Local Boundary Conditions Research Articles

Application and Implementation of Incorporating Local Boundary Conditions into Nonlocal Problems

ABSTRACTWe study nonlocal equations from the area of peridynamics, an instance of nonlocal wave equation, and nonlocal diffusion on bounded domains whose governing equations contain a convolution operator based on integrals. We generalize the notion of convolution to accommodate local boundary conditions. On a bounded domain, the classical operator with local boundary conditions has a purely discrete spectrum, and hence, provides a Hilbert basis. We define an abstract convolution operator using this Hilbert basis, thereby automatically satisfying local boundary conditions. The main goal in this paper is twofold: apply the concept of abstract convolution operator to nonlocal problems and carry out a numerical study of the resulting operators. We study the corresponding initial value problems with prominent boundary conditions such as periodic, antiperiodic, Neumann, and Dirichlet. To connect to the standard convolution, we give an integral representation of the abstract convolution operator. For discretization, we use a weak formulation based on a Galerkin projection and use piecewise polynomials on each element which allows discontinuities of the approximate solution at the element borders. We study convergence order of solutions with respect to polynomial order and observe optimal convergence. We depict the solutions for each boundary condition.

Modeling of the in situ state of stress in elastic layered rock subject to stress and strain-driven tectonic forces

Abstract. In this study we describe and compare eight different strategies to predict the depth variation of stress within a layered rock formation. This reveals the inherent uncertainties in stress prediction from elastic properties and stress measurements, as well as the geologic implications of the different models. The predictive strategies are based on well log data and in some cases on in situ stress measurements, combined with the weight of the overburden rock, the pore pressure, the depth variation in rock properties, and tectonic effects. We contrast and compare stresses predicted purely using theoretical models with those constrained by in situ measurements. We also explore the role of the applied boundary conditions that mimic two fundamental models of tectonic effects, namely the stress- or strain-driven models. In both models, layer-to-layer tectonic stress variations are added to initial predictions due to vertical variation in rock elasticity, consistent with natural observations, yet describe very different controlling mechanisms. Layer-to-layer stress variations are caused by either local elastic strain accommodation for the strain-driven model, or stress transfers for the stress-driven model. As a consequence, stress predictions can depend strongly on the implemented prediction philosophy and the underlying implicit and explicit assumptions, even for media with identical elastic parameters and stress measurements. This implies that stress predictions have large uncertainties, even if local measurements and boundary conditions are honored.

Estimation of filament temperature and adhesion development in fused deposition techniques

This work presents an analytical solution to the transient heat conduction developing during filament deposition in Fused Deposition Techniques (FDT), which is coupled to a routine that activates or deactivates all relevant local boundary conditions depending on part geometry, operating conditions and deposition strategy. Boundary conditions include contact between filament segments, between filament segments and the support, as well as heat transfer with the environment. The resulting MatLab code comprises an adhesion criterion that is used to estimate whether contiguous filament segments will adhere adequately to each other prior to solidification. Predicted and experimental data for the filament surface temperature showed very good agreement. Also, adhesion predictions were in accordance with the results of real peel-like tests. The practical potential of this calculation tool is demonstrated by an application example.

Sensitivity analysis of climate change impacts on dune erosion: case study for the Dutch Holland coast

Climate change could have large implications for the management of dune-fringed coasts. Sea level rise and changes in storm wave and surge characteristics could lead to enhanced dune erosion and hence a decrease in safety levels. Here, we use the process-based model XBeach to quantify the impact of sea level rise and changing hydrodynamic boundary conditions on the magnitude of future dune erosion at two locations along the Dutch coast. We find a linear relation between sea level rise and dune erosion volume, the exact linear relation being dependent on the local hydrodynamical boundary conditions. The process driving higher erosion appears to be sea level rise, allowing waves to attack the dune at a higher level. Additional simulations illustrate that a change in the offshore wave angle, potentially produced by changes in storm tracks, could influence the erosion volume with the same order of magnitude as sea level rise. Finally, simulations with different mitigation options (i.e., sand nourishments) illustrate the strong effect of the location of the added sand to the reduction in the dune erosion volume.

An iteratively adaptive multi-scale finite element method for elliptic PDEs with rough coefficients

We propose an iteratively adaptive Multi-scale Finite Element Method (MsFEM) for elliptic PDEs with rough coefficients. The choice of the local boundary conditions for the multi-sale basis functions determines the accuracy of the MsFEM numerical solution, and one needs to incorporate the global information of the elliptic equation into the local boundary conditions of the multi-scale basis functions to recover the underlying fine-mesh solution of the equation. In our proposed iteratively adaptive method, we achieve this global-to-local information transfer through the combination of coarse-mesh solving using adaptive multi-scale basis functions and fine-mesh smoothing operations. In each iteration step, we first update the multi-scale basis functions based on the approximate numerical solutions of the previous iteration steps, and obtain the coarse-mesh approximate solution using a Galerkin projection. Then we apply several steps of smoothing operations to the coarse-mesh approximate solution on the underlying fine mesh to get the updated approximate numerical solution. The proposed algorithm can be viewed as a nonlinear two-level multi-grid method with the restriction and prolongation operators adapted to the approximate numerical solutions of the previous iteration steps. Convergence analysis of the proposed algorithm is carried out under the framework of two-level multi-grid method, and the harmonic coordinates are employed to establish the approximation property of the adaptive multi-scale basis functions. We demonstrate the efficiency of our proposed multi-scale methods through several numerical examples including a multi-scale coefficient problem, a high-contrast interface problem, and a convection-dominated diffusion problem.

Role of the pore interconnectivity on the dielectric, switching and tunability properties of PZTN ceramics

The influence of phase interconnectivity in porous Pb0.988(Zr0.52Ti0.48)0.976Nb0.024O3 ceramics on the structural, dielectric and tunability properties is reported. Porous ceramics with the same porosity level, but with different types of pore interconnectivity (0−3) and (3-3) type have been prepared by using dry and wet mixing of perovskite powder with starch porosity former agent. By comparison to the dense sintered ceramic (predominantly tetragonal), high porosity tends to favour the formation of monoclinic phase on the expense of the tetragonal one. A strong decrease of the effective permittivity with respect with the dense ceramic in both the porous structures was found, with lower values for the (3-3) connectivity. To describe the observed role of the phase interconnectivity on the dielectric properties, Finite Element Modeling calculations were performed. The local boundary conditions for such specific configurations generate a local field inhomogeneity, with a broad field distribution and decrease of the average field, which is more pronounced in the (3-3) porous configuration. As result, a strong decrease of the effective permittivity, switching polarization, loop area and rectangularity factor are predicted for this configuration, and confirmed by high-field experiments (tunability ε(E) and P(E) ferroelectric loops). This study shows that not only the pore amount, but the pore interconnectivity has a major role on the electrical properties and well-designed phase interconnectivity may provide useful functional properties.

ЧИСЛЕННЫЙ МЕТОД РЕШЕНИЯ ОДНОЙ НЕЛОКАЛЬНОЙ ЗАДАЧИ ТРУБОПРОВОДНОГО ТРАНСПОРТА ВЯЗКИХ ЖИДКОСТЕЙ

Х.М. Гамзаев, Азербайджанский государственный университет нефти и промышленности, г. Баку, Азербайджан E-mail: xan.h@rambler.ru. K.M. Gamzaev Azerbaijan State Oil and Industry University, Baku, Azerbaijan E-mail: xan.h@rambler.ru

Exponential boundary stabilization for nonlinear wave equations with localized damping and nonlinear boundary condition

Let \begin{document}$ D\subset R^{d}$\end{document} be a bounded domain in the \begin{document}$d- $\end{document} dimensional Euclidian space \begin{document}$R^{d} $\end{document} with smooth boundary $Γ=\partial D.$ In this paper we consider exponential boundary stabilization for weak solutions to the wave equation with nonlinear boundary condition: \begin{document}$\left\{ \begin{gathered}u_{tt}(t)-ρ(t)Δ u(t)+b(x)u_{t}(t)=f(u(t)), u(t)=0 \text{on }Γ_{0}×(0,T), \dfrac{\partial u(t)}{\partialν}+γ(u_{t}(t))=0 \text{on }Γ _{1}×(0,T), u(0)=u_{0},u_{t}(0)=u_{1},\end{gathered} \right.$ \end{document} where \begin{document}$\left\| {{u_0}} \right\| \begin{document}$ E(0) where \begin{document}$λ_{β}, $\end{document} \begin{document}$d_{β} $\end{document} are defined in (21), (22) and \begin{document}$Γ=Γ_{0}\cupΓ_{1} $\end{document} and \begin{document}$\bar{Γ}_{0}\cap\bar{Γ}_{1}=φ. $\end{document}

Hyperbolicity and Vitali properties of unbounded domains in Banach spaces

Let $\Om$ be a unbounded domain in a Banach space. In this work, we wish to impose {\it local conditions} on boundary point of $\Om$ (including the point at infinity) that guarantee complete hyperbolic of $\Om.$ We also search for local boundary conditions so that Vitali properties hold true for $\Om.$ These properties might be considered as analogues of the taut property in the finite dimensional case.

High order local absorbing boundary conditions for acoustic waves in terms of farfield expansions

We devise a new high order local absorbing boundary condition (ABC) for radiating problems and scattering of time-harmonic acoustic waves from obstacles of arbitrary shape. By introducing an artificial boundary S enclosing the scatterer, the original unbounded domain Ω is decomposed into a bounded computational domain Ω− and an exterior unbounded domain Ω+. Then, we define interface conditions at the artificial boundary S, from truncated versions of the well-known Wilcox and Karp farfield expansion representations of the exact solution in the exterior region Ω+. As a result, we obtain a new local absorbing boundary condition (ABC) for a bounded problem on Ω−, which effectively accounts for the outgoing behavior of the scattered field. Contrary to the low order absorbing conditions previously defined, the error at the artificial boundary induced by this novel ABC can be easily reduced to reach any accuracy within the limits of the computational resources. We accomplish this by simply adding as many terms as needed to the truncated farfield expansions of Wilcox or Karp. The convergence of these expansions guarantees that the order of approximation of the new ABC can be increased arbitrarily without having to enlarge the radius of the artificial boundary. We include numerical results in two and three dimensions which demonstrate the improved accuracy and simplicity of this new formulation when compared to other absorbing boundary conditions.

Analysis of the Slab Temperature, Thermal Stresses and Fractures Computed with the Implementation of Local and Average Boundary Conditions in the Secondary Cooling Zones

Abstract The numerical simulations of the temperature fields have been accomplished for slab casting made of a low carbon steel. The casting process of slab of 1500 mm in width and 225 mm in height has been modeled. Two types of boundary condition models of heat transfer have been employed in numerical simulations. The heat transfer coefficient in the first boundary condition model was calculated from the formula which takes into account the slab surface temperature and water flow rate in each secondary cooling zone. The second boundary condition model defines the heat transfer coefficient around each water spray nozzle. The temperature fields resulting from the average in zones water flow rate and from the nozzles arrangement have been compared. The thermal stresses and deformations resulted from such temperature field have given higher values of fracture criterion at slab corners.

Double absorbing boundaries for finite-difference time-domain electromagnetics

We describe the implementation of optimal local radiation boundary condition sequences for second order finite difference approximations to Maxwell's equations and the scalar wave equation using the double absorbing boundary formulation. Numerical experiments are presented which demonstrate that the design accuracy of the boundary conditions is achieved and, for comparable effort, exceeds that of a convolution perfectly matched layer with reasonably chosen parameters. An advantage of the proposed approach is that parameters can be chosen using an accurate a priori error bound.

A Monte Carlo Study on the Effect of Structural and Operating Parameters on the Water Distribution within the Microporous Layer and the Catalyst Layer of PEM Fuel Cells

Depending on the operating conditions, the overall performance and durability of polymer electrolyte membrane fuel cells (PEMFC) can be strongly influenced by liquid water saturation of pores within the gas diffusion layer (GDL). Water agglomerations in the GDL may occur due to material morphology, surface characteristics and local boundary conditions. In the catalyst layer (CL) and microporous layer (MPL) these parameters are considerably different than those of the fibre substrate. The MPL and CL feature by orders of magnitude smaller pores than the substrate and the boundary conditions, e.g. temperature and relative humidity, are different than those in the substrate near the bipolar plate. Thus, different physical behavior within each layer can be observed, e.g. local liquid sorption. Therefore, finding an optimal strategy regarding water management requires knowledge on virtually all the factors involved. To approach this target, we are using and further developing a 3D Monte Carlo (MC) model which simulates water distribution within the GDL, employing the real GDL structure and operating conditions, exploited respectively from tomography and computational fluid dynamics (CFD) studies1. The 3D images are obtained from an assembled fuel cell and therefore, the results include compression and inhomogeneity information of the GDL. Furthermore, recent progress of the model has enabled simulations on the length scales relevant for the MPL and CL in addition to the fibre substrate. GDL substrates usually have an average pore size in the micrometer scale, whereas this value for the MPLs and CLs may go down to the nanometer scale. Therefore, a complete simulation of the water distribution in GDL includes working on at least two scales. Although obtaining accurate results on all three sub-layers is equally important, MPL and CL are the more challenging ones from the point of view of both structure and surface properties acquisition. Furthermore, simulating relevant domains within the material requires expensive computational effort. In this study, we report first results on MC simulations of the water distribution within the MPL and CL pores for different operating conditions. The 3D reconstruction of the material is obtained from the focused ion beam - scanning electron microscope (FIB-SEM) tomography2. By using atomic layer deposition (ALD) of ZnO, a good material contrast between pores and solid particles is achieved and therefore a reliable binarization can be obtained3. The voxel size is set to 45 nm in order to achieve both, an accurate computation with respect to the applied equations and an appropriately sized simulation domain. Our MC model provides the water distribution within the actual MPL and CL structures based on the local boundary conditions. The results show how the water distribution depends on parameters such as contact angle, porosity, local temperature and relative humidity. An example of simulation results is illustrated in the Figure 1.

Measurement of Local Thermal Deformations in Heterogeneous Microstructures via SEM Imaging with Digital Image Correlation

It is challenging to measure accurately and with high spatial resolution the local thermal strains in heterogeneous microstructures due of the complex nature of the thermal deformations and local boundary conditions. In the enclosed study, a digital image correlation (DIC) based, thermal strain mapping technique is described that is able to probe thermal deformations with sub-micron spatial resolution and sub-nanometer displacement accuracy for both homogeneous and heterogeneous materials, including cross-sections of IC packages. The full-field thermal deformation maps of different materials within a nanostructured IC chip cross-section are established from room temperature up to 160 °C, uncovering the heterogeneous nature of the specimen while accurately measuring the highly non-uniform displacement and strain fields across the multiple material constituents. As described in this work, the DIC-enabled technique is capable of high resolution mapping of local thermo-mechanical deformations in heterogeneous materials, providing a methodology that can improve our understanding of complex material systems under controlled thermal-environmental conditions.

LPMLE3: A novel 1‐D approach to study water flow in streambeds using heat as a tracer

AbstractWe introduce LPMLE3, a new 1‐D approach to quantify vertical water flow components at streambeds using temperature data collected in different depths. LPMLE3 solves the partial differential equation for coupled water flow and heat transport in the frequency domain. Unlike other 1‐D approaches it does not assume a semi‐infinite halfspace with the location of the lower boundary condition approaching infinity. Instead, it uses local upper and lower boundary conditions. As such, the streambed can be divided into finite subdomains bound at the top and bottom by a temperature‐time series. Information from a third temperature sensor within each subdomain is then used for parameter estimation. LPMLE3 applies a low order local polynomial to separate periodic and transient parts (including the noise contributions) of a temperature‐time series and calculates the frequency response of each subdomain to a known temperature input at the streambed top. A maximum‐likelihood estimator is used to estimate the vertical component of water flow, thermal diffusivity, and their uncertainties for each streambed subdomain and provides information regarding model quality. We tested the method on synthetic temperature data generated with the numerical model STRIVE and demonstrate how the vertical flow component can be quantified for field data collected in a Belgian stream. We show that by using the results in additional analyses, nonvertical flow components could be identified and by making certain assumptions they could be quantified for each subdomain. LPMLE3 performed well on both simulated and field data and can be considered a valuable addition to the existing 1‐D methods.

Approximate solution of linear and nonlinear fractional differential equations under m-point local and nonlocal boundary conditions

This paper investigates a computational method to find an approximation to the solution of fractional differential equations subject to local and nonlocal m-point boundary conditions. The method that we will employ is a variant of the spectral method which is based on the normalized Bernstein polynomials and its operational matrices. Operational matrices that we will developed in this paper have the ability to convert fractional differential equations together with its nonlocal boundary conditions to a system of easily solvable algebraic equations. Some test problems are presented to illustrate the efficiency, accuracy, and applicability of the proposed method.

Higher spectral flow for Dirac operators with local boundary conditions

We consider a one parameter family [Formula: see text] of families of fiberwise twisted Dirac type operators on a fibration with the typical fiber an even dimensional compact manifold with boundary, which verifies [Formula: see text] with [Formula: see text] being a smooth map from the fibration to a unitary group [Formula: see text]. For each [Formula: see text], we impose on [Formula: see text] a certain fixed local elliptic boundary condition [Formula: see text] and get a self-adjoint extension [Formula: see text]. Under the assumption that [Formula: see text] has vanishing [Formula: see text]-index bundle, we establish a formula for the higher spectral flow of [Formula: see text], [Formula: see text]. Our result generalizes a recent result of [A. Gorokhovsky and M. Lesch, On the spectral flow for Dirac operators with local boundary conditions, Int. Math. Res. Not. IMRN (2015) 8036–8051.] to the families case.

On a class of nonlocal wave equations from applications

We study equations from the area of peridynamics, which is a nonlocal extension of elasticity. The governing equations form a system of nonlocal wave equations. We take a novel approach by applying operator theory methods in a systematic way. On the unbounded domain ℝn, we present three main results. As main result 1, we find that the governing operator is a bounded function of the governing operator of classical elasticity. As main result 2, a consequence of main result 1, we prove that the peridynamic solutions strongly converge to the classical solutions by utilizing, for the first time, strong resolvent convergence. In addition, main result 1 allows us to incorporate local boundary conditions, in particular, into peridynamics. This avenue of research is developed in companion papers, providing a remedy for boundary effects. As main result 3, employing spherical Bessel functions, we give a new practical series representation of the solution which allows straightforward numerical treatment with symbolic computation.

Blow-up problems for the heat equation with a local nonlinear Neumann boundary condition

This paper estimates the blow-up time for the heat equation ut=Δu with a local nonlinear Neumann boundary condition: The normal derivative ∂u/∂n=uq on Γ1, one piece of the boundary, while on the rest part of the boundary, ∂u/∂n=0. The motivation of the study is the partial damage to the insulation on the surface of space shuttles caused by high speed flying subjects. We show the finite time blow-up of the solution and estimate both upper and lower bounds of the blow-up time in terms of the area of Γ1. In many other work, they need the convexity of the domain Ω and only consider the problem with Γ1=∂Ω. In this paper, we remove the convexity condition and only require ∂Ω to be C2. In addition, we deal with the local nonlinearity, namely Γ1 can be just part of ∂Ω.

3D filling simulation of micro- and nanostructures in comparison to iso- and variothermal injection moulding trials

Flow simulations can cut down both costs and time for the development of injection moulded polymer parts with functional surfaces used in life science and optical applications. We simulated the polymer melt flow into 3D micro- and nanostructures with Moldflow and Comsol and compared the results to real iso- and variothermal injection moulding trials below, at and above the transition temperature of the polymer. By adjusting the heat transfer coefficient and the transition temperature in the simulation it was possible to achieve good correlation with experimental findings at different processing conditions (mould temperature, injection velocity) for two polymers, namely polymethylmethacrylate and amorphous polyamide. The macroscopic model can be scaled down in volume and number of elements to save computational time for microstructure simulation and to enable first and foremost the nanostructure simulation, as long as local boundary conditions such as flow front speed are transferred correctly. The heat transfer boundary condition used in Moldflow was further evaluated in Comsol. Results showed that the heat transfer coefficient needs to be increased compared to macroscopic moulding in order to represent interfacial polymer/mould effects correctly. The transition temperature is most important in the packing phase for variothermal injection moulding.