment plans. Results: An AC electric field of amplitude 1V/cm–2V/cm in the 100kHz–200kHz range was previously found to be the most effective for arresting mitotic cells. In this range, however, the dielectric constant is very large and varies considerably between tissue layers. These large variations cause large impedances in layers of, e.g., fat and bone, i.e., these layers effectively “shield” deeper tissue layers against E‐fields applied in the normal direction. This requires large external voltages on surface electrodes to penetrate and achieve the required 1V/cm–2V/cm at deeper tissue layers, especially in human treatments. These large impedances are avoided with an E‐field applied in the tangential direction. The tangential fields are also patient size independent, greatly simplifying treatment planning. Results from several simulations comparing methods are presented. Conclusions: This work shows that an external applied device can produce an effective TTF at reasonably safe voltages if the applied field is not applied normally to the tissue layers.