Free Vertices Research Articles

Iterative-solver convergence for loop-star and loop-tree decompositions in method-of-moments solutions of the electric-field integral equation

Method-of-moments (MoM) solutions of the electric-field integral equation using Rao-Wilton-Glisson (RWG) basis functions suffer from the so-called low-frequency breakdown. Introduction of loop-tree or loop-star decompositions of the basis functions can effectively solve this problem, and a number of papers have been published discussing various aspects with respect to these techniques. Several papers imply that loop-tree or loop-star decompositions may help to improve iterative-solver convergence for the solution of the resulting linear-equation systems. Since only a few results with respect to this issue are available, a study of the frequency-dependent iterative-solver convergence for RWG, loop-tree, and loop-star basis functions was performed. Two metallic scattering objects, with meshes comprising up to 21060 unknowns, were considered. RWG functions were found to provide the best convergence behavior, as long as the frequency considered was high enough to prevent the low-frequency breakdown. Among the loop-tree and loop-star bases, the loop-tree functions were found to be superior to the loop-star functions. The loop-tree functions resulted in good and stable convergence behavior if the number of subdivisions per wavelength was larger than a few hundred. Moreover, it is shown that the so-called loop-tree decomposition can also be viewed as a loop-cotree decomposition if an alternative tree of edges connecting the free vertices of the mesh is constructed.

Constraint-based design for 3D shapes

This paper proposes an ATMS-based geometric reasoning system for feature-based 3D solid modeling. Here, every feature is described by a set of geometric constraints such as distances between edges and angles between faces. The system has to evaluate the constraints in order to determine the attributes of all the geometric elements in the features. Therefore, the modeling process can be considered as a constraint satisfaction problem. Our ATMS-based approach overcomes two serious drawbacks of conventional rule-based approaches: inefficiency and poor conflict handling. For the first problem, a state reduction method, represented as an ATMS justification, resolves the problem of combinatorial explosion in the rules' pattern matching. Here, intermediate states are defined by the degree of freedom: the determined geometric elements have zero degrees, and free faces, edges, and vertices have three, four, and three degrees, respectively. Each constraint invocation reduces the degree; that is, it increases the level of determinacy of the status. For the second problem, the ATMS's label update propagation mechanism resolves conflicts of constraints. It distinguishes conflicting situations from redundant or under-constrained ones, and the minimum diagnosis technique detects which constraint causes the conflict. The use of an ATMS as a propositional reasoning function has various advantages over rule-based systems, such as avoidance of infinite loops and reasoning without pattern matching. The paper also considers the computational efficiency of our approach and proves its practicality by presenting data on an actual product.

A Mixed-Valence Molybdenum Monophosphate with a Three-Dimensional Framework: LiMo2O3(PO4)2

A new mixed-valence monophosphate possessing a unique structure type, LiMo2O3(PO4)2, has been synthesized. It crystallizes in theP21/mspace group witha= 7.771 (1) Å,b= 12.687 (2) Å,c= 8.445 (2) Å, β = 106.94 (1)°. Its three-dimensional [Mo2P2O11]∞framework consists of two kinds of [Mo2P2O15]∞chains. The first kind of chain running along [101] is built up from “Mo2PO13” units linked through single PO4tetrahedra; the second kind of chain, running along [100] is similar to that encountered in W2O3(PO4)2. This framework is also described by the three-dimensional assemblage of [MoPO8]∞chains, running along b. One recognizes two sorts of small tunnels running along b and a, respectively. The second kind of tunnel exhibits a similarity with that observed in the brownmillerite structure. Another novelty of this structure deals with the existence of two coordinations for lithium: tetrahedral and octahedral. The octahedral lithium ions obstruct the tunnels running along a and share two oxygen atoms with Mo(1) corresponding to the free vertices of the MoO6octahedra; the tetrahedral lithium ions share one oxygen atom with Mo(2), and one oxygen with Mo(3) corresponding to the free vertex of the Mo(3)O6octahedron.

On the maximal number of certain subgraphs inK r -free graphs

Given two graphsH andG, letH(G) denote the number of subgraphs ofG isomorphic toH. We prove that ifH is a bipartite graph with a one-factor, then for every triangle-free graphG withn verticesH(G) ≤ H(T 2(n)), whereT 2(n) denotes the complete bipartite graph ofn vertices whose colour classes are as equal as possible. We also prove that ifK is a completet-partite graph ofm vertices,r > t, n ? max(m, r ? 1), then there exists a complete (r ? 1)-partite graphG* withn vertices such thatK(G) ≤ K(G*) holds for everyK r -free graphG withn vertices. In particular, in the class of allK r -free graphs withn vertices the complete balanced (r ? 1)-partite graphT r?1(n) has the largest number of subgraphs isomorphic toK t (t < r),C 4,K 2,3. These generalize some theorems of Turan, Erdos and Sauer.

Three-dimensional branching-type models of flow through porous media

The total path followed by the various fractions of a spot of dye injected into a porous medium through which flow is occurring is considered to be represented by various types of three-dimensional branching configurations (“graphs”), the mass of the dye being distributed on the “free vertices” of the graphs. The graphs are generated according to two different cyclic random procedures. Using each procedure one can obtain samples of graphs generated in a given number of cycles. Statistics is done over such samples of graphs. One class of models gives the following relation between the coefficient of longitudinal dispersion D and the average pore velocity v, and between the coefficient of lateral dispersion D lat and v: D∼v 1.2∼v which are in good agreement with the experimental results.