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By John Stuart Beeteson

Visualizing Magnetic Fields: Numerical Equation Solvers in motion presents a whole description of the idea at the back of a brand new strategy, a close dialogue of the methods of fixing the equations (including a software program visualization of the answer algorithms), the applying software program itself, and the complete resource code. most significantly, there's a succinct, easy-to-follow description of every approach within the code. The physicist Michael Faraday acknowledged that the examine of magnetic traces of strength was once significantly influential in best him to formulate lots of these suggestions which are now so basic to our glossy global, proving to him their "great application in addition to fertility." Michael Faraday may well basically visualize those traces in his imagination and, inspite of glossy desktops to assist us, it's been very dear and time eating to devise strains of strength in magnetic fields. * review of the physics of magnetic fields * whole description of the speculation at the back of the hot approach * evaluation of the idea of Gaussian removal and Conjugate Gradient equation solvers (including a software program program to visualize the algorithms in motion) * program software program itself, the full-source code and a undeniable English description of every method within the code

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Extra resources for Visualising magnetic fields: numerical equation solvers in action

Example text

E. 9 simultaneous equations: Rh1 Rv1 0 Rh4 Rv5 3 Rh2 Rv2 Rh3 1 Rh5 Rv6 2 Rv4 5 Rv8 Rv3 Rh6 4 Rv7 V Rh7 Rv9 Rh8 6 Rv10 7 Rh10 Rh11 Rh9 Rv11 8 Rh12 Figure 3-11: 3 × 3 resistor mesh with single branch voltage. e. Rh1 + Rv2 + Rh4 + Rv1. e. Rh2 + Rv2 + Rh5 + Rv3. e. Rv2. e. again Rv2. and so on. V1 etc. refer to the loop voltage of mesh 1 and so on. Now some of these coefficient terms, for example R38, are zero, since there is no mutual resistor shared between mesh 3 and mesh 8, and many of the loop voltages are zero also.

These iterations are repeated until the 32 Numerical algorithm theory procedure determines so that no further improvements are possible. There is no filling-in of matrix terms with the conjugate gradient method, and therefore storage is primarily limited to three diagonals (remembering that the two lower diagonals are identical to the upper two) and the loop voltages and currents. The actual algorithm will be presented first, together with a discussion of the use of double and single precision arithmetic.

E. again Rv2. and so on. V1 etc. refer to the loop voltage of mesh 1 and so on. Now some of these coefficient terms, for example R38, are zero, since there is no mutual resistor shared between mesh 3 and mesh 8, and many of the loop voltages are zero also. The equations then reduce to a sparser form: R00I0 – R01I1 0 –R10I0 + R11I1 – R12I2 0 –R30I0 0 – R21I1 + R22I2 0 – R41I1 – R03I3 0 0 0 – R14I4 0 0 0 0 0 = 0 0 0 0 0 = 0 0 0 = 0 0 0 = 0 0 – R25I5 0 + R33I3 – R34I4 0 – R43I3 + R44I4 – R45I5 – R52I2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 – R63I3 0 – R54I4 + R55I5 0 – R74I4 0 0 – R36I6 0 0 – R47I7 0 = V – R58I8 = 0 0 0 0 + R66I6 – R67I7 0 – R76I6 + R77I7 – R78I8 = –V – R85I5 0 = – R87I7 + R88I8 = 0 With the zero terms in the equations a diagonal matrix form emerges, with the mesh self-resistance as the centre diagonal, the mesh vertical resistor mutual resistance as the next diagonal (upper or lower) and the mesh horizontal resistor mutual resistance as the outer diagonal (upper or lower).

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