Explicit Boundary Conditions Research Articles

Catastrophic caldera-forming eruptions II: The subordinate role of magma buoyancy as an eruption trigger

Recent analytical investigations have suggested that magma buoyancy is critical for triggering catastrophic caldera forming eruptions. Through detailed assessment of these approaches, we illustrate how analytical models have been misapplied for investigating buoyancy and are, therefore, incorrect and inconclusive. Nevertheless, the hypothesis that buoyancy is the critical trigger for larger eruptions warrants further investigation. As such, we utilize viscoelastic finite element models that incorporate buoyancy to test overpressure evolution and mechanical failure in the roof due to the coalescence of large buoyant magma bodies for two model cases. In the first case, we mimic empirical approaches and include buoyancy as an explicit boundary condition. In the second set of models, buoyancy is calculated implicitly due to the density contrast between the magma in the reservoir and the host rock. Results from these numerical experiments indicate that buoyancy promotes only minimal overpressurization of large silicic magma reservoirs (<0.1MPa). Furthermore, no Mode-1 tensile failure is predicted along the magma chamber boundary due to buoyancy in large reservoirs. Rather, compressional stresses are observed due to buoyant magma focusing away from the edges of the reservoir and toward the center. Given the shortcomings of the analytical implementations and the results from the numerical experiments, we conclude that buoyancy does not provide an eruption triggering mechanism for large silicic systems. Therefore, correlations of buoyancy with magma residence times, the eruption frequency–volume relationship, and the dimensions of calderas are re-assessed. We find a causal relationship with magma reservoir volume that implicates the mechanical conditions of the host rock as a primary control on eruption frequency. As magma reservoirs grow in size (>100km3) they surpass a rheological threshold where their subsequent evolution is controlled by host rock mechanics. Consequently, this results in a thermomechanical division between small systems that are triggered “internally” by magmatic processes and large systems that are triggered “externally” by faulting related to roof uplift or tectonism. Finally, critical assessment of recent analytical approaches illustrates that care must be used when applying previously derived analytical solutions to ensure that assumptions used in the original formulation are not violated during application to new geologic problems.

A high-order Discontinuous Galerkin Chimera method for laminar and turbulent flows

In this study we present a Chimera scheme for the Discontinuous Galerkin method for the compressible Navier–Stokes equations as well as the Reynolds-Averaged Navier–Stokes (RANS) equations. For the turbulent simulations we use the Spalart–Allmaras one-equation turbulence model to close the RANS system. Focus of the study is the implementation of the Chimera scheme with a detailed description of all the necessary parts of the method: hole-cutting, definition of an interpolation operator and adaption of the time integration scheme. Concerning the time integration scheme the differences between explicit and implicit Chimera boundary conditions are elaborated for an implicit time-integration scheme solved with a Newton-GMRES method. The accuracy of the implemented method is tested with the method of manufactured solutions and an inviscid simulation of a Gaussian bump. The robustness and reliability is then assessed with a circular cylinder (Re=40) and a NACA0012 airfoil (Re=2.88·106) at different angles of attack. Both cases are run with a single grid and a Chimera grid version and compared with each other and with experimental and numerical reference data.

Integral Equation-Based Nonoverlapping DDM Using the Explicit Boundary Condition

A new integral equation-based nonoverlapping domain decomposition method (NDDM) is proposed for analyzing electromagnetic (EM) scattering from electrically large PEC objects whose MLFMA models are too large for the user’s computer to accommodate. The integral equation is first built on the entire PEC surface, and the entire surface is partitioned into some nonoverlapping subsurfaces. The RWG functions are used to expand the surface current on the entire surface. Then, each full-RWG function across some boundary curve is split into two half-RWG functions whose unknown coefficients are constrained by the boundary current continuity (using the explicit boundary condition). The convergence of the outer-iterative scheme of the proposed NDDM is investigated theoretically and demonstrated to be very good, and hence a very large problem can be effectively solved in an ordinary computer. Some numerical examples are provided to demonstrate the correctness and robustness of the proposed method, and to compare the proposed method with the existing overlapping DDMs having similar attributes.

DOMAIN DECOMPOSITION METHOD FOR SINGULARLY PERTURBED DIFFERENTIAL DIFFERENCE EQUATIONS WITH LAYER BEHAVIOR

In this paper, a domain decomposition method has been presented for solving singularly perturbed differential difference equations with delay as well as advances whose solution exhibits boundary layer behavior. By introducing a terminal point, the original problem is divided into inner and outer region problems. An implicit terminal boundary condition at the terminal point has been determined. The outer region problem with the implicit boundary condition is solved and produces an explicit boundary condition for the inner region problem. Then, the modified inner region problem (using the stretching transformation) is solved as a two-point boundary value problem. Fourth order stable central difference method has been used to solve both the inner and outer region problems. The proposed method is iterative on the terminal point. To demonstrate the applicability of the method, some numerical examples have been solved for different values of the perturbation parameter, delay and advance parameters. The stability and convergence of the scheme has also investigated

Integrated and Explicit Boundary Conditions of Electromagnetic Fields at Arbitrary Interfaces between Two Anisotropic Media

This paper derives two new integrated and explicit boundary conditions, named the “explicit normal version” and “explicit tangential versions” respectively for electromagnetic fields at an arbitrary interface between two anisotropic media. The new versions combine two implicit boundary equations into a single explicit matrix formula and reveal the boundary values linked by a 3 × 3 matrix, which depends on the interface topography and model property tensors. We analytically demonstrate the new versions equivalent to the common implicit boundary conditions and their application to transformation of the boundary values in the boundary integral equations. We also give two synthetic examples that show recovery of the boundary values on a hill and a ridge, and highlight the advantage of the new versions of being a simpler and more straightforward method to compute the electromagnetic boundary values.

Transition‐metal dichalcogenides for spintronic applications

Spin‐orbit splitting in transition‐metal dichalcogenide monolayers is investigated on the basis of density‐functional theory within explicit two‐dimensional periodic boundary conditions. The spin‐orbit splitting reaches few hundred meV and increases with the size of the metal and chalcogen atoms, resulting in nearly 500 meV for WTe2. Furthermore, we find that similar to the band gap, spin‐orbit splitting changes drastically under tensile strain. In centrosymmetric transition metal dichalcogenide bilayers, spin‐orbit splitting is suppressed by the inversion symmetry. However, it could be induced if the inversion symmetry is explicitly broken, e.g. by a potential gradient normal to the plane, as it is present in heterobilayers (Rashba‐splitting). In such systems, the spin‐orbit splitting could be as large as for the heavier monolayer that forms heterobilayer. These properties of transition metal dichalcogenide materials suggest them for potential applications in opto‐, spin‐ and straintronics.

Stability analysis of Crank–Nicolson and Euler schemes for time-dependent diffusion equations

In this paper, we study the stability of the Crank–Nicolson and Euler schemes for time-dependent diffusion coefficient equations on a staggered grid with explicit and implicit approximations to the Dirichlet boundary conditions. Using the matrix representation for the numerical scheme and boundary conditions it is shown that for implicit boundary conditions the Crank–Nicolson scheme is unrestrictedly stable while it becomes conditionally stable for explicit boundary conditions. Numerical examples are provided illustrating this behavior. For the Euler schemes the results are similar to those for the constant coefficient case. The implicit Euler with implicit or explicit boundary conditions is unrestrictedly stable while the explicit Euler with explicit boundary conditions presents the usual stability restriction on the time step.

Terminal boundary-value technique for solving singularly perturbed delay differential equations

A terminal boundary-value technique is presented for solving singularly perturbed delay differential equations, the solutions of which exhibit layer behaviour. By introducing a terminal point, the original problem is divided into inner and outer region problems. An implicit terminal boundary condition at the terminal point was determined. The outer region problem with the implicit boundary condition was solved and produces an explicit boundary condition for the inner region problem. Then, the modified inner region problem (using the stretching transformation) is solved as a two-point boundary value problem. The second-order finite difference scheme was used to solve both the inner and outer region problems. The proposed method is iterative on the terminal point. To validate the efficiency of the method, some model examples were solved. The stability and convergence of the scheme was also investigated.

Method of adiabatic modes in research of smoothly irregular integrated optical waveguides: zero approximation

We consider the application of the method of adiabatic waveguide modes for calculating the propagation of electromagnetic radiation in three-dimensional (3D) irregular integrated optical waveguides. The method of adiabatic modes takes into account a three-dimensional distribution of quasi-waveguide modes and explicit (‘inclined’) tangential boundary conditions. The possibilities of the method are demonstrated on the example of numerical research of two major elements of integrated optics: a waveguide of ‘horn’ type and a thin-film generalised waveguide Luneburg lens by the methods of adiabatic modes and comparative waveguides.

Ostéotomies métatarsiennes distales percutanées : suivi rétrospectif de 50 cas

Modeling the propagation of nonlinear free surface waves is a challenging task but inevitable in studying nearshore wave propagation processes. More recently, meshless methods are emerging as potential tools for studying water wave related problems, in particular when there is a flow separation. In this study, smoothed particle hydrodynamics (SPH) has been shown to be useful to study the propagation of nonlinear water waves generated in a numerical flume. The problems considered address various aspects associated with long time simulation and highly nonlinear waves such as bore formation and wave breaking. An explicit dynamic free surface boundary condition has been imposed in the SPH model for steady long time simulation. A correction has been prescribed for the mass of the ghost particles at the geometric singular zones for the present application. The developed model is validated with FEM based simulation and other meshless methods. The split-up of a single solitary wave as it crosses continental shelf is simulated by the present method.

General solution for calculating polarization electric fields in the auroral ionosphere and application examples

[1] We devise an approach to calculate the polarization electric field in the ionosphere, when the ionospheric conductances, the primary (modeled) or the total (measured) electric field, and the Cowling efficiency are given. In contrast to previous studies, our approach is a general solution which is not limited to specific geometrical setups, and all parameters may have any kind of spatial dependence. The solution technique is based on spherical elementary current (vector) systems (SECS). This way, we avoid the need to specify explicit boundary conditions for the searched polarization electric field of its potential which would be required if the problem was solved in a differential equation approach. Instead, we solve an algebraic matrix equation, and the implicit boundary condition that the divergence of the polarization electric field vanishes outside our analysis area is sufficient. In order to illustrate our theory, we then apply it to two simple models of auroral electrodynamic situations, the first being a mesoscale strong conductance enhancement in the early morning sector within a relatively weak southward primary electric field, and a morning sector auroral arc with only a weak conductance enhancement, but a large southward primary electric field at the poleward flank of the arc. While the significance of the polarization electric field for maximum Cowling efficiency is large for the first case, it is rather minor for the second one. Both models show that the polarization electric field effect may not only change the magnitude of the current systems but also their overall geometry. Furthermore, the polarization electric field may extend into regions where the primary electric field is small, thus even dominating the total electric field in these regions. For the first model case, the total Joule heating integrated over the analysis area decreases by a factor of about 4 for maximum Cowling efficiency as compared to the case of vanishing Cowling efficiency. Furthermore, for this case the resulting total electric field structurally shows a strong resemblance to that frequently observed during auroral omega band events.

Boundary feedback control of 2-D shallow water equations

Local explicit feedback boundary conditions are given for the stabilization in \(L^2-\)norm of the 2-D shallow water model. The proposed method is based on symmetrization of the flux matrices of the linearized model and analysis of the Riemann invariants. The non-conservative 2-D shallow water equations are linearized around the target steady state sub-critical flow. The established feedback control laws guarantee a decay of the energy of the perturbation model. We present numerical simulations to demonstrate how the proposed controller works for the linearized as well as nonlinear shallow water problem.

SPH Simulation of shallow water wavepropagation

Modeling the propagation of nonlinear free surface waves is a challenging task but inevitable in studying nearshore wave propagation processes. More recently, meshless methods are emerging as potential tools for studying water wave related problems, in particular when there is a flow separation. In this study, smoothed particle hydrodynamics (SPH) has been shown to be useful to study the propagation of nonlinear water waves generated in a numerical flume. The problems considered address various aspects associated with long time simulation and highly nonlinear waves such as bore formation and wave breaking. An explicit dynamic free surface boundary condition has been imposed in the SPH model for steady long time simulation. A correction has been prescribed for the mass of the ghost particles at the geometric singular zones for the present application. The developed model is validated with FEM based simulation and other meshless methods. The split-up of a single solitary wave as it crosses continental shelf is simulated by the present method.

A Galerkin Radial Basis Function Method for the Schrödinger Equation

In this article, we consider the discretization of the time-dependent Schrödinger equation using radial basis functions (RBFs). We formulate the discretized problem over an unbounded domain without imposing explicit boundary conditions. Since we can show that time stability of the discretization is not guaranteed for an RBF-collocation method, we propose to employ a Galerkin ansatz instead. For Gaussians, it is shown that exponential convergence is obtained up to a point where a systematic error from the domain where no basis functions are centered takes over. The choice of the shape parameter and of the resolved region is studied numerically. Compared to the Fourier method with periodic boundary conditions, the basis functions can be centered in a smaller domain which gives increased accuracy with the same number of points.

Almost exact boundary condition for one-dimensional Schrödinger equations

An explicit local boundary condition is proposed for finite-domain simulations of the linear Schrödinger equation on an unbounded domain. Based on an exact boundary condition in terms of the Bessel functions, it takes a simple form with 16 neighboring grid points, and it involves no empirical parameter. While the computing load is rather low, the proposed boundary condition is effective in reflection suppression, comparable to the exact convolution treatments. An extension to nonlinear Schrödinger equations is also proposed. Numerical comparisons clearly demonstrate the effectiveness of this ALmost EXact (ALEX) boundary condition for both the linear and the cubic nonlinear Schrödinger equations.

Numerical validation of the volume penalization method in three-dimensional pseudo-spectral simulations

The penalization method is used to simulate the effect of boundaries on fluid flows without body-fitted mesh. It is introduced here in a three-dimensional Fourier pseudo-spectral code with periodic boundary conditions. We propose an implicit implementation for the penalization term, thus getting rid of the strong stability constraint stemming from the stiffness of the penalized equations. The convergence of this implicit penalized method is shown by comparison with solutions of Navier–Stokes equations with explicit boundary conditions. Different test-cases are used, among which a comparison with channel flow simulations with a Chebyshev–Fourier pseudo-spectral code in which the projection base automatically ensures the no-slip boundary condition and the divergence-free condition. We consider inertial or internal gravity waves reflection on solid walls, hence assessing the correct account of pressure coupling for wave-propagation and reflection by the implicit penalized algorithm, and the common benchmark of a Gaussian vortex ring impacting on a normal flat plane solid boundary. Numerical convergence is characterized by increasing the grid resolution and by reducing the penalization parameter η.

Decoupling algorithms from schedules for easy optimization of image processing pipelines

Using existing programming tools, writing high-performance image processing code requires sacrificing readability, portability, and modularity. We argue that this is a consequence of conflating what computations define the algorithm , with decisions about storage and the order of computation. We refer to these latter two concerns as the schedule , including choices of tiling, fusion, recomputation vs. storage, vectorization, and parallelism. We propose a representation for feed-forward imaging pipelines that separates the algorithm from its schedule, enabling high-performance without sacrificing code clarity. This decoupling simplifies the algorithm specification: images and intermediate buffers become functions over an infinite integer domain, with no explicit storage or boundary conditions. Imaging pipelines are compositions of functions. Programmers separately specify scheduling strategies for the various functions composing the algorithm, which allows them to efficiently explore different optimizations without changing the algorithmic code. We demonstrate the power of this representation by expressing a range of recent image processing applications in an embedded domain specific language called Halide, and compiling them for ARM, x86, and GPUs. Our compiler targets SIMD units, multiple cores, and complex memory hierarchies. We demonstrate that it can handle algorithms such as a camera raw pipeline, the bilateral grid, fast local Laplacian filtering, and image segmentation. The algorithms expressed in our language are both shorter and faster than state-of-the-art implementations.

Boundary triples and quasi boundary triples for elliptic operators

AbstractThe abstract concepts of boundary triples and their recent generalizations are useful tools to parametrize the self‐adjoint and maximal dissipative/maximal accumulative extensions of formally symmetric elliptic differential expressions with the help of explicit boundary conditions. In the present note the parametrizations induced by the “natural” quasi boundary triple with the Dirichlet and Neumann trace as boundary maps are compared with the parametrizations induced by a “classical” ordinary boundary triple, where a regularized Neumann trace is used for one of the boundary maps. (© 2011 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Computation of plate wave dispersion diagrams and surface wave velocities without explicit boundary conditions

We discuss the computation of the band structure of plate waves using the plane wave expansion (PWE) method. This method is generally used to formulate eigenvalue problems to compute dispersion diagrams for solid-solid phononic crystals. We show how the free surface boundary condition can be included implicitly in the form of the PWE solution, thus leading to an efficient eigenvalue problem. This generic method for wave dispersion is non-iterative and does not require an initial guess for the solution. Furthermore, surface acoustic wave velocities can be estimated from the slowest wave for large wave vectors. Examples for a single plate and a multilayer plate are given, and extension to piezoelectric materials is discussed.

Local determination of ionospheric plasma convection from coherent scatter radar data using the SECS technique

A new technique for merging line‐of‐sight (LOS) data of the ionospheric plasma convection velocity, as obtained from coherent scatter radars, into a full velocity vector field on a sphere is presented. This technique is based on the expansion into Spherical Elementary Current Systems (SECS) which have been successfully applied to many other problems in ionosphere‐magnetosphere physics. Despite their name mentioning currents for historical reasons, SECS can be used as basis functions for any continuously differentiable vector field on a sphere. In contrast to the traditional modeling of the radar data with spherical harmonics over the whole auroral zone, the new technique does not require any “a priori” model input but relies solely on the measured data, nor does it need any explicit boundary conditions to be specified. The new technique is designed to be applied locally to areas where sufficient radar backscatter exists. The analysis area that satisfies this condition may have any shape and is not limited to, e.g., spherical caps. A test with synthetic data shows that the method performs excellently (less than 5% relative error) if 25% or more of the optimal coverage of input data are actually available as backscatter data, with respect to the scale on which the results for the velocity vector field are desired to be obtained. Still if only 10% of the optimal coverage of input data are available, the technique performs fairly well with a relative error of ∼12%. A second test with real LOS input data from the SuperDARN radars shows that on such a local area with sufficient backscatter, our new technique is able to reproduce mesoscale details of the LOS data significantly better than the current standard analysis based on the technique of Ruohoniemi and Baker (1998) which processes the radar data on the whole auroral zone. While the new technique is presented here for the application with LOS radar data, it can be applied for merging any kind of vector component data on a sphere to a vector field.