SUMMARYThe material point method (MPM) has demonstrated itself as a computationally effective particle method for solving solid mechanics problems involving large deformations and/or fragmentation of structures, which are sometimes problematic for finite element methods (FEMs). However, similar to most methods that employ mixed Lagrangian (particle) and Eulerian strategies, analysis of the method is not straightforward. The lack of an analysis framework for MPM, as is found in FEMs, makes it challenging to explain anomalies found in its employment and makes it difficult to propose methodology improvements with predictable outcomes.In this paper we present an analysis of the quadrature errors found in the computation of (material) internal force in MPM and use this analysis to direct proposed improvements. In particular, we demonstrate that lack of regularity in the grid functions used for representing the solution to the equations of motion can hamper spatial convergence of the method. We propose the use of a quadratic B-spline basis for representing solutions on the grid, and we demonstrate computationally and explain theoretically why such a small change can have a significant impact on the reduction in the internal force quadrature error (and corresponding 'grid crossing error') often experienced when using MPM.

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.

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.

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Over the last decade block-structured adaptive mesh refinement (SAMR) has found increasing use in large, publicly available codes and frameworks. SAMR frameworks have evolved along different paths. Some have stayed focused on specific domain areas, others have pursued a more general functionality, provid- ing the building blocks for a larger variety of applications. In this survey paper we examine a representative set of SAMR packages and SAMR-based codes that have been in existence for half a decade or more, have a reasonably sized and active user base outside of their home institutions, and are publicly available.The set consists of a mix of SAMR packages and application codes that cover a broad range of scientific domains. We look at their high-level frameworks, and their approach to dealing with the advent of radical changes in hardware architecture. The codes included in this survey are BoxLib, Cactus, Chombo, Enzo, FLASH, and Uintah.

SUMMARYA new numerical method for Nwogu's (ASCE Journal of Waterway, Port, Coastal and Ocean Engineering 1993; 119:618) two-dimensional extended Boussinesq equations is presented using a linear triangular ÿnite element spatial discretization coupled with a sophisticated adaptive time integration package. The authors have previously presented a ÿnite element method for the one-dimensional form of these equations (M. Walkley and M. Berzins (International Journal for Numerical Methods in Fluids 1999; 29 (2):143)) and this paper describes the extension of these ideas to the two-dimensional equations and the application of the method to complex geometries using unstructured triangular grids. Computational results are presented for two standard test problems and a realistic harbour model. Copyright ?

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