Numerical analysis of finite volume schemes for population balance equations

Magdeburg, Univ., Fak. für Mathematik, Diss., 2011

Stability of preconditioned finite volume schemes at low Mach numbers

In [4], Guillard and Viozat propose a finite volume method for the simulation of inviscid steady as well as unsteady flows at low Mach numbers, based on a preconditioning technique. The scheme satisfies the results of a single scale asymptotic analysis in a discrete sense and comprises the advantage that this can be derived by a slight modification of the dissipation term within the numerical flux function. Unfortunately, it can be observed by numerical experiments that the preconditioned approach combined with an explicit time integration scheme turns out to be unstable if the time step Dt does not satisfy the requirement to be O(M2) as the Mach number M tends to zero, whereas the corresponding standard method remains stable up to Dt=O(M), M to 0, which results from the well-known CFL-condition. We present a comprehensive mathematical substantiation of this numerical phenomenon by means of a von Neumann stability analysis, which reveals that in contrast to the standard approach, the dissipation matrix of the preconditioned numerical flux function possesses an eigenvalue growing like M-2 as M tends to zero, thus causing the diminishment of the stability region of the explicit scheme. Thereby, we present statements for both the standard preconditioner used by Guillard and Viozat [4] and the more general one due to Turkel [21]. The theoretical results are after wards confirmed by numerical experiments.

Numerical simulation of tunnel fires using preconditioned finite volume schemes

This article is concerned with the numerical simulation of flows at low Mach numbers which are subject to the gravitational force and strong heat sources. As a specific example for such flows, a fire event in a car tunnel will be considered in detail. The low Mach flow is treated with a preconditioning technique allowing the computation of unsteady flows, while the source terms for gravitation and heat are incorporated via operator splitting. It is shown that a first order discretization in space is not able to compute the buoyancy forces properly on reasonable grids. The feasibility of the method is demonstrated on several test cases.

IMEX finite volume methods for the shallow water equations

Die Flachwassergleichungen (SWE) sind ein hyperbolisches System von Bilanzgleichungen, die adäquate Approximationen an groß-skalige Strömungen der Ozeane, Flüsse und der Atmosphäre liefern. Dabei werden Masse und Impuls erhalten. Wir unterscheiden zwei charakteristische Geschwindigkeiten: die Advektionsgeschwindigkeit, d.h. die Geschwindigkeit des Massentransports, und die Geschwindigkeit von Schwerewellen, d.h. die Geschwindigkeit der Oberflächenwellen, die Energie und Impuls tragen. Die Froude-Zahl ist eine Kennzahl und ist durch das Verhältnis der Referenzadvektionsgeschwindigkeit zu der Referenzgeschwindigkeit der Schwerewellen gegeben. Für die oben genannten Anwendungen ist sie typischerweise sehr klein, z.B. 0.01. Zeit-explizite Finite-Volume-Verfahren werden am öftersten zur numerischen Berechnung hyperbolischer Bilanzgleichungen benutzt. Daher muss die CFL-Stabilitätsbedingung eingehalten werden und das Zeitinkrement ist ungefähr proportional zu der Froude-Zahl. Deswegen entsteht bei kleinen Froude-Zahlen, etwa kleiner als 0.2, ein hoher Rechenaufwand. Ferner sind die numerischen Lösungen dissipativ. Es ist allgemein bekannt, dass die Lösungen der SWE gegen die Lösungen der Seegleichungen/ Froude-Zahl Null SWE für Froude-Zahl gegen Null konvergieren, falls adäquate Bedingungen erfüllt sind. In diesem Grenzwertprozess ändern die Gleichungen ihren Typ von hyperbolisch zu hyperbolisch.-elliptisch. Ferner kann bei kleinen Froude-Zahlen die Konvergenzordnung sinken oder das numerische Verfahren zusammenbrechen. Insbesondere wurde bei zeit-expliziten Verfahren falsches asymptotisches Verhalten (bzgl. der Froude-Zahl) beobachtet, das diese Effekte verursachen könnte.Ozeanographische und atmosphärische Strömungen sind typischerweise kleine Störungen eines unterliegenden Equilibriumzustandes. Wir möchten, dass numerische Verfahren für Bilanzgleichungen gewisse Equilibriumzustände exakt erhalten, sonst können künstliche Strömungen vom Verfahren erzeugt werden. Daher ist die Quelltermapproximation essentiell. Numerische Verfahren die Equilibriumzustände erhalten heißen ausbalanciert.rnrnIn der vorliegenden Arbeit spalten wir die SWE in einen steifen, linearen und einen nicht-steifen Teil, um die starke Einschränkung der Zeitschritte durch die CFL-Bedingung zu umgehen. Der steife Teil wird implizit und der nicht-steife explizit approximiert. Dazu verwenden wir IMEX (implicit-explicit) Runge-Kutta und IMEX Mehrschritt-Zeitdiskretisierungen. Die Raumdiskretisierung erfolgt mittels der Finite-Volumen-Methode. Der steife Teil wird mit Hilfe von finiter Differenzen oder au eine acht mehrdimensional Art und Weise approximniert. Zur mehrdimensionalen Approximation verwenden wir approximative Evolutionsoperatoren, die alle unendlich viele Informationsausbreitungsrichtungen berücksichtigen. Die expliziten Terme werden mit gewöhnlichen numerischen Flüssen approximiert. Daher erhalten wir eine Stabilitätsbedingung analog zu einer rein advektiven Strömung, d.h. das Zeitinkrement vergrößert um den Faktor Kehrwert der Froude-Zahl. Die in dieser Arbeit hergeleiteten Verfahren sind asymptotisch erhaltend und ausbalanciert. Die asymptotischer Erhaltung stellt sicher, dass numerische Lösung das "korrekte" asymptotische Verhalten bezüglich kleiner Froude-Zahlen besitzt. Wir präsentieren Verfahren erster und zweiter Ordnung. Numerische Resultate bestätigen die Konvergenzordnung, so wie Stabilität, Ausbalanciertheit und die asymptotische Erhaltung. Insbesondere beobachten wir bei machen Verfahren, dass die Konvergenzordnung fast unabhängig von der Froude-Zahl ist.

Wavelets And Adaptive Grids For The Discontinuous Galerkin Method

In this paper, space adaptivity is introduced to control the error in the numerical solution of hyperbolic systems of conservation laws. The reference numerical scheme is a new version of the discontinuous Galerkin method, which uses an implicit diffusive term in the direction of the streamlines, for stability purposes. The decision whether to refine or to unrefine the grid in a certain location is taken according to the magnitude of wavelet coefficients, which are indicators of local smoothness of the numerical solution. Numerical solutions of the nonlinear Euler equations illustrate the efficiency of the method. © Springer 2005.

Stabilized Petrov-Galerkin methods for the convection-diffusion-reaction and the Helmholtz equations

We present two new stabilized high-resolution numerical methods for the convection–diffusion–reaction (CDR) and the Helmholtz equations respectively. The work embarks upon a priori analysis of some consistency recovery procedures for some stabilization methods belonging to the Petrov–Galerkin framework. It was found that the use of some standard practices (e.g. M-Matrices theory) for the design of essentially non-oscillatory numerical methods is not feasible when consistency recovery methods are employed. Hence, with respect to convective stabilization, such recovery methods are not preferred. Next, we present the design of a high-resolution Petrov–Galerkin (HRPG) method for the 1D CDR problem. The problem is studied from a fresh point of view, including practical implications on the formulation of the maximum principle, M-Matrices theory, monotonicity and total variation diminishing (TVD) finite volume schemes. The current method is next in line to earlier methods that may be viewed as an upwinding plus a discontinuity-capturing operator. Finally, some remarks are made on the extension of the HRPG method to multidimensions. Next, we present a new numerical scheme for the Helmholtz equation resulting in quasi-exact solutions. The focus is on the approximation of the solution to the Helmholtz equation in the interior of the domain using compact stencils. Piecewise linear/bilinear polynomial interpolation are considered on a structured mesh/grid. The only a priori requirement is to provide a mesh/grid resolution of at least eight elements per wavelength. No stabilization parameters are involved in the definition of the scheme. The scheme consists of taking the average of the equation stencils obtained by the standard Galerkin finite element method and the classical finite difference method. Dispersion analysis in 1D and 2D illustrate the quasi-exact properties of this scheme. Finally, some remarks are made on the extension of the scheme to unstructured meshes by designing a method within the Petrov–Galerkin framework.

Numerical simulation of drop impact and jet buckling problems using the eXtended Pom-Pom model

This work presents numerical simulations of two fluid flow problems involving moving free surfaces: the impacting drop and fluid jet buckling. The viscoelastic model used in these simulations is the eXtended Pom-Pom (XPP) model. To validate the code, numerical predictions of the drop impact problem for Newtonian and Oldroyd-B fluids are presented and compared with other methods. In particular, a benchmark on numerical simulations for a XPP drop impacting on a rigid plate is performed for a wide range of the relevant parameters. Finally, to provide an additional application of free surface flows of XPP fluids, the viscous jet buckling problem is simulated and discussed. (C) 2011 Elsevier B.V. All rights reserved.

HORMONAL RESPONSES TO DIFFERENT RESISTANCE EXERCISE SCHEMES OF SIMILAR TOTAL VOLUME

Uchida, MC, Crewther, BT, Ugrinowitsch, C, Bacurau, RFP, Moriscot, AS, and Aoki, MS. J Strength Cond Res 23(7): 2003-2008, 2009-This study assessed the effect of different resistance exercise scheme (RES) designs of similar total of load lifted on the responses of testosterone, cortisol, and creatine kinase (CK). Twenty-seven healthy males performed 1 of 4 bench press workouts described by the 1 repetition maximum (1RM) load: 4 sets of maximum repetitions at 50%-1RM (50%-1RM RES), 5 sets of maximum repetitions at 75%-1RM (75%-1RM RES), 10 sets of maximum repetitions at 90%-1RM (90%1RM RES), or 8 sets of maximum repetitions at 110%-1RM (110%-1RM RES). Each RES was equated by the total volume of load lifted (repetitions x sets x load). Blood samples, collected pre-exercise (Pre) and post-exercise (Post) at 1 and 24 hours (24 h), were analyzed for total and free testosterone, total cortisol, and CK. In general, testosterone and cortisol showed little change within or between the different RES (p > 0.05), possibly because of the relatively low volume lifted and/ or the small muscle mass activated by the bench press exercise. Cortisol was elevated after the 75%-1RM RES at the Post sample, with this response also exceeding the other RES (p < 0.05). The 24 h CK response was also elevated after the 75%-1RM RES (p < 0.05), thereby suggesting greater training strain for the same volume of load. These results confirm previous recommendations regarding the prescription of resistance exercise and the importance of total volume as a stimulus for activating the endocrine system and achieving long-term adaptation.

Airframe sound simulation based on staggered-grid higher order schemes and finite volume CFD methods

Abstract not available

SALIVARY HORMONE AND IMMUNE RESPONSES TO THREE RESISTANCE EXERCISE SCHEMES IN ELITE FEMALE ATHLETES

Nunes, JA, Crewther, BT, Ugrinowitsch, C, Tricoli, V, Viveiros, L, de Rose Jr, D, and Aoki, MS. Salivary hormone and immune responses to three resistance exercise schemes in elite female athletes J Strength Cond Res 25(8): 2322-2327, 2011-This study examined the salivary hormone and immune responses of elite female athletes to 3 different resistance exercise schemes. Fourteen female basketball players each performed an endurance scheme (ES-4 sets of 12 reps, 60% of 1 repetition maximum (1RM) load, 1-minute rest periods), a strength-hypertrophy scheme (SHS-1 set of 5RM, 1 set of 4RM, 1 set of 3RM, 1 set of 2RM, and 1set of 1RM with 3-minute rest periods, followed by 3 sets of 10RM with 2-minute rest periods) and a power scheme (PS-3 sets of 10 reps, 50% 1RM load, 3-minute rest periods) using the same exercises (bench press, squat, and biceps curl). Saliva samples were collected at 07:30 hours, pre-exercise (Pre) at 09:30 hours, postexercise (Post), and at 17:30 hours. Matching samples were also taken on a nonexercising control day. The samples were analyzed for testosterone, cortisol (C), and immunoglobulin A concentrations. The total volume of load lifted differed among the 3 schemes (SHS > ES > PS, p < 0.05). Postexercise C concentrations increased after all schemes, compared to control values (p < 0.05). In the SHS, the postexercise C response was also greater than pre-exercise data (p < 0.05). The current findings confirm that high-volume resistance exercise schemes can stimulate greater C secretion because of higher metabolic demand. In terms of practical applications, acute changes in C may be used to evaluate the metabolic demands of different resistance exercise schemes, or as a tool for monitoring training strain.

The effect of the number of biopsy cores on the concordance between prostate biopsy and prostatectomy Gleason score - A prostate volume-controlled study

Context.-Studies analyzing the concordance of biopsy and radical prostatectomy (RP) Gleason scores have limitations. Some included 2 or more centers, used historical controls from the early prostate specific antigen era or lacked a clear definition of the biopsy schemes. Furthermore, most did not control the results for prostate volume. Objective.-To confirm whether prediction of RP Gleason score can be optimized by taking more biopsy cores in a contemporary series of patients, with pathologic samples analyzed by the same pathologist, and controlling these results for prostate volume. Design.-The study comprised a retrospective case-control analysis of 393 patients with prostate cancer treated with RP. Patients were divided into 3 groups: those in group 1 underwent a 6-core biopsy; group 2, an 8-core biopsy; and group 3, a 10 or more-core biopsy. Concordance rates between biopsy and RP Gleason scores, as well as the rates of undergrading and overgrading, were determined for each biopsy scheme. Results.-Concordance rates were 60.9%, 58.3%, and 64.6% for patients from groups 1, 2, and 3, respectively (P = .18). When we analyzed patients with prostate volumes of less than 50 cm(3), concordance rates were 58.3%, 58.3%, and 65.1% for each group, respectively (P = .03). Among patients with prostate volumes of 50 cm3 or more, concordance rates were 70%, 58.1%, and 63.6%, respectively (P = .66). Conclusions.-Taking 10 or more cores can improve the prediction of RP Gleason score in patients with prostate volumes of less than 50 cm3. For patients with prostate volumes of 50 cm3 or more, increasing the biopsy cores to 10 or more did not improve prediction of RP Gleason score.

An iterative Multiscale Finite-Volume algorithm converging to the exact solution

The multiscale finite volume (MsFV) method has been developed to efficiently solve large heterogeneous problems (elliptic or parabolic); it is usually employed for pressure equations and delivers conservative flux fields to be used in transport problems. The method essentially relies on the hypothesis that the (fine-scale) problem can be reasonably described by a set of local solutions coupled by a conservative global (coarse-scale) problem. In most cases, the boundary conditions assigned for the local problems are satisfactory and the approximate conservative fluxes provided by the method are accurate. In numerically challenging cases, however, a more accurate localization is required to obtain a good approximation of the fine-scale solution. In this paper we develop a procedure to iteratively improve the boundary conditions of the local problems. The algorithm relies on the data structure of the MsFV method and employs a Krylov-subspace projection method to obtain an unconditionally stable scheme and accelerate convergence. Two variants are considered: in the first, only the MsFV operator is used; in the second, the MsFV operator is combined in a two-step method with an operator derived from the problem solved to construct the conservative flux field. The resulting iterative MsFV algorithms allow arbitrary reduction of the solution error without compromising the construction of a conservative flux field, which is guaranteed at any iteration. Since it converges to the exact solution, the method can be regarded as a linear solver. In this context, the schemes proposed here can be viewed as preconditioned versions of the Generalized Minimal Residual method (GMRES), with a very peculiar characteristic that the residual on the coarse grid is zero at any iteration (thus conservative fluxes can be obtained).

A Comparative Study of Convective Parameterization Schemes in WRF-NMM Model

Severe local storms, including tornadoes, damaging hail and wind gusts, frequently occur over the eastern and northeastern states of India during the pre-monsoon season (March-May). Forecasting thunderstorms is one of the most difficult tasks in weather prediction, due to their rather small spatial and temporal extension and the inherent non-linearity of their dynamics and physics. In this paper, sensitivity experiments are conducted with the WRF-NMM model to test the impact of convective parameterization schemes on simulating severe thunderstorms that occurred over Kolkata on 20 May 2006 and 21 May 2007 and validated the model results with observation. In addition, a simulation without convective parameterization scheme was performed for each case to determine if the model could simulate the convection explicitly. A statistical analysis based on mean absolute error, root mean square error and correlation coefficient is performed for comparisons between the simulated and observed data with different convective schemes. This study shows that the prediction of thunderstorm affected parameters is sensitive to convective schemes. The Grell-Devenyi cloud ensemble convective scheme is well simulated the thunderstorm activities in terms of time, intensity and the region of occurrence of the events as compared to other convective schemes and also explicit scheme

A high-order arbitrarily unstructured finite-volume model of the global atmosphere: tests solving the shallow-water equations

Simulations of the global atmosphere for weather and climate forecasting require fast and accurate solutions and so operational models use high-order finite differences on regular structured grids. This precludes the use of local refinement; techniques allowing local refinement are either expensive (eg. high-order finite element techniques) or have reduced accuracy at changes in resolution (eg. unstructured finite-volume with linear differencing). We present solutions of the shallow-water equations for westerly flow over a mid-latitude mountain from a finite-volume model written using OpenFOAM. A second/third-order accurate differencing scheme is applied on arbitrarily unstructured meshes made up of various shapes and refinement patterns. The results are as accurate as equivalent resolution spectral methods. Using lower order differencing reduces accuracy at a refinement pattern which allows errors from refinement of the mountain to accumulate and reduces the global accuracy over a 15 day simulation. We have therefore introduced a scheme which fits a 2D cubic polynomial approximately on a stencil around each cell. Using this scheme means that refinement of the mountain improves the accuracy after a 15 day simulation. This is a more severe test of local mesh refinement for global simulations than has been presented but a realistic test if these techniques are to be used operationally. These efficient, high-order schemes may make it possible for local mesh refinement to be used by weather and climate forecast models.

Is there a relationship between the total volume of load lifted in bench press exercise and the rating of perceived exertion?

Aim. The purpose of the study was to investigate the relationship between the total volume of load lifted (TVLL) and the rating of perceived exertion (RPE) measures during different resistance training (RT) schemes using the bench press exercise. Methods. The present study was divided into two experiments. In the first experiment, 18 healthy men performed three different RT schemes: a strength oriented scheme (SS), a muscular endurance oriented scheme (ES) and a hypertrophy oriented scheme (HS). TVLL was calculated for each scheme. Mean-RPE and session-RPE were assessed. In the second experiment, 23 men performed two resistance exercise bouts at different intensities (50 %-1RM and 75%-1RM) with matched TVLL. Mean-RPE and session-RPE were also assessed. Results. SS and HS showed higher TVLL and greater RPE scores as compared to ES (P<0.05). No significant difference was observed between SS and HS. It was verified significant positive correlations between TVLL and session-RPE (SS r=0.63, HS r=0.64, ES r=0.56; P<0.05), and between mean-RPE and TVLL (SS r=0.55, HS r=0.52, ES r=0.47; P<0.05) for all schemes. No differences were observed for mean-RPE, session-RPE and TVLL between the 50%-1RM and 75%1RM. Significant positive relationships between TVLL and session-RPE (50 %-1RM r=0.61, 75 %-1RM r=0.66; p<0.05) and between TVLL and mean-RPE (50 %-1RM r=0.51, 75%1RM r=0.49; P<0.05) were observed. Conclusion. The results of this study have shown that the TVLL in RT influences RPE measures. These findings corroborates the existence of a relationship between total work performed (external training load) and perception of effort (internal training load).