On the extension problem for accretive differential operators

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.

Within the context of a pressure-based finite volume method for simulating incompressible flow involving open boundaries, asolution procedure without the need for explicit boundary conditions at any open boundary is proposed. The sole information required in solving the discretized momentum and pressure-linked equations is the flow rate allocated to each of the open boundaries within the solution domain. The methodology isdemonstrated by simulating the laminar flow of an incompressible Newtonian fluid in a duct with a 90 degree branch.

<p style='text-indent:20px;'>In this paper, the chaotic oscillations of the initial-boundary value problem of linear hyperbolic partial differential equation (PDE) with variable coefficients are investigated, where both ends of boundary conditions are nonlinear implicit boundary conditions (IBCs). It separately considers that IBCs can be expressed by general nonlinear boundary conditions (NBCs) and cannot be expressed by explicit boundary conditions (EBCs). Finally, numerical examples verify the effectiveness of theoretical prediction.</p>

Calculating the unknown coupled multiphysics fields from the known boundary condition is of great significance in computational physics. Existing classical algorithms are usually time consuming and resource demanding. Explosive growth in deep learning (DL) has provided a substitutive way to accelerate the calculation process by fully making use of the parallel computing capability of the graphics processing unit. In this work, a cascaded DL framework is presented to compute the coupled multiphysics fields related to electrical potential, temperature, velocity, etc. The whole framework is comprised of an input module and several Fourier modules, which achieves to acquire the coupled fields from the explicit boundary conditions. Compared with conventional networks, the Fourier network emerges consistent average error among different resolutions which saves training costs. As a result, a fully trained network can obtain the expected physics fields in real time, offering rosy prospects for practical scenarios.

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

This paper extends the generating function approach to the state constrained linear quadratic optimal control problem by utilizing penalties. The original constrained problem is approximated by an unconstrained problem with a novel penalty function by which the latter problem can be solved via generating function method based on Taylor series approximation. In view of this, the optimal solution can be given as the state feedback control with explicit boundary conditions and pre-computed coefficients. This structure is useful in the repetitive on-line computation for optimal solutions satisfying a number of different boundary conditions. Examples demonstrate the effectiveness of the developed method.

A derivation of ordinary QED perturbation theory is given in the Schrödinger picture. The path integral is restricted to fields in a finite time interval, ti ≤ t ≤ tf, with explicit boundary conditions given by vacuum state functionals at the initial and final times. The free photon vacuum functional is uniquely specified by the requirement of stationarity in time. For the free electron vacuum further conditions need to be imposed, e.g., the occupation of all negative energy states. The calculations are carried out using a discretized time interval of finite spacing dt, and exact results (to all orders in dt) are obtained for the free generating functionals in Coulomb gauge. Hence Feynman diagrams may be evaluated before the limits of continuous and infinite time are taken, with 1/dt and 1/(tf − ti) providing, respectively, an ultraviolet and an infrared cut-off on the energy integrals. The method may also be used for studying the time evolution of interacting states, in particular of bound states.

The incidence of the numerical resolution and the blockage effect are investigated in an embedding method for solving bidimensional bluff body flows. This method consists of using an artificial boundary instead of imposing exact conditions on the body surface. It requires us to define a blur frontier ratio and a blockage effect ratio. The blockage effect ratio is found using the mean flow of a circular cylinder directly. The blur frontier ratio is obtained by comparison of the present method with another numerical method where explicit boundary conditions on the body are imposed. For this ratio, the investigations are based on the flow past a square cylinder which discard the uncertainty on the surface of the body for the embedding method. Hence, the two factors allow the transformations of the Strouhal and the Reynolds numbers for the flow past a circular cylinder. The universal Strouhal-Reynolds number relationship of the circular cylinder is finally recovered.

Liquid metals are prospective coolant and working media for fission and fusion applications. In latter case they will be affected by very high magnetic fields. By now there is no possibility to recreate these conditions directly, and research is being made using numerical simulations and model liquids. Special care is taken to the possible thermogravitational phenomena in the presence of magnetic field. Temperature and temperature fluctuations are used to describe the flow structure as they can be obtained from experiments with high accuracy and localization. Area of intense fluctuations existence was explored using several experimental facilities including a new HELME Facility with magnetic fields of up to 2.7 Tesla. Area where fluctuations occur and specific of transition regions have been explored for the round vertical pipes with down flow and uniform heating (Grashof number of up to 1.2·108) as a basic configuration with explicit boundary conditions.

An accurate solution of the wave equation at a fluid-solid interface requires a correct implementation of the boundary condition. Boundary conditions at fluid-solid interface require continuity of the normal component of particle velocity and traction, whereas the tangential components vanish. A main challenge is to model interface waves, namely, the Scholte and leaky Rayleigh waves. This study uses a nodal discontinuous Galerkin (dG) finite-element method with the medium discretized using an unstructured uniform triangular meshes. The natural boundary conditions in the dG method are implemented by (1) using an explicit upwind numerical flux and (2) by using an implicit penalty flux and setting the modulus of rigidity of the acoustic medium to zero. The accuracy of these methods is evaluated by comparing the numerical solutions with analytical ones, with source and receiver at and away from the interface. The study shows that the solutions obtained from the explicit and implicit boundary conditions provide the correct results. The stability of the dG scheme is determined by the numerical flux, which also implements the boundary conditions by unifying the numerical solution at shared edges of the elements in an energy stable manner.

We present a solution of the extended Phillips-Kato extension problem about existence and parametrization of all accretive (*)-extensions (with the exit into triplets of rigged Hilbert spaces) of a densely defined non-negative operator.I n particular, the analogs of the von Neumann and Friedrichs theorems for existence of non-negative self-adjoint (*)-extensions are obtained. Relying on these results we introduce the extremal classes of Stieltjes and inverse Stieltjes functions and show that each function from these classes can be realized as the impedance function of an L-system.I t is proved that in this case the realizing L-system contains an accretive operator and, in case of Stieltjes functions, an accretive (*)-extension.M oreover, we establish the connection between the above-mentioned classes and the Friedrichs and Kre”n-von Neumann extremal non-negative extensions.

This work extends the algebraic expression of influence coefficients developed for one-dimensional aquifer models to a two-dimensional (2-D) case. First, the partial differential equation governing the flow in a 2-D semi-confined aquifer is discretized using a finite difference scheme. This results in a system of discrete equations presented in the form of water balance equations associated with a network of interconnected compartments centred on the grid nodes. The foregoing system is transformed into a series of uncoupled 1-D equations stated in terms of some generalized hydraulic head for which they are also solved. Second, the original hydraulic head is recovered from the generalized one via an appropriate linear transformation. Whence, the algebraic expression making the hydraulic head explicit versus sources and boundary conditions is derived. This discrete expression, mapped onto its continuous counterpart, helps to deduce an algebraic form of the inter-compartment influence coefficients. Finally, a comparison with the analytical Green function is carried out.

The ring test has been devised to measure friction paratmeters under extreme pressure conditions as in metal forming processes. However, gravely erroneous assumptions are used to develop and interpret the test. The continuum difinition of stress is violated not only by the test theory but also by the current concepts of friction and lubrication. The author has proposed a general solution for the contact problem based primarily on the continuum definition of stress as a local quantity. An experimental procedure, following Bridgman’s shear-compression fracture tests is proposed to measure frictional properties in a new format. The resulting information is introduced in the analysis as explicit displacement or traction boundary conditions. The incremental nature of the nonlinear boundary value problem involving the deforming media in contact is conveniently used to generate a discretized approximation to the state of stress and deformation along the contact surface at the end of each solution increment. An important consequence of this approach is that the symmetry of the stiffness matrix in the finite element analysis is insured, resulting in considerable computational savings. Another advantage of the proposed method is the elimination of non-physical corner stress singularities. Moreover, recognition of the local nature of contact stresses implies that the proposed method is applicable to any geometry of contact.

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.

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.