Adjoint goal-based error norms for adaptive mesh ocean modelling

Flow in the world's oceans occurs at a wide range of spatial scales, from a fraction of a metre up to many thousands of kilometers. In particular, regions of intense flow are often highly localised, for example, western boundary currents, equatorial jets, overflows and convective plumes. Conventional numerical ocean models generally use static meshes. The use of dynamically-adaptive meshes has many potential advantages but needs to be guided by an error measure reflecting the underlying physics. A method of defining an error measure to guide an adaptive meshing algorithm for unstructured tetrahedral finite elements, utilizing an adjoint or goal-based method, is described here. This method is based upon a functional, encompassing important features of the flow structure. The sensitivity of this functional, with respect to the solution variables, is used as the basis from which an error measure is derived. This error measure acts to predict those areas of the domain where resolution should be changed. A barotropic wind driven gyre problem is used to demonstrate the capabilities of the method. The overall objective of this work is to develop robust error measures for use in an oceanographic context which will ensure areas of fine mesh resolution are used only where and when they are required. (c) 2006 Elsevier Ltd. All rights reserved.

Why high seed densities within buried mesh bags may overestimate depletion rates of soil seed banks

1. Estimates of seed bank depletion rates are essential for modelling and management of plant populations. The seed bag burial method is often used to measure seed mortality in the soil. However, the density of seeds within seed bags is higher than densities in natural seed banks, which may elevate levels of pathogens and influence seed mortality. The aim of this study was to quantify the effects of fungi and seed density within buried mesh bags on the mortality of seeds. Striga hermonthica was chosen as the study species because it has been widely studied but different methods for measuring seed mortality in the soil have yielded contradictory estimates. 2. Seed bags were buried in soil and exhumed at regular time intervals to monitor mortality of the seeds in three field experiments during two rainy seasons. The effect of fungal activity on seed mortality was evaluated in a fungi exclusion experiment. Differences in seed-to-seed interaction were obtained by using two and four densities within the seed bags in consecutive years. Densities were created by mixing 1000 seeds with 0, 10, 100 or 1000 g of coarse sand. 3. The mortality rate was significantly lower when fungi were excluded, indicating the possible role of pathogenic fungi. 4. Decreasing the density of seeds in bags significantly reduced seed mortality, most probably because of decreased seed-to-seed contamination by pathogenic fungi. 5. Synthesis and applications. Models of plant populations in general and annual weeds in particular often use values from the literature for seed bank depletion rates. These depletion rates have often been estimated by the seed bag burial method, yet seed density within seed bags may be unrealistically high. Consequently, estimates of seed mortality rates may be too high because of an overestimation of the effects of soil or seed-borne pathogens. Species that have been classified from such studies as having short-lived seed banks may need to be re-assessed using realistic densities either within seed bags or otherwise. Similarly, models of seed bank dynamics based on such overestimated depletion rates may lead to incorrect conclusions regarding the seed banks and, perhaps, the management of weeds and rare species.

A non-uniform mesh scheme for compressible flow

A non-uniform mesh scheme is presented for the computation of compressible flows governed by the Euler equations of gas dynamics. The scheme is based on flux-difference splitting and represents an extension of a similar scheme designed for uniform meshes. The numerical results demonstrate that little, if any, spurious oscillation occurs as a result of the non-uniformity of the mesh; and importantly, shock speeds are computed correctly.

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.

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.

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.

Template-based quadrilateral mesh generation from imaging data

This paper describes a novel template-based meshing approach for generating good quality quadrilateral meshes from 2D digital images. This approach builds upon an existing image-based mesh generation technique called Imeshp, which enables us to create a segmented triangle mesh from an image without the need for an image segmentation step. Our approach generates a quadrilateral mesh using an indirect scheme, which converts the segmented triangle mesh created by the initial steps of the Imesh technique into a quadrilateral one. The triangle-to-quadrilateral conversion makes use of template meshes of triangles. To ensure good element quality, the conversion step is followed by a smoothing step, which is based on a new optimization-based procedure. We show several examples of meshes generated by our approach, and present a thorough experimental evaluation of the quality of the meshes given as examples.