Martin Berzins

A method for the spatial discretization of parabolic equations in one space variable

This paper is concerned with the design of a spatial discretization method for polar and nonpolar parabolic equations in one space variable. A new spatial discretization method suitable for use in a library program is derived. The relationship to other methods is explored. Truncation error analysis and numerical examples are used to illustrate the accuracy of the new algorithm and to compare it with other recent codes.

An unstructured finite-volume algorithm for predicting flow in rivers and estuaries

Abstract A numerical algorithm is presented for the solution of geometrically challenging two-dimensional river and estuary flows, based on an adaptive triangular tessellation of the flow domains of interest. The governing, shallow water, equations are discretised using a finite volume approach embodying variable step time integrators, to yield a method that is second order accurate in both space and time. An approximate Riemann solver is used to determine flow directionality in conjunction with an effective means of dealing with wetting and drying at the boundaries. The approach is capable of handling complex flow domains and yielding solutions for which errors are controlled automatically by the use of spatial re-gridding and time stepping based on local error estimates. Its range of applicability is demonstrated through considering several problems involving super/sub-critical flow, wetting/drying, culminating in the solution of a complete estuary problem.

Selective laser sintering of an amorphous polymer—simulations and experiments

Abstract Thermal and powder densification modelling of the selective laser sintering of amorphous polycarbonate is reported. Three strategies have been investigated: analytical, adaptive mesh finite difference and fixed mesh finite element. A comparison between the three and experimental results is used to evaluate their ability reliably to predict the behaviour of the physical process. The finite difference and finite element approaches are the only ones that automatically deal with the non-linearities of the physical process that arise from the variation in the thermal properties of the polymer with density during sintering, but the analytical model has some value, provided appropriate mean values are used for thermal properties. Analysis shows that the densification and linear accuracies due to sintering are most sensitive to changes in the activation energy and heat capacity of the polymer, with a second level of sensitivities that includes powder bed density and powder layer thickness. Simulations of the manufacture of hollow cylinders and T-pieces show feature distortions due to excessive depth of sintering at downward facing surfaces in the powder bed. In addition to supporting the modelling, the experiments draw attention to the importance of sintering machine hardware and software controls.

Developing software for time-dependent problems using the method of lines and differential-algebraic integrators

Abstract The method of lines is one of the most powerful tools for the solution of time-dependent coupled ODE/PDE systems. The attraction of this method is that the complex systems of coupled ordinary and partial differential equations arising in mathematical modelling can be solved by using the sophisticated software which has been developed for initial value differential-algebraic equations. The sprint software of Berzins, Dew and Furzeland [2] has been developed specifically for the method of lines. This software contains a selection of spatial discretisation methods, time integrators and linear algebra routines. These components together with utility routines for spatial remeshing and discontinuity detection form an open-ended “tool-kit” for the method of lines. The purpose of the paper is to use the sprint software to illustrate some of the issues that arise in the use and development of algorithms and software which employs the method of lines.

Dynamic load‐balancing for PDE solvers on adaptive unstructured meshes

Modern PDE solvers written for time-dependent problems increasingly employ adaptive unstructured meshes (Flaherty et al., 1989) in order to both increase efficiency and control the numerical error. If a distributed memory parallel computer is to be used, there arises the significant problem of dividing the domain equally amongst the processors whilst minimising the inter-subdomain dependencies. A number of graph-based algorithms have recently been proposed for steady-state calculations. The paper considers an extension to such methods which renders them more suitable for time-dependent problems in which the mesh may be changed frequently.

A 3D UNSTRUCTURED MESH ADAPTATION ALGORITHM FOR TIME-DEPENDENT SHOCK-DOMINATED PROBLEMS

SUMMARY In this paper we present a tetrahedron-based, h-refinement-type algorithm for the solution of problems in 3D gas dynamics using unstructured mesh adaptation. The mesh adaptation algorithm is coupled to a cell-centred, Riemann problem-based, finite volume scheme of the MUSCL type, employing an approximate Riemann solver. The adaptive scheme is then used to compute the diffraction of shock waves around a box section corner for subsonic and supersonic post-shock flow. In the subsonic case, preliminary measurements of vortex filament speed and vortical Mach number are in broad quantitative agreement with known theoretical results. # 1997 by John Wiley & Sons, Ltd.

On the use of adaptive gridding methods for modelling chemical transport from multi-scale sources

This paper investigates the solution of atmospheric reaction/flow problems using time-dependent adaptive mesh gridding techniques. Preliminary studies of time varying problems in two space dimensions related to the effects of power station emissions on regional ozone levels have been carried out. The results show the importance of using adaptive grids in order to represent the interaction of the plume with background air over large distances. The adaptive mesh reveals features of cross wind concentration profiles which would not be shown using the standard mesh sizes adopted in regional atmospheric calculations. As the level of adaptivity increases, and the mesh becomes locally refined in regions of large spatial error, the total and peak ozone concentrations change quite significantly. The results demonstrate that the level of error which can result from using fixed or telescopic grid approaches for spatially inhomogeneous source patterns may be significantly reduced by the use of adaptive meshes.

A FINITE ELEMENT METHOD FOR THE ONE-DIMENSIONAL EXTENDED BOUSSINESQ EQUATIONS

SUMMARY A new finite element method for Nwogu’s (O. Nwogu, ASCE J. Waterw., Port, Coast., Ocean Eng., 119, 618‐638 (1993)) one-dimensional extended Boussinesq equations is presented using a linear element spatial discretisation method coupled with a sophisticated adaptive time integration package. The accuracy of the scheme is compared to that of an existing finite difference method (G. Wei and J.T. Kirby, ASCE J. Waterw., Port, Coast., Ocean Eng., 121, 251‐261 (1995)) by considering the truncation error at a node. Numerical tests with solitary and regular waves propagating in variable depth environments are compared with theoretical and experimental data. The accuracy of the results confirms the analytical prediction and shows that the new approach competes well with existing finite difference methods. The finite element formulation is shown to enable the method to be extended to irregular meshes in one dimension and has the potential to allow for extension to the important practical case of unstructured triangular meshes in two dimensions. This latter case is discussed. Copyright

A Solution-Based Triangular and Tetrahedral Mesh Quality Indicator

A new mesh quality measure for triangular and tetrahedral meshes is presented. This mesh quality measure is based on both geometrical and solution information and is derived by considering the error when linear triangular and tetrahedral elements are used to approximate a quadratic function. The new measure is shown to be related to existing measures of mesh quality but with the advantage that local solution information in the form of scaled derivatives along edges is taken into account.

Positive cell-centered finite volume discretization methods for hyperbolic equations on irregular meshes

The conditions sufficient to ensure positivity and linearity preservation for a cell-centered finite volume scheme for time-dependent hyperbolic equations using irregular one-dimensional and triangular two-dimensional meshes are derived. The conditions require standard flux limiters to be modified and also involve possible constraints on the meshes. The accuracy of this finite volume scheme is considered and is illustrated by two simple numerical examples.