No description available.

No description available.

No description available.

Some numerical simulations of multi-scale physical phenomena consume a significant amount of computational resources, since their domains are discretized on high resolution meshes. An enormous wastage of these resources occurs in refinement of sections of the domain where computation of the solution does not require high resolutions. This problem is effectively addressed by adaptive mesh refinement (AMR), a technique of local refinement of a mesh only in sections where needed, thus allowing concentration of effort where it is required. Sections of the domain needing high resolution are generally determined by means of a criterion which may vary depending on the nature of the problem. Fairly straightforward criteria could include comparing the solution to a threshold or the gradient of a solution, that is, its local rate of change to a threshold. While the former criterion is not particularly rigorous and hardly ever represents a physical phenomenon of interest, it is simple to implement. However, the gradient criterion is not as simple to implement as a direct comparison of values, but it is still quick and a good indicator of the effectiveness of the AMR technique. The objective of this thesis is to arrive at an adaptive mesh refinement algorithm for a finite difference scheme using a quadtree decomposition approach. In the AMR algorithm developed, a mesh of increasingly fine resolution permits high resolution computation in sub-domains of interest and low resolution in others. In this thesis work, the gradient of the solution has been considered as the criterion determining the regions of the domain needing refinement. Initial tests using the AMR algorithm demonstrate that the paradigm adopted has considerable promise for a variety of research problems. The tests performed thus far depict that the quantity of computational resources consumed is significantly less while maintaining the quality of the solution. Analysis included comparison of results obtained with analytical solutions for four test problems, as well as a thorough study of a contemporary problem in solid mechanics.

There has been much confusion evident in the literature in terms of the influence of impurity elements on grain refinement of magnesium alloys. This thesis addresses how impurity elements such as iron, manganese, carbon and beryllium affect grain refinement in magnesium alloys. The thesis starts with an investigation into the effect of the uptake of iron on grain refinement of Mg-Zr alloys. The highly detrimental influence of the uptake of iron on grain refinement in Mg-Zr alloys has been confirmed. The gradual loss of grain refinement of Mg-Zr alloys partly arises from the consumption of Zr by the formation of Fe2Zr via the reaction between soluble Zr and Fe picked up from mild steel crucibles. (Settling of undissolved Zr particles also partly attributes to the gradual loss of grain refinement.) The morphological evolution of Zr-rich cores from circular to rosette-like has been reported here for the first time. In contrast to the detrimental effect in Mg-Zr alloys, a positive effect of iron has been observed in grain refinement of Mg-Al based alloys. The addition of iron in the form of anhydrous FeCl3 produces significant grain refinement of high-purity Mg-Al alloys. Obvious grain refinement was also achieved through the uptake of iron from steel crucible surfaces; however, the addition of Fe powder in the form of an ALTABTM Fe75 powder compact (75%Fe, 15%Al and 10% Na-free flux) did not give rise to grain refinement. The results obtained from both the grain refinement tests conducted in aluminium titanite crucibles and an ultra-low carbon 316L stainless crucible indicate that the grain refinement of Mg-Al alloys by iron inoculation has little to do with the Al4C3 hypothesis. The nucleant particles have been clarified to be Fe- and Al-rich intermetallics. The effect of manganese on the grain refinement of high purity Mg-Al based alloys and commercially available AZ31 alloys has been investigated using an Al-60%Mn master alloy splatter at 730 „aC in aluminium titanite crucibles. Grain refinement was readily achievable in these alloys. Electron microprobe analyses revealed that prior to the addition of extra manganese the majority of the intermetallic particles found in AZ31 are of the Al8Mn5 type. However, after the addition of extra manganese in the range of 0.1% to 1.0%, the predominant group of intermetallic particles changed to the metastable AlMn type. This leads to a hypothesis that the metastable AlMn intermetallic particles are more effective than Al8Mn5 as nucleation sites for magnesium grains. The hypothesis was supported by the observation that a long period of holding at 730 „aC led to an increase in grain size, due probably to the transformation of the metastable AlMn to the stable Al8Mn5. Native grain refinement in magnesium alloys has been clarified. Based on the fact that native grain refinement is an exclusive feature of high purity Mg-Al alloys, it is hypothesized that Al4C3 particles act as nucleation centres. This is also the mechanism of carbon grain refinement of Mg-Al alloys. A trace of beryllium leads to dramatic grain coarsening in Mg-Al alloys at normal cooling rates. Apart from Mg-Al alloys, a trace of beryllium also causes considerable grain coarsening in Mg-Zn, Mg-Ca, Mg-Ce, Mg-Nd and also hinders grain refinement of magnesium alloys by Zr. Modelling grain refinement to predict the final grain size has been made on the basis of understanding of existing models. The modified model has resolved a fundamental gap in the relative grain size model using a more universal expression of solute concentration in the liquid.

No description available.

Two refinement techniques to generate solution adaptive meshes have been developed. Both techniques utilize arbitrary polyhedra (general elements) to constrain the propagation of refinement. A face-based approach that produces isotropic refinement and a combined element- and edge-based approach that produces anisotropic refinement are presented. Refinement is triggered through sensors that use a shock detection algorithm or error estimation based on the smoothness of the reconstructed solution variables. The basic algorithms as well as specific implementation issues are presented. The advantages and disadvantages of the different methods are discussed and illustrated through a set of synthetic and realistic test cases. It is shown that general elements can be employed effectively in solution adaptive meshes generated using refinement.

Two efficiency-based grid refinement strategies are investigated for adaptive finite element solution of partial differential equations. In each refinement step, the elements are ordered in terms of decreasing local error, and the optimal fraction of elements to be refined is deter- mined based on e±ciency measures that take both error reduction and work into account. The goal is to reach a pre-specified bound on the global error with a minimal amount of work. Two efficiency measures are discussed, 'work times error' and 'accuracy per computational cost'. The resulting refinement strategies are first compared for a one-dimensional model problem that may have a singularity. Modified versions of the efficiency strategies are proposed for the singular case, and the resulting adaptive methods are compared with a threshold-based refinement strategy. Next, the efficiency strategies are applied to the case of hp-refinement for the one-dimensional model problem. The use of the efficiency-based refinement strategies is then explored for problems with spatial dimension greater than one. The work times error strategy is inefficient when the spatial dimension, d, is larger than the finite element order, p, but the accuracy per computational cost strategy provides an efficient refinement mechanism for any combination of d and p.

Two efficiency-based grid refinement strategies are investigated for adaptive finite element solution of partial differential equations. In each refinement step, the elements are ordered in terms of decreasing local error, and the optimal fraction of elements to be refined is deter- mined based on e±ciency measures that take both error reduction and work into account. The goal is to reach a pre-specified bound on the global error with a minimal amount of work. Two efficiency measures are discussed, 'work times error' and 'accuracy per computational cost'. The resulting refinement strategies are first compared for a one-dimensional model problem that may have a singularity. Modified versions of the efficiency strategies are proposed for the singular case, and the resulting adaptive methods are compared with a threshold-based refinement strategy. Next, the efficiency strategies are applied to the case of hp-refinement for the one-dimensional model problem. The use of the efficiency-based refinement strategies is then explored for problems with spatial dimension greater than one. The work times error strategy is inefficient when the spatial dimension, d, is larger than the finite element order, p, but the accuracy per computational cost strategy provides an efficient refinement mechanism for any combination of d and p.

Specification animation has become a popular technique in industry, particularly for validation in model-based design processes. Animation tools provide the ability to explore and visualise the behaviour of a model without needing to study its internal workings. Formal refinement techniques should also be of interest to industry since they support verifiably correct transformations of system models towards implementation. So far, however, refinement techniques are not widely used. Their application requires a high degree of mathematical skill, even with the currently available tool support. Better tool support is needed to make refinement techniques accessible to industry. In this thesis we investigate the application of existing specification animation and visualisation tools to problems in refinement theory. We show how animation and visualisation can be used to support verification, by refinement, and validation, by comparing the behaviour of a refined specification against its abstract specification. Such techniques can be used to explain and/or improve the understanding of a refinement and to check for the presence of errors in a refinement, for example, before attempting a proof. In the most challenging cases, data refinements, the designer needs to supply an abstraction relation in order to prove the refinement. We initially assume that an abstraction relation is provided as an input to the verification and validation tasks. However, finding abstraction relations is hard, and is currently a matter of trial and error. We therefore study the problem of finding abstraction relations. We show that, if an abstraction relation exists, there is always a unique weakest abstraction relation and at least one minimal abstraction relation, and we describe algorithms for finding both the weakest abstraction relation and minimal abstraction relations. These algorithms can be applied to small finite-state systems to produce abstraction relations in terms of explicit values of state variables. We then investigate a symbolic algorithm for finding abstraction relations, which can be applied to systems with infinite states, to produce abstraction relations in predicate form. The theory and the algorithms we develop thus make it possible for us to extend our animation-based verification and validation techniques so that they can be used without providing a complete abstraction relation. Additionally our extended techniques can help a designer construct an abstraction relation or check a proposed one.