conductor free space FIGURE 44.3 Edge variables for a tetrahedron element, h ,=(p2-p1)/. r)=(-)|Hr2) (44.10) 4: where R is the distance between the source and field point, respectively. For problems with linear materials Eq (44.10)reduces to integrations over the bounding surfaces of materials in terms of the magnetic scalar d1/R (44.11) R dr Equation(44.11)can be solved numerically by the boundary element method(BEM) in which the active surfaces are discretized into elements. The advantages of integral formulations compared to the standard differential approach using finite elements are(a)only active regions need to be discretized,(b) the far field boundary condition is automatically taken into account, and(c) the fields recovered from the solution are usually very smooth. Unfortunately, the computational costs rapidly escalate as the problem size increases because of the complexity of the system coefficients and because the resulting matrix is fully populated, whereas the differential approach the coefficients are simple and the matrix is sparse, allowing the exploitation of fast equation solution methods. 44.4 Modern design environment The most common system used in software packages is one in which the pre-processor includes data input, model building, and mesh(element)generation. Although fully automated meshing is now a practical possibility it needs to be combined with error estimation in order to allow the generation of optimal meshes. This approach is now common for 2-D systems and is available in many 3-D systems. Figure 44.4 illustrates a field simulation environment in which the solution processor includes an adaptive mesh generator controlled by a posteriori error estimation. This avoids the costly and essentially heuristic task of mesh generation which, in the past had to be performed by the designer. Furthermore a modern system should have solid modeling capabilities driven by parametric data allowing the user to specify the appropriate engineering quantities, e. g, in the case of a solid cylinder the radius and length are all that is needed to specify the geometry at some predefined location. The system should also be supported by a database which is compliant with evolving standards such as STEP(Standard for the Exchange of Product data-ISo 10303 [Owen, 1993))thus allowing data communi- cation between other system The environment illustrated in Fig. 44. 4 also shows tools for automatic optimization that are now becoming feasible in industrial design applications. Both deterministic and stochastic methods for minimizing constrained bjective functions of the design space have been developed Russenschuck, 1996). It must be emphasized, however, that the use of optimizing methods is only part of the total problem of design. For example, the process of automatic synthesis based on design rules and engineering C 2000 by CRC Press LLC© 2000 by CRC Press LLC (44.10) where R is the distance between the source and field point, respectively. For problems with linear materials Eq. (44.10) reduces to integrations over the bounding surfaces of materials in terms of the magnetic scalar potential, i.e., (44.11) Equation (44.11) can be solved numerically by the boundary element method (BEM) in which the active surfaces are discretized into elements. The advantages of integral formulations compared to the standard differential approach using finite elements are (a) only active regions need to be discretized, (b) the far field boundary condition is automatically taken into account, and (c) the fields recovered from the solution are usually very smooth. Unfortunately, the computational costs rapidly escalate as the problem size increases because of the complexity of the system coefficients and because the resulting matrix is fully populated, whereas in the differential approach the coefficients are simple and the matrix is sparse, allowing the exploitation of fast equation solution methods. 44.4 Modern Design Environment The most common system used in software packages is one in which the pre-processor includes data input, model building, and mesh (element) generation.Although fully automated meshing is now a practical possibility it needs to be combined with error estimation in order to allow the generation of optimal meshes. This approach is now common for 2-D systems and is available in many 3-D systems. Figure 44.4 illustrates a field simulation environment in which the solution processor includes an adaptive mesh generator controlled by a posteriori error estimation. This avoids the costly and essentially heuristic task of mesh generation which, in the past, had to be performed by the designer. Furthermore a modern system should have solid modeling capabilities driven by parametric data allowing the user to specify the appropriate engineering quantities, e.g., in the case of a solid cylinder the radius and length are all that is needed to specify the geometry at some predefined location. The system should also be supported by a database which is compliant with evolving standards such as STEP (Standard for the Exchange of Product data-ISO 10303 [Owen, 1993]) thus allowing data communi￾cation between other systems. The environment illustrated in Fig. 44.4 also shows tools for automatic optimization that are now becoming feasible in industrial design applications. Both deterministic and stochastic methods for minimizing constrained objective functions of the design space have been developed for electromagnetic applications (For a review see Russenschuck, 1996). It must be emphasized, however, that the use of optimizing methods is only part of the total problem of design. For example, the process of automatic synthesis based on design rules and engineering FIGURE 44.3 Edge variables for a tetrahedron element, h1 = (F2 – F1)/l. M r( ) = - ( ) H (r¢ - — M × — Ê Ë Á ˆ ¯ ˜ È Î Í Í ˘ ˚ ˙ ˙ Ú m p 1 1 s R ) d 4 1 W W 4 1 1 pf ¶f ¶ f ¶ ¶ = - - Ê Ë Á ˆ ¯ Ú ˜ R n R n d / G G