Two-dimensional Setting Research Articles

The Oberbeck–Boussinesq system with non-local boundary conditions

We consider the Oberbeck–Boussinesq system with non-local boundary conditions arising as a singular limit of the full Navier–Stokes–Fourier system in the regime of low Mach and low Froude numbers. The existence of strong solutions is shown on a maximal time interval [ 0 , T m a x ) [0, T_{\mathrm {max}}) . Moreover, T m a x = ∞ T_{\mathrm {max}} = \infty in the two-dimensional setting.

Comparison of Two Aspects of a PDE Model for Biological Network Formation

We compare the solutions of two systems of partial differential equations (PDEs), seen as two different interpretations of the same model which describes the formation of complex biological networks. Both approaches take into account the time evolution of the medium flowing through the network, and we compute the solution of an elliptic–parabolic PDE system for the conductivity vector m, the conductivity tensor C and the pressure p. We use finite differences schemes in a uniform Cartesian grid in a spatially two-dimensional setting to solve the two systems, where the parabolic equation is solved using a semi-implicit scheme in time. Since the conductivity vector and tensor also appear in the Poisson equation for the pressure p, the elliptic equation depends implicitly on time. For this reason, we compute the solution of three linear systems in the case of the conductivity vector m∈R2 and four linear systems in the case of the symmetric conductivity tensor C∈R2×2 at each time step. To accelerate the simulations, we make use of the Alternating Direction Implicit (ADI) method. The role of the parameters is important for obtaining detailed solutions. We provide numerous tests with various values of the parameters involved to determine the differences in the solutions of the two systems.

Numerical solution of the two-dimensional Steigmann–Ogden model of material surface with a boundary

In this paper, we consider an infinite isotropic elastic matrix that contains a material surface described by the Steigmann–Ogden theory, where the material surface possesses a boundary. In two-dimensional setting, the surface represents a material curve with endpoints. The presence of the surface tension introduces non-linearity in the problem since a superposition with respect to the far-field load is not allowed. We review the governing equations, the jump conditions, and the conditions at the surface tips. For solving the problem in the plane strain setting, the displacements in the matrix are sought in the complex variables form of a single layer elastic potential. The density of the potential represents the jump in complex tractions across the surface. Exact expressions for the elastic fields in the matrix are provided in terms of complex integral representations. The problem of the surface located along a straight segment is considered in detail. Its solution is reduced to that of the system of boundary integral equations using the approximations of the boundary data that involve the series of Chebyshev’s polynomials. We present numerical examples to illustrate the influence of dimensionless parameters on the solution.

Development and implementation of a micromechanically motivated cohesive zone model for ductile fracture

Gaining popularity after its coupling with the finite element method, cohesive zone modeling has been used extensively to model fracture, especially in delamination problems. Its constitutive relations, i.e. traction-separation laws, are mostly derived phenomenologically without considering the underlying physical mechanisms of crack initiation and propagation. The approach could potentially be used for ductile fracture as well where the micromechanics of the phenomenon is explained by nucleation, growth and coalescence of pores. In this context, the objective of this work is to develop and implement a physically motivated cohesive zone modeling framework for ductile fracture in metallic materials. In order to accomplish this, a micromechanically motivated traction-separation relation which considers the growth of a physical pore is developed. Tractions are directly represented as a function of pore fraction, and its evolution is driven by separations. The model is implemented as an intrinsic cohesive zone element in a two-dimensional setting. Implementation steps and methodology including the finite element framework are presented in detail for mode-I, mode-II and mixed-mode fracture cases. The derivation of the mixed-mode case leads to a novel yield function representation of tractions and separations, instead of an explicit expression. Hence, an incremental implicit elasto-plastic numerical integration scheme is utilized to solve mixed-mode system of equations required for crack initiation and propagation. The framework is implemented as a user element subroutine in Abaqus (UEL) and the numerical simulations are conducted with compact tension (CT) and single edge notch (SEN) specimens to test the implementation and influence of the micromechanical parameters such as pore size, shape and spacing on the ductile crack initiation and propagation. The work is concluded by presenting an outlook for the capability of the model to predict crack path.

Ray theory for elastic wave propagation in graded metamaterials

We present a ray theory for modeling elastic wave propagation in spatially graded mechanical metamaterials. Wave propagation in periodic metamaterials has been well studied, motivated by their beneficial wave steering and bandgap properties. By contrast, comparably little work has explored wave propagation in spatially graded metamaterials despite the increased design opportunities, largely due to the lack of efficient modeling techniques. We develop a ray theory to model waves in graded metamaterials based on high-frequency asymptotics and the assumption of local periodicity. This work builds upon the well-developed ray theories that are fundamental in a wide range of fields, from optics to seismology. Our derivations produce a practical framework for computing approximate wave fields in graded metamaterials. Ray trajectories are computed by independently solving a system of ordinary differential equations for each ray, requiring only knowledge of local dispersion relations throughout the metamaterial, which vary smoothly in space due to grading. Equations for the wave amplitude along rays are also derived in the two-dimensional setting. We show that the form of the ray tracing equations are nearly identical to those for smooth solids in seismic ray theory, with the primary difference being the dispersion relations. A numerical framework for computing ray solutions is demonstrated on a mass–spring network with analytical dispersion relations as well as a truss metamaterial that requires the numerical evaluation of dispersion relations. Through these examples, we demonstrate that ray theory provides an efficient means of studying the fascinating behavior of waves in graded metamaterials such as wave guiding along curved trajectories.

Numerical modeling of solute convection in the system of two porous layers with different permeabilities

The numerical simulation of the dynamics of a liquid with solute of high concentration in two horizontal layers of a porous medium with different permeability coefficients is carried out. The dynamics of the solute is modeled within the framework of a nonlinear MIM model that takes into account the adhesion and separation process at the skeleton of a porous medium. The modeling was carried out in a two-dimensional setting. The mobile (moving together with the carrier fluid) and immobile (deposited on the skeleton) phases are modeled by the finite volume method with adaptive mesh refinement algorithm. At each point in space, functions of volumetric saturation are set, and a kinematic equation is solved that describes the dynamics of adhesion and separation. The effective permeability of a porous medium depends on the saturation function of the immobile phase and is calculated using the Kozeny-Karman formula. The obtained numerical data are compared with the results of a linear analysis of stability, and the nonlinear regimes of concentration convection and the distribution of impurities at high supercriticalities are studied. In the case of high ratio of permeability coefficients of upper and lower layers, convection occurs locally in a more permeable sublayer, even with a decrease in its thickness. With an increase in supercriticality the penetration of the convective flow into the poorly permeable sublayer and the transition from local convection to large-scale convection are observed.

Factors Affecting Spatial Autocorrelation in Residential Property Prices

Within housing literature, the presence of spatial autocorrelation (S.A.) in housing prices is typically examined horizontally in a two-dimensional setting. However, in the context of apartment buildings, there is also a vertical component of S.A. for housing units located on different floor levels. This paper therefore explores the determinants of both horizontal and vertical S.A. within residential property prices. First, we posit that S.A. in housing prices is a consequence of the price discovery process of real estate, in which property traders acquire price information from recent market transactions (i.e., comparables) to value a subject property. Furthermore, we contend that the extent to which property traders rely on comparables to determine housing prices is governed by the liquidity and volatility conditions of the market, which in turn affects the magnitude of the S.A. By developing and testing several spatial autoregressive hedonic models using open market transaction data for the Hong Kong residential property market, we find that market liquidity tends to increase both vertical and horizontal S.A., whilst market volatility is more prone to increase vertical S.A. but depress horizontal S.A.

Offline Evaluation of Recommender Systems in a User Interface With Multiple Carousels.

Many video-on-demand and music streaming services provide the user with a page consisting of several recommendation lists, i.e., widgets or swipeable carousels, each built with specific criteria (e.g., most recent, TV series, etc.). Finding efficient strategies to select which carousels to display is an active research topic of great industrial interest. In this setting, the overall quality of the recommendations of a new algorithm cannot be assessed by measuring solely its individual recommendation quality. Rather, it should be evaluated in a context where other recommendation lists are already available, to account for how they complement each other. The traditional offline evaluation protocol however does not take this into account. To address this limitation, we propose an offline evaluation protocol for a carousel setting in which the recommendation quality of a model is measured by how much it improves upon that of an already available set of carousels. We also propose to extend ranking metrics to the two-dimensional carousel setting in order to account for a known position bias, i.e., users will not explore the lists sequentially, but rather concentrate on the top-left corner of the screen. Finally, we describe and evaluate two strategies for the ranking of carousels in a scenario where the technique used to generate the two-dimensional layout is agnostic on the algorithms used to generate each carousel. We report experiments on publicly available datasets in the movie domain to show how the relative effectiveness of several recommendation models compares. Our results indicate that under a carousel setting the ranking of the algorithms changes sometimes significantly. Furthermore, when selecting the optimal carousel layout accounting for the two dimensional layout of the user interface leads to very different selections.

Numerical approximation of optimal convex and rotationally symmetric shapes for an eigenvalue problem arising in optimal insulation

We are interested in the optimization of convex domains under a PDE constraint. Due to the difficulties of approximating convex domains in R3, the restriction to rotationally symmetric domains is used to reduce shape optimization problems to a two-dimensional setting. For the optimization of an eigenvalue arising in a problem of optimal insulation, the existence of an optimal domain is proven. An algorithm is proposed that can be applied to general shape optimization problems under the geometric constraints of convexity and rotational symmetry. The approximated optimal domains for the eigenvalue problem in optimal insulation are discussed.

The Evolution of Cooperation in Two-Dimensional Mobile Populations with Random and Strategic Dispersal

We investigate the effect of the environment dimensionality and different dispersal strategies on the evolution of cooperation in a finite structured population of mobile individuals. We consider a population consisting of cooperators and free-riders residing on a two-dimensional lattice with periodic boundaries. Individuals explore the environment according to one of the four dispersal strategies and interact with each other via a public goods game. The population evolves according to a birth–death–birth process with the fitness of the individuals deriving from the game-induced payouts. We found that the outcomes of the strategic dispersal strategies in the two-dimensional setting are identical to the outcomes in the one-dimensional setting. The random dispersal strategy, not surprisingly, resulted in the worst outcome for cooperators.

Formation of local and global currents in a toroidal Bose-Einstein condensate via an inhomogeneous artificial gauge field

We study the effects of a position-dependent artificial gauge field on an atomic Bose-Einstein condensate in quasi-one-dimensional and two-dimensional ring settings. The inhomogeneous artificial gauge field can induce global and local currents in the Bose-Einstein condensate via phase gradients along the ring and vortices, respectively. We observe two different regimes in the system depending on the radial size of the ring and the strength of the gauge field. For weak artificial gauge fields, the angular momentum increases, as expected, in a quantized manner; however, for stronger values of the fields, the angular momentum exhibits a linear (nonquantized) behavior. We also characterize the angular momentum for noncylindrically symmetric traps.

Phase field-finite element analysis of magnetic-induced deformation in ferromagnetic shape memory alloy

A phase field theory based model considering non-uniform demagnetizing field within ferromagnetic shape memory alloy and free space is established and realized by finite element method in a two-dimensional setting to save huge storage spaces and computational times. The martensite reorientation (contain nucleation and growth) from the stress field-favored martensite variant to magnetic field-favored martensite variant is numerically simulated. The simulation reasonably depicts the essence of magnetic-induced martensite reorientation and magnetization change for NiMnGa ferromagnetic shape memory alloy and their dependence on the loading level. Furthermore, the evolution and interaction of martensite domain structure, magnetic domain structure, magnetic potential and demagnetizing field are analyzed. And the dominant driving force generated by minimization of total Helmholtz free energy for loading level dependent martensite reorientation and magnetization change are revealed by addressing the influence of loading level. The simulation results show that the local demagnetizing field within ferromagnetic shape memory alloy and free space is the key to the local nucleation of new martensite variant. The dominant driving force of local nucleation and growth of new martensite variant are completely different: the local nucleation of new martensite variant is driven by local high magnetocrystalline anisotropy energy caused by local demagnetizing field; the local high Landau-type double well potential energy caused by local nucleation of new martensite variant has the largest impact on growth of the new martensite variant. The local high magnetocrystalline anisotropy energy caused by local nucleation and growth of new martensite variant has the largest impact on local rapid magnetization change within new martensite variant domain. When it comes to high loading level, the inhibition of martensite reorientation is attributed to the elastic strain energy, and the approximately uniform magnetization change is driven by Zeeman energy.

Lubricated viscous gravity currents of power-law fluids. Part 1. Self-similar flow regimes

We examine the gravity-driven flow of thin films of viscous fluid of power-law rheology, lubricated by another power-law viscous fluid from below. Such flows are relevant to a range of geophysical and industrial settings, including the flow of ice sheets and fast-flowing ice streams lubricated from below by a layer of subglacial till. We model both layers using lubrication theory in two-dimensional and axisymmetric settings, in the limit in which vertical shear provides the dominant resistance to the flow. The flow is self-similar if the power-law exponents, describing the rheology of the two layers of fluid, are equal. We examine the similarity solutions in both geometries and describe the flow in terms of four distinct flow regimes ranging from thin films of viscous fluid coating a more viscous fluid from above, to thin layers of fluid lubricating a more viscous fluid from below. In contrast to the former scenario, a thin film of a low-viscosity fluid strikingly alters the dynamics of a more viscous fluid when it lubricates it from below: the overlying layer thins, the upper surface gradients lessen and most of the shear is confined to the lower layer. Such features amplify, and this flow regime becomes increasingly dominant when the viscous fluids are shear thinning, like the deformation of glacial ice on the large scale. This flow regime is most relevant to the flow of lubricated ice sheets, which thin and accelerate, forming fast-flowing ice streams, when they are well lubricated from below.

Multi-Cue Kinetic Model with Non-Local Sensing for Cell Migration on a Fiber Network with Chemotaxis

Cells perform directed motion in response to external stimuli that they detect by sensing the environment with their membrane protrusions. Precisely, several biochemical and biophysical cues give rise to tactic migration in the direction of their specific targets. Thus, this defines a multi-cue environment in which cells have to sort and combine different, and potentially competitive, stimuli. We propose a non-local kinetic model for cell migration in which cell polarization is influenced simultaneously by two external factors: contact guidance and chemotaxis. We propose two different sensing strategies, and we analyze the two resulting transport kinetic models by recovering the appropriate macroscopic limit in different regimes, in order to observe how the cell size, with respect to the variation of both external fields, influences the overall behavior. This analysis shows the importance of dealing with hyperbolic models, rather than drift-diffusion ones. Moreover, we numerically integrate the kinetic transport equations in a two-dimensional setting in order to investigate qualitatively various scenarios. Finally, we show how our setting is able to reproduce some experimental results concerning the influence of topographical and chemical cues in directing cell motility.

A polynomial-augmented RBF collocation method using fictitious centres for solving the Cahn–Hilliard equation

In this paper, we consider the radial basis function collocation method with fictitious centres for solving the Cahn–Hilliard equation in one-dimensional and two-dimensional settings. Radial basis functions with polynomial augmentation are used in derivative approximations to overcome the issue of the stagnation error and to avoid the hassle of determining the shape parameter. Meanwhile, we implement a semi-implicit technique for temporal discretization, which combines the stability of the implicit method and the simplicity of the explicit method. Numerical experiments show that the hybrid bases significantly improve the accuracy of the radial basis function collocation method and its stability in terms of the shape parameter c.

A realizable filtered intrusive polynomial moment method

Intrusive uncertainty quantification methods for hyperbolic problems exhibit spurious oscillations at shocks, which leads to a significant reduction of the overall approximation quality. Furthermore, a challenging task is to preserve hyperbolicity of the gPC moment system. An intrusive method which guarantees hyperbolicity is the intrusive polynomial moment (IPM) method, which performs the gPC expansion on the entropy variables. The method, while still being subject to oscillations, requires solving a convex optimization problem in every spatial cell and every time step.The aim of this work is to mitigate oscillations in the IPM solution by applying filters. Filters reduce oscillations by damping high order gPC coefficients. Naive filtering, however, may lead to unrealizable moments, which means that the IPM optimization problem does not have a solution and the method breaks down. In this paper, we propose and analyze two separate strategies to guarantee the existence of a solution to the IPM problem. First, we propose a filter which maintains realizability by being constructed from an underlying Fokker–Planck equation. Second, we regularize the IPM optimization problem to be able to cope with non-realizable gPC coefficients. Consequently, standard filters can be applied to the regularized IPM method. We demonstrate numerical results for the two strategies by investigating the Euler equations with uncertain shock structures in one- and two-dimensional spatial settings. We are able to show a significant reduction of spurious oscillations by the proposed filters.

Material-separating regularizer for multi-energy x-ray tomography

Dual-energy x-ray tomography is considered in a context where the target under imaging consists of two distinct materials. The materials are assumed to be possibly intertwined in space, but at any given location there is only one material present. Further, two x-ray energies are chosen so that there is a clear difference in the spectral dependence of the attenuation coefficients of the two materials. A novel regularizer is presented for the inverse problem of reconstructing separate tomographic images for the two materials. A combination of two things, (a) non-negativity constraint, and (b) penalty term containing the inner product between the two material images, promotes the presence of at most one material in a given pixel. A preconditioned interior point method is derived for the minimization of the regularization functional. Numerical tests with digital phantoms suggest that the new algorithm outperforms the baseline method, joint total variation regularization, in terms of correctly material-characterized pixels. While the method is tested only in a two-dimensional setting with two materials and two energies, the approach readily generalizes to three dimensions and more materials. The number of materials just needs to match the number of energies used in imaging.

Switching Cost Aware Rounding for Relaxations of Mixed-Integer Optimal Control Problems: The 2-D Case

This article is concerned with a recently proposed switching cost aware rounding (SCARP) strategy in the combinatorial integral approximation for mixed-integer optimal control problems (MIOCPs). We consider the case of a control variable that is discrete-valued and distributed on a two-dimensional domain. While the theoretical results from the one-dimensional case directly apply to the multidimensional setting, the structure of the cost function in the graph-based rounding computation is significantly more involved in the two-dimensional case. We describe a set up of the computational graph and the traversal algorithm underlying the SCARP strategy that enable a transfer to the two-dimensional setting. We demonstrate the SCARP strategy in this two-dimensional setting using the example of a MIOCP from topology optimization. We compare the graph-based approach to a ground truth computation using an integer linear programming (ILP) solver. The graph-based approach becomes computationally intractable for medium grid sizes. We show that the one-dimensional SCARP algorithm can be employed on a serialization of the grid cells in these cases and still provides an efficient heuristic that yields superior performance compared with that of other rounding heuristics such as sum-up rounding (SUR).

Two-dimensional equilibrium configurations in Korteweg fluids

In this paper, after reviewing the form of the constitutive equations for a third grade Korteweg fluid, recently derived by means of an extended Liu procedure, an equilibrium problem is investigated. By considering a two-dimensional setting, a single nonlinear elliptic equation is derived such that the equilibrium conditions are identically satisfied. Such an equation is discussed both analytically and numerically. Moreover, by considering a particular boundary value problem of Dirichlet type, some preliminary numerical solutions are presented.

Tensor PDE model of biological network formation

We study an elliptic-parabolic system of partial differential equations describing formation of biological network structures. The model takes into consideration the evolution of the permeability tensor under the influence of a diffusion term, representing randomness in the material structure, a decay term describing metabolic cost and a pressure force. A Darcy's law type equation describes the pressure field. In the spatially two-dimensional setting, we present a constructive, formal derivation of the PDE system from the discrete network formation model in the refinement limit of a sequence of unstructured triangulations. Moreover, we show that the PDE system is a formal $L^2$-gradient flow of an energy functional with biological interpretation, and study its convexity properties. For the case when the energy functional is convex, we construct unique global weak solutions of the PDE system. Finally, we construct steady state solutions in one- and multi-dimensional settings and discuss their stability properties.