By Gilbert Strang
This booklet provides the whole diversity of computational technology and engineering -- the equations, numerical tools, and algorithms with MATLABÂ® codes. the writer has taught this fabric to millions of engineers and scientists. The booklet is solution-based and never formula-based: it covers utilized linear algebra and quick solvers, differential equations with finite modifications and finite parts, Fourier research, optimization, and more.
Contents bankruptcy 1: utilized Linear Algebra; bankruptcy 2: A Framework for utilized arithmetic; bankruptcy three: Boundary price difficulties; bankruptcy four: Fourier sequence and Integrals; bankruptcy five: Analytic features; bankruptcy 6: preliminary price difficulties; bankruptcy 7: fixing huge platforms; bankruptcy eight: Optimization and minimal rules.
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The implementation of this algorithm starts by defining two integer arrays: CODE and INDEX. The array CODE holds the identification codes [computed from Eq. (22)], while INDEX contains the corresponding value of k. Both CODE and INDEX are sorted together so that the elements of CODE are rearranged to be in an ascending order. Once generated and properly sorted, these arrays can be used to find the index k (to identify k) for grid coordinates (i, j) in the following manner: 1. Given i and j, compute code ϭ i и Ny ϩ j.
Similarly, at nodes 36 (i, j ϭ 4, 5) and 37 (i, j ϭ 5, 5), Eq. (16) becomes Y5i φ37 + Y3i φ35 + Y6j φ44 + Y4j φ25 − Y4,5 φ36 = 0 (18) Y4i φ36 + Y4j φ26 − Y5,5 φ37 = −Y6iV0 − Y6j V0 = V37 (19) and where in Eq. (19) the known quantities (the potentials on the surface of the cylinder at nodes 5, 6 and 6, 5) were moved to the right-hand side. Similar equations can be obtained at the remaining free grid nodes where the potential is to be determined. Once Eq. (16) has been enforced everywhere within the grid, the resulting set of equations can be combined into the following matrix form: −Y2,2 Y3j 0 · 0 Y3i 0 · 0 0 0 0 · · · · 0 0 0 0 0 0 Y4j 0 · Y5i −Y5,5 0 0 · · 0 · Y6j 0 0 · · · · · 0 Y4j 0 0 0 · 0 φ1 φ2 · · · φ12 · φ25 φ26 · · φ35 φ36 φ37 · · · φ44 · φ92 0 · Y3i 0 −Y4,5 Y4i 0 · = · 0 V 37 · 0 (20) 545 which can be written more compactly as [Y ][φ] = [V0 ] (21) Clearly, the coefficient matrix of the above system of equations is very sparse, containing few nonzero elements.
A virtual surface Sv is defined near the actual grid truncation boundary. The electrostatic potential due to charged objects, enclosed within Sv, is computed using the regular FDM algorithm. Subsequently, it is used to calculate the surface charge density and surface magnetic current, which are proportional to the normal and tangential components of the electric field on Sv. Once the equivalent sources are known, the potential between the virtual surface and the grid truncation boundary can be readily calculated (for details see Ref.
12.Computational Science and Engineering by Gilbert Strang