Basis adaptation for the stochastic nonlinear Poisson–Boltzmann equation

Operator splitting ADI schemes for pseudo-time coupled nonlinear solvation simulations

The nonlinear Poisson–Boltzmann equation and its linearization are introduced together with methods of their solution. Models of electroosmotic flow are reviewed. Expressions for the electrostatic component of disjoining pressure are derived and applied to the problem of film drainage in the presence of electric charges. Electrowetting on dielectric and its applications are discussed.

A microfluidic pumping flow model driven by electro-osmosis mechanisms is developed to analyze the flow characteristics of aqueous electrolytes. The pumping model is designed based on a single propagative rhythmic membrane contraction applied on the upper wall of a microchannel. The flow lubrication theory coupled with a nonlinear Poisson–Boltzmann equation is used to model the microchannel unsteady creeping flow and to describe the distribution of the electric potential across the electric double layer. A generic solution is obtained for the Poisson–Boltzmann equation without the Debye–Hückel linearization. The effects of zeta potential, Debye length, and electric field on the potential distribution, pressure distribution, velocity profiles, shear stress, and net flow rate are computed and interpreted in detail. The results have shown that this electrokinetic membrane pumping model can be used to understand microlevel transport phenomena in various physiological systems. The proposed model can also be integrated with other microfluidic devices for moving microvolume of liquids in artificial capillaries used in modern biomedical applications.

Fluid–Fluid Correlation through a Model Charged Membrane: Analytical Results

The application of polynomial chaos expansions (PCEs) to the propagation of uncertainties in stochastic dynamical models is well-known to face challenging issues. The accuracy of PCEs degenerates quickly in time. Thus maintaining a sufficient level of long-term accuracy requires the use of high-order polynomials. In numerous cases, it is even infeasible to obtain accurate metamodels with regular PCEs because PCEs cannot represent the dynamics. To overcome the problem, an original numerical approach was recently proposed that combines PCEs and non-linear autoregressive with exogenous input (NARX) models, which are a universal tool in the field of system identification. The approach relies on using NARX models to mimic the dynamical behavior of the system and dealing with the uncertainties using PCEs. The PC-NARX model was built by means of heuristic genetic algorithms. This paper aims at introducing the least angle regression (LARS) technique for computing PC-NARX models, which consists of solving two linear regression problems. The proposed approach is validated with structural mechanics case studies, in which uncertainties arising from both structures and excitations are taken into account. Comparison with Monte Carlo simulation and regular PCEs is also carried out to demonstrate the effectiveness of the proposed approach.

The study of nonstationary rarefied gas flows is currently paid much attention. Such interest to these problems is caused by the creation of pulsed jets used for the deposition of thin films and special coatings on solid surfaces. However the problems of nonstationary rarefied gas flows have not been studied sufficiently fully because of their large computational complexity. In this paper the computational aspects of investigating the nonstationary flows of a reflected gas from a wall and flowing through a suddenly formed gap is considering. The objective of this study is to analyze the possible numerical kinetic approaches for solving such nonstationary problems and to identify the difficulties encountered in solving.When studying the gas flows in strong rarefaction regimes one should consider the Boltzmann kinetic equation, but its numerical implementation is rather laborious. In order to use more simple approaches based for example on approximation kinetic equations (Ellipsoidal-Statistical model, Shakhov model), it is important to estimate the difference of the solutions of the model equations and the Boltzmann equation. For this purpose two auxiliary problems are considered: reflection of the gas flow from the wall and outflow of the free jet into the rarefied background gas. Numerical solution of these problems shows a weak dependence of the solution on the type of the collision operator in the rarefied region, but a strong dependence on the velocity grid step . The detailed velocity grid is necessary to avoid non-monotonous behavior of macroparameters caused by the “ray effect”. To reduce numerical costs on detailed grid a hybrid method based on the synthesis of model equation and the Boltzmann equation is proposed. Such approach can be promising since it reduces the domain in which the Boltzmann collision integral should be used.The results presented in this paper were obtained using two different software packages Unified Flow Solver (UFS) [13] and Nesvetay 3D [14-15]. Note that UFS uses the discrete ordinate method for velocity space on a uniform grid and a hierarchical adaptive mesh refinement in physical space. The possibility of calculating both the Boltzmann equation and model equations is realized. The Nesvetay 3D complex was created to solve the Shakhov model equation, (S-model) and makes it possible to calculate on non-structured non uniform grids in velocity and physical spaces.Translated from Russian. Original text: Mathematics and Mathematical Modeling. 2018. no. 4. Pp. 27-44.

The Poisson–Boltzmann equation (PBE) is a fundamental implicit solvent continuum model for calculating the electrostatic potential of large ionic solvated biomolecules. However, its numerical solution encounters severe challenges arising from its strong singularity and nonlinearity. In (P. Benner, V. Khoromskaia, B. Khoromskij, C. Kweyu, and M. Stein, “Regularization of Poisson-Boltzmann type equations with singular source terms using the range-separated tensor format,” SIAM J. Sci. Comput., vol. 43, no. 1, pp. A415–A445, 2021; C. Kweyu, V. Khoromskaia, B. Khoromskij, M. Stein, and P. Benner, “Solution decomposition for the nonlinear Poisson-Boltzmann equation using the range-separated tensor format,” arXiv:2109.14073, 2021), the effect of strong singularities was eliminated by applying the range-separated (RS) canonical tensor format (P. Benner, V. Khoromskaia, and B. N. Khoromskij, “Range-separated tensor format for many-particle modeling,” SIAM J. Sci. Comput., vol. 40, no. 2, pp. A1034–A1062, 2018; B. N. Khoromskij, “Range-separated tensor representation of the discretized multidimensional Dirac delta and elliptic operator inverse,” J. Comput. Phys., vol. 401, p. 108998, 2020) to construct a solution decomposition scheme for the PBE. The RS tensor format allows deriving a smooth approximation to the Dirac delta distribution in order to obtain a regularized PBE (RPBE) model. However, solving the RPBE is still computationally demanding due to its high dimension N $\mathcal{N}$ , where N $\mathcal{N}$ is always in the millions. In this study, we propose to apply the reduced basis method (RBM) and the (discrete) empirical interpolation method ((D)EIM) to the RPBE in order to construct a reduced order model (ROM) of low dimension N ≪ N $N\ll \mathcal{N}$ , whose solution accurately approximates the nonlinear RPBE. The long-range potential can be obtained by lifting the ROM solution back to the N $\mathcal{N}$ -space while the short-range potential is directly precomputed analytically, thanks to the RS tensor format. The sum of both provides the total electrostatic potential. The main computational benefit is the avoidance of computing the numerical approximation of the singular electrostatic potential. We demonstrate in the numerical experiments, the accuracy and efficacy of the reduced basis (RB) approximation to the nonlinear RPBE (NRPBE) solution and the corresponding computational savings over the classical nonlinear PBE (NPBE) as well as over the RBM being applied to the classical NPBE.

The small-angle x-ray scattering technique was applied to measure the distribution of Cs/sup +/ counterions around the surface of cylindrical micelles. The scattering intensity distribution was compared with that calculated by use of the analytical solutions of a nonlinear Poisson-Boltzmann (PB) equation in a cell model. Quantitative agreement was obtained for the cases without added salt. The addition of other monovalent salts did not modify the counterion distribution significantly in accordance with the prediction of the numerical solution of the PB equation.

Advanced Plume Analysis Software for gas measurementTheodore Hilgeman and Lewis SmithNorthrop Grumman CorporationAdvanced Technology and Development CenterBethpage, NY 11714ABSTRACTPassive remote determination of the presence and quantity of gaseous chemicals is of interest for a variety of applications,but has been technically difficult to accomplish because of challenging measurement and analysis difficulties. This paperdescribes progress in the development of infrared plume analysis software. To unravel the atmosphere, background, andthermal contrast relationships it has been found necessary to obtain both spatial and spectral information and to have a set ofcodes which take these phenomena into account. Measured data were used along with model analyses to develop AdvancedPlume Analysis Software (APAS) applicable to calculating how accurately amounts of gas can be determined from overheadwith current and possible future sensor technology. The APAS suite of codes incorporates sensor noise, plume contrast andatmospheric effects modules to arrive at a comparative measure ofgas plume detection accuracy. The application ofthe APAScodes to prediction of detection accuracy for several candidate sensors is shown.keywords: plume analysis, plume software, gas detection, sensor noise, atmospheric effects1. BACKGROUNDThe complex phenomenology ofgases in an uncontrolled environment makes passive remote determination of thepresence and quantity of gaseous chemicals technically difficult to accomplish. In Figure 1, the illustration ofthe smoke stackplume is typical ofthe types of difficulties to be overcome. The gases of interest may be hot, just coming out of a stack, orthey may have cooled to ambient temperature. They may be seen against a cooler (or hotter) background with good contrast,or seen against a nearly equal temperature background with poor contrast. The intervening atmosphere introduces artifacts ofabsorption and emission which perturb the signal. The sensor is part ofthe measurement, and its noise properties have asignificant impact on the sensitivity ofdetennining the amount ofgas present. Accounting for these relationships requiresmodeling both spatial and spectral information. This investigation utilizes model analyses to address the question of howaccurately amounts of gas can be determined passively from overhead with current and possible future infrared sensortechnology.Figure 2 illustrates a gas measurement spectrum, a result of complex phenomena and thus multiple parameters. With theuse ofAPAS, spectra such as the spectrum illustrated here of a hot CO2 plume behind cold atmospheric H20 can be invertedto yield underlying conditions. Alternatively, in this paper, we apply APAS to the forward problem to estimate detectioncapability given various gas amounts and sensor parameters.2. APAS MODELSThe models in APAS calculate sensor noise, plume radiance and temperature distribution, atmospheric absorption andemission, and background signatures from emission and reflection. Sensor noise models are available for Michelson Fouriertransform spectrometers (FTS), Fabry-Perot spectrometers and diffraction grating spectrometers. The SIN relationshipdepends on the target (plume) size relative to the diffraction limited size for the particular telescope and range. Threeconditions can apply -subpixel, diffraction limited, and superpixel. In the subpixel case, the plume size is smaller than thediffraction limited pixel. In this case, the plume behaves like a point source. For this case, the S/N for the FTS is given by

We consider a spatially homogeneous and isotropic gas consisting of hard-sphere molecules. A vector representation of the scattering kernel is used to adapt the original Boltzmann equation to the idealized geometrical situation. By means of an expansion of the distribution function in terms of Laguerre polynomials this scalar Boltzmann equation is transformed to a set of moment equations. All “algebraized” collision integrals can be evaluated analytically. We discuss the truncation of the moment equations necessary for the practical application of this method. The eigenvalues of the linearized relaxation problem show a good convergence with respect to the truncation index.

Based on Boltzmann transport equation, the drift-diffusion, hydrodynamic, and Monte-Carlo physical models are accurately developed. The model equations are self-consistently solved with Poisson equation, and with Schrodinger equation when quantization effects take place, in one and two-dimensions to characterize the operation and optimize the structure of mm-wave devices. The effects of the devices dimensions, biasing conditions and operating frequencies on the accuracy of the obtained model (simulator) results are thoroughly investigated. Based on physical understanding of the models, the simulation results are analyzed and conclusions are drawn to fully determine the limits at which a certain device simulator can be accurately and efficiently used to characterize the noise behaviour of mm-wave devices.

Self-heating has emerged as a critical bottleneck to scaling in modern transistors. In simulating heat conduction in these devices, it is important to account for the granularity of phonon transport since electron-phonon scattering occurs preferentially to select phonon groups. However, a complete accounting for phonon dispersion, polarization and scattering is very expensive if the Boltzmann transport equation (BTE) is used. Moreover, difficulties with convergence are encountered when the phonon Knudsen number becomes small. In this paper we simulate a two-dimensional bulk MOSFET hotspot problem using a partially-implicit hybrid BTE-Fourier solver which is significantly less expensive than a full BTE solution, and which shows excellent convergence characteristics. Volumetric heat generation from electron-phonon collisions is taken from a Monte Carlo simulation of electron transport and serves as a heat source term in the governing transport equations. The hybrid solver is shown to perform well in this highly non-equilibrium situation, matching the solutions obtained from a pure all-BTE solution, but at significantly lower computational cost. The paper establishes that this new model and solution methodology are viable for the simulation of thermal transport in other emerging transistor designs and in other nanotechnology applications as well.

Dependences of electric characteristics on the technological parameters of the field-effect Hall sensors based on SOI structures (FEHS-SOI) are discussed. The manufacturing process for the formation of a magnetosensitive structure comprising the field-effect Hall sensor based on a MOSFET in the SOI structures was simulated. Electrical characteristics of the device were calculated and the optimization research devoted to the influence of process parameters on the FEHS-SOI voltage-current characteristics and sensitivity was made.

Nanowire sensor calibration and performance evaluation in microfluidic flow velocity monitoring

Numerical solution of parquet equations for the Anderson impurity model