Tangential Boundary Research Articles

On the existence of Rayleigh waves with full impedance boundary condition

The existence of Rayleigh waves (propagating in isotropic elastic half-spaces) with the tangential and normal impedance boundary conditions was investigated. It has been shown that for the tangential impedance boundary condition (TIBC), there always exists a unique Rayleigh wave, while for the normal impedance boundary condition (NIBC), there exists a domain (of impedance and material parameters) in which exactly one Rayleigh wave is possible and outside this domain a Rayleigh wave is impossible. In this paper, we consider the existence of Rayleigh waves with the full impedance boundary condition (FIBC) that contains both TIBC and NIBC. It is shown that the existence picture of Rayleigh waves for this general case is more complicated. It contains domain for which exactly one Rayleigh wave exists, domain where a Rayleigh wave is impossible, and domain for which all three possibilities may occur: two Rayleigh waves exist, one Rayleigh wave exists, and no Rayleigh wave exists at all. The obtained existence results recover the existence results established previously for the cases of TIBC and NIBC. The formulas for the Rayleigh wave velocity are derived. As these formulas are totally explicit, they are very useful in various practical applications, especially in the non-destructive evaluation of the mechanical properties of structures. In order to establish the existence results and derive formulas for the Rayleigh wave velocity, the complex function method, which is based on the Cauchy-type integrals, is employed.

Γ-convergence of the Ginzburg-Landau functional with tangential boundary conditions

A classical result in the study of Ginzburg-Landau equations is that, for Dirichlet or Neumann boundary conditions, if a sequence of functions has energy uniformly bounded on a logarithmic scale then we can find a subsequence whose Jacobians are convergent in suitable dual spaces and whose renormalized energy is at least the sum of absolute degrees of vortices. However, the corresponding question for the case of tangential or normal boundary conditions has not been considered. In addition, the question of convergence of up to the boundary is not very well understood. Here, we consider these questions for a bounded, connected, open set of R2 with C2,1 boundary.

Photoinduced Transformation of the Orientational Structure of a Chiral Nematic under Planar-Conical Anchoring

The photo-induced change in the orientation structure of the chiral nematic based on LN-396 doped with chiral dopants S811 and cChD has been investigated. It is shown that the use of planar-conical boundary conditions makes it possible to realise a smooth variation of the structure twist angle by changing the helix pitch. It is shown that the twisting of the structure of the studied chiral nematic to an angle of more than 240° leads to undulations when the conical anchoring is specified by the PiBMA polymer. The PtBMA polymer, on the other hand, specifies tangential boundary conditions for the chiral nematic under study, allowing the director to rotate easily in the plane of the sample. The results presented may be promising for the development of liquid crystal systems with finely tunable polar and azimuthal angles of the director.

Asymptotically correct isoenergetic formulation of geometrically nonlinear anisotropic plates

In the present work, a new displacement-based Equivalent Single Layer (ESL) plate theory has been developed using the Variational Asymptotic Method (VAM) and the principle of isoenergetics. In contrast to the existing assumption-based ESL plate theories, the proposed VAM-based plate theory is devoid of any presuppositions. The VAM decouples the 3D plate problem into a 1D through the thickness analysis and a 2D planar problem. Closed form solutions have been obtained for the 1D problem in terms of 2D displacement variables and their higher-order derivatives. The complexity involving higher-order derivatives of 2D displacement variables has been simplified through the isoenergetic principle ensuring a similar rate of convergence for strains as well as displacements. Solutions for various field variables have been compared against the benchmark problems available in the literature and 3D FEA. These comparisons demonstrate the accuracy, efficiency, and versatility of the present methodology. Key contributions of the present work are (a) First principles-based derivation of the reduced-order 2D plate model from the 3D model energy. (b) The plane stress condition as well as the quadratic variation of transverse shear stress and strain are natural outcomes of the present mathematical framework. (c) This leads to zero tangential traction boundary conditions on the plate surface. (d) The effectiveness of capturing higher-order behavior at lower-order results in simplified analysis, reduced computational complexity and increased efficiency.

On Tangential Boundary Behavior of Functions from Hardy Type Spaces

On Tangential Boundary Behavior of Functions from Hardy Type Spaces

Construction of [formula omitted] spatial interpolants with prescribed arc lengths

In this paper the problem of constructing spatial G2 continuous Pythagorean-hodograph (PH) spline curves, that interpolate points and frame data, and in addition have the prescribed arc-length, is addressed. The interpolation scheme is completely local and can be directly applied for motion design applications. Each spline segment is defined as a PH biarc curve of degree 7 satisfying super-smoothness conditions at the biarc’s joint point. The biarc is expressed in a closed form with additional free parameters, where one of them is determined by the length constraint. The selection of the remaining free parameters is suggested, that allows the existence of the solution of the length interpolation equation for any prescribed length and any ratio between norms of boundary tangents. By the proposed automatic procedure for computing the frame and velocity quaternions from the first and second order derivative vectors, the paper presents a direct generalization of the construction done for planar curves to spatial ones. Several numerical examples are provided to illustrate the proposed method and to show its good performance, also when a spline construction in considered.

Scattering characteristics of a perfect electromagnetic conducting cylindrical object covered with a general Bi-isotropic layer

Abstract An analytic theory has been developed for the scattering of electromagnetic plane wave from a perfect electromagnetic conducting (PEMC) cylindrical object coated with a general bi-isotropic (BI) material. The proposed problem has been solved using cylindrical vector wave function expansion approach along with the application of tangential boundary conditions. Analytic expressions of the scattering coefficients have been derived in their simplest forms. It is seen that by proper selection of admittance of PEMC core, electromagnetic parameters of BI coating, and coating thickness, one can optimize the scattering characteristics for specific applications. It is shown that the specific types of BI and strong chiral-coated PEMC cylinders having certain coating thicknesses can be used to significantly enhance the co-polarized forward scattering while keeping the cross-polarized forward scattering very small. Such types of enhanced co-polarized forward scattering are preferred in point-to-point communication. Some interesting features have been discussed where co-polarized and cross-polarized backscattering may be suppressed, which find applications in radar engineering problems and stealth technology.

The Rayleigh–Bénard problem for water with maximum density effects

Linear stability and weakly nonlinear stability analyses are developed for Rayleigh–Bénard convection in water near 3.98 °C subject to isothermal boundary conditions. The density–temperature relationship (equation of state) is approximated by a cubic polynomial, including linear, quadratic, and cubic terms. The continuity equation, the Navier–Stokes momentum equation, the equation of state, and the energy equation constitute the governing system. Linear stability analysis is used to investigate how the maximum density property of water affects the onset of convective instability and the choice of unstable wave number for four different types of boundary conditions. Then, a weakly nonlinear stability study is done using the spectral Fourier method for isothermal tangential stress-free boundary conditions to quantify the heat transport of the system and demonstrate the transition from regular/periodic convection to chaotic convection. A Stuart-Ginzburg–Landau equation is obtained using the multiscale expansion method. Streamlines and isotherms are presented and analyzed. The influence of maximum density has been shown to delay the onset of instability and is, therefore, a stabilizing mechanism for thermal instability. Due to the maximum density, the onset of chaotic convection is also delayed. Among four different boundaries, the impermeable rigid boundaries require the highest Rayleigh number for instability to begin. Increasing boundary temperatures advance the onset of chaotic convection and improve the heat transport situation.

Scattering of aggregated multi-layered biological cells by Bessel beams

Interactions between collective multi-layered cells and an off-axis high-order Bessel beam (HOBB) are investigated. The generalized Lorenz-Mie theory is applied to derive the expansion of HOBB. Based on the additional theorem, multiple scattering of collective multi-layered nanoparticles is obtained by considering the tangential continuous boundary conditions. The present theory and codes are proven to be effective by comparing with the simulations obtained from the computer simulation technology (CST) software. Numerical results concerning the effects of beam order, beam conical angle, spherical layer number, core radius and outer layer radius, outer layer refractive index and the spherical number on the scattering of various types of aggregated multi-layered particles are displayed in detail, which may provide critical support for analytically understanding the optical scattering characteristics of aggregated multi-layered biological cells of complex shapes and may find important applications in manipulating multi-layered biological structures.

On minimizers of the 2D Ginzburg–Landau energy with tangential anchoring

We analyze Ginzburg–Landau minimization problems in two dimensions with either a “strong or weak” tangential boundary condition. These problems are motivated by experiments in liquid crystals with boundary defects. The imposition of classical boundary conditions on a liquid crystal surface is referred to as “strong anchoring”, while “weak anchoring” refers to the inclusion of boundary energy terms. In the singular limit when the correlation length tends to zero, we show that boundary defects will be observed for weak anchoring, while both boundary and interior vortices are possible for strong anchoring.

Abstract P5-02-09: Quantitative analysis of fiber-level collagen features in H&E whole-slide images predicts neoadjuvant therapy response in patients with HER2+ breast cancer

Abstract Background: Neoadjuvant treatment (NAT) combining chemotherapy and HER2-targeted agents is frequently administered to HER2-positive (HER2+) breast cancer (BC) patients, with some experiencing a pathological complete response (pCR) and others having residual disease measured by the residual cancer burden (RCB) score. Here, we use a physics-guided machine learning (ML)-based approach to extract fiber-level collagen features from hematoxylin and eosin (H&E)-stained whole slide images (WSIs) and identify collagen-related associations with treatment response in HER2+ patients receiving NAT. Methods: Clinical data and specimens from stage II-III HER2+ BC patients enrolled on the De-escalation to Adjuvant Antibodies Post-pCR to Neoadjuvant THP (DAPHNe; NCT03716180) clinical trial and treated with neoadjuvant paclitaxel/trastuzumab/pertuzumab were analyzed. An ML-based model trained to identify regions of BC tissue as invasive carcinoma, ductal carcinoma in situ (DCIS), diffuse inflammatory infiltrate, stroma, necrosis, or normal tissue was deployed on WSIs of H&E-stained diagnostic core needle biopsies (N=89) to generate tissue overlays. Additional tissue areas were computed from the tissue model predictions using heatmap transformation, including tumor nests (continuous regions predicted as invasive cancer epithelium or DCIS), tumor nest borders (stromal region boundaries 10 μm from tumor nests), and bulk tumor borders (stromal region boundaries 300 μm from aggregated tumor nests). A separate ML-based model trained to identify fiber-level collagen features in WSIs of H&E-stained specimens was also deployed to generate collagen overlays. A fiber feature extraction pipeline was utilized to characterize properties of all identified collagen fibers in the WSI (on the order of hundreds of thousands per slide), including length, width, tortuosity, and angle. These fiber features were then assessed based on their position within the tumor (e.g. relative to the tumor nest border). Combinatorial features (e.g. angle of fibers with respect to tumor boundary) were then explored univariately for associations (N=609) with treatment response. Patients with pCR (RCB=0; N=53) were considered responders, while all other cases (RCBI-III; N=36) were designated non-responders. Due to the small size of the cohort analyzed here, raw p-values are reported. Results: Using estrogen receptor status as a clinical covariate, a logistic regression-based univariate analysis of 609 collagen-associated features revealed six features to strongly associate with pCR (p< 0.05, AUC≥0.75; Table 1). Notable feature themes were identified: 1) fiber tortuosity in tumor nest borders and tumor borders, 2) angle of fibers in tumor border with respect to tumor boundary, and 3) distribution patterns of fiber width in tumor nest borders. The presence of fibers perpendicular to tumor boundary tangents was negatively associated with pCR, as was higher fiber tortuosity and thickness in tumor nest borders. Conclusions: Improved prediction of response to NAT in patients with BC is needed to determine appropriate treatment strategies for each patient. Here, using ML-based models to identify tissue features and collagen fibers, we identify collagen-associated features, measured directly from WSIs of H&E-stained diagnostic BC biopsies, that negatively correlate with pCR. Additional development of this strategy, including the addition of cell identification models and known clinical information, is underway to further refine this novel predictive model. Citation Format: Tan H. Nguyen, Mohammad Mirzadeh, Aaditya Prakash, Emma L. Krause, Jun Zhang, Michael Pyle, Esther R. Ogayo, Harry C. Cramer, Busem Binboga Kurt, Jacqueline Brosnan-Cashman, Michael G. Drage, Stuart Schnitt, Andrew H. Beck, Michael Montalto, Ilan Wapinski, Laura Chambre, Sara Tolaney, Adrienne Waks, Justin Lee, Elizabeth A. Mittendorf. Quantitative analysis of fiber-level collagen features in H&E whole-slide images predicts neoadjuvant therapy response in patients with HER2+ breast cancer [abstract]. In: Proceedings of the 2022 San Antonio Breast Cancer Symposium; 2022 Dec 6-10; San Antonio, TX. Philadelphia (PA): AACR; Cancer Res 2023;83(5 Suppl):Abstract nr P5-02-09.

An SBP-SAT FDTD Subgridding Method Using Staggered Yee’s Grids Without Modifying Field Components for TM Analysis

A summation-by-parts simultaneous approximation term (SBP-SAT) finite-difference time-domain (FDTD) subgridding method is proposed to model geometrically fine structures in this article. Compared with other works, the proposed SBP-SAT FDTD method uses the staggered Yee’s grid without adding or modifying any field components through field extrapolation on the boundaries to make the discrete operators satisfy the SBP property. The accuracy of extrapolation keeps consistency with that of the second-order finite-difference scheme near the boundaries. In addition, the SATs are used to weakly enforce the tangential boundary conditions between multiple mesh blocks with different mesh sizes. With carefully designed interpolation matrices and selected free parameters of the SATs, no dissipation occurs in the whole computational domain. Therefore, its long-time stability is theoretically guaranteed. Four numerical examples are carried out to validate its effectiveness. Results show that the proposed SBP-SAT FDTD subgridding method is stable, accurate, efficient, and easy to implement based on the existing FDTD codes with only a few modifications.

A new global divergence free and pressure-robust HDG method for tangential boundary control of Stokes equations

In Gong et al. (2020), we proposed an HDG method to approximate the solution of a tangential boundary control problem for the Stokes equations and obtained an optimal convergence rate for the optimal control that reflects its global regularity. However, the error estimates depend on the pressure, and the velocity is not divergence free. The importance of pressure-robust numerical methods for fluids was addressed by John et al. (2017). In this work, we devise a new HDG method to approximate the solution of the Stokes tangential boundary control problem; the HDG method is also of independent interest for solving the Stokes equations. This scheme yields a H(div) conforming, globally divergence free, and pressure-robust solution. To the best of our knowledge, this is the first time such a numerical scheme has been obtained for an optimal boundary control problem for the Stokes equations. We also provide numerical experiments to show the performance of the new HDG method and the advantage over the non pressure-robust scheme.

On One Tangential Boundary Value Problem of the Field Theory on a Plane

One tangential boundary value problem of the field theory on a plane for the system of Poisson equations is studied. The peculiarity of the problem is that the boundary condition posed on the desired solution in nonlocal in nature. A weak solution of the problem formulated is to be found. In view of the fact that boundary condition posed on the problem formulated is nonlocal in nature, an appropriate space of functions is introduced, among which the solution of the problem is sought. Since the problem is invariant with regard to the additive vector constant, the search for a solution narrows down to the space of functions having an average integral value equal to zero in the specified domain. The weak solution of the problem formulated is defined, and it is shown that it is well-posed in nature. The main result of the work is the theorem of the existence and uniqueness of a weak solution of the problem. The proof is carried out in a few stages, and an example for the case of a rectangular domain is given.

A compactness result for the div-curl system with inhomogeneous mixed boundary conditions for bounded Lipschitz domains and some applications

For a bounded Lipschitz domain with Lipschitz interface we show the following compactness theorem: Any mathsf {L}_{}^{2}-bounded sequence of vector fields with mathsf {L}_{}^{2}-bounded rotations and mathsf {L}_{}^{2}-bounded divergences as well as mathsf {L}_{}^{2}-bounded tangential traces on one part of the boundary and mathsf {L}_{}^{2}-bounded normal traces on the other part of the boundary, contains a strongly mathsf {L}_{}^{2}-convergent subsequence. This generalises recent results for homogeneous mixed boundary conditions in Bauer et al. (SIAM J Math Anal 48(4):2912-2943, 2016) Bauer et al. (in: Maxwell’s Equations: Analysis and Numerics (Radon Series on Computational and Applied Mathematics 24), De Gruyter, pp. 77-104, 2019). As applications we present a related Friedrichs/Poincaré type estimate, a div-curl lemma, and show that the Maxwell operator with mixed tangential and impedance boundary conditions (Robin type boundary conditions) has compact resolvents.

Birefringence-modulated total internal reflection in liquid crystal shells

The combination of anisotropic boundary conditions and topological constraints acting on a spherical shell of nematic liquid crystal confined between aqueous phases gives rise to peculiar but well-defined configurations of the director field, and thus of the optic axis that defines the impact of the nematic birefringence. While the resulting optics of nematic shells has been extensively investigated in transmission, studies of the reflection behavior are scarce. Here we show that nematic shells exhibit specific light guiding paths mediated by birefringence-modulated total internal reflection (TIR) within the shell. With stabilizers promoting tangential boundary conditions, shells show immobile antipodal spots revealing the locations of maximum effective refractive index, but their intensity is modulated by the polarization of the illuminating light. With normal-aligning stabilizers, shells instead show bright arcs separated by dark spots, and these follow the rotation of the polarization of the illuminating light. Reflection polarizing microscopy thus offers a valuable complement to the more common characterization in transmission, adding data that can be helpful for accurately mapping out director fields in shells of any liquid crystal phase. Moreover, the TIR-mediated light guiding paths may offer interesting handles to localize photopolymerization of reactive liquid crystal shells or to dynamically modulate the response of light-triggered liquid crystal elastomer shell actuators.

Defect absorption and emission for p-atic liquid crystals on cones.

We investigate the ground-state configurations of two-dimensional liquid crystals with p-fold rotational symmetry (p-atics) on fixed curved surfaces. We focus on the intrinsic geometry and show that isothermal coordinates are particularly convenient as they explicitly encode a geometric contribution to the elastic potential. In the special case of a cone with half-angle β, the apex develops an effective topological charge of -χ, where 2πχ=2π(1-sinβ) is the deficit angle of the cone, and a topological defect of charge σ behaves as if it had an effective topological charge Q_{eff}=(σ-σ^{2}/2) when interacting with the apex. The effective charge of the apex leads to defect absorption and emission at the cone apex as the deficit angle of the cone is varied. For total topological defect charge 1, e.g., imposed by tangential boundary conditions at the edge, we find that for a disk the ground-state configuration consists of p defects each of charge +1/p lying equally spaced on a concentric ring of radius d=(p-1/3p-1)^{1/2p}R, where R is the radius of the disk. In the case of a cone with tangential boundary conditions at the base, we find three types of ground-state configurations as a function of cone angle: (i) for sharp cones, all of the +1/p defects are absorbed by the apex; (ii) at intermediate cone angles, some of the +1/p defects are absorbed by the apex and the rest lie equally spaced along a concentric ring on the flank; and (iii) for nearly flat cones, all of the +1/p defects lie equally spaced along a concentric ring on the flank. Here the defect positions and the absorption transitions depend intricately on p and the deficit angle, which we analytically compute. We check these results with numerical simulations for a set of commensurate cone angles and find excellent agreement.

Light Interaction with Cluster Chiral Nanostructures by High-Order Bessel Beam

Interactions between cluster chiral nanoparticles and a high-order Bessel beam (HOBB) with arbitrary illuminations are investigated. The generalized Lorenz–Mie theory (GLMT) is applied to derive the expansions of HOBB. Based on the additional theorem, multiple scattering results of cluster chiral nanoparticles are obtained by taking into account the tangential continuous boundary conditions. The present theory and codes proved to be effective when confronted with the simulations obtained from the Computer Simulation Technology (CST) software. Numerical results concerning the effects of beam order, beam conical angle, incident angles, beam polarization state, the chirality, and the material loss on the scattering of various types of aggregated chiral particles are displayed in detail, including the linearly chiral sphere chain, the chiral cube array, and the complex models composed of aggregated chiral spheres. This study may provide critical support to analytically understand the optical scattering characteristics with aggregated chiral particles of complex shapes, and may find important applications in manipulating collective chiral particles.

Beltrami’s completeness for 𝕃p symmetric matrix fields

Geymonat extended Gurtin’s result on Beltrami’s completeness to [Formula: see text]-case. The first objective of this paper is to generalize Geymonat’s result to [Formula: see text]-case with tangential and normal boundary conditions. Maggiani et al. proposed an extension of Beltrami’s-type decomposition for symmetric matrix fields in [Formula: see text] when [Formula: see text] is simply connected domain and of class [Formula: see text]. The second objective of this paper is to present more extensions of the above decomposition when the domain [Formula: see text] is only of class [Formula: see text] and not necessarily simply connected. The third objective of this paper is to use the previous characterizations of matrix fields to solve some variants of the bi-Laplacian problem.

Construction of [formula omitted] planar Hermite interpolants with prescribed arc lengths

In this paper we address the problem of constructing G2 planar Pythagorean–hodograph (PH) spline curves, that interpolate points, tangent directions and curvatures, and have prescribed arc-length. The interpolation scheme is completely local. Each spline segment is defined as a PH biarc curve of degree 7, which results in having a closed form solution of the G2 interpolation equations depending on four free parameters. By fixing two of them to zero, it is proven that the length constraint can be satisfied for any data and any chosen ratio between the two boundary tangents. Length interpolation equation reduces to one algebraic equation with four solutions in general. To select the best one, the value of the bending energy is observed. Several numerical examples are provided to illustrate the obtained theoretical results and to numerically confirm that the approximation order is 5.