High-order Solver Research Articles

High-order numerical methods for nonlinear acoustics: A Fourier Continuation approach

Dispersion errors, which result from the use of low-order numerical methods in wave-propagation and transport problems, can have a devastating impact on the accuracy of acoustic simulations. These errors are especially problematic in settings containing nonlinear acoustic waves that propagate many times their fundamental wavelength. In these cases, the use of high-order numerical schemes is vital for the accurate and efficient evaluation of the acoustic field. We present a class of high-order time-domain solvers for the treatment of nonlinear acoustic propagation problems. These solvers are based on the Fourier Continuation method, which produces rapidly-converging Fourier series expansions of non-periodic functions (thereby avoiding the Gibbs phenomenon), and are capable of accurately and efficiently simulating nonlinear acoustic fields in large, complex domains.

Well-conditioned boundary integral equations for two- dimensional sound-hard scattering problems in domains with corners

We present several well-posed, well-conditioned direct and indirect integral equation formulations for the solution of two-dimensional acoustic scattering problems with Neumann boundary conditions in domains with corners. We focus mainly on Direct Regularized Combined Field Integral Equations (DCFIE-R) formulations whose name reflects that (1) they consist of combinations of direct boundary integral equations of the second-kind and first-kind integral equations which are preconditioned on the left by coercive boundary single-layer operators, and (2) their unknowns are physical quantities, i.e the total field on the boundary of the scatterer. The DCFIE-R equations are shown to be uniquely solvable in appropriate function spaces under certain assumptions on the coupling parameter. Using Calderon’s identities and the fact that the unknowns are bounded in the neighborhood of the corners, the integral operators that enter the DCFIE-R formulations are recast in a form that involves integral operators that are expressed by convergent integrals only. The polynomially-graded mesh quadrature introduced by Kress [30] enables the high-order resolution of the weak singularities of the kernels of the integral operators and the singularities in the derivatives of the unknowns in the vicinity of the corners. This approach is shown to lead to an efficient, high-order Nystrom method capable of producing solutions of sound-hard scattering problems in domains with corners which require small numbers of Krylov subspace iterations throughout the frequency spectrum. We present a variety of numerical results that support our claims.

Biomedical applications and diagnostics of atmospheric pressure plasma

Numerous applications of non-equilibrium (cold, low temperature) plasmas require those plasmas to operate at atmospheric pressure. Achieving non-equilibrium at atmospheric pressure is difficult since the ionization growth is very fast at such a high pressure. High degree of ionization on the other hand enables transfer of energy between electrons and ions and further heating of the background neutral gas through collisions between ions and neutrals. Thus, all schemes to produce non-equilibrium plasmas revolve around some form of control of ionization growth. Diagnostics of atmospheric pressure plasmas is difficult and some of the techniques cannot be employed at all. The difficulties stem mostly from the small size. Optical emission spectroscopy and laser absorption spectroscopy require very high resolution in order to resolve the anatomy of the discharges. Mass analysis is not normally applicable for atmospheric pressure plasmas, but recently systems with triple differential pumping have been developed that allow analysis of plasma chemistry at atmospheric pressures which is essential for numerous applications. Application of such systems is, however, not free from problems. Applications in biomedicine require minimum heating of the ambient air. The gas temperature should not exceed 40 °C to avoid thermal damage to the living tissues. Thus, plasmas should operate at very low powers and power control is essential. We developed unique derivative probes that allow control of power well below 1 W and studied four different sources, including dielectric barrier discharges, plasma needle, atmospheric pressure jet and micro atmospheric pressure jet. The jet operates in plasma bullet regime if proper conditions are met. Finally, we cover results on treatment of bacteria and human cells as well as treatment of plants by plasmas. Localized delivery of active species by plasmas may lead to a number of medical procedures that may also involve removal of bacteria, fungi and spores.

Correlative Real-Time and Super Resolution Imaging of Mitochondrial Dynamics

Mitochondria are energy producing organelles that play essential functions in all aspects of cell biology. In most mammalian cells mitochondria exist in the form of a highly dynamic interconnected network. Live cell imaging experiments revealed that this network undergoes dramatic shape re-organization as motor proteins actively pull the mitochondrial membrane. In addition, mitochondria that exist as individual organelles can be actively transported from one location to another. Mitochondrial shape changes and their active transport involve microtubules and play key roles in mitochondrial function However, several questions remain unanswered regarding the dynamic behavior of mitochondria: how often do mitochondria change tracks as they are transported along microtubules, how do the underlying microtubule tracks influence the observed shape changes and can mitochondrial membrane be pulled along multiple microtubule tracks.In order to relate the dynamic events of the mitochondrial network to the underlying cytoskeleton, we developed a sequential live-cell and super-resolution imaging approach. Live cell imaging allowed us to track the movement of mitochondria and characterize their dynamic behavior. We could stop the dynamics at a specific time point by in situ fixation on the microscope stage. Subsequent immuno-staining followed by super-resolution imaging using Stochastic Optical Reconstruction Microscopy (STORM) allowed us to obtain a high resolution (∼20 nm) image of the underlying microtubule network. The microtubule network was stabilized through treatment with low concentrations of drugs (nocodazole and taxol) and did not change appreciably during live cell imaging. As a result, we could overlay the dynamic behavior of mitochondria with the high resolution images of microtubule tracks by using fiduciary markers for alignment. This approach allowed us to correlate mitochondrial dynamics and the microtubule tracks at high resolution in order to answer the questions that we pose above.

A p-multigrid spectral difference method for two-dimensional unsteady incompressible Navier–Stokes equations

This paper presents the development of a 2D high-order solver with spectral difference method for unsteady incompressible Navier–Stokes equations accelerated by a p-multigrid method. This solver is designed for unstructured quadrilateral elements. Time-marching methods cannot be applied directly to incompressible flows because the governing equations are not hyperbolic. An artificial compressibility method (ACM) is employed in order to treat the inviscid fluxes using the traditional characteristics-based schemes. The viscous fluxes are computed using the averaging approach (Sun et al., 2007; Kopriva, 1998) [29,12]. A dual time stepping scheme is implemented to deal with physical time marching. A p-multigrid method is implemented (Liang et al., 2009) [16] in conjunction with the dual time stepping method for convergence acceleration. The incompressible SD (ISD) method added with the ACM (SD-ACM) is able to accurately simulate 2D steady and unsteady viscous flows.

CERES in higher-order logic

We define a generalization CERESω of the first-order cut-elimination method CERES to higher-order logic. At the core of CERESω lies the computation of an (unsatisfiable) set of sequents CS(π) (the characteristic sequent set) from a proof π of a sequent S. A refutation of CS(π) in a higher-order resolution calculus can be used to transform cut-free parts of π (the proof projections) into a cut-free proof of S. An example illustrates the method and shows that CERESω can produce meaningful cut-free proofs in mathematics that traditional cut-elimination methods cannot reach.

A two-dimensional morphological model based on a next generation circulation solver I: Formulation and validation

This paper presents an effective morphological modeling system with new module designs. A two dimensional (2D) morphological model which contains grid generation module, circulation module, wave module, transport module and bed evolution module has been developed and validated. The grid generation module is based on unstructured triangular grid with second order accuracy achieved through cell reconstruction using multiple least square methods. The circulation module is based on semi-implicit Eulerian–Lagrangian method, which is stable and free from Courant–Friedrichs–Lewy (CFL) constraint. Using the cell reconstruction, the circulation module relaxes the grid orthogonality requirement of previous similar circulation solvers. In the wave module, unstructured SWAN model was incorporated to simulate the effects of current–wave interaction simultaneously. The transport module is developed with finite-volume method with high-order resolution. The modeling of suspended sediment transport and updating of morphological evolution is contained in the sediment-morphology module. The morphological modeling system is referred as CurWaC2D-Sed. Following the detailed description of each module, a number of theoretical and laboratory validation studies were conducted. The CurWaC2D-Sed model has shown to perform well in all those validation tests. The applications of this model to the field cases are presented in a companion paper.

Numerical study of radiative heat transfer effects on a complex configuration of rack storage fire

This work describes the application and the performance of a new radiation model in CFD calculations for the simulation of thermal radiation transfer effects on a fire scenario. A 3D Cartesian coordinates radiative heat transfer procedure based on coupling of the FTn finite volume method (FTnFVM) with the bounded high-order resolution CLAM scheme is developed. The narrow-band based weighted-sum-of-gray-gases (NB-WSGG) model is applied to take account of nongray effects by CO 2, H 2O and soot. To treat irregular boundaries, the present model used the blocked-off-region procedure. This radiation code is implemented in the Fire Dynamics Simulator (FDS), a Computational-Fluid-Dynamics-based fire model, where a the combustion is represented by means of the mixture fraction with a single step chemical reaction model and the Large Eddy Simulation (LES) is used to model the dissipative processes. Computational results with and without radiation effects are compared against available experimental data and quasi-steady state law correlations of in-rack storage fire, which consists a complex configuration of double tri-wall corrugated paper cartons placed onto a wood pallet. Sensibility analyses of spatial and angular grids demonstrate the improvements due to the FTnFVM and to the CLAM scheme in the configuration studied. Results show that the simulations of the flame height, the gas temperature and the gas velocity are strongly influenced by thermal radiation. Overall, simulations predicted closer profiles to the experimental results only when the nongray-sooting radiation model was incorporated and an over-prediction of the gas temperature and the flame height is found when radiation is neglected. A sensibility analysis has shown that the flame characteristics are strongly affected by the soot yield.

High-order resolution Eulerian–Lagrangian simulations of particle dispersion in the accelerated flow behind a moving shock

This paper presents a computational study of the two-dimensional particle-laden flow developments of bronze particle clouds in the accelerated flow behind a moving normal shock. Particle clouds with a particle volume concentration of 4% are arranged initially in a rectangular, triangular and circular shape. Simulations are performed with a recently developed high-order resolution Eulerian–Lagrangian method that approximates the Euler equations governing the gas dynamics with the improved high order weighted essentially non-oscillatory (WENO-Z) scheme, while individual particles are traced in the Lagrangian frame using high-order time integration schemes. Reflected shocks form ahead of all the cloud shapes. The detached shock in front of the triangular cloud is weakest. At later times, the wake behind the cloud becomes unstable, and a two-dimensional vortex-dominated wake forms. Separated shear layers at the edges of the clouds pull particles initially out of the clouds that are consequently transported along the shear layers. Since flows separated trivially at sharp corners, particles are mostly transported out of the cloud into the flow at the sharp front corner of the rectangular cloud and at the trailing corner of the triangular cloud. Particles are transported smoothly out of the circular cloud, since it lacks sharp corners. At late times, the accelerated flow behind the running shock disperses the particles in cross-stream direction the most for the circular cloud, followed by the rectangular cloud and the triangular cloud.

A Study of Curved Boundary Representations for 2D High Order Euler Solvers

The present study discusses several approaches of representing curve boundaries with high-order Euler solvers. The aim is to minimize the solution entropy error. The traditional boundary representation based on interpolation polynomials is presented first. A new boundary representation based on the Bezier curve is then developed and analyzed. Several numerical tests are conducted to compare both representations. The new approach is shown to have several advantages for complicated geometries.

A kinetic flux-vector splitting method for the shallow water magnetohydrodynamics

A kinetic flux-vector splitting (KFVS) scheme for the shallow water magnetohydrodynamic (SWMHD) equations in one- and two-space dimensions is formulated and applied. These equations model the dynamics of a thin layer of nearly incompressible and electrically conducting fluids for which the evolution is nearly two-dimensional with magnetic equilibrium in the third direction. The proposed numerical scheme is based on the direct splitting of macroscopic flux functions of the SWMHD equations. In two-space dimensions the scheme is derived in a usual dimensionally split manner; that is, the formulae for the fluxes can be used along each coordinate direction. The high-order resolution of the scheme is achieved by using a MUSCL-type initial reconstruction and Runge–Kutta time stepping method. Both one- and two-dimensional test computations are presented. For validation, the results of KFVS scheme are compared with those obtained from the space–time conservation element and solution element (CE/SE) method. The accuracy, efficiency and simplicity of the KFVS scheme demonstrate its potential in modeling SWMHD equations.

Three-dimensional visualization of in vitro cultivated chondrocytes inside porous gelatine scaffolds: A tomographic approach

Synchrotron radiation-based microcomputed tomography (SR-μCT) has become a valuable tool in the structural characterization of different types of materials, achieving volumetric details with micrometre resolution. Biomedical research dealing with porous polymeric biomaterials is one of the research fields which can benefit greatly from the use of SR-μCT. This study demonstrates that current experimental set-ups at synchrotron beamlines achieve a sufficiently high resolution in order to visualize the positions of individual cartilage cells cultivated on porous gelatine scaffolds made by a freeze-structuring technique. Depending on the processing parameters, the pore morphology of the scaffolds investigated was changed from large-pore sized but non-ordered structures to highly directional and fine pored. The cell-seeded scaffolds were stained with a combined Au/Ag stain to enhance the absorption contrast in SR-μCT. While only some cells showed enhanced absorption contrast, most cells did not show any difference in contrast to the surrounding scaffold and were consequently not detectable using conventional greyscale threshold methods. Therefore, using an image-based three-dimensional segmentation tool on the tomographic data revealed a multitude of non-stained cells. In addition, the SR-μCT data were compared with data obtained from scanning electron microscopy, energy dispersive X-ray spectroscopy and histology, while further linking the initial cell density measured via a MTT assay to the pore size as determined by SR-μCT.

Resolution is Cut-Free

In this article, we show that the extension of the resolution proof system to deduction modulo is equivalent to the cut-free fragment of the sequent calculus modulo. The result is obtained through a syntactic translation, without using any cut-elimination procedure. Additionally, we show Skolem theorem and inversion/focusing results. Thanks to the expressiveness of deduction modulo, all these results also apply to the cases of higher-order resolution, Peano's arithmetic and Gentzen's LK.

Obtaining and Verifying High-Order Unstructured Finite Volume Solutions to the Euler Equations

Interest in high-order unstructured-mesh finite volume methods continues to grow, as researchers are attracted both to the possibilities of high-accuracy solutions on coarse meshes and to applications for which high-order accuracy is all but mandatory, including large-eddy simulations for complex geometries. A major obstacle for those interested in developing a high-order solver, however, is that there are many more details that must be exactly right for a nominally-high-order scheme to be genuinely-high-order than for a second-order scheme to be second-order. In many cases, these details are either omitted or glossed over in the literature. This paper attempts to close this gap by illuminating a number of obscure, or often overlooked, aspects of correctly implementing a high-order-accurate unstructured-mesh finite volume solver. We discuss reconstruction, flux integration, curved-boundary treatment, and demonstrating order of accuracy. In each case, we describe test cases that are appropriate for confirming the correct behavior of a high-order code, as well as demonstrating some of the common pitfalls in implementation and showing their effects. We hope that this paper will serve other researchers as a practical guide to creating their own high-order-accurate solvers.

High precision measurement of surface cracks using an optical system

In this contribution an advanced technique to measure the width of surface cracks in concrete is presented. The goal is to provide a system for laboratory applications with a high resolution in order to be able to estimate the stabilization of surface cracks in the range of 0–250 µm during laboratory loading tests. Using digital imaging and suitable image processing algorithms, an automated system for the determination of crack widths was developed. A 4 megapixel monochrome digital camera was used in combination with a special telecentric lens, in order to achieve a resolution of 1 µm per pixel. The main challenge in this work was to develop a robust image processing algorithm, which determines the surface crack width under changing environmental conditions, especially in terms of ambient light.

High order resolution of the Maxwell–Fokker–Planck–Landau model intended for ICF applications

A high order, deterministic direct numerical method is proposed for the non-relativistic 2Dx×3Dv Vlasov–Maxwell system, coupled with Fokker–Planck–Landau collision operators. The magnetic field is perpendicular to the 2Dx plane surface of computation, whereas the electric fields occur in this plane. Such a system is devoted to modelling of electron transport and energy deposition in the general frame of Inertial Confinement Fusion applications. It is able to describe the kinetics of the plasma electrons in the nonlocal equilibrium regime, and permits to consider a large anisotropy degree of the distribution function. We develop specific methods and approaches for validation, that might be used in other fields where couplings between equations, multiscale physics, and high dimensionality are involved. Fast algorithms are employed, which makes this direct approach computationally affordable for simulations of hundreds of collisional times.

Application of the Finite-Volume Method to Study the Effects of Baffles on Radiative Heat Transfer in Complex Enclosures

A finite-volume radiation model for participating gray media in 2-D and 3-D complex rectangular enclosures with obstacles is developed. The step and the bounded high-order resolution curved-line advection method (CLAM) schemes are examined. Using the blocked-off-region procedure, the present model is capable of predicting radiative heat transfer in enclosures with obstructions and baffles. In order to validate the formulations derived here a square cavity with one or three baffles and finned internal cylinder, then a three-dimensional complex heat recuperator of a pilot plant of biomass pyrolysis with obstructions and baffles, are studied. It should be pointed out that the developed code using the CLAM scheme is accurate and convenient for computational thermal calculations. For the considered heat recuperator, the presence of baffles enhances radiative heat flux and contributes to the increase of the mean medium temperature.

Simulation of bipolar charge transport with trapping and recombination in polymeric insulators using Runge–Kutta discontinuous Galerkin method

The widely used QUICKEST method with ULTIMATE flux limiter is not capable of solving the charge transport problems with a very steep wavefront accurately, due to the wide stencil adopted. Furthermore, the splitting process of separating the convection and the reaction terms in the method introduces additional errors. To solve such problems accurately, a novel numerical method based on the Runge–Kutta discontinuous Galerkin (RKDG) method is introduced in this paper, which has high-order resolution and weak correlation between cells. The bipolar charge transport under dc voltage in solid dielectrics with trapping and recombination is simulated using this new method. The results of charge profiles provided by the method are obviously different from the simulation results in the existing literature. The method was verified by problems with analytical solution and experimental observations.

Electronic spectroscopy of carbon chains

Gas phase electronic spectra allow direct comparisons between laboratory data and the wealth of astrophysical measurements that have been tabulated. Over the past decade a number of assorted carbon chain systems have been spectroscopically investigated, yielding an understanding of the transitions and trends apparent among their homologous series. This rigorous program begins with trapping unstable carbon chain ions and radicals in neon matrices, where origin bands and vibrational data are obtained. From these results one can then seek out the gas phase transitions in higher resolution in order to more accurately determine assignments and spectroscopic constants. To this purpose, laser spectroscopic methods are employed, including pulsed and cw cavity ringdown, resonant enhanced multiphoton ionisation, and more recently, a two-photon two-colour excitation/fragmentation approach to investigate trapped ions. From a comparison involving the electronic spectra of a number of carbon chain systems measured in the laboratory with diffuse interstellar band (DIB) absorptions, one concludes that species involving only a handful of carbon atoms can not be the carriers. However, larger carbon chains and ring species still remain viable. In this review, we consider which systems require further investigation, including these longer species, metal-capped chains and other cationic radicals.

Optimal corrections for digitization and quantification effects in angle measure technique (AMT) texture analysis

AbstractThe effects of digitization and quantification in one‐ and two‐dimensional signals on angle measure technique (AMT) texture analysis are described in order to find optimal corrections. AMT analysis with varying parameters has been carried out on simulated and real images as well on time‐series data. All images and signals are of high resolution in order to study these effects with particularly attention to the low(est), local scales. Quantification effects are studied by multiplying original images or signals by varying scale factor values, K with posterior round‐up, while digitization effects were realized by applying various resampling schemes for images and signals using bilinear interpolation. Results lead to general conclusions regarding possible operational regimes for correction of these adverse effects as a function of optimal scale‐ and sampling‐factors during AMT pre‐processing. Copyright © 2008 John Wiley & Sons, Ltd.