Interface Problems Research Articles

Analysis and computation of a weak Galerkin scheme for solving the 2D/3D stationary Stokes interface problems with high-order elements

Abstract In this paper, we present a high-order weak Galerkin finite element method (WG-FEM) for solving the stationary Stokes interface problems with discontinuous velocity and pressure in ℝ d , d = 2, 3. This WG method is equipped with stable finite elements consisting of usual polynomials of degree k ⩾ 1 for the velocity and polynomials of degree k − 1 for the pressure, both are discontinuous. Optimal convergence rates of order k + 1 for the velocity and order k for the pressure are established in L 2-norm on hybrid meshes. Numerical experiments verify the expected order of accuracy for both two-dimensional and three-dimensional examples. Moreover, numerically it is shown that the proposed WG algorithm is able to accommodate geometrically complicated and very irregular interfaces having sharp edges, cusps, and tips.

WELL-POSEDNESS OF A NONLINEAR INTERFACE PROBLEM DESCRIBED BY HEMIVARIATIONAL INEQUALITIES VIA BOUNDARY INTEGRAL OPERATORS

This paper is devoted to the well-posedness of a novel nonlinear interface problem on an unbounded domain with nonmonotone set-valued transmission conditions. This interface problem involves a nonlinear monotone partial differential equation in the interior domain and the Laplacian in the exterior domain. Such a scalar interface problem models nonmonotone frictional contact of elastic infinite media. The variational formulation of the interface problem leads to a hemivariational inequality (HVI), which however lives on the unbounded domain, and thus cannot be analyzed in a reflexive Banach space setting. Boundary integral methods lead to another HVI that is amenable to functional analytic methods using standard Sobolev spaces on the interior domain and Sobolev spaces of fractional order on the coupling boundary. Broadening the scope of the paper, we consider extended real-valued HVIs augmented by convex extended real-valued functions. Under a smallness hypothesis, we provide existence and uniqueness results and, moreover, establish a stability result for extended real-valued HVIs with respect to the extended real-valued function as a parameter. Based on the latter general stability result, we provide various stability results for the interface problem, as well as the stability of a related bilateral obstacle interface problem with respect to the obstacles.

Thermo-elastic Characteristics of Thermal Barrier Coating in a Gas Turbine Combustion Chamber with an Intermediate FGM Layer

This study is focused on the analysis of thermos-elastic characteristics of a gas turbine combustion chamber, where the inner surface experiences a higher temperature while the outer region of the wall should be thermally conductive to dissipate the heat of combustion. Using a single material is not suitable to ensure the above two requirements simultaneously. Instead of using a single material, a layered combustion chamber with high-temperature material at the inner region and thermally conductive material at the outer region can be a solution. This produces the problem of a sharp interface where disbonding occurs due to high thermal stresses. The problem of sharp interface can be eliminated by using a functionally graded material (FGM) layer in between the inner and outer layers of the combustion chamber. The present study considers a gas turbine combustion chamber consisting of three layers: the inner layer is used as a thermal barrier coating, the intermediate layer is a smooth transition from the inner layer to the outer layer, and the outer layer is a thermally conductive material. ANSYS simulation is used for the analysis of thermos-elastic characteristics of the combustion chamber with homogeneous and FGM intermediate layers. The effect of linear and nonlinear material distributions in the FGM layer on the thermos-elastic characteristics is studied. A comparison of results reveals that thermal stress smoothly changes from the inner layer to the outer layer in the case of the FGM intermediate layer compared to the homogeneous intermediate layer. This suggests that a combustion chamber with an FGM intermediate layer is more reliable than a homogeneous intermediate layer.

On Well-posedness of the Two-dimensional MHD–Maxwell Free Interface Problem

On Well-posedness of the Two-dimensional MHD–Maxwell Free Interface Problem

Research Progress of β-Ketoenamine-Linked Covalent Organic Frameworks for Photocatalytic Hydrogen Evolution

β-Ketoamide covalent organic frameworks (COFs), also named Tp-COFs, are considered to be a milestone material in the history of photocatalysts because of their excellent visible-light absorption, high crystallinity, ultra-high stability and structural diversity. In recent years, a large number of Tp-COFs and their composites have been successfully constructed based on molecular or composite engineering strategies, and exhibited splendid photocatalytic water splitting activity. In comparison with a composite strategy, the molecular engineering technique effectively avoids interface problems by designing and preparing frameworks at the molecular level. Therefore, it is necessary to timely summarize the construction of Tp-COF photocatalysts based on the molecular engineering strategy, so as to provide some theoretical basis and enlightenment for the subsequent development of high-performance Tp-COFs. Finally, the shortcomings and challenges of this technique and personal views on the further development of Tp-COFs are presented.

On optimal control in a nonlinear interface problem described by hemivariational inequalities

Abstract This article is devoted to the existence of optimal controls in various control problems associated with a novel nonlinear interface problem on an unbounded domain with non-monotone set-valued transmission conditions. This interface problem involves a nonlinear monotone partial differential equation in the interior domain and the Laplacian in the exterior domain. Such a scalar interface problem models non-monotone frictional contact of elastic infinite media. The variational formulation of the interface problem leads to a hemivariational inequality (HVI), which, however, lives on the unbounded domain, and thus cannot be analyzed in a reflexive Banach space setting. Boundary integral methods lead to another HVI that is amenable to functional analytic methods using standard Sobolev spaces on the interior domain and Sobolev spaces of fractional order on the coupling boundary. Broadening the scope of this article, we consider extended real-valued HVIs augmented by convex extended real-valued functions. Under a smallness hypothesis, we provide existence and uniqueness results; moreover, we establish a stability result with respect to the extended real-valued function as a parameter. Based on the latter stability result, we prove the existence of optimal controls for four kinds of optimal control problems: distributed control on the bounded domain, boundary control, simultaneous boundary-distributed control governed by the interface problem, and control of the obstacle driven by a related bilateral obstacle interface problem.

Interphase design enabling stable cycling of all-solid-state lithium metal batteries by in-situ X-ray photoelectron spectroscopy lithium metal sputtering

Interface compatibility is one of the key issues for all-solid-state lithium metal batteries (ASSLMBs). The interface problems include the formation of lithiophobic phases (e.g., Li2CO3) on the surface of the solid electrolytes to worsen the solid-solid interface contact and the continuous parasitic reactions between the highly reactive lithium metal and the electrolyte during cycling. In this work, the in-situ XPS (X-ray Photoelectron Spectroscopy) characterization is used to study the interface between lithium metal and Li1.5Al0.5Ge1.5(PO4)3 (LAGP) solid electrolyte using different modification strategies. An ultra-thin composite modification layer of 6 nm Au-6 nm LiF is deposited on the LAGP, which is called C-LAGP. The Li|C-LAGP|Li symmetric cell can be stably cycled for more than 800 h at room temperature at the current density of 0.1 mA cm−2, and the critical current density can reach 1 mA cm−2. The LiFePO4|C-LAGP|Li solid-state battery delivers good cycling stability and high reversible capacity at room temperature. All these good electrochemical performances are resulted from the well-constructed interfacial structure at the solid electrolyte/Li contact. This work provides new ideas and characterization methods for the design of artificial modification layers to solve the interface issues for all-solid-state lithium metal batteries.

Stability of the rarefaction wave in the singular limit of a sharp interface problem for the compressible Navier-Stokes/Allen-Cahn system

Stability of the rarefaction wave in the singular limit of a sharp interface problem for the compressible Navier-Stokes/Allen-Cahn system

A novel surface-derivative-free of jumps AIIM with triangulated surfaces for 3D Helmholtz interface problems

Triangular surface-based 3D IIM (Immersed Interface Method) algorithms face major challenges due to the need to calculate surface derivative of jumps. This paper proposes a fast, easy-to-implement, surface-derivative-free of jumps, augmented IIM (AIIM) with triangulated surfaces for 3D Helmholtz interface problems for the first time, which combines the simplified AIIM with domain decomposed and embedding techniques. The computational domain is divided into sub-domains along the interface and the solutions of sub-domains are continuously extended into larger regular domains by embedding. The jumps in normal derivative of solution along the interfaces in the extended domains are introduced as unknowns to impose the original jump relations. The original problem is simplified into Helmholtz interface problems with constant coefficients by coupling them with the augmented equation, which is then solved using fast simplified AIIM. This approach eliminates the need to compute surface derivatives of jumps, making implementation of 3D IIM based on triangulated surfaces fairly simple. Numerical results demonstrate that the algorithm is efficient and can achieve the overall second-order accuracy.

Numerical assessment of hyperbolic type double interface problems via Haar wavelets

In this manuscript, we have addressed wave propagation challenges within heterogeneous media and hyperbolic interface model. A hybrid numerical approach is introduced for solving these problems, combining the finite difference method and Haar wavelet collocation method. The method entails approximating the second-order space partial derivative through a truncated Haar wavelet series, while the temporal derivative is approximated using finite difference method. For linear hyperbolic interface model, the resulting algebraic systems are solved using the Gauss elimination technique. In the case of non-linear problems, the nonlinearity is addressed through the quasi-Newton linearization formula. To assess the accuracy of the proposed technique, we compute the computational rate of convergence, root mean square errors and maximum absolute errors employing various collocation points. The proposed method perform very well and produces a stable solution if sharp transitions exists in the solution space or if there is a discontinuity between initial and boundary conditions, whereas, the other existing method loses its accuracy in such cases. The numerical experiments, stability and rate of convergence both theoretical and computational, confirm the accuracy and diverse applicability of the method.

Dynamic anti-planar characteristics of circular holes in laminated structures under the action of line source

Based on the complex variable function method and the wave function expansion method, the dynamic anti-plane characteristics of the circular hole in the laminated structure under line source excitation were studied. To deal with the interface problem within the laminated structure, the idea of “fit” is adopted to construct continuity conditions that satisfy displacement and stress. The circular hole boundary stress-free condition is used, the wave field is constructed using the mirror method, and the unknown coefficients of the scattered wave field are solved through Fourier series expansion, and finally the total wave field in the laminated structure is obtained. Taking the dynamic stress distribution of a circular hole in a laminated structure as a specific calculation example, the effects of changes in line source position, layer thickness, and medium hard-soft ratio on the dynamic stress concentration coefficient around the circular hole are discussed. The results show that when the line source load is applied directly above the circular hole and the lower layer is narrow, the dynamic stress concentration coefficient will have a sudden change compared with other situations. When other conditions are the same, under low, medium, and high-frequency wave fields, the dynamic stress concentration caused by different soft-to-hard ratios shows completely different rules. These rules can be used to better detect defects in laminated structures.

The least squares finite element method for elasticity interface problem on unfitted mesh

In this paper, we propose and analyze the least squares finite element methods for the linear elasticity interface problem in the stress-displacement system on unfitted meshes. We consider the cases that the interface is $C^2$ or polygonal, and the exact solution $(\bsigma, \bu)$ belongs to $H^s(\div; \Omega_0 \cup \Omega_1) \times H^{1+s}(\Omega_0 \cup \Omega_1)$ with $s > 1/2$. Two types of least squares functionals are defined to seek the numerical solution. The first is defined by simply applying the $L^2$ norm least squares principle, and requires the condition $s \geq 1$. The second is defined with a discrete minus norm, which is related to the inner product in $H^{-1/2}(\Gamma)$. The use of this discrete minus norm results in a method of optimal convergence rates and allows the exact solution has the regularity of any $s > 1/2$. The stability near the interface for both methods is guaranteed by the ghost penalty bilinear forms and we can derive the robust condition number estimates. The convergence rates under $L^2$ norm and the energy norm are derived for both methods. We illustrate the accuracy and the robustness of the proposed methods by a series of numerical experiments for test problems in two and three dimensions.

Surface-subsurface filtration transport with seawater intrusion: multidomain mixed variational evolution problems

Surface-subsurface filtration transport with seawater intrusion phenomena are formulated and variationally analyzed, as coupled multimedia mixed pairs of free boundary interface problems. Physically, multidomain subsurface mixed velocity-pressure fractional Darcian flow models coupled with surface evolution Stokesian mixed flows are considered. Specifically, two-phase air-fresh water above the sea level and fresh water-seawater characterizations are considered. Internal boundary synchronizing transmission conditions of multidomain nonoverlapping decompositions are modeled in terms of variational Lagrangian dual subpotential maximal monotone inclusions. Similarly, filtration transport coupling interface transmission constraints are implemented by mass flux-velocity-pressure Lagrange dual multipliers as solutions of subpotential subdifferential equations.

Cryogenic flip-chip interconnection for silicon qubit devices

Interfacing qubits with peripheral control circuitry poses one of the major common challenges toward realization of large-scale quantum computation. Spin qubits in silicon quantum dots (QDs)are particularly promising for scaling up, owing to the potential benefits from the know-how of the semiconductor industry. In this paper, we focus on the interposer technique as one of the potential solutions for the quantum–classical interface problem and report DC and RF characterization of a silicon QD device mounted on an interposer. We demonstrate flip-chip interconnection with the qubit device down to 4.2 K by observing Coulomb diamonds. We furthermore propose and demonstrate a laser-cut technique to disconnect peripheral circuits no longer in need. These results may pave the way toward system-on-a-chip quantum–classical integration for future quantum processors.

A unified immersed finite element error analysis for one-dimensional interface problems

It has been known that the traditional scaling argument cannot be directly applied to the error analysis of immersed finite elements (IFE) because, in general, the spaces on the reference element associated with the IFE spaces on different interface elements via the standard affine mapping are not the same. By analyzing a mapping from the involved Sobolev space to the IFE space, this article is able to extend the scaling argument framework to the error estimation for the approximation capability of a class of IFE spaces in one spatial dimension. As demonstrations of the versatility of this unified error analysis framework, the manuscript applies the proposed scaling argument to obtain optimal IFE error estimates for a typical first-order linear hyperbolic interface problem, a second-order elliptic interface problem, and the fourth-order Euler-Bernoulli beam interface problem, respectively.

Initial‐boundary value and interface problems on the real half line for the fractional advection–diffusion‐type equation

We use the unified transform method (UTM) to solve initial‐boundary value problems for the fractional advection–diffusion‐type equation (FADE) on the real half line. We generalize this equation using the modified definition of the Atangana–Baleanu fractional derivative of order in order to satisfy the initial condition. A solution methodology is proposed when the UTM is implemented in fractional differential equations with boundary conditions of Dirichlet and Robin type, in particular when using the modified definitions of fractional operators with non‐singular kernel. In addition, an interface problem is stated and solved in the adjacent domains and , where perfect contact continuity conditions are imposed. The exact solutions obtained include as a particular case, the diffusion and advection–diffusion equations with integer‐order derivatives. Finally, representative curves for the solution are shown by varying the fractional order.

A Petrov–Galerkin immersed finite element method for steady Navier–Stokes interface problem with non-homogeneous jump conditions

This paper develops a lowest-order Petrov–Galerkin immersed Q1−Q0 method for solving Stokes interface problem. We make use of a uniform, interface-unfitted Cartesian mesh. An immersed Petrov–Galerkin formulation is presented, where the test spaces are conventional finite element spaces and the solution spaces satisfying the jump conditions. For the Stokes and Navier–Stokes interface problem, simple stabilized items are introduced. The nonlinear convective term is treated using Picard’s iteration. Extensive numerical experiments validate the feasibility and optimal convergence order for the Petrov–Galerkin immersed Q1−Q0 scheme both with homogeneous and non-homogeneous jumps.

Effective interface conditions for a porous medium type problem

Motivated by biological applications on tumour invasion through thin membranes, we study a porous medium type equation where the density of the cell population evolves under Darcy’s law, assuming continuity of both the density and flux velocity on the thin membrane which separates two domains. The drastically different scales and mobility rates between the membrane and the adjacent tissues lead to consider the limit as the thickness of the membrane approaches zero. We are interested in recovering the effective interface problem and the transmission conditions on the limiting zero-thickness surface, formally derived by Chaplain et al. (2019), which are compatible with nonlinear generalized Kedem–Katchalsky ones. Our analysis relies on a priori estimates and compactness arguments as well as on the construction of a suitable extension operator, which allows us to deal with the degeneracy of the mobility rate in the membrane, as its thickness tends to zero.

Entering the Next Dimension: A Review of 3D User Interfaces for Virtual Reality

In the last decade, the use of virtual reality (VR) technologies has been increasing, as head-mounted display devices, such as the Meta Quest series, have become commercially accessible to everyday consumers. There are plenty of software applications developed for VR, ranging from games to serious training. Despite the fast-growing emergence of novel and natural gesture-based interface components, traditional windows, icons, menus, and pointer user interfaces still remain popular in these implementations. Therefore, navigating and interacting within 3-dimensional environments can still be challenging in virtual reality. This body of work will serve as a reference to the literature survey on 3-dimensional user interfaces and their implementations, and categorize them to reiterate a set of design recommendations for implementing 3-dimensional user interfaces in virtual reality. While questions revolve around the need for a 3-dimensional user interface if a 2-dimensional user interface suffices, we cannot ignore the capabilities and benefits offered when utilising 3-dimensional space. Unlike other work, this work proposes a novel metaphor-guided quadrant model to delineate the 3-dimensional user interface problem in an end-to-end manner. To achieve this, over 30 unique articles selected using a meta-analyses methodology were reviewed. The result highlights the dimensional semantics of the inputs and dimensional outcomes of the virtual targets. Notable reiterated design recommendations would match devices to interaction and then to the tasks in 3-dimensional environments, reducing the degrees of freedom, imposing constraints when possible, and not implementing one single best interaction technique. The results of this survey will be beneficial for virtual reality simulation designers or researchers attempting to determine the appropriate model when developing user interfaces for virtual reality content.

Construction of Conductive Polymer Coatings onto Flexible PDMS Foam Composites with Exceptional Mechanical Robustness for Sensitive Strain Sensing Applications

AbstractFlexible piezoresistive‐sensing materials with high sensitivity and stable sensing signals are highly required to meet the accurate detecting requirement for human motion. Herein, a conductive poly(3,4‐ethylenedioxythiophene): poly(styrenesulfonate) / polydimethylsiloxane foam (P:P@p‐PSF) composite with strong interfacial action is designed. The porous structures and good interface combination not only show outstanding mechanical flexibility and reliability but also possess high sensitivity at a relatively wide strain range. The P:P@p‐PSF sensor achieves extreme sensitivity (Gauge Factor) of 6.25 in the subtle strain range of 1%–8%. Furthermore, the sensor forms a highly interconnected conductive network induced by the serious deformation of elastic‐interconnect pores, thus providing extremely sensitive sensing behavior for a relatively wide strain range (97.4% resistance change rate at 60% compressive strain). Moreover, the sensor presents repeatable stability and good thermal adaptation, which would meet the critical requirements of subtle vital signs, human motion monitoring, and so on. This work supplies insight into the design of a new flexible sensor material to overcome the weak interface problem and the flexible mismatch between conductive filler and matrix, showing great application potential in the field of electronic skin.