Numerical Error Prediction and its applications in CFD using tau-estimation

Nowadays, Computational Fluid Dynamics (CFD) solvers are widely used within the industry to model fluid flow phenomenons. Several fluid flow model equations have been employed in the last decades to simulate and predict forces acting, for example, on different aircraft configurations. Computational time and accuracy are strongly dependent on the fluid flow model equation and the spatial dimension of the problem considered. While simple models based on perfect flows, like panel methods or potential flow models can be very fast to solve, they usually suffer from a poor accuracy in order to simulate real flows (transonic, viscous). On the other hand, more complex models such as the full Navier- Stokes equations provide high fidelity predictions but at a much higher computational cost. Thus, a good compromise between accuracy and computational time has to be fixed for engineering applications. A discretisation technique widely used within the industry is the so-called Finite Volume approach on unstructured meshes. This technique spatially discretises the flow motion equations onto a set of elements which form a mesh, a discrete representation of the continuous domain. Using this approach, for a given flow model equation, the accuracy and computational time mainly depend on the distribution of nodes forming the mesh. Therefore, a good compromise between accuracy and computational time might be obtained by carefully defining the mesh. However, defining an optimal mesh for complex flows and geometries requires a very high level expertize in fluid mechanics and numerical analysis, and in most cases a simple guess of regions of the computational domain which might affect the most the accuracy is impossible. Thus, it is desirable to have an automatized remeshing tool, which is more flexible with unstructured meshes than its structured counterpart. However, adaptive methods currently in use still have an opened question: how to efficiently drive the adaptation ? Pioneering sensors based on flow features generally suffer from a lack of reliability, so in the last decade more effort has been made in developing numerical error-based sensors, like for instance the adjoint-based adaptation sensors. While very efficient at adapting meshes for a given functional output, the latter method is very expensive as it requires to solve a dual set of equations and computes the sensor on an embedded mesh. Therefore, it would be desirable to develop a more affordable numerical error estimation method. The current work aims at estimating the truncation error, which arises when discretising a partial differential equation. These are the higher order terms neglected in the construction of the numerical scheme. The truncation error provides very useful information as it is strongly related to the flow model equation and its discretisation. On one hand, it is a very reliable measure of the quality of the mesh, therefore very useful in order to drive a mesh adaptation procedure. On the other hand, it is strongly linked to the flow model equation, so that a careful estimation actually gives information on how well a given equation is solved, which may be useful in the context of _ -extrapolation or zonal modelling. The following work is organized as follows: Chap. 1 contains a short review of mesh adaptation techniques as well as numerical error prediction. In the first section, Sec. 1.1, the basic refinement strategies are reviewed and the main contribution to structured and unstructured mesh adaptation are presented. Sec. 1.2 introduces the definitions of errors encountered when solving Computational Fluid Dynamics problems and reviews the most common approaches to predict them. Chap. 2 is devoted to the mathematical formulation of truncation error estimation in the context of finite volume methodology, as well as a complete verification procedure. Several features are studied, such as the influence of grid non-uniformities, non-linearity, boundary conditions and non-converged numerical solutions. This verification part has been submitted and accepted for publication in the Journal of Computational Physics. Chap. 3 presents a mesh adaptation algorithm based on truncation error estimates and compares the results to a feature-based and an adjoint-based sensor (in collaboration with Jorge Ponsín, INTA). Two- and three-dimensional cases relevant for validation in the aeronautical industry are considered. This part has been submitted and accepted in the AIAA Journal. An extension to Reynolds Averaged Navier- Stokes equations is also included, where _ -estimation-based mesh adaptation and _ -extrapolation are applied to viscous wing profiles. The latter has been submitted in the Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering. Keywords: mesh adaptation, numerical error prediction, finite volume Hoy en día, la Dinámica de Fluidos Computacional (CFD) es ampliamente utilizada dentro de la industria para obtener información sobre fenómenos fluidos. La Dinámica de Fluidos Computacional considera distintas modelizaciones de las ecuaciones fluidas (Potencial, Euler, Navier-Stokes, etc) para simular y predecir las fuerzas que actúan, por ejemplo, sobre una configuración de aeronave. El tiempo de cálculo y la precisión en la solución depende en gran medida de los modelos utilizados, así como de la dimensión espacial del problema considerado. Mientras que modelos simples basados en flujos perfectos, como modelos de flujos potenciales, se pueden resolver rápidamente, por lo general aducen de una baja precisión a la hora de simular flujos reales (viscosos, transónicos, etc). Por otro lado, modelos más complejos tales como el conjunto de ecuaciones de Navier-Stokes proporcionan predicciones de alta fidelidad, a expensas de un coste computacional mucho más elevado. Por lo tanto, en términos de aplicaciones de ingeniería se debe fijar un buen compromiso entre precisión y tiempo de cálculo. Una técnica de discretización ampliamente utilizada en la industria es el método de los Volúmenes Finitos en mallas no estructuradas. Esta técnica discretiza espacialmente las ecuaciones del movimiento del flujo sobre un conjunto de elementos que forman una malla, una representación discreta del dominio continuo. Utilizando este enfoque, para una ecuación de flujo dado, la precisión y el tiempo computacional dependen principalmente de la distribución de los nodos que forman la malla. Por consiguiente, un buen compromiso entre precisión y tiempo de cálculo se podría obtener definiendo cuidadosamente la malla, concentrando sus elementos en aquellas zonas donde sea estrictamente necesario. Sin embargo, la definición de una malla óptima para corrientes y geometrías complejas requiere un nivel muy alto de experiencia en la mecánica de fluidos y el análisis numérico, así como un conocimiento previo de la solución. Aspecto que en la mayoría de los casos no está disponible. Por tanto, es deseable tener una herramienta que permita adaptar los elementos de malla de forma automática, acorde a la solución fluida (remallado). Esta herramienta es generalmente más flexible en mallas no estructuradas que con su homóloga estructurada. No obstante, los métodos de adaptación actualmente en uso todavía dejan una pregunta abierta: cómo conducir de manera eficiente la adaptación. Sensores pioneros basados en las características del flujo en general, adolecen de una falta de fiabilidad, por lo que en la última década se han realizado grandes esfuerzos en el desarrollo numérico de sensores basados en el error, como por ejemplo los sensores basados en el adjunto. A pesar de ser muy eficientes en la adaptación de mallas para un determinado funcional, este último método resulta muy costoso, pues requiere resolver un doble conjunto de ecuaciones: la solución y su adjunta. Por tanto, es deseable desarrollar un método numérico de estimación de error más asequible. El presente trabajo tiene como objetivo estimar el error local de truncación, que aparece cuando se discretiza una ecuación en derivadas parciales. Estos son los términos de orden superior olvidados en la construcción del esquema numérico. El error de truncación proporciona una información muy útil sobre la solución: es una medida muy fiable de la calidad de la malla, obteniendo información que permite llevar a cabo un procedimiento de adaptación de malla. Está fuertemente relacionado al modelo matemático fluido, de modo que una estimación precisa garantiza la idoneidad de dicho modelo en un campo fluido, lo que puede ser útil en el contexto de modelado zonal. Por último, permite mejorar la precisión de la solución resolviendo un nuevo sistema donde el error local actúa como término fuente (_ -extrapolación). El presenta trabajo se organiza de la siguiente manera: Cap. 1 contiene una breve reseña de las técnicas de adaptación de malla, así como de los métodos de predicción de los errores numéricos. En la primera sección, Sec. 1.1, se examinan las estrategias básicas de refinamiento y se presenta la principal contribución a la adaptación de malla estructurada y no estructurada. Sec 1.2 introduce las definiciones de los errores encontrados en la resolución de problemas de Dinámica Computacional de Fluidos y se examinan los enfoques más comunes para predecirlos. Cap. 2 está dedicado a la formulación matemática de la estimación del error de truncación en el contexto de la metodología de Volúmenes Finitos, así como a un procedimiento de verificación completo. Se estudian varias características que influyen en su estimación: la influencia de la falta de uniformidad de la malla, el efecto de las no linealidades del modelo matemático, diferentes condiciones de contorno y soluciones numéricas no convergidas. Esta parte de verificación ha sido presentada y aceptada para su publicación en el Journal of Computational Physics. Cap. 3 presenta un algoritmo de adaptación de malla basado en la estimación del error de truncación y compara los resultados con sensores de featured-based y adjointbased (en colaboración con Jorge Ponsín del INTA). Se consideran casos en dos y tres dimensiones, relevantes para la validación en la industria aeronáutica. Este trabajo ha sido presentado y aceptado en el AIAA Journal. También se incluye una extensión de estos métodos a las ecuaciones RANS (Reynolds Average Navier- Stokes), en donde adaptación de malla basada en _ y _ -extrapolación son aplicados a perfiles con viscosidad de alas. Este último trabajo se ha presentado en los Actas de la Institución de Ingenieros Mecánicos, Parte G: Journal of Aerospace Engineering. Palabras clave: adaptación de malla, predicción del error numérico, volúmenes finitos

Allocation of 14C assimilated in late spring to tissue and biochemical stem components of cork oak (Quercus suber L.) over the seasons

Carbon distribution in the stem of 2-year-old cork oak plants was studied by 14CO2 pulse labeling in late spring in order to trace the allocation of photoassimilates to tissue and biochemical stem components of cork oak. The fate of 14C photoassimilated carbon was followed during two periods: the first 72 h (short-term study) and the first 52 weeks (long-term study) after the 14CO2 photosynthetic assimilation. The results showed that 14C allocation to stem tissues was dependent on the time passed since photoassimilation and on the season of the year. In the first 3 h all 14C was found in the polar extractives. After 3 h, it started to be allocated to other stem fractions. In 1 day, 14C was allocated mostly to vascular cambium and, to a lesser extent, to primary phloem; no presence of 14C was recorded for the periderm. However, translocation of 14C to phellem was observed from 1 week after 14CO2 pulse labeling. The phellogen was not completely active in its entire circumference at labeling, unlike the vascular cambium; this was the tissue that accumulated most photoassimilated 14C at the earliest sampling. The fraction of leaf-assimilated 14C that was used by the stem peaked at 57% 1 week after 14CO2 plant exposure. The time lag between C photoassimilation and suberin accumulation was ∼8 h, but the most active period for suberin accumulation was between 3 and 7 days. Suberin, which represented only 1.77% of the stem weight, acted as a highly effective sink for the carbon photoassimilated in late spring since suberin specific radioactivity was much higher than for any other stem component as early as only 1 week after 14C plant labeling. This trend was maintained throughout the whole experiment. The examination of microautoradiographs taken over 1 year provided a new method for quantifying xylem growth. Using this approach it was found that there was more secondary xylem growth in late spring than in other times of the year, because the calculated average cell division time was much shorter.

The C-terminal SET domains of ALL-1 and TRITHORAX interact with the INI1 and SNR1 proteins, components of the SWI/SNF complex

The ALL-1 gene was discovered by virtue of its involvement in human acute leukemia. Its Drosophila homolog trithorax (trx) is a member of the trx-Polycomb gene family, which maintains correct spatial expression of the Antennapedia and bithorax complexes during embryogenesis. The C-terminal SET domain of ALL-1 and TRITHORAX (TRX) is a 150-aa motif, highly conserved during evolution. We performed yeast two hybrid screening of Drosophila cDNA library and detected interaction between a TRX polypeptide spanning SET and the SNR1 protein. SNR1 is a product of snr1, which is classified as a trx group gene. We found parallel interaction in yeast between the SET domain of ALL-1 and the human homolog of SNR1, INI1 (hSNF5). These results were confirmed by in vitro binding studies and by demonstrating coimmunoprecipitation of the proteins from cultured cells and/or transgenic flies. Epitope-tagged SNR1 was detected at discrete sites on larval salivary gland polytene chromosomes, and these sites colocalized with around one-half of TRX binding sites. Because SNR1 and INI1 are constituents of the SWI/SNF complex, which acts to remodel chromatin and consequently to activate transcription, the interactions we observed suggest a mechanism by which the SWI/SNF complex is recruited to ALL-1/trx targets through physical interactions between the C-terminal domains of ALL-1 and TRX and INI1/SNR1.

Eps15 Is Recruited to the Plasma Membrane upon Epidermal Growth Factor Receptor Activation and Localizes to Components of the Endocytic Pathway during Receptor Internalization

Eps15 is a substrate for the tyrosine kinase of the epidermal growth factor receptor (EGFR) and is characterized by the presence of a novel protein:protein interaction domain, the EH domain. Eps15 also stably binds the clathrin adaptor protein complex AP-2. Previous work demonstrated an essential role for eps15 in receptor-mediated endocytosis. In this study we show that, upon activation of the EGFR kinase, eps15 undergoes dramatic relocalization consisting of 1) initial relocalization to the plasma membrane and 2) subsequent colocalization with the EGFR in various intracellular compartments of the endocytic pathway, with the notable exclusion of coated vesicles. Relocalization of eps15 is independent of its binding to the EGFR or of binding of the receptor to AP-2. Furthermore, eps15 appears to undergo tyrosine phosphorylation both at the plasma membrane and in a nocodazole-sensitive compartment, suggesting sustained phosphorylation in endocytic compartments. Our results are consistent with a model in which eps15 undergoes cycles of association:dissociation with membranes and suggest multiple roles for this protein in the endocytic pathway.

Histones H3 and H4 are components of upstream activation factor required for the high-level transcription of yeast rDNA by RNA polymerase I

RNA polymerase I (Pol I) transcription in the yeast Saccharomyces cerevisiae is greatly stimulated in vivo and in vitro by the multiprotein complex, upstream activation factor (UAF). UAF binds tightly to the upstream element of the rDNA promoter, such that once bound (in vitro), UAF does not readily exchange onto a competing template. Of the polypeptides previously identified in purified UAF, three are encoded by genes required for Pol I transcription in vivo: RRN5, RRN9, and RRN10. Two others, p30 and p18, have remained uncharacterized. We report here that the N-terminal amino acid sequence, its mobility in gel electrophoresis, and the immunoreactivity of p18 shows that it is histone H3. In addition, histone H4 was found in UAF, and myc-tagged histone H4 could be used to affinity-purify UAF. Histones H2A and H2B were not detectable in UAF. These results suggest that histones H3 and H4 probably account for the strong binding of UAF to DNA and may offer a means by which general nuclear regulatory signals could be transmitted to Pol I.

Yeast global transcriptional regulators Sin4 and Rgr1 are components of mediator complex/RNA polymerase II holoenzyme.

Sin4 and Rgr1 proteins, previously shown by genetic studies to play both positive and negative roles in the transcriptional regulation of many genes, are identified here as components of mediator and RNA polymerase II holoenzyme complexes. Results with Sin4 deletion and Rgr1 truncation strains indicate the association of these proteins in a subcomplex comprising Sin4, Rgr1, Gal11, and a 50-kDa polypeptide. Taken together with the previous genetic evidence, our findings point to a role of the mediator in repression as well as in transcriptional activation.

Combinatorial association and abundance of components of interferon-stimulated gene factor 3 dictate the selectivity of interferon responses.

Genes containing the interferon-stimulated response element (ISRE) enhancer have been characterized as transcriptionally responsive primarily to type I interferons (IFN alpha/beta). Induction is due to activation of a multimeric transcription factor, interferon-stimulated gene factor 3 (ISGF3), which is activated by IFN alpha/beta but not by IFN gamma. We found that ISRE-containing genes were induced by IFN gamma as well as by IFN alpha in Vero cells. The IFN gamma response was dependent on the ISRE and was accentuated by preexposure of cells to IFN alpha, a treatment that increases the abundance of ISGF3 components. Overexpression of ISGF3 polypeptides showed that the IFN gamma response depended on the DNA-binding protein ISGF3 gamma (p48) as well as on the 91-kDa protein STAT91 (Stat1 alpha). The transcriptional response to IFN alpha required the 113-kDa protein STAT113 (Stat2) in addition to STAT91 and p48. Mutant fibrosarcoma cells deficient in each component of ISGF3 were used to confirm that IFN gamma induction of an ISRE reporter required p48 and STAT91, but not STAT113. A complex containing p48 and phosphorylated STAT91 but lacking STAT113 bound the ISRE in vitro. IFN gamma-induced activation of this complex, preferentially formed at high concentrations of p48 and STAT91, may explain some of the overlapping responses to IFN alpha and IFN gamma.

Concentrations of interstitial water, methane, and other sediment components related to microorganisms at DSDP Holes 96-618 and 96-619

Radiolabeled products were formed from labeled substrates during anaerobic incubation of sediments from Sites 618, 619, and 622. One set of experiments formed 14CO2, 14CH4, and 35SH2 from 2-14C-acetate and 35S-sulfate; a second set formed 14CH4 from 14C-methylamine or 14C-trimethylamine. Levels of 14CO2 and 35S2 formed were two to three orders of magnitude greater than 14CH4. Production of 14CH4 by Deep Sea Drilling Project (DSDP) sediments was four to five orders of magnitude less than that formed by anoxic San Francisco Bay sediment. However, incubation of Site 622 sediment slurries under H2 demonstrated production of small quantities of CH4. These results indicate that DSDP sediments recovered from 4 to 167 m sub-bottom (age 85,000-110,000 yr.) harbor potential microbial activity which includes sulfate reducers and methanogens. Analysis of pore waters from these DSDP sites indicates that bacterial substrates (acetate, methylated amines) were present.

Analysis of Melanoma Onset: Assessing Familial Aggregation by Using Estimating Equations and Fitting Variance Components via Bayesian Random Effects Models

We investigate whether relative contributions of genetic and shared environmental factors are associated with an increased risk in melanoma. Data from the Queensland Familial Melanoma Project comprising 15,907 subjects arising from 1912 families were analyzed to estimate the additive genetic, common and unique environmental contributions to variation in the age at onset of melanoma. Two complementary approaches for analyzing correlated time-to-onset family data were considered: the generalized estimating equations (GEE) method in which one can estimate relationship-specific dependence simultaneously with regression coefficients that describe the average population response to changing covariates; and a subject-specific Bayesian mixed model in which heterogeneity in regression parameters is explicitly modeled and the different components of variation may be estimated directly. The proportional hazards and Weibull models were utilized, as both produce natural frameworks for estimating relative risks while adjusting for simultaneous effects of other covariates. A simple Markov Chain Monte Carlo method for covariate imputation of missing data was used and the actual implementation of the Bayesian model was based on Gibbs sampling using the free ware package BUGS. In addition, we also used a Bayesian model to investigate the relative contribution of genetic and environmental effects on the expression of naevi and freckles, which are known risk factors for melanoma.

Using urea dilution to standardise cellular and non-cellular components of pleural and bronchoalveolar lavage (BAL) fluids in the cat

A technique to standardise the analysis of cellular and non-cellular components in epithelial lining fluid (ELF) collected during saline lavage of pulmonary and pleural cavities was developed using the urea dilution method. Bronchoalveolar lavage (BAL) and pleural lavage (PL) fluids were collected from 12 clinically healthy cats. Total and differential cell counts in BAL fluid were within normal ranges for the cat, while cell Counts in PL fluid were assumed to be normal based on clinical health during examination, auscultation and lactate dehydrogenase (LDH) activities being comparable with other species. The major clinical implication of this study was that nucleated cell counts within feline ELF could not be predicted from analysis of lavage fluid which suggests that calculation of the proportion of ELF in lavage fluid by the urea dilution method may be necessary to avoid misdiagnosis of health or disease in pulmonary or pleural cavities. (C) 2005 ESFM and AAFP. Published by Elsevier Ltd. All rights reserved.

Using distributions as statistical evidence in well-structured and ill-structured problems

Research has suggested that understanding in well-structured settings often does not transfer to the everyday, less-structured problems encountered outside of school. Little is known, beyond anecdotal evidence, about how teachers' consideration of distributions as evidence in well-structured settings compares with their use in ill-structured problem contexts. A qualitative study of preservice secondary teachers examined their use of distributions as evidence in four tasks of varying complexity and ill-structuredness. Results suggest that teachers' incorporation of distributions in well-structured settings does not imply that they will be incorporated in less structured problems (and vice-versa). Implications for research and teaching are discussed.