A moving-mesh finite element method and its application to the numerical solution of phase-change problems

A distributed Lagrangian moving-mesh finite element method is applied to problems involving changes of phase. The algorithm uses a distributed conservation principle to determine nodal mesh velocities, which are then used to move the nodes. The nodal values are obtained from an ALE (Arbitrary Lagrangian-Eulerian) equation, which represents a generalization of the original algorithm presented in Applied Numerical Mathematics, 54:450--469 (2005). Having described the details of the generalized algorithm it is validated on two test cases from the original paper and is then applied to one-phase and, for the first time, two-phase Stefan problems in one and two space dimensions, paying particular attention to the implementation of the interface boundary conditions. Results are presented to demonstrate the accuracy and the effectiveness of the method, including comparisons against analytical solutions where available.

Velocity-based moving mesh methods for nonlinear partial differential equations

This article describes a number of velocity-based moving mesh numerical methods formultidimensional nonlinear time-dependent partial differential equations (PDEs). It consists of a short historical review followed by a detailed description of a recently developed multidimensional moving mesh finite element method based on conservation. Finite element algorithms are derived for both mass-conserving and non mass-conserving problems, and results shown for a number of multidimensional nonlinear test problems, including the second order porous medium equation and the fourth order thin film equation as well as a two-phase problem. Further applications and extensions are referenced.

Age-related changes in the functional neuroanatomy of overt speech production

Alterations of existing neural networks during healthy aging, resulting in behavioral deficits and changes in brain activity, have been described for cognitive, motor, and sensory functions. To investigate age-related changes in the neural circuitry underlying overt non-lexical speech production, functional MRI was performed in 14 healthy younger (21–32 years) and 14 healthy older individuals (62–84 years). The experimental task involved the acoustically cued overt production of the vowel /a/ and the polysyllabic utterance /pataka/. In younger and older individuals, overt speech production was associated with the activation of a widespread articulo-phonological network, including the primary motor cortex, the supplementary motor area, the cingulate motor areas, and the posterior superior temporal cortex, similar in the /a/ and /pataka/ condition. An analysis of variance with the factors age and condition revealed a significant main effect of age. Irrespective of the experimental condition, significantly greater activation was found in the bilateral posterior superior temporal cortex, the posterior temporal plane, and the transverse temporal gyri in younger compared to older individuals. Significantly greater activation was found in the bilateral middle temporal gyri, medial frontal gyri, middle frontal gyri, and inferior frontal gyri in older vs. younger individuals. The analysis of variance did not reveal a significant main effect of condition and no significant interaction of age and condition. These results suggest a complex reorganization of neural networks dedicated to the production of speech during healthy aging.

The functional neuroanatomy of the evolving parent-infant relationship

Infant survival and the development of secure and cooperative relationships are central to the future of the species. In humans, this relies heavily on the evolving early parent–infant social and affective relationship. While much is known about the behavioural and psychological components of this relationship, relatively little is known about the underlying functional neuroanatomy. Affective and social neuroscience has helped to describe the main adult brain networks involved, but has so far engaged very little with developmental findings. In this review, we seek to highlight future avenues for research by providing a coherent framework for describing the parent–infant relationship over the first 18 months. We provide an outline of the evolving nature of the relationship, starting with basic orienting and recognition processes, and culminating in the infant's attainment of higher socio-emotional and cognitive capacities. Key social and affective interactions, such as communication, cooperative play and the establishment of specific attachments propel the development of the parent–infant relationship. We summarise our current knowledge of the developing infant brain in terms of structure and function, and how these relate to the emergent abilities necessary for the formation of a secure and cooperative relationship with parents or other caregivers. Important roles have been found for brain regions including the orbitofrontal, cingulate, and insular cortices in parent–infant interactions, but it has become clear that much more information is needed about the developmental time course and connectivity of these regions.

Prediction of a typhoon using a fine-mesh NWP model

The ECMWF operational grid point model (with a resolution of 1.875° of latitude and longitude) and its limited area version (with a resolution of !0.47° of latitude and longitude) with boundary values from the global model have been used to study the simulation of the typhoon Tip. The fine-mesh model was capable of simulating the main structural features of the typhoon and predicting a fall in central pressure of 60 mb in 3 days. The structure of the forecast typhoon, with a warm core (maximum potential temperature anomaly 17 K). intense swirling wind (maximum 55 m s-1 at 850 mb) and spiralling precipitation patterns is characteristic of a tropical cyclone. Comparison with the lower resolution forecast shows that the horizontal resolution is a determining factor in predicting not only the structure and intensity but even the movement of these vortices. However, an accurate and refined initial analysis is considered to be a prerequisite for a correct forecast of this phenomenon.

Numerical conformal mapping and mesh generation for polygonal and multiply-connected regions

Details are given of a boundary-fitted mesh generation method for use in modelling free surface flow and water quality. A numerical method has been developed for generating conformal meshes for curvilinear polygonal and multiply-connected regions. The method is based on the Cauchy-Riemann conditions for the analytic function and is able to map a curvilinear polygonal region directly onto a regular polygonal region, with horizontal and vertical sides. A set of equations have been derived for determining the lengths of these sides and the least-squares method has been used in solving the equations. Several numerical examples are presented to illustrate the method.

A depth-integrated 2D coastal and estuarine model with conformal boundary-fitted mesh generation

Details are given of the development and application of a 2D depth-integrated, conformal boundary-fitted, curvilinear model for predicting the depth-mean velocity field and the spatial concentration distribution in estuarine and coastal waters. A numerical method for conformal mesh generation, based on a boundary integral equation formulation, has been developed. By this method a general polygonal region with curved edges can be mapped onto a regular polygonal region with the same number of horizontal and vertical straight edges and a multiply connected region can be mapped onto a regular region with the same connectivity. A stretching transformation on the conformally generated mesh has also been used to provide greater detail where it is needed close to the coast, with larger mesh sizes further offshore, thereby minimizing the computing effort whilst maximizing accuracy. The curvilinear hydrodynamic and solute model has been developed based on a robust rectilinear model. The hydrodynamic equations are approximated using the ADI finite difference scheme with a staggered grid and the solute transport equation is approximated using a modified QUICK scheme. Three numerical examples have been chosen to test the curvilinear model, with an emphasis placed on complex practical applications

A moving mesh approach for modelling avascular tumour growth

A key step in many numerical schemes for time-dependent partial differential equations with moving boundaries is to rescale the problem to a fixed numerical mesh. An alternative approach is to use a moving mesh that can be adapted to focus on specific features of the model. In this paper we present and discuss two different velocity-based moving mesh methods applied to a two-phase model of avascular tumour growth formulated by Breward et al. (2002) J. Math. Biol. 45(2), 125-152. Each method has one moving node which tracks the moving boundary. The first moving mesh method uses a mesh velocity proportional to the boundary velocity. The second moving mesh method uses local conservation of volume fraction of cells (masses). Our results demonstrate that these moving mesh methods produce accurate results, offering higher resolution where desired whilst preserving the balance of fluxes and sources in the governing equations.

Neuroanatomy of individual differences in language in adult males with autism

One potential source of heterogeneity within autism spectrum conditions (ASC) is language development and ability. In 80 high-functioning male adults with ASC, we tested if variations in developmental and current structural language are associated with current neuroanatomy. Groups with and without language delay differed behaviorally in early social reciprocity, current language, but not current autistic features. Language delay was associated with larger total gray matter (GM) volume, smaller relative volume at bilateral insula, ventral basal ganglia, and right superior, middle, and polar temporal structures, and larger relative volume at pons and medulla oblongata in adulthood. Despite this heterogeneity, those with and without language delay showed significant commonality in morphometric features when contrasted with matched neurotypical individuals (n = 57). In ASC, better current language was associated with increased GM volume in bilateral temporal pole, superior temporal regions, dorsolateral fronto-parietal and cerebellar structures, and increased white matter volume in distributed frontal and insular regions. Furthermore, current language–neuroanatomy correlation patterns were similar across subgroups with or without language delay. High-functioning adult males with ASC show neuroanatomical variations associated with both developmental and current language characteristics. This underscores the importance of including both developmental and current language as specifiers for ASC, to help clarify heterogeneity.

Fast three dimensional r-adaptive mesh redistribution

This paper describes a fast and reliable method for redistributing a computational mesh in three dimensions which can generate a complex three dimensional mesh without any problems due to mesh tangling. The method relies on a three dimensional implementation of the parabolic Monge–Ampère (PMA) technique, for finding an optimally transported mesh. The method for implementing PMA is described in detail and applied to both static and dynamic mesh redistribution problems, studying both the convergence and the computational cost of the algorithm. The algorithm is applied to a series of problems of increasing complexity. In particular very regular meshes are generated to resolve real meteorological features (derived from a weather forecasting model covering the UK area) in grids with over 2×107 degrees of freedom. The PMA method computes these grids in times commensurate with those required for operational weather forecasting.

Mapping Educational Specialist knowHow (MESH): the building of a knowledge management system for research informed practice

In this invited article the authors present an evaluative report on the development of the MESHGuides project (http://www.meshguides.org/). MESHGuides’ objective is to provide education with an international knowledge management system. MESHGuides were conceived as research summaries for supporting teachers’ in developing evidence-based practice. Their aim is to enhance teachers’ capacity to engage actively with research in their own classrooms. The original thinking for MESH arose from the work of UK-based academics Professor Marilyn Leask and Dr Sarah Younie in response to a desire, which has recently gathered momentum in the UK, for the development of a more research-informed teaching profession and for the establishment of an on-line platform to support evidence-based practice (DfE, 2015; Leask and Younie 2001; OECD 2009). The focus of this article is on how the MESHGuides project was conceived and structured, the technical systems supporting it and the practical reality for academics and teachers of composing and using MESHGuides. The project and the guides are in the early stages of development, and discussion indicates future possibilities for more global engagement with this knowledge management system.

Mesh adaptation on the sphere using optimal transport and the numerical solution of a Monge-Ampère type equation

An equation of Monge-Ampère type has, for the first time, been solved numerically on the surface of the sphere in order to generate optimally transported (OT) meshes, equidistributed with respect to a monitor function. Optimal transport generates meshes that keep the same connectivity as the original mesh, making them suitable for r-adaptive simulations, in which the equations of motion can be solved in a moving frame of reference in order to avoid mapping the solution between old and new meshes and to avoid load balancing problems on parallel computers. The semi-implicit solution of the Monge-Ampère type equation involves a new linearisation of the Hessian term, and exponential maps are used to map from old to new meshes on the sphere. The determinant of the Hessian is evaluated as the change in volume between old and new mesh cells, rather than using numerical approximations to the gradients. OT meshes are generated to compare with centroidal Voronoi tesselations on the sphere and are found to have advantages and disadvantages; OT equidistribution is more accurate, the number of iterations to convergence is independent of the mesh size, face skewness is reduced and the connectivity does not change. However anisotropy is higher and the OT meshes are non-orthogonal. It is shown that optimal transport on the sphere leads to meshes that do not tangle. However, tangling can be introduced by numerical errors in calculating the gradient of the mesh potential. Methods for alleviating this problem are explored. Finally, OT meshes are generated using observed precipitation as a monitor function, in order to demonstrate the potential power of the technique.