Nonlocal Boundary Conditions Are Applied to the Analysis of Curve Equations

The traditional curve equation solution method has a low accuracy, so the non-local boundary conditions are applied to the curve equation solution. Firstly, the solution coordinate system is established, and then the key parameters are determined to solve the curve equation. Finally, the curve equation is solved by combining the non-local boundary conditions. The experiment proves that the method of this design is more accurate than the traditional method in solving simple curve equation or complex curve equation.

Functional Subdivisions of the Cerebellum in Naturalistic Paradigm Functional Magnetic Resonance Imaging

Compelling evidence has suggested that the human cerebellum is engaged in a wide range of cognitive tasks besides traditional opinions of motor control, and it is organized into a set of distinct functional subregions. The existing model-driven cerebellum parcellations through resting-state functional MRI (rsfMRI) and task-fMRI are relatively coarse, introducing challenges in resolving the functions of the cerebellum especially when the brain is exposed to naturalistic environments. The current study took the advantages of the naturalistic paradigm (i.e., movie viewing) fMRI (nfMRI) to derive fine parcellations via a data-driven dual-regression-like sparse representation framework. The parcellations were quantitatively evaluated by functional homogeneity, and global and local boundary confidence. In addition, the differences of cerebellum–cerebrum functional connectivities between rsfMRI and nfMRI for some exemplar parcellations were compared to provide qualitatively functional validations. Our experimental results demonstrated that the proposed study successfully identified distinct subregions of the cerebellum. This fine parcellation may serve as a complementary solution to existing cerebellum parcellations, providing an alternative template for exploring neural activities of the cerebellum in naturalistic environments.

Adaptive opaque façades and their potential to reduce thermal energy use in residential buildings: A simulation-based evaluation

Adaptive façades are a promising choice to achieve comfortable low-energy buildings. Their effective performance is highly dependent on the local boundary conditions of each application and on the way the dynamic properties are controlled. The evaluation of whole building performance through building performance simulation can be useful to understand the potential of different Adaptive opaque façades (AOF) in a specific context. This paper evaluates through dynamic simulations promising design solutions of AOF for a residential building use in six different climates. It quantifies the total delivered thermal energy of 15 typologies of AOFs which consist of alternative adaptation strategies: (i) variation of solar absorptance of the cladding, (ii) variation of the convective heat transfer of air cavities and (iii) adaptive insulation strategies. For the first time, it also quantifies the performance of AOF which combine more than one adaptation strategy. The results show that the variation of the heat transfer by means of Adaptive Insulation components has the most significant impact on the reduction of the thermal energy use. The variation of the solar absorptance has also a significant positive impact when reducing heating consumption, but only if this adaptation strategy is actively controlled and combined with Adaptive Insulation components.

An Energy Conservative hp-method for Liouville’s Equation of Geometrical Optics

Liouville’s equation on phase space in geometrical optics describes the evolution of an energy distribution through an optical system, which is discontinuous across optical interfaces. The discontinuous Galerkin spectral element method is conservative and can achieve higher order of convergence locally, making it a suitable method for this equation. When dealing with optical interfaces in phase space, non-local boundary conditions arise. Besides being a difficulty in itself, these non-local boundary conditions must also satisfy energy conservation constraints. To this end, we introduce an energy conservative treatment of optical interfaces. Numerical experiments are performed to prove that the method obeys energy conservation. Furthermore, the method is compared to the industry standard ray tracing. The numerical experiments show that the discontinuous Galerkin spectral element method outperforms ray tracing by reducing the computation time by up to three orders of magnitude for an error of $$10^{-6}$$ 10 - 6 .

Existence of solutions and continuous and semi-discrete stability estimates for 3D/0D coupled systems modelling airflows and blood flows

In this paper we analyse geometric multiscale models arising in the description of physiological flows such as blood flow in arteries or air flow in the bronchial tree. The geometrical complexity of the networks in which air/blood flows lead to a classical decomposition in two areas: a truncated 3D geometry corresponding to the largest contribution of the domain, and a 0D part connected to the 3D part, modelling air/blood flows in smaller airways/vessels. The fluid in the 3D part is described by the Stokes or the Navier-Stokes system which is coupled to 0D models or {\em so-called} Windkessel models. The resulting Navier-Stokes-Windkessel coupled system involves Neumann non-local boundary conditions that depends on the considered applications. We first show that the different types of Windkessel models share a similar formalism. Next we derive existence results and stability estimates for the continuous coupled Stokes-Windkessel or Navier-Stokes-Windkessel problem as well as stability estimates for the semi-discretized systems with either implicit or explicit coupling. In all the calculations, we pay a special attention to the dependancy of the various constants and smallness conditions on the data with respect to the physical and numerical parameters. In particular we exhibit different kinds of behavior depending on the considered 0D model. Moreover even if no energy estimates can be derived in energy norms for the Navier-Stokes-Windkessel system, leading to possible and observed numerical instabilities for large applied pressures, we show that stability estimates for both the continuous and semi-discrete problems, can be obtained in appropriate norms for small enough data by introducing a new well chosen Stokes-like operator. These sufficient stability conditions on the data may give a hint on the order of magnitude of the data enabling stable computations without stabilization method for the problem. Numerical simulations illustrate some of the theoretical results.

Spline Collocation for Multi-Term Fractional Integro-Differential Equations with Weakly Singular Kernels

We consider general linear multi-term Caputo fractional integro-differential equations with weakly singular kernels subject to local or non-local boundary conditions. Using an integral equation reformulation of the proposed problem, we first study the existence, uniqueness and regularity of the exact solution. Based on the obtained regularity properties and spline collocation techniques, the numerical solution of the problem is discussed. Optimal global convergence estimates are derived and a superconvergence result for a special choice of grid and collocation parameters is given. A numerical illustration is also presented.

Kinematic dynamos in triaxial ellipsoids

Planetary magnetic fields are generated by motions of electrically conducting fluids in their interiors. The dynamo problem has thus received much attention in spherical geometries, even though planetary bodies are non-spherical. To go beyond the spherical assumption, we develop an algorithm that exploits a fully spectral description of the magnetic field in triaxial ellipsoids to solve the induction equation with local boundary conditions (i.e. pseudo-vacuum or perfectly conducting boundaries). We use the method to compute the free-decay magnetic modes and to solve the kinematic dynamo problem for prescribed flows. The new method is thoroughly compared with analytical solutions and standard finite-element computations, which are also used to model an insulating exterior. We obtain dynamo magnetic fields at low magnetic Reynolds numbers in ellipsoids, which could be used as simple benchmarks for future dynamo studies in such geometries. We finally discuss how the magnetic boundary conditions can modify the dynamo onset, showing that a perfectly conducting boundary can strongly weaken dynamo action, whereas pseudo-vacuum and insulating boundaries often give similar results.

Parameter uniform numerical methods for singularly perturbed delay parabolic differential equations with non-local boundary condition

The motive of this paper is, to develop accurate and parameter uniform numerical method for solving singularly perturbed delay parabolic differential equation with non-local boundary condition exhibiting parabolic boundary layers. Also, the delay term that occurs in the space variable gives rise to interior layer. Fitted operator finite difference method on uniform mesh that uses the procedures of method of line for spatial discretization and backward Euler method for the resulting system of initial value problems in temporal direction is considered. To treat the non-local boundary condition, Simpsons rule is applied. The stability and parameter uniform convergence for the proposed method are proved. To validate the applicability of the scheme, numerical examples are presented and solved for different values of the perturbation parameter. The method is shown to be accurate of O(h2 + △t) . Finally, conclusion of the work is included at the end.

On the Crank-Nicolson difference scheme for the time-dependent source identification problem

In this study the source identification problem for the one-dimensional Schr¨odinger equation with non-local boundary conditions is considered. A second order of accuracy Crank-Nicolson difference scheme for the numerical solution of the differential problem is presented. Stability estimates are proved for the solution of this difference scheme. Numerical results are given.