Domains problems

In most engineering problems the boundary of the problem domain includes curved sections. The discretization of domains with curved boundaries using meshes that consist of elements with straight sides inevitably involves some error. This type of discretization error can obviously be reduced by mesh refinements. However, in general, it cannot be entirely eliminated unless finite elements which themselves have curved sides are used. 

The discretization of a problem domain into a finite element mesh consisting of randomly sized triangular elements is shown in Figure 2,1. In the coarse mesh shown there are relatively large gaps between the actual domain boundary and the boundary of the mesh and hence the overall discretization error is expected to be large. 

The main consequence of the discretization of a problem domain into finite elements is that within each element, unknown functions can be approximated using interpolation procedures. 

Figure 5.22 Problem domain in the micro-mechanical analysis of the particulate polymer composite...

Hybrid grids are used for very complex geometries where combination of structured mesh segments joined by zones of unstructured mesh can provide the best approach for discretization of the problem domain. The flexibility gained by combining structured and unstructured mesh segments also provides a facility to improve accuracy of the numerical solutions for field problems of a complicated nature. Figure 6.3 shows an example of this type of computational mesh. 

Using these procedures it is always possible to generate smooth internal divisions. Therefore they offer the advantage of preventing the extension of the exterior boundary discontinuities to the inside of the problem domain. 

Delaunay method - in this method the computational grid is essentially constructed by connecting a specified set of points in the problem domain. The connection of these points should, however, be based on specific rules to avoid unacceptable discreti2ations. To avoid breakthrough of the domain boundary it may be necessary to adjust (e.g. add) boundary points (Liseikin, 1999). 

Subdivision or discretization of the flow domain into cells or elements. There are methods, called boundary element methods, in which the surface of the flow domain, rather than the volume, is discretized, but the vast majority of CFD work uses volume discretization. Discretization produces a set of grid lines or cuives which define a mesh and a set of nodes at which the flow variables are to be calculated. The equations of motion are solved approximately on a domain defined by the grid. Curvilinear or body-fitted coordinate system grids may be used to ensure that the discretized domain accurately represents the true problem domain. 

The range of problems that probabilistie teehniques ean be applied to is vast, basieally anywhere where variability dominates that problem domain. If the eompo-nent is eritieal and if the parameters are not well known, then their uneertainty must be ineluded in the analysis. Under these sorts of requirements, it is essential to quantify the reliability and safety of engineering eomponents, and probabilistie analysis must be performed (Weber and Penny, 1991). In terms of SSI analysis, the main applieation modes are ... 

The type of theories we will be using to prove dominance and/or equivalence of solutions will not be specific to the particular problem domain, but will rely on more general features of the problem formulation. Thus, for our flowshop example, we will not rely on the fact that we are dealing with processing times, end-times, or start-times, to formulate the general theory. The general theory will be in terms of sufficient statements about the underlying mathematical relationships, as described in Section III. 

The applications concern classification as well as quantification problems. In Table 44.3 some examples are given in both problem domains. 

The knowledge domain is an area within what is usually a much broader problem domain. 

As shown in Figure 1.36, Catalysis addresses three levels of modeling the problem domain or business, the component or system specification (externally visible behavior), and the internal design of the component or system (internal structure and behavior). 

Establish problem domain terminology U nderstand business process, roles, collaborations Build as-is and to-be models... 

These techniques apply at the level of a business process, problem domain description, software component specification, design, or implementation. Part V describes how to apply these techniques at different levels in a systematic process. It says more about the process of discovering objects and of achieving continuity and traceability from problem domain to code. 

Similarly, sketches of situations or phenomena in the problem domain are encouraged. Use informal or formal notations, including drawings and rich pictures, tables, and snapshots. 

Business model A (section of a) document that describes the business or problem domain, precisely capturing object types, attributes, invariants, and actions. This document may itself be structured based on... 

Pattern 6.1, The 00 Golden Rule (Seamlessness or Continuity) This shows how to achieve one of the most important benefits of objects a seamless path from problem domain to code. 

Build Many Projects on the Same Model. A problem domain model is useful across more than a single project. But don t take this as a reason for perfecting a model before building your first system see Pattern 14.2, Make a Business Model. 

The resulting system relates better to the end users because it deals in the terms they are familiar with (assuming a reasonable problem domain model) and traces back to the... 

Moreover, there is little flexibility and reuse at the level of individual problem domain objects. Each object plays roles in different collaborations a far more likely unit of reuse is the collaboration or its manifestation as a framework in code (see Chapter 11, Reuse and Pluggable Design Frameworks in Code). 

Sometimes the greatest improvements in reuse and flexibility come by adopting a significantly more abstract (or even orthogonal) view of problem domain terms. Use refinement and frameworks so that you don t lose traceability to domain terms. 

You get only so much reuse and flexibility by adopting the most straightforward model of the problem domain. The terms have evolved to be specific to that domain and may be inconsistent with terms used elsewhere. 

---