Non‐weak/strong solutions in gas dynamics: a C11 p‐version STLSFEF in Eulerian frame of reference using ρ, u,p primitive variables

This paper presents a space–time least squares finite element formulation of one‐dimensional transient Navier–Stokes equations (governing differential equations: GDE) for compressible flow in Eulerian frame of reference using ρ, u, p as primitive variables with C¹¹ type p‐version hierarchical interpolations in space and time. Time marching procedure is utilized to compute time evolutions for all values of time. For high speed gas dynamics the C¹¹ type interpolations in space and time possess the same orders of continuity in space and time as the GDE. It is demonstrated that with this approach accurate numerical solutions of Navier–Stokes equations are possible without any assumptions or approximations. In the approach presented here SUPG, SUPG/DC, SUPG/DC/LS operators are neither used nor needed. Time accurate numerical simulations show resolution of shock structure (i.e. shock speed, shock relations and shock width) to be in excellent agreement with the analytical solutions. The role of diffusion i.e. viscosity (physical or artificial) and thermal conductivity on shock structure is demonstrated. Riemann shock tube is used as a model problem. True time evolutions are reported beginning with the first time step until steady shock conditions are achieved. In this approach, when the computed error functionals become zero (computationally), the computed non‐weak solutions have characteristics as those of the strong solutions of the gas dynamics equations. Copyright © 2001 John Wiley & Sons, Ltd.     

On the computation of steady‐state compressible flows using a discontinuous Galerkin method

Computation of compressible steady‐state flows using a high‐order discontinuous Galerkin finite element method is presented in this paper. An accurate representation of the boundary normals based on the definition of the geometries is used for imposing solid wall boundary conditions for curved geometries. Particular attention is given to the impact and importance of slope limiters on the solution accuracy for flows with strong discontinuities. A physics‐based shock detector is introduced to effectively make a distinction between a smooth extremum and a shock wave. A recently developed, fast, low‐storage p‐multigrid method is used for solving the governing compressible Euler equations to obtain steady‐state solutions. The method is applied to compute a variety of compressible flow problems on unstructured grids. Numerical experiments for a wide range of flow conditions in both 2D and 3D configurations are presented to demonstrate the accuracy of the developed discontinuous Galerkin method for computing compressible steady‐state flows. Copyright © 2007 John Wiley & Sons, Ltd.     

Implicit boundary method for analysis using uniform B‐spline basis and structured grid

Several analysis techniques such as extended finite element method (X‐FEM) have been developed recently, which use structured grid for the analysis. Implicit boundary method uses implicit equations of the boundary to apply boundary conditions in X‐FEM framework using structured grids. Solution structures for test and trial functions are constructed using implicit equations such that the boundary conditions are satisfied even if there are no nodes on the boundary. In this paper, this method is applied for analysis using uniform B‐spline basis defined over a structured grid. Solution structures that are C¹ or C² continuous throughout the analysis domain can be constructed using B‐spline basis functions. As a structured grid does not conform to the geometry of the analysis domain, the boundaries of the analysis domain are defined independently using equations of the boundary curves/surfaces. Compared with conforming mesh, it is easier to generate structured grids that overlap the geometry and the elements in the grid are regular shaped and undistorted. Numerical examples are presented to demonstrate the performance of these B‐spline elements. The results are compared with analytical solutions as well as with traditional finite element solutions. Convergence studies for several examples show that B‐spline elements provide accurate solutions with fewer elements and nodes compared with traditional FEM. They also provide continuous stress and strain in the analysis domain, thus eliminating the need for smoothing stress/strain results. Copyright © 2008 John Wiley & Sons, Ltd.     

Effects of Water and Different Solutes on Carbon‐Nanotube Low‐Voltage Field‐Effect Transistors

Semiconducting single‐walled carbon nanotubes (swCNTs) are a promising class of materials for emerging applications. In particular, they are demonstrated to possess excellent biosensing capabilities, and are poised to address existing challenges in sensor reliability, sensitivity, and selectivity. This work focuses on swCNT field‐effect transistors (FETs) employing rubbery double‐layer capacitive dielectric poly(vinylidene fluoride‐co‐hexafluoropropylene). These devices exhibit small device‐to‐device variation as well as high current output at low voltages (<0.5 V), making them compatible with most physiological liquids. Using this platform, the swCNT devices are directly exposed to aqueous solutions containing different solutes to characterize their effects on FET current–voltage (FET I–V) characteristics. Clear deviation from ideal characteristics is observed when swCNTs are directly contacted by water. Such changes are attributed to strong interactions between water molecules and sp²‐hybridized carbon structures. Selective response to Hg²⁺ is discussed along with reversible pH effect using two distinct device geometries. Additionally, the influence of aqueous ammonium/ammonia in direct contact with the swCNTs is investigated. Understanding the FET I–V characteristics of low‐voltage swCNT FETs may provide insights for future development of stable, reliable, and selective biosensor systems.     

A block iterative finite element algorithm for numerical solution of the steady–state,compressible Navier–Stokes equations

An iterative method for numerically solving the time independent Navier–Stokes equations for viscous compressible flows is presented. The method is based upon partial application of the Gauss–Seidel principle in block form to the systems of the non-linear algebraic equations which arise in construction of finite element (Galerkin) models approximating solutions of fluid dynamic problems. The C⁰-cubic element on triangles is employed for function approximation. Computational results for a free shear flow at Re = 1000 indicate significant achievement of economy in iterative convergence rate over finite element and finite difference models which employ the customary time dependent equations and symptotic time marching procedure to steady solution. Numerical results are in excellent agreement with those obtained for the same test problem employing time marching finite element and finite difference solution techniques.     

Finite element analysis of time‐dependent semi‐infinite wave‐guides with high‐order boundary treatment

A new finite element (FE) scheme is proposed for the solution of time‐dependent semi‐infinite wave‐guide problems, in dispersive or non‐dispersive media. The semi‐infinite domain is truncated via an artificial boundary ??, and a high‐order non‐reflecting boundary condition (NRBC), based on the Higdon non‐reflecting operators, is developed and applied on ??. The new NRBC does not involve any high derivatives beyond second order, but its order of accuracy is as high as one desires. It involves some parameters which are chosen automatically as a pre‐process. A C⁰ semi‐discrete FE formulation incorporating this NRBC is constructed for the problem in the finite domain bounded by ??. Augmented and split versions of this FE formulation are proposed. The semi‐discrete system of equations is solved by the Newmark time‐integration scheme. Numerical examples concerning dispersive waves in a semi‐infinite wave guide are used to demonstrate the performance of the new method. Copyright © 2003 John Wiley & Sons, Ltd.     

Non‐local dispersive model for wave propagation in heterogeneous media: one‐dimensional case

Non‐local dispersive model for wave propagation in heterogeneous media is derived from the higher‐order mathematical homogenization theory with multiple spatial and temporal scales. In addition to the usual space–time co‐ordinates, a fast spatial scale and a slow temporal scale are introduced to account for rapid spatial fluctuations of material properties as well as to capture the long‐term behaviour of the homogenized solution. By combining various order homogenized equations of motion the slow time dependence is eliminated giving rise to the fourth‐order differential equation, also known as a ‘bad’ Boussinesq problem. Regularization procedures are then introduced to construct the so‐called ‘good’ Boussinesq problem, where the need for C¹ continuity is eliminated. Numerical examples are presented to validate the present formulation. Copyright © 2002 John Wiley & Sons, Ltd.     

Hierarchical high‐order conforming C1 bases for quadrangular and triangular finite elements

We present three new sets of C¹ hierarchical high‐order tensor‐product bases for conforming finite elements. The first basis is a high‐order extension of the Bogner–Fox–Schmit basis. The edge and face functions are constructed using a combination of cubic Hermite and Jacobi polynomials with C¹ global continuity on the common edges of elements. The second basis uses the tensor product of fifth‐order Hermite polynomials and high‐order functions and achieves global C¹ continuity for meshes of quadrilaterals and C² continuity on the element vertices. The third basis for triangles is also constructed using the tensor product of one‐dimensional functions defined in barycentric coordinates. It also has global C¹ continuity on edges and C² continuity on vertices. A patch test is applied to the three considered elements. Projection and plate problems with smooth fabricated solutions are solved, and the performance of the h‐ and p‐refinements are evaluated by comparing the approximation errors in the L₂‐ and energy norms. A plate with singularity is then studied, and h‐ and p‐refinements are analysed. Finally, a transient problem with implicit time integration is considered. The results show exponential convergence rates with increasing polynomial order for the triangular and quadrilateral meshes of non‐distorted and distorted elements. Copyright © 2016 John Wiley & Sons, Ltd.     

Fourier‐based numerical approximation of the Weertman equation for moving dislocations

This work addresses the numerical approximation of solutions to a dimensionless form of the Weertman equation, which models a steadily moving dislocation and is an important extension (with advection term) of the celebrated Peierls‐Nabarro equation for a static dislocation. It belongs to the class of nonlinear reaction‐advection‐diffusion integro‐differential equations with Cauchy‐type kernel, thus involving an integration over an unbounded domain. In the Weertman problem, the unknowns are the shape of the core of the dislocation and the dislocation velocity. The proposed numerical method rests on a time‐dependent formulation that admits the Weertman equation as its long‐time limit. Key features are (1) time iterations are conducted by means of a new, robust, and inexpensive Preconditioned Collocation Scheme in the Fourier domain, which allows for explicit time evolution but amounts to implicit time integration, thus allowing for large time steps; (2) as the integration over the unbounded domain induces a solution with slowly decaying tails of important influence on the overall dislocation shape, the action of the operators at play is evaluated with exact asymptotic estimates of the tails, combined with discrete Fourier transform operations on a finite computational box of size L; (3) a specific device is developed to compute the moving solution in a comoving frame, to minimize the effects of the finite‐box approximation. Applications illustrate the efficiency of the approach for different types of nonlinearities, with systematic assessment of numerical errors. Converged numerical results are found insensitive to the time step, and scaling laws for the combined dependence of the numerical error with respect to L and to the spatial step size are obtained. The method proves fast and accurate and could be applied to a wide variety of equations with moving fronts as solutions; notably, Weertman‐type equations with the Cauchy‐type kernel replaced by a fractional Laplacian.     

Scaled boundary finite‐element analysis of a non‐homogeneous elastic half‐space

As a result of stresses experienced during and after the deposition phase, a soil strata of uniform material generally exhibits an increase in elastic stiffness with depth. The immediate settlement of foundations on deep soil deposits and the resultant stress state within the soil mass may be most accurately calculated if this increase in stiffness with depth is taken into account. This paper presents an axisymmetric formulation of the scaled boundary finite‐element method and incorporates non‐homogeneous elasticity into the method. The variation of Young's modulus (E) with depth (z) is assumed to take the form E=m_Ez^α, where m_E is a constant and αis the non‐homogeneity parameter. Results are presented and compared to analytical solutions for the settlement profiles of rigid and flexible circular footings on an elastic half‐space, under pure vertical load with αvarying between zero and one, and an example demonstrating the versatility and practicality of the method is also presented. Known analytical solutions are accurately represented and new insight regarding displacement fields in a non‐homogeneous elastic half‐space is gained. Copyright © 2003 John Wiley & Sons, Ltd.     

k‐version of finite element method in gas dynamics: higher‐order global differentiability numerical solutions

In this paper, we consider and examine alternate finite element computational strategies for time‐dependent Navier–Stokes equations describing high‐speed compressible flows with shocks in a viscous and conducting medium, with the ultimate objective of establishing the desired features of a general mathematical and computational framework for such initial value problems (IVP) in which: (a) the numerically computed solutions are in agreement with the physics of evolution described by the governing differential equations (GDEs) i.e. the IVP, (b) the solutions are admissible in the non‐discretized form of the GDEs in the pointwise sense (i.e. anywhere and everywhere) in the entire space–time domain, and hence in the integrated sense as well, (c) the numerical approximations progressively approach the same global differentiability in space and time as the theoretical solutions, (d) it is possible to time march the solutions (this is essential for efficiency as well as ensuring desired accuracy of the computed solution for the current increment of time, i.e. to minimize the error build up in the time marching process), (e) the computational process is unconditionally stable and non‐degenerate regardless of the choice of discretization, nature of approximations and their global differentiability and the dimensionless parameters influencing the physics of the process, (f) there are no issues of stability, CFL number limitations and (g) the mathematical and computational methodology is independent of the nature of the space–time differential operators. We consider one‐dimensional compressible flow in a viscous and conducting medium with shocks as model problems to illustrate various features of the general mathematical and computational framework used here and to demonstrate that the proposed framework is general and is applicable to all IVP. The Riemann shock tube with a single diaphragm serves as a model problem. The specific details presented in the paper discuss: (1) Choice of the form of the GDEs, i.e. strong form or weak form. (2) Various choices of variables. The paper establishes and considers density, velocity and temperature as variables of choice. (3) Details of the space–time least squares (LS) integral forms (meritorious over all others in all aspects) are presented and choice of approximation spaces are discussed. (4) In all numerical studies we consider a viscous and conducting medium with ideal gas law, however results are also presented for non‐conducting medium. Extension of this work to real gas models will be presented in a separate paper. It is worth noting that when the medium is viscous and conducting, the solutions of gas dynamics equations are analytic. (5) It is also significant to note that upwinding methods based on addition of artificial diffusion such as SUPG, SUPG/DC, SUPG/DC/LS and their many variations are neither needed nor used in this present work. (6) Numerical studies are aimed at resolving the localized details of the shock structure, i.e. shock relations, shock width, shock speed, etc. as well as the over all global behaviour of the solution in the entire space–time domain. (7) Numerical studies are presented for Riemann shock tube for high Mach number flows with special emphasis also on time accuracy of the evolution which is ensured by requiring that the approximations for each increment of time satisfy non‐discretized form of the GDEs in the pointwise sense, and hence in the integrated sense as well. (8) Comparisons are made with published results as well as theoretical solutions (when possible). It is established that space–time least squares processes are the only processes that yield variationally consistent space–time integral forms, and hence unconditionally non‐degenerate space–time computational processes, which when considered in higher‐order scalar product spaces provide the desired mathematical framework in which progressively higher‐order global differentiability solutions in space and time yield the same characteristics as the theoretical solutions of the IVP in all aspects. Copyright © 2006 John Wiley & Sons, Ltd.     

General time‐dependent C(t) and J(t) estimation equations for elastic‐plastic‐creep fracture mechanics analysis

This paper presents equations for estimating the crack tip characterizing parameters C(t) and J(t), for general elastic‐plastic‐creep conditions where the power‐law creep and plasticity stress exponents differ, by modifying the plasticity correction term in published equations. The plasticity correction term in the newly proposed equations is given in terms of the initial elastic‐plastic and steady‐state creep stress fields. The predicted C(t) and J(t) results are validated by comparison with systematic elastic‐plastic‐creep FE results. Good agreement with the FE results is found.     

A cut‐cell non‐conforming Cartesian mesh method for compressible and incompressible flow

This paper details a multigrid‐accelerated cut‐cell non‐conforming Cartesian mesh methodology for the modelling of inviscid compressible and incompressible flow. This is done via a single equation set that describes sub‐, trans‐, and supersonic flows. Cut‐cell technology is developed to furnish body‐fitted meshes with an overlapping mesh as starting point, and in a manner which is insensitive to surface definition inconsistencies. Spatial discretization is effected via an edge‐based vertex‐centred finite volume method. An alternative dual‐mesh construction strategy, similar to the cell‐centred method, is developed. Incompressibility is dealt with via an artificial compressibility algorithm, and stabilization achieved with artificial dissipation. In compressible flow, shocks are captured via pressure switch‐activated upwinding. The solution process is accelerated with full approximation storage (FAS) multigrid where coarse meshes are generated automatically via a volume agglomeration methodology. This is the first time that the proposed discretization and solution methods are employed to solve a single compressible–incompressible equation set on cut‐cell Cartesian meshes. The developed technology is validated by numerical experiments. The standard discretization and alternative methods were found equivalent in accuracy and computational cost. The multigrid implementation achieved decreases in CPU time of up to one order of magnitude. Copyright © 2007 John Wiley & Sons, Ltd.     

High‐Performance Li–SeSx All‐Solid‐State Lithium Batteries

All‐solid‐state Li–S batteries are promising candidates for next‐generation energy‐storage systems considering their high energy density and high safety. However, their development is hindered by the sluggish electrochemical kinetics and low S utilization due to high interfacial resistance and the electronic insulating nature of S. Herein, Se is introduced into S cathodes by forming SeS_x solid solutions to modify the electronic and ionic conductivities and ultimately enhance cathode utilization in all‐solid‐state lithium batteries (ASSLBs). Theoretical calculations confirm the redistribution of electron densities after introducing Se. The interfacial ionic conductivities of all achieved SeS_x–Li₃PS₄ (x = 3, 2, 1, and 0.33) composites are 10^?6 S cm^?1. Stable and highly reversible SeS_x cathodes for sulfide‐based ASSLBs can be developed. Surprisingly, the SeS₂/Li₁₀GeP₂S₁₂–Li₃PS₄/Li solid‐state cells exhibit excellent performance and deliver a high capacity over 1100 mAh g^?1 (98.5% of its theoretical capacity) at 50 mA g^?1 and remained highly stable for 100 cycles. Moreover, high loading cells can achieve high areal capacities up to 12.6 mAh cm^?2. This research deepens the understanding of Se–S solid solution chemistry in ASSLB systems and offers a new strategy to achieve high‐performance S‐based cathodes for application in ASSLBs.     

Singular‐ended spline interpolation for two‐dimensional boundary element analysis

A method of interpolation of the boundary variables that uses spline functions associated with singular elements is presented. This method can be used in boundary element method analysis of 2‐D problems that have points where the boundary variables present singular behaviour. Singular‐ended splines based on cubic splines and Overhauser splines are developed. The former provides C²‐continuity and the latter C¹‐continuity across element edges. The potentialities of the methodology are demonstrated analysing the dynamic response of a 2‐D rigid footing interacting with a half‐space. It is shown that, for a given number of elements at the soil–foundation interface, the singular‐ended spline interpolation increases substantially the displacement convergence rate and delivers smoother traction distributions. Copyright © 2000 John Wiley & Sons, Ltd.     

A three‐dimensional C1 finite element for gradient elasticity

In gradient elasticity strain gradient terms appear in the expression of virtual work, leading to the need for C¹ continuous interpolation in finite element discretizations of the displacement field only. Employing such interpolation is generally avoided in favour of the alternative methods that interpolate other quantities as well as displacement, due to the scarcity of C¹ finite elements and their perceived computational cost. In this context, the lack of three‐dimensional C¹ elements is of particular concern. In this paper we present a new C¹ hexahedral element which, to the best of our knowledge, is the first three‐dimensional C¹ element ever constructed. It is shown to pass the single element and patch tests, and to give excellent rates of convergence in benchmark boundary value problems of gradient elasticity. It is further shown that C¹ elements are not necessarily more computationally expensive than alternative approaches, and it is argued that they may be more efficient in providing good‐quality solutions. Copyright © 2008 John Wiley & Sons, Ltd.     

Highly Conductive Ti3C2Tx MXene Hybrid Fibers for Flexible and Elastic Fiber‐Shaped Supercapacitors

Fiber‐shaped supercapacitors (FSCs) are promising energy storage solutions for powering miniaturized or wearable electronics. However, the scalable fabrication of fiber electrodes with high electrical conductivity and excellent energy storage performance for use in FSCs remains a challenge. Here, an easily scalable one‐step wet‐spinning approach is reported to fabricate highly conductive fibers using hybrid formulations of Ti₃C₂T_x MXene nanosheets and poly(3,4‐ethylenedioxythiophene):polystyrene sulfonate. This approach produces fibers with a record conductivity of ≈1489 S cm^?1, which is about five times higher than other reported Ti₃C₂T_x MXene‐based fibers (up to ≈290 S cm^?1). The hybrid fiber at ≈70 wt% MXene shows a high volumetric capacitance (≈614.5 F cm^?3 at 5 mV s^?1) and an excellent rate performance (≈375.2 F cm^?3 at 1000 mV s^?1). When assembled into a free‐standing FSC, the energy and power densities of the device reach ≈7.13 Wh cm^?3 and ≈8249 mW cm^?3, respectively. The excellent strength and flexibility of the hybrid fibers allow them to be wrapped on a silicone elastomer fiber to achieve an elastic FSC with 96% capacitance retention when cyclically stretched to 100% strain. This work demonstrates the potential of MXene‐based fiber electrodes and their scalable production for fiber‐based energy storage applications.     

Controllable,Wide‐Ranging n‐Doping and p‐Doping of Monolayer Group 6 Transition‐Metal Disulfides and Diselenides

Developing processes to controllably dope transition‐metal dichalcogenides (TMDs) is critical for optical and electrical applications. Here, molecular reductants and oxidants are introduced onto monolayer TMDs, specifically MoS₂, WS₂, MoSe₂, and WSe₂. Doping is achieved by exposing the TMD surface to solutions of pentamethylrhodocene dimer as the reductant (n‐dopant) and “Magic Blue,” [N(C₆H₄‐p‐Br)₃]SbCl₆, as the oxidant (p‐dopant). Current–voltage characteristics of field‐effect transistors show that, regardless of their initial transport behavior, all four TMDs can be used in either p‐ or n‐channel devices when appropriately doped. The extent of doping can be controlled by varying the concentration of dopant solutions and treatment time, and, in some cases, both nondegenerate and degenerate regimes are accessible. For all four TMD materials, the photoluminescence intensity; for all four materials the PL intensity is enhanced with p‐doping but reduced with n‐doping. Raman and X‐ray photoelectron spectroscopy (XPS) also provide insight into the underlying physical mechanism by which the molecular dopants react with the monolayer. Estimates of changes of carrier density from electrical, PL, and XPS results are compared. Overall a simple and effective route to tailor the electrical and optical properties of TMDs is demonstrated.     

On the stability and convergence of a Galerkin reduced order model (ROM) of compressible flow with solid wall and far‐field boundary treatment

A reduced order model (ROM) based on the proper orthogonal decomposition (POD)/Galerkin projection method is proposed as an alternative discretization of the linearized compressible Euler equations. It is shown that the numerical stability of the ROM is intimately tied to the choice of inner product used to define the Galerkin projection. For the linearized compressible Euler equations, a symmetry transformation motivates the construction of a weighted L² inner product that guarantees certain stability bounds satisfied by the ROM. Sufficient conditions for well‐posedness and stability of the present Galerkin projection method applied to a general linear hyperbolic initial boundary value problem (IBVP) are stated and proven. Well‐posed and stable far‐field and solid wall boundary conditions are formulated for the linearized compressible Euler ROM using these more general results. A convergence analysis employing a stable penalty‐like formulation of the boundary conditions reveals that the ROM solution converges to the exact solution with refinement of both the numerical solution used to generate the ROM and of the POD basis. An a priori error estimate for the computed ROM solution is derived, and examined using a numerical test case. Published in 2010 by John Wiley & Sons, Ltd.     

Tri‐s‐triazine‐Based Crystalline Carbon Nitride Nanosheets for an Improved Hydrogen Evolution

Tri‐s‐triazine‐based crystalline carbon nitride nanosheets (CCNNSs) have been successfully extracted via a conventional and cost‐effective sonication–centrifugation process. These CCNNSs possess a highly defined and unambiguous structure with minimal thickness, large aspect ratios, homogeneous tri‐s‐triazine‐based units, and high crystallinity. These tri‐s‐triazine‐based CCNNSs show significantly enhanced photocatalytic hydrogen generation activity under visible light than g‐C₃N₄, poly (triazine imide)/Li⁺ Cl^–, and bulk tri‐s‐triazine‐based crystalline carbon nitrides. A highly apparent quantum efficiency of 8.57% at 420 nm for hydrogen production from aqueous methanol feedstock can be achieved from tri‐s‐triazine‐based CCNNSs, exceeding most of the reported carbon nitride nanosheets. Benefiting from the inherent structure of 2D crystals, the ultrathin tri‐s‐triazine‐based CCNNSs provide a broad range of application prospects in the fields of bioimaging, and energy storage and conversion.