Consistent Dirichlet Boundary Conditions for Numerical Solution of Moving Boundary Problems

We consider the imposition of Dirichlet boundary conditions in the finite element modelling of moving boundary problems in one and two dimensions for which the total mass is prescribed. A modification of the standard linear finite element test space allows the boundary conditions to be imposed strongly whilst simultaneously conserving a discrete mass. The validity of the technique is assessed for a specific moving mesh finite element method, although the approach is more general. Numerical comparisons are carried out for mass-conserving solutions of the porous medium equation with Dirichlet boundary conditions and for a moving boundary problem with a source term and time-varying mass.

Modelling oil absorption during post-frying cooling - II: solution of the mathematical model, model testing and simulations

The mathematical models that describe the immersion-frying period and the post-frying cooling period of an infinite slab or an infinite cylinder were solved and tested. Results were successfully compared with those found in the literature or obtained experimentally, and were discussed in terms of the hypotheses and simplifications made. The models were used as the basis of a sensitivity analysis. Simulations showed that a decrease in slab thickness and core heat capacity resulted in faster crust development. On the other hand, an increase in oil temperature and boiling heat transfer coefficient between the oil and the surface of the food accelerated crust formation. The model for oil absorption during cooling was analysed using the tested post-frying cooling equation to determine the moment in which a positive pressure driving force, allowing oil suction within the pore, originated. It was found that as crust layer thickness, pore radius and ambient temperature decreased so did the time needed to start the absorption. On the other hand, as the effective convective heat transfer coefficient between the air and the surface of the slab increased the required cooling time decreased. In addition, it was found that the time needed to allow oil absorption during cooling was extremely sensitive to pore radius, indicating the importance of an accurate pore size determination in future studies.

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.

Two-point boundary value problems for linear evolution equations

We study boundary value problems for a linear evolution equation with spatial derivatives of arbitrary order, on the domain 0 < x < L, 0 < t < T, with L and T positive nite constants. We present a general method for identifying well-posed problems, as well as for constructing an explicit representation of the solution of such problems. This representation has explicit x and t dependence, and it consists of an integral in the k-complex plane and of a discrete sum. As illustrative examples we solve some two-point boundary value problems for the equations iqt + qxx = 0 and qt + qxxx = 0.

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.

A cell by cell anisotropic adaptive mesh ALE scheme for the numerical solution of the Euler equations

In this paper a cell by cell anisotropic adaptive mesh technique is added to an existing staggered mesh Lagrange plus remap finite element ALE code for the solution of the Euler equations. The quadrilateral finite elements may be subdivided isotropically or anisotropically and a hierarchical data structure is employed. An efficient computational method is proposed, which only solves on the finest level of resolution that exists for each part of the domain with disjoint or hanging nodes being used at resolution transitions. The Lagrangian, equipotential mesh relaxation and advection (solution remapping) steps are generalised so that they may be applied on the dynamic mesh. It is shown that for a radial Sod problem and a two-dimensional Riemann problem the anisotropic adaptive mesh method runs over eight times faster.

Voronoi, Delaunay, and Block-Structured Mesh Refinement for Solution of the Shallow-Water Equations on the Sphere

Alternative meshes of the sphere and adaptive mesh refinement could be immensely beneficial for weather and climate forecasts, but it is not clear how mesh refinement should be achieved. A finite-volume model that solves the shallow-water equations on any mesh of the surface of the sphere is presented. The accuracy and cost effectiveness of four quasi-uniform meshes of the sphere are compared: a cubed sphere, reduced latitude–longitude, hexagonal–icosahedral, and triangular–icosahedral. On some standard shallow-water tests, the hexagonal–icosahedral mesh performs best and the reduced latitude–longitude mesh performs well only when the flow is aligned with the mesh. The inclusion of a refined mesh over a disc-shaped region is achieved using either gradual Delaunay, gradual Voronoi, or abrupt 2:1 block-structured refinement. These refined regions can actually degrade global accuracy, presumably because of changes in wave dispersion where the mesh is highly nonuniform. However, using gradual refinement to resolve a mountain in an otherwise coarse mesh can improve accuracy for the same cost. The model prognostic variables are height and momentum collocated at cell centers, and (to remove grid-scale oscillations of the A grid) the mass flux between cells is advanced from the old momentum using the momentum equation. Quadratic and upwind biased cubic differencing methods are used as explicit corrections to a fast implicit solution that uses linear differencing.