By Jean Claude Sabonnadiere, Jean Louis Coulomb
The finite point approach (FEM) has been understood, no less than in precept, for greater than 50 years. The imperative formula on which it really is dependent has been recognized for an extended time (thanks to the paintings of Galerkin, Ritz, Courant and Hilbert,1.4 to say the main important). in spite of the fact that, the tactic couldn't be utilized in a pragmatic manner because it concerned the answer of a giant variety of linear or non-linear algebraic equations. this present day it truly is very common, due to pcs, to resolve non-linear algebraic difficulties of numerous thousand equations. the required numerical tools and programming ideas are actually an essential component of the instructing curriculum in so much engineering faculties. Mechanical engineers, faced with very complex structural difficulties, have been the 1st to exploit complicated computational equipment and excessive point languages (FORTRAN) to rework the mechanical versions into algebraic equations (1956). in recent years (1960), the FEM has been studied through utilized mathematicians and, having acquired rigorous therapy, has turn into part of the extra basic learn of partial differential equations, progressively exchanging the finite distinction process which have been thought of the common software to resolve some of these problems.
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Extra info for Finite Element Methods in CAD: Electrical and Magnetic Fields
Example text
5 Quadratic element (Lagrangian). 6 u Hermittian type elements. WesetW = l-u-v <1>2 = w(IOw - 15 w 2 + 6 w 3 + 30 uv(u + v») = uw 2 (3 - 2 w - 3 u 2 + 6 uv) <1>3 = <1>* = 2 u 2 w 2 (w + 3 v) <1>5 = uvw 2 <1>6 = <1>7 = u 2 (lOu - <1>1 vw 2 (3 - 2 w - 3 v 2 6 uv) I I 2 v2 w 2 (w 2 <1>8 + + 3 u) 15u2 + + 6u 3 + 15v 2 w) = ~ ( - 8u = -2- (6 - 4 u - 3 v - 3 v2 u2 V u2 = "4 (2 u(1 u2 V 14u 2 U)2 = -2- (- 2 <1>13 = v 2 (lOv - 15v 2 <1>14 = 2 <1>15 v2 ='2(- 8v 6u 3 + + v2 6v 3 + v u 2 v2 4 W 14v 2 u2 v3 +-2- + 3 uv) - 6v 3 uv) - 15 u 2 w) + (6 - 3 u - 4 v - 3 u 2 + 15v 2 w) - + 5 v 2 w) + 2u + <1>11 uv 2 - - 3 uv) 15u 2 w) 59 FINITE ELEMENTS AND APPROXIMA nON FUNCTIONS QUADRILATERAL ELEMENTS This type of element is very popular in the form of a curved isoparametric element because it permits the generation of approximation functions which have particular properties in a chosen direction.
If we consider the combination of the two elements in Fig. 12) we can write A(e) = AI \e) oAe ox = AA + A2 ~e) O~. 56) 47 FROM ONE DIMENSION ... TO THREE Using the same procedure we used previously, we define the transformation matrices which relate the global and local numbering: Xii) '" Xii) -B Xiii C Xiii D X l21 A X I21 'B X~2) Xl21 D ~ li 0 0 0 0 0 I 0 ~l! 13 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 <1>2A <1>2B <1>2c <1>2D ~ ~
I>~) G) = ( - h)+(3-~) x P(x - 2 h) 2 + ~) P(x - 2 h) We note that these functions are continuous, piecewise linear and piecewise differentiable. Construction of the matrices Recalling Eqn A2 = (A T)(T A). T A + (X2 AT