Exterior Boundary Research Articles

Analytical free vibration solutions of rectangular edge-cracked plates by the finite integral transform method

Analytical free vibration solutions of rectangular edge-cracked thin plates are obtained by the finite integral transform (FIT) method. While the previous studies focus on the mechanical behaviors of intact plates, this study establishes an FIT-based new scheme for analyzing the free vibration characteristics of rectangular thin plates with an edge crack. The plates may have any combined conditions of simply supported, clamped, and free boundaries. In specific solution scheme, an edge-cracked plate is decomposed into four subdomains, and a double cosine FIT is imposed on the fourth-order partial differential governing equations of each subdomain, which yields the relationship between the transformed modal deflections and naturally defined unknowns. By substituting the exterior boundary conditions of the plate, continuity conditions across the subdomains, and free conditions at the crack into the inverse transforms, the natural frequencies and unknowns are successively determined via the established homogeneous linear algebraic equations. The modal deflections are provided by further substituting the obtained unknowns into the inverse transforms. Comprehensive benchmark results for the edge-cracked plates under representative boundary conditions are shown. The results are well verified by other solution methods. A parametric study quantitatively reveals the effects of boundary conditions and crack length on the natural frequencies of edge-cracked plates. Due to the rigorous but easy-to-implement mathematical derivations, this study presents a novel solid way for exploring analytical solutions of vibration problems.

A discontinuous Galerkin method for sequences of earthquakes and aseismic slip on multiple faults using unstructured curvilinear grids

SUMMARYPhysics-based simulations provide a path to overcome the lack of observational data hampering a holistic understanding of earthquake faulting and crustal deformation across the vastly varying space–time scales governing the seismic cycle. However, simulations of sequences of earthquakes and aseismic slip (SEAS) including the complex geometries and heterogeneities of the subsurface are challenging. We present a symmetric interior penalty discontinuous Galerkin (SIPG) method to perform SEAS simulations accounting for the aforementioned challenges. Due to the discontinuous nature of the approximation, the spatial discretization natively provides a means to impose boundary and interface conditions. The method accommodates 2-D and 3-D domains, is of arbitrary order, handles subelement variations in material properties and supports isoparametric elements, that is, high-order representations of the exterior boundaries, interior material interfaces and embedded faults. We provide an open-source reference implementation, Tandem, that utilizes highly efficient kernels for evaluating the SIPG linear and bilinear forms, is inherently parallel and well suited to perform high-resolution simulations on large-scale distributed memory architectures. Additional flexibility and efficiency is provided by optionally defining the displacement evaluation via a discrete Green’s function approach, exploiting advantages of both the boundary integral and volumetric methods. The optional discrete Green’s functions are evaluated once in a pre-computation stage using algorithmically optimal and scalable sparse parallel solvers and pre-conditioners. We illustrate the characteristics of the SIPG formulation via an extensive suite of verification problems (analytic, manufactured and code comparison) for elastostatic and quasi-dynamic problems. Our verification suite demonstrates that high-order convergence of the discrete solution can be achieved in space and time and highlights the benefits of using a high-order representation of the displacement, material properties and geometries. We apply Tandem to realistic demonstration models consisting of a 2-D SEAS multifault scenario on a shallowly dipping normal fault with four curved splay faults, and a 3-D intersecting multifault scenario of elastostatic instantaneous displacement of the 2019 Ridgecrest, CA, earthquake sequence. We exploit the curvilinear geometry representation in both application examples and elucidate the importance of accurate stress (or displacement gradient) representation on-fault. This study entails several methodological novelties. We derive a sharp bound on the smallest value of the SIPG penalty ensuring stability for isotropic, elastic materials; define a new flux to incorporate embedded faults in a standard SIPG scheme; employ a hybrid multilevel pre-conditioner for the discrete elasticity problem; and demonstrate that curvilinear elements are specifically beneficial for volumetric SEAS simulations. We show that our method can be applied for solving interesting geophysical problems using massively parallel computing. Finally, this is the first time a discontinuous Galerkin method is published for the numerical simulations of SEAS, opening new avenues to pursue extreme scale 3-D SEAS simulations in the future.

Bernoulli Free Boundary Problems Under Uncertainty: The Convex Case

Abstract The present article is concerned with solving Bernoulli’s exterior free boundary problem in the case of an interior boundary that is random. We provide a new regularity result on the map that sends a parametrization of the inner boundary to a parametrization of the outer boundary. Moreover, assuming that the interior boundary is convex, also the exterior boundary is convex, which enables to identify the boundaries with support functions and to determine their expectations. We in particular construct a confidence region for the outer boundary based on Aumann’s expectation and provide a numerical method to compute it.

The vector Helmholtz equation revisited: Inverse obstacle scattering

The vector Helmholtz equation, from a mathematical point of view, provides a generalization of the time-harmonic Maxwell equations for the propagation of time-harmonic electromagnetic waves. After reviewing some classic results on the two main exterior boundary value problems for the vector Helmholtz equation, i.e., the so-called electric boundary condition and the magnetic boundary condition, we prove reciprocity results for scattering of plane waves and point sources. Then we make use of them for obtaining uniqueness results for the inverse obstacle scattering problem to determine the shape of the scatterer from knowing the far field pattern of the scattered waves and results on the so-called far field operator for the two scattering problems. These results are generalizations of corresponding results for the Maxwell equations and analogues to results for the Dirichlet and Neumann boundary condition for the scalar Helmholtz equation. We also briefly consider the extension of the so-called DB boundary condition from the Maxwell equations to the vector Helmholtz equation.

Operator estimates for homogenization of the Robin Laplacian in a perforated domain

Let ε>0 be a small parameter. We consider the domain Ωε:=Ω∖Dε, where Ω is an open domain in Rn, and Dε is a family of small balls of the radius dε=o(ε) distributed periodically with period ε. Let Δε be the Laplace operator in Ωε subject to the Robin condition ∂u∂n+γεu=0 with γε≥0 on the boundary of the holes and the Dirichlet condition on the exterior boundary. Kaizu (1985, 1989) and Brillard (1988) have shown that, under appropriate assumptions on dε and γε, the operator Δε converges in the strong resolvent sense to the sum of the Dirichlet Laplacian in Ω and a constant potential. We improve this result deriving estimates on the rate of convergence in terms of L2→L2 and L2→H1 operator norms. As a byproduct we establish the estimate on the distance between the spectra of the associated operators.

Well-Posedness for Nonlinear Parabolic Stochastic Differential Equations with Nonlinear Robin Conditions

In this paper, we present the existence and uniqueness of strong probabilistic solutions for nonlinear parabolic Stochastic Partial Differential Equations (SPDEs) with nonlinear Robin boundary conditions in a domain with holes. On the boundary of the holes, a nonlinear Robin condition is imposed, while a homogeneous Dirichlet condition is prescribed on the exterior boundary. The coefficient matrix is assumed to be symmetric, while the nonlinear random forces are assumed to satisfy some types of regularities. We use Galerkin’s approximation method, probabilistic compactness results and some results from stochastic calculus.

A hexagon-based method for polygon generalization using morphological operators

Numerous methods based on square rasters have been proposed for polygon generalization. However, these methods ignore the inconsistent distance measurement among neighborhoods of squares, which may result in an imbalanced generalization in different directions. As an alternative raster, a hexagon has consistent connectivity and isotropic neighborhoods. This study proposed a hexagon-based method for polygon generalization using morphological operators. First, we defined three generalization operators: aggregation, elimination, and line simplification, based on hexagonal morphological operations. We then used corrective operations with selection, skeleton, and exaggeration to detect, classify, and correct the unreasonably reduced narrow parts of the polygons. To assess the effectiveness of the proposed method, we conducted experiments comparing the hexagonal raster to square raster and vector data. Unlike vector-based methods in which various algorithms simplified either areal objects or exterior boundaries, the hexagon-based method performed both simplifications simultaneously. Compared to the square-based method, the results of the hexagon-based method were more balanced in all neighborhood directions, matched better with the original polygons, and had smoother simplified boundaries. Moreover, it performed with shorter running time than the square-based method, where the minimal time difference was less than 1 min, and the maximal time difference reached more than 50 mins.

Exact Internal Controllability for a Problem with Imperfect Interface

In this paper, we study the internal exact controllability for a second order linear evolution equation defined in a two-component domain. On the interface, we prescribe a jump of the solution proportional to the conormal derivatives, meanwhile a homogeneous Dirichlet condition is imposed on the exterior boundary. Due to the geometry of the domain, we apply controls through two regions which are neighborhoods of a part of the external boundary and of the whole interface, respectively. Our approach to internal exact controllability consists in proving an observability inequality by using the Lagrange multipliers method. Eventually, we apply the Hilbert Uniqueness Method, introduced by Lions, which leads to the construction of the exact control through the solution of an adjoint problem. Finally, we find a lower bound for the control time depending not only on the geometry of our domain and on the matrix of coefficients of our problem but also on the coefficient of proportionality of the jump with respect to the conormal derivatives.

Solution of the Exterior Boundary Value Problem for the Helmholtz Equation Using Overlapping Domain Decomposition

Solution of the Exterior Boundary Value Problem for the Helmholtz Equation Using Overlapping Domain Decomposition

Heat transport properties within living biological tissues with temperature-dependent thermal properties

Understanding of the heat transport within living biological tissues is crucial to effective heat treatments. The heat transport properties of living biological tissues with temperature-dependent properties are explored in this paper. Taking into account of variable physical properties, the governing equation of temperature is first derived in the context of the dual-phase-lags model (DPL). An effective method, according to the Laplace transform and a linearization technique, is then employed to solve this nonlinear governing equation. The temperature distribution of a biological tissue exposed to a pulsed heat flux on its exterior boundary, which frequently happens in various heat treatments, is predicted and analyzed. The results state that a lower temperature can be predicted when temperature dependence is considered in the heating process. The contributions of key thermal parameters are different and dependent on the ratio of phase lag and the amplitude of the exterior pulsed heat flux.

Research on temperatures located within facet coating layers along z-axis of semiconductor lasers

The critical temperatures of semiconductor laser facet coating layers were studied via measurements from three prongs on the surface (coating exterior boundary), on the facet (coating interior boundary) and along the inner cavity, with temperatures measured via infrared imaging, thermoreflectance and forward-voltage methods, respectively. We subdivided the facet microstructure temperatures and demonstrated the feasibility of facet temperature measurement via thermoreflectance for the reflected light carrying the temperature information mainly comes from the coating interior boundary. Facet reflectivity transients of an AlGaInAs/AlGaAs laser diode are traced using an optical setup featuring fiber transmission and fiber detection. The highest temperature along the z-axis was measured at the facet, with a temperature rise exceeding two times that at the surface, which confirmed the significance of the subdivision, and approaching four times that in the inner cavity. Thermal transient analysis from optical and electrical methods presents similar results; steady temperature measurements as a function of injection current indicate light absorption’s contribution to facet heating.

A Boundary Element Procedure for 3D Electromagnetic Transmission Problems with Large Conductivity

We consider the scattering of time-periodic electromagnetic fields by metallic obstacles, or the eddy current problem. In this interface problem, different sets of Maxwell equations must be solved both in the obstacle and outside it, while the tangential components of both electric and magnetic fields are continuous across the interface. We describe an asymptotic procedure, applied for large conductivity, which reflects the skin effect in metals. The key to our method is a special integral equation procedure for the exterior boundary value problems corresponding to perfect conductors. The asymptotic procedure leads to a great reduction in complexity for the numerical solution, since it involves solving only the exterior boundary value problems. Furthermore, we introduce a FEM/BEM coupling procedure for the transmission problem and consider the implementation of Galerkin’s elements for the perfect conductor problem, and present numerical experiments.

Theoretical and quantitative evaluation of hybrid PML-ABCs for seismic wave simulation

A good artificial boundary treatment in a seismic wave grid-based numerical simulation can reduce the size of the computational region and increase the computational efficiency, which is becoming increasingly important for seismic migration and waveform inversion tasks requiring hundreds or thousands of simulations. Two artificial boundary techniques are commonly used: perfectly matched layers (PMLs), which exhibit the excellent absorption performance but impose a greater computational burden by using finite layers to gradually reduce wave amplitudes; and absorbing boundary conditions (ABCs), which have the high computational efficiency but are less effective in absorption because they employ the one-way wave equation at the exterior boundary. Naturally, PMLs have been combined with ABCs to reduce the number of PMLs, thus improving the computational efficiency; many studies have proposed such hybrid PMLs. Depending on the equations from which the ABCs are derived, there are two hybrid PML variants: the PML+unstretched ABC (UABC), in which the ABC is derived from a physical equation; or the PML+stretched ABC (SABC), in which the ABC is derived from the PML equation. Even though all the previous studies concluded that hybrid PMLs can improve the absorption performance, none of them quantified how many PMLs can be removed by combining the PML with the ABC compared with the pure PML. In this paper, we systematically study the absorption performance of the two hybrid PML variants. We develop a method to distinguish the artificial reflections from the PML-interior interface and those caused by the PML exterior boundary to accurately approximate the additional absorption achieved by using the UABC and the SABC. The reflection coefficients based on a theoretical derivation and numerical tests both show that the UABC amplifies most reflections and is not recommended in any situation; conversely, the SABC can always diminish reflections, but the additional absorption achieved by the SABC is relatively poor and cannot effectively reduce the number of PMLs. In contrast, we find that simply increasing the damping parameter improves absorption better than the PML+SABC. Our results show that the improvement in absorption achieved by combining the PML with either the SABC or the UABC is not better than that obtained by simply adjusting the damping profile of the PML; thus, combining the PML with the ABC is not recommended in practice.

Mathematical Modeling of the Evolution of the Exterior Boundary in Spheroidal Tumour Growth

The present paper concerns the formulation and the evolution of the non symmetrical growth of an avascular cancerous cell colony in an analytical mathematical fashion. Although most of the existing research considers spherical tumours, here we work in the frame of a more general case of the prolate spheroidal geometry. The tumour lies inside a host spheroidal shell which provides vital nutrients, receives the debris of the dead cells and also transmittes to the tumour the pressure imposed by the surrounding on its exterior boundary. Under the aim of studying the evolution of the exterior tumour boundary, we focus on the exterior conditions under which such a geometrical structure can be sustained. To that purpose, the corresponding nutrient concentration, the inhibitor concentration and the pressure field are calculated analytically providing the necessary data for the evolution equation to be solvable. It turns out that an avascular tumour can exhibit a prolate spheroidal growth only if the external conditions for the nutrient supply and the transversally isotropic pressure field have a specific form, which is consistent with the tumour evolution. Additionally, our model exhibits a geometrical reduction to special cases and, mainly, to the spherical geometry in order to recover the existing results for the sphere.

A finite element model to predict the double hearing protector effect on an in-house acoustic test fixture.

The sound attenuation of double hearing protectors (DHPs), earplugs combined with earmuffs, generally falls short of the sum of each single protector's attenuation when used independently. This phenomenon, referred to as the DHP effect, is found to be related to structure-borne sound transmission involving the outer ear and can also be observed on acoustic test fixtures (ATFs). At present, it still remains not fully understood, and no available model can help demonstrate the associated sound transmission mechanisms. In this work, a finite element model is proposed to study the DHP effect on an ATF between 100 Hz and 5 kHz. Power balances are calculated with selected configurations of the ATF in order to (i) quantify the contribution of each sound path, and study the effects of (ii) the artificial skin and (iii) acoustic excitation on the ATF exterior boundaries. The DHP effect is shown to originate from the structure-borne sound power injected from the ATF boundaries and/or earmuff cushion. The important influence of earcanal wall vibration is highlighted when the skin is accounted for. The simulation results allow for gaining more insight into the sound transmission through a DHP/ATF system.

Efficient light coupling between conventional silicon photonic waveguides and quantum valley-Hall topological interfaces.

Robust and efficient light coupling into and out of quantum valley-Hall (QVH) topological interfaces within near-infrared frequencies is demanded in order to be integrated into practical two-dimensional (2D) optical chips. Here, we numerically demonstrate efficient light coupling between a QVH interface and a pair of input/output silicon photonic waveguides in the presence of photonic crystal line defects. When the topological QVH interface is directly end-butt coupled to the silicon waveguides, the input-to-output transmission efficiency is lower than 50% and the exterior boundaries associated with a QVH interface also cause inevitable back-reflections and high-order scatterings, further reducing the transmission efficiency. The transmission efficiency is substantially increased to 95.8% (94.3%) when photonic crystal line defects are introduced between the bridge (zigzag) QVH interface and the waveguides. The buffering line defect mode, with an effective group refractive index between the interface state and the waveguide mode will ease their mode profile conversion. The design we present here brings no fabrication complexity and may be used as a guide for future implementation of on-chip 2D topological photonics.

Deep Learning for Viscoelastic Fluids and Turbulent Diffusion

This study aims to predict the conformation stress field of viscoelastic, drag-reducing turbulent channel flow and to estimate the passive-scalar diffusion in Newtonian turbulent channel flow. In general, understanding of the nature of wall turbulence and predicting of turbulent phenomena have been based on statistical features, but here we would discuss the possibility of extracting non-trivial features from instantaneous fields with the help of machine learning. If we can extract features unique to the instantaneous field, which are lost when equalized by statistical processing, it could be possible to predict the interior (intrinsic physics) and exterior (initial and/or boundary conditions) from limited spatial, temporal information. Using deep learning of CNN such as U-Net, we demonstrated the reproduction of the conformation stress field from the instantaneous velocity field and the estimation of the diffusion point source location from the downstream concentration distribution based on instantaneous local information. We have successfully demonstrated the usefulness of instantaneous local information, which might be lost in statistics, and confirmed the possibility of using deep learning to construct a viscoelastic-fluid surrogate model and a diffusion-source estimation tool.

Application of a constructive method in solving the composite reservoir model of dual-porosity media with fractal characteristics

In this paper, we apply a constructive method to solve the extended Bessel differential equations with boundary value problem(BVP). Considering wellbore storage and skin effects, a dual-porosity composite reservoir model with fractal characteristics and three typical exterior boundary conditions is established. After symbolic comparison, the reservoir model is identical to the BVP proposed above, and the solving of the reservoir model is naturally reduced to the solving of the BVP. Based on the Laplace transform and a constructive method, the Laplacian space solutions of fracture system pressure and wellbore pressure in the two zones are obtained. The type curves of wellbore pressure and the pressure derivative are plotted with the aid of the Stehfest numerical inversion technique, the flow regimes are identified, and the effects of fractal and reservoir parameters on the type curves are analyzed. Compared with the traditional method, the similar constructive method is more efficient and fast in solving complex reservoir models, which is of great value in designing and implementing well-test analysis algorithms for complex reservoirs.

Charged stellar model with three layers

We establish new charged stellar models from the Einstein-Maxwell field equations for relativistic superdense objectsoutfitted with three layers. The core layer is described by a linear equation of state (EoS) describing quark matter, while the intermediate layer is described by a Bose-Einstein condensate EoS for Bose-Einstein condensate matter and the envelope layers satisfying a quadratic EoS for the neutron fluid. We have specified a new choice of the electric field and one of the metric potentials. It is interesting to note that the choice of electric field in this model can be set to vanish and we can regain earlier neutral models. Plots generated depict that the matter variables, gravitational potentials and other physical conditions are consistent with astrophysical studies. The interior layers and exterior boundary are also matched.

Isolation and reconstruction of cardiac mitochondria from SBEM images using a deep learning-based method

Mitochondrial morphological defects are a common feature of diseased cardiac myocytes. However, quantitative assessment of mitochondrial morphology is limited by the time-consuming manual segmentation of electron micrograph (EM) images. To advance understanding of the relation between morphological defects and dysfunction, an efficient morphological reconstruction method is desired to enable isolation and reconstruction of mitochondria from EM images. We propose a new method for isolating and reconstructing single mitochondria from serial block-face scanning EM (SBEM) images. CDeep3M, a cloud-based deep learning network for EM images, was used to segment mitochondrial interior volumes and boundaries. Post-processing was performed using both the predicted interior volume and exterior boundary to isolate and reconstruct individual mitochondria. Series of SBEM images from two separate cardiac myocytes were processed. The highest F1-score was 95% using 50 training datasets, greater than that for previously reported automated methods and comparable to manual segmentations. Accuracy of separation of individual mitochondria was 80% on a pixel basis. A total of 2315 mitochondria in the two series of SBEM images were evaluated with a mean volume of 0.78 µm3. The volume distribution was very broad and skewed; the most frequent mitochondria were 0.04–0.06 µm3, but mitochondria larger than 2.0 µm3 accounted for more than 10% of the total number. The average short-axis length was 0.47 µm. Primarily longitudinal mitochondria (0–30 degrees) were dominant (54%). This new automated segmentation and separation method can help quantitate mitochondrial morphology and improve understanding of myocyte structure–function relationships.