New Algebra as Tool for Electric Field Calculation

New Algebra as Tool for Electric Field Calculation Mirsad Kapetanović ENERGOINVEST Electric Power Institute, Sarajevo Bosnia and Herzegovina Abstract - A basic description of some topics from the new algebra (we call it Bosnian algebra or algebra of fractal vector) which deals directly with elements of space is given. Using fractal triangle as fractal space defined by fractal vector, new fully automatic 2D mesh generation technique is presented. Examples of electric field calculation of switching gap for an SF6 interrupter showing the capability of the algebra. INTRODUCTION Wide classes of physical problems are described by partial differential equations with some boundary conditions. The analytical solution is known only in a few cases on special domains. Therefore it is necessary to use some numerical methods to get an approximation of the solution. Establishing the mathematical model into a form suitable for numerical solution usually means performing discretization and solving the resulting discrete problem. Problem discretization is usually related with discretization of a space. One of the most useful numerical methods for solving partial differential equations is finite element method (FEM). Automatization of the whole computational process is the main task. The most basic requirement of an automated FEM is a fully automatic finite element mesh generation. For these purposes the mesh generator must be able to automatically, without interacting with meshing process, generate a valid finite element mesh for any geometry. Although a lot of papers deals with the term fully automatic mesh generation, mesh generators described in those papers can not satisfy mentioned requirement.

Kemo Sokolija University of Sarajevo Elektrotechnical faculty, Sarajevo Bosnia and Herzegovina Bosnian algebra, established by Mirsad Kapetanović in [1], provides a general framework for an unconventional way of fully automatic space discretization. THE IDEA OF BOSNIEN ALGEBRA Classical vector is represented by three elements of vector quantity: magnitude, direction and sense. If two vectors have the same these three elements they equal. Example on Fig.1 demonstrates that the sum c1 of two vectors a1 and b1, and the sum c2 of another two vectors a2 and b2 equal according to the definition of vector equality (c1=c2). The intuitive idea is that there is some difference between vectors c1 and c2 because theirs summands are different. Let us add the common point of summands as the fourth element of vector quantity to its sum (the point F1 to vector c1 and the point F2 to vector c2). These points we call vector’s fractal points and a vector with fractal point, fractal vector. We denote fractal vectors by boldface underline letters. Now, fractal vectors c1 and c2 unequal (c1≠c2). Geometrically, vectors c1 and c2 denote the same length AB, while fractal vectors c1 and c2 denote both

Fig.1 Geometrical interpretation of fractal point

the length AB and two different triangles ABF1 and ABF2, respectively. Bosnian algebra consider classical vector as elementary fractal vector (special case of fractal vector without fractal point). Fig.2 locates the three points A, B and F represented by three vectors rA, rB and rF. Triangle ABF represents fractal vector c on level 0 (zero). We say c is former of his left follower a and his right follower b. The level of a and b is 1. If a and b are not elementary vectors they have theirs followers, etc. All these fractal vectors are descendants of ancestor c and they have their fractal points. The level n on which descendants are elementary vectors defines the order of fractal vector. Fractal vector of order n has 2n-1 fractal points. Obviously it is very difficult deal with so many elements of vector quantity. However, if the same function defines relative position of every fractal point to start and end points of its vector, only one additional element of vector quantity, the function rF=f(rA,rB), is enough to describe space of the fractal vector. We will use the ordered list of four elements as notation for such fractal vector: c=(rA,rB,rF,n) =(rA,rB,f(rA,rB),n) (1) According to this notation its left follower is lf(c)=a=(rA,rF,f(rA,rF),n-1) (2) and its right follower is (3) rf(c)=b=(rF,rB,f(rF,rB),n-1) Fractal vector c is former of both left and right followers: fo(a)=fo(b)=c (4) Now, we will index space of the fractal vector using binary number system: c 1 10 lf(c)=lf(1) rf(c)=rf(1) 11 100 lf(lf(c))=lf(lf(1))=lf(10) rf(lf(c))=rf(lf(1))=rf(10) 101 110 lf(rf(c))=lf(rf(1))=lf(11) rf(rf(c))=rf(rf(1))=rf(11) 111 ....................................... ...... Indexes will generally be denoted by lower case letters x, y, etc. The only fractal vector on level 0

Fig.2 Fractal vector

(fractal vector c) has index 1. Formula lf(x)=10x (5) define index of left followers, and formula rf(x)=10x+1 (6) define index of right followers. Former of any fractal vector (represented by index x) is integer of quotient x/10: fo(x)=int(x/10) (7) When n→∞ than exists one-to-one correspondence between space of fractal vector c and set of natural numbers N. This very important fact gives an opportunity to deal only with indexes, meaning directly with elements of space defined by the fractal vector. In the work carried out in [1], fractal triangle defined by fractal vector is used for discretization of 2D Euclidean space. When common fractal point of followers in (1) is given by rFF=(rA+rB)/2 the fractal vector defines the selfsimilar fractal in 2D Euclidean denote the space - fractal triangle (Fig.3). Symbol properties of a fractal triangle: t= (rA,rB,rF,∞)= (8) =(rA,rB,(rF,rFF=(rA+rB)/2),∞) Note that set of all same level triangles covers triangle ABF. Number of binary index numerals predetermines the level of any descendant in the space of fractal triangle. Two-numeral indexes are on level 1, three-numeral indexes are on level 2, etc. Set of indexes may be identified with a mesh with indexes corresponding to triangles. A mesh is regular if there do not exist triangle intersections and if any neighboring couple has the same bordering side. Obviously, any set of same level triangles is regular mesh. Let x be any triangle of level n. The rules that

Fig.3 Fractal triangle

associate his three possible same level neighbor triangles with x are very simple operations on index x. All three neighbors are defined by transformation: (9) f(x,k)=lfk(fok(x))+rfk(fok(x))-x We write co(x) for the image f(x,n) of x. This is the first neighbor of x. We call it complement neighbor because x and co(x) cover their common former. Operation co(x) practically converts the last numeral of binary noted index x (1 to 0 or 0 to 1). Example: co(1011)=1010, co(1010)=1011 Second neighbour of x we call noncomplement one and write nc(x). It is the image f(x,μ) of x, where μ is odd number (3≤μ≤n) satisfying following conditions: lf(rf(foμ(x)))=foμ-2(x) or rf(lf(foμ(x)))=foμ-2(x) and lf(lf(foμ-2i(x)))=foμ-2i-2(x) or rf(rf(foμ-2i(x)))=foμ-2i-2(x), where i=1,2,3,... while 2i